MULTISCALE MODELING AND COMPUTATION OF NANO-ELECTRONIC
TRANSISTORS AND TRANSMEMBRANE PROTON CHANNELS
By
Duan Chen

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Applied Mathematics
2010

ABSTRACT
MULTISCALE MODELING AND COMPUTATION OF
NANO-ELECTRONIC TRANSISTORS AND TRANSMEMBRANE
PROTON CHANNELS
By
Duan Chen

The miniaturization of nano-scale electronic transistors, such as metal oxide semiconductor field effect transistors (MOSFETs), has given rise to a pressing demand
in the new theoretical understanding and practical tactic for dealing with quantum
mechanical effects in integrated circuits. In biology, proton dynamics and transport
across membrane proteins are of paramount importance to the normal function of
living cells. Similar physical characteristics are behind the two subjects, and model
simulations share common mathematical interests/challenges. In this thesis work,
multiscale and multiphysical models are proposed to study the mechanisms of nanotransistors and proton transport in transmembrane at the atomic level.
For nano-electronic transistors, we introduce a unified two-scale energy functional
to describe the electrons and the continuum electrostatic potential. This framework
enables us to put microscopic and macroscopic descriptions on an equal footing at
nano-scale. Additionally, this model includes layered structures and random doping
effect of nano-transistors.
For transmembrane proton channels, we describe proton dynamics quantum mechanically via a density functional approach while implicitly treat numerous solvent
molecules as a dielectric continuum. The densities of all other ions in the solvent
are assumed to obey the Boltzmann distribution. The impact of protein molecular
structure and its charge polarization on the proton transport is considered in atomic
details. We formulate a total free energy functional to include kinetic and potential
energies of protons, as well as electrostatic energy of all other ions on an equal footing.

For both nano-transistors and proton channels systems, the variational principle is employed to derive nonlinear governing equations. The Poisson-Kohn-Sham
equations are derived for nano-transistors while the generalized Poisson-Boltzmann
equation and Kohn-Sham equation are obtained for proton channels. Related numerical challenges in simulations are addressed: the matched interface and boundary
(MIB) method, the Dirichlet-to-Neumann mapping (DNM) technique, and the Krylov
subspace and preconditioner theory are introduced to improve the computational efficiency of the Poisson-type equation. The quantum transport theory is employed to
solve the Kohn-Sham equation. The Gummel iteration and relaxation technique are
utilized for overall self-consistent iterations.
Finally, applications are considered and model validations are verified by realistic
nano-transistors and transmembrane proteins. Two distinct device configurations,
a double-gate MOSFET and a four-gate MOSFET, are considered in our threedimensional numerical simulations. For these devices, the current fluctuation and
voltage threshold lowering effect induced by discrete dopants are explored. For proton
transport, a realistic channel protein, the Gramicidin A (GA) is used to demonstrate
the performance of the proposed proton channel model and validate the efficiency of
the proposed mathematical algorithms. The electrostatic characteristics of the GA
channel is analyzed with a wide range of model parameters. Proton channel conductances are studied over a number of applied voltages and reference concentrations.
Comparisons with experimental data are utilized to verify our model predictions.

ACKNOWLEDGMENT
Within the limited space, I wish to acknowledge a few amazing people who contribute
to my success. Before this, I would like to thank all my comprehensive exam and dissertation thesis defense committee members, Dr. Guowei Wei, Dr. Keith Promislow,
Dr. Moxun Tang, Dr. Changyi Wang and Dr. Zhengfang Zhou. It is their mathematical wisdom, help and personal humor that guide me through my Ph.D. career
with happiness, make me mature in academics and give my doctoral study a perfect
end.
My first thanks go to my advisor, Dr. Guowei Wei, who always has intense faith
in the capabilities of his students and encourages them to pursue the next higher goal
in research. Prof. Wei brought me into my current research five years ago and it is
his interdisciplinary knowledge, great passion and infinite patience that lead me from
a young man to a potential researcher. Besides mathematics, Prof. Wei is also an
expert in physics, chemistry, biology and engineering. Five years’ study with him has
accumulated interdisciplinary knowledge for me and given me strong confidence to
face many challenges in applied mathematics. He gave us all kinds of computational
training to strengthen our basic skills and then lead us wandering in different worlds
by bringing one and other brilliant ideas. He is also the greatest mentor from whom I
can get help when I had difficulties, challenges or lost directions in my research. Most
importantly, as a group leader, he always shares the same faith and dream with us.
Not only providing guidance in our research topics, Dr. Wei gives us supports and
help in our future research career.
I would like to thank my wife, Shaoyu, the most import person in my life for
supporting me all the years. For my career, she is one of the reasons that I always
keep myself moving forward. In my life, she is my soulmate with whom I can share my
everything. We cheer up for each other’s successes and make the happiness doubled,
while considerate consolation is given in time when one of us is frustrated. Because
iv

of her unbounded and continuous love for me, our tough PhD life in a foreign country
becomes full of fun and happiness. After my graduation, we are expecting a more
wonderful life in the future.
I also appreciate my father and mother for their continuous support and encouragement, not only in my career but also in daily life. It is my luck to be born and
raised in such a harmonic and happy family. From the first day in the college to
my study aboard, they always respect my decisions and encourage me to pursue my
interests. Every week we talk to each other through the phone or web camera, they
care about every detail of our lives, give good advise and help us as possible as they
can. They have credits in my every achievement.
My thanks also go to a lot of lovely friends. Dr. Shan Zhao in University of
Alabama, Dr. Yongcheng Zhou in Colorado State University, Dr. Weihua Geng in
University of Michigan, Dr. Sining Yu and Dr. Yuhui Sun are great researchers from
whom I have learned a lot when I was new in the group. It is their fundamental work
that my research is based on. I would like to thank other current and former members
in our group: Dr. Siyang Yang, Mr. Zhan Chen, Mrs. Qiong Zheng, Mr. Langhua
Hu, Mrs. Jin Kyoung Park, Mrs. Yajie Yu, Mrs. Weijuan Zhou, Mr. Wangheng Liu,
Mr. Manfeng Hu, Mrs. Shuailing Wang, Mr. Qi Zhao, Mr. Kelin Xia, Mr. Huibing
Zhu, for their valuable help and suggestions in my research and daily life. Our group
is like a family, we celebrate every new year’s day together and BBQ in Dr. Wei’s
house each summer. We also had wonderful trips to Maryland and Toronto. I also
thank Dr. Maureen Morton, Mr. Nick Boros and Mr. Li Yang for their generous help.
At last but not least, I thank my friends Mr. Yanming Li, who is now in University
of Michigan, Mr. Linkan Bian in Georgia Tech, Mr. Shangbang Rao in University
of North Carolina and Mr. Lening Kang in Statistics, MSU. We got to know each
other when I arrived in the U.S., the days we spent together as new comers will be
the valuable memory in my life.
v

TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Introduction to Nano-electronic transistors . . . . . . . . . . . . . . .
1.1.1 Physical background . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Review of the current models . . . . . . . . . . . . . . . . . .
1.1.3 Existing challenges . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Introduction to ion channels and proton transport . . . . . . . . . . .
1.2.1 Biological background . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Review of current models . . . . . . . . . . . . . . . . . . . . .
1.2.3 Existing challenges . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Multiscale modeling theory of nano-electronic transistors and transmembrane proton channels . . . . . . . . . . . . . . . . . . . . . . .
1.4 Implicit solvent theory for structural and electrostatic analyses of biomolecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Mathematical issues and numerical challenges in model simulations .
1.5.1 Highly accurate and efficient solver for linear and nonlinear
Poisson equations with singularities . . . . . . . . . . . . . . .
1.5.2 Quantum computation for particle transport . . . . . . . . . .
1.5.3 The self-consistent iteration . . . . . . . . . . . . . . . . . . .

1
1
1
2
5
9
9
11
13

2 Modeling and computation of nano-electronic transistors . .
2.1 Theory and models . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Multiscale energy functional on an equal footing . . . . . .
2.1.2 Electronic system . . . . . . . . . . . . . . . . . . . . . . .
Double-gate MOSFET . . . . . . . . . . . . . . . . . . . .
Four-gate MOSFET . . . . . . . . . . . . . . . . . . . . .
2.1.3 Transport system . . . . . . . . . . . . . . . . . . . . . . .
The Non-equilibrium Green’s function (NEGF) formalism
Electron density . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Electrostatic system . . . . . . . . . . . . . . . . . . . . .
Individual dopant model . . . . . . . . . . . . . . . . . . .
Silicon-insulator-interface modeling . . . . . . . . . . . . .
2.2 Numerical analysis and implementation . . . . . . . . . . . . . . .
2.2.1 Decomposition of the Poisson equation . . . . . . . . . . .
2.2.2 Numerical implementation of the self-consistent iterations
vi

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2.2.3

Analysis of the model well-posedness, convergence and iteration
efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical simulation of the nano-electronic transistors . . . . . . . .
2.3.1 Device configurations . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Simulation results for double-gate MOSFETs . . . . . . . . .
2.3.3 Simulation results for four-gate MOSFETs . . . . . . . . . . .
Conclusion remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53
60
60
63
71
75

3 Modeling and simulations of transmembrane proton channels . .
3.1 Theory and model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 General description of the model . . . . . . . . . . . . . . . .
3.1.2 Free energy components . . . . . . . . . . . . . . . . . . . . .
Electrostatic free energy in the biomolecular region . . . . . .
Electrostatic free energy in the solvent region . . . . . . . . .
Proton free energies and interactions . . . . . . . . . . . . . .
Kinetic energy. . . . . . . . . . . . . . . . . . . . . . . .
Electrostatic potential. . . . . . . . . . . . . . . . . . .
Generalized-correlation potential. . . . . . . . . . . . .
External potentials . . . . . . . . . . . . . . . . . . . .
Proton total energy functional. . . . . . . . . . . . . . .
3.1.3 Total free energy functional of the system . . . . . . . . . . .
3.1.4 Governing equations . . . . . . . . . . . . . . . . . . . . . . .
Generalized Poisson-Boltzmann equations . . . . . . . . . . .
Generalized Kohn-Sham equations . . . . . . . . . . . . . . .
3.1.5 Proton density operator for the non-hermitian Hamiltonian . .
3.1.6 Proton transport . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Computational algorithms . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Proton density structure and transport . . . . . . . . . . . . .
The solution of the generalized Kokn-Sham equation . . . . .
Boundary treatment of the transport calculation . . . . . . . .
3.2.2 Dirichlet-to-Neumann mapping for the generalize PB equation
3.2.3 The self-consistent iteration . . . . . . . . . . . . . . . . . . .
3.2.4 The work flow of the self-consistent iteration . . . . . . . . . .
3.2.5 Model parameter selection . . . . . . . . . . . . . . . . . . . .
The selection of generalized-correlation potential . . . . . . . .
Choices of the dielectric constants . . . . . . . . . . . . . . . .
Effective mass of the proton . . . . . . . . . . . . . . . . . . .
3.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Electrostatic properties of the Gramicidin A channel . . . . .
3.3.2 Conductivity properties of the Gramicidin A channel . . . . .
3.3.3 Model limitations . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Conclusion remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79
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104
107
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115
116
118
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128
129

2.3

2.4

vii

4 Structure and electrostatic analysis for bio-molecules . . . . . .
4.1 The Krylov subspace theory (KSP) and preconditioner based MIBPB
solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Review of the Poisson-Boltzmann equation . . . . . . . . . . .
4.1.2 KSP based and preconditioner accelerated MIBPB solver . . .
4.2 Application to solvation energy calculations . . . . . . . . . . . . . .
4.3 Application to salt effects on protein-protein binding . . . . . . . . .
4.4 Conclusion remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .

132

5 Thesis achievements and future work . . . . . . . . . . . . . .
5.1 On the modeling and simulations of nano electronic transistors . . .
5.1.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 On the modeling and simulations of transmembrane proton channels
and biomolecule structures . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155
156
156
158

A The MIB method . . .
A.1 Dimension splitting . .
A.2 Derivative elimination
A.3 Derivative evaluation .

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159
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B Krylov subspace method and preconditioning for the MIB scheme 173
B.1 Linear equation systems and MIB matrices . . . . . . . . . . . . . . . 173
B.2 The Krylov subspace theory and preconditioners . . . . . . . . . . . . 177
C A short user manual for the MIBPB package . . . . . . . . . . 180
C.1 Work flow of the MIBPB package . . . . . . . . . . . . . . . . . . . . 180
C.2 Work flow for the display of the surface electrostatic potential . . . . 184
Bibliography

. . . . . . . . . . . . . . . . . . . . . . . . . . 188

viii

LIST OF TABLES

2.1

Computational L∞ error of the model with MIB scheme . . . . . . .

4.1

Physical units notations . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.2

Some physical constants . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.3

Iteration numbers and CPU time for the discretization matrices of
proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

4.4

Iteration time for different combinations of KSP solvers and preconditioners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

4.5

Convergence test of MIBPB solver with a set of proteins . . . . . . . 143

4.6

Comparison of CPU time for the PETSc and the SLATEC schemes . 145

4.7

Solvation energies and CPU time for proteins . . . . . . . . . . . . . 149

4.8

Solvation energies (kcal/mol) and CPU time (second) for large proteins 150

4.9

Binding affinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

ix

59

LIST OF FIGURES

2.1

Illustration of a double-gate MOSFET with its y-direction being infinitely long. (a) Configuration of the double-gate MOSFET; (b) Computational domain. For interpretation of the references to color in this
and all other figures, the reader is referred to the electronic version of
this dissertation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

An illustration of a four-gate MOSFET (i.e., silicon nanowire transistor). (a) Configuration; (b) Cross section at y = 0 for the computational domain; (c) Cross section at x = 0. . . . . . . . . . . . . . . .

34

2.3

Work flow of the overall self-consistent iteration. . . . . . . . . . . . .

50

2.4

Computational errors in simulating a double-gate MOSFET. (a) Individual doping in the source and drain with the dopant distribution
shown in the lower panel of Fig. 2.5(a); (b) Individual doping in the
channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

Dopant distributions and iteration errors for the double-gate MOSFET. (a) Two distributions of 10 dopants. The distribution in the
lower panel may lead to a divergence in the iteration. While the distribution in the upper panel leads to convergence; (b) Relaxation-factordependent convergence behaviors for the dopant distribution shown in
the left-lower panel. . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Errors of simulating the double-gate MOSFET in the line of y = 0 and
z = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

Contour plot of the electrostatic potential (a) and its absolute errors
obtained with three sets of meshes(b, c, d) (10 dopants). . . . . . . .

57

2.2

2.5

2.6

2.7

x

2.8

Electrostatic potential energy and difference of potentials for the doublegate MOSFET. (a) Potential landscape obtained with the standard
finite difference method; (b) Difference of electrostatic potentials between computed with the standard finite difference method and with
the MIB method. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

Profiles of electrostatic potential and electron density obtained with the
continuous doping approximation. (a) Potential function; (b) Electron
density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

2.10 Profiles of electrostatic potential energy and electron density obtained
with 5 individual dopants (N = 5). (a) Potential function; (b) Electron
density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

2.11 Profiles of electrostatic potential energy and electron density obtained
with 10 individual dopants (N = 10). (a) Potential function; (b)
Electron density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

2.12 Profiles of electrostatic potential energy and electron density obtained
with 40 individual dopants (N = 40). (a) Potential function; (b)
Electron density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

2.13 Subband energies of the double-gate MOSFET obtained with individual dopant model. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

2.14 Subband energy profiles under different gate voltage biases. (a) Continuum dopant; (b) Discrete dopants (10). . . . . . . . . . . . . . . .

68

2.15 Subband energy profiles under different source/drain voltage biases.
(a) Continuum Dopant; (b) Discrete Dopants (10). . . . . . . . . . .

69

2.16 The dependence of I-V profiles on the number of individual dopants.
(a) Fluctuation of 5 dopants; (b) Fluctuation of 10 dopants; (c) Fluctuation of 40 dopants; (d) Fluctuation of different number of dopants.

70

2.17 The comparison of I-V profiles under different doping models revealing
the individual dopant induced voltage threshold lowering effect. (a)
VS/D = 0.2V; (b) VS/D = 0.4V. . . . . . . . . . . . . . . . . . . . . .

71

2.18 Subband energies for the four-gate MOSFET. The dopants break the
symmetry and energy degeneracy in the second and third subbands.

72

2.19 Cross-section of potential profiles for the four-gate MOSFET obtained
with (a) Continuum doping; (b) Discrete dopants (10). . . . . . . . .

73

2.9

xi

2.20 Current fluctuations of the four-gate MOSFET with (a) 15 dopants;
(b) 40 dopants; (c) 80 dopants; (d) Different number of dopants. . .
3.1

(a) Illustration of multiscale model of a proton channel; (b) Computational domains of the multiscale model with Ωm being the channel
molecule and membrane domain and Ωs being the solvent domain.
Here z-direction is regarded as the transport direction. . . . . . . . .

74

80

3.2

Work flow of the overall self-consistent iteration. . . . . . . . . . . . . 112

3.3

3D illustration of the Gramicidin A (GA) channel structure and surface
electrostatic potential. The negative surface electrostatics as indicated
by the intensive red color on the channel upper surface and inside the
channel pore implies that the GA selects positive ions. (a) Top view
of the GA channel; (b) Side view of the GA channel. . . . . . . . . . 117

3.4

Electrostatic potential and charge density of the GA channel along the
z-axis obtained with m = 2 and n0 = 0.1 molar (Red: ch = 20;
p
Green: ch = 40; Blue: ch = 80). (a) Electrostatic potential profiles
in channel; (b) Proton density profiles in the channel. . . . . . . . . 118

3.5

Electrostatic potential and charge density of the GA channel along the
z-axis obtained with m = 5 and n0 = 0.1 molar (Red: ch = 20;
p
Green: ch = 40; Blue: ch = 80). (a) Electrostatic potential profiles
in the channel; (b) Proton density profiles in the channel. . . . . . . 120

3.6

Electrostatic potential and charge density of the GA channel along the
z-axis obtained with m = 10 and n0 = 0.1 molar (Red: ch = 20;
p
Green: ch = 40; Blue: ch = 80). (a) Electrostatic potential profiles
in the channel; (b) Proton density profiles in the channel. . . . . . . 121

3.7

Electrostatic potential profiles of the GA channel under different ion
reference densities n0 . Red: n0 = 0.1 molar; Green: n0 = 0.2 molar;
p
p
p
Blue: n0 = 1.0 molar; Black: n0 = 2.0 molar. m = 5 and ch = 40. . 122
p
p

3.8

The total potential of the GA channel which includes electrostatic and
generalized-correlation contibutions under various voltage biases. Dielectric constants are m = 5 and ch = 30. The pH value of the
solution is 2.75. (a) Total potential of monovalent cations; (b) Total
potential of monovalent anions. . . . . . . . . . . . . . . . . . . . . . 122

3.9

The first 15 eigenvalues (the U j (z) in Eq. 3.41) of the effective potentials along the transport direction used in the transport calculations
at the voltage bias of 0.2V. . . . . . . . . . . . . . . . . . . . . . . . 123
xii

3.10 Voltage-current relation of proton translocation of GA at different concentrations. Blue dots: experimental data of Eisenman et al [59]; Red
curve: model prediction. . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.11 Conductance-concentration relation of proton translocation at a fixed
voltage. Voltage bias=0.05V; Blue dots: experimental data of Eisenman et al[59]; Red curve: model prediction. . . . . . . . . . . . . . . 126
4.1

The implicit protein-solvent system . . . . . . . . . . . . . . . . . . . 134

4.2

Condition numbers over 15 proteins (PDB IDs from protein 1 to protein
15: 1ajj, 1vii, 1cbn, 1bbl, 1fca, 1sh1, 1vjw, 1fxd, 1bpi, 1a2s, 1frd,
1svr, 1a63, 2erl, 2pde). (a) Condition numbers for unpreconditioned
(unPCed) MIBPB matrices; . . . . . . . . . . . . . . . . . . . . . . . 139

4.3

Condition numbers over 15 proteins (PDB IDs from protein 1 to protein
15: 1ajj, 1vii, 1cbn, 1bbl, 1fca, 1sh1, 1vjw, 1fxd, 1bpi, 1a2s, 1frd, 1svr,
1a63, 2erl, 2pde). (b) Comparisons of condition numbers under three
settings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

4.4

Comparison of solvation energies of proteins (From protein 1 to 19:
1ajj, 1bbl, 1vii, 1cbn, 2pde, 1sh1, 1fca, 1fxd, 1vjw, 1bor,1hpt,1bpi,
1mbg, 1r69, 1neq, 1a2s, 1svr, 1a63, 1a7m) calculated by using the
MIBPB and the APBS methods. . . . . . . . . . . . . . . . . . . . . 147

4.5

Comparison of CPU time of preconditioned (PCed) and un-preconditioned
(unPCed) MIBPB methods for 19 proteins (from protein 1 to 19: 1ajj,
1bbl, 1vii, 1cbn, 2pde, 1sh1, 1fca, 1fxd, 1vjw, 1bor, 1hpt, 1bpi, 1mbg,
1r69, 1neq, 1a2s,1 svr,1a63, 1a7m). . . . . . . . . . . . . . . . . . . . 148

4.6

(a) Binding affinity of 1emv; (b) Binding affinity of 1beb.

. . . . . . 153

C.1 Work flow of the MIBPB package . . . . . . . . . . . . . . . . . . . . 182
C.2 Visualizations of surface electrostatic potentials of protein 1beb. From
left to right: (a) Surface electrostatic potential (I = 0, h = 1.0˚); (b):
A
Surface electrostatic potential with the ionic strength I = 1 (h = 1.0˚);
A
(c): The difference of surface electrostatic potentials between in an
ionic solvent (I = 1) and in a water solvent (I = 0); (d): The difference
of surface electrostatic potentials in water (I = 0) between grid meshes
h = 1.0˚ and h = 0.5˚. . . . . . . . . . . . . . . . . . . . . . . . . . 185
A
A

xiii

Chapter 1
Introduction
1.1
1.1.1

Introduction to Nano-electronic transistors
Physical background

The continuous demand in rising the performance of electronic devices has led to
the reduced geometric dimension and supply voltage of metal oxide semiconductor
field effect transistors (MOSFETs), or complementary metal oxide semiconductors
(CMOSs), which are fundamental building blocks of large scale integrated circuits
used in almost all electronic equipments. At present, MOSFETs are designed, manufactured and operating on much less than 100nm scale. According to “International Technology Roadmap for Semiconductors (ITRS) (http://www.itrs.net/)”,
the channel length of CMOSs will be down scaled from the present 45 to about 22
nm in 2016. The down-scaling of the transistor channel length also requires simultaneous down-scaling of the gate oxide, connecting material, doping concentration and
operation voltages [55, 40, 19]. The ultimate channel length is expected to be around
16 nm. At such a channel length, most critical design parameters quickly approach
the atomic scale and associated physical limits. Many down-scaling associated devices characteristics have been studied by Vasileska, et al [4, 3, 70, 72, 90, 71, 91]
1

for MOSFET, FinFET and various other silicon on insulator (SOI) devices. The
most important feature of a nano-scale transistor is that quantum mechanical effects become significant and will dramatically impact the macroscopic quantities,
such as current-voltage characteristics and conductance. In particular, at 22 nm or
less, channel tunneling and gate leakage may devastate the classical function of the
MOSFET. Electrostatic control and suppression of quantum effects are important
issues [116, 129, 54, 43, 137, 83, 6, 51, 120, 49, 10, 150]. Nano transistors with the
range of channel lengths being around 20 nm and 10 nm are referred as “ultimate
CMOSs” and “functionally enhanced CMOSs”, respectively. Ultimate CMOSs are
the smallest CMOSs that still operate with the classical principle while severe quantum effects have to be suppressed by appropriate electrostatic potentials and designs.
Functionally enhanced CMOSs are nano-quantum transistors which utilize the fundamental properties of nature that do not have direct analogs in classical physics.
Some of these properties are quantum coherence, i.e., a possibility for a quantum
system to occupy several states simultaneously, and quantum correlation or entanglement. Presently, the majority of these quantum structures, such as nano-mechanical
resonators, quantum dots, quantum wires, single electron transistors, and similar low
dimensional structures, exist only as prototypes in research laboratories or just being
contemplated [86, 64]. The working principle and physical function of quantum devices are subjects of extensive research. Practical realization of quantum transistors
faces a number of challenges in design, test, material selection, lithography, interconnect, process integration, metrology, assembly, packaging, plus device modeling and
simulation.

1.1.2

Review of the current models

The main purpose of the device modeling is to predict device characteristics and
performance. This amounts to the understanding of transport features, including
2

current-voltage (I-V) characteristics at the source, drain or gate contacts of the device.
For ultimate MOSFETs and other nano-quantum transistors, quantum effects, such
as gate leakage and channel tunneling under various voltage settings will be of main
concerns of the modeling and simulation [96, 143]. Currently, the non-equilibrium
Green’s functions (NEGF) formalism is the main workhorse for nano-device transport
modeling [153, 74]. The NEGF formalism was originally developed by Schwinger
[136], Kadanoff and Baym [89], and has been revived recently for device modeling
[45, 96, 44, 143]. This is a general and useful formalism using the Fermi-Dirac statistics
for electrons. It allows the description of interactions, including scattering processes
of particles (i.e., electrons and phonons) and relaxation due to the surroundings.
An equivalent approach is the Dyson integral equation representation. However,
computational aspects for differential and integral equations are quite different.
Another practical transport model is the Boltzmann equation, or the BoltzmannVlasov equation, which describes the kinetic of a typical particle (e.g., electron,
phonon or photon), due to the two-body scattering with another particle and/or
external field effect [73, 22]. The inherent Boltzmann distribution can be a good
approximation to the Fermi-Dirac distribution at high temperature. Transport properties, such as current density, conductance and tunneling rate, can be computed as
expectation values of physical observables with the distribution function, the Wigner
distribution [85] or density operator [3]. In fact, the transport equation derived
from the quantum Boltzmann equation, known as the Waldmann-Snider equation
[151, 142], can provide quantum correction to the classical drift-diffusion expression.
The Waldmann-Snider equation can be derived from the BBGKY hierarchy with an
appropriate scattering closure for the two-body density operator. Other density matrix methods, such as the Master equation [82, 65], describe the time evolution of the
probability function. Assuming a continuous-time Markov process, the integrated
master equation obeys a Chapman-Kolmogorov equation. However, to describe the
3

transport of electronic devices, the transition matrix of the master equation has to
be evaluated by other reliable means. The semiconductor Bloch equation is derived
by using the slowly varying envelope and rotating wave approximation, providing a
diagonally-dominated calculation of currents and allowing a simple approximation of
scattering. Yet another approach is the Fokker-Planck equation describing the rate
change of the probability density function of a particle in terms of drift-diffusion processes [116, 129, 109]. This equation can be used to model the electron transport
in the quantum ballistic regime [113]. Additionally, Monte Carlo methods have also
been applied to electron transport [22, 4, 90]. The electron scattering effect from
the devices interface roughness was studied via an ensemble Monte Carlo device simulation technique [91]. A new scheme was proposed and applied to study the role
of the discrete impurities in the device terminal characteristics [70, 71, 72]. By using a corrected Coulomb force, this approach prevents the double-counting of the
electron-electron and electron-ion long-range interaction.
To account for the quantum effect, the electronic structure in terms of wavefunctions is required in most transport evaluations. The quantum mechanical theory is
indispensable for electron structures at nano scale. To this end, one has to select
the level of the description of the quantum system and the level of the approximation to governing equations. The description of electron structures depends on the
level of approximations and the size of the quantum system depends on the level
of sophistication of the model. Although formally a fully quantum mechanical first
principle description is desirable for a given device feature at nano scale, it normally
involves a large number of atoms, molecules and electrons. Therefore, the resulting
full scale quantum system is intractable and appropriate approximations are required.
Currently, practical models describe the dynamics of a few electrons or even a single
electron in which the indispensable essential feature of the quantum effect is retained
and different levels of approximations can still be derived depending on how the in4

teraction of the single electron with other particles is handled. At the lowest level of
approach, a single electron dynamics in a band structure of the solid is governed by
the Schr¨dinger equation, which is coupled back to the Poisson equation as charge
o
sources [104, 146, 15, 87, 99, 8, 62, 133, 13]. The interaction of many bands can also
be considered by using a general k · p method derivation of many-body Schr¨dinger
o
equation [162, 64, 106]. Recently, linear combination of bulk band (LCBB) method
[83], which relies on the expansion of the confined states in terms of periodic Bloch
wave functions, was used for a large number of atoms. Some quantum corrected classical methods, such as quantum drift-diffusion (QDD) models or Schr¨dinger-Poisson
o
drift-diffusion (SPDD) models, are employed and summarized in a unified framework
[48, 51]. The well-posedness of these models and numerical efficiency are analyzed
mathematically in the fashion of solution fixed point maps [51].

1.1.3

Existing challenges

Apart from difficulties with device fabrication and testing which typically require
nano scale resolution and high precision control, there are numerous modeling and
computational problems associated with ultimate and functionally enhanced nano
transistors as discussed by the ITRS. The essences of these problems are quantum
effects, geometric interface effects and dopant effects.
1) Quantum effects include electron confinement, resonance states, source-drain
off state quantum tunneling current, channel barrier tunneling, gate leakage, many
body correlations and channel-channel interference at nano scale. These effects are
commonly modeled by the coupled Poisson-Schr¨dinger equations. However, the
o
consistence and validity of these equations have rarely been examined. There is a
pressing need for innovative methods, models and algorithms that contribute to the
prediction and design of nano-quantum transistors whose channel lengths are in the
range of 8-22 nm.
5

The solution of the many-electron Schr¨dinger equation, including atomic inforo
mation, is extremely expensive. Semi-empirical approaches which make use of parameters from experimental data are often used. More rigorous but expensive methods
are ab-initio approaches, including the Hartree-Fock method [141] and the density
functional theory (DFT) [76, 93, 117]. The size of the system is limited when abinitio methods are used. To increase the computational capability, pseudopotential
methods can be used to remove core electrons and singularities in calculations. The
resulting quantum mechanical system is still formidably expensive to solve for nano
devices. The DFT is associated with the Kohn-Sham equation and it can be accelerated by using the linear scaling divide and conquer method [156, 101], and the tight
binding approximation [79]. In general, there is a pressing need to develop innovative
modeling strategies and efficient computational methods for realistic device problems.
To improve the accuracy of the electron structure, it is necessary to consider atomistic models. The core electrons on different atoms are essentially independent of the
state of the surrounding atoms. Therefore, only valence electrons participate effectively in interactions between atoms. Thus, the core electron states can be assumed
to be fixed and a pseudo-potential may be constructed for each atomic specie which
takes into account the effects of the nucleus and core electrons. As such, one only
needs to explicitly consider the valence electrons. Furthermore, the tight binding
model assumes that the full Hamiltonian of the system can be approximated by the
Hamiltonian of an isolated atom centered at each lattice point for a solid-state lattice
of atoms. This simplifies the formulation and offers a further saving in computational
effort.
2) Geometric interface effects refer to the impact of (layered) material variations within a device and interconnects between devices to the device performance.
These effects become crucial to ultimate CMOSs and functionally enhanced CMOSs.
For example, dielectric interfaces of metal-oxide, metal-semiconductor, and oxide6

semiconductor will induce non-ballistic transport behavior even if there is no other
interaction [91]. However, with few exceptions [103], most present simulation models
are based on simplified rectangular geometric shapes, homogeneous dielectric media,
and even reduced dimensions. The impact of realistic geometry, including gate dielectric layers and interconnects, has hardly been investigated in the past and calls
for new modeling strategies and innovative methods.
Typically, dielectric constants of different components in MOSFETs vary dramatically. For example, the dielectric constant of the silicon dioxide insulator is a few
times smaller than that of the buck silicon substrate. The ratio of dielectric constants
in different layers is also important to the device scaling. According to device scaling
physics [66], the scale length Λ of the device, for the first order approximation, depends on the insulator thickness (TI ) and the ratio ( Si / I ) of dielectric constants of
the silicon and the insulator in the way
Λ = Wdm + TI Si ,
I

where Wdm is the maximum channel depletion depth relating to the channel doping
concentration. The above theory predicts that the proper minimum design length
lies between Λ and 2Λ. It is clear that smaller value of TI and larger value of I help
device scaling. Replacing the silicon dioxide gate dielectric with a high-k material allows increased gate capacitance without the concomitant leakage effects. The proper
formation of distinct interfaces is a stringent requirement for ultimate CMOSs and
functional electronic devices to suppress leakage currents due to tunneling, as the
thickness scales much below 2 nm. Computationally, it is important to be able to
simulate the interface roughness and irregularity due to the device fabrication processes [91]. The use ofinterface description is indispensable for modeling of ultimate
and functional CMOSs.
7

3) Introducing appropriate impurity atoms (known as dopants) into a semiconductor provides electron reservoirs and can increase the electrical conductivity by many
orders of magnitude. By doping a semiconductor device, we can engineer its electrical
properties, e.g., its conductivity, electrostatic potential and its charge carrying mode.
Doping is a key to our understanding of semiconductor devices and a strategy for
the design and manufacture of desirable devices [70, 71, 72]. Doping effects are often
described by distribution functions in continuum device models without explicit consideration of individual dopant atoms and traps [25]. This continuum approach works
very well for electronic devices of large sizes but will lead to severe errors in electron
structure and transport for ultimately scaled nano devices. These errors are often seen
as statistical fluctuations [92, 63]. When the device size is reduced to 22 nm or less, it
becomes indispensable to consider individual dopant atoms and traps. However, with
few exceptions [83, 70, 71, 72], this issue has been hardly addressed in the literature.
As individual dopants are fundamental to the function of ultimate MOSFETs and
nano-quantum transistors, it is imperative to develop innovative models and efficient
methods to analyze their impact.
In continuum modeling, dopants have either been described as continuous distributions in p-n regions or been formulated as a change in the dielectric effect, leading
to different dielectric values in different p-n regions. These treatments work mostly
well for the prediction of device properties. However, when the channel length reduces
to about 10 nm, the quantum effect becomes important. Thus, each doping atom may
have a dramatic impact to the quantum state of nearby electrons. Atomistic model
for dopants becomes indispensable. Wong and Taur [155] provided a classical study
of discrete random dopants. Recently, quantum random dopant models are applied to
the channel of sub-0.1µm [9] and 25nm [84] MOSFETs for threshold voltage lowering
and fluctuations. The impact of random dopant aggregation in source and drain is
studied via the NEGF formalism [108]. It is found in these studies that doping is
8

only macroscopically controllable when the discrete microscopic dopant distribution
is also controlled. Macroscopically, identical devices may suffer from strong performance variations because of the microscopic differences. Therefore, it is important to
understand individual dopant effect in nano electronic devices.

