Q‘AEI‘D'LI 9:0 ll "'“' IIIII IIII|IIII||I I III III 31293 00551 2375 LIBRARY Michigan State University This is to certify that the thesis entitled A COMPACT NON-QUASI-STATIC MOSFET DEVICE MODEL presented by Chi-Jung Huang has been accepted towards fulfillment of the requirements for Master's degree in Electrical Engineering Major professor Date H‘q" 8g 0‘7639 MS U is an Affirmative Action/Equal Opportunity Institution MSU LIBRARIES —p—- RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES wil1 be charged if book is returned after the date stamped be1ow. A COMPACT NON-QUASI-STATIC MOSFET DEVICE MODEL By Chi-Jung Huang A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Electrical Energineering and System Sciece 1988 ABSTRACT A COMPACT NON -QUASI-STATIC MOSFET DEVICE MODEL By Chi-Jung Huang Circuit simulators whose MOSFET device models adopt the quasi- static assumption assume that the channel charge distribution can be built up instantaneously. Such quasi-static device models are becoming inade- quate as the trend of increasing the speed of MOS integrated circuits operation continues. The purpose of this thesis is to investigate the significance of the non-quasi-static (NQS) behavior of the MOSFET under various circuit application. A compact device model was developed by approximating the channel charge distribution. This device model was then implemented into a circuit simulator which was used in various cir- cuit applications to investigate the importance of the NQS effect. This work shows that for circuits with small capacitive loading or for the cir- cuits operating in the near-threshold or subthreshold regions, the Q8 assumption is inadequate for the accurate determination of the circuit tim- ing. ACKNOWLEDGMENTS I would like to express my appreciation to my academic advisor Dr. Timothy A. Grotjohn for his guidance and for his many valuable sugges- tions while reviewing my manuscript. I am also very grateful to my wife Su-Yun for her patience and encourgements. iii TABLE OF CONTENTS LIST OF TABLES .......................... ' ........................................................ vi LIST OF FIGURES ................................................................................ vi Chapter 1. Introduction ........................................................................... 1 1.1 Review of MOSFET Models ........................................ 1 1.2 Statement of Purpose .................................................... 6 1.3 Thesis Preview .............................................................. 7 Chapter 2. Formulation of the NQS Model ........................................... 9 2.1 Basic Equations of MOSFET ..................................... 10 2.1.1 Continuity Equation .......................................... 12 2.1.2 Current Transport Equation .............................. 13 2.1.3 Charge Neutrality Equation .............................. 14 2.2 Formulation of the NQS Numerical Model ............... 17 2.3 Formulation of the Compact Model ........................... 21 2.4 Boundary Conditions for the Compact Model ........... 25 2.4.1 Strong Inversion Charge Density ..................... 26 2.4.2 Weak Inversion Charge Density ...................... 27 2.4.3 Total Charge Density ....................................... 30 2.5 NQS Drain and Source Currents ................................ 32 2.6 Further Improvements on the Compact Model .......... 34 Chapter 3. Implementation of the Compact Device Model into Circuit Simulator ...................................................... 40 3.1 NQS Terminal Charges ............................................... 42 3.2 NQS Circuit Simulator Implementation ..................... 44 Chapter 4. NQS Compact Model Applications ................................... 45 4.1 Application 1: Accurate I-V Characteristics by the Charge Summing Method .......................... 45 4.2 Application 2: Transient Analysis of Resistor-MOS Transistor Circuit ........................... 46 iv 4.3 Application 3: Transient Simulation in the Subthreshold Region ............................................ 52 Chapter 5. Conclusions ......................................................................... 56 Appendix A - Changes to the TABLET User’s Manual ..................... 58 A.l MOSFET Parameter Entries .................................... 58 A2 Valid Output List for the MOSFET ........................ 59 LIST OF REFERENCES ....................................................................... 61 2.1: Normalization factors ...................................................................... 18 A.2.1: Valid output list for MOSFET ................................................... 60 LIST OF FIGURES 1. The four regions of operation of the MOSFET .................................. 2 2. MOSFET device models ...................................................................... 5 3. Schematic diagram of a N-channel MOSFET .................................. 11 4. Energy band diagram and space charge density of a MOSFET ...... 15 5. Charge densities for MOSFET upon tum-on or turn off ................. 