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SCANNING PROBE SPECTROSCOPY OF
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Morewell Gasseller

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SCANNING PROBE SPECTROSCOPY OF
INDIVIDUAL DOPANTS IN SILICON
By

Morewell Gasseller

A DISSERTATION

Submitted to
Michigan State University
In partial fulfillment of the requirements
For the degree of

DOCTOR OF PHILOSOPHY
Physics

2010

 

ABSTRACT

SCANNING PROBE SPECTROSCOPY OF INDIVIDUAL DOPANTS IN SILICON

By

Morewell Gasseller

I have applied a novel scanned probe method called Charge Accumulation Imaging
technique (CAI) to study the electronic states of individual boron dopants in silicon. In
particular, I used the Charge Accumulation Imaging technique to image boron acceptors

buried about 15 nm below the surface of a p-type silicon substrate and I was able to

determine the binding energy of the B + state on an atom-by-atom basis for 20 such
acceptors with millivolt precision. The data shows that acceptors with more distant
neighbors exhibit weaker binding than those with close by neighbors. Hence I was able to
discern both the average binding energy and the trend, which shows that binding energy
increases with decreasing nearest neighbor distance. Comparing our data to a simple
hydrogenic model that combines the effective mass approximation and configuration-
interaction calculation show a general agreement. Thus our technique has proved to be

an ideal tool to study directly individual and small clusters of dopant atoms.

© Copyright by Morewell Gasseller, 2010

This thesis is dedicated to my family.

iv

ACKNOWLEDGEMENTS

My PhD work would never have been possible without the help, support and
encouragement I have received from various people. I want to take this opportunity to
express my heartfelt thanks and gratitude to all those people. First of all, I would like to
thank my thesis adviser Professor Stuart Tessmer for his continuous advice,
encouragement and support. I thank him for giving me the chance to join his name group
and guiding me to complete my thesis work starting from scratch. The discussions I have
had with him over the years have not only helped to expand my interest in physics but
have also left an enduring impression in my mind. His encouragement and support has
been the most critical driving force throughout my student career. The advice and
positive criticism of the drafts of my thesis I got from him had a major impact on the final
outcome of this thesis. I consider myself very fortunate to have the chance to work under
his supervision.

I am also thankful to Prof. Golding for agreeing to proof read a draft of this thesis.
His advice and comments have helped to make the final outcome of this thesis better. I
am thankful to Prof. Zelevinsky, Prof. Dykman, and Prof. Tollefson for agreeing to serve
in my committee. I am very grateful to Professors Carlo Piermarocchi, Torn Kaplan and
SD Mahanti for their interest in my research. Their expert theoretical input was very
helpful in the process of understanding the interesting physics of my experimental
findings. Many thanks are due to Dr. Sven Rogge of Delft University for his
collaboration and to R. Loo and M Caymax of IMEC, Belgium for providing us with the

sample.

rt;

I would like to attribute my special thanks to all members of our group starting
with Irma Kuljanishvil who taught me many tricks which made my experiments easier.
Aleksandra Tomic, with whom I shared a lab room, was always very considerate and
helpful. I would like to say thanks to Cemil Kayis, Josh Veazey, and Christopher
Lawrence, my lab—mates and good friends who were always ready to assist not only with
the lab related tasks but any day-to-day life issues. A number of undergraduate students
passed through our lab over the years, but two of them left a lasting impression on me

because of their work rate and commitment to their projects. These two are John Raguse
and Jessica Muir. It was a great pleasure to have these two young students as my
coworkers and friends

I am very grateful to Mrs. Debbie Barrett for her constant assistance on different
occasions, always ready to give a timely advice or reminder of something important that
one may forget to do. Her professionalism and responsibility for her work are truly
remarkable. I should thank Dr. Reza Lolee for his help with the timely purchase of
helium whenever I was running my experiments. He was always flexible with us when
we have ordered too little or too much helium. I am very thankful to all the staff
members of the machine shop, particularly James Muns, Thomas Hudson, and Thomas
Palazzolo for their help and assistance during the initial stages of my research.

Lastly and by no means least, I would like to thank my wife Pauline. She has
been very supportive and patient with me during all these years, always encouraging me
and always believing in me.

This work was supported by the National Science Foundation grants DMR-

0305461 and the Institute for Quantum Sciences.

vi

Table of Contents

List of Figures ........................................................................... ix
1. Introduction ........................................................................... 1
1.1 Motivation and thesis outline ....................................................... l
1.2 Semiconductor technology ......................................................... 2
1.3 Classification of impurities ......................................................... 4
1.4 Shallow and deep level defects .................................................... 4
1.5 Effective mass approximation ....................................................... 8
1,6 Subsurface charge accumulation imaging ....................................... 12
2 Review of Previous Experiments .................................................. 16
2.1 Resonant tunneling experiment on a delta-doped silicon sample ............. 16
2.2 Charge accumulation imaging experiment on a GaAs sample ................ 21
3 Silicon Samples ........................................................................ 24
3.0 Silicon band structure ................................................................ 24
3.1 Chemical vapor deposition ......................................................... 27
3.2 Delta doped semiconductors ........................................................ 31
3.3 Secondary ion mass spectroscopy .................................................. 34
3.4 Strain issues in boron doped silicon ................................................ 36
3.5 Specifics of my sample ............................................................... 38
3.5.1 Sample growth ...................................................................... 38
3.5.2 Valence band diagram of sample .................................................. 41
4 Design and construction of microscope ........................................... 43
4.1 Scanning unit .......................................................................... 43
4.1.1 The base plate ....................................................................... 44
4.1.2 Body of scanning unit ............................................................. 48
4,2 Top-loading probe .................................................................... 51
4.3 Wiring and electronics ............................................................... 53
4.4 Noise isolation ....................................................................... 53
4.4.1 Mechanical stability of the microscope .......................................... 54
4.5 Cryogenic system ..................................................................... 56
4.6 The cryogenic charge sensor circuit ................................................ 58
5 Methods and Experimental Techniques .......................................... 65
5.0 Introduction ........................................................................... 65
5.1 Voltage Lever arm .................................................................... 65
5.2 Nulling Voltage ....................................................................... 66
5.3 Control of tip-sample separation ................................................... 67
5.4 Modes of operation of microsc0pe ................................................. 68
5.4.1 Tunneling mode ..................................................................... 71
5.4.2 Capacitance mode .................................................................. 72
5.4.3 Kelvin probe mode ................................................................. 75

vii

5.5 Procedure for analyzing the C—V curves ........................................... 75

6 Testing and calibration of microscope ............................................. 77
6.1 Calibration of scanning unit ......................................................... 77
6.2 Testing of microscope ................................................................. 81
7 Single Dopant Charging Results and Discussion ................................. 83
7.1 Introduction ........................................................................... 83
7.2 Observed capacitance curves and images .......................................... 85
7.3 Capacitance images and further analysis ........................................... 95
7.4 Determination of addition energy of individual boron dopants ................. 99
7.5 Analysis of secondary peaks ........................................................ 103
7.6 Configuration-interaction theory .................................................. 106
8 Summary and Future Directions .................................................. 114
8.1 Summary of the investigations ..................................................... 114
8.2 Future directions ..................................................................... 114
References ............................................................................... 1 14

viii

LIST OF FIGURES

Figure 1.1 Substitutional dopants in silicon. Phosphorus (P) behaves as a donor and
boron (B) as an acceptor, shown within a covalently bound Si lattice with the extra or
missing electron (line) at P or B ................................................................. 6

Figure 1.2 Energy band diagram of a typical semiconductor showing the bottom edge of
the conduction band EC and the top edge of the valence band Ev- EC and Ev are

separated by band gap energy E g ............................................................. 7

Figure 1.3 Band diagram of a typical semiconductor showing the added states due to the
presence of donor and acceptor dopants ...................................................... 7

Figure 1.4 Relationship between the envelope function F (r) and the wave function
wn (r) of a Bloch wave packet for an electron localized near a hydrogen-like impurity.

Here adenotes the lattice constant and a; is the effective Bohr radius
[19] .................................................................................................. 10

Figure 1.5 System drawing of a generic SCM with an AFM topographic control (adapted
from Ref.14). ..................................................................................... 15

Figure 2.1 (a) Secondary Ion Mass Spectroscopy profile (SIMS) of the boron
concentration in the device used by Caro and co—workers. Details of SIMS technique are
discussed in Section 3.1. (b) Valence band edge diagram of the device. The barrier

height IS¢B =AEV —EF [10 .................................................................. 19

Figure 2.2 Conductance characteristic of the device. The peaks could be clearly seen
atiIOmV. [10] ................................................................................... 20

Figure 2.3 (a) Schematic of the CA1 measurement for the Kuljanishvili experiment [6].
(b) Spectrosc0pic data showing broad peaks, labeled A, B and C. Each peak corresponds
to roughly 15 electrons. These peaks were interpreted as electrons entering clusters of
donor, effectively forming “donor molecules”. The rms amplitude of the excitation
voltage was 15 mV. (c) A subsequent measurement acquired with tip voltages indicated
by the rectangular box in (b) showing single-electron peaks. In this case, the nns
amplitude of the excitation voltage was 3.8 mV, as indicated.
...................................................................................................... 23

ix

Figure 3.0 the band structure of silicon. The conduction band minimum is away from
k=0, making silicon an indirect semiconductor. The valence band shows two degenerate
bands and the third band called the split off hole band is not shown ..................... 26

Figure 3.1 Simplified schematic diagram of a horizontal epitaxial reactor chamber
showing the wafers stacked in vertical slots and the motion of the precursor
gases .................................................................................................. 29

Figure 3.2 (a) Schematic representation of the formation of a boundary layer above the
substrate in a horizontal epitaxial reactor [28] .................................................. 30

Figure 3.3 Schematic illustrations of a semiconductor substrate and an epitaxial film
containing S-doped layer [54] .................................................................... 33

Figure 3.4 The SIMMS process. Incoming primary ion generates a variety of secondary
particles from a narrow escape region [51] ..................................................... 35

Figure 3.5 A local tensile strain is generated as a born atom substitutes into a silicon
lattice. AL is the change in the lattice constant due to the replacement of a silicon atom
with a boron atom (Adapted from reference 60) .............................................. 37

Figure 3.6 (a) Schematic of the sample showing the random distribution of boron dopants
in the delta-layer. The Silicon layer above the delta layer is 15 nm thick, the delta layer is
2mm thick, the silicon layer below the delta layer is 20mm thick and the heavily doped p+
layer is 1000 nm. (b) SIMS plot of our sample showing that the 8-layer is about 15 nm
from surface. (c) A photograph of the actual sample. The sample is a rectangular silicon
crystal of dimensions 10 x 2 x 1 mm, the gold wires shown bring the bias voltage to the
doped p+ layer. .................................................................................... 40

Figure 3.7 Schematic of the profile of the valence band edge of my sample. The heavily-
dOped bottom contact layer acts as a metallic substrate. Hole can tunnel onto the boron
atoms from this layer, as indicated by the red arrow. The gap between the tip and the
sample forms a second higher barrier, inhibiting charge from tunneling directly onto the
tip. E0 is the binding energy of the neutral boron state (B0) with one bound hole; E + is the
binding energy of the of the positive boron (3+) state with two bound holes; ¢B is the
effective work function or tunnel-barrier height ............................................. 42

Figure 4.0 (a) A blown out schematic of the scanning unit showing the main parts. The
diameter of the scanning unit is 3.00 cm and the total length of the compact assembly is
8.04 cm. The sample holder sketch shows the sample and the SIOpped ramps. It is a
stainless steel disk of outer diameter 2.70 cm and thickness 0.70 cm. The base plate
schematic shows how the four piezotubes are arranged. (b) A photograph of the scanning
unit; the door is removed to show the copper feet onto which the sample holder rests.
....................................................................................................... 45

Figure 4.1 Photograph of the base plate (diameter 3.00 cm and thickness 0.50 cm)
showing the top and bottom sides. In this photograph, only a few of the connecting wires
are shown and one carrier piezotubes is missing ............................................... 47

Fig. 4.2 A close-up image of the tip holder and sample holder. In this picture the copper
feet and the steel balls on top of two of carrier piezotubes are clearly shown. A Teflon
ring surrounds the sample holder, preventing it from making electrical contact to the body
of the scanning unit. The tip is mounted at the top corner of a GaAs chip that also holds
the sensor circuit. The sensor circuit is controlled by four coaxial wires which emerge
from small holes to the left and right of the tip holder (not discernable in the picture).
...................................................................................................... 50

Figure 4.3 A schematic of the tOp loading probe, (a) the total length of the vacuum lock
and the sliding insert is about 2 m. (b) A photograph of the top cone showing the 12 SMA
connectors attached to the top cone, (c) a photograph of the top loading probe placed on
top of the 3 He cryostat ready for positioning of scanning unit into cryostat. In (a) the top-
loading probe is fully pushed into cryostat and in (c) it is fully pulled
out .................................................................................................. 52

Figure 4.4 Schematic of the two stage vibration isolation system used in our
measurements. The hatched bars represent the stages of vibration
isolation ............................................................................................ 55

Figure 4.5 Schematics of the cryostat. The cryostat is about 60 cm wide and holds up to
60 L of liquid 4He. During the experiment, the probe resides inside the cryostat and the
scanning head is situated at the bottom of the cryostat surrounded by superconducting
magnet ............................................................................................ 57

Figure 4.6 Figure 4.6 Drain I-V characteristics, at different gate voltages VGS, of our
HEMT [70]. The circles mark the voltage range in the linear regime in which we
operated ........................................................................................... 60

Figure 4.7 (a) Schematic picture of the cross section of a HEMT [70]. (b) A photograph
of the Fujutsi model FHX35X HEMT. The size of the device is ~ 500nm across.. ....61

Figure 4.8 Schematic of the charge sensor circuit. The charge sensor circuit uses two
HEMTs, the measurement HEMT is to the right and the bias HEMT is to the left in Fig.
4.8 (a). The gate of the measurement HEMT is biased through the bias HEMT. The role
of the bias HEMT is to act as a very high resistor, preventing the charge induced on the

tip from leaking to the lefi. C3 is the standard capacitor and Vin is the rms value of the
applied AC excitation to the sample. The AC excitation applied to Cs, is of the same

xi

amplitude but 1800 degree out of phase with the excitation voltage that is applied to the
sample. The charge sensitivity of the circuit is 0.01 electron/VHz“ . . . . . . . . . . . . . . . ............63

Figure 5.1 (a) Schematic representation of a typical STM piezotube with four quadrants.
(b)The top view: the quadrants denoted with +X, +Y, -X, and —Y are used for
elctromechanical control (c) Schematic of the three can'ier piezotubes indicating motion
in a rotational sense ................................................................................. 69

Figure 5.2 Three modes of operation of the microscope: tunneling mode (right), Kelvin
probe (middle) and charge accumulation (left) ................................................. 70

Figure 5.3 (a) Schematic of Charge Imaging method. Charge entering the sample induces
image charge q on the tip, which is monitored using a cryogenic charge sensor. The

charge sensor has a sensitivity of 0.15electron / WT; . (b) An example of charging
structure measured in a GaAs-AlGaAs two-dimensional electron system using this
technique. The dark spot is an area of reduced charging which resulted from an effective
reduction in the concentration of Si dopants [9] ................................................ 73

Figure 5.4 A schematic representation of charge accumulation data. (a) Spectroscopic
data, the sharp peak represents the voltage for which an energy level in the dopant layer
is in resonance with the F errni level of substrate. (b) A representation of scanning image.
Yellow spots indicate dOpant sites (these sketches are not drawn to scale) ................ 74

Figure 6.0 (a) Schematic drawing of two successive layers of hexagonal graphite. Open
circles represent atoms that appear in an STM image. (b) Carbon atoms in the STM
image of graphite obtained at the tunneling current of 0.5nA and bias voltage of 50mV.
Scan range is 2.00 x 1.80 nm. Obtaining image (b) implies the scanning unit is
stable ................................................................................................................................ 79.

Figure 6.1 Calibration of scanner. (a) STM image of graphite obtained at the tunneling
current of 0.5nA and bias voltage of 50mV with a scan range of 1.10 x 1.25 nm. (b) STM
image of platinum sample showing the edge of one of the pits. From these images the x, y
and z sensitivity of the piezotubes is obtained. ................................................ 80

Figure 6.2 Topography images of the silicon and GaAs samples. (a) is a typical STM
image of the silicon sample surface, the scan range for this image is about 500 um x 500
nm. (b) is a cross-section along the path marked by the double arrow in (a). (c) is the
topographic image of GaAs showing the typical growth features of the GaAs surface of ~
1pm long and ~ 5 nm high. These STM images were taken at 77 K ...................... 82

xii

Fig 7.1 (a) Representative capacitance data, plotted with respect to the raw tip voltage.