1.2

Introduction to ion channels and proton transport

1.2.1

Biological background

There are a couple of seemingly conflicting fundamental requirements for a living cell
to survive and function properly: On one hand, the cell needs the protection of the
plasma membrane, which works as a potential barrier and maintains the intracellular
electrolyte composition that may be different from that of the extracellular environment. On the other hand, information communication and material exchange must
be established between the intracellular and extracellular environments for all living
cells. A wide variety of biological processes, such as signal transduction, nerve impulse
and so on, are modulated and sometimes, initiated by the intra/extra-cellular information and material exchanges. These two conflicting tasks are accomplished by ion
channels, which are proteins with pores and embedded in lipid bilayers, selectively
permitting the permeation of specific ions. Because of these important biological
roles, as well as frequently serving as the target for drug designing, ion channels have
attracted great research interest in experimental, theoretical and computational explorations. Most research activities are focused on a few ion channel properties [75]:
(i) The gating of ion channels. Ion channels are not always open or close. Based on
the mechanism controlling the open/close status, they are categorized as ligand-gated
ion channel (the channel is open only when the specific ligand is bound to the ex9

tracellular receptor domain), voltage-gated ion channel (the channel is open/close by
the regulating membrane potential) and other gating channels, such as mechanical,
sound, and thermal stimuli. It is worthwhile to point out that the present work does
not focus on the ion channel gating mechanism — channels discussed here are all
assumed open. (ii) The selectivity of the ion channel. When an ion channel is open,
it is not open to all the ion species, only certain ions can penetrate. In this sense,
ion channels are also classified by the permeating ions, such as potassium channels,
sodium channels, and proton channels, etc. (iii) The efficiency of ion conductance.
When an ion channel is open and conducts a specific ion species, the efficiency of
ion conductance is of major interest, which is measured by the current-voltage (I-V)
curve. Technological advance in the past few decades makes it possible to measure
I-V curves through a single channel for a variety of ion channels under physiological
conditions. These techniques are considerably empowered by the genetic engineering technology to identify the gating mechanism. (iv) Structural analysis. Many
channel protein structures have been discovered by X-ray crystallography, nuclear
magnetic resonance (NMR) and cryoelectron microscopy. Channel protein structural
information is deposited in the Protein Data Bank (PDB). (v) Theoretical and computational research. Abundant knowledge about ion channels accumulated by experimental means has created an excellent testbed for theoretical modeling and prediction
of ion channel transport. Various mathematical/physical models have been proposed
for numerical simulations. However, there are still many important theoretical problems in the field [37]. One of the problems concerns the dynamical detail of the ion
permeating process. Due to the relative narrowness of the pore size, the ion-water
geometry needs to be rearranged in order for ions to successfully cross the channel.
Therefore, the orientation and polarity of water molecules, the interaction between
partially dehydrated ions and fixed charges on the protein wall must be significantly
different from those under the bath condition. Another problem is the precise role
10

of quantum effects in many proton channels, such as the narrow M2 channels of Influenza A. These problems pose challenges for theoretical/mathematical modelings.
Commonly used approaches include molecular dynamics, Brownian dynamics, and
the Poisson-Nernst-Planck (PNP) equations. There are a number of excellent reviews [37, 95, 58, 132, 127, 126] for various theoretical models at a variety of levels of
descriptions and approximations.

1.2.2

Review of current models

Molecular dynamics (MD) provides one of the most detailed descriptions in modeling biomolecular systems and there are several user-friendly packages available, such
as AMBER [118], CHARMM [105], etc. In fact, MD is one of viable models which
are able to predict the ion selectivity in ion channel modeling. However, the use of
MD in modeling ion permeation is still limited. The most significant barrier for MD
applications in ion channels is the difficulties of predicting the channel conductance,
which is the primary physical observable. Extremely small time step (around 1 or
2 femto seconds) has to be employed in the numerical integration of the Newton’s
equation to obtain the necessary accuracy because the fast time scale of molecular
bond motions. Whereas, a typical channel current (with the magnitude of the order of pico Ampere) corresponds to average transit time of tens of nanoseconds for
a single ion. Therefore, the MD simulation must last around microseconds in order
to obtain sufficiently accurate conductance calculations. Due to the total simulation
time needed and the necessarily small time step, the MD computation without invoking crude approximations is still not affordable with current computers for accurate
conductance prediction. Therefore, the full scale MD simulation of ion channels is not
feasible. In practice, it is still very useful for MD simulations to obtain alternative
channel configurations, solvent polarizations, diffusion coefficients, etc., in assisting
other approaches for the transport estimation [37].
11

Brownian dynamics (BD) [36] based on the Langevin equation treats ions as explicit particles in the ion channel modeling, while describes the surrounding environments (channel proteins and lipid bilayers) implicitly by a continuum approach. In
Brownian dynamics, there are many forces which act on the target ion [37]. First,
there is a force from fixed charges in the protein and membrane, as well as the applied external field. Additionally, there is a force from the self-induced charge by an
ion on the channel boundary. When ions pass through the channel, there is always
a repelling force induced at the channel boundary against the ion motion. Finally,
there is a force from mobile ions in bath regions. Among these three components,
the force from fixed charges can be obtained by solving the Poisson equation in the
absence of mobile ions, whereas forces due to mobile ions can be evaluated by solving
the Poisson equation while switching off fixed charge and applied field, and allowing
ions to move around all the grid points. Once these Poisson equations are solved
numerically, the forces are pre-stored in the grid and ready to be used to determine
ion trajectories.
By assuming a mean-field approximation, the Poisson-Nernst-Planck (PNP) model
[140] is a continuum electro-diffusion theory which treats not only the protein, lipid
layer, bath solution as continuums, but also the ions of interest. The Poisson equation provides the electrostatic potential profile in the whole computational domain
based on charge sources from mobile ions in the solution and fixed charges in the
channel protein and lipid layer. The gradient of the electrostatic potential gives rise
to the driven force, which, together with the gradient of ion density, is used in the
Nernst-Planck equation to determine ion density flux. Therefore, the ion density
distribution is governed by both the electrostatics induced drifting and the density
gradient induced diffusion. The ion conductance is computed from the charge flux.
Obviously, both the BD and PNP models have a number of similarities in their initial
setups and computational approaches [107, 122, 57].
12

1.2.3

Existing challenges

Due to its computational efficiency, the PNP model has been widely implemented
for various ion channels [77, 80, 88, 161] embedded in different lipid bilayers [80, 24].
Many mathematical analyses, for example, derivation of the NP equation from Boltzmann equation via perturbation theory [140], asymptotic expansions of the I-V relations [1], accelerating algorithms [56] and inverse problems related to the ion selectivity [23], are also popular research topics in the field. However, the validity of the
PNP model has been questioned in many aspects, particularly for narrow ion channels
[38, 111, 69]. Arguments root from the theoretical defect that ions are treated as continuum instead of particles in the narrow channel. This continuum assumption is only
reasonable under bulk concentration condition or a channel pore with a sufficiently
large diameter. First of all, it is conceptually difficult to define ion “concentration”
when the diameter of a channel pore is comparable to that of an ion. Secondly, when
the scale is down to a couple of angstroms, non-electrostatic factors such as Brownian
motion, may become important or even dominant. The screening effect is significant
when the channel diameter is smaller than the Debye length of the realistic electrolyte.
In this situation, ion particles induce dielectric boundary charges, which result in a
dielectric self-energy (DSE) barrier. The PNP model neglects these energy barrier
factors [24, 37], and usually overestimates biological quantities of interest. The PNP
model also ignores the non-electrostatic forces and self-energy, and employs an artificially reduced diffusion coefficient (about a factor of 1/50) to fit experimental data
[77]. Several modified PNP models have been proposed, in which the ion self-energy
is obtained either by using the Poisson equation [39, 69] or the MD [107] simulation,
and is added to the Nernst-Planck equation.
Apart from the ion transport of sodium, potassium and calcium, the long range
proton transfer (LRPT) across biomembranes is also of central importance and plays a
13

major role in many biochemical processes, such as cellular respiration, ATP synthase,
photosynthesis and denitrification [94]. The LRPT is usually realized via proton
channels or proton nanowires, where water molecules are connected in a chain to
conduct protons. Two common examples of proton channels are the Gramicidin A
(GA) and the newly discovered M2 proton channel of Influenza A [134]. Theoretical
investigation has been extensively carried out and various experimental data about
the proton flux are available [122, 123, 121]. However, the main mechanism of the
LRPT is not fully understood yet [144], with the belief that protons are totally different from other ions and have larger conductance. In the Grotthuss-type mechanism
theory [112, 2], protons achieve the translocation in the channel through a succession
of hops along a single chain of hydrogen-bonded water molecules, i.e., an existing
hydrogen bonded network, compared to other ions for which the permeation occurs
mainly via hydrodynamic diffusion. The actual transfer through the hydrogen bonded
net work is usually fast and both the rearrangement of the hydrogen bonded net work
and energy barrier are considered as rate limiting factors. There is an agreement that
the aforementioned BD theory and PNP model may be expected to work well for
heavy ions but not for protons, which have lighter mass and whose transfer involves
the hydrogen bonds making and breaking. These processes need to be studied quantum mechanically. Some investigators have explored proton channels via Feynman
path integral simulations and quantum energy levels of protons are computed by the
Schr¨dinger equation [121, 123, 122]. Several theoretical models are proposed in the
o
last decade [135, 144, 21].

14

1.3

Multiscale modeling theory of nano-electronic
transistors and transmembrane proton channels

It is not appropriate to apply any single modeling strategy to interpret such complex
nano-transistor and proton channel systems. Fully atomic models such as molecular
dynamics keep the number of degrees of freedom for the model but impose unaffordable computational burden to model simulations. While for the pure continuum
approximation, many detailed information will lost. Therefore, multiscale or multiphysical modeling techniques become sharp tools to analyze these systems. Much
of the modeling effort is devoted to the proton channel because of the biological
complexity.
The ion channel system is divided into the solvent and the molecule subdomains.
The former includes the extra/intra cellular bulk aqueous environment, as well as
the solvent in the channel pore. Because of the huge number and fast rotating of
water molecules, it is computationally expensive to trace their motions individually.
Additionally, water molecules are also not the object of main interests. Therefore,
the water is implicitly treated as a structureless dielectric continuum. The molecule
subdomain consists of channel protein and lipid bilayers. The structure and property
of the channel protein is paramount to the proton permeation, so the continuum
model is not appreciated. Fortunately, the total number of atoms of the channel
protein is not high (usually up to thousands), one can record the atomic details, such
as the positions, radii and partial charges of atoms to give explicit representation for
the channel protein. On the contrast, the structure of the lipid bilayer is relatively
simpler, it can also be approximated by a dielectric continuum.
Besides the multiscale treatment for the model, different physical principles are
15

employed to study objects of different interest. The most important ion is the proton.
Due to the special transport mechanism plus unique properties, the quantum mechanics is used to illustrate the permeation process of protons. For other ion species, they
are all treated in classical mechanics and their densities are assumed to obey the
Boltzmann distribution. The motion of protons is under intensive electrostatic and
other interactions with the mobile ions in the solvent, fixed charges in the channel
protein, the water molecules, and other surrounding environments. Ion species are
modeled with various mechanisms and corresponding approximations, it is the mutual interactions among ions that recover the whole system as a reasonable approach
to the primitive biological process.
The multiscale/domain/physical idea can be also applied to the modeling of
nano-transistors. We introduce a two-scale variational framework that, upon energy
optimization, generates new self-consistently coupled Poisson-Kohn-Sham equations
which allow the easy incorporation of linear scaling tight-binding, pseudopotential,
atomic charges and dopants, divide and conquer methods. The proposed framework
puts macroscopic description of electrostatic potentials and the microscopic description of electronic structures on an equal footing at nano-scale. MOSFETs are made
of several materials, for example, the silicon as semiconductor material and the silicon
dioxide as insulator. These materials have different dielectric permittivities. Therefore, interface models and associated elliptic interface techniques are introduced to the
nano-electronic device modeling and computation. Because of the nanometer downscaling of devices and the strong confinement of the channel, electron transport must
be illustrated in a quantum mechanical way. Finally, we provide a new mathematical
model to account for the random individual dopant effect in semiconductor material.
The Dirac delta function is used to represent the dopant position and eliminate the
finite-size effect in the previous discrete dopant models.
In work of this thesis, the multiscale modeling method and multi-physical tools
16

are attempted to study nano-transistors and proton channels. Energy components
of both systems are integrated on an equal footing framework, in view of total free
energy functional. Governing equations can be derived from the energy functional
by the variational principle for both problems. Generally, one elliptic type equation
(the Poisson equation for nano-transistors and the generalized Poisson-Boltzmann
equation for proton channels) and the generalized Kohn-Sham equation are derived,
the former usually provides the electrostatic landscape of the system while the latter
describes the motion of quantum particles.

1.4

Implicit solvent theory for structural and electrostatic analyses of bio-molecules

Implicit treatment, or continuum approximation of the solvent is a ubiquitous technique of multiscale modeling for ion channel systems. It is also important for structural and electrostatic analyses of general bio-molecular systems.
Under physiological conditions, almost all important biological processes, for example, signal transduction, DNA specification, transcription, post transcription modification, translation, protein folding and protein ligand binding, occur in water which
comprises 65-90% of cellular mass. An elementary prerequisite for the quantitative
description and analysis of the above-mentioned processes is the understanding of solvation, which involves energetics of interactions between solute molecules and solvent
molecules or ions in aqueous environment. Solute-solvent interactions are typically
classified as the polar type and the non-polar type. Although widely used, this classification is arbitrary and has caveats associated with the non-unique descriptions, as
well as the intrinsic coupling between these two types of interactions. The polar type
of solute-solvent interactions is the main interest of the present work. It originates
from electrostatic effects, which play important roles in biophysics, biochemistry,
17

structural biology, electrochemistry and electrophoresis.

The solvent has a substantial volume and a significant contribution to electrostatics via numerous mobile ions. However, it is the solvated solute molecule that
is the focus of the interest in most research. As such, the solute is described in
atomic or electronic detail, while atomic details of the solvent and mobile ions are
approximated by a mean-force description and probability distribution, respectively.
This implicit solvent method can greatly reduce the computational cost of the traditional explicit solvent methods, in which a microscopic description of the solvent
is retained. Various implicit solvent models are available to describe polar solvation
[128, 138, 47, 29, 12]. The most widely-used methods are currently the generalized
Born method [53, 61, 166, 110, 29], polarizable continuum [35, 145, 81] and PoissonBoltzmann equation (PBE) [11, 97, 138, 47] models. The use of polarizable continuum
models is mostly restricted to small molecular systems. Generalized Born methods
are very fast but are only heuristic models for estimating polar solvation energies
of biomolecular structures. These methods are often used in high-throughput applications such as molecular dynamics simulations[147, 139, 60]. PBE models can
be formally derived from Maxwell’s equations [16] and offer a somewhat slower, but
more accurate way for evaluating polar solvation properties [46, 114, 14]. Additionally,
PBE techniques are often used to parameterize and assess the accuracy/performance
of generalized Born models [114, 114, 148]. Finally, unlike most generalized Born
methods, PB models provide a global solution for the electrostatic potential and field
within and around a biomolecule, therefore make them uniquely suited to visualization and other analysis [102, 18] that require global information about electrostatic
properties.
18

1.5

Mathematical issues and numerical challenges
in model simulations

Other than sharing similar modeling methodologies, implementations of both nanotransistors and proton channel models encounter common numerical challenges, which
have attracted great mathematical interests for the last decades. For example, the
multidomain and multiscale treatment of both systems results in interface problems
and delta functions in the partial differential equation. Highly accurate and efficient
numerical schemes are required to handle these singularities. Additionally, desirable
computational theory and algorithms for the scattering of quantum particle in finite
region are indispensable. Furthermore, numerical convergence and efficiency of the
self-consistent iteration need to be explored for the coupled governing equations that
are derived from the total free energy functional. The numerical challenges and
associated mathematical treatments in model implementations are outlined as follows:

1.5.1

Highly accurate and efficient solver for linear and nonlinear Poisson equations with singularities

One of the governing equations, the Poison equation for nano-transistors or the generalized Poisson-Boltzmann equation for proton channels, admits interface and delta
source singularities. These singularities in the elliptic equation are successfully solved
by the evolution of the MIBPB solver, which is a MIB algorithm based PoissonBoltzmann solver package but can be easily extended to general elliptic equations.
The MIBPB-I [163], the MIBPB-II [157] and the MIBPB-III [67] have been developed
(http://www.math.msu.edu/~wei/MIBPB/). The MIBPB-I is the first PB solver
that directly enforces the flux continuity conditions at the dielectric interface in the
biomolecular context. However, it cannot maintain its designed order of accuracy in
19

the presence of MS singularities, such as cusps and self-intersecting surfaces. This
problem was addressed in the MIBPB-II by utilizing an advanced MIB technique developed by Yu et al. [158] who offered special treatments for geometric singularities.
However, the MIBPB-II loses its accuracy when the mesh size is as large as half of the
smallest van der Waals radius, due to the interference of the interface and singular
charges. To split the singular charge part of the solution, a Dirichlet to Neumann
mapping approach [33] was designed in the MIBPB-III, which is by far the most
accurate and reliable PB solver. This new solver remains accurate at the smallest
van der Waals radius, i.e., about 1.0 ˚ grid resolution for proteins. Comparing to
A
traditional PB solvers, the MIPPB-III is a few orders of magnitude more accurate
at a given mesh size and about three times faster at a given accuracy [157, 67]. The
MIBPB is the first and still the only known second-order convergent PB solver for
the singular molecular surfaces of biomolecules, where the second order convergence
means that the accuracy of the solution improves four times when the mesh size is
halved.
The MIBPB solver serves as a powerful tool that provides the electrostatics analysis for nano-transistor and ion channel studies. Apart from the accuracy, the efficiency of the MIBPB solver is another important issue crucial to many applications.
Previous MIBPB solvers are typically slow in comparing with other standard finite
difference or finite element methods that do not invoke an interface treatment. This
comes from the trade-off of the high accuracy and convergency. Since detailed local interface information (the intersection and normal direction, etc) are included,
the matrices which represent the discretization of the elliptic operator loss their
nice properties, such as symmetry and positive definiteness, especially for complicated surfaces of biomolecules. The latest version of MIBPB solvers is equipped
with Krylove subspace (KSP) theory and preconditioner techniques to increase the
solver efficiency. Special efforts have been paid on the strategies for the selection
20

of most suitable linear system solvers for the resulting MIBPB matrices. Two linear solver libraries, the SLATEC (http://people.sc.fsu.edu/~burkardt/f_src/
slatec/slatec.html) and the PETSc (http://www.mcs.anl.gov/petsc/petsc-as/)
are considered in the exploration of linear solvers.
With the development of the MIBPB, a complete software package for solving the
Poisson-Botlzmann equation and its generalization is established for bio-molecular
electrostatics analysis. It has been applied to the molecular solvation energy calculation, salt effect based binding energy and other biological applications. The MIBPB
package is also modified and adopted in the ion channel calculation to supply the
electrostatic background of the system. Furthermore, the treatments on the interface singularities, geometric singularities and source charge singularities in MIBPB
have been implanted to the algorithms of solving Poisson equation in nano-devices
simulations.

1.5.2

Quantum computation for particle transport

To study the transport of quantum particles, the full Schr¨dinger equation for N
o
particles is computationally unfeasible because of the extremely high number of degrees of freedom. Instead, we consider the density functional theory and utilize the
generalized Kohn-Sham equation from the total energy functional. Although being
greatly simplified by the one-particle approximation, fully solving the Kohn-Sham
equation still gives computational obstacles. However, based on specific physical
properties, the Kohn-Sham operator presents distinguish behaviors under different
circumstances. Given reasonable assumptions, the computation complexity might be
reduced in further.
In usual cases, one can define three types of boundary conditions for the KohnSham equation. The first one is the infinite boundary condition, which is related to a
translation invariance of the potential. Under this circumstance or approximations,
21

the spectrum of the Kohn-Sham operator is absolute continuous and wavefunctions
can be represented by plane waves. The second type of boundary condition is the open
boundary conditions, which is for the particle transport through a position dependent potential landscape. The Kohn-Sham operator also has an absolute continuous
spectrum while the wavefunction need to be carefully solved. Finally, the confined
boundary condition is defined when a particle is confined in a given potential surface
and the probability of finding the particle out of the region is zero. Under this circumstance, the energy level of the target particle is quantized, i.e., the Kohn-Sham
operator yields a discrete spectrum.
For proton channel computation, the primary motion is the proton transport process, so the transport boundary condition should be imposed along the transport
direction, in which the channel pore is aligned. The bath regions are considered as
ion reservoirs which are assumed to have infinitely large dimension and perpendicular to the transport direction. Therefore, the Hamiltonian of the proton has an
absolutely continuous spectrum in the corresponding plane. In contrast, the channel
regions give strong confinement to protons because they are assumed not to penetrate
the channel wall. Consequently, a discrete spectrum results in the non-transport direction in the channel region. Similar treatments are also applied in the nano-devices
simulations. For MOSFETs, the major direction is also the transport direction in
which the transport boundary conditions are applied. For others, in the direction not
confined by the insulator materials, the potential may be assumed position invariance.
Thus, wavefunctions can be regarded as planewaves and corresponding energy could
be integrated out. While for directions along which the material is confined by insulators, confined boundary conditions are applied and discrete sub-bands result. In
both proton channel and nano-devices simulations, a series of 1D transport problems
eventually need to be solved after the operator splitting. Additionally, corresponding
statistical distribution functions indicate that higher energies actually contribute less
22

to the whole system. Therefore, the computation complexity of the Kohn-Sham equation is greatly reduced since the high energy components can be abandoned within
numerical tolerance. Meanwhile, because of the classification of the boundary condition or dimension decomposition of the Kohn-Sham operator, the particle density has
different structures in corresponding regions of proton system and various MOSFETs.

1.5.3

The self-consistent iteration

As introduced in earlier sections, two partial differential equations are derived from
the total free energy functional via the variational principle. The Poisson-type equations give the electrostatic background for particle transport and the Kohn-Sham
equation provides the number density of quantum particles. These two types of equations are strongly coupled. The electrostatics servers part of the potential energy
in the Hamiltonian of the Kohn-Sham equation, while the number density of quantum particles forms part of charge sources in the Poisson-type equation. Therefore,
these two equations need to be solved in a self-consistent manner till converges are
achieved. There are several interesting mathematical issues about the self-consistent
system. For example, the well-posedness, i.e., the existence and uniqueness of solutions of the PDE system needs to be carefully justified. There are several preliminary
results available in the literature, on the existence of solutions from the fixed point
mapping view [51, 120].
For the overall self-consistent iteration, directly inserting the solution of one PDE
to another usually does not give good convergence, as verified in many applications[149,
27]. Instead, the Gummel iteration [50] converts the whole self-consistent iteration
into an outer and an inner iteration, and is proved in practices to be a more efficient
way to solve the system. Essentially, it is pointed out in this thesis that the Gummel iteration is one type of Newton’s method. Therefore, various inexact Newton’s
methods could be employed to enhance the numerical efficiency. Theoretical analy23

sis on the iteration convergence of the Newton’s method is also available. Another
technique to increase the self-consistent convergence is the relaxation factor scheme
[7]. It origins from the fixed point map and converts the process of finding the fixed
point (solution) of the iteration to the equivalent process of seeking the steady state
of an ordinary differential equation (ODE). As a result, many theories and schemes of
solving ODEs can be borrowed to study the self-consistent iteration. To summarize,
other than modeling and computation, there are several interesting mathematical issues in the bio-nano system, which are worthwhile to be studied and explored in the
future research.

24

Chapter 2
Modeling and computation of
nano-electronic transistors
In this chapter we present the theoretical formulation of our model of material interface and individual dopants for nano-electronic transistors.

2.1
2.1.1

Theory and models
Multiscale energy functional on an equal footing

In the continuum mechanical description, the electric field E(r) can be expressed as
the negative gradient of the electrostatic potential u(r), i.e., E(r) = − u(r). The
standard Poisson equation can be derived from Gauss’s law describing how electric
charge can create and alter electric fields

· (r)E(r) = −

· (r) u(r) = ntotal (r)q

(2.1)

where ntotal is the free charge number density, q is the electron charge and (r) is the
permittivity. The electrostatic energy functional induced by the given free number
25

density of charge ntotal (r) can be given by
(r)
| u(r)|2 − u(r)ntotal (r)q dr.
2

EElectrostatic [u] =

(2.2)

where the integration is over R3 . The variation of EElectrostatic [u] with respect to u
via the Euler-Lagrange equation recovers the Poisson equation
δEElectrostatic [u]
⇒−
δu

· (r) u(r) − ntotal (r)q = 0.

(2.3)

In the quantum mechanical description, the electron density is given by
|Ψj (r)|2 f (Ej − µ),

n(r) =

(2.4)

j

where Ψj are the Kohn-Sham orbitals [93, 117], and

f (Ej − µ) =

1
(E −µ)/kB T
1+e j

,

(2.5)

is the Fermi-Dirac distribution with µ being the Fermi energy, kB the Boltzmann
constant and T the temperature. The electron energy functional is


EElectronic [n] =

2 f (E
j


j

−
j

− µ)
1
| Ψj (r)|2 +
2m(r)
2

Zj n(r)q 2
+ EXC [n(r)] −
|r − rj |

n(r)n(r )q 2
dr
|r − r |


Ej f (Ej − µ)|Ψj (r)|2  dr, (2.6)
j

where m(r) is the position dependent electron mass,

= h/(2π) with h being the

Planck constant, Zj is the nuclear charge at position rj , EXC [n(r)] is the exchange
correlation term and Ej are eigenvalues. Energy minimization with respect to Ψ∗ ,
j
26

which is the complex conjugate of Ψj , leads to the Kohn-Sham equation [93]

δEElectronic [n]
⇒ −
δΨ∗
j

2

·

2m

+

n(r
dr −
|r − r |



q2

)q 2

j

Zj
+ UXC [n(r)]
|r − rj |

× f (Ej − µ)Ψj (r) − Ej f (Ej − µ)Ψj (r) = 0

where UXC [n(r)] =

δEXC [n(r)]
δn(r)

(2.7)

is the exchange correlation potential. It is convenient

to cast the Kohn-Sham equation in the form of the Schr¨dinger equation
o
2

−

·

2m(r)

+ U (r) Ψj (r) = Ej Ψj (r),

(2.8)

where the potential energy U (r) includes all the interaction potential energies in Eq.
(2.7).

At nano scale, there should be a unified framework to bring the continuum mechanics and quantum mechanics on an equal footing. To establish the relation between
the Poisson equation and the Kohn-Sham equation, let set

ntotal (r)q = n(r)(−q) + nn (r)q

where we recognize that the electron charge is negative and nuclear charge is positive.
Here, the nuclear number density is given by nn (r) =

k Zk δ(r

− rk ). Then, the

solution to the Poisson equation in the free space is

u(r) = −

n(r )q
dr +
|r − r |

k

qZk
.
|r − rk |

(2.9)

where Zk and rk indicate the charge magnitude and position of kth nuclear. Therefore, we introduce a multiscale total energy functional as
27

ETotal [u, n] =

(r)
| u(r)|2 − u(r)ntotal (r)q
2


2 f (E
j

−
j

− µ)
| Ψj (r)|2 − EXC [n(r)] +
2m(r)

Ej f (Ej − µ)|Ψj (r)|2  dr. (2.10)
j

To optimize the total energy, we consider the variation of ETotal [u, n] with respect
to u
δETotal [u, n]
⇒−
δu

· (r) u(r) − ntotal (r)q = 0.

(2.11)

This is the standard Poisson equation. Similarly, by variation of ETotal [u, n(r)] with
respect to Ψ∗ , we have
j
δETotal [u, n]
⇒− −
δΨ∗
j

2

·

2m(r)

+ u(r)(−q) + UXC [n(r)] Ψj (r) + Ψj (r) = 0.
(2.12)

This is exactly the Kohn-Sham equation, Eq. (2.7), because the electrostatic potential, u(r), includes the Coulomb potential effects of both electrons and nuclei, as
shown in Eq. (2.9). Since the electron density in the Poisson equation depends on the
solution of the Kohn-Sham equation, which, in turn, depends on the solution of the
Poisson equation for the interaction potential, we have a system of self-consistently
coupled Poisson-Kohn-Sham equations. The electronic structure obtained from the
present theory will be used to evaluate device transport via the NEGF formalism
presented in Section 2.1.3.
To our knowledge, this is the first time that the coupled Poisson-Kohn-Sham
equations have been derived from the optimization of the total energy functional,
Eq. (2.10). Note that the proposed Poisson-Kohn-Sham equations differ from the
Poisson-Schr¨dinger equations commonly used in device modeling [104, 146, 15, 87,
o
99, 8, 62, 133, 13] in the following aspects: (i) The solution of the Poisson equation
28

with the nature boundary condition reproduces the correct Coulomb potential for the
Kohn-Sham equation. This consistence does not exist in the commonly used PoissonSchr¨dinger equations. (ii) The inclusion of the exchange correlation functional allows
o
the construction of various density functional approximations. (iii) While the electron
density is computed from the Kohn-Sham equation, the nuclear density is prescribed
as point charges. This framework can be used as a starting point for formulating other
linear scaling approximations, such as the pseudopotential method, density functional
tight binding method [79] and divide and conquer method [156, 101]. (iv) As the
mass is a function of position, the effective mass approximation can be generalized
to describe different band structures in different regions, including interconnects. (v)
Moreover, the present approach also enables the treatment of individual doping atoms,
defects and traps as additional charge sources, i.e.,replacing nn (r) with

β

nβ (r),

where β = nuclei,n-dopants, p-dopants and defects. The doping density functions are
discussed in Section 2.1.4. (vi) Interactions of electrons and other particles, such as
phonons, photons and excitations can also be allowed in the total energy functional.
Finally, it is emphasized that the main contribution of the proposed total energy
functional framework is that it enables the treatment of continuum mechanism and
quantum mechanics on an equal footing at nano scale and provides a unified derivation
of coupled Poisson-Kohn-Sham equations in a consistent manner.

2.1.2

Electronic system

The difficulty of solving the full-scale Kohn-Sham equation is one of the major obstacles in the modeling and simulation of nano electronic devices. In particular, the
consideration of many-body and multiband interactions is very time consuming. Typical computations are often conducted under the single electron approximation, which
can provide a reasonable account of the quantum effect in nano electronic device. By
inspecting the device geometry and possible symmetry, one can further reduce the
29

(a)

(b)

Figure 2.1: Illustration of a double-gate MOSFET with its y-direction being infinitely
long. (a) Configuration of the double-gate MOSFET; (b) Computational domain.
For interpretation of the references to color in this and all other figures, the reader is
referred to the electronic version of this dissertation.
computational dimensions and/or domain by the decomposition of the Kohn-Sham
equation and using appropriate transport models [44, 124, 120]. These aspects are
discussed with a double-gate MOSFET and a four-gate MOSFET in this subsection.
An efficient transport model, the NEGF method, is described in the next subsection.

Double-gate MOSFET
For a double-gate MOSFET [44, 124], the computational dimensions can be split into
a confined direction, a transport direction and an infinite direction. Figure 2.1 depicts
the geometric configuration and computational domain of the double-gate MOSFET.
A detailed description of the involved parameters for the double-gate MOSFET is
provided in section 2.3.1. Along the infinite direction, the potential is assumed as
translation invariant and the associated Schr¨dinger operator has an absolutely cono
tinuous spectrum. It can be solved with the plane waves. In the confined direction
where the gate voltages are applied, the associated Schr¨dinger operator which ino
30

cludes the electrostatic potential, is essentially compact, despite that the potential
may admit a finite number of point singularities. Therefore, the energy spectrum is
discrete in the confined direction. The transport direction accounts for the charge carrier motion in the channel connecting the source and drain contacts. The associated
Schr¨dinger operator also has an absolutely continuous spectrum. The corresponding
o
scattering states can be evaluated with appropriate incoming and outgoing waves,
and are subject to the potential consisting of the eigenvalues computed from the
confined direction. Without the lost of generality, we denote x, y and z directions
for transport, infinite and confined directions, respectively. Under this setting, the
wavefunction can be expressed as

Ψj,kx ,ky (x, y, z) = ψj,kx (x, z)χky (y),
where the index j denotes the jth eigenmode of the discrete spectrum and χky (y) =
eiky y is the plane wave of given wavenumber ky . As such, the Kohn-Sham equation
of a single electron can be split into two parts

2

∂ 1 ∂
∂ 1 ∂
+
+ U ψj,kx (x, z) = Ej,kx ψj,kx (x, z) (2.13)
2 ∂x mx ∂x ∂z mz ∂z
2k2
2 ∂ 1 ∂
y
−
χky (y) =
χ (y),
(2.14)
2 ∂y my ∂y
2my ky
−

2 k2

y
where Ej,kx = Ej,kx ,ky − 2my with Ej,kx ,ky being the total energy of the system,

and U (x, z) is the electrostatic potential energy satisfying the Poisson equation. The
solution of Eq. (2.13) requires the information of U .
Note that for the double-gate MOSFET, the y direction is homogeneous, we therefore can solve the Poisson equation in two dimensions. Furthermore, on the x − z
plane, due to the confinement along the z direction provided by the insulator layers,
31

combined with the assumption that the device geometry does not change significantly
along transport direction, the system yields discrete states only in the z direction.
We split the wavefunction as

ψj,kx (x, z) = ψj ψkx ,

(2.15)

where ψj (x0 ; z) is the discrete eigenstate in the z direction for a given x0 label and
ψkx (x) is a scattering state in the transport direction x. The eigenvalue problem with
the Dirichlet boundary condition is
2

−

d 1 d
+ U (x0 ; z) ψj (x0 ; z) = εj (x0 )ψj (x0 ; z), j = 1, 2, ...
2 dz mz dz

ψj (x0 ; z) = 0 on ∂ΩD

(2.16)
(2.17)

where εj (x0 ) represents the energy of the jth discrete subband at x0 . Here ∂ΩD
is the boundary of the silicon layer at where the wavefunction is forced to be zero
because of the confinement. Based on this formalism, one only needs to calculate
the quantum transport along the x-direction. We therefore end up with a scattering
problem
2

−

d 1 d
j
j
+ εj (x) ψk (x) = Ej,kx ψk (x),
x
x
2 dx mx dx

(2.18)

Since εj (x) varies along the x axis, it serves as the potential for the scattering problem.
j

The superscript in the scattering wavefunction ψk (x) indicates that the potential is
x
associated with the jth discrete subband. Similarly, the subband label j on Ej,kx
indicates the scattering potential used in Eq.(2.18). From the physical point of view,
the scattering energy is conserved during the scattering process. The transmission
and reflection coefficients can be computed based on each given initial energy Ej,kx ,

32

which is a part of the absolute continuum spectrum.
Equation (2.18) can be solved in many ways, such as time dependent and time
independent means. However, it is convenient to use the NEGF strategy [44] as
described in Section 2.1.3, which provides not only the solution to Eq. (2.18), but
also the desirable transport quantities.
To solve the Poisson equation for U (x, z), one needs to determine the electron
density n(x, z, y) according to Eq. (2.4)
j

|ψj (x0 ; z)|2 |χky (y)|2 |ψk (x)|2 f (Ej,kx ,ky − µ)

n(r) =

(2.19)

x

j,kx ,ky
j

where n(r) is homogeneous in y direction. In fact, the scattering part |ψk (x)|2
x

does not need to be explicitly evaluated in the NEGF formulation. The further
simplification of n(r) is discussed in Section 2.1.3.

Four-gate MOSFET

For a four-gate MOSFET, we denote x, y and z directions for the transport, confined and confined directions, respectively. In the confined y − z directions where
the gate voltages are applied, the associated Schr¨dinger operator is essentially como
pact, which leads to discrete energy states for the charge carrier. Figure 2.2 depicts
the geometric configuration and computational domain of the four-gate MOSFET.
A detailed description of the four-gate MOSFET parametrs can be found in Section
2.3.1. As in the double-gate case, the associated Schr¨dinger operator has an absoo
lutely continuous spectrum in the transport direction. The total energy is given by
j

Ej,kx and total wavefunction is Ψj,kx . By assuming Ψj,kx = Ψj (x : y, z)Ψk (x), we
x

33

Figure 2.2: An illustration of a four-gate MOSFET (i.e., silicon nanowire transistor).
(a) Configuration; (b) Cross section at y = 0 for the computational domain; (c) Cross
section at x = 0.