23 6. Typical I-V characteristics for a MOSFET ....................................... 29 7. Illustration of the charge-summing method ...................................... 31 8. Strong inversion region comparision ................................................. 35 9. Weak inversion region comparision .................................................. 37 10. Saturation region comparision ......................................................... 38 11. Representation of the MOSFET device .......................................... 41 12. Schematic diagram of the setup of application 1 ........................... 47 13. Application 1 - MOSFET I-V characteristics ................................. 48 14. Schematic diagram of the setup of application 2 ........................... 50 15. Application 2 - large capacitive load .............................................. 51 16. Application 2 - small capacitive load .............................................. 53 17. Application 3 - subthreshold region simulation .............................. 55 LIST OF TABLES vi CHAPTER 1 Introduction 1.1 Review of MOSFET Models MOSFET device models are used in circuit simulators to represent the device characteristics in the form of analytical mathematical equations for the regions of operation. As shown in Fig. 1, a MOSFET has four regions of operations: 1) Linear region: Both the drain and source ends of the channel are turned on. The mathematical criterion for a n-channel MOSFET in the linear region is: Vgs > Vth and O < Vds < Vdm, where Vth is the threshold voltage for Vgs to turn on the source end and Vdsa, is the voltage for Vds to pinch-off the drain end. 2) Saturation region: The drain end is pinched-off, but the source end is turned on. The criterion is: Vgs > V”, and Vds > Vdm. 3) Subthreshold / Depletion region: The drain end is pinched-off and the source end is slightly turned on. The criterion is: Vfl, + Vbs < Vgs < V, , where Vfb is the flat band voltage for the MOSFET. 1 Vds g» \ by" Saturation Region linear Region Accumulation Region Subthreshold Region » vrb+ Vb: vu: Van Fig. 1. The four regions of operation of a N-channel MOSFET. 4) Accumulation region: In this region, both the drain and source are pinched-off. The majority carriers dominant in the channel and the channel is non- conducting. The criterion is : Vgs < Vfb + Vbs. Depending on their applications, these device models may be further distinguished into the DC, AC, or transient type simulation: 1) DC: A DC simulation solves for the circuit parameters, typically the DC voltages on the nodes and the DC currents through the branches of the circuit, under constant power supplies and con- stant input voltages. In general, nodal analysis is performed to converge the initial guess of the circuit parameters to a final solu- tion. 2) AC: An AC simulation solves the steady state currents and voltages when small signals are added to DC biases. A linearized model around the DC bias is used for the nodal analysis. 3) Transient: A transient type simulation solves for the circuit behavior by applying large signals to the input nodes. A transient simulator 4 solves the dynamical behavior of the circuit. The sophistication and complexity of currently used DC, AC and tran- sient models vary widely depending on the accuracy and operation range required. Most of the currently used circuit simulators, such as Spice [1], represent the MOSFET with a lumped equivalent circuit model as seen in Fig. 2a. The lumped models adopt the Quasi-Static (QS) assumption that the charges in the channel can change immediately when the voltages on the four terminals of the MOSFET are changed. However, since channel charges can only be varied by the current flowing from the drain or the source, the Q8 assumption is not valid for fast tum-on or tum-off signals applied to the terminals of MOSFET’s. For fast signals, a distributed dev- ice model as shown in Fig. 2b should be considered. When MOS integrated circuits are operated with fast signals, the distributed or Non- Quasi-Static (NQS) effects become significant. Thus, circuit simulators using quasi-static MOSFET models need careful evaluations in the regions where the quasi-static assumption may not be valid. Recently, several attempts have been made to include the non-quasi- static behavior into the device model for the MOSFET. Two methods have been published which include NQS behavior in large signal transient simu- Cdb Cbs B (8) Gate I ___L_ _L I _L == .1. .1. .l. .1. Drain figele GIG__©E@I©I@E~=Source r " T I T T T T Substrate (b) Fig. 2. MOSFET device model of: (a) lumped equivalent device model; (b) distributed device model. lation. The first one, proposed by Mancini, Turchetti and Masetti [2], solves the transient continuity equation along the channel of MOSFET. It is observed that the numerical solution of the continuity equation describes the N QS channel charges. Unfortunately, this approach is limited to small circuits due to the long computational time needed for solving the differential equation. The second one, also by Turchetti, Mancini and Masetti [3], solves an approximating ordinary differential equation for NQS channel charges under strong inversion conditions. This CAD model, although efficient, is inadequate for the near-threshold and subthreshold region where NQS behavior is significant due to the slow diffusion mechanism which is dom- inant in the weak inversion condition. 