Two types of features were consistently observed: peaks near Vmw=-0.15 V (red arrow)
and a step near -0.30 V (green arrow). The peaks are consistent with single charges
entering the dopant layer. The step is consistent with the formation of an accumulation
layer at the sample surface. (b) Valence-band diagram (reproduced from Fig. 3.7)
showing our interpretation of the peaks as holes entering acceptor atoms, and the step as
an accumulation layer formed at the semiconductor surface ................................. 86

Figure 7.2 Schematic of capacitance technique to extract the nulling voltage Vnu”. This is
the effective zero voltage, the potential for which no electric field terminates on the tip.
The method uses the step feature that corresponds to the formation of a surface
conducting layer. The parameters dgap and or correspond to the tip-surface separation and
the lever-arm, respectively. (top) Two capacitance curves acquired at different tip
positions, plotted as a function of the raw tip-voltage. (bottom) The same two curves
plotted with a new voltage scales V*. In principle, the two curves will only coincide if
they are rescaled with respect to the nulling voltage, as indicated........... . . . . . . . . . . . . . . . . .88

Figure 7.3 (a) Two capacitance curves acquired with a blunt tip at nominal distances from
the surface dgap of 1 nm and 2 nm, as indicated. The clear step-like curve is consistent
with charge accumulating on a surface conducting layer. The data were acquired at a
temperature of 4.2 K. The solid curves are least-squares fits, using the step function
shown in Eq.5.2. For the solid Curve 1, the fitting parameters are P1=-0.63822, P2
=1.0374, P3=-0.10295, P4=0.O3251, P5=10.O7723 and P6=0.04011; for the solid Curve 2,
the fitting parameters are P1=-2.61947, P2 =1.08667, P3=-0.13956, P4=0.03600,
P5=45.71289 and P6=0.01299. (b) The same two curves normalized and plotted with
voltage scales V* for which Vm,” and or;/ or] were chosen to yield the best agreement. The
best-fit parameters are VHF-0.068 and az/oz,=l .295 ..................................................... 90

Figure 7.4 (a) Two capacitance curves acquired with a sharp tip at nominal distances

from the surface dgap of 1 nm and 2 nm, as indicated. The 1 nm curve is shifted vertically
by 5 aF for clarity. The curve displays clear peak and step-like structure, indicated by the
red and green arrows, respectively. The data were acquired at a temperature of 4.2 K. (b)
The same two curves normalized and plotted with voltage scales V*. In this case a
nulling voltage of 0.068 V is used, consistent with the blunt-tip data set (Fig. 5.6); or;/ or]
was chosen to yield the best agreement, which in this case is or;/oq=1.200. We see that a
reasonably good fit is achieved. The same tip was used for all subsequent measurements
presented in this thesis ........................................................................... 94

Figure 7.5 (a) Representative capacitance data reproduced from Fig. 7.1. Here we have
compensated for the contact potential by displaying the voltage axis with respect to V,,-,,.
(b) A representative 70 x 50 nm capacitance image, acquired by fixing the voltage at V,,-,,
=—0.076 V while scanning. The data are unfiltered and unprocessed, other than a standard

xiii

 

line-wise correction to remove drift effects in the vertical (slow scan) direction. Bright
high-capacitance features are indicated by the red dots. (c) The capacitance profile of an
individual dopant; r=0 corresponds to the center of the bright spot. To reduce scatter, we
have averaged three paths as indicated. This averaging does not change the general shape
compared to an individual path through the dopant. In this case the measured curve
(black dots) is well described by a Lorentzian-squared [1+(r/15)2]'2 (blue curve) ........ 96

Figure 7.6 Data set showing an integrated capacitance image and spectroscopy. The
image is to the left. The plots on the right display C—V curves in charge units acquired at
the indicated locations, A and B. To emphasize the peak structure, I have subtracted a
background slope of ~4eN, similar to the gray line I Fig. 75(3). The solid curves are fits
from the model based on the assumption that each peak corresponds to a single hole
entering the system (as discussed in Sec. 7.4). The top panel shows two curves nominally
acquired at the same location A, but with a time delay of 16 hours. The top curve is
shifted vertically by 0.12e for clarity. The curves show similar peak structure, but some
of the features have shifted in voltage by as much as 10 mV. These lead to ~1 meV shifts
in the extracted quantum energies (as discussed in Sec. 7.4). The arrow indicates the
tallest or primary peak. The lower panel shows the measurement and fit for location B.
We see that the C—V curve exhibits two prominent peaks of nearly equal magnitude. This
most likely arises from two closely spaced acceptors at location B ...................... 98

Figure 7.7 (a) Data show three distinct peaks that are relatively isolated. The measured a
parameters for the peaks labeled 1, 2 and 3 are 0.067, 0.0833 and 0.0971 respectively. (b)
A histogram of the quantum state energies E, extracted from measurements of 20
distinct peaks. The mean value and standard deviation of the distribution are 4.3 meV and
3.1 meV, respectively. (c) A table showing the peak voltages, a parameters and the
extracted E+ for all of the peaks ................................................................ 101

Figure 7.8 Estimating the nearest neighbor distance. (a) Tables of the fitting parameter, a
for the C-chrves at locations A (top curve) and B shown in Fig. 7.3. The primary peak,
P, corresponds to the resonance with the greatest lever arm; other peaks are referred to as
secondary, S. The bottom two rows show aS/ap for each secondary peak and the
estimated distances to the respective acceptors, based on Eq. 7.5. (b) The capacitance
image with triangles marking the likely locations of dopants. The hollow triangles
correspond to dopants most likely too far from locations A and B to contribute to the
respective C-V curves. The dashed circles are of radius 25 nm to show the approximate
area of interaction for the tip at the two locations ........................................................... 105

Figure 7.9 Hole binding energy versus nearest-neighbor distance. A simple modelling
method uses the effective mass approximation and the configuration-ineraction method.

At large separations the molecule can hold three holes, similar to the H- state, but for

xiv

 

small separations only two holes can be accommodated. The intuitive picture is that the
neutral system can polarize and weakly bind the third hole, but this is prohibited for small
separations for which the direct Coulomb repulsion dominates ............................ 107

Figure 7.10 Hole binding energy versus nearest-neighbor distance. Primary—peak binding
energy plotted as a function of nearest-neighbor distance R. The parameters were
extracted from measurements of dopants at different locations, including the data from
Fig. 7.6 A and B. The error bars show the estimated uncertainty of the energy and
distance. Also shown in blue is a least-squares linear fit. The inset shows in red the
results of a simple-hydrogenic model which incorporates two neutral acceptors and finds

the binding energy of a third-hole; E + is positive if the third charge results in a lower
electronic energy for the system, i.e., if the third hole is bound ............................. 109

Figure 7.11 The ground state wave firnction potential well. (a) The two potential are well-
separated compared to the wave function width of an isolated well. (b) The two potentials
are close together compared to the width of an isolated well. This state has a lower
energy due to reduced gradient .................................................................. 110

Figure 7.12 Statistical nearest neighbor distances for the boron dopants in our system,
calculated using the nominal grth density of boron dOpants in our system ............ 112

XV

Chapter 1

Introduction

1.1 Motivation and thesis outline

As the size of conventional semiconductor components is reduced to nanometer scales,
the improvement in their performance is determined by ever fewer numbers of impurity
atoms known as dopants. The ultimate limit is a single dopant, and numerous devices
have been proposed based on manipulating the charge and spin of small numbers of
dopant atoms [1, 2, 3, 4, 5]. Identifying the position and quantum state of an individual
dOpant in a semiconductor is a key step and a difficult technical challenge. Here we
present measurements that extend the reach of scanned-probe methods to discern the
properties of individual dopants in silicon, tens of nanometers below the surface. Using a
capacitance-based approach we have both spatially—resolved individual subsurface boron
acceptors and detected spectroscopically single holes entering and leaving these atoms.
By directly measuring the quantum levels and the influence of dopant interaction, this
method represents a valuable tool for the development of future atomic scale
semiconductor devices.

This thesis is organized as follows: In the introduction I give a brief outline of the
current trends in semiconductor device technology including the difficulties and
challenges it faces as the device size approaches the quantum limit. A brief introduction
to the effective mass approximation theory of shallow dopants in bulk semiconductors is
also given. Finally an overview of scanning probe techniques as a tool to probe and
manipulate small clusters of dopants is discussed. Chapter 2 represents a brief review of

two experiments that inspired this project. One experiment was done by Kuljanishvili and

coworkers [6, 7, 8]. In this study they were able to observe donor layer charging in a
GaAs heterostructured sample using a technique called subsurface charge accumulation

imaging (CAI) [9]. The other experiment was performed by Caro and coworkers [10],

who observed resonant tunneling through the B + state of boron dopants in a silicon
sample. Both experiments are concerned with the characterization of dopants at the
atomic level. Chapter 3 reviews some general aspects of doped silicon samples and
describes specifically the details of the sample that I used, which is a silicon crystal
doped with boron. Chapter 4 describes the design and construction of my CAI
microscope and gives the details of the other apparatus used in my experiments. Chapter
5 describes the methods and experimental techniques and Chapter 6 describes the testing
and calibration of the CAI microscope. Results, analysis and discussions are presented in
Chapter 7. Concluding remarks and possible directions for further work are given in

Chapter 8.

1.2 Semiconductor technology

Semiconductors refer to a class of materials whose electrical properties are between
conductors and insulators. Semiconductor devices are critical components in the
electronics industry and they affect people in all facets of life. The industry has sustained
a rapid growth in the past 40 years [11]. Over these years the industry has taken
advantage of the fact that decreasing the device size not only reduces cost but also
improves performance in terms of speed, functionality and power dissipation. This trend
is often referred to as Moore's law, an expression coined in 1965 that prescribes that the

number of transistors on a silicon chip will double every two years. Maintaining this

growth for as long as possible is the primary goal of many of those involved in
semiconductor device technology, from university researchers to industry.

However it now appears that the industry is rapidly approaching the formidable
quantum limit consisting of an unprecedented number of technical challenges, which
threaten the continuation of this miniaturization process. The current line widths in state-
of-the-art microprocessors are as small as ten nanometers, for which quantum mechanical
effects become apparent and in some cases dominant. The key atoms that control
electrical activity in semiconductor devices are called dopants; these are deliberately
added impurity atoms that supply electrons (from donors) or holes (from acceptors) to the
semiconductor. With line widths on the nanometer scale a few misplaced dopants can
completely change the way the device is supposed to work [12]. Hence, the physical
properties of these systems can no longer be determined by simply scaling down from
large sizes. New techniques are required for characterization of such small devices.
Projections of future device performance and key challenges to be overcome are
discussed in a biannual Semiconductor Industry Association (SIA) document called the
“National Technology Roadmap for Semiconductors” [13]. In the 2007 version the SIA
identified two and three-dimensional dopant profiling as key needs for future device
technology. Scanning capacitance microscopy (SCM) provides a direct method for
mapping the dopant distribution in a semiconductor device on a 10 nm scale [14]. This
capability is critical for the development, optimization and understanding of future

nanometer size devices [1.5].

1.3 Classification of impurities
The electronic effects of dopant atoms are easily described when they are from adjacent
columns of the periodic table. For example, phosphorous has one more electron in the
outer shell available for chemical bonding than silicon. When a phosphorous atom is
inserted into a crystal of silicon atoms, the extra donated electron can move in the crystal
lattice and carry an electric current. Because the electron carries a negative charge, using
phosphorous as a dopant produces an n-type semiconductor. On the other hand, boron has
one electron fewer than silicon; so boron atoms can accept electrons form the host
crystal, leaving spaces or holes where the electrons could have been. These holes are also
free to move and when they do they act like positive charges moving through the solid.
Hence, boron-doped silicon is a p—type semiconductor. Fig. 1.1 shows a sketch of p-type
and n-type silicon.
1.4 Shallow and deep level defects
In a semiconductor, the energy region that is free of electronic states and separates the
valence band from the conduction band is called the band gap. Fig. 1.2 shows the
energy band diagram of a typical semiconductor. Dopants modify the electrical
conductivity of semiconductors by adding states in this band gap. These added states
provide electrical charge carriers (electrons for donors or holes for acceptors) to the
conduction or valence band respectively. Fig. 1.3 shows the band diagram of a doped
semiconductor.

Shallow donors or acceptors have energy levels within a few tens of millielectron
volts from the respective band edges, whereas deep defects reside closer to the center of

the band gap [16]. Deep levels have highly localized wave functions whereas shallow

level wave functions are more extended. The boron acceptors in my experiment are

shallow level acceptors.

lllll lllll

__Si :Si :Si :Si :Si ““" ——-Si :Si :Si :Si :Si "'7'

 

 

 

 

 

 

ll ll II II ll
-LQI._I._L{-=I.— __.—..:..._.:.._
ll/l ll llllllllll
__..Si : P:—_Si:-:sr :Si -- ---Si:: B__Si:Si:Si""'
l I l I l
(a) (b)

Figure 1.1 Substitutional dopants in silicon. Phosphorus (P) behaves as a donor and

boron (B) as an acceptor, shown schematically within a covalently bound Si lattice with
the extra or missing electron (line) at P or B.

Conduction band

 

Eg Band gap

 

 

,/’./’//’./’//’,/’,/f//}//f/" 13V

Valence Band

the conduction band EC and the top edge of the valence band EV. EC and E,, are
separated by the band gap energy E g .

Conduction band

i 13C

........................‘...........

 

Donor ionization energy

Acceptor ionization energy

4 EV

 

Valence Band

Figure 1.3 Band diagram of a typical semiconductor showing the added states due the
presence of donor and acceptor dopants.

1.5 The effective mass approximation

Here we discuss the effective mass approximation (EMA) to describe the electronic
structure of shallow level dopants. The electronic structure of shallow impurity states in
semiconductors has been widely investigated with EMA [17, 18]. The dopant atom is
modeled essentially as a single charge with one or more electrons or holes bound to it
[19]. The eigenstates of this donor or acceptor can be described approximately as those of
a hydrogen atom with a modified field reduced by the dielectric constant, K, and with an
electron or hole of effective mass m*. The combined effect of the effective mass and
dielectric constant causes the charge to be bound much less tightly than the actual
hydrogen atom with an effective radius of several lattice constants.

The key assumptions of EMA are that the electron or hole wave functions of
shallow dopants extend over several lattice constants of the host semiconductor and that
short wavelength oscillations are not important. With respect to spatial coordinate r,
where F0 is the location of the dopant, the wave function y/(r) can be expressed as a
product: y/(r) =F(r)¢(r), where ¢(r) is a function that oscillates with a wavelength
comparable to the lattice spacing of the host crystal, a, and F (r) is an envelope function,
as shown in Fig. 1.4. The envelope function satisfies an appropriately modified
hydrogen-like Schrodinger equation.

Two important parameters for EMA are the effective Bohr radius a; and the
effective Rydberg energy R I, both of which are defined with respect to the ground state of

a single electron (or hole) bound to an isolated donor (or acceptor): a; gives the size of

the envelope wave function and R‘ gives the binding energy. These parameters can be

expressed as

 

 

e 2 . 00 a
m e m
and
2
e
R. = . ,
87280an

where so is the permittivity of free space, his the reduced Planck constant, m0 is the free
electron mass and a0 is the hydrogen atom Bohr radius, a0=0.592A. For common
semiconductors such as silicon and gallium-arsenide, a; is typically l-lO nm and R. is
typically 5-100 meV. This model has been confirmed to hold approximately over a broad

range of electrical and optical studies of a vast number of dopants in numerous

semiconductor materials [20].

s
.t.

\
\
A
I
l
I
I
[OI-Innocence

 

 

 

 

 

 

 

 

 

l,’ [I [I {7“

.«n’ F i ITTm-L .-_—
“Wu JUU r
i U U U

»§a§«- Wt)

 

Figure 1.4 Relationship between the envelope function F (r) and the wave function u/(r)
for the effective mass approximation. A shallow dopant is located at r=0; a denotes the

lattice constant of the host semiconductor and a; is the effective Bohr radius [19].

10

With regard to the silicon system, in general the effective mass approximation for
acceptors does not give as good a description as it does for donors. As discussed in
Section 3.0, the valence band in silicon consists of two degenerate bands and a third spin-
split band. The two degenerate bands interact to distort the ideal Spherical symmetry and
the parabolic distribution upon which the effective mass is based. This distortion leads to
a poor agreement between theoretically derived and measured properties of the holes [21,
22, 23]. For example, for an isolated boron acceptor with one bound hole, EMA
calculations yield R*=37.l meV for the ground-state binding energy; however
measurements give 45.7 meV, a discrepancy of 20% [10, 20, 21].

For my experiment, I probed holes tunneling into boron acceptors in a silicon
host. Both light and heavy holes tunnel. Therefore, an average mass for tunneling, which
takes into account the directionality of the tunneling process actually applies here.
However, due to the bending of the bands and the uncertainty in the Fermi energy, a

determination of this mass is not trivial. Therefore following the method of Ref. [10], I

estimated an effective Bohr radius from the empirical relation agNZi. = 0.26 , where N BI

is the critical hole carrier density at the metal—to—insulator transition point. This relation is

referred to as the Mott criterion for the metal-to-insulator transition. For this system,
N19,. =4.0 x 10m cm'3, which gives an effective Bohr radius (1;: 1.6 nm.

In light of the above considerations, clearly there are many approximations
involved with applying EMA to my system. While EMA-based modeling may be used as
a guide for our understanding of long-wavelength effects, we should keep in mind that

we do not expect quantitative agreement measurements to a precision of a few percent.

11

1.6 Subsurface charge accumulation imaging

A number of different techniques have been developed for two-dimensional dopant
profiling. These techniques include scanning probe microscopy (SPM) and other
electron/ion beam approaches. See Reference 14 for a detailed review. Of all these
techniques, SPM has been identified as a leading approach for obtaining two-
dimensional dopant profiles of semiconductor devices and is currently being modified
and adapted to a variety of problems by many researchers and industrial users.