34

therefore can split the Kohn-Sham equation as
2

−

2

∂ 1 ∂
∂ 1 ∂
+
∂y my ∂y ∂z mz ∂z

+ U (x0 ; y, z) ψj = εj (x0 )ψj

(2.20)

ψj (x0 ; y, z) = 0 on ∂ΩD ;
2

−

∂ 1 ∂
j
j
+ εj (x) ψk (x) = Ej,kx ψk (x),
x
x
2 ∂x mx ∂x

(2.21)

where εj (x0 ) is the jth eigenvalue of the 2D problem at position x0 , and ψj =
j

ψj (x0 ; y, z) is the corresponding eigenfunction. Here ψk (x) is the scattering wavex
function associated with the scattering potential εj (x) and energy Ej,kx . The transport equation (2.21) is the same as that of Eq. (2.18) and can be solved with the
j

2 k2

x
NEGF method. For a given set of scattering energies, Ek = 2mx , the Poisson
x

equation (2.11) is solved in 3D and its electronic density source term is
j

|ψj (x0 ; y, z)|2 |ψk (x)|2 f (Ej,kx − µ).

n(r) =

x

(2.22)

j,kx

The further evaluation of the density is discussed in Section 2.1.3.
In both double-gate and four-gate MOSFET calculations, it is possible to consider subband interactions by using a variety of combinations of the discrete energy
levels. As such, subband states belonged to different levels are used along the x axis.
Obviously, this approach can lead to improved transport properties and an increase
in computational cost. However, high energy modes, particularly those modes whose
energies are significantly higher than the scattering barrier, do not play much role in
the transport calculation.

2.1.3

Transport system

This section briefly describes the NEGF formalism in a setting that is consistent
with the double-gate MOSFET and four-gate MOSFET studied in the present work.
35

Additionally, a detailed description of the electron density is also given.

The Non-equilibrium Green’s function (NEGF) formalism

Without the loss of generality, we consider the NEGF formalism in a multichannel
setting in R3−l . Here l is the number of non-scattering dimensions, it equals 2 for
double-gate and four-gate MOSFETs discussed in the last subsections. We define the
whole open system on the domain Ω = ΩD ∪ (

α Ωα ),

which consists of the device

domain ΩD and the union of (multiple) contact domains Ωα , such as the source and
drain. Let Γα = ΩD ∩ Ωα denotes the intersection boundaries of the device domain
and contacts. Here, Ωα may extend to infinity but only the ΩD (or plus small portion
of Ωα ) is the computational domain of interest. In the framework of the NEGF, the
Green’s operator (function) on ΩD is defined as the inverse operator
G−1 (E) = EI − H = EI − H 0 −

Σα

(2.23)

α

where E is the total energy of the scattering system, I is the identity operator,
H = H0 +

αΣ

α

is the full scattering Hamiltonian and H 0 is the Hamiltonian

of the single charge carrier associated with the scattering potential. In order to
reduce the computational cost, the infinite (or large) contact domain Ωα needs to be
chopped off and is restricted on the domain of interest ΩD . Each of the self-energy
operators Σα is solely defined on the corresponding Γα and reveals the coupling effect
of the contacts to the device [44]. In practice, Σα takes different forms for different
numerical discretizations, and more details can be found in Ref. [83]. In the position
representation
H0 = −

2

2

·

1
m(r)

+ Uscat (r),
36

r ∈ R3−l ,

(2.24)

where m(r) is the space dependent effective mass of the charge carriers and Uscat (r)
is the interaction potential for scattering and Uscat (r) = εj (x) for double-gate and
four-gate MOSFETs discussed in the last subsections.
Once the Green’s function/operator is defined, all quantities of interest can be
calculated. Among these quantities, the scattering wavefunction is given by

ψE = lim iεG(E + iε)φE ,
ε→0

(2.25)

where φE is the incoming wavefunction of energy E. The scattering wavefunction
satisfies the Schr¨dinger equation HψE = EψE . Additionally, the non-equilibrium
o
charge carrier density operator is given by
f (Hfull − µα )Aα (E).

ρ=

(2.26)

α

In the NEGF theory, each contact is assumed in equilibrium state and α is for contact index, µα is the contact Fermi level for each contact, and f is the Fermi-Dirac
distribution. Here Hfull = H + Hns is the full Hamiltonian of the electron system
with Hns being the Hamiltonian of the non-scattering system. The spectral function
Aα (E) in Eq. (2.26) is given by
Aα (E) = G(E)Γα (E)G† (E),

(2.27)

where Γα (E) is the broadening operator, which reflects the dissipative effects on the
transport from contact region Ωα , and is defined by
Γα (E) = i[Σα (E) − (Σα (E))† ].

(2.28)

Moreover, the number density of charge in the system is given by the position
37

representation of the density operator (2.26)

n(r) =< r|ρ|r >,

(2.29)

where < ·| and |· > are Dirac notations.
Furthermore, the current in the quantum device from the source (S) to drain (D)
is calculated via

I=

−q
TrTDS (E)[f (Hfull − µS ) − f (Hfull − µD )]
h

(2.30)

where µS and µD are the Fermi levels of source and drain, respectively. The Tr is the
trace and TDS (E) is the transmission operator.
TDS (E) = ΓD (E)G(E)ΓS (E)G† (E).

(2.31)

Finally, the number density contributed from the scattering process can also be computed by using the NEGF formalism. This aspect is elaborated in the next subsection.

Electron density
Electron density is an important quantity in the coupled Poisson- Kohn-Sham theory
and is used in the Poisson equation to compute the potential. The general expression
of the density is given in Eq. (2.29). Due to the subband decomposition, the evaluation of density becomes slightly subtle. One needs to distinguish the subband degree
of freedom, the scattering degree of freedom, and the degree of freedom due to the infinity dimensions. In particular, the energy associated with infinity dimensions should
be integrated. Whereas the discrete subband energies serving as the potentials for
the scattering system and the discrete subband states are summed over. We illustrate
the density evaluation in the double-gate MOSFET and the four-gate MOSFET.
38

As shown in Eq. (2.29), the electron density is given by the position representation of the density operator nscat (r) =< r|ρ|r >. For the double-gate MOSFET, the
y direction is infinity and homogeneous. The total energy Ej,kx ,ky is distributed to
the plane wave, the eigenstate, and the scattering kinetic energy. Since the eigenstate
energy acts as the scattering potential energy, the total energy available to the scattering system in Eq. (2.23) is Ej,kx (x). The energy associated with the plane wave
can be integrated. To evaluate ρ, we need to use some notation and identities
1
|ψ >< ψE |dE = 1,
2π R E

< r|ψE >= ψE (r),

(2.32)

and
g(k) →
k

2
(2π)d

g(k)(dk)d ,

(2.33)

for a function g(k). We therefore write the electron density of the double-gate MOSFET as
j

|ψj (x0 ; z)|2 |χky (y)|2 |ψk (x)|2 f (Ej,kx ,ky − µ)

n(r) =

x

(2.34)

j,kx ,ky
j

|ψj (x0 ; z)|2 |ψk (x)|2 f (Ej,kx +

=

x

j,kx ky
j

|ψj (x0 ; z)|2 |ψk (x)|2 P j

=

x

2k2
y

2my

− µ)

(2.35)
(2.36)

j,kx
j

|ψj (x0 ; z)|2 nscat (x, y)

=

(2.37)

j

where P j =

2 k2

y
ky f (Ej,kx + 2my − µ). Since ky is continuous, we should change the

39

summation into an integral

Pj =

2k2
y

f (Ej,kx +

2my

ky

1
2

2my kB T

1
= √
π

2

− µ) =

2k2
1 ∞
y
− µ)dky
f (Ej,kx +
π −∞
2my

F 1 (µ − Ej,kx )
−

(2.38)

(2.39)

2

1
where F 1 (µ − Ej,kx ) is the Fermi integral of order − 2 given by
−
2

1

∞
1
x− 2
F 1 (y) = √
dx.
−2
π 0 1 + ex−y/kB T

(2.40)

from Eq. (2.29), we have
j

|ψj (x0 ; z)|2 |ψk (x)|2 P j

n(r) =

x

(2.41)

j,kx

1
= √
π
×

2my kT

1
2

|ψj (x0 ; z)|2

2

(2.42)

j

1
F 1 (µα − Ej,kx )Aα (Ej,kx )dEj,kx .
−2
2π R α

(2.43)

From this expression one can identify that

j
nscat (x, y)

1
=√
π

2my kB T
2

1
2

1
F 1 (µα −Ej,kx )Aα (Ej,kx )dEj,kx (2.44)
−2
2π R α

At each given location along the x direction, Ej,kx is given by Ej,kx (x) = εj (x) +
2 k2

x
Ekx = εj (x) + 2mx . Here, εj (x) are the eigenvalues of the 1D confined system.

Based on this expression and Eq.(2.15), the density n(r) given in Eq. (2.41) can
be calculated. The Kohn-Sham equation and the Poisson equation are completely
coupled and their solutions have to be pursued iteratively.
40

For the four-gate MOSFET, the density can be evaluated as the follow
j

|ψj (x0 ; y, z)|2 |ψk (x)|2 f (Ej,kx − µ)

n(r) =

x

(2.45)

j,kx
j

|ψj (x0 ; y, z)|2 nscat (x),

=

(2.46)

j
j

where nscat (x) is given by
j

nscat (x) =

1
f (Ej,kx − µα )Aα (Ej,kx )dEj,kx ,
2π R α

(2.47)

2 k2

x
where Ej,kx (x) = εj (x) + Ekx = εj (x) + 2mx . Here, εj (x) are the eigenvalues of the

2D confined systems, which change over different position x in the transport direction.
Note that in Eqs. (2.42) and (2.47), the integrations over energy Ej,kx are carried
out at each given position x.

2.1.4

Electrostatic system

Individual dopant model
In earlier individual models, the discrete dopants are approximated by either dilated
Gaussian functions [84] or constant charges supported by small regions. All these
models are parameter-dependent. In this paper, we propose a point doping model for
individual dopants and define the doping density as

cjβ δ(r − rjβ ),

nβ (r) =

(2.48)

j

where where β = n-dopants and p-dopants, cβ are charges of doping atoms and
δ(r − rjβ ) is the Dirac delta function at position rjβ . Theoretically, this doping
charge source can be added to the total energy functional, Eq. (2.10). This model
41

provides a better description for microscopic dopants. In fact, it has a connection
to the usual Gaussian function model characterized by the influence domain (σβ ) of
each dopant

nβ,σ (r) =
β

cjβ
2
(2πσβ )3/2
j

e

2
−(r−rjβ )2 /2σβ σβ →0
−→

cjβ δ(r − rjβ ).

nβ (r) =

(2.49)

j

The point doping model is recovered if the influence domains are set to zero. Computationally, the delta functions give rise to unbounded electrostatic values locally
and is numerically difficult to deal with. This singularity can be alleviated by the
Dirichlet to Neumann mapping (DNM) method, which analytically resolves the delta
functions and leads to a set of flux jump conditions at the interface. The effective
use of the DNM method requires the careful enforcement of additional flux jump
conditions. To this end, the MIB framework developed for the Poisson equation will
be used. Tens of thousands of atoms can be handled in this approach.

Silicon-insulator-interface modeling
The recognition of material interfaces in MOSFETs implies the acknowledgment of
discontinuous material properties or coefficients across the interfaces. This has profound consequences in the well-posedness and numerical convergence of the Poisson
equation. For simplicity, the electron is assumed as the majority of charge carriers
and then the hole density is neglected. In the present work, we consider the Poisson
equation of the form

−

·(

u) = ρ(r) in Ω

(2.50)

[u] = uχΩ − uχΩ
= 0 along ΓSi/SiO
Si
SiO2
2

(2.51)

[ un ] =

Si un χΩSi

− SiO2 un χΩ
= 0 along ΓSi/SiO ,
SiO2
2
42

(2.52)

where


MA

MD

cjA δ(r − rjA )

cjD δ(r − rjD ) −

ρ(r) = q ND − NA − n(r) +



j

j

and un = ( u) · n with n being the interface normal direction, letters “A” and “D”
denote acceptor and donor, respectively, and MA and MD are numbers of discrete
acceptors and donors, respectively. Being a multiscale model, some doping regions in
the device may still be modeled as continuum, and thus the continuous doping functions NA and ND are reserved for these specific parts. This treatment is reasonable.
For example, when the doping in the channel is small and the system is dominated
with electron ballistic transport, the continuum doping treatment is a good approximation. Another case is that when the voltage threshold lowering effect is studied,
individual dopants are used for the channel region while continuum doping treatment
can be used in the source and drain regions. The computational domain Ω is divided
into silicon (Si) region ΩSi and silicon dioxide (SiO2 ) insulator layers ΩSiO2 . The interface is defined as ΓSi/SiO , i.e., Ω = ΩSi ∪ΩSiO2 , ΓSi/SiO = ΩSi ∩ΩSiO2 . It follows
2

2

that the dielectric constant

is set to as Si and SiO2 in corresponding regions. The

solution of the Poisson equation u(r) is restricted to the Si and SiO2 regions, and
denoted as uχΩ

Si

and uχΩ

SiO2

respectively, where χΩ is the characteristic function

on set Ω. Models (2.51) and (2.52) indicate that the Poisson equation is subject to
the jump conditions along the Si/SiO2 interface, where the jump conditions reveals
the continuities of the potential landscape and its flux.
Although the jump conditions are trivial in physical sense, if no specific numerical
algorithm is applied, the discontinuity induced non-smoothness in the solution will
be smeared. As such, the numerical scheme is of low accuracy and convergence. A
novel numerical scheme, the matched interface and boundary (MIB) method will be
illustrated in the next section. The boundary conditions of the Poisson equation
43

are the following: it takes Dirichlet boundary conditions where the gate voltages are
applied and Neumann boundary conditions for the rest of the device.

2.2

Numerical analysis and implementation

Multigate MOSFETs have become an important means to alleviate channel tunneling
and gate leaking in ultimate CMOSs. In this section, we consider two multigate
MOSFETs, the double-gate MOSFET and the four-gate MOSFET. The proposed
interface model and individual doping model are evaluated based on these two types
of MOSFETs. Some mathematical algorithms and numerical analysis are presented
in the section.

2.2.1

Decomposition of the Poisson equation

In this section, the Dirichlet-to-Neumann mapping (DNM) formalism is introduced to
achieve an accurate and efficient account of randomly distributed dopants formulated
in Section 2.1.4.
The proposed interface and individual dopant models (2.50) treat the dopants
as Dirac delta functions that pose computational difficulties. As an approximation,
delta functions can be distributed to the neighboring grid points [157]. However,
due to the application of the interface techniques, the interference of the interface
treatment and the distributed delta functions leads to the reduction of accuracy to
the first order. As such, the combination of an interface technique and the Dirichletto-Neumann mapping (DNM) strategy becomes necessary [67]. This combination can
substantially improve computational accuracy and the speed of convergence. The
essence of the DNM technique is to split the solution into certain parts, such that the
singular source terms can be accounted in one part of the solution analytically. Such
a treatment will in general produce an additional Neumann condition either on the
44

interface or on the boundary.

In the present problem of nonlinear iterations, we can further take the advantage
of the solution splitting to accelerate the convergence of the iterations. It is also
noted that in the Poisson equation, only the electron density n(r) is directly involved
in the loop of self-consistent iterations and it is a regular function. Therefore, we first
decompose the solution of the Poisson equation into two parts, a slow varying part
and a fast varying part: u = uslow + ufast . The fast varying part is up-dated during
the iterations and solves


 −




·(

ufast ) = −qn(r) in Ω

ufast = uvoltage



 fast
 u
n =0

(2.53)

on ΓSiO /Gate
2
on other boundary

where ΓSiO /Gate is the interface between the insulator and the metal contact. At
2
the Si/SiO2 interface ΓSi/SiO , Eq. (2.53) is subject to the jump conditions
2
[ufast ] = ufast χΩ

Si

− ufast χΩ

SiO2

along ΓSi/SiO

=0

(2.54)

along ΓSi/SiO .

(2.55)

2

[ ufast ] = Si ufast χΩ − SiO2 ufast χΩ
=0
n
n
n
Si
SiO
2

2

The slow varying part uslow solves


 −




·(

uslow ) = ρ(r) in Ω

uslow = 0



 slow
 u
=0
n

on ΓSiO /Gate
2
other boundary
45

(2.56)

subject to jump conditions
[uslow ] = uslow χΩ

Si

− uslow χΩ

SiO2

=0

along ΓSi/SiO

(2.57)

along ΓSi/SiO .

(2.58)

2

[ uslow ] = Si uslow χΩ − SiO2 uslow χΩ
=0
n
n
n
Si
SiO

2

2

Here uslow is solely contributed from the fixed doping terms, either continuous doping
or individual dopants. It is noticed that jump conditions in both (2.53) and (2.56)
are decoupled. Therefore, solutions uslow and ufast are decoupled too. As such, we
only need to undate ufast during the nonlinear iterations.

For a given n(r), the system (2.53) with its jump condition is readily to be solved
with the MIB method. Whereas, uslow should be further decomposed as
uslow = ucont + udisc
= ucont χΩ + ucont χΩ
Si

SiO2

+ udisc χΩ + udisc χΩ

= ucont χΩ + (ucont + udisc )χΩ
Si

Si

SiO2

+ udisc χΩ

SiO2

Si

where ucont and udisc are the solution associated with continuous doping and individual doping, respectively.
ucont χΩ + (ucont + udisc )χΩ
Si



 −




·(

Since there is no doping in ΩSiO2 , we set u1 =

SiO2

and u2 = udisc χΩ . Here u1 solves
Si

u1 ) = q [ND − NA ] in Ω

u1 = 0



 1
 u =0
n

on ΓSiO /Gate
2
on other
46

(2.59)

subject to jump conditions
[u1 ] = u1 χΩ − u1 χΩ
Si

SiO2

=0

along ΓSi/SiO

(2.60)

along ΓSi/SiO .
2

(2.61)

2

[ u1 ] = Si u1 χΩ − SiO2 u2 χΩ
= −φ
n
n
n
Si
SiO
2

Similarly, u2 solves

 − ·(




2
 u =0


 2
 u =0
n

u2 ) = q

MD
cjD δ(r − rjD ) −
j

MA
cjA δ(r − rjA )
j

in Ω
on Gate
on other
(2.62)

subject to the jump conditions
[u2 ] = u2 χΩ

Si

− u2 χΩ

SiO2

along ΓSi/SiO
2

=0

=φ
[ u2 ] = Si u2 χΩ − SiO2 u1 χΩ
n
n
n
Si
SiO
2

(2.63)

along ΓSi/SiO .
2

(2.64)

In this manner, we have decomposed uslow into two systems (2.59) and (2.62) with
corresponding boundary conditions and jump conditions (2.60-2.61) and (2.63-2.64),
respectively. The boundary conditions appear trivial, following the homogeneous
Dirichlet and Neumann boundary conditions from uslow . However, it can be seen
that the jump conditions, specifically the flux jump conditions in Eqs. (2.61) and
(2.64), have been revised. They need to be carefully evaluated.
Since u2 is zero on ΩSiO2 , we can restrict u2 on ΩSi . As such, interfaces Si/SiO2
play the role of boundaries, where the homogeneous Dirichlet boundary conditions
are applied, which also results in homogeneous jump conditions in Eqs. (2.60) and
(2.63). However, one cannot generally have the homogeneous Dirichlet condition and
the Neumann boundary condition simultaneously. The Neumann boundary of u2
on Si/SiO2 interfaces creates nonhomogeneous jump conditions in Eqs. (2.61) and
47

(2.64), which is denoted by φ and is computed as the follow.
For u2 = u2 χΩ , it can be written as u2 for simplicity and it satisfies:
Si


 − ∆u2 = q


Si


2
 u =0


 2
 u =0
n

MD
cjD δ(r − rjD ) −
j

MA
cjA δ(r − rjA )
j

in ΩSi
on ΓSi/SiO

2

on other.
(2.65)

To solve equation (2.65), we set u2 = u∗ + u0 , in which

u∗

MA

=

MD

qcjA

2π Si
j

ln(|r − rj |) −

qcjD

2π Si
j

ln(|r − rj |)

(2.66)

for 2D simulation, and

u∗

MA

=−
j

qcjA

1
+
4π Si |r − rj |

MD
j

qcjD

1
4π Si |r − rj |

(2.67)

for 3D cases. Finally, u0 solves the harmonic equation with the corresponding boundary condition



 −∆u0 = 0 in ΩSi


u0 = −u∗ on ΓSi/SiO

2


 0
 u = −u∗ on Γ
n
n
Si/Source ∪ ΓSi/Drain .

(2.68)

It follows that the jump φ in Eqs. (2.61) and (2.64) reads
φ = Si (u∗ + u0 ).
n
n

(2.69)

Note that Eqs. (2.66) and (2.67) are actually the fundamental solutions of the
Laplacian operator with the δ function source in an unbounded domain. Harmonic
equation (2.68) is used to restrict the fundamental solution in the bounded domain
48

and match the boundary conditions. This procedure of rendering a Neumann boundary condition u∗ + u0 on the interface from the original Dirichlet boundary condition
n
n
u∗ is called Dirichlet-to-Neumann mapping. Meanwhile, the φ = Si (u∗ + u0 ) is used
n
n
as the jump at Si/SiO2 interfaces of the flux of u2 , if the whole domainΩ = ΩSi ∪ΩSiO2
is considered. It is easy to identify that the δ functions for individual dopants are exactly treated without any approximation by this decomposition of the whole problem
in to sub-systems (2.53), (2.59) and (2.62) with corresponding boundary conditions.
Other than the naturally induced homogeneous interface jump conditions, the decomposition of the problem also introduces nonhomogeneous jump conditions. Systems
(2.53), (2.59) and (2.62) are typical interface problems and are to be solved by the
MIB method.

2.2.2

Numerical implementation of the self-consistent iterations

To achieve efficient convergence, we present an inner-outer iteration procedure in this
section. The Gummel iterative scheme was proposed in [50] for solving nonlinear
coupled equations in all kinds of device applications. The numerical implementation
of the iteration scheme for the nano device simulation is provided below.
• Step 0 (Solution of uslow ) : This step is out of the main iteration loop to solve
for uslow , which is related to the fixed discrete and continuous doping functions.
u∗ , the singular part of uslow , is calculated by using Eq. (2.66) for 2D or Eq.
(2.67) for 3D. The Harmonic part u0 is solved on the silicon region by using Eq.
(2.68). Therefore, u∗ + u0 is the part of uslow from the discrete doping, and
their corresponding interface jump φ is derived via Eq. (2.69). The continuous
doping part, u1 , is solved from via Eq. (2.59) and the corresponding jump
condition (2.60). Finally one obtains uslow = u1 + u∗ + u0 .
49

Solve for uslow
Solve for ufast
l

Inner iteration

Outer iteration

ul = ufast + uslow
l
l

Solve for proton electron density nl (r)

Get qusi Fermi level ζl+1 (r)

No

Iteration converged?

Yes

Calculate current

Figure 2.3: Work flow of the overall self-consistent iteration.

50

• Step 1 (Inner iteration for the Poisson equation): Given the quasi-Fermi level
function in lth step, ζl , and the calculated uslow , solve the nonlinear Poisson
equation for ufast
l

 − · ( ufast ) = −qn F (ζ − uslow − ufast )

0 1/2 l

l
l



[ufast ] = ufast χΩ − ufast χΩ
=0
l
l
SiO2
Si
 l




fast · nχ
fast · nχ
 [ ufast · n] =
Si ul
Ω − SiO2 ul
Ω
l
Si

(2.70)
SiO2

=0

where n0 is the intrinsic density-of-state of an electron system
1
n0 = √
2

mkB T 3/2
π 2

(2.71)

where F1/2 (x) is the Fermi-Dirac integral of order 1/2, which takes the form of
∞
2
y 1/2 dy
F1/2 (x) = √
.
π 0 1 + ey−qx/kB T

(2.72)

Here, ζ(r) is the quasi-Fermi potential and will be used as an index function
to determine the convergence. The initial value ζ0 (r) is obtained via a linear
interpolation of the source and drain Fermi levels (voltages) over the channel.
This initial guess is found to be very effective in our numerical simulations.

• Step 2 (Inner iteration for the Kohn-Sham equation): The outcome of the Step
1 is the electronic potential at lth step, ul (r) = uslow (r) + ufast (r). Along the
l
transport direction (say x-direction), we slice the electronic potential for each
fixed x0 , and solve the eigenvalue problem either in 1D for the double-gate
51

MOSFET or in 2D for the four-gate MOSFET









2

−

2 + U (x ; r) ψ l (x ; r) = εl (x )ψ l (x ; r), j = 1, 2, ...,
l 0
j 0 j 0
j 0
2m r

(2.73)

n−1
l
 ψj (x0 ; r) = 0 on ∂ΩD





 r ∈ Rn−1 , n = 2, 3
l
where Ul (r) = ul (r)(−q). Results from this step are set of ψj and εl .
j

• Step 3 (Update density and quasi-Fermi potential): With the available of subband energies εl (x), we calculate the 3D electron density nl (r) by using Eq.
j
(2.42) or Eq. (2.46) at the lth step. Once nl (r) is obtained, one can update
the quasi-Fermi level to the (l + 1)th step ζl+1 by inverting the expression
nl (r) = n0 F1/2 (ζl+1 (r) − ul (r))
−1
ζl+1 (r) = ul (r) + F1/2 (nl (r)/n0 ).

(2.74)

• Step 4 (Convergence check): The convergence is checked by the criterion of
||ζl+1 (r)−ζl (r)|| < ε, where ε is a given small positive number. If the inequality
is satisfied, one calculates the current via Eq. (2.30), otherwise go to Step 1.
Figure 2.3 gives the work flow of the present self-consistent iteration scheme.
Remark 1: The reason of a nonlinear Poisson equation applied here is that, during
the outer self-consistent iteration loop, the fluctuation of n(r) may undermine the
iteration convergence, according to [149]. The use of the Fermi-Dirac integral will
average or normalize this kind of fluctuations and thus lead to efficient convergence.
This scheme is known as the Gummel iteration [50]when Boltzmann statistics is
applied.
Remark 2: There is no analytical form for the Fermi-Dirac integral of order 1/2.
Ref. [98] provides a very accurate approximation to the integral by polynomials. It
52

−1

Continum
5 dopants
10 dopants
40 dopants

−2

Outer Error (log10)

Outer Error (log10)

−1

−3
−4
−5

4

6
n Iteration Step
th

−2
−3
−4
−5
−6

8

(a)

Continum
5 dopants
10 dopants
40 dopants

4

6
n Iteration Step
th

8

(b)

Figure 2.4: Computational errors in simulating a double-gate MOSFET. (a) Individual doping in the source and drain with the dopant distribution shown in the lower
panel of Fig. 2.5(a); (b) Individual doping in the channel.
is based on these approximations in this algorithm that all the related calculations
are carried out.
Remark 3: Once the nonlinear Poisson equation is employed, it is noted that the
initial guess for the whole self-consistent iteration is the quasi-Fermi potential ζ(r).
The choice of the initial guess is based on the physical assumption that ζ(r) are
constants of the voltages at the source/drain and taken as a linear interpolation of
two constants along the channel. This initial guess is found to be very effective in
our numerical simulations.

2.2.3

Analysis of the model well-posedness, convergence and
iteration efficiency

Before numerical simulations are implemented under various physical conditions, some
analyses on the robustness and efficiency of numerical algorithms are presented, based
on the double-gate MOSFET for simplicity. The detailed device configurations and
parameter values are presented in next section.
53

The well-posedness of the numerical implementation consists of the analysis of
the outer iteration loop and inner solution of the nonlinear Poisson equation. We
define the Ks , Us and Ns as the spaces to which the quasi-Fermi functions ζ(r),
electrostatic function u(r), and electron density n(r) belong, respectively. For the
whole self-consistent Poisson-NEGF system, it can be interpreted as the application
of the fixed point map T : Ks → Ks to the quasi-Fermi potential function
ζ(r) = T (ζ(r)).

(2.75)

To characterize the details of the map T : Ks → Ks , we have the operator L :
Ks → Us , which indicates the action of the inner iteration (2.70) from the quasi-Fermi
potential to the system potential landscape. Then it is followed by G : Us → Ns ,
the procedure of using the NEGF scheme and subband decomposition to calculate
−1
the electron density from potential. Finally the map F1/2 : Ns → Ks represents the

step that recovers the new quasi-Fermi level from the obtained electron density. The
composition of the actions of the above operators yields the definition of the operator
T , representing the outer iteration
−1
T := F1/2 ◦ G ◦ L .

(2.76)

We define a closed convex set
Ks = {ζ ∈ L∞ (Ω) : C2 ≤ ζ(r) ≤ C1 , a.e. in Ω}

(2.77)

where C1 and C2 are lower and upper bounds of ζ in Ω. By analyzing the continuities
of these operators and assuming the continuous selection hypothesis, one can prove
the existence of the fixed point of the map T by the Schauder fixed point theorem
[51].
54

3
2
1
0
5

Relative Error (log10)

Thickness
Thickness

3
2
1
0
5

Channel
10
15
20
Transport Direction (nm)

25

Channel
10
15
20
Transport Direction (nm)

−1
−1.5
−2
−2.5
−3

25

(a)

α=1.0
α=0.3
10

20
30
40
Iteration Step

50

(b)

Figure 2.5: Dopant distributions and iteration errors for the double-gate MOSFET.
(a) Two distributions of 10 dopants. The distribution in the lower panel may lead
to a divergence in the iteration. While the distribution in the upper panel leads to
convergence; (b) Relaxation-factor-dependent convergence behaviors for the dopant
distribution shown in the left-lower panel.

Another way to check the existence of the solution is the equivalence of the problem
with the Poisson-Schr¨dinger system [120]. The existence of the Poisson-Schr¨dinger
o
o
system is proved in [119] and the solution is unique with the constraint of the positivity
of carrier concentration.
For the nonlinear Poisson equation, the approximation of the Fermi-Dirac integral
has no stronger nonlinearity than the polynomial of order 2 [98]. In this approximation, writing the equation in a variational form, standardly checking the Palais-Smale
condition and applying the Soblov embedding theorem, one can easily get the wellposedness analysis.
During the numerical implementation, the Newton-Raphson method guaranteed
the convergence of the inner iteration, by picking a reasonable quasi-Fermi level. For
the outer iteration, the nonlinearity is too complicated to analyze and no specific analyzing scheme has been designed so far. Many simple potential-density self-consistent
loops are employed and proved numerically efficient in many engineering applications.
55

Potential Absolute Error (eV)

0.04
Coarse Grid
Moderate Grid
Fine Grid

0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0

5

10

15

20

25

30

Transport Direction (nm)

Figure 2.6: Errors of simulating the double-gate MOSFET in the line of y = 0 and
z = 0.

In our model, the convergence of the scheme needs to be re-examined since the introduction of the interface and delta function in the Poisson equation significantly
reduce the regularity of the solution.
Figure 2.4 records the outer loop iteration of the system. The left one lists the
situation when discrete dopants occur in source and drain ends, and the right one is
for individual dopants in the channel. Results for a different number of dopants are
compared to the continuous doping model. The horizontal axis is for the iteration loop
steps. The first two steps are considered as starting steps and therefore skipped. The
vertical axis is the log10 of the absolute convergence error of the potential. Generally,
one can conclude that the convergence pattern of the individual dopants model follows
that of the continuous model. It is found that the convergence is slight better for the
situation that there are dopants in the channel, the reason might be that the gate
voltage is applied on the channel part. The δ functions cause high variation of the
potential landscape and Dirichlet boundary condition gives stronger suppress to these
56

5
−0.4

4

−0.6

3

−0.8

2

−1

1

−1.2

0
0

−1.4

Confined Direction (nm)

Confined Direction (nm)

5

4

2

(a) Potential Landscape
5

4

0.08

3

0.06

2

0.04

1

0.02
10
20
30
Transport Direction (nm)

Confined Direction (nm)

Confined Direction (nm)

10
20
30
Transport Direction (nm)

0

(b) Error at coarse grid

5

0
0

0.1

1
0
0

10
20
30
Transport Direction (nm)

0.2

3

4

(c) Error at moderate grid

0.02

3
2

0.01

1
0
0

0

0.03

10
20
30
Transport Direction (nm)

0

(d) Error at fine grid

Figure 2.7: Contour plot of the electrostatic potential (a) and its absolute errors
obtained with three sets of meshes(b, c, d) (10 dopants).

potential variations than the Neumann boundary condition does.
In our numerical experiments, it is also noticed that for certain number of randomly located dopants, some specific dopant configurations fail the simple self consistent outer iteration loop. The reason of the convergence failure, although not
completely clear, might be the relative positions of the discrete dopants. Locally
crowded individual dopants may lead to local significant variation of the potential
landscape and undermine the convergence efficiency. Although the map (2.75) issues
a fixed point, it promises no contraction property. Therefore the usually used outer
57

iteration
−1
ζ n+1 = F1/2 ◦ G ◦ L (ζ n )

(2.78)

may not converge. It is difficult to verify the contraction or construct contract mapping based on these operators because of the complex nonlinearity of the NEGF
calculation G . Figure 2.5(a) reveals these situations: The upper panel is one of the
position distribution of 10 dopants that one can easily reach the steady state. The
convergence behavior of the position configuration in upper panel of Fig. 2.5(a) can
be found in Fig. 2.4(a). However, the distribution in the lower panel of Fig. 2.5(a)
may lead to convergence failures. It can be seen that in the lower panel, dopants
are very crowdedly distributed near the left top corner in the source. To deal with
this numerical difficulty, we convert Eq. (2.78) into the steady-state problem of an
ordinary differential equation (ODE) [7]
∂ζ
−1
= F1/2 ◦ G ◦ L (ζ) − ζ.
∂t

(2.79)

Therefore many ODE related techniques such as the Runge-Kutta method can be used
to improve the convergence properties. One simple treatment is the discretization of
Eq. (2.79) as
ζ n+1 − ζ n
−1
= F1/2 ◦ G ◦ L (ζ n ) − ζ n ,
α

(2.80)

which leads to a self-consistent iteration with a relaxation factor α
−1
ζ ∗ = F1/2 ◦ G ◦ L (ζ n )

ζ n+1 = αζ ∗ + (1 − α)ζ n .