1.2 Statement of Purpose The major purpose of this work is to efficiently include the subthres- hold current in an NQS model and to make the necessary improvements or modifications to the model so that all regions are accurately simulated. By implementing this compact NQS MOSFET model into a circuit simulator, the significance of the NQS behavior for the MOSFET in each region of operation can be evaluated. Additionally, the importance of the NQS effect 7 in MOSFET device models can be investigated in various circuit applica- tions. 1.3 Thesis Preview In chapter two, the equations of this compact NQS MOSFET device model are developed. By appropriately solving the continuity equation, the numerical model of Turchetti’s work [2] is formulated in Section 2.2. The weighted residue method is then applied to the numerical model to obtain an ordinary differential equation which can be efficiently implemented into a form for computer solution (Section 2.3). The method of calculating the carrier concentrations at each end of the channel is described in Section 2.4. Some further improvements made by comparing to an exact numerical solution are then explained in Section 2.6. The formulation of the NQS circuit simulator is described in chapter three. First, the expressions for the charges on the four terminals of the MOSFET are derived in Section 3.1. The implementation of the terminal charges along with other device parameters into the TABLET (Table model based Transient simulator) program [4] is detailed in Section 3.2. Various applications using this simulator are investigated in chapter four. Comparision between NQS simulations, QS simulations and meas- urements are made to verify the results of the simulations. 8 The significance of NQS effects on the MOSFET device models is concluded in chapter five. CHAPTER 2 Formulation of the NQS Model In this chapter, the formulation of the compact N QS model is presented. Since the quasi-static behavior of an intrinsic, long channel NMOS device has been thoroughly studied [6], the NQS model is developed using sirniliar assumptions as the quasi-static model. Important assumptions inlude the gradual channel approximation and depletion approximation. Formulation of the NQS model in this way will permit comparision of the Q8 and NQS device models. Brews’s charge sheet model [7] which compresses the inversion layer of minority carriers into a conducting plane of zero thickness has been well verified in most applications. The NQS model adopts such a charge- sheet formulation since it leads to simple algebraic equations. The basic equations of the MOSFET device are reviewed in Section 2.1. Section 2.2 develops the NQS numerical model which solves for the channel charge behavior in terms of a partial differential equation. In Sec- tion 2.3, the compact NQS model which approximates the channel charge distribution is formulated. Section 2.4 describes an efficient procedure of solving the boundary conditions for the compact model. Non-quasi-static drain current and source current expressions are derived in Section 2.5.. 10 The last section presents further improvements which are made to the currents. 2.1 Basic Equations of MOSFET The geometry of a N-channel MOSFET is shown in Fig. 3. Due to the attraction force of the gate voltage, the minority carriers (electrons) form an inversion layer at the oxide-semiconductor interface. Electrons, which flow in the channel due to the drift or diffusion mechanism, contribute to the conduction current. The x axis is penetrating into the semiconductor, the y axis is parallel to the flow of the electrons, and the z axis is transverse to the electron flow. The width and length of the channel are given by Z and L respectively. The NQS model solves for the current and the charge density at each point in the channel with respect to time. The three primary equations for- mulated to describe the NQS behavior of the MOSFET are: 1) The continuity equation which relates the N QS charge and the NQS current. 2) The current transport equation which relates the current with the potential and the charge density. 3) The charge neutrality equation which relates the mobile charge density with the bulk and gate charge density. 11 Vg Vd Metal Electrode p- type / Fig. 3. Schematic digram of a N-channel MOSFET. 12 These equations are derived in the following sections. 