Scanning probe microscopy is based on the interaction between an ultra sharp tip
and a sample. Measurements described in this thesis are based on a special type of the
scanned probe technique. This technique can resolve features down to the atomic level.
There are many types of SPM techniques, including scanning tunneling microscopy
(STM), scanning capacitance microscopy (SCM), atomic force microscopy (AFM),
force modulation microscopy and many others. In the case of scanning capacitance
microscopy, a conducting tip is scanned across a sample and small changes in the tip-
sample capacitance are measured. These changes can be caused by topographic changes,
dielectric variations or by local variations in carrier density. The volume probed by
SCM is determined by several factors, including the tip radius, the gap between tip and
surface, the local dielectric constant and the voltage between the tip and sample [24, 25].
Below I will give a brief review of the development and adaptations of SCM techniques
up to our charge accumulation imaging (CAI) technique.

One of the early scanning capacitance microscopes was developed by Mately
and Blanc in 1984 [26]. Their microscope demonstrated the capability of detecting

variations of surface topography on the order of 0.3 run over areas of the order of 2.5

12

umz. This great sensitivity to height changes is achieved by the placement of a
capacitance probe some 20 nm above the sample surface. In 1988 Bugg and King

demonstrated imaging on a scale of 2 pm lateral resolution with an unguided scanning

system similar to SCM [27]. Williams and co-workers then demonstrated capacitive
imaging on a 25 nm scale and showed that the limit for capacitive imaging was below
10 nm. Various other systems were developed as described in reference [28]. Central to
all these systems is the fact that capacitive interaction is used to control the height of the
tip above the surface during scan. This was accomplished by modulating the height of
the tip at a fixed frequency and measuring the capacitance change at that frequency with
a lock-in amplifier. Since the capacitance change increases as the tip approaches the
sample, a feedback loop is used to adjust the height of the tip to keep the capacitance
changes constant as the tip is scanned over the surface. A generic example of SCM
system with a force feedback is shown in Fig. 1.5.

In order to achieve high spatial resolution, ultra high capacitance sensitivity is

required. The typical tip/sample capacitance is in the atto-farad range (1 0’18]r ). While
it is extremely difficult to measure such a small static capacitance, measuring a dynamic

change in capacitance on this scale is possible. In fact the first sensor used in SCM was

capable of measuring capacitance changes on the order of 10—21 F [29].

The techniques mentioned above are predominantly room temperature
measurements. A special type of SCM called charge accumulation imaging (CAI) has
the unique advantage of being able to operate at low temperatures. Low temperature
operation enables the probing of quantum systems with energy level spacing on the

millivolt scale. Moreover, the technique enables the direct imaging of electronic systems

13

“ Ian "._‘ 3"“.

buried tens of nanometers beneath the surface of semiconductor materials. An
adaptation of this technique is the one we used for measurements described in this

thesis. This method is described in detail in Chapter 5.

l4

Capacitance sensor

E _J

AFMIM l——-l [

 

 

 

 

 

 

 

 

 

Tip
- '-'-°"~'.'.:.:-:.:.:-:.:-:o:-:o:-:.:7zT?:-:-:-:-1
OXIde -—b:::2::.:~:4:4:-2-:-:2-Igi-I-Z'Z-2-2;3'2-I;I-1;2-2-2;
. Depletion @
Silicon

 

 

 

 

Figure 1.5 System drawing of a generic SCM with an atomic force microscopy
topographic control system (adapted from Ref. 14)

Chapter 2

Review of Previous Experiments

The next generation semiconductor devices will be so small that their functionality will
likely be determined by the presence of only a few dopant atoms. Hence the need and
ability to control and detect single dopants with atomic resolution and identify their
electronic states has become vital [30, 31, 32, 33, 34, 35]. In this chapter we will review
two experiments that probed dopant atoms with atomic precision. One experiment was
done by Kuljanishvili and coworkers [6]. This experiment successfully detected donor
layer charging in a GaAs heterostructured sample. The broad spectroscopic peaks
obtained there were attributed to dozens of electrons tunneling into clusters of donors.
The other experiment by Caro and coworkers measured resonant tunneling through an
acceptor quantum level in a boron-doped silicon sample. Related to these two
experiments are optical experiments on single impurity states. For example, a

photoconductance spectroscopy experiment on silicon samples with low doping density

by Burger and Lassman measured the binding energy of an isolated 8+ ion to be 2.0

meV [36, 37, 38].

2.1 Resonant tunneling experiment on a delta-doped silicon sample

Resonant tunneling experiments offer an opportunity for detailed investigations of both
the tunneling process through an impurity state and the local properties of a two
dimensional electron gas (2DEG) [39, 40, 41]. Caro and coworkers used a resonant
tunneling device with a metal-insulator-metal structure to probe the characteristic of a

delta-doped layer. Delta-doping refers to dopants confined approximately to a single

16

atomic layer, as opposed to bulk doping where the dopants are everywhere in the

material. The layering sequence of the device was:
p+Si(500nm)/ p—Si(20nm)/ (3 / p—Si(20nm)/ p+(500nm).
Boron was the dopant for the bottom and top p+ layers and the 8 layer. The 1r)+ layers

were heavily doped at 1.0x10l9 cm'3 and hence acted metallic, whereas the p- layers
were intrinsic silicon and formed barriers. The areal density of boron in the delta layer
was 1.7x10ll cm'z, which gives an average spacing of 25 nm between boron atoms. The
doping profile of the device using secondary ion mass spectroscopy (SIMS) is shown in
Fig. 2.1 (a). The 5 layer, the barrier and the metallic layers are clearly discernible on the

diagram. The profile of the valence band edge of the device is shown in Fig. 2.1 (b),
where E0 is the ground state energy of the neutral boron atom (BO) and E + is the binding
energy of the B+ state. The B+ state is the acceptor counterpart of the generally known

D— state. It forms when a second hole is weakly bound to a neutral boron atom. The
SIMS method and details of delta doping are further discussed in Chapter 3.

Fig. 2.2 shows the differential conductance of such a device as function of the
bias voltage. Measurements were taken for temperatures in the range of 0.5 K to 12.5K.
A tunneling resonance is clearly seen as a peak atilOmV. The silicon wafer was

patterned into micron-size mesas; hence the transport current can interact with millions of

boron dopants. The heavily doped p+ layers at the bottom and top of the sample acted

as the contact layers to which wire leads were attached. The conductance peak shown in

Fig. 2.2 is attributed to resonant tunneling through the B+ states of the boron impurities

in the 5 layer. The resonant peaks cannot be due to tunneling through E0 because it is too

17

deep below the Fermi level so that it is permanently occupied for the voltage ranges
permitted by this measurement. Resonant tunneling refers to tunneling in which the
electron transmission coefficient through a structure is sharply peaked about certain

energies. This happens when the electron has an intermediate state to tunnel into. In this

case the B+ state provides the intermediate step. Since the measured resonance comes
from the 5 layer, equal parts of the bias drop across the barriers at either side of the

potential well associated with the boron impurity. Thus the resonance voltage is

Vres = 2[¢B — E+]/e . With ¢B =1 1.7 meV and the measured value of Vm =10 mV, the

value of E + is found to be 6.7 meV.

18

 

1 r 1 1

1019 .__ we —

 
    

 

 

 

 

 

 

 

 

 

 

5,? I. :
3, 1018 1‘ : a)
z” t f I
1017 r I I
o .- V I fit r q
s: w. I I -- 5+ bl
g -10 'EF . . z... -
v A
>. -20 ~ EV ~
9 .
w _ ..
5 '3“ Ev...“ v
-40 r- l .1- Eu l 1 '1
20 40 60 80 100
depth (nm)

Figure 2.1 (a) Secondary Ion Mass Spectroscopy profile (SIMS) of the boron
concentration in the device used by Caro and co-workers. Details of SIMS technique are
discussed in Section 3.1. (b) Valence band edge diagram of the device. The barrier

height IS¢B = My -EF [IO].

19

 

Glms]

 

 

 

 

V [mV]

Figure 2.2 Conductance characteristic of the resonant tunneling device used by Caro and
co-workers. The peaks are clearly seen at :10 mV [10].

20

.1:

The main result from this experiment is the observation of resonant tunneling
through B+ states of the boron dopants in the delta layer. From their measurements Care
and coworkers determined that the binding energy E + of the extra hole on an isolated

B+ ion is 6.7 meV. This value deviates from the 2.0 meV which was obtained from
photo conductance spectroscopy experiments on Si samples with low doping levels [42,
43, 44, 45]. The most plausible explanation for this discrepancy is that at this
concentration, the boron dopants cannot be considered to be isolated. This issue will be

the major focus of my experiment.

2.2 Charge accumulation imaging (CAI) experiment on a GaAs sample

Charge accumulation imaging (CAI) is a powerful tool to probe mobile charges in many
experiments. The method will be described in detail in Chapter 5. Kuljanishvili and
coworkers used CAI to probe the electron addition spectra of a system of silicon donors
in AlGaAs. Fig. 2.3 (a) shows the schematic of the CA1 method for this experiment,
which essentially measures the capacitance between the sample and a sharp metal tip.
For this measurement, the key sample parameters are the surface-to-delta layer distance
of 60 nm and the dopant density within the delta layer of 1.25x10l6 m'z. The key tip
parameters are the radius of curvature of the apex of 50 nm and its distance from the
sample surface of about 1 nm. The measurement had rather poor spatial resolution of
about 60 nm; in other words, with the tip at a fixed location, it interacted with a circular
area of the donor layer of radius ~60 nm, which contained about 150 dopants. Fig. 2.3 (b)
shows the representative data for this measurement. The capacitance curves consistently

showed three broad peaks over a tip voltage range of O-l.0 V. Fine structure

21

measurements on the millivolt scale showed sharp peaks which are consistent with
single-electron peaks. These are shown in Fig. 2.3 (c). The broader peaks labeled A, B
and C are interpreted as bundles of ~15 electrons entering “donor molecules” formed by
nearest-neighbor silicon atoms.

For both of these experiments discussed in this chapter, the electronic structure of
the dopants is inferred from spectroscopic measurements of systems which give the
average behavior of 100 to several million dopants. However, as devices size shrink
knowledge of the behavior of small numbers of dopants become increasingly vital. In my
experiment I combine both spectroscopic single-electron measurements and actual

imaging of dopants to examine the electronic structure of individual dopants one by one.

22

[arbnnits]

Donor
Layer

V

 

 

 

 

 

x’ . e {I . . . ‘ I
- . fi-J-L—J- .-‘.- .-- -: -.-.- us. -
Base electrode

 

 

EV.
:r‘f/ ‘

Si
Donors

 

[our]

iM iii/VIN!

 

 

chc= 3. 8mV .

‘ V
1

”firm” Mflvfit‘

 

 

 

_l
as: ° I" II “I
0.00 0.50 1.0 0 100 200
Tip voltage [V] Tip Voltage [V1

Figure 2.3 (a) Schematic of the CA1 measurement for the Kuljanishvili experiment [6].
(b) Spectroscopic data showing broad peaks, labeled A, B and C. Each peak corresponds
to roughly 15 electrons. These peaks were interpreted as electrons entering clusters of
donor, effectively forming “donor molecules”. The rrns amplitude of the excitation
voltage was 15 mV. (c) A subsequent measurement acquired with tip voltages indicated
by the rectangular box in (b) showing single-electron peaks. In this case, the rms
amplitude of the excitation voltage was 3.8 mV, as indicated.

23

Chapter 3

Silicon Samples

Silicon is the best known and most frequently used semiconductor [46]. Under exposure
to oxygen a silicon surface oxidizes to form silicon dioxide (SiOz). The stability of the
native oxide and silicon’s large band gap led to it being the dominant semiconductor
material used in electronic devices [47]. The sample used in my experiment was grown
by chemical vapor deposition on a silicon substrate. The dopant in all the layers is boron
which forms an electron acceptor. This means that my sample is a p-type semiconductor.
In this chapter I will first discuss the silicon band structure and then talk about the general
aspects of silicon samples such as growth by chemical vapor deposition, delta doping and
dopant profiling by secondary ion mass spectroscopy. After this I will give specific

details of the sample used in my experiment.

3.0 Silicon band structure

The electronic properties of silicon are completely determined by the comparatively
small number of electrons excited into the conduction band and holes left behind in the
valence band. The electrons will be found in the levels near the conduction band minima
while the holes are confined in the neighborhood of the valence band maxima. The band
structure of silicon, which is the energy (E) versus wave vector (k) diagram, is shown in
Fig. 3.0. The conduction band has three minima, the lowest of which is away from k=0
(called the T point) along the [100] direction. Hence silicon is said to be an indirect
semiconductor. The electrons in this minimum have a 6-fold degeneracy. With regard to

holes, there are two degenerate valence band maxima, both located at k=0 which are

24

‘IJIFZ‘

spherically symmetric. These two bands give rise to light holes and heavy holes. There is
also a third band (not shown in the Fig 3.0) about 44 meV below the valence band
maximum. This band is separated from the other two by the spin-orbit coupling. At
temperatures on the order of room temperature (kBT=0.025 eV) it too can be a significant

source of carriers.

25

1"“

 

Conduction
Electrons

 

 

 

fix I Light and Heavy holes
I

-3I-

 

 

 

I

5 0 5

LIIII] c—- I‘ —) I [100] X
K(nm")

 

Figure 3.0 The band structure of silicon. The horizontal axis shows the k=0 point (I') and
two directions in k-space: [1 l l] (L) and [100] (X). The energy band gap of 1.12 eV is
indicated. The conduction band minimum is displaced in k-space away from the valence
band maximum, making silicon an indirect semiconductor. The valence band shows two
degenerate bands and a third band called the split-off hole band (not shown).

26

3.1. Chemical vapor deposition

My sample was grown using chemical vapor deposition (CVD); in this section I briefly
describe the method. CVD is a synthesis process in which the chemical constituents react
in the vapor phase near or on a heated substrate to form a solid deposit. The technique is
used for depositing thin films of a large variety of materials. The substrate is placed
inside a reactor to which a number of precursor gases are supplied. Fig. 3.1 shows a
simplified schematic picture of a CVD horizontal epitaxial reactor chamber.

A CVD reaction is governed by the thermodynamics and the growth kinetics of
the system. The thermodynamics of the system determines the direction the reaction is
going to proceed while the grth kinetics defines the transport process and determines
the rate control mechanisms [48]. Chemical thermodynamics deals with the interrelation
of various forms of energy and the transfer of the energy from one chemical system to
another. In the case of CVD, this transfer occurs when the gaseous compounds react to
form the solid deposit. A desired CVD reaction will only take place if the
thermodynamics is favorable. This happens when the free energy of the reaction is
negative [49].

Another important aspect of CVD is the growth kinetics and mass transport
mechanisms. In a CVD process the precursor gases are transported from the location
where the gases are supplied to the surface on which deposition must occur. The rate of
deposition can be controlled by the surface reaction rate or the mass transport or diffusion
rate. The boundary layer model has been widely accepted for the description of mass
transport from bulk gas to the substrate surfaces [50, 51, 52, 53]. Fig. 3.2 shows the

formation of a boundary layer above the substrate in a horizontal epitaxial reactor. The

27

gas velocity is zero at the substrate and rapidly increases to a value identical to the
velocity of the bulk gas. The distance in which the gas velocity increases from zero to the
bulk value is referred to as the thickness of the boundary layer a. This layer thickness is
determined by the flow dynamics of the gas mixture in the reactor. In open tube reactors,

such as the one shown in Fig. 3.1, the gas flows by forced convection.

28

 

><

 

 

 

 

 

Wafers Convection around wafers
«g. ,___f Diffusion between wafers
— L

Exhaust cg. a... 4 Gas
1 inlet

r—-—————-—-—————————fi

Cantilever
—

 

 

 

 

 

Resistive heaters

 

Figure 3.] Simplified schematic diagram of a horizontal eprtaxral reactor chamber
showing the wafers stacked in vertical slots and the motion of the precursor gases.

29

.I'“"‘T

 

‘ GAS FLOW VELOCITY

  
 
  

BOUNDARY LAYER
SUBSTRATE

o I

Figure 3.2 Schematic representation of the formation of a boundary layer of thickness 6
above the substrate in a horizontal epitaxial reactor [52].

30

3.2 Delta-doped semiconductors
In delta doping, the dopant atoms are confined to a single atomic layer in the host
semiconductor. Such a situation is illustrated schematically in Fig. 3.3. A S-doping

profile is characterized by the location of the dopant plane and the density of the doping

atoms in the plane. Assuming that the dopants are located at z = zd in the x-y plane, then

the doping profile is given by
N(z) = N206(z — zd),

where the two-dimensional (2D) density denoted by N 20 is the number of doping atoms
in the plane per cmz.

In Section 3.1 we discussed the growth of samples by chemical vapor deposition
(CVD). Here we summarize briefly the basic procedure for the growth of 8-doped
samples. The first step involves the suspension of the epitaxial growth of the
semiconductor. The surface should be atomically flat with minimal atomic terrace steps.
In the second stage this flat surface is exposed to either a flux of elemental doping atoms
or to a flux of the doping precursor gas. The doping atoms would then form bonds with
the host semiconductor atoms. Ideally, all doping atoms occupy substitutional lattice sites
on the semiconductor surface. In the final stage of 5—doping the deposition of the dopants
is stopped and the epitaxial growth of the host semiconductor is resumed.