(2.81)

The traditionally used outer loop iteration actually is the special case of Eq. (2.81)
with α = 1. By carefully choosing the relax factor α, one can reach the steady

58

Table 2.1: Computational L∞ error of the model with MIB scheme
5 dopants error order 10 dopants error order 40 dopants error order
Coarse
0.350
Coarse
0.301
Coarse
0.218
Moderate 0.222 0.7
Moderate 0.123 1.3
Moderate 0.133 0.7
Fine
0.089 1.3
Fine
0.038 1.7
Fine
0.047 1.5
state (fix point) by the self-consistent iteration for arbitrarily distributed individual
dopants.
Figure 2.5(b) compares the convergence of the self-consistent iterations with different relaxation factors corresponding to the situation in the upper panel of Fig.
2.5(a). It indicates that α = 1 does not work for the convergence loop, while α = 0.3
leads to the convergence of the electron current within 0.2% relative error in around
50 steps.
One can easily come to the conclusion that although smaller relaxation factors
promise the convergence, they result in more iteration steps. The exact reason of the
position-dependent convergence and the choice of the optimal relaxation factor need
to be further analyzed in the future.
The numerical solutions of the system are supposed to converge to the real solutions as meshing grids get smaller and smaller. The standard finite difference method
is of second order convergence for the inner iteration. However, the strong outer nonlinearity may ruin this theoretical rate. Moreover, the discontinuity of the dielectric
constant and δ source function singularities in the Poisson equation further reduce the
regularity of the solution. The standard FD method will not maintain its second order
convergence. It may even diverge. The MIB scheme is designed not only for facilitating the Dirichlet-to-Neumann mapping but also for maintaining the high order convergence in interface problems. For the following convergence analysis, the numerical
result obtained with a fine resolution of hx = 0.075nm, hz = 0.025nm is considered as
a reference solution, where hx and hz are for the grid step along the transport and confined direction, respectively. Three sets of mesh sizes of (hx , hz ) =(0.15nm, 0.05nm),
59

(0.3nm, 0.1nm), and (0.6nm,0.2nm) are denoted as fine, moderate and coarse, respectively.
Figure 2.6 gives the errors under different grids resolutions in the central line of the
silicon layer. Numerical results suggest the convergence of the present MIB method.
Furthermore, we examine the 2D errors of the simulation. Figure 2.7 presents the
2D errors for the situation when the dopant number is 10 in both the source and the
drain.
Combined with Figs. 2.6 and 2.7, one can conclude that the major convergent errors occur at the junctions of discrete and continuous doping regions. Errors decrease
as the grid is refined.
Finally, the L∞ norm error (Eh ) is considered and the convergence order is defined
as log2 (Eh /Eh/2 ) for three sets of meshes. The numerical convergence orders are
calculated for several cases with different dopants numbers and listed in Table 2.1.
The targeted second order of the MIB method is not achieved, partially because the
strong nonlinearity of the coupled equation system.

2.3

Numerical simulation of the nano-electronic
transistors

2.3.1

Device configurations

In the following sections, the proposed model and numerical implementation are used
to examine the effects of random individual dopants and material interface. The impact of individual dopant random fluctuation was first recognized by Hoeneisen and
Mead in 1970s and has been studied for many years via semiclassical or quantum
mechanical means. Among the quantum models, Martinez et al [108] explored the
impact of random dopant aggregation in the source and drain. Jiang et al [84] stud60

ied the gate threshold voltage lowering and fluctuation induced by random dopants
in the channel. These studies are based on the smooth function approximation of
individual dopants. The results from this treatment depend on empirical parameters
and discretization mesh sizes. By using the Dirichlet-to-Neumann mapping, it can
be found in the numerical simulation that the Dirac function model proposed in this
work treats the individual dopants exactly, is of parameter free and does not require
fine grids.
Multi-gate MOSFETs will play a dominant role in quantum devices because their
ability to suppress channel tunneling and gate leaking effects. A variety of related
investigations are carried out for such device models. We consider multi-gate MOSFETs in this work. Figure 2.1(a) gives an illustration of the 2D double-gate MOSFET
and Fig. 2.1(b) is the corresponding computational domain. All relevant components
of the device are presented in the graph. The x-direction is taken as the transport direction, the z-direction as the confined direction and all physical profiles are assumed
invariant along y-direction. In the gray region, the source and drain are heavily doped
while the channel is assumed near ballistic, in which case no doping is imposed. It is
noticed from Fig. 2.1 (b) that the dielectric constants are different in the silicon and
silicon dioxide layers, which are separated by the interfaces. For the Poisson equation on the computational domain, Dirichlet boundary conditions are taken at the
double-gate region (EF and GH). For other boundaries (AE, FB, BC, CH, DG and
AD), homogeneous Neumann boundary conditions are employed. The parameters of
the device are the following: The total length (AD) of the device is 30 nm, with 10nm
for each doping area and channel.The thickness of the silicon layer is 3nm, while the
upper and lower silicon dioxide layers are 1nm each.
Figure 2.2(a) is the 3D sketch of the silicon nanowire transistor (SNWT), which is
a MOSFET with all-around gates. The x-direction is the transport direction, while
the other two are confined directions, where insulator layers exist and gate voltages
61

are applied. In our simulation, for simplicity, the cross-section of the SNWT is taken
as a square, so it is also called a four-gate MOSFET. Figure 2.2(b) gives the crosssection of the SNWT in the y-direction (y = 0). It is similar to the structure of
the planar double-MOSFET. Figure 2.2(c) presents the cross-section of the SNWT
in x-direction (x = 0). The total length of the device is 30 nm, with 10nm for each
doping area and the channel. The thickness of the silicon layer is 3nm. The thickness
of the upper and lower silicon dioxide layers is 1nm each. Treatments that are similar
to those for the double-gate MOSFET are taken for the boundary condition. If the
slice has gate all around it, the Dirichlet boundary condition is imposed, otherwise
homogeneous Neumann boundary condition is imposed.
The parameters of the device are the following: The source and drain voltage
bias is VDS = 0.4V, the double gate voltage is VG = 0.4V. Three subbands are
accounted for electron density calculations. In the double-gate MOSFET simulation,
the electron effective mass is taken as mx = 0.50m0 in the transport direction and
as mz = 0.20m0 in the confined direction. The dielectric constants for silicon layer
and silicon dioxide layer are Si = 11.7 0 and SiO2 = 3.9 0 , respectively, where
0

= 8.85 × 10−12 Fm−1 . The reference continuous n-doping is taken as ND (r) =

2 × 1020 /cm3 in the source and drain. For individual dopants in channel tests, the
p-doping is take as NA (r) = −1 × 1020 /cm3 . Room temperature of T = 300K is
assumed.
The strategies of placing individual dopants are as following: The discrete dopants
are located for the 5nm long region right before the channel in the source contact and
5nm long region right after the channel in the drain contact. These regions are
called discrete regions. The rest areas of the source/drain are self-averaging areas
with continuum doping because a sufficient number of doping is necessary [108]. The
number of individual dopants MD and each dopant charge quantity cj are chosen to
62

match the self-averaging doping concentration in the sense of an integration

ΩD k

ck δ(r − rk )dr =

ΩD

ND (r)dr,

(2.82)

where ND (r) is the continuous doping. Individual dopants are randomly and evenly
distributed in the discrete region, i.e. the x- and y-coordinates of the dopants are
independently generated by a uniform quasi-random number generator.

2.3.2

Simulation results for double-gate MOSFETs

In this section, we investigate the impact of the proposed interface model and the
random individual doping model using the double-gate MOSFET. Before proceeding
to the numerical results of the discrete dopant model, we show in Fig. 2.8(a) the
landscape of the device electrostatic potential calculated from the standard finite
difference method without the interface technique. The difference of the electrostatic
potentials between obtained from the standard finite difference method and from the
MIB method is plotted in Fig. 2.8(b). It is seen that two methods have about 3-4%
of differences. The largest difference occurs around the interface region near insulator
layers. We believe that the MIB scheme gives a more accurate calculation of potential
landscape. Although the advantage of the MIB scheme over the standard FD scheme
is not that tremendous in this case because of the relatively simple interface geometry,
the MIB technique also has the ability of handling more complex interfaces which may
occur in the future MOSFET design. Additionally, the DNM treatment of the discrete
dopants can not be realized without the MIB scheme.
We next study the physical profiles of fluctuations under different amount of
dopants and positions. The individual dopants in source or drain are anticipated
to produce discrete aggregation effects: The on-state current fluctuates due to individual dopants and their relative positions. The fluctuations are not directly caused
63

0

0.04
0.03
eV

eV

−0.5
−1
−1.5
4
nm

0.02
0.01

2

0
4
0 0

10
nm

20

30

(a)

nm

2

0 0

10
nm

20

(b)

Figure 2.8: Electrostatic potential energy and difference of potentials for the doublegate MOSFET. (a) Potential landscape obtained with the standard finite difference
method; (b) Difference of electrostatic potentials between computed with the standard
finite difference method and with the MIB method.

(a)

(b)

Figure 2.9: Profiles of electrostatic potential and electron density obtained with the
continuous doping approximation. (a) Potential function; (b) Electron density.

64

30

(a)

(b)

Figure 2.10: Profiles of electrostatic potential energy and electron density obtained
with 5 individual dopants (N = 5). (a) Potential function; (b) Electron density.

by the spatially resolved individual dopant atoms but mesoscopically by the selfconsistent electrostatic potential. Physical parameters are taken as stated in the
previous section. To validate the individual dopant model, we employ the constraint
of Eq. (2.82). This constraint may appear unphysical. However, it is useful in verifying the validity of the present model. It is expected that simulation profiles of discrete
dopants will converge to those of the continuous doping when the number of dopants
is large. The reason is that as the number of dopants increases, the amplitude of each
dopant becomes smaller and the doping becomes evenly distributed.
For a comparison, Fig. 2.9 illustrates the steady-state potential and electron density obtained using the continuum doping approximation. It is predictable that the
presence of the individual dopants will drag down the potential landscape and therefore trap the electron where a discrete dopant presents. This fact is revealed in our
simulations: if there were no individual dopants in the region 5-10nm and 20-25nm
along the x-direction, the potential landscape and electron density there would be
similar to those in the channel (10-20nm in the transport direction). Due to the
presence of discrete dopants, the potential landscape is dragged down and becomes
65

(a)

(b)

Figure 2.11: Profiles of electrostatic potential energy and electron density obtained
with 10 individual dopants (N = 10). (a) Potential function; (b) Electron density.

(a)

(b)

Figure 2.12: Profiles of electrostatic potential energy and electron density obtained
with 40 individual dopants (N = 40). (a) Potential function; (b) Electron density.

66

Subband 1
Subband 2
Subband 3

Subband Energy (eV)

2
1.5
1
0.5
0
−0.5
−1
0

5

10

15

20

25

30

Transport Direction (nm)
Figure 2.13: Subband energies of the double-gate MOSFET obtained with individual
dopant model.

flat, as that presented in the continuous doping region (0-5nm or 25-30nm in the
transport direction). As a consequence, the electron probability distribution (density) is increased near the discrete dopant site, comparing to the distribution profile
of the continuous regions. Figures 2.10, 2.11 and 2.12 depict the potential landscapes
and electron densities obtained with 5, 10 and 40 discrete dopants, respectively. Furthermore, it is observed that the more individual dopants presented in the field, the
smoother the potential is, and the closer to the continuum doping model the present
discrete doping will be. This is consistent with our expectation.
Detailed physical profiles of the device can be demonstrated by the subband energy. Figure 2.13 provides one of the subband energy profiles of the double-gate
MOSFET with discrete dopants in the source and drain. The electron density is
calculated based on the first three subband energies. Due to the Fermi-Dirac distribution, the higher the subband is, the less it contributes to the total amount of the
electron density. From the subband energies profiles one can also find the flatting out
67

0

VGS=0.3

Subband Energy (eV)

Subband Energy (eV)

0

VGS=0.4

−0.2

VGS=0.5
VGS=0.6

−0.4
−0.6
−0.8
0

10

20

Transport Direction (nm)

VGS=0.3
VGS=0.4

−0.2

VGS=0.5
VGS=0.6

−0.4
−0.6
−0.8

30

0

(a)

10

20

Transport Direction (nm)

30

(b)

Figure 2.14: Subband energy profiles under different gate voltage biases. (a) Continuum dopant; (b) Discrete dopants (10).

effect of the individual dopants to the potential in the discrete region. Different from
the continuous doping, the “flatting” is oscillatory.
Figure 2.14 illustrates the first subband energy profiles of the double-gate MOSFET at different gate voltages. The increase in the gate voltage will push down the
potential barrier in the device channel and thus lead to larger charge current. In this
study the source/drain bias is fixed under 0.4V.
Figure 2.15 shows the first subband energy of the double-gate MOSFET under
different source/drain voltage biases. The differences in the subband profiles at two
ends reflect the voltage biases. The width of potential barrier is reduced when the
voltage bias is increased. The increase of the voltage bias leads to the increase of
charge current in the double-gate MOSFET. The gate voltage is fixed at 0.4V in this
study.
Macroscopically, the individual dopants introduce the fluctuation in on-state currents. Figures 2.16(a), (b) and (c) demonstrate the current fluctuations with 5, 10
and 40 dopants, respectively. It can be observed that the fluctuation is about 10%
when there are 5 dopants in different positions. The fluctuation decreases gradually
68

Subband Energy (eV)

Subband Energy (eV)

−0.2

−0.4
VGS=0.3
−0.6

VGS=0.4
VGS=0.5

−0.8
0

V

=0.6

GS

10

20

Transport Direction (nm)

30

−0.2

−0.4

−0.6

V

=0.4

GS

VGS=0.5
−0.8
0

(a)

VGS=0.3

VGS=0.6
10

20

Transport Direction (nm)

30

(b)

Figure 2.15: Subband energy profiles under different source/drain voltage biases. (a)
Continuum Dopant; (b) Discrete Dopants (10).

when there are more dopants. From Fig. 2.16(d), one can draw the conclusion that
the fluctuation is small when there is a large number of discrete dopants with small
charge amplitude, as expected.
Another interesting aspect of the individual dopants is the lowering effect of the
device voltage threshold. The voltage threshold of a MOSFET is defined as the gate
voltage applied in order to generate the inversion layer between the insulator layer
and the silicon layer, then electron current follows. The presence of the atomistic, or
individual dopants has an effect of lowering the voltage threshold. In order to check
the ability of our model to detect this effect, we introduce the p-type doping to the
equivalent concentration of −2×1019 /cm3 in the channel and compare the I-V curves
obtained form the continuous model and individual dopants model.
To obtain the I-V curve presented in Fig. 2.17, we set the source/drain voltage
biases as 0.2V (left) and 0.4V (right), and gradually increase the gate voltage from 0
to 0.4V. It is clear from Fig. 2.17 that, to attain the same amount current, the present
individual dopants model requires less amount of gate voltage than the continuous
model does. State differently, the same gate voltage gives a higher charge current in
69

1500
1200
1000
Continuum
Distribution 1
Distribution 2
Distribution 3
Distribution 4
Distribution 5

500

0
0

I (A/m)

I (A/m)

1000

0.1

0.2

0.3

800
Continuum
Distribution 1
Distribution 2
Distribution 3
Distribution 4
Distribution 5

600
400
200
0
0

0.4

0.1

0.2

VDS (V)

(a)

0.4

(b)

1200

1200

1000

1000

800
Continuum
Distribution 1
Distribution 2
Distribution 3
Distribution 4
Distribution 5

600
400
200
0
0

I (A/m)

I (A/m)

0.3

VDS (V)

0.1

0.2

0.3

800
Continuum
N=100
N=20
N=60
N=40
N=80

600
400
200
0
0

0.4

0.1

0.2

VDS (V)

(c)

0.3

0.4

VDS (V)

(d)

Figure 2.16: The dependence of I-V profiles on the number of individual dopants.
(a) Fluctuation of 5 dopants; (b) Fluctuation of 10 dopants; (c) Fluctuation of 40
dopants; (d) Fluctuation of different number of dopants.

70

300

I (A/m)

200
150
100

250
200

I (A/m)

250

300
Continuum
N=18
N=20
N=15
N=40
N=5

150
100

50

50

0

Continuum
N=18
N=20
N=15
N=40
N=5

0

0

0.1

0.2

0.3

0.4

0

0.1

VGate (V)

0.2

0.3

0.4

VGate (V)

(a)

(b)

Figure 2.17: The comparison of I-V profiles under different doping models revealing
the individual dopant induced voltage threshold lowering effect. (a) VS/D = 0.2V;
(b) VS/D = 0.4V.
the present individual model than that in the continuous model. This fact reveals the
individual dopant induced voltage threshold lowering effect.
It is noted that without using the constraint of Eq. (2.82), the impact of individual
dopants and impurity can be very significant for small nano devices. The proposed
Dirac delta function model can be used for the further investigation of this aspect.

2.3.3

Simulation results for four-gate MOSFETs

In order to achieve good device performance, more effective gate control strategy is
required for nano-scale MOSFETs. In this section, the four-gate MOSFET, one kind
of silicon nanowire transistors (SNWTs) which resemble multi-gate or gate-all-around
devices, is explored. The transport profile of an electron in the SNWT is similar to
that of the double-gate MOSFET and is simulated via the NEGF as well. While being
different from the planar double-gate MOSFET, the charge and potential profiles
in the transverse direction are no longer invariant. 3D simulations of the SNWT
with discrete dopants are implemented in this section by using the technique of the
71

Subband Energy (eV)

0.5

0

Subband 1
Subband 2
Subband 3
Subband 4
−0.5
0

5

10

15

20

25

30

Transport Direction (nm)
Figure 2.18: Subband energies for the four-gate MOSFET. The dopants break the
symmetry and energy degeneracy in the second and third subbands.

subband decomposition. The MIB method and the Dirichlet-to-Neumann mapping in
3D versions described in the previous sections are employed. The configuration of the
SNWT is given in Fig. 2.2 and the detailed parameters are almost the same as those
in the double-gate MOSFET, except that the y-direction width is 5nm, with 3nm for
the silicon layer and 1nm for each SiO2 layer. The effective mass in the y-direction
is taken as mz = 0.318m0 .
The main difference of the present simulation and the simulation of the doublegate MOSFET is that one needs to solve the Schr¨dinger eigenvalue problem in 2D
o
for each slice of the device. Additionally, the Green’s formula for δ− functions takes
a different form, as discussed in an earlier section.
Main results from simulating the four-gate MOSFET are quite similar to those of
simulating the double-gate MOSFET. For example, the relation between the subband
energy and S/D bias (or gate voltage) are generally similar to those of the doublegate MOSFET. Figure 2.18 shows the first 4 subband energies of the SNWT with 15
72

3

2.5

2.5

−0.2
2
−0.25

1.5
1

−0.3

0.5
0
0

Silicon Width (nm)

Silicon Width (nm)

3

−0.35
1

2

3

−0.15
−0.2

2

−0.25
−0.3

1.5

−0.35
1

−0.4
−0.45

0.5

−0.5
0
0

1

2

Silicon Thickness (nm)

(a)

3

Silicon Thickness (nm)

(b)

Figure 2.19: Cross-section of potential profiles for the four-gate MOSFET obtained
with (a) Continuum doping; (b) Discrete dopants (10).

discrete dopants in source and drain under 0.4V S/D bias and gate voltages. Because
a 2D eigenvalue problem is solved, the second and third subbands are almost identical
(they are identical for continuous dopants). The fourth and fifth subbands are very
similar too. Only the fourth subband is plotted.
Figure 2.19 presents the potential profile in the four-gate MOSFET at certain slice
in the source region. It is found from Fig. 2.19(a) that the profile is symmetric due to
the uniform doping and the symmetry of the device. While due to the presence of the
randomly distributed individual dopants, the potential profile is no longer symmetric.
Finally, as in the last section, the current fluctuation of the four-gate MOSFET
with individual dopants is studied and the I-V characteristics is presented in Fig.
2.20. For comparison to previous section , the current is re-scaled to current density
according to the device geometry. Figures 2.20(a), (b) and (c) are I-V curves of the
SNWT with 15, 40 and 80 dopants in different positions, respectively. While Fig.
2.20(d) shows the fluctuations with different numbers of dopants. For each number
of dopants, the current values are averaged out with different positions. It can be
concluded that as the number of individual dopants increases, the I-V curve is getting
73

600

500

500

400

Continuum
Distribution 1
Distribution 2
Distribution 3
Distribution 4
Distribution 5

300
200
100
0
0

I (A/m)

700

600

I (A/m)

700

0.1

0.2

0.3

400

Continuum
Distribution 1
Distribution 2
Distribution 3
Distribution 4
Distribution 5

300
200
100
0
0

0.4

0.1

VDS (V)

(a)

0.3

0.4

(b)
700

600

600

500

500

400

Continuum
Distribution 1
Distribution 2
Distribution 3
Distribution 4
Distribution 5

300
200
100
0
0

I (A/m)

700

I (A/m)

0.2

VDS (V)

0.1

0.2

0.3

400
Continuum
<N>=15
<N>=40
<N>=80
<N>=100

300
200
100
0
0

0.4

VDS (V)

0.1

0.2

0.3

0.4

VDS (V)

(c)

(d)

Figure 2.20: Current fluctuations of the four-gate MOSFET with (a) 15 dopants; (b)
40 dopants; (c) 80 dopants; (d) Different number of dopants.

74

closer to the profile of the continuous doping. This verifies the validity of our modeling
for the same reason stated in the last section.

2.4

Conclusion remarks

This work presents mathematical modeling and computational algorithms to nanoscale electronic devices, or ultimate metal oxide semiconductor field effect transistors
(MOSFETs). MOSFETs are the elementary building blocks of integrated circuits,
such as microcircuits, microchips, or silicon chips, which are used in almost all electronic devices or equipments presently in use and have revolutionized the world of
electronics in the past few decades. For years, MOSFETs are designed and functioning according to classical mechanical laws. The continuous miniaturization of nano
transistors has led to a new era in design, manufacture, modeling and simulation
of nano electronic devices where the quantum effects start to play a very important role. The benefits from the success of these new devices can be tremendous:
On one hand, the new generation of ultimate MOSFETs will dramatically advance
current semiconductor technology and still operate with the classical principle after suppressing severe quantum effects . On the other hand, functionally enhanced
MOSFETs, or nano-quantum transistors, that utilize the fundamental properties of
nature which do not have direct analogs in classical physics, will lead to new breakthroughs in device science and technology. However, enormous degrees of freedom of
nano electronic devices make their first principle quantum mechanical modeling and
simulation essentially intractable. Efficient mathematical modeling, approximation,
and computation for this class of problems promise an important topic in applied
and computational mathematics. This work addresses a few important issues in the
modeling and simulation of nano electronic devices.
A simple and perhaps the most popular model is the effective mass description
75

of a single electron in band structures governed by the Kohn-Sham equation with an
electrostatic potential solved from the Poisson equation. Even at this level of description, current modeling and simulation hardly take into account the significant impact
of individual dopants, irregular geometric designs and material interfaces, due to the
complexity and challenge of the problem. Additionally, inconsistence in many approximations undermines their performance. The present work addresses the above mentioned problems. First, we introduce a two-scale variational framework that, upon energy optimization, generates new self-consistently coupled Poisson-Kohn-Sham equations, or Poisson-Schr¨dinger equations. In this framework, the quantum dynamics
o
of microscopic particles is determined by a nano-scale environment, which models
the continuum electrostatics governed by the classical Maxwell’s equation. As such,
classical theory and quantum mechanics are put on an equal footing at nano-scale.
Additionally, the magic and art of semiconductors are, by and large, due to the manipulation of layered structures. Such structures have great impact in electrostatic
potentials and thus, electronic properties. In this work, we introduce material interfaces in coupled Poisson- Kohn-Sham equations to properly describe the interface
effect to the electrostatic potential. Finally, random dopants and impurity have a
dramatic impact to the electronic structures and electronic transport. We present a
new individual dopant model by utilizing the Dirac delta function. Unlike previous
individual dopant models [155, 9, 108, 84] that depend on tunable parameters, the
proposed model is free of computational parameters.
This work also introduces two computational algorithms for the simulation of
nano electronic devices. An efficient elliptic interface method, the matched interface
and boundary (MIB) method [160, 165, 164, 159, 158], is employed for solving the
Poisson equation with semiconductor interfaces. Although the MIB method is of
arbitrarily high-order accuracy in principle, we only utilize the second order MIB
scheme in the present work. However, due to the strong nonlinearity of the coupled
76

Poisson- Kohn-Sham equations, the numerical order was found to be about 1.5 in the
present Gummel-like iteration. The other computational algorithm employed in the
present work is the Dirichlet to Neumann mapping (DNM). This approach provides a
rigorous treatment of the singular charges sources in the Poisson equation due to the
individual dopant model proposed in the present work. The basic idea of the DNM
is to separate the singular part of the solution from the regular part, and then solve
the singular part analytically. The analytical treatment of the singular part gives rise
to a Neumann interface condition or boundary condition for the regular part of the
Poisson equation. By splitting the Poisson equation, we have taken the advantage
that the amplitudes of the both discrete dopants and continuum doping are fixed,
while those of the electronic density is changing during the Gummel-like iteration.
The DNM technique works well in the present device simulation.
Two multi-gate MOSFET systems, a double-gate MOSFET and a four-gate MOSFET, are considered in the numerical simulations of the present work. Both problems are modeled in the three-dimensional (3D) setting. The subband decomposition
approach is utilized to simplify the computation of the electronic structures. The
non-equilibrium Green’s functions (NEGF) formalism is employed for the description
of electronic transport. The convergence and well-posedness of the present models
are analyzed. A relaxation technique is developed to improve the convergence of the
iteration scheme which is subject to possible numerical divergence induced by the
charge aggregation of randomly distributed individual dopants. In our double-gate
MOSFET simulations, the basic characteristics and the quantum effect of the I-V
curves are similar to those in the literature. The impact of randomly distributed
individual dopants to electronic structure and transport is studied. In particular, individual dopant induced voltage threshold lowering effect is clearly demonstrated. In
our four-gate MOSFET simulations, individual dopants effectively break the symmetry of the device. Due to the 2D quantum confinement, the density of quantum states
77

that are relevant to the electronic transport become larger than that of the doublegate MOSFET. Effects of multigate voltage settings and irregular device geometry
are under our investigation.

78

Chapter 3
Modeling and simulations of
transmembrane proton channels
In this chapter we provide the theoretical formulation of our model of quantum dynamics in continuum.

3.1
3.1.1

Theory and model
General description of the model

An ion channel system is complex in terms of biological structure, dynamics and
proton transport. Our goal is to model the dynamics and to predict the transport.
To this end, we propose a multiscale, multiphysics and multidomain model. The
computational domain Ω is divided into two subdomains, i.e., the solvent domain
Ωs consisting of the extracellular/intracellular solvent regions and the ion channel or
channel pore region, and the biomolecular subdomain Ωm including the membrane
protein(s) as well as lipid bilayers. Therefore, we have Ω = Ωs ∪Ωm . A detailed graph
of these subdomains is given in Fig. 3.1. The interface Γ between solvent-membrane
protein is defined by the molecular surface generated by the MSMS software package
79

(a)

(b)

Figure 3.1: (a) Illustration of multiscale model of a proton channel; (b) Computational
domains of the multiscale model with Ωm being the channel molecule and membrane
domain and Ωs being the solvent domain. Here z-direction is regarded as the transport
direction.

[131]. It is interesting to note that the physics in each subdomain is very different and
there are multiphysics phenomena even in a single subdomain. For the biomolecular
subdomain, the membrane protein and lipid bilayer structural data are either generated for molecular dynamics simulations, or downloaded from the Protein Data Bank
(PDB) which are collected from X-ray crystallography or nuclear magnetic resonance
(NMR) experiments. The force field parameters, such as atomic van der Waals radii
and point charges, are obtained from the CHARMM force field [105]. This structural
information is utilized in solving the Poisson-type equation for the electrostatic potential. The electrostatic potential distribution near the channel pore is crucial to
the channel selectivity, gating, and ion conductance. The interactions between the
channel protein and transmission channel ions are accounted in the present model.
In the solvent subdomain, there are three types of materials, ions of interest (i.e.,
protons), all other ion species and water molecules. In this system, the charge-charge
80

interactions contribute to the predominate potential energy landscape. Whereas, the
strength of other interactions, such as ion-water dipolar interactions, water-water
interactions and molecular van der Waals interactions, is much weaker than that
of direct charge-charge interactions. This feature provides us a ground to take a
multiscale approach to the multiphysics situation in ion channel dynamics. To reduce
the number of degrees of freedom, we treat solvent (water) molecules as continuum
background or bath. The formation of ion and water clusters and possible ion-water
correlations are modeled partially as a dielectric constant effect and partially as a
generalized-correlation potential effect. Except for the ions of interest, other ions
usually have a small population in the channel pore of a selective channel. Whereas
in the bath region, all ions are essentially in a quasi-equilibrium state and their
densities are well described by the Boltzmann distribution except for at the solventmembrane protein interface. Near the solvent-membrane protein interface, the density
distribution of ions might be better described by the density functional theory of
solution, or integral equations, in which the dispersion interaction between solvent
and solute can be better accounted. This effect is modeled as generalized-correlation
potential effect in the present work.
The physics in the channel pore region is of central interest and is sharply different from those of other regions. The ions of interest are selected as those which have
significant population inside the channel region. There are many evidences which indicate the quantum mechanical behavior of proton transfer in biomolecular systems and
proton channels [41, 20]. The first reason is the small mass of a proton which enhances
the quantum tunneling effect in the proton transport. Additionally, a narrow channel
morphology in many proton channels, such as the Influenza A M2 proton channel
[28, 134] leads to severe quantum confinement, which consequently promotes quantum effects. Finally, hydrogen-bonded chain (proton nanowire) of water molecules assisted proton translocation is quantum mechanical in origin [121, 123, 122]. Although
81

theoretical models were proposed in the last decades [135, 144, 21], the detailed mechanism of proton dynamics and transport is not fully understood. For these reasons,
we treat protons quantum mechanically via a scattering formalism which describes
how a quantum mechanical proton scatters through electrostatic and generalizedcorrelation potential fields. The electrostatic potentials include interactions between
protons represented by a self-consistent mean field approximation, the interactions
between protons and fixed ions from membrane proteins and lipid bilayers, and the
interactions between protons and other ion species. The generalized-correlation potential is due to the impacts of the continuum solvent, the van der Waals interaction
between the solvent and biomolecules, the effect of ion-water clusters, disperion effect, and possible break-down of hydrogen-bonded chain in a narrow channel, etc.
We utilize a total energy functional framework [154, 27, 31] to incorporate quantum
mechanical description and continuum description. Coupled Kohn-Sham equation
for the proton dynamics and Poisson-Boltzmann equation for the electrostatic potential are derived from the variational principle. Solutions to these coupled equations
give rise to proton structure dynamics, and transport in the ion-channel process,
which describes how a quantum mechanical proton scatters through electrostatic and
generalized-correlation potential fields.

3.1.2

Free energy components

This subsection describes various free energy components in our multiscale model of
quantum dynamics in continuum. In order to give a clear description, Fig. 3.1(a) is
reduced to a sketch in Fig. 3.1(b) in x − z cross section, where the z direction represents the proton transport direction: the system is restricted to a rectangular cuboid
with appropriate size and partitioned into two different computational domains. The
82

permittivity (r) has different values in two domains


 s (r) ∀r ∈ Ωs
(r) =
.

 m (r) ∀r ∈ Ωm

(3.1)

Since both the membrane and channel protein are treated with same dielectric
medium, the interface between them is erased and a constant dielectric constant is
assumed on Ωm . On the contrast, the solvent in the bath regions and in the channel pore have different biological characteristics. Therefore the position dependent
dielectric constant is imposed on the solvent domain Ωs . In fact, s (r) in the channel
region can differ much from that in the bulk region. The detailed discussion about
the dielectric constants is given in Section 3.2.5.

Electrostatic free energy in the biomolecular region

The biomolecular region consists of membrane protein and lipid bilayer. Their structures determine the channel selectivity and gating efficiency. In the present treatment,
we assume that structures of membrane protein and lipid bilayer are given and do
not change during the ion transport process. This is certainly an approximation and
will be easily removed in our future work by a combination of the present formulation with MD simulations [154]. Without structural cooperation, the biomolecules
still significantly contribute to ion dynamics and transport by electrostatic interactions. The fixed charges in the channel protein and nearby lipid bilayers determine
the fundamental characteristics of the channel and provide the primary environment
for ions’ permeation. Since the total number of fixed charges is not too large (i.e., up
to thousands), with the assumption that the positions of them are essentially fixed,
the explicit discrete description is actually affordable. In this sense, they serve as a
83

source term in the electrostatic potential calculation
Na

Qi δ(r − ri )

ρf (r) =

(3.2)

i=1

where Na is the total number of fixed charges, Qi and ri are the point charge and
position of the ith atom. Therefore, the electrostatic free energy in biomolecular
domain is given by
m (r)

GMol [Φ, n] =

2

| Φ|2 − Φρf dr,

(3.3)

where Φ(r) is the electrostatic potential and is defined on the whole domain Ωs ∪ Ωm .

Electrostatic free energy in the solvent region
The ions in the solvent region also contribute to the electrostatic potential. Protons
and other ion species are treated in different manners. Let us denote the proton
number density in the solvent region as n(r) and the charge density as ρp = qn(r),
q is the elementary charge or charge carried by a single proton. The charge density
serves as a source term in the electrostatic free energy.
In the solvent region, particularly, in the extracellular and intracellular solvent
regions, apart from ions of interest, there are many other ions. In the present model,
all other ions are treated in a different manner from the ions of interest. Specifically,
no detailed description is given to individual ions except for the ions of interest.
However, other ions contribute considerably to the electrostatic property of the whole
system. To account for their electrostatic effort, we describe other ions by using the
Boltzmann distribution. The charge density of other ions is given by
Nc

ρ =

Nc

j

−qj (Φ(r)−VExt )/kB T
,

qj n0 e
j

qj nj (r) =
j

84

(3.4)

where Nc is the total number of other ion species, n0 and qj are the bulk constant
j
−qj (Φ(r)−VExt )/kB T

density and charge of the jth ion species. Here nj = n0 e
j

is the

number density of jth ion species, it can be noticed that the Boltzmann distribution of the other ion species with respect to the potential has been modified with
the generalized chemical potential VExt , which represents the effects of the chemical
potential of jth ion species and the external electric field [125, 127].
The corresponding electrostatic free energy in the solvent region is given by
s (r)

GSol [Φ] =

2
Nc

+ kB T

| Φ(r)|2 − Φ(r)ρp (r)

qj (Φ(r)−VExt )
−
kB T
n0 e
j




− 1 dr

(3.5)

j

Note that the electrostatic free energy of other ions in Eq. (3.5) is similar in spirit to
Sharp and Honig [138], Gilson et al [68], Chen et al [31] and Wei [154].

Proton free energies and interactions

The solvent region might admit a number of ion species, of which a full quantum model
can be technically complicated and computationally time consuming. We therefore
only treat the ions of interest, i.e., protons, quantum mechanically and assume a
continuum description of other ion species. To simplify the problem further, we
consider a generalized density functional theory for protons.

Kinetic energy. The proton density operator nH is given by
nH = e−(H−EExt )/kB T .
85

(3.6)

where H is the Hamiltonian of the system and EExt is the external electrical field
energy. We define the proton density n(r) as

n(r) = r|nH |r =

|ΨE (r)|2 e−(E−EExt )/kB T dE,

(3.7)

where ΨE and E are the wavefunction and corresponding energy associated with H.
The Boltzmann statistics is adopted in the present work. The kinetic energy is given
p2
by 2m(r) where p is the momentum and m is proton effective mass. In the coordinate

representation, the kinetic energy of protons can be given as
2 e−(H−EExt )/kB T

2m(r)

| ΨE (r)|2 dEdr,

(3.8)

where the Boltzmann factor weights different energy contributions.

Electrostatic potential. Protons have a number of electrostatic interactions. First,
protons interact repulsively among themselves

UIon−Ion (r) =

q 2 n(r)n(r )
dr .
(r)|r − r |

1
2

(3.9)

These interactions lead to a term that is nonlinear in density n and the resulting
equations are to be solved iteratively.
Additionally, interactions between protons in the solvent and fixed charges in
biomolecules are described as
Na

UIon−Fix (r) =
i=1

qn(r)Qi
.
(r)|r − ri |

(3.10)

This contribution can be handled by the so called Dirichlet to Neumann mapping
approach [27].
86

Finally, interactions between protons and other ion species are of the form
Nc

UIon−Other (r) =
j=1

qqj n(r)nj (r )
(r)|r − r |

dr .