The density of doping atoms in the monolayer can be determined by the flux of
dopant atoms in the doping precursor and the time that the semiconductor surface is
exposed to the dopant flux. In practice, for most 5-doped semiconductors, the dopants are

not perfectly confined to a single atomic layer. There is often a distribution on the depth

31

of the 5-layer. Surface roughness and other processes such as diffusion, drift and

segregation may contribute to the doping redistribution.

32

.-
iiii

!!!!!

IIIIIII

Epitaxial
layer

} Substrate

Figure 3.3 Schematic illustrations of a semiconductor substrate and an epitaxial film
containing a delta-doped layer [54].

lllll

IIIII
OOOOOO

    

Lattice

33

3.3 Secondary ion mass spectroscopy

The doping profile of any delta—doped sample can be measured by secondary ion mass
spectroscopy (SIMS). In SIMS, primary ions with energies between 1 and 15 keV are
made to impinge on the surface of the material to be analyzed. Part of the energy of the
primary beam goes into ejecting atoms and clusters of atoms from the surface of the
material under consideration (Fig. 3.4). A small fraction of these sputtered-off particles
leave the sample either positively or negatively charged. These secondary ions are then
identified using a mass spectrometer and counted as a function of their mass. Over time
the ejected atoms leave a pit in the sample. Hence data collected as secondary ion counts
versus time can be converted to concentration versus depth [55, 56, 57]. An example of a

SIMS plot is shown in Fig. 2.1(a).

34

~ .i"v— --
Irmat- _.

Primary Ion Secondary
Particles: Atoms, Ions, Clusters

o
\ o Q30
\ 00 0;;0
0'
\ / /
\ / /
O O 0 001300 (5 O 0
Escape 0 O 009 O 00 00
Region 0 O O O 00
01000 ‘OQO oo o'ooo'b oo Cascade
Dopant “0:104:000 0000000 Mixing
Layer 0 o 050(200 0'0 o o0 Regm

00000 00000
000000 0000

Recoil
Implantation

Figure 3.4 The SIMS process. An incoming primary ion generates a variety of secondary
particles from a narrow escape region [54].

35

3.4 Strain issues in boron doped silicon

As strain can lead to energy shifts in semiconductors, its effects on the B-doped samples
studied in this thesis should be considered. Under a strain, the symmetry of a crystal is
lowered in general, resulting in the lifting of degeneracies associated with the energy
levels of dopant atoms [58, 59]. Substitutional boron dopants in silicon exert strain in the
following way: As the smaller boron atom replaces a silicon atom, there is a tendency for
the lattice to contract locally and change the unit cell. However, the silicon lattice resists
contracting, resulting in a tensile strain, as shown in Fig. 3.5 [60, 61]. The amount of this
kind of strain in the lattice is dependent upon the density of the boron dopants. In our
sample the boron dopants make up only 0.02% of the lattice atoms [62]. Interstitial
impurity defects in silicon also introduce strain to the lattice. For example, impurity
defects such as oxygen or carbon can create some local strain which may cause the
shifting of the ground state energy of the boron dopants [58, 59, 63]. Possible effects of

strain on our measurements are discussed in Chapter 7.

36

 

0 Silicon

. Boron

Figure 3.5 A local tensile strain is generated as a boron atom substitutes into a silicon
lattice. AL is the change in the lattice constant due to the replacement of a silicon atom
with a boron atom. (Figure adapted from Reference 60.)

37

3.5 Specifics of my sample

The silicon sample used in this experiment was grown by Roger Loo and Matty Caymax
at the IMEC facility in Belgium [62, 64]. Boron is the dopant for the bottom layer and the
delta—layer. The active structure was deposited by chemical vapor deposition in an ASM

Epsilon 2000 reactor on a Si (001) substrate. It is a layered structure whose active part is
(top to bottom) Si( I5nm )/ 5/ Si( 20nm )/ p+Si( 500nm), where Si denotes intrinsic

undoped silicon and p+Si denotes boron doped silicon with a dopant density of 1.0 x

1019 cm'3; as shown in Fig. 3.6 (b). The overall dimensions of the sample are 10 x 2 x 1
mm. Fig. 3.6 (a) shows a schematic of the sample showing the boron dopants randomly

distributed in the delta—layer and Fig. 3.6 (c) shows a photograph of the sample.

3.5.1 Sample growth

The sample was grown at 700° C at a pressure of 40 torr using hydrogen as the carrier
gas. B2H6 was used as the dopant precursor gas. The dopant atoms were introduced using
the vapor phase doping technique (VPD) [62]. In this technique the dopant atoms are
deposited on the Si surface through the thermal decomposition of the gaseous precursor

(B2H6) in a standard chemical vapor decomposition reactor. The dopant dose increases

linearly with dopant flow and time. With regard to silicon growth, the bottom p+ layer

~500 nm was grown with SiH4; for the other layers SiClez was used. The 5-layer has an
areal density of 1.7 x 10H cm'2 and is almost centered in a 35 nm layer of intrinsic
silicon. This areal density corresponds to a mean distance of ~25 nm between the boron

dopants. The width of the S-layer is about 2 nm wide and it has a peak bulk

38

4"“

concentration of ~5 x 10'7 cm'3. The p + layer is heavily doped, N 310 = 1019 cm'3 and

serves as a bottom contact layer, where N i!) is the 3D density of the boron acceptor

atoms. The deping profile of our sample was measured by secondary ion mass
spectrosc0py (SIMS). This measurement was done by the sample growers and is shown
in Fig. 3.6 (b).

The background contamination of oxygen or carbon during growth was not
measured for our particular sample. However the sample growers carried out weekly
systematic process control tests (SPC—tests) during production. For samples grown under
identical conditions to our sample, the measured background contamination of oxygen

and carbon is far below the SIMS detection limit [64].

39

 

8 concentration (cm'3)

5 20 40 60
Depth 2 (nm)

Figure 3.6 (a) Schematic of my sample showing the random distribution of boron dopants
in the delta-layer as blue dots. The area of interaction with the tip is indicated in red. The
silicon layer above the delta layer is 15 nm thick (1); the silicon layer below the delta
layer is 20 nm thick (2) and the heavily-doped p+ substrate layer is 1000 nm thick (3). (b)
SIMS plot of our sample showing that the 8—layer is about 15 nm from surface. (0) A
photograph of the actual sample. The sample is a rectangular silicon crystal of
dimensions 10 x 2 x 1 mm; the gold wires shown bring the bias voltage to the heavily-
doped p+ layer.

40

3.5.2 Valence band diagram of sample
A boron dopant is an acceptor atom in silicon. In other words, when a boron atom

replaces a silicon atom in the lattice, it readily gains an electron, or equivalently, loses a

hole; this state is denoted as the 3‘ state. Can the hole be reunited with the acceptor?
The answer is “yes”. In fact the negative B ion can bind one or two holes. When
occupied by one hole, the acceptor becomes neutral and the energy of the hole can be
approximated as a hydrogen-like atom. The ground state in this case is known as BO state.
The binding energy of the hole in the B” state is E0 = 45.7 meV [10, 20] this state is
sufficiently deep such that it is far below the Fermi level of the bottom contact layer. The
system can also accommodate a second hole; this hole will experience a repulsive energy

with the first hole, thus forming a state with reduced binding energy, E I, known as the BI
state. The goal of my experiment is to probe locally holes entering the B+ state. (This

state is the acceptor counterpart of the donor state known the D_ state.) The state can be

described with the following equation: B0 + h = B“.

Figure 3.7 shows a semi-quantitative schematic of the expected valence band for

our sample relative to the Fermi levels of the tip and bottom contact layer, labeled as

metallic substrate. We see that the weakly-bound B+ state is expected to be only a few
meV from the Fermi level of the substrate. An important parameter is the effective

tunneling barrier height though which the hole must tunnel. This parameter is denoted as
¢B in Fig. 3.7. It is given by the valence-band contribution to the band-gap narrowing of

the bottom contact layer, minus the Fermi energy Ep. The value is expected to be 11.7

meV [10, 65].

41

Vacuum
Harrier

 

 

 

Tip Accumulation \lclzlllic
A" Well substrate
(I)B=10meV
.__._ EF
'6Vtip
E
Z '_ Ev
E0 —

 

Figure 3.7 Schematic of the profile of the valence band edge of my sample. The heavily-
doped bottom contact layer acts as a metallic substrate. Holes can tunnel onto the boron
atoms from this layer, as indicated by the red arrow. The gap between the tip and the
sample forms a second higher barrier, inhibiting charge from tunneling directly onto the
tip. 50 is the binding energy of the neutral boron state (3”) with one bound hole; E+ is the
binding energy of the of the positive boron (3“) state with two bound holes; ¢B is the
effective work firnction or tunnel-banier height.

42

Chapter 4

Design and Construction of Microscope

This chapter describes the design and construction of the scanning unit and top loading
probe of my CAI microscope. This construction was done with the help of staff at the
physics machine shop. The design of my scanner is based on the Besocke-type scanning
tunneling microscope. The CAI microscope can be operated in three different modes.
These modes, which will be discussed in detail in Chapter 5, are the STM mode, the
capacitance mode and the Kelvin probe mode.

The main components of the scanning unit are the base plate, body, door and the
sample holder, all of which are made from stainless steel. The main goal of the design is
to have a microscope that is stable and has little thermal drift. The scanning unit is
compact and rigid which makes it less sensitive to external vibrations. The combined
effect of mechanical vibrations and thermal drift of the system must be extremely small
to allow the tip to stay above a single dopant atom for many hours. The scanning unit is
rigidly attached to the top loading probe. The top loading probe can be divided into three
main parts; the vacuum lock, the sliding insert and the top cone. These parts are also

constructed from stainless steel.

4.1 Scanning unit

This section describes the scanning unit in detail. Fig. 4.0 (a) shows a blown out drawing

of the scanning unit and Fig. 4.0 (b) is a photograph.

43

r‘2-Wflo- .«

4.1.1 The base plate

The base plate is a rigid steel plate of diameter 3.00 cm and thickness 0.50 cm. At the top
side of this plate we machined four sockets that hold four identical piezotubes. Each
piezotube has four quadrants to which electric signals could be applied, as described in
detail in Section 5.2. On one end, the piezotubes are soldered to the bottom plate while
on the other end they are free and can be actuated in x, y and 2 directions by applying
appropriate electric signals. We used indium solder to bond the piezotubes to the base
plate. Fig. 4.0 (a) shows the base plate and the arrangement of the piezotubes on the base
plate.

The three outer piezotubes are arranged in a triangular configuration and act as
the sample carriers. The fourth piezotube is located at the center of the triangle and acts
as the scanning piezotube. We used custom-made piezotubes made from lead-zirconium-
titanate (type PZT-SA). The tubes are of length 1.91 cm, diameter 0.48 cm, with a wall-
thickness of 0.09 cm. The scanning and carrier piezotubes are all of equal length. Equal
length piezotubes are important for thermal stability of the scanning unit, as they expand
and contract identically as the temperature changes.

The free end of each of the three carrier piezotubes are soldered to copper feet
with stainless steel balls attached to them. We used a stycast mixture for bonding the steel
balls to the copper feet. The balls are highly polished for low friction performance during
the approach of the sample. The scanning piezotube has a miniature L-shaped copper cap
soldered to it. This copper piece holds the sensor circuit chip with the tip rigidly attached
to it. Details of the sensor circuit will be given in Section 4.6. On each of the four

piezotubes we soldered six thin Kapton-coated copper wires which carry electrical

44

W”. ' v‘!
A .J

Sample 1%;
/v ' .

3 Sloped ramp
3 Sample holder

  

(a) Connecting flange

I» f, ' Body

7 I 8 Stainless steel
, 7 ball

I->:-_' I Door
, , ' Copper foot

  
 

Piezotube
Base plate

1cm

.. ‘— Screws

Figure 4.0 (a) A blown out schematic of the scanning unit showing the main parts. The
diameter of the scanning unit is 3.00 cm and the total length of the compact assembly is
8.04 cm. The sample holder sketch shows the sample and the slopped ramps. It is a
stainless steel disk of outer diameter 2.70 cm and thickness 0.70 cm. The base plate
schematic shows how the four piezotubes are arranged. (b) A photograph of the scanning
unit; the door is removed to show the copper feet onto which the sample holder rests.

45

 

signals. Four of the wires are for each of the four outer quadrants; the fifth wire is for the
interior, which is held at ground, and the sixth wire goes to the copper foot on top of the
piezotube. For the three carrier piezotubes, this sixth wire brings the bias voltage to the
sample, whereas on the scanning piezotube the wire is connected to ground. Great care
must be taken when soldering the wires to the piezotube quadrants. Any stray soldering
flux that lands on the space between the quadrants is a potential source of an electric
short and must be thoroughly cleaned. Fig. 4.1 is a photograph of the top and bottom
sides of the base plate. The top and bottom of the base plate has grooves that are 0.30 cm
deep into which all the 24 wires from the piezotubes are neatly packed. The wires are
then routed into a vertical groove on the side of the body of the scanning unit and

ultimately run the length of the probe to the top cone.

46

 

Figure 4.1 Photograph of the base plate (diameter 3.00 cm and thickness 0.50 cm)
showing the top and bottom sides. In this photograph, only a few of the connecting wires
are shown and one carrier piezotubes is missing.

47

4.1.2 Body of scanning unit
The base plate is attached with three screws to the body, which is a solid cylindrical-
shaped piece as shown in Fig. 4.0 (a). Due to the compact design of the scanning unit, it
can be difficult to route and attach the four coaxial wires that control the sensor circuit.
In my design the body is tailored to provide easy access to the sensor circuit and to these
wires. The length of the body is 7.54 cm and the inside diameter is 3.10 cm. In Fig. 4.2
two of the cooper feet on the carrier piezotubes are clearly shown and the L-shaped
copper foot of the scanning piezotube is also visible. The small GaAs chip that holds the
sensor circuit and the tip is shown together with the four wires that connect the sensor to
the control electronics.

The sample holder is a stainless steel disk of thickness 0.70 cm and outer diameter
2.70 cm which rests on the three carrier piezotubes. It has a Teflon ring on the outside
which prevents electrical contact between the sample and the body. On the bottom of the
sample holder are three sloped ramps as shown in Fig. 4.0 (a). Each ramp is tilted an
amount of about two degrees with respect to the horizontal. Fig. 4.2 shows a picture of
the sample holder sitting on top of the carrier piezotubes. Also shown in the picture are
the Teflon ring and two 0.3 cm deep grooves on opposite’s sides of the sample holder.
The purpose of the grooves is to provide for easy handling of the sample holder using a
two-pronged fork. To keep the sample holder in place and also for the purposes of
protecting the sensor circuit, we machined a stainless steel door which is screwed onto

the body. The door is shown in Fig. 4.0 (b).

48

When all the different parts of the scanning unit are assembled, the whole
scanning unit is then rigidly attached by a connecting flange to a 2.00 m long top-loading

probe for subsequent positioning of the unit inside the 3 He cryostat.

49

Sample holder
Teflon ring

TIP Copper foot

 

Tip holder

Fig. 4.2 A close-up image of the tip holder and sample holder. In this picture the copper
feet and the steel balls on top of two of carrier piezotubes are clearly shown. A Teflon
ring surrounds the sample holder, preventing it from making electrical contact to the body
of the scanning unit. The tip is mounted at the top comer of a GaAs chip that also holds
the sensor circuit. The sensor circuit is controlled by four coaxial wires which emerge
from small holes to the left and right of the tip holder (not discemable in the picture).

50

4.2 Top-loading probe

The top-loading probe can be divided into three main parts: a vacuum lock, a sliding
insert and a top cone. Fig. 4.3 (a) shows a schematic of the probe. The vacuum lock is a
stainless steel cylinder of length 47.50 cm and inner diameter 5.00 cm. On the top end of
the vacuum lock there is a sliding seal and a pumping port. The seal ensures that no air
enters during the sliding of the insert. At the bottom end of the vacuum lock there is an o-
ring that locks to the gate valve of the commercially purchased 3 He cryostat. A sliding
insert of length 2.00 m and inner diameter 1.30 cm runs through the vacuum lock. The
seaming unit is attached to the bottom end of this sliding insert. Just above the scanning
unit there is an important part of the top loading probe. This is a copper disk called the 1
K pot wedge. Its purpose is to make a good thermal contact between the scanning unit
and the 1 K pot. Details of the 1 K pot will be given in Section 4.5. On top of the probe
is a cone which houses 36 electrical feedthroughs (a 24 pin Fisher connector and 12
coaxial SMA connectors). Fig. 4.3 (b) and Fig. 4.3 (0) show photographs of the cone

and the top loading probe respectively.

51

Coaxial SMA connectors

Top cone

24—pin
Connector

 

=l"\ Slidingseal

Pumping port

Vacuum
Lock

 

 

(C)

 

 

 

<-— Gate valve

‘— l Kpot wedge

F ‘— Scanning Unit
'5

Figure 4.3 (a) Schematic of the top loading probe. The total length of the vacuum lock
and the sliding insert is about 2 m. (b) A photograph of the top cone showing the 12 SMA
connectors attached to it. (c) A photograph of the top loading probe placed on top of the

3He cryostat ready for positioning of scanning unit into cryostat.

 

52

4.3 Wiring and electronics

The wiring of the system is divided is into three parts: outside cables connecting
microscope to control electronics, wires in the top-loading probe and wires in the
scanning unit. Since wires connected to the microscope should not transmit vibrations,
all the outside wires are thin and flexible. The 24 wires that go to the piezotubes are
Constantine wires of thickness 0.6 mm. They are all put together in the form of a 12 twin-
pair ribbon cable that starts at the Fisher connector on the side of the cone and terminate
at a connector inside the vacuum lock. From this connector, thin Kapton-coated coppers
wires were used to make the connection to the piezotubes. The wires for the tunneling
current and sensor circuit connections are all shielded stainless steel coaxial cables. They
start from SMA connectors at the top of the cone and terminate at some connectors inside
the probe. All together there are 28 wires that go into the scanning unit. The intermediate
connectors inside the probe allow the scan head to be dismounted without the disruption
of any cables. Our microscope is operated using a commercially purchased scanning

control and data acquisition system from RHK Technology, Inc.

4.4 Noise isolation

Noise isolation is one of the main challenges we faced in our measurements. The noise
can be electrical or mechanical. With regard to electrical noise, I carefully shielded the
charge sensor control and measurement leads, all of which are coaxial; moreover ground
loops were eliminated from the microscope and control system. The following section

addresses the issue of mechanical noise.

53

4.4.1 Mechanical stability of the microscope
Mechanical noise, e.g. vibrations, originates from external building vibrations as well as
acoustic waves in the air. When operating the microscope in tunneling mode the
mechanical vibrations should not produce variations in tunneling current of more than a
few percent of the set value. This requires tip-sample mechanical stability of much better
than ~ 0.1 A. Our scanning unit design coupled with two stages of vibration isolation
achieves this stability. The compact scanning unit design gives high normal mode
vibration frequencies of roughly 2 kHz which are not easily excited by room vibrations.
Moreover, the two-stage vibration isolation platform that supports the system decouples
the scanning unit from floor vibrations. The first stage of the isolation platform consists
of four pressurized rubber springs and the second stage consists of four optical table legs
as shown in Fig. 4.4. These two stages result in the lowest normal-mode vibration
frequency of the platform of about one hertz. This is sufficiently far from room vibrations
~10-20 Hz. Acoustic vibrations transmitted through the air also affect the stability of the
tunneling current. In order to counter this, the room in which the experiments were
performed is cushioned with sound absorbing padding on the walls. The padding consists
of patterned sheets of foam rubber material arranged in checkerboard pattern, which
significantly attenuates reverberation.

These precautions allowed me to eliminate any extra noise while performing the
capacitance measurements. In other words, I achieved noise levels consistent with the

intrinsic noise of the charge sensor amplifier, as discussed further in Chapter 7.

54

Air springs

    
  

".2”. n. y s t: l. . 3 :1
a 5A.: AJ than.

 

E. j
m, a. -. .w» - .nn»: - w
“223 'R 1.3.1,?! ”3:23:13; {53“
«H ' I ls ' '

><

w

 

 

r’ Optical table legs

 

 

 

 

?

Scanning unit

 

 

 

 

 

 

 

Figure 4.4 Schematic of the two-stage vibration isolation system used in our
measurements. The hatched bars represent two rigid platforms.

55

4.5 Cryogenic system
The 3 He cryogenic system we used for our experiments can be operated at room
temperature as well as other selected low temperatures. Most of the data discussed here
were obtained at the liquid 4He boiling temperature of 4.2 K. However some preliminary
and testing data was obtained at room temperature and at 77 K. The probe which carries
the scanning unit, the sample and the charge sensor is loaded onto the top of the 3H6
cryostat and is inserted slowly. The process of lowering the probe takes approximately 5-
6 hours. It is important to protect the fragile ceramic material of the piezotubes from
thermal stress; otherwise the piezotubes may break or become depolarized. The
schematic picture of the cryostat is shown in Fig. 4.5

The principle of operation of a 3 He cryostat is relatively simple. The whole
cryostat is cooled in stages from room temperature to 4.2 K. First, we cool the cryostat
down to 77 K with liquid nitrogen. Then we remove the liquid nitrogen and fill the
cryostat with liquid 4H6. The sample space is kept under vacuum (~4.7 x10'6 mbar)
during the cool-down procedure and for the subsequent measurements. Outside the
sample space and attached to its outer wall is a small donut like chamber called the l K
pot. The 1 K pot is filled with liquid 4He and it can be pumped to maintain a temperature
of about one Kelvin. For most of my experiments it was maintained at a pressure of one
atmosphere and a temperature of 4.2 K. This allowed me to eliminate vibrations induced
by the mechanical pump. The sliding insert has a copper wedged plate that makes good
thermal contact with the 1 K pot (see Fig. 4.3 (a)). The temperature of the sample

equilibrates to that of the l K pot, 4.2 K for most of my measurements.

56

Probe entry port Transfer tube entry

Port

   
  

Gate valve

Helium exhaust
Port

Magnet
//

,' /'

 

 

 

 

 

2 Liquid delivered to
l ' ' \J\ Bottom of LHe

- 7-... -x ' >\ reservoir

\‘\ /

 

Figure 4.5 Schematic of the cryostat. The cryostat is about 60 cm wide and 1.5 m long
and can hold up to 60 L of liquid “He. During the experiment, the probe resides inside the
cryostat and the scanning head is situated at the bottom of the cryostat surrounded by
superconducting magnet.

57

4.6 The cryogenic charge sensor circuit

At the heart of the charge accumulation imaging technique is the cryogenic charge
sensor. Our charge sensor is made from high electron mobility transistors (HEMTs),
which are field-effect transistors. The HEMTs that we used are unpackaged commercial
devices (model FHX35X) manufactured by Fujitsu. The active channel is formed at the
interface of GaAs and AlGaAs. In general to allow conduction, semiconductors need to
be doped with impurities to generate mobile electrons in the channel. However the
presence of the dopants lowers the mobility, as the charges collide with impurities that
were used to generate them in the first place. The HEMT use modulation doping to
resolve this contradiction. This is accomplished using the heterojunction of a highly-
doped supply layer of AlGaAs and a non-doped channel layer of GaAs. The electrons
generated in the n-type AlGaAs move into the GaAs layer, falling in the potential well
created by the different band gaps of the two materials. The electrons are confined in this
quantum well, typically ~20 nm away from the doped AlGaAs. The effect of this is to
create a very thin layer of highly mobile conducting electrons. This layer is called a two-
dimensional electron gas.

As with all other types of FETs, a voltage applied to the gate alters the
conductivity of this layer. We chose the HEMT as our active device due to their superior
low temperature performance as well as low input capacitance (channel-gate mutual
capacitance). HEMTs do not suffer from carrier freeze out at low temperatures because
the carriers are not thermally generated. Field-effect transistors are typically operated in
saturation mode for which the current is approximately independent of voltage. However

the power dissipated in this regime is typically in the miliwatt range. This would be large

58

enough to heat up the sample surroundings and may cause thermal instabilities. We
operate our circuit in the low voltage regime (ohmic region) far from saturation, indicated
by the circle in Fig. 4.6. In this region, the power dissipation is decreased by 2-3 orders of
magnitude compared to saturation mode operation [66]. For my typical Operating
parameters the power dissipated is about 0.3 uW. Figure 4.6 shows the drain I-V
characteristics of our HEMTs, indicating the regime in which we operated. Fig. 4.7 (a)
shows the schematic picture of the cross section of the HEMT and Fig. 4.7 (b) is a
photograph of the device.

Our charge sensor circuit uses two HEMTs, the measurement HEMT and the bias
HEMT, as shown in Fig. 4.8 (a). The gate of the measurement HEMT is biased through
the bias HEMT. The role of the bias HEMT is to act as a very high resistor, preventing
the charge induced on the tip from leaking to the left. The inclusion of the bias HEMT is
the major improvement from the original design, which used an on-chip thin film resistor
[67, 68, 69]. The bias HEMT couples less than 0.1 pF of additional capacitance to the tip
electrode. In contrast the thin-film resistor added 1.0 pF of stray capacitance; this

significantly reduces the Circuit’s gain, as discussed on the following pages.

59

 

 

 

 

 

 

 

 

 

€40

E ///"’“""

a 20 ////

a .. //.._//

D 6
1 2

Drain-source Voltage (V)

 

VGs [V]

0.0

Figure 4.6 Drain-source I-V characteristics at different gate voltages VGs of our HEMT
[70]. The circles mark the voltage range in the linear regime in which we operate the

device.

 

 

 

 

. *‘ Ohmic contact
n-Alea]-xAS
Undoped GaAs
\
'E . . . z; E L
‘ Semr-msulatmg Cllectrirlr ayer
(100) GaAs substrate

 

 

 

(b)

 

Figure 4.7 (a) Schematic picture of the cross section of a HEMT [70]. (b) A photograph
of the Fujutsi model FHX35X HEMT. The size of the device is ~ 500 um across.

61

The signal of interest which I want to measure originates from charge moving
below the apex of the tip in response to an excitation voltage Vexc applied to the base of
the sample. The standard capacitor C 5 is used to subtract away the stray (or background)
capacitance, arising mostly from the electric fields from the sample which do not
terminate on the apex of the tip. This is accomplished by applying an AC voltage,
Vbalance to CS. Vbalance is of the same amplitude as Vexc but exactly 1800 out of phase.
The typical value of the C5 is about 20 tF (femtofarad). We can model our circuit with a
simple voltage divider as shown in Fig. 4.8 (b). We assume that bias HEMT is tuned to
assure that no current leaks to the left. If we consider now only the measurement HEMT,

we see that the circuit acts like a simple voltage divider. Here Vin is the input voltage, in
our case Vexc , and Vow is proportional to the measured voltage me/G. Here G is a gain
factor.

To construct the voltage divider equation, we label C in as the input capacitance,
the capacitance between the gate and the source-drain channel of the HEMT. C mm is

the tip-sample capacitance, proportional to the tip charge, which is the quantity that we

need to measure. We can write the following expression:

Vout : Vmeas _ 22 41

V,- GVexc 21+22’

 

 

 

1 l
where Z = —— and Z = .
l . C 2 .
“0 'meas lein

62

(a) (b)
w Vbalance Vin

a! £5

HEMT - -- HEMT 2'

(”J—L. ,_..- To lock-in

VCXC Vou

VCXC

 

 

 

‘ Vmeas

 

 

 

. T... H .

 

 

 

1‘

Sample

Figure 4.8 (a) Schematic of the charge sensor circuit. The charge sensor circuit uses two
HEMTs, the measurement HEMT is to the right and the bias HEMT is to the left. The
gate of the measurement HEMT is biased through the bias HEMT. The role of the bias
HEMT is to act as a very high resistor, preventing the charge induced on the tip from
leaking to the left. C5 is the standard capacitor and Vin is the rms value of the applied
AC excitation to the sample. The AC excitation applied to C5, is of the same amplitude
but 1800 degree out of phase with the excitation voltage that is applied to the sample. The

charge sensitivity of the circuit is 0.15 electron/VH2. (b) The sensor circuit can be
modeled as a simple voltage divider.

63

Cmeas is typically about 1 aF, whereas Cm is a much larger 0.4 pF. Hence the typical
value for Z; is much smaller than Z1 therefore we can neglect Z2 in the denominator of
above equation. The voltage measured by the lock-in amplifier V meas is given by the

following relationship.

Cmeas . 42

 

Vm eas = C’ Vexc .
m

The gain of the circuit G depends on the electronics external to the circuit. For my

setup G is 1.0; therefore I can write

Vmeas z Cmeas 43

Vexc Ci
The key point is that low input capacitance means larger signal. The input capacitance of
the HEMT is only 0.3 pF. It is important to keep in mind that the stray capacitance adds
to this parameter. In other words, the geometric self—capacitance of the entire device must
be less than ~0.l pF. This sets the 1 mm as the size limit of the tip plus circuit system.
Finally, it is straightforward to convert from the measured capacitance to the rrns charge

induced on the tip:
qfip : CmeasVexc —_- VmeasCin a 4.4

where we have assumed an equality for Eq. 4.3.

Our circuit is ultimately limited in sensitivity by the Johnson noise in the bias
HEMT, which gives a charge sensitivity of 0.15 electrons/VH2 at 4.2 K [66]. However, if
the input capacitance is too large it directly reduces the value of the signal so that other

sources of noise will start to dominate, decreasing the sensitivity.

64

Chapter 5
Methods and Experimental Techniques

5.0 Introduction

In my experiment, I study the electronic structures of dopants which are buried about 15
nm below the surface of silicon. The usual STM method cannot probe subsurface
dopants. However a modification to the STM, utilizing the same scanning unit, can be
used to probe structures below the surface. This technique is called Charge Accumulation
Imaging (CAI) [9, 71, 72, 73, 74]. The measurement basically consists of monitoring the
image charge induced on a metallic tip due to an excitation voltage applied to the sample.
Unlike the STM tip, our tip is connected to a charge sensor. Some fraction of electric
field lines emerging from charges entering the subsurface dopants are terminated on the
apex of the sharp tip, hence inducing charges of opposite sign on the tip. In this way
charge flowing in and out of the sample induces charge to flow in and out of the tip. By
monitoring this signal, we

can locally map out the charging of the dopants inside the dopant layer. This chapter

describes the methods and experimental techniques used in all our measurements.

5.1 Voltage lever arm

An important aspect of the CA1 measurement is the voltage lever arm of the sample (also
referred to as the a-parameter). Changing the voltage of the tip relative to the substrate
shifts the potential in the dopant layer. However the potential shift is less than the full
voltage applied to the tip. The value of the lever arm parameter depends on geometry of

the sample and it sets the reduction factor between the voltage applied to the tip and the

65

voltage actually felt at the dopant site. As a simplified example, let’s suppose that the
dielectric constant of the material is one and that a d0pant is located below the tip and

midway between the substrate and the tip. Lastly, let’s assume here that the tip acts as a
parallel plate. Now if one volt is applied to the tip, i.e., V,,-p=l V, with the substrate

grounded, the potential at the dopant is Vdopan,=0.5 V. In this case, one-half of the voltage
drop occurs between the tip and the dopant and one-half of the drop occurs between the
dopant and the substrate. Hence the voltage lever arm is a = Vdopam/ V,,-,, = 1/2.

A realistic model must consider that the dopant is actually immersed in a
dielectric which partially screens the electric field. With regard to the tip, this electrode
can be modeled as a sharp needle with a radius of curvature at the apex of tens of
nanometers, positioned 1-2 nm above the dielectric surface. Such a model is considered
in detail by Tessmer and Kuljanishvili [75]. Applying the model to my sample, given the
Si dielectric constant of 11.5, the 15 nm depth of the dopant layer, and the 20 nm
separation between the dopant layer and substrate electrode, yields 0: ~ 0.15. The precise
value of or depends on the distance between the tip and the dopant in question, and the
size of the gap between the tip and the surface. (The lever arm is 11.5 times more
sensitive to variations of the gap compared to distance variations within the sample;
essentially the absence of the dielectric in the gap means the electric field gradient is 11.5

times greater.)

5.2 N ulling voltage
Another key parameter of the system is the contact potential between the tungsten tip and

silicon sample which arises from the work function difference of the two materials. This

66

parameter represents an offset voltage for which we must compensate. For clarity, I shall
refer to the parameter as the nulling voltage, me. This is the effective zero voltage of the
tip for which there is no electric field terminating on it. In principle, Vnull can be
calculated by taking the difference of work function measurements for the two materials,
which can be found in the literature. (For my system, this approach yields a nulling
voltage of about 280 mV). However, in practice for layered semiconductors the quantity
must be measured as it can vary considerably depending on the density of permanently

trapped charges [76]. Our measurement of the nulling voltage is discussed in Section 7.2.

5.3 Control of tip-sample separation

The motion of the sample holder with respect to the tip, which is attached to the central
piezotube, is achieved by the applying a saw tooth wave to the appropriate quadrants of
the carrier piezotubes. During the fast part of the pulse the sample holder slides with
respect to the steel balls due to inertia. During the slow part of the saw tooth pulse the
sample holder moves along with the supporting elements due to the fiiction between the
ramps and the steel balls. Depending on the signs of the pulses applied to the different
quadrants, the sample holder can be made to move in the vertical and horizontal
directions. The amplitude and frequency of the saw tooth signal control the step size of
the sample motion from a few nm up to a mm per step. Coarse movement on a scale of
millimeters can be achieved horizontally. Coarse movement on a scale of hundreds of
microns can be achieved in the vertically direction by applying the saw-tooth in a
rotational sense, causing the sample holder to rotate. Figure 5.1 shows the quadrants of

piezotubes and the sequence for vertical motion.

67

5.4 Modes of operation of microscope

Our microscope can operate in three totally independent modes. These are the tunneling
mode, capacitance mode and Kelvin probe mode. Figure 5.2 shows a schematic of the
three modes. Our microscope is designed to operate inside a standard 3He refrigerator
manufactured by Oxford Instruments. All capacitance mode measurements were done at
a temperature of 4.2 K as described in Section 4.5 (some calibration and testing
measurements were done at 77 K). For every data run we start by scanning the tip in the
tunneling mode to check that the surface is clean and there are no major electronic
defects. If any defects or dirt on the surface are found, we can easily move the tip to a

different part of the surface.