(3.11)

where the other ionic densities are determined from the continuum Boltzmann distribution in the solvent region with a given profile of electrostatic potential as shown in
Eq. (3.4). Therefore, the electrostatic potential energy functional of protons is

[UIon−Ion (r) + UIon−Fix (r) + UIon−Other (r)] dr.

Generalized-correlation potential. The electrostatic potential plays a dominant
role in the ion transport process. However, generalized-correlation effects are also important to ion conductance efficiency. Sometimes, generalized-correlation effects can
even determine the channel selectivity. Generalized-correlation effects physically originate from van der Waals interactions, dispersion interactions, ion-water dipolar interactions, ion-water cluster formation/dissociation, temperature and entropy effects,
etc. For example, one of generalized-correlation effects is an energy barrier to the ion
transport due to the change in the solvation environment from the bulk water to a
relatively dry channel pore. However, due to the lack of a comprehensive understanding of the ion behavior in channel region, the modeling of generalized-correlations is
less quantitative, compared to the electrostatic modeling. In the Brownian dynamics
model and the PNP theory, these generalized-correlation effects are encapsulated in
the relaxation time and diffusion coefficients, respectively, which are obtained from
experimental data and tuned in a reasonable biological range to predict new results.
In the present work, we consider a reduced model for generalized-correlation potential energy. We assume that generalized-correlation potential is also a functional of
the local ion density n(r) and the density gradient
87

n, i.e., UGC [n, n]. It includes

two contributions: One is the interaction among the target ions themselves, which
represents those short range interactions and possible collisions; the other is the interaction between the ion and the surrounding water molecules, which may include
many-body ion-water collisions and dehydration effects. In an analogous structure of
energy (3.9), the former should be a quadratic form while the latter is a linear form
like Eq. (3.10) of the ion density n(r). Based on these considerations, we assume
that the generalized-correlation potential energy functional has the following form
∂UGC [n,
∂n
where the

n]

dr =

VGC [n]dr =

(αn(r)kB T + VIon−sur [n]) dr,

(3.12)

n dependence has been omitted for simplicity. The first term of Eq. (3.12)

is linear in the ion density. Intuitively, if more ions exist in the system, the possibility
of the ion-ion generalized-correlation interaction is higher. The energy resulting from
the ion-surrounding interaction is simply modeled as energy VIon−sur [n], which can
be considered as related to the density and relaxation time of ions. The range of
VIon−sur [n] value is discussed in Section 3.2.5. Here α is a relative weighting parameter
for balancing the contribution of two components in the overall UGC [n].

External potentials Since the extracellular and intracellular surroundings can be
infinitely large, it is impossible to include them in a detailed description. In the
present work, we make appropriate truncation of the surrounding system. As such,
the interaction of channel protons with extracellular and intracellular surroundings
are described by external potentials UExter . In addition to the truncation effect,
the external potentials also describe the experimental conditions such as the effect of
given extracellular and intracellular bulk concentrations. We denote channel potential
88

energy functional as

UExter [n]dr =

VExter (r)n(r)dr =

[VExtra (r)n(r) + VIntra (r)n(r)] dr (3.13)

where VExtra n(r) and VIntra n(r) are extracellular and intracellular positions, respectively. Because much of extracellular and intracellular surrounding is not explicitly
described, VExter must be non-hermitian. This aspect is discussed in Section 3.1.5.

Proton total energy functional. The total proton potential consists of electrostatic, generalized-correlation and external potentials

U (r) = UElec (r) + UGC (r) + UExter (r)
1
=
2

q 2 n(r)n(r )
dr +
(r)|r − r |

Na
i=1

qn(r)Qi
+
(r)|r − ri |

Nc
j=1

qqj n(r)nj (r )
(r)|r − r |

+ UGC [n(r)] + UExter [n(r)].

dr
(3.14)

Thus, the total free energy functional of protons includes kinetic and potential contributions

GIon [Φ, n] =

2 e−(E−EExt )/kB T

2m(r)

| ΨE (r)|2 dE + U (r) dr,

(3.15)

where each kinetic energy term is weighted by the Boltzmann distribution, which is
similar to the treatment in our recent work [27].

3.1.3

Total free energy functional of the system

To understand the behavior of protons and their interactions, we consider a total free
energy functional that includes all significant kinetic and potential energies. Similar
energy framework has been developed in our recent work for biomolecular systems
89

and nano-electronic devices [154, 27, 31]. The total free energy functional of the
present system is given by the combination of the electrostatic energy of the system
and the quantum mechanical energy of protons. However, it is important to avoid
double counting when one constructs the total energy functional [154]. For the present
system, it is interesting to note that had the charge sources qn(r ) +
r)+

Nc
j=1 qj nj (r

1
qn(r)Φ(r) =
2

Na
i=1 Qi δ(r −

) been independent of Φ, we would have

q 2 n(r)n(r )
dr +
(r)|r − r |

Na
i=1

qn(r)Qi
+
(r)|r − ri |

Nc
j=1

qqj n(r)nj (r )
(r)|r − r |

dr (3.16)

in a homogeneous dielectric medium. Therefore, the charge source for the electrostatic potential also serves the electrostatic potential energy for protons. With this
consideration, we propose the total free energy functional

GTotal [Φ, n] =




qj (Φ−VExt )
Nc
 (r)
−
kB T

| Φ|2 − Φ(ρp + ρf ) + kB T
n0 e
− 1
j
 2
j

2 e−(E−EExt )/kB T

−

2m(r)
−λ

| ΨE (r)|2 dE + UGC [n] + UExter [n]

e−(E−EExt )/kB T |ΨE (r)|2 dE − Np

dr.

(3.17)

where the last term in Eq. (3.17) is the Lagrange multiplier for the constraint of
proton density. The quantity Np is the total number of protons in the system, i.e.,

n(r)dr = Np ,

(3.18)

Ωs

However, in most experimental set-ups, one does not know Np . Instead, the bulk
concentration or the bulk number density, n0 , is given. When the solvent domain is
p
90

sufficiently large compared to the channel pore region, one has two approximations
Np ∼ n0
= p

Ωs

e−q(Φ(r)−VExt )/kB T dr ∼ n0
= p

dr,

(3.19)

Ωs

where the second approximation is a crude estimation.
The energy functional (3.17) is a truly multiphysical and multiscale framework
that contains the continuum approximation for solvent and membrane while explicitly
takes into account for the channel protein in a discrete fashion. More importantly, it
mixes the classical theory and quantum mechanical descriptions on an equal footing.
Note that Eq. (3.17) is a typical minimization-maximization problem, where the
electrostatic free energy is to be minimized while the kinetic energy of protons is to
be maximized. Fortunately, this situation does not create a problem as the optimization of the total free energy functional is achieved with two governing equations as
described in the next section.

3.1.4

Governing equations

The present system has two unknown functions: the electrostatic potential Φ and
the wavefunction ΨE . All other functions either are to be explicitly given or depend
on Φ and ΨE . The governing equations for Φ and ΨE are to be derived from the
free energy functional by variational principle via the Euler-Lagrange equation. This
multiscale variational framework approach was developed in our recently work [154,
27]. It offers successful predictions of the solvation free energies of proteins and small
compounds[31, 32].

Generalized Poisson-Boltzmann equations
The total free energy functional given above determines the density distribution and
dynamics of protons. The governing equation for electrostatic potential can be derived
91

by the variation of the functional with respect to the potential Φ
δGTotal [Φ, n]
=⇒ −
δΦ

· ( (r) Φ(r)) = ρp (r) + ρf (r) + ρ (r).

(3.20)

Equation (3.20) is a generalized Poisson-Boltzmann (GPB) equation describing the
electrostatic potential generated from three types of charge sources: the ions of interest, other ions species in the solvent described by the continuum approximation
and the fixed point charges in biomolecules. This equation is not closed because n(r)
needs to be evaluated from another governing equation.
A special case of Eq. (3.20) is also very interesting. Let us assume that all ions in
the system are described either by fixed point charges from biomolecules, or by the
continuum treatment. Therefore, the system is closed and we arrive at the classical
Poisson-Boltzmann equation

−

where ρs (r) =

· ( (r) Φ(r)) = ρf (r) + ρs (r),

Nc
j=1 qj nj (r),

(3.21)

and Nc is for all ions in the continuum solvent.

Generalized Kohn-Sham equations
In the present multiscale model, the density n of protons in Eq. (3.20) is governed by
generalized Kohn-Sham equations. This set of equations is obtained by the variation
of the total free energy functional with respect to wavefunction Ψ∗
E
δGTotal [Φ, n]
=⇒ −
δΨ∗
E

2

·

2m(r)

ΨE (r) + V (r)ΨE (r) = λΨE (r) = EΨE (r) (3.22)

where the multiplier λ is chosen as the eigenvalue E and

V (r) = qΦ(r) + VGC [n] + VExter (r)
92

is the effective potential, which includes electrostatic, generalized-correlation and external interactions. The effective potential is discussed in Section 3.1.2.
Equation (3.22) appears to be the conventional Kohn-Sham equation. However,
there are some important differences. First, the exchange-correlation potential, which
is crucial to electrons, is not presented in Eq. (3.7). The origin of the exchangecorrelation potential is from the Fermi-Dirac distribution, spin and many other unknown effects. In the present theory, we use the generalized-correlation potential to
represent many unaccounted effects. We assume the Boltzmann statistics for the ions
of interest at ambient temperature. Additionally, we define the density as in Eq.
(3.7), instead of the conventional choice for electrons: nelectron (r) =

j

|Ψj (r)|2 .

This definition is partially due to the Boltzmann statistics and partially due to the
spectrum of the present Kohn-Sham operator, which is bounded from below. Technically, the Hamiltonian of the generalized Kohn-Sham equation (3.22) has not only
discrete spectra, but also absolute continuum spectrum. As such, a Boltzmann factor
in the density definition is indispensable. Finally, unlike the conventional Kohn-Sham
equation, the present generalized Kohn-Sham equation is not a closed one. It is inherently coupled to the generalized Poisson-Boltzmann equation (3.20). This coupled
Kohn-Sham and Poisson-Boltzmann system endows us the flexibility to deal with
complex multiphysics in a multiscale fashion — the quantum dynamics in continuum.

3.1.5

Proton density operator for the non-hermitian Hamiltonian

As mentioned earlier, the external potential has a non-hermitian component to describe the interaction with truncated extracellular and intracellular surroundings. Let
93

us explicitly separate the anti-hermitian (or skew hermitian) components
h
ah
VExtra = VExtra + VExtra ,

h
ah
VIntra = VIntra + VIntra ,

(3.23)

where
1
†
h
Vα = (Vα + Vα ),
2

1
†
ah
Vα = (Vα − Vα ),
2

α = Extra, Intra.

(3.24)

The non-hermitian parts of the external potentials describe the relaxation effect or
spectral line shape broadening due to the interaction with the surroundings. Accordingly, we split the Hamiltonian as
ah
ah
H = H h + V ah = H h + VExtra + VIntra .

(3.25)

We first note that the density of protons can be further given by

nH =

e−(E−EExt )/kB T δ(E − H)dE.

(3.26)

In this work, we define the spectral operator δ(E − H) as

δ(E − H) =

i
1
1
lim lim
−
2π ε→0 V ah →0 E − (H − iε) E − (H − iε)†

(3.27)

We therefore approximate the proton density operator by

nH =

i
2π

e−(E−EExt )/kB T G(E) − G† (E) dE,

(3.28)

where G is the Green’s function (operator)
G(E) = (E − H)−1 .

94

(3.29)

We therefore arrive at a useful expression for the proton density

nH =

i
π

e−(E−EExt )/kB T

ah
G(E)Vα G† (E) dE

(3.30)

α

i
=
π α

ah
e−(E−Eα )/kB T G(E)Vα G† (E)dE,

(3.31)

where α = Extra, Intra, EExtra and EIntra are the external electrical field energies at
extracellular and intracellular electrodes, respectively. Note that EExt behaves like
an operator such that its value is chosen according to the nearest external interaction.
Equation (3.31) provides an appropriate expression for computing the total proton
density.

3.1.6

Proton transport

Typically, external electrical field is applied as the difference of electrical potentials,
(EExtra /q − EIntra /q). The experimental measurements are given as the current and
voltage curve, or the so called I-V curve. Therefore, a major goal of our theoretical
model is to provide predictions of the current under different external voltages. The
current in the standard quantum mechanics is given by

I = qTr
= q

1
n v † + vnH
2 H
2mi

Ψ∗ (r)
E

(3.32)
ΨE (r) − ΨE (r)

Ψ∗ (r)
E

1
where Tr is the trace operation and 2 nH v † + vnH

−

e

(E−EExt )
kB T
drdE

(3.33)

is the symmetrized current

operator with v being the velocity vector. Equation (3.33) requires the evaluation of
the full scattering wavefunction ΨE (r). The spatial derivative can be carried out at
a location consistent with the specific feature of the external electrical field EExt .
An alternative current expression can be given by examining the transition rates
95

due to the anti-hermitian parts of the external interaction potential. Let us evaluate
ah
the transition rate according to the interaction potential VExtra

I = q
=

†
1 1
ah
ah
Tr nH VExtra + VExtra nH
i
2

q
Tr
h

†

ah
ah
G(E)Vα G† (E) VExtra

e−(E−EExt )/kB T

dE

α
ah
G(E)Vα G† (E)dE

ah
VExtra e−(E−EExt )/kB T

+

(3.34)

(3.35)

α

Now we need to make a decision for EExt because each term involves two interaction
potentials. In this work, we systematically choose EExt according to the nearest
external interaction

I =

q
Tr
h

e−(E−Eα )/kB T

ah
ah
G(E)Vα G† (E) VExtra
ah
G(E)Vα G† (E)dE

α
E−EExtra
−
kB T
e


=

q
Tr
h

dE

α

ah
VExtra e−(E−EExtra )/kB T

+

†

ah
ah
GVIntra G† VExtra

(3.36)


E−EIntra
−
kB T
 dE
−e

(3.37)

ah
Similarly, we obtain a current expression by using the interaction potential VIntra
†
1 1
ah
ah
Tr nH VIntra + VIntra nH
i
2


E−EIntra
E−EExtra
−
−
q
ah
ah
kB T
kB T
 dE
=
Tr GVExtra G† VIntra e
−e
h

I = q

(3.38)

Equations (3.37) and/or (3.38) can be used for current evaluations under different
external electrical field strengths and concentrations.

96

3.2

Computational algorithms

The implementation of the theoretical model described in Section 3.1.4 involves a
number of computational issues. The present section is devoted to the computational
implementation of our quantum dynamics in continuum model.

3.2.1

Proton density structure and transport

Proton density structure concerns the solution of the generalized Kohn-Sham equation whereas the proton transport offers the current-voltage curves, which are to be
compared with experimental measurement. This subsection describes the solution
strategy of the generalized Kohn-Sham equation and theoretical prediction of experimental data.

The solution of the generalized Kokn-Sham equation
Typically, solving the full-scale Kohn-Sham equation can be a major obstacle in the
simulation. Due to the fact that biological characteristics for each subdomain of the
ion channel system are quite different and the Kohn-Sham operator will have distinct
properties correspondingly. In this subsection, we make use of various decomposition
schemes to reduce the computational complexity in solving Eq. (3.22).
Motions of quantum particles in the present system can be generally classified into
three categories: scattering along transport directions, confined motion and free motion. The channel pore direction (i.e., the z direction) is designated as the transport
direction, in which protons cross the transmembrane protein or scatter back to the
solvent. Along the z direction, the Kohn-Sham operator has an absolutely continuous spectrum. In the x − y directions, the Kohn-Sham equation possesses different
behaviors. In the extracellular and intracellular regions where the solvent domains
are sufficiently large, proton motions are essentially unconfined in the x − y direc97

tions. They undergo intensive electrostatic and generalized-correlation interactions
although the system can be regarded as near the equilibrium. The associated KohnSham operator for protons also has an absolutely continuous spectrum. In contrast,
in channel pore region, the protons are confined in x − y plane by the channel wall. In
the confined plane, the Kohn-Sham operator is essentially compact and has a discrete
spectrum. For two different regions, formulations and corresponding treatments of
the proton density are different.
The proton density structure in the channel pore is crucial to the proton transport.
Whereas, the behavior of protons in the bath is relatively less important. Therefore, as
a good approximation, we can truncate the computational domain in the bath regions.
Consequently, the Kohn-Sham operator becomes compact for all x − y directions and
has discrete eigenvalues. As a good approximation for many ion channels, we split
the total wavefunction ΨE (r) as
j

ΨE (r) = ψ j (x, y; z)ψk (z)

(3.39)

where ψ j (x, y; z) is the j-th eigen-mode in the confined directions at a specific location
j

z, and ψk (z) is the wavefunction along the transport direction, with transport wave
number k. Under this circumstance, it is convenience to relabel the total energy E as
j

Ek , where j and k are related to the energies for confined and transport directions,
respectively. If the mode-mode interaction along the confined direction is neglected,
j

it is easy to verify that ψ j and ψk satisfy the following decomposed Kohn-Sham
equations,
2

−

2

∂ 1 ∂
∂ 1 ∂
+
∂x mx ∂x ∂y my ∂y

+ V (x, y; z) ψ j = U j (z)ψ j

(3.40)

ψ j (x, y; z) = 0 on ∂ΩD (z);
2

−

∂ 1 ∂
j
j j
+ U j (z) ψk (z) = Ek ψk (z),
2 ∂z mz ∂z
98

j = 1, 2, · · · ,

(3.41)

where V (x, y; z) is the restriction of the potential operator V (x, y, z) at position z,
U j (z) is the jth eigenvalue of the 2D problem at position z, and ψ j = ψ j (x, y; z) is
j

the corresponding eigenfunction. Here ψk (z) is the scattering wavefunction associated
with the scattering potential U j (z). Here ∂ΩD (z) is the boundary for the cross section
at z. The transport equation (3.41) can be solved as a scattering problem. Finally
the proton density (3.7) can be modified as
j

|ψ j (x, y; z)|2 |ψk (z)|2 e

n(r) =

j
−(E −EExt )/kB T
j
k
dEk

j

.
=

j

|ψ j (x, y; z)|2 nscat (z).

(3.42)

j

Equation (3.42) only gives the symbolic proton density structures for an unspecified
EExt . More detailed consideration of EExt requires the further treatment of the
scattering boundary conditions as shown in Sections 3.1.5 and 3.1.6. However, the
2D wavefunction |ψ j (x, y; z)|2 in Eq. (3.42) can be evaluated from the Kohn-Sham
equation (3.40). The solution to this equation is quite standard — it is just the
eigenvalue problem of an equation of elliptic type. While to solve the transport
problem, as indicated in the theory, one needs to find appropriate expressions of
the non-hermitian external operators. The corresponding computational aspects are
presented in the next subsection.

Boundary treatment of the transport calculation

Although the quantum confinement Eq. (3.40) only happens in finite channel region,
the transport problem Eq. (3.41) is associated with infinitely large surroundings, in
principle. Since the same procedure is used to solve Eq. (3.41) for different j, let us
99

drop the j label
2

−

∂ 1 ∂
+U
2 ∂z mz ∂z

ψk (z) = Eψk (z),

z ∈ (−∞, ∞),

(3.43)

2 ∂ 1 ∂
where − 2 ∂z mz ∂z + U is the scattering Hamiltonian and E is the scattering energy.

In practical computations, the extracellular and intracellular surroundings have to be
truncated. Suppose [zExtra , zIntra ] is the finite transport interval of interest and the
regions (−∞, zExtra ) and (zIntra , ∞) are assumed as infinitely long extracellular and
intracellular environments. We assume that in regions (−∞, zExtra ) and (zIntra , ∞),
the interaction potential U is independent of position due to the spatial average of
homogenization type over the large scale. Consequently, Eq. (3.43) admits planewave
solutions asymptotically. For instance, if one considers the wavefunctions ψk (z) in
the extracellular environment, it has the following form
ψk (z) = eikz + rm e−ikz if z ∈ (−∞, zExtra )
ψk (z) = tm

eikz

(3.44)

if z ∈ (zIntra , ∞)

where rm and tm are reflection and transmission coefficients, respectively. Given the
specific formulation of the wavefunction in the extracellular bath, Eq. (3.44) can be
employed as boundary conditions of Eq. (3.43) to obtain the proton density originated
from the extracellular part. Similar boundary conditions for the intracellular part can
be derived in the same fashion.
Suppose that the interval [zExtra , zIntra ] is discretized as zExtra = z1 , z2 , ..., zN =
Intra, where N is the total number of grid points and the grid size is denoted as
∆z = (z2 − z1 )/N . For simplicity, let t =

2
2mz (∆z)2

, then for interior points zi , (i =

2, ..., N − 1), the discretization of Eq. (3.43) is quite standard by the finite difference
method
−tψi−1 + (2t + Ui − E)ψi − tψi+1 = 0
100

(3.45)

where ψi represents the numerical solution of ψk (zi ) and Ui is for U (zi ). For the
discretization at boundary point z1 , we first define a fictitious function value of ψ(z)
on z0 , the point ahead of z1 as ψ0 , then the discretization at z1 is

−tψ0 + (2t + Ui − E)ψ1 − tψ2 = 0.

(3.46)

Now one needs to determine the fictitious value ψ0 in terms of ψi , (i = 1, 2, ..., N ).
From the boundary condition (3.44), we have
ψ0 = eik0 z0 + rm e−ik0 z0
ψ1 = eik1 z1 + rm e−ik1 z1 .

(3.47)

In fact, we have k0 = k1 since the free motion of the wave in the asymptotic regions.
( k )2

1
We can denote k0 and k1 by k1 with 2mz = E − U1 . By this notation, we have

ψ0 − ψ1 eik1 ∆z = eik1 z0 − eik1 (z1 +∆z)
= eik1 (z1 −∆z) − eik1 (z1 +∆z) .

(3.48)

Inserting Eq. (3.48) into Eq. (3.46), one yields
−tψ1 eik1 ∆z + (2t + U1 − E)ψ1 − tψ2 = −2ti sin (k1 ∆z)eik1 z1 .

(3.49)

Applying the same strategy for ψN and fictitious function value ψN +1 , we have
ψN +1 − ψN eikN ∆z = tm eikN zN +1 − tm eikN zN eikN ∆z = 0,

where

( kN )2
2mz

(3.50)

= E − UN and further
−tψN −1 + (2t + UN − E)ψN − tψN eikN ∆z = 0.
101

(3.51)

Follow the same boundary treatment for the intracellular environment, the whole
system is discretized in vector and matrix forms as the following
G−1 ΨExtra = (Hs − EI)Ψ = bExtra

(3.52)

where ΨExtra = (ψ1 , ψ2 , ..., ψN )T , I is the identity matrix of dimension N × N and


ik1 ∆z
−t
...
 2t + U1 − te


−t
2t + U2 −t
s=
H

.
.
.

.
.
.
.
.
.


0
...
...

...
...
..
.

0
0
.
.
.

−t 2t + UN − teikN ∆z











. (3.53)

N ×N

Here bExtra is the source term for the incoming waves from the extracellular surroundings
bExtra = (2ti sin (k1 ∆z)eik1 z1 , 0, . . . , 0)T .

(3.54)

The wavefunction ΨExtra can be written as

ΨExtra = GbExtra .

(3.55)

†

Let ΨExtra be the complex conjugate of ΨExtra . We have


2 0 ...
 [2t sin (k1 ∆z)]


0
... ...

†
ΨExtra ΨExtra † = GbExtra bExtra G† = G 
.
.
.

.
.
.
.
.
.


0
... ...


... 0 

... 0  †

. G .
... . 
. 

... 0
(3.56)

Similar derivation can be carried out for the wavefunction ΨIntra related to intracel102

lular surroundings,


 0 0 ...

 0 ... ...
†
† = Gb
† = G
ΨIntra ΨIntra
 . .
Intra bIntra G
.
 . .
.
.
 . .

0 ... ...

...

0

...
...

0
.
.
.

. . . [2t sin (kN ∆z)]2





 †
G .



(3.57)

Therefore, the total density matrix is

D=

1
2π

e−(E−Eα )/kB T Gbα bα † G† dk,

α = Extra, Intra.

(3.58)

α

Use the relation
dE = d

2k
( k)2
+0=
dk
2m
m

(3.59)

to change the above integral into that with respect to energy E, and use the simple
limit sin (k∆z)/(k∆z) → 1 as ∆z → 0, the above integral can be easily revised as

D=

where

i
π∆z

ah
e−(E−Eα ) GVα G† dE,

α = Extra, Intra,

α



 −it sin (k1 ∆z) 0 . . .


0
... ...

ah
VExtra = 
.
.
.

.
.
.
.
.
.


0
... ...
and

(3.60)






ah
VIntra = 




0

0

... ...

0 ... ... ...
. .
. ..
. .
.
.
. .
.

... 0 

... 0 


.. . 
. . 
.

... 0

0
0
.
.
.

0 . . . . . . . . . −it sin (kN ∆z)
103



(3.61)






.




(3.62)

It is clear that VExtra and VIntra are the non-hermitian components in the external
ah
potential Eq. (3.23) that are introduced to truncate the surroundings. Since Vα

is solely nonzero for one entry in the matrix and this fact is independent of the
ah
discretization, it is easy to verify that lim∆z→0 ||Vα || = 0, as required in Eq. (3.27).

Obviously, Eq. (3.60) is actually the discretization form of Eq. (3.31). Finally,
the scattering number density is calculated as

nscat (z) = diag(D).

3.2.2

(3.63)

Dirichlet-to-Neumann mapping for the generalize PB
equation

Considering Eq. (3.4) and expression (3.20), the generalized Poisson-Boltzmann equation is
Nc

Na

−

· ( (r) Φ(r)) = qn(r) +

Qi δ(r − ri ) +

−
qj n0 e
j

qj (Φ−VExt )
kB T

(3.64)

j=1

i=1

Recall the fact that the electrostatic potential Φ(r) is defined throughout the domain
Ω, which is inhomogeneous with respect to the dielectric constant (r). Therefore,
we need to physically impose the continuity matching conditions at the interface Γ
of two adjunctive subregions. The continuity matching conditions are given as
[Φ]|Γ = Φ+ (r) − Φ− (r) = 0,
[

Φ · n]|Γ = + Φ+ (r) · n − − Φ− (r) · n = 0

(3.65)
(3.66)

where superscripts “+ ” and “− ” represent the limiting values of a certain function at
two sides of interface Γ, and n is the unit outward normal direction of Γ. Equations
(3.65) and (3.66) guarantee the continuities of the potential function and its flux.
104

Theoretically, Eq. (3.64) admits the boundary condition Φ(∞) = 0 at the infinity.
However, in practical computation, a finite domain is used and appropriate boundary
conditions need to be imposed at the domain boundary ∂Ω. In our studies, the channel
protein and the associated membrane are embedded in a rectangular cuboid with
appropriate sizes. It is very nature to apply the Dirichlet boundary conditions along
the electrode portions of the rectangular cuboid boundary, while for the remainder
of the boundary, we apply the Neumann boundary condition (i.e., the zero normal
electric field conditions).

Physically, the generalized Poisson-Boltzmann equation (3.64) has two types of
charge source terms, i.e., the fixed charges given by the delta functions, and the
unsteady charges. Therefore, it is wise to treat these source terms separately such
that when we keep updating the unsteady source term, we just need to compute the
effect of the fixed charge source term once. Mathematically, the solution of Eq. (3.64)
has a singular part due to the delta function (i.e., fixed charges) which may cause
computational problems. Thus, we should treat the regular part and the singular
part of the solution differently [67]

¯ ˜
Φ=Φ+Φ

(3.67)

¯
˜
where Φ and Φ denote the singular part and regular part of Φ, respectively. More
¯
specifically, Φ should correspond to the singular delta function term and vanish out˜
side the protein and membrane domain Ωm , while Φ is defined in the whole domain.
¯
By this consideration, we split Φ(r) as
¯
Φ(r) = Φ∗ (r) + Φ0 (r)
105

(3.68)

where
Φ∗ (r) =

Na

Qi
m |r − ri |

i=1

(3.69)

¯
represents the Coulomb’s potential from the protein fixed charges. Since Φ(r) is
required to vanish outside the Ωm as well as the boundary ∂Ωm , the Φ∗ (r) should be
corrected by Φ0 (r), which is a harmonic function on Ωm and
Φ0 (r) = −Φ∗ (r),

∀r ∈ ∂Ωm .

(3.70)

˜
For the regular part Φ, we can take the advantage of the fact that n0 is zero in Ωm ,
j
and have the following equation and interface jump conditions:
Nc

−

·

˜
(r) Φ(r) −

−
qj n0 e
j

˜
qj (Φ−VExt )
kB T

= qn(r)

(3.71)

j=1

˜
[Φ]Γ = 0
[

(3.72)

˜
¯
Φ · n]|Γ = −[ Φ · n]|Γ

(3.73)

Through Eqs. (3.67) to (3.71), the electrostatic potential Φ is decomposed into a
˜
singular part and a regular part. It should be noted that it is Φ that is coupled to the
¯
Kohn-Sham equation since Φ is solely nonzero in the protein and membrane region.
The effect of the fixed charges is actually first mapped on the ∂Ωm in a Dirichlet sense
(Eq. (3.70)) and reflected into the solvent region in a Neumann manner (Eq. (3.73))
at the solvent-protein interface Γ. This Dirichlet-to-Neumann mapping (DNM) analytically takes care of the Dirac delta functions and is successfully employed in various
applications [67, 27]. The trade-off of this treatment is that one has to solve an elliptic
equation (3.71) with non-homogeneous interface jump conditions.
Traditional finite difference or finite element methods fail to come up with highorder accuracy and convergence in solving Eq. (3.71) due to geometric singularities in
106

the molecular surface [131] and the need to enforce the interface conditions (3.72) and
(3.73). The matched interface and boundary (MIB) method has been developed for
elliptic equations with complex interfaces, geometric singularity, and singular charges
[158, 159, 67, 165, 163, 26]. It offers second-order accuracy and convergence in solving
the Poisson-Boltzmann equation with biomolecular context [159, 67, 163, 26]. Therefore, the combination of DNM and MIB provides a robust and efficient solution to
the generalized PB equation with second-order accuracy and convergence, even for
complex channel protein geometries.

3.2.3

The self-consistent iteration

In this section we analyze the self-consistent iteration between the generalized PB
equation and the Kohn-Sham equation. To focus on the essential idea, Eq. (3.71) is
symbolically written as
˜
˜
LΦ + F (Φ) = ρp ,

(3.74)

˜
where Φ and ρp represent the electrostatic potential energy and proton charge density,
˜
L represents the linear part of the GPB equation while the F (Φ) is the nonlinear
part. Simply substituting the quantity ρp into Eq. (3.74) does not offer a clue
about the iteration convergence analysis and efficiency. The Gummel iteration [50]
proposed in semiconductor device applications was verified practically that it works
well for a similar self-consistent iteration problem. The idea of the Gummel iteration
is described below.
˜
The proton charge density ρp and the electrostatics potential Φ are assumed to
have the following intrinsic connection

˜
ρp (r) = F (Φ(r), EExt ),
107

(3.75)

˜
˜
where F (Φ, EExt ) = qn0 e−(qΦ−EExt )/kB T is the Boltzmann function and n0 is the

reference number density of the protons. Equation (3.75) represents the relation
between the electrostatic potential and the particle density in the equilibrium state.
However, the relation does not hold any more at non-equilibrium. Nevertheless, we
can extend EExt to a function defined over the entire domain EExt (r) such that
˜
ρp (r) = F (Φ(r), EExt (r)). The intermediate values of EExt (r) can be easily found
˜
once ρp and Φ(r) are available. Based on this argument, Eq. (3.74) is written as a
new nonlinear equation
˜
˜
˜
LΦ + F (Φ) = F (Φ, EExt ).

(3.76)

We need to linearize Eq. (3.76) appropriately. Note that

˜
F (Φ, EExt ) = −

q
kB T

˜
F (Φ, EExt ) = −

q
kB T

ρp

˜
˜
with F (Φ, EExt ) being the Fr´chet derivatives of F with respect to Φ. Similarly,
e
˜
F (Φ) can be evaluated.
l
˜
˜
Suppose Φl , EExt and ρl are the values of Φ, EExt and ρp at lth step iteration,
p

then the Newton’s method for solving Eq. (3.76) is naturally reduced to the Gummel
iteration:
˜
L + F (Φl ) +

q

˜
˜
˜
ρl ∆Φl = ρl − LΦl − F (Φl )
p
kB T p

(3.77)

˜
˜
˜
˜
where we update Φl+1 as Φl+1 = Φl + λ∆Φl and 0 < λ ≤ 1 is chosen through a line
search to guarantee
˜
˜
˜
˜
||LΦl+1 + F (Φl+1 ) − ρl+1 || < ||LΦl + F (Φl ) − ρl ||.
p
p

(3.78)

l+1
˜
Once Φl+1 and ρl+1 are obtained, EExt can be modified, and whole iteration can
p

continue till the convergence is achieved. It is worthwhile to point out that in order
to improve numerical efficiency, Eq. (3.77) can be solved by applying various inexact
108

Newton’s methods. There is plenty of literature about the convergence order discussion so it is necessary for us to generalize the Gummel iteration to the Newton’s
method.
Another technique to enhance the self-consistent convergence is the relaxation
method [27]. If we define the Ks , Us and Ns as the spaces which the external potential
˜
EExt (r), electrostatics Φ(r) and proton charge density ρ(r) belong to, respectively.
For the whole iteration of the generalized Poisson-Boltzmann Kohn-Sham system, it
can be interpreted as the application of the fixed point map T on any of the above
spaces, say T : Us → Us for the electrostatics
˜
˜
Φ(r) = T (Φ(r)).

(3.79)

To characterize the details of the map T , we denote the operator G : Us → Ns ,
which indicates the process of using the Kohn-Sham equation to solve for proton
charge density based on the electrostatic potential. Such a process is followed by
˜
F −1 : Ns → Ks , which updates EExt (r) by ρp (r) and Φ(r). Finally L : Ks → Us
represents solving the nonlinear GPB equation. The combination of all the above
operations yields the definition of the operator T , which shows the outer iteration
T := L ◦ F −1 ◦ G

(3.80)

˜
˜
Φl+1 = L ◦ F −1 ◦ G (Φl ).

(3.81)

and

The relaxation scheme converts Eq. (3.81) into the steady-state problem of an ordinary differential equation (ODE)
˜
∂Φ
˜
˜
= L ◦ F −1 ◦ G (Φ) − Φ.
∂t
109

(3.82)

Therefore many ODE related techniques such as the Runge-Kutta method can be used
to improve the convergence properties. One simple treatment is the discretization of
Eq. (3.82) as
˜
˜
Φl+1 − Φl
˜
˜
= L ◦ F −1 ◦ G (Φn ) − Φn ,
β

(3.83)

which leads to a self-consistent iteration with a relaxation factor β [27, 31]
˜
˜
Φ = L ◦ F −1 ◦ G (Φn )
˜
˜
˜
Φn+1 = β Φ + (1 − β)Φn .

(3.84)

The traditionally used outer loop iteration actually is the special case of Eq. (3.84)
with β = 1. By carefully choosing the relax factor β, one can reach the steady state
(fix point) by self-consistent iterations.