68

 

 

 

 

 

 

 

 

Figure 5.1 (a) Schematic representation of a typical scanning probe piezotube with four
quadrants. (b)The top view. The quadrants denoted with +X, +Y, —-X, and —Y are used for
mechanical control. (0) Schematic of the three carrier piezotubes indicating motion in a
rotational sense.

69

 

 

 

 

 

 

 

 

 

Charge Kelvin Tunneling
Accumulation Probe

Figure 5 .2 Three modes of operation of the microscope: charge accumulation mode
(left), Kelvin-probe mode (middle) and tunneling mode (right).

70

5.4.1 Tunneling mode

The microscope can be operated in the conventional STM mode by suspending all AC
voltages. To do this we bypass the charge sensor and attach the tip wire to a standard
STM DC current-to-voltage converter. In the surface topographic imaging mode, the tip
is scanned over the surface of the sample while the tunneling current is maintained
constant. A fixed bias voltage is applied and the desired tunneling current is set. As the
tip is scanned across the surface of the sample, variations in the topography of the sample
and the underlying electronic structure affect the tunneling current [77, 78]. In order to
keep the pre-selected value of the tunneling current constant, a feedback loop provided by
the microscope control electronics is used to adjust the position of the tip above the
sample surface. The feedback voltage, corresponding to the vertical position of the tip is
recorded, reflecting the surface topography. The set current and the set bias voltage
depend on the nature of the sample. The tunneling current is typically set to a range of
0.1-1.0 nA for graphite or silicon samples.

The tunneling mode proves useful for finding any defects or absorbed particles on
the surface of the sample. It can also accurately determine the tilt of the sample plane
relative to the horizontal so that one may apply the appropriate plane compensation
parameters to the scan piezotube to correct for this effect. This is particularly important
when we wish to proceed to the capacitance mode since in this mode the microscope
scans at a constant tip height with the feedback loop turned off. Any uncompensated

slope may cause the tip to crash into the sample.

71

5.4.2 Capacitance mode

Charge Accumulation Imaging technique is done in the capacitance mode. With the tip
positioned a few nanometers from the sample, the sample and tip system can be thought
of as a parallel plate capacitor. Any movement of charge in the sample results in a
changing image charge on the tip, which we monitor. Figure 5.3 shows aspects of the
CA1 technique.

Positioning of the tip is first achieved in tunneling mode, where a constant tip
sample distance is controlled by a tunneling current feedback 100p. Once the tip is in
range (i.e. a tunneling current is observed), we use the z off-set of our piezotubes to
move the tip away from the sample for about 30—40 nm and then move it laterally
sufficiently far from the tunneling location to an unperturbed location. Then with the
feedback off, the tip is brought closer to the surface using manual fine approach. The
measurement proceeds by recording the signal in phase with the AC excitation voltage
with the aid of a lock-in-amplifier. If the measurement is obtained while the tip is
scanned close to the sample surface one obtains a charging map. The dopants sites can
be in principle marked by the increase in image charge as shown in the sketch of Fig. 5.4
(b).

We may also carry out a different measurement by keeping the tip at a fixed
position above the surface while varying the bias voltage. This measurement yields
capacitance-voltage curves. The resonance voltages will be found as peaks in the
charging as a function of the bias voltage. A sketch of a resonance peak is shown in Fig.
5.4 (a). To be clear, our plotted C-V curves represent the change in capacitance as a

function of voltage. In other words, we subtract away the background capacitance from

72

 

 

Charge (arb.units)

 

 

 

Metallic layer

Figure 5.3 (a) Schematic of Charge Imaging method. Charge entering the sample induces
image charge q,,-p on the tip, which is monitored using a cryogenic charge sensor. The
charge sensor has a sensitivity of 0.15 electrons/VH2. (b) An example of charging
structure measured in a GaAs-AlGaAs two-dimensional electron system using this
technique. The dark spot is an area of reduced charging which resulted from an effective
reduction in the concentration of Si dopants [9].

73

.-
'-

(a) (b)

1

 

Capacitance

 

Vtip

Figure 5.4 A schematic representation of the anticipated charge accumulation data. (a)
Spectroscopic data. The sharp peak represents the voltage for which an energy level in
the dopant layer is in resonance with the Fermi level of substrate. (b) A representation of
scanning capacitance image with the tip voltage fixed near the peak voltage. Yellow
indicates a high capacitance and dark red indicates a relatively low signal. We anticipate
that distinct high-capacitance features will mark the locations of the subsurface dopant
sites within the plane.

74

the plotted curves using the bridge circuit described in Chapter 4. To compensate for
electronic noise and drift effects in both types of measurements, several curves and
capacitance images are averaged together to achieve an acceptable signal-to-noise ratio.
At the end of every capacitance image data run we convert to the tunneling mode to study
the surface of the sample; this allows us to check if there are any features in the

topography that may lead to artifacts in the capacitance images.

5.4.3 Kelvin probe mode
The Kelvin probe measurement is a technique used to measure the contact potential
between two surfaces that are brought in close proximity [9, 79, 80]. This method

requires the tip to oscillate in order to modulate the tip-sample separation; this in turn

modulates the tip’s charge giving a signal Aq,,-p. Ideally, one can find VnulI from the tip

voltage for which Aq,,-p vanishes. Unfortunately, for my system I have found that the stray

electric field emanating from the electrodes of the oscillating piezoelectric tube results in
an artifact in the measurement. As an alternative, we have developed a method to
measure the nulling voltage that does not require the tip to oscillate. Instead, we rely on

the step structure that appears in the capacitance measurement, as described in Section

7.2.

5.5 Procedure for analyzing the C-V curves

As part of my analysis, I compare the capacitance peaks in the C-V curves with
calculated fitting curves. These fitting curves are based on single-electron capacitance
peaks, which are essentially the convolution of semi-elliptical peaks with the derivative

of the Fermi distribution function scaled by the lever arm parameter a [72, 81, 82]. As

75

mentioned in the previous section, the measurement here utilized an amplifier output
low-pass filter with time constant 1:, which tends to further broaden the peaks. This can be
incorporated into a calculation by performing a simple integration. To generate the
fitting curves, we used a modeling routine that superposes 4-6 single-electron peaks and
then integrates the resulting curve appropriately. The excitation amplitude, temperature

and time constant were fixed parameters, set to the experimental values of Vac = 3.7 mV

V peak

tip were considered as free and hence

and T=4.2 K, 1 =3.0 s. The parameters aand

varied to achieve the best fit. We find that this procedure reproduces the measured curves

well. (See for example, Fig. 7.6 in Chapter 7.)

76

Chapter 6
Testing and Calibratidn of the Microscope
6.1 Calibration of the scanning unit
After putting our scanning unit together, the first thing to do is to test if the microscope
can achieve simple STM atomic resolution. A graphite sample was used for this purpose.
Obtaining atomic resolution images of graphite demonstrates that the scanning unit is
stable. Graphite crystals consist of layers of carbon atoms. In each layer, carbon atoms
are covalently bonded into hexagons, while different layers are connected together by
weak van der Waals forces. Carbon atoms on the graphite surface appear in the STM
images as a triangular lattice which contains every other atom, not as honeycomb rings as
would be expected. This asymmetry was explained by Tomanek as a purely electronic
effect due to the interlayer interactions [83]. Essentially there are two nonequivalent
carbon atom types in each layer. Carbon atoms of one type have neighbors directly
beneath them in the underlying atomic layer as shown in Fig. 6.0 (a). Carbon atoms of the
second type do not have neighbors directly beneath them. Band structure calculations
predict that atoms visible in the STM image are ones that do not have the subsurface
atom beneath them. In Fig. 6.0 (a), open circles represent atoms that appear in an STM
image. Fig. 6.0 (b) shows carbon atoms in the STM image of graphite I obtained at a
tunneling current of 0.5 nA and bias voltage of 50 mV. The scan range is 2.00 x 1.80 nm.
This image clearly demonstrates that our scanning unit is stable at the atomic level.

Once atomic resolution is obtained, the next step is to calibrate the scanning
unit's scan range. The purpose of the calibration is to know the sensitivity of the

piezotubes. This calibration was done at room temperature with the system in the

77

 

tunneling mode. For the lateral sensitivity of the piezotubes graphite samples were used.
An STM image of graphite with good atomic resolution is used to obtain the correct
value for the x and y scan range, knowing that graphite plane lattice constant is 2.46 A.
The STM image of graphite used for the x and y calibration is shown in Fig. 6.1 (a). This
image was obtained at the tunneling current of 0.5 nA and bias voltage of 60 mV. The
scan range was 1.10 x 1.25 nm. For a particular STM image, the applied voltage to the
scanning piezotube for motion in the lateral direction is known. The lateral sensitivity of
the piezotube is then determined as a measure of lateral displacement per unit volt.

A standard calibration sample, provided by Veeco Instruments was used for
vertical 2 calibration. This sample is silicon patterned into a mesh of squared pits and
covered with platinum. The value used for z calibration is the actual depth of the platinum
pits which is 180 nm. The squared pits are 5 pm in size, with the same size of separation
between them. As the room temperature scan range of the scanning unit is 5 pm, we can
only hope to find an edge of the pits as shown in Fig. 6.1 (b). Using the same method as
for the lateral calibration, the z sensitivity of the piezotubes can be obtained. The
sensitivity values I obtained for our scanning unit at room temperature are: x sensitivity =
29.31 nm/V, y sensitivity = 36.22 rim/V and z sensitivity = 1.04 nm/V. As the piezo
response is known to be reduced by a factor of 1/ 5.60 at helium temperatures [84], from
these values we were able to determine that the sensitivities at 4.2 K as 5.23 nm/V, 6.46

nm/V and 0.19 nm/V respectively for the x, y and 2 directions.

78

 

(a) (b)

Figure 6.0 (a) Schematic drawing of two successive layers of hexagonal graphite. Open
solid circles represent atoms that appear in an STM image. (b) Carbon atoms in the STM
image of graphite obtained at the turmeling current of 0.5 nA and bias voltage of 50 mV.
The scan range is 2.00 x 1.80 nm. Obtaining image (b) demonstrates the scanning unit is
sufficiently stable.

79

 

(a) (b)

Figure 6.1 Images used for calibration of scanner. (a) An STM image of graphite
obtained at the tunneling current of 0.5 nA and bias voltage of 50mV with a scan range of
1.10 x 1.25 nm. (b) STM image of platinum sample showing the edge of one of the pits.
From these images the x, y and z sensitivity of the piezotubes were obtained.

80

6.2 Testing of microscope

Before I started work on my sample at 4.2 K, I performed some preliminary tests at 77 K
using my sample and also a GaAs sample. For my sample the first test was to scan the
surface at various places in STM mode in order to find the best place to do the
capacitance measurements. From these measurements we see that the sample surface is
generally flat with some small undulations of less than 5 nm. Figure 6.2 (a) shows a
typical topography image of my sample. The scan range for this image is about 500 nm x
500 nm. Fig. 6.2 (b) is a cross-section along the path marked by the double arrow in Fig.
6.2 (a). I also scanned the surface of a GaAs sample, shown in Fig. 6.2 (b), in order to
compare the topography of the two samples. The topographic image of GaAs show the
typical growth features in the form of surface mounds which are ~ 1 pm long and ~ 5 nm
high. These STM images were taken at 77 K.

I also performed the capacitancevmode C-V measurements at 77 K. When I
averaged as many as 32 curves 1 could discern peaks and a capacitance step; these
features were similar to data obtained at 4.2K, shown in the next chapter. Although the
data were relatively noisy and contained some artifacts related to tip instabilities, the
curves indicated that the experiment was working, encouraging me to perform the

measurement at 4.2 K.

81

 

Topography nm

mm

   

 

(C)

Figure 6.2 Topography images of silicon and GaAs samples. (a) A typical STM image of
the silicon sample surface, the scan range for this image is about 500 nm x 500 nm. (b) A
cross-section along the path marked by the double arrow in (a). (c) The topographic
image of GaAs showing the typical growth features of the GaAs surface of ~ 1 pm long
and ~ 5 nm high. These STM images were taken at 77 K.

82

Chapter 7

Single Dopant Charging Results and Discussions

7.1 Introduction

We start here by reviewing the procedure for performing my measurements. All
measurements were performed at 4.2 K which was achieved by direct immersion of the
whole microscope in liquid 4He. A chemically-etched tungsten tip of ~25 nm radius was
positioned within a few nanometers of the sample surface. Positioning of the tip is first
achieved in tunneling mode, where a constant tip sample distance is controlled by a
tunneling current feedback loop. Once the tip is in range (i.e. a tunneling current is
observed), we use the z off-set of our piezo tubes to pull the tip away from the sample for
about 30-40 nm and then move it laterally sufficiently far from the tunneling location to
an unperturbed location. Then with the feedback off, the tip is brought closer to the

surface using manual fine approach.

With the system in capacitance mode, the capacitance curves (C-V) are acquired
by positioning the tip about 1 nm from sample surface and holding it at the fixed location
while sweeping the tip voltage. To acquire the capacitance images, we scan the tip over
the sample surface while maintaining a fixed bias voltage. This bias voltage is selected to
correspond to the peaks that appear on the capacitance curve. To compensate for noise
and drift effects in both types of measurements, several curves and images are averaged

together to achieve an acceptable signal-to-noise ratio.

As described in Chapter 4, our sensor circuit includes a bridge that allows us to
subtract away the background mutual capacitance of the tip and sample, which is

approximately 20 fF. This means that our plotted C-V curves represents the change in

83

capacitance as a function of voltage. As a reminder, we can convert from the measured

capacitance C me“ to the charge induced on the tip using the equation,

q tip : Cmeas Vexc a

where Vexc is the excitation voltage applied to the sample.

In the analyses that follow we refer to charges as holes entering and exiting states
of the acceptor atoms. However, this is an interpretation of the data; the measurement is
not sensitive to the sign of the charge carrier. Moreover, conceptually a hole entering an
acceptor is equivalent to an electron exiting the state. With regard to the noise in the
images and C-V curves, the charge sensor itself attains a sensitivity of 0.15 eNHz. As
will be shown on the following pages, I find that vibrations in my system are sufficiently
small such that the achieved noise level is consistent with this value. An experimental
challenge for these measurements is the long data acquisition time required to achieve
single-hole sensitivity. Essentially the rate of data acquisition must be of order 1 s". For
the data presented here, we used a rate of 0.33 points per second for the CV curves, and
~5 pixels per second for the capacitance images. In contrast to STM methods, we do not
use feedback to maintain the tip position. This puts a stringent demand on the stability of

the tip position.

84

7.2 Observed capacitance curves and determination of the nulling voltage
C-V curves acquired with our probe consistently showed the two types of features

indicated by the arrows in Fig. 7.1 (a), which is a representative capacitance curve. The

horizontal axis is the raw tip voltage Vmw, which is the actual voltage on the tip with

respect to the substrate, without compensating for the contact potential (see Sec. 5.2).
Two types of structures are evident: The green arrow marks a step structure that
reproducibly occurred near -0.30 V, with little dependence on the lateral position of the
tip. The red arrow indicates a peak structure, which was found to occur in the voltage
range of -0.150 i 0.075 V. We find the magnitude and voltage of these peaks varies as
the tip is moved from one position to another.

Before analyzing the peaks, we turn our attention to the step structure, which is
similar in appearance to capacitance steps observed in previous capacitance
measurements [6, 72]. We interpret the step structure (green arrow in Fig. 7.1) as the
capacitance increase due to the accumulation of charge in a surface potential well.
Although a detailed description of the formation of this layer is complicated by the
fringing nature of the tip’s electric field, we can estimate the threshold potential using a
parallel-plate picture. The first charge will enter the surface accumulation layer at a tip

potential of
Vtip : Vraw _ Vnull = —(¢B + A)/ae , 7.1

where V,,,,,, accounts for the contact potential, 9 is the elementary charge, A is the

quantum level spacing between the bottom of the well and the first hole state to appear,

and a in this case is the voltage lever arm with respect to the surface (see Sec. 5.1).

85

 

Capacitance (arb.units)
30.x“... ‘

 

 

A_ A

0.0 —0.1 -0.2 —0.3 -0.4
Vraw (V)

(b) Vacuum barrier
\lt‘lilll lL‘
TIP ~. ACCllm. well Suhxnulc

(DB: 10 meV
\ E}:

 

  
     

'evtip __L R’ /

 

Ev

Fig 7.1 (a) Representative capacitance data, plotted with respect to the raw tip voltage.
Two types of features were consistently observed: peaks near Vmw=-0.15 V (red arrow)
and a step near -0.30 V (green arrow). The peaks are consistent with single charges
entering the dopant layer. The step is consistent with the formation of an accumulation
layer at the sample surface. (b) Valence-band diagram (reproduced from Fig. 3.7)
showing our interpretation of the peaks as holes entering acceptor atoms, and the step as
an accumulation layer formed at the semiconductor surface.

86

We expect ato be about 0.15 directly below the apex [75, 81], and ¢3==l 1.7 meV [10].
With regard to A, the accumulation layer will initially form a quantum dot below the apex
of the tip of radius ~25 nm, determined mostly by the tip radius. From a simple estimate
based on a quantum box of this size, and using a hole effective mass of half of the free
electron mass, we find A :10 meV. These values give a threshold voltage of about -150
mV. As Vn-p increase further, the dot will grow in size and more charges will enter. As a
result, a conducting surface layer is expected to form, giving rise to the characteristic
step-like C-V curve [81].