3.2.4

The work flow of the self-consistent iteration

In previous sections algorithms and related mathematical treatments for solving the
GPB equation and the Kohn-Sham equation individually are introduced. Here we
assemble all the components together and give a main work flow for the numerical
simulation of these coupled equations.
• Step 0. Preparation. All the necessary preparations for the whole loop are
accomplished in this step, which include:
– 1. The channel protein of interest is downloaded from the Protein Data
Bank. The partial charges, positions, radii of all atoms as well as molecular surfaces are determined by CHARMM force field [105] and related
software packages, such as PDB2PQR, see Section 3.3 for detail. The
prepared channel structure and surface are then embedded in a proper
computational domain.
110

¯
– 2. Use Eqs. (3.69) and (3.70) to solve for Φ, then the quantity in Eq. (3.73)
is obtained. Implement the DNM and the MIB schemes to discretize the
Laplace operator as matrix L.
• Step 1. Solving the generalized PB equations (3.71) and (3.73). Given ρm
p
(taken an initial guess if m = 0), use the inexact Newton’s method, Eq. (3.77)
˜
and Eq. (3.78) to obtain Φm . Note that the index l in Eq. (3.77) is for the
Newton’s method or inner iteration and the index m is for the outer or whole
self-consistent iteration loop.
• Step 2. Solving the Kohn-Sham equation. The solution of the Kohn-Sham
equation consists of two parts, the eigenvalue problem and the scattering prob˜
lem with the evaluated electrostatic potential energy operator U = q Φm .
– 1. Solving the eigenvalue problem Eq. (3.40).
– 2. Solving the transport problem Eq. (3.41).
– 3. Assembling the total charge density ρm+1 by Eqs. (3.42) and (3.63).
p
˜
˜
˜
• Step 3. Convergence check. Go to Step 1 to obtain Φm+1 , if ||Φm+1 − Φm || <
ε1 and ||ρm+1 − ρm || < ε2 , where ε1 and ε2 are predefined error tolerances,
p
p
then go to Step 4; otherwise go to Step 2.
• Step 4. Current calculation by Eq. (3.37).
Figure 3.2 gives an explicit illustration of the above work flow.

3.2.5

Model parameter selection

The selection of generalized-correlation potential
Generalized-correlation effects are important to ion conductance efficiency. Unfortunately, it is expensive to give a full quantitative description for UGC [n]. In current
111

Channel protein preparation

¯
Solve for Φ

Outer iteration

˜
Solve for Φm

Inner iteration

Solve for proton charge density ρm
p

No

Iteration converged?

Yes

Calculate current

Figure 3.2: Work flow of the overall self-consistent iteration.

existing models, such as PNP based ones, the generalized correlation is integrated as
an overall effect and represented implicitly by the phenomenologically reduced diffusion coefficients in the channel pore region. While in BD based models, the effect
of generalized correlations is included in the ion friction factor, which is also related
to the diffusion coefficient by Einstein’s relation [95]. All these treatments indicate
that UGC [n] should be related to the diffusion coefficient of ions, which is a physical observable. Based on this discussion, we ignore all detailed components while
describe the generalized-correlation interactions as one effective, overall component
in the mean field manner. As indicated by Eq. (3.12), the UGC [n] is also a density
functional of the n(r), and the first term represents the connection between UGC [n]
and given reference ion density. It is quite obvious that α is a tunable parameter.
Here we focus on how to choose parameter VIon−sur .
For a simple start, let this energy be related to the relaxation time τ of an ion by
VIon−sur = /2τ , according to the Einstein’s relation D = kB T τ /m, where D and m
112

are the diffusion coefficient and mass of the particle. Then the energy VIon−sur can
be given by
VIon−sur =

kB T
2mD

(3.85)

for protons. With a appropriate proton mass and the diffusion coefficient in the
bath, one yields VIon−sur ≈ 3.4kB T . However, the value of diffusion coefficient in
the channel is commonly believed reduced, but is inconclusive due to the variation of
the channel pore structure diameters and solvation conditions. According to Table
1 of Ref. [42], proton diffusion coefficients reduce to 1/2 to 1/7 of that in the bath
condition in various lipid layers. We take the resulting reduction accordingly in the
channel region. This argument gives the UGC a range of 6 ∼ 20kB T .

Choices of the dielectric constants

The Poisson equation describes the electrostatic potential function due to existence
of free charges. The left hand side of the Poisson equation can be written as
− 2 Φ(r) +

· P (r)

(3.86)

P (r) is the polarization field vector which describes the density of permanent or
induced electric dipole moments in a dielectric material. For an isotropic medium
that has linear response, the polarization field can be defined by

P (r) = χE(r) = −χ Φ(r)

(3.87)

where χ(r) = (r) − 1 is the dielectric susceptibility of the medium. Then Eq. (3.86)
can be written as
−

· (r) Φ(r).
113

(3.88)

Therefore, the permittivity (r), which is also called dielectric constant, represents
the polarizability of the medium. In biomolecular calculations, (r) is generally assumed as piecewise constants in most applications. It is noted that in charge neutral molecules, electric polarization corresponds to the rearrangement of electrons in
molecules. In most popular force field packages, some of the polarizations of a charge
neutral macromolecule are often treated as partial charges located at the centers of
individual atoms. These partial charges give rise to most of the fixed charge source
term ρf in the generalized Poisson-Boltzmann equation. Due to this treatment of the
polarization effect, a relatively small (r) value is normally assigned to the biomolecular domain. For example, when calculating the solvation energy of proteins, (r) is
set to 1 or 2 for the biomolecular domain while 80 for the solvent domain. These
values are commonly accepted and vary in only small ranges for different purposes.
However, in the application of ion channels, choices of dielectric constants in different
regions of interest are worthwhile to be carefully explored.
First, although the ion permeation is a dynamical process, dielectric constants
are all assumed time independent due to the fact that the electrolytic solution is a
fast relaxing bath, i.e., the relaxation time of the solvent water is extremely short.
Secondly, the dielectric constants are approximated as piecewise constants for computational simplicity. In the bulk concentration, a widely used dielectric constant as 80,
which is the experimental measurement at room temperature for water. The value
of

is usually set to 1 or 2 in the protein domain, which partially accounts for the

field-induced atomic polarization of the protein. However, two features about protein
structures are neglected in the continuum approximation for ion channels and should
be partially compensated by the dielectric constant of the channel protein. One is the
re-organization of the protein and water in extremely confined channels and the other
is the protein’s response to ion’s presence in the channel, since the ion permeation
takes places at the same time scale. Therefore, in order to encapsulate these features
114

in a continuum model with a single dielectric coefficient, the value of (r) for channel
proteins is suggested to be greater than 2.
There are also some issues in assigning the dielectric coefficient for the aqueous
region in the ion channel. A general conclusion is that (r) in the bulk aqueous region
should be much higher that that in the channel region. The main reason is the high
confinement of the channel geometry. In ion channel pores which are usually very
narrow, water molecules are highly ordered, and their motions are restricted, so are
their response to external fields. Therefore, the value of (r) should be much smaller
than 80, and can be as small as 3 for a dry channel pore. However, these extreme
value do not work well in practical computations. In fact, the dielectric coefficient in
the channel pore region is still taken as 80 in most existing models despite the above
arguments. In the present work, (r) values are set to be smaller than 80, but are not
too small in order to model the biological environment.

Effective mass of the proton
The choice of effective mass m(r) of the particle in the total Hamiltonian H as in Eq.
(3.22) is an important issue to be discussed. The concept of effective mass origins
from the solid state physics, which describes the response of the charge carrier to
the electric or magnetic fields when quantum mechanism is applied. It is defined by
analogy with Newton’s second law but in the quantum mechanical framework

m=

2

d2 E
dk 2

−1

(3.89)

where E and k are the energy and the corresponding wavenumber of the particle,
respectively. Generally the effective mass is chosen in the range of 0.001 or 10 times
the real mass of the particle and depends on the material and the experimental
condition. However, little research has been done, to our knowledge, on the choice of
115

the effective mass of protons in proton channels or proton experiments. In the present
model, we describe protons by quantum mechanics while treat many other particles
by classical mechanics and/or continuum description. Therefore, an effective mass
approximation is appropriate for our model. We set effective mass m(r) as a model
parameter and its value is chosen from 0.01 to 1.0 time of the real proton mass.

3.3

Numerical simulations

In this section, the validity of the proposed model and related performance analysis
are presented based on a specific channel protein, the Gramicidin A (GA, PDB code:
1MAG). The GA channel protein is obtained from the soil bacterial species Bacillus
brevis and is one of the best studied molecular channels, both structurally and functionally. In a bilayer membrane, the GA is dimers and consists of two head-to-head
β-helical parts. Each part of the dimer has the sequence of FOR-VAL-GLY-ALADLE-ALA-DVA-VAL-DVA-TRP-DLE-TRP-DLE-TRP-DLE-TRP-ETA, and forms a
narrow pore of about 4˚ in diameter and 25˚ in length. It appears to select small
A
A
monovalent cations, bind bivalent cations, while reject anions. In our approach,
the GA structure is downloaded from the PDB, and the pdb file is processed by
the PDB2PQR [52], in which the radii and partial charges are adopted from the
CHARMM force field values [105]. The molecular surface of the GA is generated via
the MSMS package [131] with water probe radius 1.3 ˚ and density 10. Figure 3.3
A
gives an illustration of the GA in a 3D display of the structure, surface and electrostatics distribution. From Fig. 3.3(a), one can see that a complete channel pore is
formed after the generation of the molecular surface. Although the GA is neutral
in general, its surface electrostatics is negatively distributed near the channel mouth
as indicated by the red color. Furthermore, as shown in Fig. 3.3(b), the inner part
of the channel pore is also intensively negatively charged. This fact indicates the
116

(a)

(b)

Figure 3.3: 3D illustration of the Gramicidin A (GA) channel structure and surface
electrostatic potential. The negative surface electrostatics as indicated by the intensive red color on the channel upper surface and inside the channel pore implies that
the GA selects positive ions. (a) Top view of the GA channel; (b) Side view of the
GA channel.

selectivity of GA channel to positive ions. Having prepared the GA structure and
surface, the channel pore is aligned to z-direction. The simulation grid resolution is
taken as 0.5˚. Under this discretization all the grid points are classified as either in
A
the solvent domain or in the molecular domain. Furthermore, the molecular surface is
projected on each layer along the transport direction to determine the beginning and
the end of the channel respectively, by the first layer and the last layer on which closed
projections can be found. An artificial membrane slab is added along the transport
direction between the beginning and end of the channel, see Fig. 3.1(b).
117

−1

15

εch=20
εch=40

−2

Ion Density (M)

Electrostatic Potential (kBT/q)

0

εch=80

−3
−4
−5
−20

−10
0
10
20
Channel Direction (Angstrom)

(a) Electrostatics

10

εch=20
εch=40
εch=80

5

0
−20

−10
0
10
20
Channel Direction (Angstrom)

(b) Ion density

0
Figure 3.4: Electrostatic potential and charge density of the GA channel along the
z-axis obtained with m = 2 and n0 = 0.1 molar (Red: ch = 20; Green: ch = 40;
p
Blue: ch = 80). (a) Electrostatic potential profiles in channel; (b) Proton density
profiles in the channel.

3.3.1

Electrostatic properties of the Gramicidin A channel

This subsection presents the electrostatic analysis of the GA channel over a wide
range of (r) values in the present model. At the atomic level, the motion of an ion
when it is passing through the channel is determined by a number of factors, such as
electrostatic interactions and generalized-correlation interactions. The electrostatic
interactions include the Columbic interactions between ions, and between ions and
fixed charges of the channel. The generalized-correlation interactions consist of ionion excluded volume effects, the thermal fluctuation of the solvent, van der Waals
interactions, and other short range interactions such as the frequent collisions and
associations between water molecules and ions. One more factor is the structural
cooperation of the channel protein during ion permeation. In the present model, the
quantitative description of electrostatic interactions is the major ingredient while the
degrees of freedom of generalized-correlation interactions are suppressed to reduce
the computational cost.
118

The electrostatics of the channel system depends on the dielectric constants. In the
present work, we carefully test the effect of dielectric constants within an appropriate
biological range in order to obtain a reasonable prediction. It is also worth checking
the dependence or changing trend of the electrostatics upon these parameters for
model training and validity verification. Before the transport problem is simulated,
the mathematical algorithms, choices of dielectric constants are carefully examined
via the generalized Poisson-Boltzmann equation.
As discussed earlier, m (r) is given as a constant in Ωm and its value is tested
over a range. However, s (r) is strongly position dependent, having different values
in the bulk solvent and the channel pore. For simplicity, we take s (r) as piecewise
constants, i.e., impose a constant value denoted as bath in the bulk solvent, whereas
another for the channel pore denoted as ch . There is no controversy upon the choice
of bath = 80, which is employed in all the following simulations. Figures 3.4-3.6
display the electrostatic potential profiles and (positive) ion density in GA protein
with various combinations of ch and m within the range discussed in the earlier
section. The reference ion density is taken as 0.1 molar.
All quantities in Figs. 3.4-3.6 are averaged on each cross section along the channel
axis. The vertical dash lines in these figures indicate the entrance (left) and exit
(right) of the channel. The GA protein is overall neutral in charge, but possesses
a negative environment in the channel region and this fact leads to potential well.
Near the entrance and the exit of the channel, there are two local potential minima
(the valley near the dash line) and a major barrier in the middle of the channel.
Accordingly, for the density profile, there are two peaks at the positions where two
energy minima present and the density is lower in the middle of the channel. These
electrostatic profiles agree with the biological properties of the GA channel.
For each fixed m , the magnitude of the electrostatic potentials responds directly
to the change of ch value, as showed in Fig. 3.4(a). When the ch decreases from 80,
119

−1

8

εch=20
εch=40

Ion Density (M)

Electrostatic Potential (kBT/q)

0

εch=80

−2
−3

6

εch=20
εch=40
εch=80

4
2

−4
−20

−10
0
10
20
Channel Direction (Angstrom)

(a) Electrostatics

0
−20

−10
0
10
20
Channel Direction (Angstrom)

(b) Ion density

Figure 3.5: Electrostatic potential and charge density of the GA channel along the
z-axis obtained with m = 5 and n0 = 0.1 molar (Red: ch = 20; Green: ch = 40;
p
Blue: ch = 80). (a) Electrostatic potential profiles in the channel; (b) Proton density
profiles in the channel.

which is the commonly used value for the solvent, to the lower values suggested by
biological observations, the contrast between the energy wells near the entrance/exit
and the barrier in the middle becomes sharper. This phenomenon verifies the impact
of ch value and leads us to prefer the lower value in our model. For the ion density
profile shown in Fig. 3.4(b), the changes in the peaks with respect to the changes
of ch are very clear. As ch doubles, the magnitudes of the density at the peaks
decrease half accordingly.
The impact of m can be examined by fixing ch , i.e., checking the same color
curves throughout Figs. 3.4-3.6. It can be found out that changes in m do not
affect the potential structure but solely change the magnitudes. When m increases,
the absolute value of electrostatic potential decreases, and consequently the proton
density becomes smaller.
Figure 3.7 depicts the electrostatics profile change with respect to reference proton
densities at a certain combination of dielectric constants ( m = 5 and ch = 40). It is
easy to see that the higher the proton reference concentration, the higher the sources
120

−1

εch=80

εch=40

Ion Density (M)

Electrostatic Potential (kBT/q)

0

5

εch=20

−2
−3
−20

−10
0
10
20
Channel Direction (Angstrom)

(a) Electrostatics

εch=20

4

εch=40

3

εch=80

2
1

0
−20

−10
0
10
20
Channel Direction (Angstrom)

(b) Ion density

Figure 3.6: Electrostatic potential and charge density of the GA channel along the
z-axis obtained with m = 10 and n0 = 0.1 molar (Red: ch = 20; Green: ch = 40;
p
Blue: ch = 80). (a) Electrostatic potential profiles in the channel; (b) Proton density
profiles in the channel.
in the Poisson equation and the results in electrostatic potential profiles are.

3.3.2

Conductivity properties of the Gramicidin A channel

The mechanism of the selectivity of the GA channel can be easily explained in view
of the overall potential landscape. Figure 3.8 shows the total effective potential with
both the electrostatic and generalized-correlation contributions. Figure 3.8(a) is for
the monovalent cations while Fig. 3.8(b) is for monovalent anions. According to the
previous discussion, the generalized-correlation potential serves as an energy barrier
while the GA protein provides a negatively charged environment for cations in the
channel region. Two energy components with opposite signs cancel each other and
result in an overall potential landscape that permeates a monovalent cation. However,
the overall potential gives rise to a huge barrier for the anions since the positive
generalized-correlation potential adds up with the positive electrostatic potential, as
Fig. 3.8(b) shows.
Conductance reveals the efficiency of the ion channel transport of some specific
121

Electrostatic Potential (kBT/q)

0.5
0
−0.5
−1
−1.5
−2
−2.5
0.1M
0.2M
1.0M
2.0M

−3
−3.5
−4
−20

−10

0

10

20

Channel Direction (Angstrom)

5
0
0.05V
0.10V
0.15V
0.20V

−5
−10
−20

−10
0
10
20
Channel Direction (Angstrom)

(a) For monovalent cations

Total Effective Potential (k T/q)
B

Total Effective Potential (k T/q)
B

Figure 3.7: Electrostatic potential profiles of the GA channel under different ion
reference densities n0 . Red: n0 = 0.1 molar; Green: n0 = 0.2 molar; Blue: n0 = 1.0
p
p
p
p
0 = 2.0 molar.
molar; Black: np
m = 5 and ch = 40.

20
10
0.05V
0.10V
0.15V
0.20V
−10
−20
−10
0
10
20
Channel Direction (Angstrom)
0

(b) For monovalent anions

Figure 3.8: The total potential of the GA channel which includes electrostatic and
generalized-correlation contibutions under various voltage biases. Dielectric constants
are m = 5 and ch = 30. The pH value of the solution is 2.75. (a) Total potential of
monovalent cations; (b) Total potential of monovalent anions.
122

Transport Eigenvalues ((kBT/q)

30
25
20
15
10
5
0
−5
−10

−10

−5

0

5

10

Channel Direction (Angstrom)
Figure 3.9: The first 15 eigenvalues (the U j (z) in Eq. 3.41) of the effective potentials
along the transport direction used in the transport calculations at the voltage bias of
0.2V.

123

ions. Due to the fast development of experimental technologies in the past several
decades, the single-channel conductance can be measured and becomes one of the
prevalent descriptor of the channel function. The simulation of channel conductance
mainly focuses on calculating the channel current within the physiological ranges
of membrane potentials (i.e., −0.2V < V < 0.2V) and bath concentrations (up to
molars). The channel conducting current is measured at the scale of pico-Ampere
(pA) for ion channels. The corresponding characteristics of channel conductance is
observed at the scale of pico-Siemens (pS) and is recorded in the voltage-current (I-V)
curves and concentration-current (C-C) curves. Based on experimental observations,
the I-V curves are expected to be in linear or sub-linear form while the C-C curves
are supposed to exhibit saturation behavior, i.e., when the concentration increases,
the conductance increases linearly at beginning and then becomes saturated later on.
The conductivity of the proton channel mainly depends on the proton scattering
process. Thus we first present the effective potential profile along the transport
direction. Figure 3.9 depicts the first 15 effective potential eigenvalues (i.e., U j (z) in
Eq. (3.41)) used in the current calculation under the voltage bias of 0.2V. Similarly,
the channel region is presented between two black dash lines. The channel region
is essentially confined by the protein surface and a tube-like pore is formed. As
displayed in Fig. 3.9, the potential energy profile in the channel pore region has
discrete eigenstates, due to the small area confinement at each cross section and the
light mass of the proton. For each specific location along the transport direction, the
discrete ascending energies correspond to the eigenvalues of the operator in Eq. (3.40).
In theory, the total number of the eigenvalues is infinite, but is finite in practical
computations, and depends on the discretization of the cross section. In principle, all
the eigenvalues should be accounted in computations. However, numerically, due to
the Boltzmann distribution, higher energy components contribute little in the total
transport quantity. In practise, only a few low lying eigenvalues need to be included
124

0.8

pH=3.75

Proton Current (pA)

Proton Current (pA)

0.08
0.06
0.04
0.02
0
0

Simulation Curve
Experimental Data
0.05

0.1

0.15

pH=2.75
0.6
0.4
0.2
0
0

0.2

Transmembrane Voltage (V)

Simulation Curve
Experimental Data
0.05

0.1

0.15

0.2

Transmembrane Voltage (V)

(a) pH=3.75

(b) pH=2.75

Figure 3.10: Voltage-current relation of proton translocation of GA at different concentrations. Blue dots: experimental data of Eisenman et al [59]; Red curve: model
prediction.

in numerical simulations. In our case, the first 15 eigenstates are sufficient to obtain
a good degree of convergence in calculating the proton density and current.
Figure 3.10 illustrates the simulation results of the present multiscale model for
proton transport, compared with the experimental data from the literature [135] for
the GA channel. The blue dots in each figure represent the available experimental
observations for certain voltage biases while the red curves are our model predictions
calculated with sufficiently many voltage samples. The model parameters are chosen
to match the experimental data but all of the choices are taken within the range of
physical measurements. The dielectric coefficients are taken as m = 5, ch = 30
and bath = 80, according to the discussion in previous sections. To determine the
generalized correlations, the diffusion coefficients of protons are taken as 3.6 × 10−9
m2 /s in the channel, less than a half of the value in the bulk environment, and
the relative weighting parameter is set to α = 0.03. Taking into account above
considerations, we can conclude that experimental data and the present predictions
agree quite well and this agreement verifies the validity of our quantum dynamics in
125

Log

10

conductance (pS)

4
3
2

Voltage=0.05V

1
0
Simulation Curve
Experimental Data

−1
−5

−4

−3

−2

−1

0

1

+

Log10[H ] (M)
Figure 3.11: Conductance-concentration relation of proton translocation at a fixed
voltage. Voltage bias=0.05V; Blue dots: experimental data of Eisenman et al[59];
Red curve: model prediction.

126

continuum model.
Apart from I-V curves, there are also experimental data available about the
conductance-concentration relation (C-C curve) of the proton transport under given
voltages. Figure 3.11 displays such a relationship with a comparison between experimental data and model predictions. At a given voltage bias, the conductance
of the channel is calculated with various proton concentrations as indicated by the
horizontal axis. Using the same set of parameters as those in Fig. 3.10, the computed conductance-concentration relation also agrees fairly well with experimental
data. At lower proton concentrations (i.e., pH value being greater than 2), the agreement between our prediction and experimental data is quite good. At relatively
higher concentrations, although the numerical simulations slightly overestimate the
observed conductance, the conductance saturation against the concentration can still
be observed in simulations and it corresponds to the sub-linear characteristics or the
flat tail of the C-C curve.
The experimental data used in this work are reported by Eisenman et al[59]
and are also employed to verify another proton transport model by Schumaker et
al. [135]. There are other experimental data on proton conductance available [5,
42, 34] but under different experimental conditions. First, the experimental data
provided by Cukierman et al. [42] offer proton conductions recorded with natural Gramicidin A and with its Dioxolane-Linked dimer in different lipid bi-layers
(phosphatidylethanolamine-phosphatidylcholine, or PEPC and glycerymonooleate, or
GMO). Their experimental studies were carried out for low (9.8 mM) and high (1578
mM) proton concentrations against the transmembrane voltages. Additionally, in
another piece of work [34], the attenuation of proton transfer in Gramicidin water
wires by phosphoethanolamine was investigated and a number of I-V curves were
provided. It is impossible to fit all the experimental data by a single group of parameters because of the difference in experimental conditions and lipid membrane types.
127

Nevertheless, it can be observed that our simulation curves under the current set
of parameters have shown similar qualitative shapes. Therefore, the present model
can fit to these experimental data by slightly adjusting model parameters to reflect
the different experimental conditions. Finally, Akeson and Deamer [5] also reported
I-V curves of proton conductance for the F1 F0 ATPases studies. In their results, a
severe saturating or sublinear character is found for proton concentration of 10 mM
and there were an obvious superlinear pattern for 1.0 M hydrogen chloride (HCl).
Our model can not capture these characteristics by just tuning the parameter values.
In fact, this set of experimental data was also found difficult for another theoretical
model of proton transport [135].

3.3.3

Model limitations

Based on the multiscale approximation, the present model captures the most important factors which have large impacts on the proton permeation. Meanwhile,
the quantum treatment of protons provides a potential analysis tool to account
for the quantum behavior in proton channel transport and proton translocation in
biomolecules. The setup of the present model roots from essential biophysical principles with reasonable approximations, and thus the numerical simulations give considerably good agreements with experimental data under appropriate choices of model
parameters. However, this model also has a number of limitations, which are to be
studied further in the future. First, in this model, the channel protein is assumed
to be rigid, i.e. it does not response to the permeation of ions. This is not true in
real situation and the configurational change of the channel protein has been found
to have fairly important impact on the ion permeation process. Although the omitted ion-protein interaction has been somehow compensated implicitly by adjusting
the dielectric constants, this interaction can not be fully accounted unless more sophisticated models, such as the multiscale molecular dynamics [154], are invoked.
128

Additionally, the plasma membrane where the channel protein is embedded is simplified. There are various types of membranes, some of them have dipoles and others
have charges. In our model, the membrane is just approximated by the uniformly
distributed dielectric medium and the charges or dipole effects are neglected. However, there is no essential difficulty to improve this aspect in our model. Point charges
from membranes can be added in the present model. Otherwise, a position dependent
dielectric constants for the biomolecular region can also represent the charge effects
in the membrane. Finally, the other limitation of the present model is the simplified
local density approximation of generalized correlations, which reduces the number
of the degrees of freedom, although. Compared to the electrostatic potential, the
generalized-correlation potential plays a less important role in general. However, it
may be of crucial importance for channel selectivity in certain situations. Therefore,
an emergent task of our future work is to come up with more quantitative modeling
of generalized-correlation interactions meanwhile without significantly increasing the
number of degrees of freedom. Local spin density approximation, local density gradient approximation and general linear scaling approaches are under our consideration.

3.4

Conclusion remarks

Proton dynamics and transport across membrane proteins are of paramount importance to the normal function of living cells. Although there are a variety of excellent
theoretical models and efficient computational methods for ion channels in general,
most commonly used models are much less successful when they are applied to the
proton channel transport due to the unique characteristics of protons. It is commonly
believed that to a certain extent, proton channels demonstrate quantum mechanical properties such as the translocation as shown in the Grotthuss-type mechanism
[112, 2]. However, the exact role of quantum mechanics in the atomic mechanism
129

of proton channels is still unclear despite of a number of elegant theories in the literature, partly due to the complexity of ion channel systems. The present paper
introduces a quantum dynamics in continuum (QDC) model for the prediction and
analysis of proton density distribution and conductance in proton channels. Our essential ideas are as follows. First, protons behave quantum mechanically due to their
light masses and channel geometric confinement in protein channels. Therefore, a
quantum mechanical treatment of protons is necessary. Additionally, since the primary interactions in proton channels are of ion-ion electrostatic type and the van der
Waals type of interactions involve less energy, a dielectric continuum treatment of
solvent medium may provide a reasonable approximation to the effect of numerous
solvent molecules. Most importantly, this treatment dramatically reduces the dimensionality of the problem. As such, our approach is called a QDC model. Moreover,
since the atomic detail of the protein structure serves as a physical boundary for
proton dynamics and transport, the present model returns molecular surface to separate the continuum solvent domain from the discrete charge domain of the protein.
Finally, densities of all other ions and counterions in the solvent are described by the
Boltzmann distribution, which is a quasi-equilibrium description as the electrostatic
potential varies during the process of protons permeating the membrane.
We propose a multiscale variational paradigm to accommodate the aforementioned aspects in a unified framework. The total free energy functional encompasses
the kinetic and potential energies of protons, and the electrostatic energies of ions
and fixed charges in the channel system. The first variation is carried out via the
Euler-Lagrange equation to derive the governing equations for the system. A generalized Poisson-Boltzmann equation is obtained for the electrostatic potential while a
generalized Kohn-Sham equation is resulted for the state of protons in the system.
The solution to these two coupled nonlinear equations leads to the desirable electrostatic distribution and proton density profile in the channel system. Expressions for
130

proton density and proton flux across the membrane are derived from fundamental
principles.
The computation of the proposed coupled equations involves a number of mathematical issues, such as the linearization of coupled nonlinear partial differential equations (PDEs) using the Gummel iterations and/or inexact Newton iterations, and
the solution of elliptic PDEs with discontinuous coefficients (i.e., piecewise dielectric constants), singular sources (i.e., Dirac delta functions for protein charges), and
nonsmooth interfaces (i.e., geometric singularities). In the present work, we utilize
the Dirichlet to Neumann mapping method to take care of singular charges, and the
matched interface and boundary (MIB) method to accurately handle the discontinuous coefficients and geometric singularities.
The Gramicidin A (GA) channel protein, a popular protein structure, is employed
in our numerical studies to demonstrate the performance of the proposed QDC model.
We give a detailed discussion about the rational for model parameter selections. The
electrostatic property of the GA channel is analyzed with the proposed model against
a large number of model parameters. Proton transport properties, i.e., the current
voltage (I-V) curves, are investigated over a large number of combinations of applied
voltages and reference bulk concentrations. Our model predictions are compared
with experimental data, which validates the present QDC model. Finally, we provide
detailed discussion of model limitations and possible future improvements.

131

Chapter 4

Structure and electrostatic analysis
for bio-molecules

In both simulations of nano-electronic transistors and proton channels, the highly
accurate and efficient calculation of electrostatics is of paramount importance. The
electrostatic model in nano-transistors is relatively straightforward because of the
simple system components. However, the biological ion channel system requires careful treatment due to the complicated protein structures and inhomogeneous environments. As reviewed in the introduction, the implicit solvent theory is applied to model
and analyze the electrostatics for bio-molecules, and the MIBPB solver serves as a
powerful tool to solve the generalized Poisson-Boltzmann equation that provides the
electrostatics background for proton channel. This chapter is devoted to the overview
of the Poisson-Boltzmann equation, detailed description of the efficient analysis of the
MIBPB solver and its various applications.
132

4.1

The Krylov subspace theory (KSP) and preconditioner based MIBPB solver

In this section, we first give the overview of the Poisson-Boltzmann equation and
related numerical challenges, then the MIBPB solver and its efficiency analysis are
demonstrated. This fills the lacked computational detail in the previous chapter because the GPB equation is the natural extension of the Poisson-Boltzmann equation.

4.1.1

Review of the Poisson-Boltzmann equation

In the implicit solvent model, the solvent is treated as a continuous medium while the
description for solute is kept at the atomic level. The electrostatic potential Φ of a
solvent-solute system can be determined by the Poisson-Boltzmann equation(PBE) in
a regular domain Ω whose dimension usually has the order from 10˚3 to 500˚3 for
A
A
biomolecular applications. Figure 4.1 gives the sketch of the protein-solute system and
the computational domain. The protein region and the solvent region are denoted as
Ω1 and Ω2 , respectively. Naturally the whole computational domain is Ω = Ω1

Ω2 ,

and the molecular surface is labeled as Γ. For simplicity, the ion-exclusive layer is
ignored in the present model. Although mobile ions in the solvent are explicitly
indicated in the figure, the whole solvent region is actually modeled by an implicit
continuum.
Under these assumptions, if one consider the 1:1 electrolyte for simplicity, the
Poisson-Boltzmann equation reads:

−

· ( (r)

Φ(r)) + κ2 (r)
¯

kB T
q

sinh

qΦ(r)
kB T

Na

Qi δ(r − ri )

= 4π

(4.1)

i=1

boundary condition of Eq.(4.1) depends on various applications. The q is the elementary charge, kB is the Boltzmann constant and T is the temperature,
133

and κ are
¯

Solvent
Mobile Ions
M bil I
Fixed Charges
Molecule

Ω1

Γ

Ω2

Figure 4.1: The implicit protein-solvent system

Abbr.
esu
c
cm/m
s
K
mol
dyn
erg
l
molar

Label
Na
kB
q
h
˚
A

Table 4.1: Physical units notations
Meaning
Represents
Equivalent expressions
statcoulomb
charge
fundamental unit
coulomb
charge
fundamental unit
centermeter/meter distance
fundamental unit
second
time
fundamental unit
Kelvin
Temperature fundamental unit
mole
Quantity
fundamental unit
dyne
force
esu2 /cm2
erg
energy
dyn · cm
liter
volume
1000cm3
mole per liter
concentration mol/l

Table 4.2: Some physical constants
Name
Values in CGS unit
Values in SI unit
23 (no unit) Avagadro’s number
6.022045 × 10
Boltzmann’s constant 1.380662 × 10−16 erg/K 8.617343 × 10−5 eV /K
Fundamental charge
4.803204 × 10−10 esu
1.602176 × 10−19 c
Planck’s constant
6.626068 × 10−27 erg · s
4.135667 × 10−15 eV · s
−8 cm
Angstrom
10
10−10 m

134

dielectric constant and modified Debye-H¨ckel screening function describing the ion
u
strength, respectively. Here Qi is the fixed charge in the protein and ri denotes the
position of the fixed charge, and Na is the total number of fractional charges.
Eq.(4.1) is presented in the Gauss unit system, in which the units and physical
constants are provided in Table 4.1 and 4.2. Usually the electrostatics Φ(r) is scaled
to be dimensionless by u = qΦ/kB T , consequently, the Eq.(4.1) is revised as
Na

−

· ( (r)

u(r)) + κ2 (r) sinh(u(r))
¯

zi δ(r − ri )

=C

(4.2)

i=1

The constant C = 4πq 2 /kB T results from dimensionless procedure, and zi = Qi /q is
the charge fraction of the fixed charge.
The hyperbolic term sinh(u(r)) takes into account the salt effect with the Boltzmann distribution theory in the equilibrium state. Therefore, Eq. (4.2) is a nonlinear partial differential equation (PDE) of elliptic type. Such a nonlinear term
can be linearized under the weak potential approximation, i.e, when u(r)

1,

sinh(u(r)) ∼ u(r). Thus the linear approximation of Eq. (4.2) is
Na

−

· ( (r)

u(x)) + κ2 (r)u(r)
¯

zi δ(r − ri )

=C

(4.3)

i=1

Typically, for biomolecular systems of given ranges of temperature and ionic
strength, the PBE is solved with the following coefficient bounds [78]

1 ≤ (r) ≤ 80
0 ≤ κ2 ≤ 127
¯
5249 ≤ C ≤ 10500
− 1 ≤ zi ≤ 1.
135

The spatial-dependent coefficients (x) and κ(x) are discontinuous across the
molecular surface. It is a challenge to solve such an elliptic equation with high
accuracy because the regularity of its solution is reduced due to the interface and
geometric singularity. For this class of problems, numerical accuracy and convergence rate are typically low without special interface treatments. Another challenge
is the singular source term which contains many Delta functions, which are infinity
at their spatial locations. Accurate approximation to the point-supported singular
functions is an important topic in computational mathematics. The above two difficulties hinder the accurate numerical solution to the PB equation. To maintain a
given accuracy, the grid spacing of the discretization has to be sufficiently small because of the low regularity of the solution. On the other hand, a small grid spacing
implies millions of variables even for a middle-size protein. For example, the cube
embedding a 2800-atom protein may have a dimension of 50 × 50 × 50(˚3 ), which
A
leads to 1 × 106 variables if the resolution is 0.5˚. This gives rise to a major obstacle
A
for PB applications, especially for the calculation of thermodynamic properties via
either the molecular dynamics or pre-equilibrium approaches. The MIB method is
introduced and applied to the solution of the PBE. The source term singularity is
removed by the DNM. In Ref. [67], the solution u(r) of the PBE is decomposed into
a singular part and a regular part. The singular part of the solution comes from
singular delta functions, and is obtained analytically as the Green’s function. As
a consequence, this separation generates an extra Neumann jump condition at the
interface for the regular part. Therefore, after the separation, one only needs to solve
the remaining homogeneous Poisson-Boltzmann equation subject to corresponding
Neumann jump conditions at the interface. This procedure is also called Dirichlet to
Neumann mapping. Consequently, truly second-order accurate solution to the PBE
with molecular surfaces and singular charges can be obtained with a relatively large
grid spacing [67].
136

4.1.2

KSP based and preconditioner accelerated MIBPB solver

The discretization of the nonlinear PB equation results in a nonlinear equation system
of the form
F (Uh ) = Lh Uh + N (Uh ) − fh = 0,

(4.4)

where h is the discretization resolution, Lh and fh represent the matrix and right hand
side generated via the MIB and DNM schemes, Uh = [u1 , u2 , ...ui , ...]T is the solution
vector. The nonlinear term N (·) is diagonal and N (Uh ) = [Ni (ui )] = [¯ 2 sinh(ui )].
κ
The inexact-Newton method is perhaps one of the most efficient ways to solve nonlinear system (4.4). If the h is dropped, it reads

F (Un )Vn = −F (Un ) + Rn ,
Un+1 = Un + Vn ,

||Rn ||
≤ ηn
||F (Un )||

(4.5)
(4.6)

where n is the iteration step, F is the Jacobian matrix ∂Fi (U )/∂uj and takes
the form F (U ) = L + N (U ). Here N is the Jacobian matrix of N (U ) and is also
diagonal Ni (U ) = Ni (ui ) = κ2 cosh(ui ). It is easy to see that the inexact-Newton
¯
method is a two-layer iterative algorithm. The correction term Vn in outer iteration
(4.6) is considered as a rough solution of inner iteration (4.5). The scheme converges
linearly when ηn , the ratio of the residual Rn between the function value F (Un ), is
less than 1, and converges super-linearly as the sequence ηn has the property that
limn→∞ ηn = 0.
The overall numerical efficiency of solving the nonlinear system strong depends
on the efficiency of solving the linear system (4.5), which in further depends on the
complexity of the matrix Lh . It is worth pointing out that in standard FDMs, the
matrix Lh only depends on the grid resolution and the dielectric constants. However,
137

in the MIBPB scheme, the structure of Lh also depends on the molecular surface
of a specific protein. Due to this reason, we also call Lh the matrix of a protein
for simplicity. The MIB and DNM successfully overcome the equation singularities
and promise a high accuracy convergence order by taking into account all the local
interface information. However, as a trade-off, the structure of matrix Lh is much
more complicated than that from standard FDMs. Specifically, the matrix loses
symmetry and may not be positive-definite any more. The lose of these properties will
lead to more computational time and memory. Therefore, the selection of appropriate
linear solvers becomes subtle when computational efficiency is sought as well.
The review of several basic linear solvers are summarized in Appendix B. However, the matrices from the MIB and the DNM scheme can barely take any advantage
from the described methods due to their complicated structures. Therefore, we put
our emphasis on choosing other methods and accelerating techniques. In Appendix
B, we also include a brief description of Krylov subspace (KSP) techniques. Based on
the KSP theory, proper linear solvers and acceleration techniques (preconditioners)
are chosen and compared in this section for the numerical efficiency of MIBPB linear
systems. Two KSP solvers, the stabilized biconjugate gradient method (BiCG) and
the generalized minimal residual method (GMRES), are potentially effective iterative
solvers for matrices with general structures. Several preconditioning strategies, the
Jacobi preconditioner (JAC), the blocked Jacobi preconditioner (BJAC) and the incomplete LU factorization preconditioner (ILU) are available to incorporate with the
two solvers to accelerate the solution of the linear system.
Matrices generated from a set of proteins are employed to test the performance
of various KSP solver-preconditioner combinations. For each matrix, the condition
number, linear system iteration number and iteration time are used to characterize
numerical efficiency. All these measurements of matrices are analyzed numerically by
the PETSc (http://www.mcs.anl.gov/petsc/petsc-as/). The grid resolution is
138

Figure 4.2: Condition numbers over 15 proteins (PDB IDs from protein 1 to protein
15: 1ajj, 1vii, 1cbn, 1bbl, 1fca, 1sh1, 1vjw, 1fxd, 1bpi, 1a2s, 1frd, 1svr, 1a63, 2erl,
2pde). (a) Condition numbers for unpreconditioned (unPCed) MIBPB matrices;
taken as 1.0˚ in the following tests unless otherwise specified. The stopping criterion
A
of all KSP solvers are taken as 1 × 10−6 in order to get more accurate solutions while
in practical biological applications the criterion can be relaxed to 1 × 10−3 to save
CPU time but satisfactory results are also achieved.
First of all, the matrix condition numbers are examined. The condition number
can predict the level of difficulty in solving the linear system before it is really solved.
The magnitude of the condition number of a matrix generated via the MIB and DNM
scheme depends on the size and structure of a biomolecule. More specifically, under
the same grid resolution, a molecule which has a larger number of atoms needs a
larger computational domain and a larger matrix size. Meanwhile, a molecule which
has a more complex surface geometry leads to more interface conditions and a larger
139

Figure 4.3: Condition numbers over 15 proteins (PDB IDs from protein 1 to protein
15: 1ajj, 1vii, 1cbn, 1bbl, 1fca, 1sh1, 1vjw, 1fxd, 1bpi, 1a2s, 1frd, 1svr, 1a63, 2erl,
2pde). (b) Comparisons of condition numbers under three settings.

140

matrix size. Both cases contribute to higher condition numbers. Therefore, the size
and complexity of a biomolecule usually affect the numerical efficiency of the MIBPB
solver.
Figure 4.2 presents condition numbers of matrices corresponding to 15 protein
structures and indicates the numerical difficulties of solution without proper acceleration techniques. The horizontal axis lists proteins. Protein data bank (PDB)
identification numbers (IDs) are listed in the figure. The numbers of atoms of these
proteins range from 500 to 2000. Discretizing the PBE with the MIB scheme, without
any preconditioner (PC) applied, the condition numbers are usually in the order of
104 , about one order larger than those of the matrices generated from the standard
FD discretization, i.e., without the interface treatment. This is expected because
the use of the molecular surface as the interface and all included local information
around the interface, the MIBPB matrices do not maintain the symmetry and are
not positive definite. The MIB matrices generally have larger condition numbers and
require more CPU time [163, 157, 67].
By using of preconditioners (PCs), the magnitudes of condition numbers of MIBPB
matrices are significantly reduced to less than one hundred, as shown in the circle plot
of Fig. 4.3. The triangle plot in Fig. 4.3 gives the condition number magnitudes of
the matrices from standard FDMs without PC, revealing the huge differences among
different treatments. The circle and dot plots are condition number magnitudes for
matrices with PC, from both the MIB scheme and the FDM, respectively. Interestingly, it can be concluded that the condition number magnitudes of two schemes are
reduced to almost the same level by using the PCs. We can safely say that the difficulty of solving the linear system generated from the interface based MIBPB scheme
is actually comparable to that from standard FD discretization. Under almost the
same numerical efficiency, MIB scheme and DNM are able to obtain higher accuracy
because all the local geometry information of the molecular surface has been taken
141

Table 4.3: Iteration numbers and CPU time for the discretization matrices of proteins
Proteins
Preconditioned iteration
Un-Preconditioned iteration
ID
atoms number time condition #
number time condition #
1mbg
903
19
0.3
40
5404
54
118900
18
0.3
40
5400
58
250400
1r69
997
20
0.3
30
2152
23
138850
1bor
832
1vii
596
17
0.2
42
3963
28
4963
19
0.3
39
7084
80
35637
1fxd
824
2erl
573
17
0.2
29
4858
36
14223
23
0.6
61
10000
156
24981
1a2s
1272

into account.
Quantitatively, for a specific KSP solver such as GMRES, the iteration numbers
and computing time of linear systems for 7 proteins are listed in Table 4.3. It is
well-known that condition numbers can only be mathematically estimated for large
matrices, then the listed condition numbers calculated by PETSc solvers may not
be exact. Despite this fact, we can still have a sense from the numbers how the PC
significantly reduces the difficulties of solving the linear systems.
As stated earlier, two KSP iterative methods, the stabilized BiCG and the GMRES, are associated with three types of preconditioners, JAC, BJAC and ILU. Table
4.4 compares the effect of combinations of these KSP solvers and preconditioners. For
different preconditioning strategies, since the ways of counting iteration numbers are
different, only the iteration time for each combination is listed in the table. Sample
proteins of various sizes are presented in this table, from small size (less than 1000
atoms), middle size (1000-3000 atoms) to large size (around 8000 atoms). It can be
concluded that the GMRES performs slightly better than the stabilized BiCG does
for small-sized proteins but stabilized BiCG takes the lead in middle- and big-sized
proteins. Among the three kinds of preconditioners, the BJAC and the ILU almost
have the same effects and are slightly better than the JAC. Therefore, the combination of stabilized BiCG and BJAC is recommended and set as the default option in
142

Table 4.4: Iteration time for different combinations of KSP solvers and preconditioners
Protein
BiCG
GMRES
ID
atom BJAC ILU JAC
BJAC ILU JAC
1ajj
519
0.24 0.24 0.26
0.23 0.23 0.38
1vjw 828
0.29 0.29 0.35
0.26 0.26 0.44
0.56 0.56 0.51
0.57 0.56 0.84
1a2s 1272
1a7m 2809
1.69 1.67 1.77
1.93 1.91 2.85
3.90 3.88 4.48
4.70 4.65 7.19
1f6w 8243

Table 4.5: Convergence test of MIBPB solver with a set of proteins
Proteins
Error
Order
Error
Order
˚ h=0.5˚
˚
ID
h=1.0A
A
h=0.25A
1ajj
6.52E-2 1.13E-2
2.52
1.76E-3
2.68
1a23
1.026E1 1.72E-1
2.57
2.74E-2
2.65
1b4l
1.19E-1 1.25E-2
3.25
2.07E-3
2.59
1bbl
1.32E-1 1.86E-2
2.82
1.81E-3
3.36
9.44E-2 1.31E-2
2.84
1.97E-3
2.73
1bor
1.20E-1 1.20E-2
3.31
1.78E-3
2.76
1fca
1frd
7.93E-2 1.24E-2
2.67
2.02E-3
2.61
1fxd
7.66E-2 1.19E-2
2.68
2.00E-3
2.57
8.05E-2 1.37E-2
2.50
1.77E-3
2.90
1hpt
1mbg
1.35E-1 1.08E-2
3.64
1.69E-3
2.67
8.52E-2 1.27E-2
2.74
1.83E-3
2.79
1neq
1r69
7.95E-2 1.15E-2
2.39
1.92E-3
2.96
7.94E-2 1.17E-2
2.21
1.94E-3
3.13
1svr
1uxc
7.55E-2 1.27E-2
2.57
2.02E-3
2.65
7.22E-2 1.20E-2
2.58
2.23E-3
2.43
1vjw
2pde
1.12E-1 1.64E-2
2.77
5.46E-3
1.58

the MIBPB package.
As indicated at the beginning, all the mathematical algorithms and techniques are
enforced to maintain the high order convergence of the MIBPB solver. Table 4.5 lists
the numerical evidence of the second order convergence through a set of given protein
surfaces, atomic coordinates, radii and charges, where protein surfaces are generated
by MSMS, and the standard CHARMM force field parameters are used. A special
analytical solution was designed and given in [67] for the convergence order check
of all proteins. In this table, the numerical error is defined as ||unum − uexact ||L∞ ,
h
143

where unum is the numerical solution of the PBE at grid resolution h while uexact
h
is the designed exact solution. The numerical experiments are implemented under
resolutions h = 1.0˚, 0.5˚ and 0.25˚. The numerical error is supposed to be reduced
A
A
A
by four times as the grid size is halved and this is clearly demonstrated in the table.

The above mentioned tests are carried out in conjunction with the PETSc software
package, whose installation may not be so straightforward. An alternative is to use
the SLATEC, which is easier to implement and also includes tens of linear system
solvers with different preconditioners. To compare the performance of the PETSc
and the SLATEC, we show the computation time of ten methods in the SLATEC
for five proteins, whose atom number varies from five hundreds to eight thousands
in Table 4.6. All methods are listed in the form: preconditioner/solver. Here GS,
DS, BiGS, and OM represent the Gauss-Seidel, the diagonal scaling, the biconjugate
gradient squared method, and the orthomin sparse iterative method, respectively.
The combination of the ILU/BiCG is used in the PETSc. From the table, one can
see that the iteration time of the PETSc is slightly shorter than that of most solvers in
the SLATEC for small-sized proteins. The last column of the table lists the averaged
CPU time for the PETSc and solvers in SLATEC. The averaged time, which in some
sense could reflect the abilities of solvers for proteins in various sizes, is the sum of
the CPU time for each corresponding protein and weighted by the atom number.
By checking the averaged CPU time one can generally conclude that the ILU/BiCG
of the PETSC takes less iteration time than the SLATEC schemes do. Moreover,
according to our experience, the PETSc is more stable than the SLATEC for large
proteins. However, the SLATEC can be easily incorporated in the MIBPB package.
144

Table 4.6: Comparison of CPU time
Protein ID
1ajj 1vjw
519
828
Atoms
PETSc
0.235 0.272
GS/GS
0.866 1.222
0.523 0.883
ILU/ILU
DS/BiCG
0.331 0.467
0.262 0.401
ILU/BiCG
DS/BiGS
0.243 0.313
0.187 0.393
ILU/BiGS
DS/OM
0.206 0.420
0.179 0.291
ILU/OM
DS/GMRES 0.417 0.559
ILU/GMRES 0.198 0.279

4.2

for the
1a2s
1272
0.529
2.225
1.344
1.041
0.701
0.602
0.410
0.496
0.389
0.999
0.439

PETSc
1a7m
2809
1.729
9.512
5.854
3.140
2.038
2.900
1.433
3.338
1.25
3.856
1.615

and the SLATEC schemes
2ade Averaged
8344 CPU time
3.777
2.72
55.016
35.58
32.479
21.07
14.015
9.27
7.846
5.27
8.879
6.05
6.575
4.34
21.993
14.08
5.993
3.95
26.262
16.84
7.685
5.05

Application to solvation energy calculations

One of the most important applications of the PBE model is solvation energy calculations for the protein-solvent systems. In this section, solvation energies of 28 proteins
are calculated and compared with popular PBE solvers to examine the feasibility, usefulness and robustness of the linear solver in the MIBPB package. These proteins have
a wide range of numbers of atoms, from around 500 up to 10,000. The corresponding
spatial dimensions extend from about 30˚×30˚×30˚ to almost 100˚×100˚×100˚.
A
A
A
A
A
A
Among these calculations, the dielectric constant is set to 1 for proteins and 80 for
the solvent. The ion strength κ is set to zero because no ion is considered for the
moment.
The calculation of electrostatic solvation energy ∆Gelec is to sum all the fixed
charges {qi } of the solute in the solvent, weighted by the reaction field potential
φrf (x):
∆Gelec =

1
2

Qi Φrf (xi )

(4.7)

i

where xi is the position of each charge. Based on the continuum electrostatics,
the reaction field potential is the difference between the electrostatic potential in
145

the solvent environment Φs (x) and the reference electrostatic potential Φref (x), i.e,
Φrf (x) = Φs (x) − Φref (x). Here Φrf (x) can be computed by solving the PBE twice
with corresponding settings. Specifically, Φref (x) is calculated by setting a uniform
dielectric constant in the whole computational domain, while Φs (x) is calculated by
setting the dielectric constants for solute and solvent differently. Therefore, Φref (x)
can be obtained by the standard linear PB equation with the Dirichlet boundary
condition via the standard finite difference or FFT methods but Φs (x) is solved by
using the MIBPB algorithm.
The performance of the MIBPB method for calculating solvation energies has already been examined in our previous work [67]. It has been shown that the MIBPB
solver has high accuracy and good convergence order because of the use of interface
treatments but has relatively low numerical efficiency due to the absence of appropriate matrix acceleration techniques. The MIBPB matrix requires longer CPU time
to solve. The Krylov theory and associated preconditioners discussed in the present
work make the MIBPB solver more efficient. Here the new results are presented for
various proteins.
Figure 4.4 gives the comparison of the calculated solvation energies from the
MIBPB and the APBS packages. It is seen that the solvation energies calculated from
the MIBPB agree very well with those from the APBS. The mesh sizes of h = 1˚ is
A
used in the MIBPB and h = 0.25˚ is used in the APBS methods, respectively. The
A
reader is referred to Ref. [67] for a more detailed comparison among the MIBPB, the
APBS and the PBEQ methods.
Once the preconditioning techniques are applied, the required CPU time is significantly reduced. Figure 4.5 illustrates the differences of the CPU time needed to
calculate solvation energies for nineteen moderately large proteins at three different
grid resolutions. The solid lines are the CPU time for preconditioned (PCed) systems
and dashed lines are for unpreconditioned (unPCed) systems. It can be concluded
146

Figure 4.4: Comparison of solvation energies of proteins (From protein 1 to 19: 1ajj,
1bbl, 1vii, 1cbn, 2pde, 1sh1, 1fca, 1fxd, 1vjw, 1bor,1hpt,1bpi, 1mbg, 1r69, 1neq, 1a2s,
1svr, 1a63, 1a7m) calculated by using the MIBPB and the APBS methods.

147

Figure 4.5: Comparison of CPU time of preconditioned (PCed) and un-preconditioned
(unPCed) MIBPB methods for 19 proteins (from protein 1 to 19: 1ajj, 1bbl, 1vii,
1cbn, 2pde, 1sh1, 1fca, 1fxd, 1vjw, 1bor, 1hpt, 1bpi, 1mbg, 1r69, 1neq, 1a2s,1
svr,1a63, 1a7m).

that at each grid resolution, preconditioners can save more than half of the overall
CPU time.
Table 4.7 lists the results for the tested proteins at different grid resolutions and
compares the values with the original MIBPB-III scheme in terms of solvation energies
and CPU time. For each protein case from different grid resolutions, the CPU time
increases in nonuniform pattern from less than 10 seconds for h =1.0˚, several tens
A
of seconds for h =0.5˚, to a few hundreds of seconds for h =0.25˚. Note that there
A
A
is an eight times increase in the number of unknowns when the mesh size is halved.
The increase in the CPU time is roughly linear with the increase in the number of
unknowns.
It is found that, at resolution of 0.25˚ , the results from the MIBPB+KSP and
A
148

Table 4.7: Solvation energies and CPU time for proteins
Solvation energy (kcal/mol)
CPU time (second)
MIBPB+KSP
MIBPB
MIBPB+KSP
MIBPB
h(˚)
A
1.0
0.5
0.25
0.25
1.0 0.5 0.25
0.25
1ajj
-1141.9 -1136.3 -1136.6 -1137.2
2.3 15
85
273
2pde
-826.2 -819.9 -817.2
-820.9
3.4 17 108
283
1vii
-914.6 -901.5 -902.8
-901.2
2.8 15
86
221
1cbn
-311.1 -303.6 -303.7
-303.8
2.9 16 102
277
1bor
-858.8 -854.3 -857.9
-853.7
4.5 24 143
377
1bbl
-998.3 -986.9 -988.7
-986.8
2.8 16 106
298
1fca
-1215.5 -1199.9 -1200.0 -1200.1
3.5 19 109
292
1uxc -1157.6 -1138.1 -1139.1 -1138.7
4.2 21 127
347
1sh1
-755.7 -728.0 -751.4
-753.3
3.3 12 109
300
1mbg -1368.4 -1349.8 -1352.4 -1346.1
4.8 25 142
378
1ptq
-893.5 -871.8 -872.2
-873.1
3.9 22 133
376
1vjw -1250.9 -1236.9 -1236.9 -1237.9
4.1 26 120
315
1fxd -3309.7 -3299.7 -3301.6 -3300.0
4.2 31 138
338
1r69 -1111.0 -1086.5 -1087.9 -1089.5
5.5 32 154
419
1hpt
-827.3 -810.9 -812.7
-814.3
4.9 26 141
322
1bpi -1320.8 -1298.9 -1301.3 -1301.9
5.4 50 164
469
1a2s -1928.8 -1913.1 -1913.6 -1913.5
9.6 47 242
780
1frd
2879.8 -2851.4 -2856.3 -2851.9
10.8 51 284
707
1svr
-1741.6 -1709.8 -1710.7 -1711.2
11.1 57 301
779
1neq -1765.6 -1729.1 -1732.7 -1730.1
9.1 50 289
804
1a63 -2420.8 -2371.2 -2370.2 -2373.5
22 113 550
1376
2erl
-964.2 -948.2 -949.3
-948.8
2.3 15 101
276

from the original MIBPB-III have less than 0.1% disagreement. This is due to the
use of different convergence norms in the KSP solvers and the regular solver. The
solvation energy calculations show a correct convergence tendency. The values from
resolutions of 0.25˚ and 0.5˚ are pretty close, while calculations at h = 0.25˚ cost
A
A
A
much more CPU time. Therefore, we can conclude that grid resolution between
˚
0.5A and 1.0˚ is sufficient for the calculation and can guarantee the accuracy.
A
Table 4.8 shows the robustness and efficiency of the MIBPB package for calculating
solvation energies of large proteins which exceed 7000 atoms. For time efficiency, all
the calculations are carried out at the grid resolution of h = 1.0˚. Note that the
A
reliability of these solvation free energies has been cross-checked with other popular
149

Table 4.8: Solvation energies (kcal/mol) and CPU time (second) for large proteins
Protein
MIBPB (h = 1.0˚)
A
ID
atoms Solvation energy CPU time (s)
1cbg 7838
-5659.9
181
1c4k 11439 -9901.9
398
1e24 7776
-9506.4
231
1f6w 8243
-5611.2
225
1po7 7796
-5471.2
206

PB solvers. The reported CPU time can be used as a reference.

4.3

Application to salt effects on protein-protein
binding

In this section, the ability of the MIBPB package to solve the nonlinear PBE is
tested by solvent salt effect on protein-protein binding. The nonlinear PBE has
had considerable success in describing biomelocular electrostatics with salt effects
on the binding of ligands, peptides and proteins to nucleic acids, membranes and
proteins. The binding free energies reflect the non specific salt dependence of the
formation of macromolecular complex and the measurement is the binding affinity.
Some experimental data are available and the binding affinity is calculated as the
ratio between salt dependent binding energy ∆∆Gel (I) at a specific salt strength
I and the natural logarithm of I. In the present work, we have implemented the
nonlinear version of the PBE solver in the MIBPB package. Simulation results are
obtained by varying the ionic strengths.
The binding energy (∆Gel ) has several components while the one related to electrostatics is the difference of the electrostatic free energies of the complex and each
150

of its free molecules [17]
∆Gel (I) = GAB (I) − GA (I) − GB (I),
el
el
el

(4.8)

where GAB (I), GA (I) and GB (I) represent the electrostatic free energies of the comel
el
el
plex AB, component A and component B, respectively, at a given ionic strength I.
The electrostatic free energy can be further split into three components

Gel (I) = Gcoul + Grxn + Gsalt (I),

(4.9)

where Gcoul is the Coulomb energy calculated in a homogeneous medium, Grxn is the
corrected reaction field energy and Gsalt (I) is the electrostatic energy contributed by
mobile ions. Among the three terms in Eq. (4.9), only Gsalt (I) is salt dependent.
Thus, the salt dependence of the binding free energy ∆∆Gel (I) is electrostatic component of the binding energy in Eq. (4.8) calculated at some salt strength I minus
the one calculated at the zero salt concentration [17]

∆∆Gel (I) = ∆Gel (I) − ∆Gel (I = 0)
= GAB (I) − GAB (I = 0)
el
el
− GA (I) − GA (I = 0)
el
el
− GB (I) − GB (I = 0) ,
el
el

(4.10)

where various energy terms are calculated at different ionic strengths by using the
MIBPB package.
To verify our nonlinear solver, one hetero-dimeric and one homo-dimeric complexes
are studied in this work. The experiments on these two protein complexes can be
found in [130, 152] and biological features(1emv and 1beb) are listed in Table 4.9.
151

The first four columns describe the properties of proteins and the last two columns are
the slopes (binding affinity) of the lines in Fig. 4.6. It can be seen in a quantitative
view that the slopes obtained from experiments and simulations are very close to
each other. The calculations were performed assuming that all Arg, Asp, Glu and
Lys residues are ionized in both free and bound states. The results are obtained with
dielectric constants of 2 and 80 for the solute and the continuum solvent, respectively.
The parameter κ2 is determined by:
κ2 =

8π 2 Na q 2
1000kB T

I

(4.11)

where q, kB T are the same as those defined in Eq. (4.2), Na is the Avogadro’s
number. After a simple derivation, κ2 is given by
−2

κ2 = 8.486902807˚
A

I

(4.12)

for T = 298K. Here the ion strength I is in the unit of molar.
Figure 4.6 depicts the experimental and calculated salt dependence of the binding
free energies ∆∆Gel (I) for the two complexes. The ∆∆Gel (I) are plotted against
the logarithm of the salt strength I. The salt dependence is assumed as in a linear
pattern therefore the least square fitting line is applied to calculate the binding affinity,
which is the slope of the line. It should be explained that experimental data dots
for ∆∆Gel (I) are read from the graphs in Ref. [17], while the fitting line slope is
explicitly given based on the experimental data with error bars. The diamond points
and solid line are experimental data and the corresponding fitting line, respectively.
The circle points and dashed line are numerical stimulations.
In the homo-dimeric complex, the experimentally observed binding free energies
decrease with the increase of ionic strength, while for the hetero-diemric complex, the
152

8

(b)
Binding Free Energy (kcal/mol)

Binding Free Energy (kcal/mol)

(a)
MIBPB (h=1.0 A)
Experimental data

6
4
2

0
−3

−2

Ln(I)

−1

0

−4

MIBPB (h=1.0 A)
Experimental data

−6
−8
−10
−12
−14

−4

−2
Ln(I)

0

Figure 4.6: (a) Binding affinity of 1emv; (b) Binding affinity of 1beb.

Complex
E9Dnase-Im9 (10)
(B-A)
Lactoglobulin
dimer (57)(A-B)

Table 4.9: Binding affinity
PDB Complex Surface
Charge of the
˚2 ) free monomers
ID
charge
area (A

Exp.
data

MIBPB
h = 1.0˚
A

1emv

-3

1465

B=+5; A=-8

2.17

2.42

1beb

+26

1167

A=B=+13

-1.62

-1.90

experimental measurement had detected an increase. Our results obtained from the
MIBPB package reproduced these observations. The calculated magnitudes of the
slopes of the salt dependence are in quite good agreement with experimental results
within the range of errors, as the fitting lines are almost parallel. The discrepancies
between the experimental data and simulation results are also expected: the PBE,
no matter in linear or full nonlinear form, only gives the ideal situation of the solutesolvent system but many details, such as the “protein conformation change, pKa
shifts upon complexation or possible ionization states”, are lacking [17]. Despite
these approximation, the application of PBE for static protein structures is suggested
in general by these good agreements with experimental data.
.
153

4.4

Conclusion remarks

The MIBPB has built in advanced interface techniques that are able to deal with
discontinuous solvent-solute interfaces and geometric singularities of molecular surfaces. Therefore, it provides reliable electrostatic potentials at the mesh of about 1˚,
A
whereas traditional methods have to resort to about 0.25˚ mesh resolution to achieve
A
a similar level of reliability. In the present work, we further equip the MIBPB solver
with advanced Krylov subspace techniques to accelerate the speed of the convergence
of solving linear equation systems originated from the MIBPB discretization. The
performances of various solver-preconditioner combinations have been carefully examined through mathematical analysis and numerical experiments. Dramatic reductions in condition numbers are found when appropriate preconditioners are utilized.
Upon the use of appropriate combinations of preconditioners and solvers, significant
reductions in the CPU time are found in solving the PB equation for large proteins.
Both the PETSc and the SLATEC are employed in the present MIBPB package to
speed up the convergence rate of the iterations of the linear systems. The PETSc
package is found to be more reliable and efficient. In the present work, the structure
preparation of proteins is conducted via the PDB2PQR software package, while the
MSMS software package is utilized for the molecular surface generation. Additionally, the nonlinear MIBPB solver has been developed in the present work. This is
achieved via the standard inexact Newton method, assisted by the Krylov subspace
acceleration techniques. The present nonlinear MIBPB solver has been tested and
applied to the salt dependence analysis of protein-protein binding interactions. Our
results of binding affinities are compared with experimental data.

154

Chapter 5
Thesis achievements and future
work
The major contribution of this thesis is the mathematical modeling and highly efficient computations for nano-electronic transistors and transmembrane proton channels. The former subject includes electronic semi-conductor devices that have continuous demand nowadays in rising the performance and reducing the dimension;
while the latter belongs to the study of ion channels in molecule-based biology. The
two topics share similar physical characteristics, attract same mathematical interests
and encounter common simulation challenges. In both fields, motions of quantum
particles(electrons, protons) are studied under intensive electrostatic and generalcorrelation potential interactions at nano-scale. With reasonable assumptions, energy components are assembled on an equal-footing energy functional from which
governing equations are derived. For the two governing equations, the generalized
Poisson-Boltzmann equation and the generalized Kohn-Sham equations, there are a
number of singularities and numerical challenges that attracted numerous mathematical interests in recent decades. Corresponding mathematical algorithms are adopted
to overcome these difficulties in simulations.
155

Most of the materials of this thesis are from the following publications:
• Duan Chen and Guo-wei Wei, “Quantum dynamics in continuum for proton
channel transport.”, submitted to Journal of computational Physics
• Duan Chen and Guo-Wei Wei, “Modeling and simulation of electronic structure, material interface and random doping in nano-electronic devices” Journal
of Computational Physics, 229, Vol. 229, pp. 4431-4460, 2010.
• Duan Chen, Zhan Chen, Changjun Chen, Weihua Geng and Guo-Wei Wei,
“MIBPB: A software package for electrostatic analysis”, Journal of Computational Chemistry, in press , 2010
• Duan Chen, G.W. Wei, X. Cong and Ge Wang, “Computational methods
for optical molecular imaging”, Commun. Numer. Methods Engng, vol. 25,
pp.1137-1161, 2008

5.1

On the modeling and simulations of nano electronic transistors

5.1.1

Contributions

This thesis contributes to the studies on nano-MOSFETs with the following aspects:
First of all, it is the first time that a two scale energy functional that includes continuum mechanism and quantum mechanics on an equal footing has been proposed,
and the coupled Poisson-Kohn-Sham equations are derived from the optimization of
the total energy functional. The macroscopic electrostatics energy functional is defined on the inhomogeneous silicon and silicon dioxide system, while the motion of
major charge carriers (electrons in this work) is microscopically described through
156

interactions between electrons and continuum/discrete dopants as well as the selfinteractions.
Secondly, two practical issues are addressed in the current work. One is the material interface problem, the ratio of dielectric constants of the silicon and the insulator
has a great impact on the device performance. The recognition of material interfaces
in MOSFETs implies the acknowledgment of discontinuous material properties or coefficients across the interfaces. The interface effect study is naturally contained in
our Poisson equation with discontinuous coefficients. Another topic is the individual
dopant effect. Electrical properties of the nano-transistor, i.e., the conductivity, electrostatic potential and its charge carrying mode can be manipulated easily by doping
strategies. In continuum modeling, dopants have always been described as continuous
distributions. These treatments work mostly well for the prediction of device properties. However, when the channel length reduces to nano-scale, the quantum effect
becomes important. Thus, each doping atom may have a dramatic impact on the
quantum state of nearby electrons. Therefore, the atomistic model for dopants becomes indispensable. This work presents a new individual dopant model by utilizing
the Dirac delta function.
Corresponding to the aforementioned issues, this work introduces two computational algorithms for the simulation of nano-electronic devices. An efficient elliptic
interface method, the matched interface and boundary (MIB) method, is employed
for solving the Poisson equation with semiconductor interfaces. The other computational algorithm employed in the present work is the Dirichlet-to-Neumann mapping
(DNM). This approach provides a rigorous treatment of the singular charge sources
in the Poisson equation due to the individual dopant model proposed in the present
work. Necessary numerical analysis are presented. The second order MIB scheme is
utilized in the present work; however, due to the strong nonlinearity of the coupled
Poisson-Kohn-Sham equations, the numerical order was found to be about 1.5 in
157

the present Gummel-like iteration. Self-consistent iteration difficulties under certain
circumstances are overcome by a relaxation scheme.
Finally, two multi-gate MOSFET systems, a double-gate MOSFET and a fourgate MOSFET, are considered in the numerical simulations of the present work.
Both problems are modeled in the three-dimensional (3D) setting. In our doublegate MOSFET simulations, the basic characteristics and the quantum effect of the
I-V curves are similar to those in the literature. The impact of randomly distributed
individual dopants on electronic structure and transport is studied. In particular,
individual dopants induced voltage threshold lowering effect is clearly demonstrated.
In our four-gate MOSFET simulations, individual dopants effectively break the symmetry of the device. Due to the 2D quantum confinement, the density of quantum
states that are relevant to the electronic transport become larger than that of the
double-gate MOSFET.

5.1.2

Future work

In our current study, nano-transistors are studied with relatively simplified assumptions. For our future work, the modeling and simulation can be improved as follows;
1. Currently the electron transport is assumed in the ballistic limit, i.e., the
transport of electrons is assumed to have no interaction with other objects. Real
devices actually operate below this limit. The interaction of electrons with their surroundings is called scattering. Important scattering mechanisms include the surface
roughness scattering, electron-electron scattering, impurity-electron scattering and
phonon-electron scattering. A sophisticated model needs to be set up to take into
account these effects.
2. The existing theory is for general nano-transistor modeling while our numerical
implementations, or specifically, the transport simulations are only suitable for simple
MOSFET geometries. In fact, there are a wide variety of geometric configurations
158

for MOSFETs for different performance requirements. Therefore, the Kohn-Sham
equation decomposition technique must be dropped. Alternatively, an efficient full
3D algorithm for transport needs to be developed.

5.2

On the modeling and simulations of transmembrane proton channels and biomolecule structures

5.2.1

Contributions

Although there are a variety of excellent theoretical models and efficient computational methods for ion channels in general, most commonly used models are much
less successful when they are applied to proton channel transport due to the special
features of proton channels. Our present work on proton channel contributes a quantum dynamics in continuum (QDC) model for the prediction and analysis of proton
density distribution and conductance in proton channels.
First we consider a necessary quantum mechanical treatment of protons due to
their light mass and channel geometric confinement in protein channels. Additionally,
primary ion-ion electrostatic interactions in proton channels are given top priority,
and then a dielectric continuum treatment of solvent medium works as a reasonable
approximation to the effect of numerous solvent molecules. Furthermore, densities
of all other ions and counter-ions in the solvent are described by the Boltzmann
distribution, which is a quasi-equilibrium description as the electrostatic potential
varies during the process of a proton permeating through the membrane.
We proposed a multiscale variational paradigm to accommodate the aforementioned aspects in a unified framework. The total free energy functional encompasses
159

the kinetic and potential energies of protons, and the electrostatic energies of ions
and fixed charges in the channel system. The first variation is carried out via the
Euler-Lagrange equation to derive the governing equations for the system. A generalized Poisson-Boltzmann equation is obtained for the electrostatic potential while a
generalized Kohn-Sham equation results for the orbitals of protons in the system. The
solution to these two coupled nonlinear equations leads to the desired electrostatic
distribution and proton density profile in the channel. Expressions for proton density
and proton current across the membrane are derived from fundamental principles.
A complete, accurate and efficient Poisson-Boltzmann solver MIBPB, is developed for electrostatics analysis in molecular biology. This solver serves as a powerful
tool in proton transport calculation and other solvation applications. The previous
MIBPB-I, the MIBPB-II and the MIBPB-III are equipped with matched interface and
boundary (MIB) scheme and Dirichlet-to-Neumann mapping (DNM) method. They
overcome numerical challenges of discontinuous coefficients (i.e., piecewise dielectric
constants), singular sources (i.e., Dirac delta functions for protein charges), and nonsmooth interfaces (i.e., geometric singularities). In this thesis, the Krylov subspace
theory and preconditioner methods are utilized to enhance the solver efficiency, the
inexact Newton’s method based on these two techniques is included to extend the
PB solver to a nonlinear one. Detailed solver processes that contain protein structure
preparation, PB calculation and solution visualization are provided.