We can extract Vm,” for my sample and tip by comparing two or more C-V curves
that are acquired with the tip positioned at different distances from the sample surface.
This is shown schematically in Fig. 7.2. For example, suppose for Curve 1 the tip is a
distance from the surface of (AW/=20 nm and for Curve 2 the tip is a distance of
dgap3=l.0 nm. From a parallel-plate picture, we expect that the lever arm parameters
follow a;/a,=(d+i<dgap,)/(d+r<dg(,p3), where d is the distance from the surface to the
conducting substrate and K is the dielectric constant. Plugging in the values for my
sample, we get ag/a,=(35 nm + ll.5*2.0 nm)/(35 nm + 11.5*1.0 nm)= 1.25. Hence the
threshold voltage will be higher for Curve 1 compared to Curve 2 by a factor of 1.25.

Of course, due to the fringing field emanating from the tip, we do not expect the
parallel-plate expression to strictly apply. However, the lever-arm parameter must
decrease as the tip distance increases, and the threshold voltage will increase accordingly.
More generally, the entire C-V curve will expand with respect to voltage as the tip
distance increases. If we account for this scale factor, the functional form of the curve

should not change as long as the tip-surface distance is small compared to the radius of

87

 

0.8 l-

  
 
   
 

0.6 L
Curve 1 at dgap.

   

Normallized capacitance
C
4:.
I

 

 

 

 

 

 

 

1
I
0.2 _ I
I
0.0 r- I
L
Vnull
Raw tip voltage Vraw
1 0 -
on
‘c’
E 0.8 —
'6
m
g 0 6 L
o .
'8 — V =Vraw‘vnull
.5 04 —
i 0
g 02 _ '— V =0-i/ (12 (Vmw-Vnun)
2
0.0 — |
l
0
Rescaled tip voltage V‘

Figure 7.2 Schematic of capacitance technique to extract the nulling voltage Vnull- This is
the effective zero voltage, the potential for which no electric field terminates on the tip.
The method uses the step feature that corresponds to the formation of a surface
conducting layer. The parameters dgap and 0: correspond to the tip-surface separation and
the lever-arm, respectively. (top) Two capacitance curves acquired at different tip
positions, plotted as a function of the raw tip-voltage. (bottom) The same two curves
plotted with a new voltage scales V*. In principle, the two curves will only coincide if
they are rescaled with respect to the nulling voltage, as indicated.

88

curvature of the tip and the distance from the surface to the substrate electrode. Because

the effective tip voltage is V,,—,, =Vm,,.— ”an, the voltage expansion should be centered on

VHP =Vmw- Vm,” =0, or with respect to the raw tip voltage, Vm,,.=V,mu. In other words, if

we compare C—V curves acquired at different tip distances, with voltage plotted on the
horizontal axis, we expect to find similar curves except that they are scaled horizontally
with respect to the nulling voltage. This is illustrated in Fig. 7.2.

To extract the most accurate value for the nulling voltage from my measurements,
I focus on data acquired with an especially blunt tungsten tip. In this case, the amplitudes
of the C-V curves were eight times greater in magnitude than the curves acquired with
the sharpest tips, yielding a favorable signal-to-noise ratio. The large amplitude indicates
a radius of curvature of tip’s apex of ~120 nm. Fig. 7.3 (a) shows two especially stable
curves acquired with this tip at distances of 2 nm and 1 nm, as indicated. (Other data
were also acquired at different distances with this tip, however these curves exhibit more
scatter).

As a guide to determine the scaling factor, the two curves of Fig. 7.3(a) are fit

with the following step-function:

P1 "P2
+60/403 )/r>4

 

y = 1’2 + + P5‘(V — P02, 7.2

l

where y is the normalized capacitance, V is the tip voltage, and the PS are free
parameters, varied to achieve the best least—squares fit. The first term is a constant; the

second term has the same form as the familiar Fermi function; the third term is parabola.

89

I

Figure 7.3 (a) Two capacitance curves acquired with a blunt tip at nominal distances from
the surface dgap of 1 nm and 2 nm, as indicated. The clear step—like curve is consistent
with charge accumulating on a surface conducting layer. The data were acquired at a
temperature of 4.2 K. The solid curves are least-squares fits, using the step function
shown in Eq. 7.2. For the solid Curve 1, the fitting parameters are P1=-0.63822, P2
=1.0374, P3=-O.10295, P4=0.03251, P5=10.07723 and P6=0.04011; for the solid Curve 2,
the fitting parameters are P,=—2.6l947, P2 =1.08667, P3=-0.l3956, P4=0.03600,
P5=45.71289 and P6=0.01299. (b) The same two curves normalized and plotted with
voltage scales V“ for which Vm,” and a2/ or] were chosen to yield the best agreement. The
best-fit parameters are Vnu1F-0.068 and az/a.=l .295.

 

 

 

 

. a... Lien; - r fir * i
a 1‘60 .Curve l,d=2nm .— i
v [ j . .1; ‘ i” my"?
.'.' a!“
0 12° "- a {A
o t .1 5,.» i J
.5 . r i
.. c-
“ 40 L " ‘r i
o i- .—' 9'5“ ‘
. 3’ J ' 1
o t- 173654” Li“
—0.10 -0.15 -020 -0.25 -0.30 -0.35 0.40
Raw tip voltage Vmw [V]
(a)

90

Fig 7.3 continued

 

 

 

 

 

 

 

r 1* —r I fi r ' I ‘ I
o Curve 1, d= 2 nm ‘
q, ' .Curve l,d =2 nm 7
CC) 1 I 2 _ _ -Step-function fit,- Step-function fit _
‘3 ____ ‘t
I:
Q d
(U .
Q- .1
‘3
U 4
'5 ..
a:
.2 ‘
.5 .
E -
8 _ .
2 0.0 1" l _ L . r . r . r -
-0I1 -0I20 -0I30
Raw tip voltage Vraw [V]
(b)
I I I I T I I I l
p t .
8 p "0. V =Vraw+0.068 _
t
c 1 -2 _ _ v =1.295 (v,,,.+0.068)

g ..
U r
3 o s :
m ' .
U

U 0"
g .

= 0.4 ‘
cu .

E _
L

O .

2 0.0 .

 

 

 

-0.05 -0. 10 -0. 15 -0.20 -0.25 -0.30
Rescaled tip voltage V. [V]

(C)

91

We include the third term to allow for asymmetry about the inflection point and thus
produce a better fit. (If we use only the best-fit Fermi function without the third term and
then apply the same procedure to find nulling voltage, in the end we arrive at a nearly
identical value for VnulI-) We see from Fig. 7.3 (b) that the data are well-fit with this
function, shown as the solid curves.

Figure 7.3 (c) shows the data with the voltages shifted and rescaled, following
Fig. 7.2, so that the two curves coincide. The best fit is achieved with a nulling voltage of

V,,u”=-0.068 V, as shown. To estimate the accuracy of this value, we consider the fact

that if we try to fit the two curves with Vnu11=0.048 V or Vm,”=0.088 V, then for the best

fit that can be achieved the solid curves have a maximum deviation of about 0.20 in the

normalized units. This is comparable to the average scatter in the data. To be specific, we

 

define the rrns scatter of the measured data M as J—1172(M, — y,)2 , where M,- is an

individual data point, y,- is the corresponding point from the step-function curve, and N is
the number of data points. For both of the curves this equals 0.018 in the normalized
units. Hence we estimate the uncertainty in the nulling voltage to be 0.020 V; i.e.,
V,,,,,,=-0.068 i 0.020 V. The 20 mV uncertainty in the nulling voltage is a source of
uncertainty for the E+ values extracted from my data. However, due to the lever arm
factor which scales the tip potential to the dopant layer, the 20 mV contributes about 2
meV to the uncertainty in E+. This is comparable to other sources of uncertainly, as
discussed in Sec. 7.4.

Returning to Fig. 7.3(b) we see the best fit was achieved with a scale factor of

ag/a,=l.295. This is close to the value of 1.25 expected from the parallel-plate

92

expression. The parallel plate model also predicts that the capacitance should scale with
the lever arm. Hence in the un—normalized units, Curve 2 should be 25% taller than Curve
1; however it is actually only 10% taller. This discrepancy reflects the limitations of the
parallel-plate model applied to the tip-sample system.

All subsequent data that I will present in this thesis were acquired with a sharper
tungsten tip with a radius of curvature at the apex ~25 nm. Here I show data acquired
with this sharper tip, and I give more details of the data acquisition and processing. Fig.
7.4 shows data similar to Fig. 7.3, acquired with the sharp tip held at the indicated
distances from the surface. Each curve displays the average capacitance of eight
consecutive voltage sweeps. The individual data points for each sweep were acquired
with a three-second time increment, which was equal to the time constant setting of the
amplifier output filter. Hence the effective averaging time for each data point in Figure
7.4 is 24 seconds. No additional filtering or processing was performed. Unless otherwise
noted, capacitance data presented in this thesis were acquired with a similar procedure,
with a nominal tip-surface distance of 1 nm. The curves display clear peak and step-like
structure, indicated by the red and green arrows, respectively. We see that a reasonable
agreement is achieved if the two curves are plotted with rescaled voltages, with respect to
a nulling voltage of V,,.,,, = -0.068 V; this is consistent with the nulling voltage found with
the blunt-tip data set.

With regard to the noise level, we see from Fig. 7.4 that the scatter in the data is
about 1 aF, which corresponds in charge units to about 0.03 electrons (this is found by
simply scaling with the 3.7 mV excitation amplitude). For comparison, the noise level of

our charge sensor is 0.15 electrons/VH2. For this measurement, we therefore expect a

93

 

 

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0.05 -0.15 -0.25 -0.35 -0.45
Raw tip voltage V... [V]
2.0 T—r 'fi rffi Tr—‘fifi fit I?
(D P g .0.-
g 1.3 i ,v =v.,.. +0.068 -.° 3 (b)
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5 0.0 - 3:4» 1
z -0.2 rigim 1 4_1 g .4 .....
0.00 -0.10 -0.20 -0.30 -0.40

Raw tip voltage VIt [V]

Figure 7.4 (a) Two capacitance curves acquired with a sharp tip at nominal distances
from the surface dsap of 1 nm and 2 nm, as indicated. The 1 nm curve is shifted vertically
by 5 aF for clarity. The curve displays clear peak and step-like structure, indicated by the
red and green arrows, respectively. The data were acquired at a temperature of 4.2 K. (b)
The same two curves normalized and plotted with voltage scales V*. In this case a
nulling voltage of 0.068 V is used, consistent with the blunt-tip data set (Fig. 5.6); or;/ on
was chosen to yield the best agreement, which in this case is (1.2/(11:1.200. We see that a
reasonably good fit is achieved. The same tip was used for all subsequent measurements
presented in this thesis.

94

noise level of about 0.15/(V24) e = 0.031 e, in agreement with the approximate scatter in
the data. In comparison to the blunt-tip data set, these data exhibit a considerably reduced
signal-to—noise ratio. For this reason, for the nulling voltage parameter, I will rely on the

analysis presented in Fig. 7.3 which yielded a value of V,,,,1r=-0.068 i 0.020 V.

Section 7.3 Capacitance images and further analysis

Figure 7.5 (a) reproduces the representative data from Fig. 7.1; in this case we have
compensated for the contact potential by displaying the voltage axis with respect to V,,-,,.
Fig. 7.5 (b) shows a representative 70 x 50 nm capacitance image, acquired by fixing the

voltage at V“-p =-0.076 V while scanning the tip. The data are unfiltered and minimally

processed (a standard line—wise correction was performed to remove drift effects in the
vertical direction). Bright high-capacitance features are clearly present, as indicated by
the red dots.

Do the bright spots correspond to boron dopants? If we consider the density of
boron dopants in the delta-layer, 1.7 x 10H cm'z, then the number of dopants expected in
a 70 x 50 nm area is about five. Indeed, here we observe six bright spots. Several similar
capacitance images were acquired and confirm that the number of bright features
approximately agrees with the expected number of boron acceptors. Moreover,
comparisons to topographical images of the surface, acquired by using our method in
tunneling microscopy mode, consistently show no correlation between the surface

topography and the capacitance features. Fig. 7.5 (c) shows the cross section of a

95

 

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g 80.6 1
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"2°03“ L, _ L _ ,4
-30 -20 -10 0 10 20 30
70x50nm r(nm)

Figure 7.5 (a) Representative capacitance data reproduced from Fig. 7.1. Here we have
compensated for the contact potential by displaying the voltage axis with respect to V,,-,,.
(b) A representative 70 x 50 nm capacitance image, acquired by fixing the voltage at V,,-,,
=-0.076 V while scanning. The data are unfiltered and unprocessed, other than a standard
line-wise correction to remove drift effects in the vertical (slow scan) direction. Bright
high-capacitance features are indicated by the red dots. (c) The capacitance profile of an
individual dopant; r=0 corresponds to the center of the bright spot. To reduce scatter, we
have averaged three paths as indicated. This averaging does not change the general shape
compared to an individual path through the dopant. In this case the measured curve
(black dots) is well described by a Lorentzian—squared [1+(r/15)2]'2 (blue curve).

96

relatively isolated bright spot. We see that the feature is well reproduced by the square of
a Lorentzian peak of width 15 nm, [1+(r/ 15 nm)2]'l. This functional form arises
essentially from a convolution of two Lorentzians [15, 75, 85] and is consistent with
charge entering a single atom. Hence considering both the CV curves and capacitance

image data, we conclude that we are observing the B+ state of individual boron dopants;

the peaks in the C-V curves correspond to the binding energies of the B+ state of the
boron dopants and the imaged high-capacitance bright spots mark the locations of the
boron dopants. We find that these peaks shift on the scale of tens of millivolts and often
resembled multiple peaks, depending on the location of the probe.

Figure 7.6 shows a data set for which I obtained a high-quality capacitance image
and subsequently acquired several C-V curves at two locations within the imaged area.
The scatter plots to the right show the measured peak structure at the high-capacitance
features A and B, as indicated. The C—V curves marked 1 and 2 show data taken at the
same place (A) but separated by 16 hours. The lower curve was taken when the tip was
moved to location marked B. The solid curves show calculated fits based on single-
electron capacitance peaks, introduced in Sec. 5.5, which are essentially the convolution
of semi-elliptical peaks with the derivative of the Fermi function [72]. In these C-V
curves, we focus on the peak structure. To examine the details of the peaks, we have
subtracted away a linear background from each curve, similar to the grey line shown in

Fig. 7.5 (a). The vertical axis is converted to charge and displayed in units of e.

97

 

 

 

 

 

 

 

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£0.00 . ° 3.2....
i: " r‘ _‘.- _"-'__
100x85nm 0.00 -0.05 -0.10-0.15
Vtip [V]

Figure 7.6 Data set showing an integrated capacitance image and spectroscopy. The
image is to the left. The plots on the right display C-V curves in charge units acquired at
the indicated locations, A and B. To emphasize the peak structure, I have subtracted a fits
fiom the model based on the assumption that each peak corresponds to a single hole
entering the system (as discussed in Sec. 7.4). The top panel shows two curves nominally
acquired at the same location A, but with a time delay of 16 hours. The top curve is
shifted vertically by 0.12e for clarity. The curves show similar peak structure, but some
of the features have shifted in voltage by as much as 10 mV. These lead to ~l meV shifts
in the extracted quantum energies (as discussed in Sec. 7.4). The arrow indicates the
tallest or primary peak. The lower panel shows the measurement and fit for location B.
We see that the C-V curve exhibits two prominent peaks of nearly equal magnitude. This
most likely arises from two closely spaced acceptors at location B.

98

7 .4 Determination of addition energy of individual boron dopants

To analyze the peak structure, we consider a quantum state with energy E + with
respect to the valence band edge (see Fig. 7.1(b)). The resonance tip voltage for this state

will be

Vngeak AM, 7,,
are

this is a similar expression to Eq. 7.1, which described the threshold voltage for the

capacitance step. In this case or is the lever arm parameter with respect to the particular

dopant. To extract the values of E + from our data we used a parameters determined for
each peak by its observed charge magnitude. In other words, we perform a case-by-case
measurement of the lever arm; this is desirable as the parameter depends on both the

lateral (x, y) and vertical (2) distance between the dopant and the apex of the tip.

Specifically, we scale the vertical axis of the CV curve by Vexc to convert to charge

units, and then examine the peak height, which must equal are due to the quantization of
charge [15, 86].

As an example, consider the three peaks shown in Fig. 7.7(a) which were taken
from three different locations; these peaks are similar to the ones shown in Fig. 7.6,
except that they are relatively well isolated from neighboring peaks. The measured a
parameters for the peaks labeled 1, 2 and 3 are 0.067, 0.0833 and 0.0971 respectively and
the extracted E+ energies from Equation 7.3 are 10.26 meV, 6.65 meV and -0.49 meV
respectively. By probing different areas on the sample, we have observed 20 peaks of
sufficient quality to apply this procedure. Fig. 7.7(b) shows the resulting histogram of

the extracted energies, which have a mean value of E+ = 4.3 i 3.1 meV, where the

99

uncertainty is determined by the standard deviation. This compares well to the average B +
energies of 6.7 meV observed for a similar sample by Caro and coworkers [10]. Fig.
7.7(c) is a table showing the peak voltages, 0: parameters and the extracted E for all the

20 distinct peaks.

100

(C)

 

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.0 .
UI

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No. of resonances
A

 

    
     
 

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Median = 5 .0 meV
Mean = 4.3 meV
Std.dev = 3.1 meV

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-2.5 0.0 2.5 5.0 7.5 10.0
E+ lmeVl

Vpmk o E+(me mek a E+(meV)
(V) V) (V)
-0.02 l 6 0.0667 10.26 -0.0926 0.0714 5.09
-0.0266 0.0909 9.28 -0.0980 0.0755 4.30
-0.0353 0.0952 8.33 -0.0996 0.0952 2.21
-0.0496 0.1026 6.62 -0.1 171 0.0556 5.19
-0.0576 0.1 136 5.15 -0. 1206 0.0800 2.05
-0.0606 0.0833 6.65 -0. l 256 0.0971 -0.49
-0.0638 0.1053 4.98 -0. 1275 0.1000 -1 .04
-0.0706 0.0870 5.56 -0. 1291 0.0833 0.94
-0.0736 0.1563 0.20 -0. 1416 0.0667 2.26
-0.0846 0.1000 3.24 -.1591 0.0400 5.34

 

 

 

 

 

 

 

101

Figure 7.7 (a) Data showing three distinct peaks that are relatively isolated. The measured
0: parameters for the peaks labeled 1, 2 and 3 are 0.067, 0.0833 and 0.0971, respectively.
(b) A histogram of the quantum state energies E, extracted from measurements of 20
distinct peaks. The mean value and standard deviation of the distribution are 4.3 meV and
3.1 meV, respectively. (c) A table showing the peak voltages, a parameters and the
extracted E+ for all of the peaks.

 

As to the widths of the peaks, we find that these are indeed consistent with the model for

single charges as described below. For unfiltered data, four parameters determine the

functional form of single-electron peaks: the excitation amplitude Vexc , the temperature

T, the local lever arm a, and the eak volta e V Peak . The measurement here utilized an
p g up

amplifier output low-pass filter with time constant I, which tends to further broaden the
peaks. This can be incorporated into a calculation by performing a simple integration.
The fitting curves were generated as discussed in Section 5.5. The parameters aand

V peak

up were considered as free and hence varied to achieve the best fit. We see that this

procedure reproduces the measured curves well. For example, the t0p solid curve of Fig.

7.6 (location A) invokes five peaks for which Vtgak ranges from -30 mV to -130 mV.

The lever-arm varies significantly from peak to peak; or=0.156 for the primary peak (red
arrow) and a=0.09l for the relatively short leftmost peak. We interpret the primary peak
as a charge entering an acceptor directly below the tip’s apex whereas the other peaks
correspond to the charging of more distant acceptors within the plane.

To see the reproducibility of our data, we go back to Fig. 7.6 where two curves
labeled (1) and (2) are shown which were nominally acquired at the same location A, but
with a time delay of 16 hours. We see that the curves have similar features, but the
amplitudes and voltages of the peaks have changed somewhat. We attribute these
changes to drift of the actual tip location, which alters the lever arm parameter for each

acceptor atom. Applying Equation 7.2 to each curve to extract the energy of the acceptor

state corresponding to the primary peak, we find E A (1) =0.2 meV and EE( 2 )=1.05

102

meV. This comparison indicates that our analysis method is accurate to about 2 meV.
The lower panel shows the measurement and fit for location B. We see that the CV curve
exhibits two prominent peaks of nearly equal magnitude, marked by two arrows. A

careful analysis indicates that the peak on the left is the primary peak in that it is closer to

the tip’s apex; the extracted energy is E} =3.24 meV. The extracted energy from the

right peak is 2.21 meV. We attribute the double peak to two closely spaced dopants near
location B. In contrast, the dopants in the vicinity of A are likely more dispersed. This

idea is explored in detail below.

7.5 Analysis of secondary peaks
For the data sets that show multiple peaks, thus far we have implied that the focus is on
the tallest or primary peak, which corresponds to the dopant closest to the tip’s apex.
However, it is equally useful to examine the other peaks as well; these secondary peaks
likely originate from neighboring acceptors not directly below the tip. In general, due to
the decreased lever arm, more distant dopants will exhibit charging peaks of reduced
magnitude and of greater peak voltage. As shown in Ref. [75], the lever arm for a
subsurface dopant laterally displaced from the tip’s apex is given by a Lorentzian
function
a(r)/a0= [1+(r/w) 2]", 7.4
where r=0 corresponds to the location in the dopant plane directly below the apex, ao=or
(0), and w is approximately equal to the depth of the underlying conducting layer, which
is 35 nm for our sample. If we assume that r20 for the primary peak, we can rewrite the

expression as

103

f, ““1

a ’r 2 Fl
0}) W

where at}; is the primary peak lever arm and as is the lever arm corresponding to a

particular secondary peak. Fig. 7.8 (a) displays a table of the lever-arm fitting parameter,
a for the CV curves of Fig. 7.6 at locations A (curve 1) and B. The bottom two rows of
the table in Fig. 7.8 show Q's/a}? for each secondary peak and the estimated distances to
the respective acceptors, based on Equation 7.5. In Fig. 7.8 (b) the triangles mark the
likely locations of dopants based on the local maxima of the capacitance image. The
dashed circles show the approximate area of interaction for the tip while positioned at A
and B. A comparison of the distances of the triangles from the center of the respective
circles and the distances extracted from the CV curves shown in the bottom row of the
table shows rough agreement. The agreement is especially good, to a precision of about 5
nm, for location B, for which the extracted distances are all 25 nm or less.

In order to gain more insight into the physics involved when two dopants are
close together, in the following section we compared our data to a simple hydrogenic
model that combines the effective-mass approximation and configuration-interaction

calculations of the electronic energies of a two acceptor molecule.

104

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Peak P s s s s
CurveB 3:6
Peaks mg“ 0.156 0.114 0.097 0.091 0.071

0.5/(1p ....... 0.73 0.62 0.58 0.46

Dist.from

F(nm) ....... 21 27 30 38

(a)

CurveB 36:: P S S S S S
Peaks

:3“ 0.100 0.095 0.087 0.080 0.067 0.067

0.3/(1p ....... 0.95 0.87 0.80 0.67 0.67

Dist.from

Hum) ........ 8 14 18 25 25

('1)

2‘1 11m

 

Figure 7.8 Estimating the nearest neighbor distance. (a) Tables of the fitting parameter, a
for the C-chrves at locations A (top curve) and B shown in Fig. 7.3. The primary peak,
P, corresponds to the resonance with the greatest lever arm; other peaks are referred to as
secondary, S. The bottom two rows show as/ap for each secondary peak and the
estimated distances to the respective acceptors, based on Eq. 7.5. (b) The capacitance
image with triangles marking the likely locations of dopants. The hollow triangles
correspond to dopants most likely too far from locations A and B to contribute to the
respective C-V curves. The dashed circles are of radius 25 nm to show the approximate
area of interaction for the tip at the two locations

105

7.6 Configuration-interaction calculation

To explore theoretically the effect of neigboring dopants on the B + state, Professor J. F.
Harrison of the Chemistry Department at Michigan State University calculated the energy
needed to add a third hole to a system of two neutral acceptors separated by a distance R.

The two acceptor model is shown in Fig. 7.9. The model incorporates two neutral

acceptors and finds the binding energy of a third—hole; E + is positive if the third charge
results in a lower electronic energy for the system, i.e., if the third hole is bound.

The calculations were performed using the configuration-interaction method in
the effective mass approximation [87, 88, 89]. As discussed in Sec. 1.5, EMA is very
much an approximate method for my system. While EMA-based modeling may be used
as a guide for our understanding of long-wavelength effects, we do not expect a high
degree of quantitative agreement between modeling and experiment.

The results of this calculation are shown as the red curve in the insert Fig. 7.10.
The data points in Fig. 7.10 were extracted from measurements of dopants at different
locations, including the data from A and B in Fig. 7.6. The blue line is the best fit line for
our primary peak quantum energies plotted as a function of nearest-neighbor distance R.
The error bars show the estimated uncertainty of the energy and distance. With regard to
energy uncertainty, the error bars do not include the contribution from the nulling voltage
(see Sec. 7.2), which gives a systematic 2 mV uncertainty to the labels on the vertical
axis. For the calculation (red), for R>9a0*, we show the asymptotic solution of E+, which

equals that of an isolated acceptor with two holes, equivalent to the binding energy of the

H _ ion. We see that E + is predicted to be positive for separations R>3a0, indicating that

the third hole is bound. In the range of separations 5ao*<R<l6ao* we see that the

106

Hydrogenic model

11

 

Figure 7.9 Schematic of our modeling method which uses the efi'ective mass
approximation and the configuration-interaction method. At large separations the
molecule can hold three holes, similar to the H- state, but for small separations only two
holes can be accommodated. The intuitive picture is that the neutral system can polarize
and weakly bind the third hole; but this is prohibited for small separations for which the
direct Coulomb repulsion dominates.

107

calculated curve has a small negative slope. Our data are consistent with this trend, and
qualitatively consistent with the resonant tunneling measurements by Caro and
coworkers,. However the data show greater-than-predicted slope compared to our model.
Moreover, the calculation shows a maximum near R=5a0* which is not apparent in the
measured data.

The behavior for large separations, R>5a0*, can be understood as a consequence

of basic quantum-mechanics of molecules. Considering our measurement range of
505 < R <16a(', , for R >> 100; , each B0 is effectively isolated; here it is well established

that an additional hole experiences a weak attraction to each of the neutral acceptors due
to polarization. Hence either one can bind the third hole to form the B + state. For the

isolated B + the size of the wave function is large as the root mean square distance of the
hole to the acceptor is F, = 5.80 a0* [10]. If we now allow the two neutral acceptors to be

separated by R~10 a0 *, the third hole has some probability to be found on either location.
Hence the corresponding wavefunction resembles a molecule with two peaks centered at
each atom. Fig. 7.1 1(a) shows an analogous system where the state is represented by the
ground-state wave function of two delta-function potentials. As the separation decreases,
the gradient of the wavefunction decreases along the line joining the two atoms as shown
in Fig. 7.11(b). Schrédinger’s equation immediately tells us that effect will decrease the
energy, or correspondingly increase the binding energy of the state. However at
sufficiently close separations this behaviour breaks down as the system will be poorly
described as a two-acceptor molecule, instead behaving more as a single atom with

double the nuclear charge [90].

108

 

 

  

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0.12
r AI
45‘}
3L ,1.
. a}:
0+

6r

 

 

 

 

 

 

 

 

 

 

 

J
0 i' + Measurement 1
t — Linear fit 1
_2 1 A . . r r A . . . r r . . m r .
10 20 30

Distance to nearest neighbor R (nm)

Figure 7.10 E binding energy versus nearest-neighbor distance. Primary-peak binding
energy plotted as a function of nearest-neighbor distance R. The parameters were
extracted from measurements of dopants at different locations, including the data from
Fig. 7.6 A and B. The error bars show the estimated uncertainty of the energy and
distance. Also shown in blue is a least-squares linear fit. The inset shows in red the
results of a simple-hydrogenic model which incorporates two neutral acceptors and finds

the binding energy of a third-hole; E + is positive if the third charge results in a lower
electronic energy for the system, i.e., if the third hole is bound.

109

Steep gradient Reduced gradient

1" ‘i/

 

 

 

i— 1 : —1L f 1 x
-a a -a a
(a) (b)

Figure 7.11 The ground state wave function potential well. (a) The two potentials are
well-separated compared to the wave firnction width of an isolated well. (b) The two
potentials are close together compared to the width of an isolated well. This state has a
lower energy due to reduced gradient.

110

To further compare the calculations to our measurements, it is instructive to
calculate the statistical nearest neighbor distances for the acceptor atoms in our sample.
Fig. 7.12 shows the statistical nearest neighbor distances for donors randomly dispersed
within a 2D layer. Nearest neighbor distances essential follow the Poissonian
distributions [78]. For any randomly selected acceptor atom, it can be shown that the

probability to find its mth nearest neighbor at a distance between R and R+dR is

(nkszH
(m — l)!

 

27er exp(—7rR2 p)dR 7-6

where p is the 2D density. For the curves shown in Fig. 7.9 we used the nominal acceptor

density of our sample, ,0 =1 .7 x 10H cm'z.

The statistically expected mean distance of
the first nearest neighbor is ~12 nm. Professor Harrison’s calculations show that the
average binding energy of the acceptors with a nearest neighbor close to the statistically
expected mean distance is about 4 meV. This agrees very well with the mean value of the
twenty resonances of 4.3 meV, shown in Fig. 7.7 (b).

Returning to the trend shown in Fig. 7.10 of decreasing binding energy with
nearest neighbor distance, it is possible that the greater-than-expected slope may result
from strain effects. To be specific, my measurements are consistent with an increase in
E + binding energy of about 4 meV as the nearest neighbor distance decreases by about 20

nm; however, our calculations show only about a 1 meV effect. As introduced in Section

3.4, both the substitutional boron acceptor atoms and any interstitial defects such as

111

 

E

a 0'06 Meandistances
a I“:12.lnm
.2 0,04 2m:18.2nm
‘ 3'“: 22.7nm
o 4"“:26. 6mm
.5 0.02

I!

E

h

0

z 0.00

 

 

 

 

0 10 20 30 40 50

Distance r of nearest neighbors (nm)

Figure 7.12 Statistical nearest neighbor distances for the boron dopants in our system,
calculated using the nominal grth density of boron dopants in our system.

112

oxygen or carbon can create strain that will affect the binding energies [58, 59, 63, 91]. In
particular, Calvet and coworkers observed a ground state energy splitting of about 0.8
meV for the first hole on an isolated boron acceptor atom in silicon [58]. Their analysis
indicated that the splitting of the excited states would be comparable to that of the ground
state. Hence it is likely that the binding energy of additional holes, e.g. the B+ state,
would be similarly affected. Although the density of the acceptor atoms in this sample
was two orders of magnitude higher than in my sample, it is plausible that an individual
acceptor in my sample with an especially close boron neighbor or defect could
experience a meV-scale increase in E+ energy. To the best of our knowledge, there is no

complete theory for the effect of strain on the BL state.

In summary, I have applied a scanning probe method to image and discern the
energy levels of subsurface boron acceptors in silicon. The analysis supports the picture
of increasing B+ binding energies due to interactions with neighboring dopants.
Moreover, we find rough agreement between the experiment and a model calculation in
the range of separations between 5510* and l6a0*. The level of agreement between our
measurements and calculated curve is not surprising in light of the highly simplified
nature of the model, which incorporates only one neighbor and neglects effects arising
from the host crystalline potential such as the multivalley structure of the Si conduction

band and strain. Additional comments on this matter are included in Section 8.2.

113

Chapter 8

Summary and Future Directions

8.1 Summary

The major technical innovation of this thesis is the demonstration that individual
subsurface dopants can be both imaged and characterized electronically at the single-
electron level. Specifically, I have used the Charge Accumulation Imaging technique to

image boron acceptors buried about 15 nm below the surface of a silicon crystal, and I

was able to determine the binding energy of the B+ state on an atom—by-atom basis for
20 such acceptors with millivolt precision. The data show that acceptors with more
distant neighbors tend to exhibit weaker binding than those with close neighbors. The
data were compared to a simple hydrogenic model that combines the effective mass

approximation and configuration-interaction calculations.

8.2 Future directions

We believe this work opens the door to a plethora of atomic-scale experimental
studies of subsurface dopants. Our very simple theoretical model neglected, among other
things, effects arising from the periodic crystalline potential, the multivalley structure of
the Si and strain. Indeed we expect such considerations to become more pronounced for
dopants separated by less than a few nanometers, in which case the separation becomes
comparable to several lattice constants. Recent calculations by the Das Sarma group
focusing on closely spaced pairs of P donors in Si predict that the precise separation and
orientation of the dopant pair with respect to the crystalline axis are crucial parameters in

determining the ground state wave functions [92, 93].

114

Thus far we have probed systems for which the lateral positions of the dopants are
random. The logical future direction for this research is to probe controlled arrangements
of donor or acceptor atoms. Adjacent clusters will be separated by a distance greater than
the spatial resolution of the probe and much greater than the effective Bohr radius. This
way measurement will effectively probe isolated donor or acceptor molecules instead of
ensembles of many donor molecules. The main purpose would be to study the basic
quantum mechanics governing the addition spectra of such systems. The motivation for
this work is three-fold. Firstly, exciting applications beyond the well-known Kane
quantum computer [93, 94, 95] have emerged for which single dopant atoms form the
functional part of the device. For example, Lansbergen and coworkers have recently
constructed three—dimensional field-effect transistors (FinFETs) for which the electron of
a single donor atom could be made to hybridize with a nearby quantum well in a
controlled way [1]. Secondly, as conventional state-of-the-art transistors scale down to
below the 32 nm level, understanding the role of individual dopants and the interactions
with neighbors is becoming increasingly important [96, 97, 98]. Thirdly, the ability to
probe the quantum levels of single dopant atoms and dopant-dopant interactions at the
atomic level paves the way for the development of new experiments and theories of few-

body quantum mechanical systems.

Preparations for this work have already started in our group. This effort will
feature collaboration with Professor Michelle Simmons of the University of New Wales
in Australia. Professor Simmons’s grOUp has developed a method to control the
placement of subsurface phosphorus donors in silicon based on STM lithography and in

situ molecular beam epitaxy [99, 100].

115

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