5.2.2

Future work

Our model is based on several assumptions which may not hold in realistic biological
processes, I intend to continue working on the modeling and simulation of ion channel
problems with further details, which may include the following aspects:
1. In the current model set up, the plasma membrane is modeled as a uniform
dielectric continuum. While in realistic conditions, there are various types of mem160

branes, some of which have dipoles and others of which have charges. There will
be no essential difficulty to improve our model in this aspect for the future work.
Point charges from membranes can be added in the present model. Otherwise, a
position dependent dielectric constant for the biomolecular region can also represent
the charge effects in the membrane;
2. The configurational cooperation of the channel protein needs to be included
in the model in a more explicit way. At this stage, the channel structure is assumed
to be rigid and the missed ion-protein interaction is somehow compensated by the
dielectric constant. In our future work, the multiscale molecular dynamics may be
adopted to remove this “frozen” structure assumption.
3. A major limitation of the present model origins from the oversimplified treatment of the general-correlation potential. Comparing to the electrostatic potential,
the general-correlation potential is solely related to the relaxation time of particles in
the channel. It is more phenomenological and lacks detailed information of ion-water
interactions. Therefore, a major task in our future work is to collect the literature
regarding the chemical/physical details about ion dehydration and hydrogen-bond
breaking/making processes, to use reasonable mathematical formulation, and to eventually establish more quantitative modeling of the general-correlation potential.

161

APPENDIX

162

Appendix A
The MIB method
In this section, the MIB method is discussed in general Rd dimensional space. Without loss of generality, assume that the computational domain is a compact rectangle

[a1 , b1 ] × [a2 , b2 ] × ... × [ad , bd ]

and uniformly discretized with grid spacing h for simplicity. Then for j ∈ {1, 2, ..., d},
Nj =

(bj −aj )
h

is the number of gird points along the xj direction. Since the MIB is a

dimension splitting method for approximating the differential operator
d
j=1

∂
∂xj

β

∂
∂xj

,

only the discretization of a specific direction, say xj ∗ , is illustrated for simplicity.
Discretizations of other directions can be easily derived in a similar way.
Let e1 = (1, 0, 0, ..., 0) and ej = (0, ..., 0, 1, 0, ..., 0) be respectively the 1st and
j th standard coordinate vectors of Rd space and the d-tuple E = (c1 , c2 , ..., cd ) be
the index of a specific grid point. We denote a non-boundary grid point as XE
(i.e., cj ∈ {2, 3, .., Nj − 1}), and its function value as uE = u(XE ). If XE is near
the interface, its discretization scheme depends whether it is a regular point or an
163

irregular point according to the designed convergence order [164]. At a regular point
where the discretization scheme does not involve any grid point across the interface,
the approximation for the differential operator is the standard central finite difference
scheme

∂
∂xj ∗

β

∂u
∂xj ∗

β+ 1
+ β+ 1
β+ 1
E− 2 ej ∗
E+ 2 ej ∗
E− 2 ej ∗
=
uE−e ∗ −
uE
j
h2
h2
β+ 1
E+ 2 ej ∗
uE+e ∗ + O(h2 ).
+
j
h2

(A.1)

(A.2)

The MIB scheme is applied at an irregular point where the discretization scheme
involves at least one grid point across the interface and it will be described in three
major steps in the following three subsections.

A.1

Dimension splitting

The MIB method takes the dimension splitting approach which reduces a multidimensional interface problem into 1D ones. Consequently, the MIB method simplifies
the local topological relation near an interface, which is crucial for 3D problems with
complex interface geometries. The MIB method provides special schemes around each
intersecting point of the given interface and prescribed meshlines. Therefore, at each
intersecting point which is not a grid point, there is one meshline, say xj ∗ , that is a
primary direction. In a second order MIB scheme, if the interface Γ passes between
two grid points XE and XE+e ∗ , then XE and XE+e ∗ are a pair of irregular points.
j
j
For simplicity, we denote the consecutive grid points XE+ke ∗ (k ∈ {..., −1, 0, 1, ...})
j
on the xj ∗ axis as i + k and function values u(XE+ke ∗ ) on these grid points as ui+k .
j

Fictitious values f1 and f2 are constructed as smoothly extended function values
of u+ and u− on both sides of the interface along the xj ∗ direction. The fictitious
164

value f1 is the extension of the solution in the left domain to the grid point i + 1,
and f2 is the extension of the solution in the right domain to the grid point i, while
the vertical line indicates the interface location. Unlike the original function value u
which might be discontinuous, both u+ (left) and u− (right) are well-behaved across
the interface. With the assistance of fictitious values, u± are matched by the jump
conditions at the interface by uniform interpolation.
Assume X0 ∈ Rd is the position where the interface intersects the mesh line. We
discretize the interface jump conditions as the follows,
[u] = u+ − u− = φ(X0 )

(A.3)

where
+,0

+,0

+,0

u+ = w−1 ui−1 + w0 ui + w1 f1 + O(h3 )
−,0

−,0

+,0

u− = w−1 f2 + w0 ui+1 + w1 ui+2 + O(h3 )

(A.4)

and
[βun ] = β + u+ · n − β − u− · n


+ u+ − β − u−
x1 
 β x1


 β + u+ − β − u− 
x2
x2 





...


 · n = ϕ(X0 )
= 
 + +
− u− 
 β ux ∗ − β x ∗ 

j
j 




...




β + u+ − β − u−
x
x
d

±,l

Here wk

(A.5)

d

(l ∈ {0, 1}, k ∈ {−1, 0, 1}) are interpolation weights which can be easily

calculated by interpolations. The superscripts ± present the two subdomains, and
165

0, 1 are for the interpolation and the first order derivative respectively, and the set of
subscripts −1, 0, 1 is the index of grid points.
In jump condition (A.5), the first derivatives for all directions and from the two
domains are all involved due to the

u± = (u±1 , u±2 , ..., u± )T . Among them, u± ∗
x
x
xd
xj

can be easily obtained by interpolation with function values and fictitious values
+,1

+,1

+,1

u+ ∗ = w−1 ui−1 + w0 ui + w1 f1 + O(h2 )
xj
−,1

−,1

+,1

u− ∗ = w−1 f2 + w0 ui+1 + w1 ui+2 + O(h2 ).
xj

(A.6)

The evaluation of u± by the interpolation formulation for all j = 1, 2, ..., d and j = j ∗
x
j

is presented in later sections A.3.
Symbolically, we assume u± are solved and the fictitious values f1 and f2 can
x
j

totally determined by equations (A.3)(A.5) Once fictitious values f1 and f2 are de∂
termined, modified discretizations for ∂x ∗
j

+

∂u
β ∂x ∗
j

at grid points ui and ui+1 are

given by

∂
∂xj ∗

∂ +
β+
u
∂xj ∗

β+ 1
i−

β+ 1 + β+ 1
i+ 2
i− 2
2u
=
ui +
i−1 −
h2
h2

β+ 1
i+
h2

2f

1

+ O(h)

(A.7)

and

∂
∂xj ∗

∂ −
β−
u
∂xj ∗

β− 1
i−

β− 1 + β− 1
i+ 2
2 f − i− 2
=
ui+1 +
2
h2
h2

β− 1
i+
h2

2u
i+2

+ O(h).

(A.8)

Methods for the determination of f1 and f2 are described in the next two subsections.
Remarks: (i) Grid points i − 1 and i are required to be in the same subdomain
and so are i + 1 and i + 2. The assumption can always be satisfied by refining the
grid mesh for a smooth interface. For sharp-edged interfaces, the grid refinement
might not work. A special MIB scheme has been developed to deal with interface
166

singularities in [159].
(ii) From the discussion above, the 2nd order MIB scheme can be generalized to
higher order ones (4th and 6th order) by extending the solution to more fictitious
values near the interface. In the MIB method, this is done by repeatedly using lowest
order jump conditions instead of creating higher order derivatives.
(iii) In Ref. [164], the fictitious values f1 and f2 have been shown to be of O(h3 ),
which guarantees the first order local truncation error and the global second order
convergence of the MIB scheme.

A.2

Derivative elimination

In general, at every intersecting point of the interface and mesh lines, there are two
original interface conditions and d − 1 additional first order interface conditions for
an elliptic interface in Rd . These d + 1 interface conditions involve 2d first order
derivatives. In the MIB method, we simultaneously determine as fewer fictitious
values around an intersecting point as possible so that we have the maximal flexibility
in avoiding the determination of many first order derivatives, which are often difficult
to evaluate due to geometric constraints. Nevertheless, we have to determine at
least two fictitious values so that both of the original two interface conditions can be
implemented at least indirectly. Consequently, we determine only two fictitious values
around an intersecting point and thus, use d−1 interface conditions to eliminate d−1
first order derivatives. The remaining d + 1 derivatives are to be approximated using
appropriate grid function values near the intersecting point.
Specifically, jump conditions (A.3) and (A.5) are employed to determine two fictitious values f1 and f2 . However the first order derivatives along all directions on each
subdomain are involved in Eq. (A.5). Two derivatives along the primary direction
xj ∗ can be approximated by Eq. (A.6). For 2d − 2 derivatives along other directions,
167

i.e., u± (j = 1, 2, ..., d, j = j ∗ ), d−1 interface conditions, which are obtained by differx
j

entiating Eq. (A.3) along tangential directions, are used together with Eq. (A.5) to
eliminate d − 1 derivatives. There are flexibilities in the selection of d − 1 derivatives
from 2d − 2 derivatives. A general principle is to avoid evaluating those derivatives
that are difficult to compute due to local geometric constraint and to optimize the
resulting matrix of linear algebraic equations. In most cases, we normally evaluate
one of u± , either u+ or u− .
xj
xj
xj

A closed interface Γ in Rd can be considered as a Rd−1 manifold embedded in a
Rd space. Consider a map Φ from Rd−1 to Rd :

Φ : (˜1 , x2 , ..., xd−1 ) → (˜1 , x2 , ..., xd−1 , I(˜1 , x2 , ..., xd−1 )),
x ˜
˜
x ˜
˜
x ˜
˜

(A.9)

where I is a hyper-surface function. The tangent space of Φ is given as

Ts Φ = span(Φx1 , Φx2 , ..., Φx
˜
˜
˜

d−1

) =: span(t1 , t2 , ..., td−1 ),

(A.10)

where xk (∀k ∈ {1, 2, ..., d − 1}) are manifold parameters, and Φx are derivatives of
˜
˜
k

Φ with respect to xk . The tangent vectors tk are not necessarily orthonormal to each
˜
other. However, in 2D and 3D practices, these tangent vectors can be orthonormal
with appropriate parametrization of the manifold. From Eq. (A.3), the gradient jump
at the interface can be decomposed into d − 1 tangential jump conditions
φ(x + tj ) − φ(x)
,
h
h→0

[ u] · tj = lim

168

j = 1, 2, ..., d − 1.

(A.11)

With the jump condition (A.5), a d × 2d system results in


 


+
u  
=

−

u






(P + , −P − ) 

φ · t1





φ · t2 


,
...



φ · td−1 

ϕ

(A.12)

where


tT
1



 tT
2


+=
P
 ...

 T
 td−1

β + nT










,








 tT
2


−=
P
 ...

 T
 td−1

β − nT

tT
1







.






(A.13)

In Eq. (A.12), only one of each remaining u± is to be approximated. Therefore, d−1
x
j

variables need to be eliminated and it is always possible to eliminate d − 1 variables
from d independent equations. Since the tangent vectors and the normal vector are
linearly independent, there is a well defined equation after the elimination. Without
loss of generality, assume u+ is to be eliminated from the system and the equation
xj
left from the elimination is
d

c1 u+ ∗
xj

+ c2 u − ∗
xj

d−1

+
j=3,j=j ∗

cj u−
xj

cj φ · tj + cd ϕ,
˜
˜

=

(A.14)

j=1

where cj and cj are coefficients after the elimination. Finally, Equations. (A.3) and
˜
(A.14) are used to determine the two fictitious values f1 and f2 .

169

A.3

Derivative evaluation

After the elimination of d − 1 derivatives, we have to evaluate d − 1 derivatives. This
is a difficult task for complex interfaces and is a challenging task for interfaces with
geometric singularities. The evaluations are pursued along xj (j = 1, 2, ..., d, j = j ∗ )
direction but in general not along any xj axis. In 2D, the evaluation must compute
in the xj ∗ -xj plane. In higher dimensions, the evaluation can be pursued with more
flexibilities. In general, we evaluate a derivative in a specific xj -xi plane so that
the resulting matrix is relatively diagonal and symmetric with respect to the given
geometric constraint. Let us assume that xj∗ -xj be such a choice and denote the consecutive grid points XE+me ∗ +nej along xj ∗ axis and xj axis as (i+m, j +n), and the
j
function values u(XE+me ∗ +nej ) as ui+m,j+n , where m, n ∈ {..., −2, −1, 0, 1, 2, ...}.
j
When the irregular points are off the interface, three auxiliary function values
uo,E , uo,1 and uo,2 along the auxiliary line xj ∗ = xo are needed to approximate u− ,
xj
where xo is the j ∗ component of Xo . Hence, u− is approximated by the first order
x
j

derivative scheme
u− = p1 uo,E + p2 uo,1 + p3 uo,2 ,
x
j

(A.15)

where p1 , p2 and p3 are derivative weights.
Unfortunately, uo,E , uo,1 and uo,2 are not available since they do not locate on
any grid point. Therefore, they are to be obtained from function values on other
nearby grid points on the corresponding mesh lines. The auxiliary value uo,E locates
between the irregular points (i, j) and (i + 1, j), which can be obtained with fictitious
values
−,0

−,0

−,0

uo,E = (w−1 , w0 , w1 ) · (f2 , ui+1,j , ui+2,j )T .

(A.16)

However, the auxiliary values uo,1 and uo,2 have to be interpolated by other nearby
170

values
−,0

−,0

−,0

˜
˜
uo,1 = (w−1 , w0 , w1 ) · (ui+1,j+1 , ui+2,j+1 , ui+3,j+1 )T
˜
−,0

−,0

−,0

uo,2 = (w−1 , w0 , w1 ) · (ui,j+2 , ui+1,j+2 , ui+2,j+2 )T .
¯
¯
¯

(A.17)

Note that all the grid points used in interpolating the auxiliary points have to
be in the same subdomain. By the assumption that after eliminating the u+ and
xj
approximating u− , all the gird points used to interpolate uo,1 and uo,2 should be in
x
j

the Ω− domain (there is no restriction for uo,E since fictitious values are involved).
Therefore, the grid points (i, j + 2), (i + 1, j + 2) and (i + 2, j + 2) are the closest grid
points to interpolate uo,2 . While for uo,1 , the gird point (i, j +1) is in the Ω+ domain,
and the closest grid points for uo,1 are (i + 1, j + 1), (i + 2, j + 1) and (i + 3, j + 1).
Therefore, the final expression for u− is:
x
j




− = [p , p , p ] 
ux
1 2 3 
j




−,0
w−1

−,0
w0

−,0
w1

0

0

0

0

w−1
˜

0

0

0

0

−,0

0
−,0

0
−,0

w0
˜

w1
˜

0

0

0
0
−,0

w−1
¯

0
0
−,0

w0
¯

0
0
−,0

w1
¯







× [f2 , ui+1,j , ui+2,j , ui+1,j+1 , ui+2,j+1 , ui+3,j+1 , ui,j+2 , ui+1,j+2 , ui+2,j+2 ].
From the above equation we can see that u− is actually a linear combination of some
x
j

unknown function values on grid points around the irregular point ui,j . By repeating
the same procedure to other coupled xj ∗ -xj planes, all similarly structured u− can
xj
be determined. Together with jump condition (A.5), two unknown fictitious values
f1 and f2 can be determined.
If the target irregular point is right on the interface, the calculation of fictitious
values is the similar. At the end of this paper, applications of the MIB scheme are
given. The 2D MIB scheme is useful to interface problems in plastic membrane,
171

electromagnetic wave propagation, etc. When d = 2, the following normal vector and
tangential vector are usually implemented as

n = (cos θ, sin θ)
t = (− sin θ, cos θ)

cos θ 
 − sin θ
P± = 
,
β ± cos θ β ± sin θ


(A.18)

where θ is the angle between the positive direction of the x-axis and the normal vector
of the interface. It is easy to show that n and t are orthonormal. For the MIB scheme
used in the implicit solvent model for electrostatics analysis in molecular biology or
in biomedical image computation, we consider a 3D MIB scheme. The normal vector
and tangential vectors can be chosen as:

n = (sin ψ cos θ, sin ψ sin θ, cos ψ)
t1 = (− cos ψ cos θ, − cos ψ sin θ, sin ψ)

P±



=



t2 = (− sin θ, cos θ, 0)

− cos ψ cos θ − cos ψ sin θ
sin ψ 

,
− sin θ
cos θ
0


± sin ψ cos θ β ± sin ψ sin θ β ± cos ψ
β

(A.19)

where θ and ψ are the azimuth and zenith angles with respect to the normal direction.
The explicit expressions of the fictitious values are referred to Refs. [165, 159].

172

Appendix B
Krylov subspace method and
preconditioning for the MIB
scheme
B.1

Linear equation systems and MIB matrices

A system of linear algebraic equations is formed after discretizing the elliptic equation

Lh uh = fh

(B.1)

where Lh is a real non-singular n by n matrix under grid spacing h, uh is the numerical
solution vector and fh is the source term vector. The matrix Lh is viewed as a linear
operator mapping Rn into Rn , the space Rn being a linear space equipped with an
inner-product (·, ·) inducing a norm || · || defined as follows
n

ui vi , ||u|| = (u, u)1/2 , ∀u, v ∈ Rn .

(u, v) =
i=1

where ui represents the i-th component of the vector u.
173

Generally, systems of linear algebraic equations are commonly solved by using
direct methods and iterative methods. Direct methods, such as Gaussian elimination,
and LU decomposition work for general matrices with arbitrary structure but require
large computer memory. Therefore, they are not computationally efficient and hence
unsuitable for solving the 3D PB model of biomolecules, even for small proteins.
Some of the iterative methods such as Richardson, Jacobi, Gauss-Seidel and SOR
iterations, also work well for general structured matrices but they are barely employed
due to the reduced robustness for large protein system. The classic linear iteration
methods for solving Eq.(B.1) can be viewed as the following form
j+1

uh

j

j

= uh − BLh uh + Bfh ,

(B.2)

where B is matrix approximating L−1 in some sense. Different construction of matrix
h
B results in a different iterative method. The necessary and sufficient condition for
the convergence of algorithm (B.2) is that the spectrum ρ of the error propagation
operator must be smaller than 1, i.e., E = Ih − BLh and ρ(E) < 1 [115], where Ih
is the identity operator associated with the grid resolution h. The smaller value of
ρ(E) indicates the better convergence of the method. The spectra of this family of
iteration methods can be expressed as ρ(E) = 1 − O(h2 ), which implies that as grid
spacing gets smaller, these methods converge more and more slowly. This property
severely restricts the wide applications of these methods for large linear systems.
The conjugate gradient (CG) method is a very efficient iterative method if the
matrix is symmetric and positive definite. Actually it is the main workhorse of most
popular PBE solvers since the matrices from the standard FDMs or FEMs satisfy
these good properties. The multigrid (MG) method is an accelerating technique and
can be applied in combination with any of commonly used solvers. Using a hierarchy
of discretizations, MG shifts the computation between coarser and finer grids by
174

extrapolation and restriction, and thus accelerates the convergence. It is almost the
fastest accelerating technique known so far and applied in many popular PB solvers,
such as the APBS.
Unfortunately, the matrix Lh from the MIB can barely take advantages from
the described methods due to its complicated structure. For the discretization of the
Laplace operator in the PBE by standard FDMs, each grid point except the boundary
ones takes the following form:

−

·(

u)|x=xi ,yj ,zk = c0 ui,j,k + c1 ui−1,j,k + c2 ui+1,j,k
+ c3 ui,j−1,k + c4 ui,j+1,k

(B.3)

+ c5 ui,j,k−1 + c6 ui,j,k+1

where i, j, k represent the discretization indices along the x, y, z directions, respectively. The coefficients cm , m = 0, 1, ...6 only depend
symmetric structure of Eq. (B.3) and the facts

and grid spacing h. The

6
m=0 cm

= 0 and c1 = c2 = c3 =

c4 = c5 = c6 make the whole matrix symmetric and positive definite.
However, since the MIB scheme takes into account the interface treatment and at
all the irregular grid points near the interface, discetizations are modified. For the
simplest case, assume that only one fictitious point is needed and without the loss of
generality, the modification is in the form:

−

·(

u)|x=xi ,yj ,zk = c0 ui,j,k + c1 f ∗ + c2 ui+1,j,k
+ c3 ui,j−1,k + c4 ui,j+1,k

(B.4)

+ c5 ui,j,k−1 + c6 ui,j,k+1
Note that the fictitious value f ∗ is used in Eq. (B.4) for the smooth extension of the
function. The fictitious value f ∗ can further be expanded as the linear combination
175

of the unknown function values.
M

f∗ =
m=1

cm ui ,j ,k
˜
m m m

(B.5)

where ui ,j ,k is the nearby function values around ui,j,k , c˜ , m = 1, 2, ..., M are
m
m m m
the corresponding coefficients. Usually M = 10 in second order MIB scheme for
a smooth interface but could be bigger for interface with singularities. The choice
of ui ,j ,k and calculation of cm totally depend on the local information of the
˜
m m m
interface. The introduction of the fictitious values gives high accuracy for the interface problems but also ruins the good properties such as symmetry and positivedefiniteness of the overall matrix.

To solve the matrices generated from the MIB scheme, the direct methods and
regular iterative methods will be ruled out from the beginning due to the poor convergence for huge systems. The CG method also does not work because the unpredictably
general matrix structures. Meanwhile, the direct application of the multigrid method,
which is an important accelerating technique, also has a potential problem due to the
shift of irregular point locations during grid refinement cycles. Ref [30] showed the
poor behavior of the algebraic multigrid method (AMD) and proposed a new multigrid scheme based on the local interface problem but the interpolation operator at
the interface will cost much extra work.

Therefore, we put more emphasis on looking for suitable solvers and accelerating
techniques in the Krylov subspace theory. Stabilized biconjugate gradient method
(BiCG) and generalized minimal residual method (GMRES) are two examples in
Krylov subspace methods, which deal with the general nonsingular matrix that does
not have to be symmetric and positive definite. In the following section, the Krylov
subspace (KS) methods, idea of preconditioners are briefly introduced.
176

B.2

The Krylov subspace theory and preconditioners

Suppose u0 is an initial guess for the solution u in system

Lh uh = fh

(B.6)

and defines the initial residual r0 = f − Lu0 . For notation simplicity, the subscript
h is dropped here. As shown in Ref. [100], the Krylov subspace can be derived from
the following projection method. The mth iteration um , m = 1, 2, ... is of the form
um ∈ r0 + Sm ,

(B.7)

where Sm is some m-dimensional space, called the search space. Strictly speaking, Eq.
(B.7) is an abused notation, it means that um can be decomposed as the residual r0
and an element in space Sm . Because of m degrees of freedom, a total of m constraints
is required to make um unique. This is achieved by choosing an m-dimensional space
Cm , called the constraint space, and by requiring that the mth residual is orthogonal
to that space, i.e.,
rm = f − Lum ∈ r0 + LSm ,

rm ⊥Cm .

(B.8)

Here the orthogonality is in the sense of the inner product in the Euclidean space.
If the space Sm is defined as the Krylov subspace Km (L, r0 ), i,e,
Sm = Km (L, r0 ) ≡ span{r0 , Lr0 , ..., Lm−1 r0 },

m = 1, 2, ...,

(B.9)

then the projection method is the so-called Krylov subspace method. More specifi177

cally, if Cm = Sm , it is the Galerkin method, which includes the CG method and its
generalizations, and if Cm = LSm , it yields the GMRES. These are the basic idea of
Krylov subspace methods.
For the convergence analysis, note that conditions (B.7) and (B.8) imply that the
error u − um and the residual rm can be written in the polynomial form
u − um = pm (L)(u − um ),

rm = pm (L)r0 ,

(B.10)

where pm is a polynomial of degree at most m and with value one at the origin. Ref.
[100] gives the error bound for Krylov subspace methods
||u − um ||
≤ min max |p(λk )|,
p∈πm k
||u − u0 ||

(B.11)

where πm denotes the set of polynomials of degree at most m and with value one
at the origin, λk are the eigenvalues of the matrix L. It can be concluded from Eq.
(B.11) that the convergence behavior of the Krylov subspace methods is completely
determined by their spectra. However, as indicated in Ref. [100], it is always difficult to really evaluate the upper bound. Alternatively, it states that the condition
number of the matrix is a criteria which although, only partially reveals the practice
convergence behavior but is easier to calculate. For matrix L, the condition number
is defined as the ratio of the extreme eigenvalues or spectra

ς=

λmax
.
λmin

(B.12)

Since the rate of the convergence of Krylov projection methods for a particular
linear system is strongly dependent on its spectrum, preconditioner is typically used to
alter the spectrum and hence accelerate the convergence rate of iterative techniques.
178

Preconditioner can be applied to system (B.6) by
−1
−1
−1
ML LMR (MR u) = ML f,

(B.13)

where ML and MR denote the left and right precondition matrices. Usually if MR =
I, the left preconditioned results and the residual is given by
−1
−1
rL = ML f − ML Lu.

(B.14)

Properly preconditioned matrix M −1 L may significantly reduce the condition number
of L, hence the rate of convergence is accelerated. The commonly used precondition
strategies are Jacobi preconditioner, block preconditioner and incomplete LU factorization. However, preconditioning a large sparse system is an empirical exercise.
Different preconditioners work better for different kinds of problems. In practice,
the combinations of different Krylov subspace solvers and preconditioners are investigated, and the rate of convergence is analyzed via the spectra of preconditioned and
un-preconditioned matrices.

179

Appendix C
A short user manual for the
MIBPB package
C.1

Work flow of the MIBPB package

The MIBPB solver package incorporates with two packages to accomplish the electrostatic potential calculation. First, molecular structures are prepared via Python
software package PDB2PQR (http://pdb2pqr.sourceforge.net/): it accomplishes
many common tasks of preparing structures for continuum electrostatic calculations,
such as adding a limited number of missing heavy atoms to biomolecular structures,
determining side-chain pKa s, placing missing hydrogens, etc.
Users can either submit the protein PDB index to the online server (http://
pdb2pqr.sourceforge.net/) or download the executable file to prepare the molecular structure.
Once the molecular structure is prepared, the computational domain Ω will be
automatically generated based on the coordinates of the protein atoms: first a smallest
cuboid that contains the protein will be calculated and then each length of the cuboid
is symmetrically extend at two ends by 5 to 10˚, depending on the protein size. This
A
180

strategy usually employed in many FDMs is verified to be reasonable in practices and
also the extension of the cuboid can be customized easily. The larger size of Ω is of
course closer to real biological situation. However, the solution of the PBE is not
sensitive to this change while the computational cost will be increased.
Additionally, the geometry of the molecular surface used in the MIB scheme is generated by the MSMS (http://www.scripps.edu/~sanner/html/msms_home.html).
Given the information of the coordinates and radius of each atom in the molecule,
surfaces are generated at given water probe radius in a triangulation form. The intersection of each triangle with the meshing lines and the normal direction extracted
from the surface information are key ingredients of the MIBPB scheme. For the
MSMS parameter, the water molecule probe radius is recommended as 1.4 and the
vertex density is 10. These parameters are enough to generate the molecular surface
with good quality, various 3D Cartesian grid resolutions in current use can obtain
necessary surface information under this setting.
There are two options for choosing KS solvers and preconditioners in solving
MIBPB matrices. One is to use the SLATEC, which has been incorporated in our
MIBPB package. The other way is to use the PETSc. According to our tests, the
PETSc is generally more stable and reliable than the SLATEC, particularly for large
proteins. It needs to be pre-installed by the user if one chooses the PETSc matrix
acceleration option.
The current MIBPB package offers half stand-alone solvers in which users have
to prepare the molecular structures and generate the surfaces on their own with
desired parameters. The package also has one-step solvers which have integrated
all the steps with default parameter settings. Either the half stand-alone or onestep solver is further classified into linear solver and nonlinear solver. Therefore,
there are in total four executable MIBPB files in the package. Additionally, two
other auxiliary small Perl scripts, the pqr2xyzr.pl and dat2dx.pl are included in the
181

Download PDB file
YES
1 step?
No
Structure preparation via
PDB2PQR.py
Surface preparation via
pqr2xyzr.pl
Structure generation via
MSMS
YES

MIBPB4.1.1

Linear
solver?
No

Linear
solver?
YES

MIBPB4.2.1

MIBPB4.1.2

No

MIBPB4.2.2

Figure C.1: Work flow of the MIBPB package
MIBPB package to accomplish the molecular surface preparation and ultimate data
visualization. Figure (C.1) is the work flow of the MIBPB package usage. Users are
referred to http://www.math.msu.edu/~wei/MIBPB/ for detailed instructions.
For a clearer demonstration, we use one specific protein example to illustrate the
procedure. Protein with ID 1ajj is assumed to have been downloaded from the Protein
Data Bank and saved as file 1ajj.pdb.
1. Prepare the protein structure
• Input file: 1ajj.pdb
• Command line: python pdb2pqr.py –ff=CHARMM 1ajj.pdb 1ajj apbs.pqr
• Output file: 1ajj apbs.pqr.
182

• Remark: For full usage of pdb2pqr.py, users are referred to the corresponding link.
2. Molecular surface preparation
• Input file: 1ajj apbs.pqr
• Command line: pqr2xyzr 1ajj
• Output files: 1ajj.xyzr, 1ajj.pqr
• Remark: 1ajj.xyzr file stores the coordinates and radii of the atoms in
the protein, 1ajj.pqr stores the coordinates and partial charges. They are
necessary files for the MSMS to generate molecular surfaces.
3. Molecular surface generation
• Input files: 1ajj.xyzr, 1ajj.pqr
• Command line: msms -if 1ajj.xyzr -prob 1.4 -de 10 -of 1ajj
• Output files: 1ajj.vert, 1ajj.face. Now the molecular surface is generated in
the triangulation form. The vertices and normal direction of each triangle
are saved in these files.
• Remark: water probe radius and triangulation density are set as default
values 1.4 and 10, respectively. They are adjustable parameters.
4. MIBPB implementation
• Linear solver: mibpb4.1.1 1ajj eps1=1 eps2=80 h=0.5
• Nonlinear solver: mibpb4.2.1 1ajj eps1=1 eps2=80 kappa=1.0 h=0.5
• Output file: 1ajj pbe.dat
• Remark: Above command lines give the standard format. Parameters are
adjustable.
183

C.2

Work flow for the display of the surface electrostatic potential

After the electrostatic potential file is obtained by running the MIBPB solver, we
can display it on the molecular surface by using the VMD (http://www.ks.uiuc.
edu/Research/vmd/), a molecular visualization program. We are able to visualize the
potential distribution on the surface by implementing a file transformation via the Perl
script dat2dx.pl. Moreover, by taking the difference of surface electrostatic potentials
under different grid resolutions h, we are also able to check the convergence of the
solutions and therefore suggest a proper grid resolution for balancing high numerical
accuracy and efficiency. The procedure is shown as the following.
1. Visualization file preparation.
• MIBPB package generates output file [pdbname] pbe.dat, in which the electrostatic potentials on grid points of the protein-solvent system are stored.
Before displaying the electrostatic potential on the molecular surface, one
needs to use dat2dx.pl script to transform the data file to the [pdbname].dx
file.
• For example, for protein 1ajj, one gets 1ajj pbe.dat file from the MIBPB
package. Then use the command: dat2dx.pl 1ajj [dcel] [xleft] [xright] [yleft]
[yright] [zleft] [zright]
where [dcel] is the mesh size (we assume a uniform mesh). Here [xleft],[xright],
[yleft], [yright], [zleft] and [zright] prescribe the span of computational domain in x, y, z direction, respectively. Here xleft, xright, yleft, yright, zleft,
and zright should be the same as those used in calculating the potential.
2. Molecular surface drawing
184

Figure C.2: Visualizations of surface electrostatic potentials of protein 1beb. From
left to right: (a) Surface electrostatic potential (I = 0, h = 1.0˚); (b): Surface
A
˚); (c): The difference
electrostatic potential with the ionic strength I = 1 (h = 1.0A
of surface electrostatic potentials between in an ionic solvent (I = 1) and in a water
solvent (I = 0); (d): The difference of surface electrostatic potentials in water (I = 0)
between grid meshes h = 1.0˚ and h = 0.5˚.
A
A
• Load the PDB data file into the VMD
• Set drawing parameters in the Graphical Representation window: choose
the “Volume” option for coloring method and the “Surf” option for drawing
method.
3. Surface electrostatic potential drawing
• Load the [pdbname].dx format potential file into the VMD. In the Molecular
File Browser window, load [pdbname].dx file for the same protein as that
in molecular surface (instead of for new molecular).
• Set drawing parameters in the same Graphical Representation window as
that in the second step. Choose the “Volume” option for coloring method
and the “Surf” option for drawing method. Adjust the Color Scale Data
Range to see different color effects.
185

Figure C.2 illustrates the visualization of electrostatic potential calculated from
the MIBPB package, using protein 1beb as an example. The potentials calculated
via both the linear MIBPB solver and the nonlinear MIBPB solver are plotted on
the molecular surface via the VMD through the above procedure. Figure C.2(a)
displays the potential distribution on the surface of protein 1beb when the solvent
is water. In this case the linear MIBPB solver is implemented because κ is set as
¯
zero. While Fig. C.2(b) presents the potential distribution when κ2 = 8.48, in which
case the nonlinear MIBPB solver is employed. These two calculations are carried
˚
out when grid resolution h is taken as 1.0A. Figure C.2(c) gives the difference of
electrostatics in (a) and (b), from which the salt effect on electrostatic distribution
may be observed. Figure C.2(d) reveals the potential difference in solvent when the
calculations are under resolutions h = 1.0˚ and 0.5˚, i.e. the error |φh −φh/2 |. It can
A
A
be found that the error is almost zero around the molecular surface, this fact indicates
˚
that at h = 1.0A, the result is accurate enough so that reducing grid resolution to
˚
0.5A does not give too much improvement. Mathematically speaking, the result is
almost convergent between mesh size 1.0˚ and 0.5˚, which is the recommended grid
A
A
resolution range in the MIBPB package.

186

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