2.1.1 Continuity Equation For the case of the one dimensional current flow, the continuity equa- tion may be expressed as an(x,z,t) = _1_ 8K .2.» at 4 3y assuming that the recombination, and generation are negligible. By integrating over the current flow cross section, i.e. II3, —dxd= I I i-a’dxd 4 3y one obtains, a "' a z 5; (I qn(x,y,t)dx = 7510;) (2.1a) where xi is the intrinsic point where the minority carrier density is negligi- ble. I(y,t) is the current flowing through the cross section at point y of the channel. Additionally, Qn(y,t) is defined as the charge density per unit area at point y of the channel for time t, i.e. xi QnO’t) = I-qn(X,y,t)dx- (2.1b) 0 Combining (2.1a) and (2.1b), one may refonnulate the continuity equation as a - _a_ — ZEQnGJ) — ay 10’”)- (22) 13 Equation (2.2) is the continuity equation relating the charge density Qn(y,t) and the current I(y,t) at point y and time t. 2.1.2 Current Transport Equation Consider both the drift and diffusion currents, the electron current density at the (x,y) position for a time t can be expressed as: 1,.(x.y.:) = qu.n(x,y.t)E,(y.t) + «rad—"(33M (2.3) where n(x,y,t) is the electron concentration per unit volume, 1.1,, is the mob- lity of the electrons which is assumed to be constant, and E), is the electric field in the y direction. Since the charge-sheet formulation compresses the electron layer into a surface charge, E), can be written as a . 5,6,!) = - 4,33 t) where IDS is the potential at the oxide-semiconductor interface. Also, by the Einstein relationship kT D" = —p.n q where k represents the Boltzmann’s constant, equation (2.3) can be written as J .(x.y.:>- - —qu,.n(x.y.t> —¢—a3(yy’+ ’0 +q—g" 9L3? (2.4) where [3 = 7ch7' 14 The total device current flowing through the active cross section at each point of the channel can be obtained by integrating the current den- sity over the cross section of current flow, i.e. Z x,- I(y,t) = I [01,,(x,y)dxdz. (2.5) 0 Substituting (2.4) into (2.5), one obtains xi 8 s t)dx axi 10.!) = Zn" I-qn(x.y.t) ¢ 0 +--i —]qn(x.y.t)dx . 0 3)’ +53? 0 By (2.1b), it follows that 10.0 =Zu. Q..—— ”3:43- -33-Q-3-:-y¥2. (2.6) Equation (2.6) lays the foundation of the NQS model, since the current at point y of the channel at time t can now be expressed as an explicit function of surface potential and charge density. From (2.6), the drift current and diffusion current can be identified as the first and second term [8]. 2.1.3 Charge Neutrality Equation Fig. 4a and Fig. 4b show the energy band diagram and the space charges of the charge-sheet formulation at an arbitrary point y of the chan- nel. When the inversion layer is tumed-on by the gate bias and a non-zero drain bias is present for an enhancement mode MOSFET, the quasi-fermi 15 Metal Oxide Semiconductor Q8 Fig. 4. At point y of the channel when the MOSFET is turned-on. (a) Energy band diagram. (b) Space charge den- sity. 16 electron potential E FN and the quasi-fermi hole potential E FF splits. The splitting of the electron and hole quasi-fermi potential is given by EFF - EFN = qV(y,t). ¢s(y,t) can then be expressed as: My.» = V6¢>'+ 2% . The voltage across the oxide is VG - VFB - tbs, and the charge density per unit area on the gate is QgW) = Cox ( V60) - Vpa - ¢,.(y.t) ) (2.7) where Cox is the oxide capacitance per unit area and VFB is the flat band voltage for this device. Charge density per unit area for the bulk charge can be found by assuming that no majority carriers are present in the depletion region, i.e. QbO’st) = - qNAW where N A is the doping of the substrate and W is the width of the deple- tion region. By solving Poisson’s equation for the bulk charge region, one may find the expression for W to be W: V421: [¢’ - VB] where as is the dielectric constant of the semiconductor. Therefore, the bulk charge density can be expressed as QbO’t) = — {ZESQNA (4)30,” _ V30» A more exact solution is [9] 17 QbCW) = - VZESqNA ¢,(y.t) - V30) - 1/[3 ). (2.8) The charge neutrality equation in the charge-sheet model requires that Q1309” = — Q8090 - QbU’t) or, from (2.7), (2.8) Que“) = " Cox( VG“) " VFB ' (ISO’J) ) +Cox W 41.6.0 - V30) - up (2.9) \IZESqN h 7t=—. were C ox 2.2 Formulation of the NQS Numerical Model For simplicity, all physical quantities are normalized according to the Table 2.1 [2]. The three basic equations derived in Section 2.1 are refor- mulated as: Normalized Continuity Equation: 6 _ 22.8. 52.01) - 1.2 3yIo,» (2.2m Normalized Current Transport Equation: a¢s(y’t) J- aQnO'J) + By I3 3y Normalized Charge Neutrality Equation: (2.6n) [(y’t) = — Qn(yet) QnOst) = VG“) - VFB - ¢s(y:t) - Nagy,» - V30) - up (2.9n) By taking the spatial derivative with respect to y on both sides of the 18 Physical quantities Normalized factors QnO’J) - Cox y L I(y,t) un Z Cox / L Table 2.1 Normalization factors. 19 charge neutrality equation (2.9n) and rearranging the terms, one obtains, 34,3090 1 A -1 aQnU’t) + . 32 241w.» - V30) — 1/I3 32 By substituting this expression into the current transport equation (2.6n), the non-quasi-static current along the channel can be written as ' (2.6.0 1 ' 89.0.» I(y,t) = + — . (2.10) 1+ 2 B 8” who,» - V30) - l/B Equation (2.10) can be combined with continuity equation (2.2n) to obtain a partial differential equation which describes the non-quasi-static charge density Qn(y,t) as .2- r QnO’J) + _1_ ‘ aQn(y,t) 3y 1+ 1 B By New) - V30) - 1/B J _ Lil _ “n a: Qn(y,t). (2.11) Also, by solving equation (2.9n), ¢s(y,t) can be expressed as 2 $30,!) = V50) - VFB - QnO’J) + 2‘5- l l. 2 1 2' '7» (‘2') + V00) " V173 " QnO’J) - V30) " F 0-”) Equations (2.11) and (2.12) constitute the equations from which Qn(y,t) can be solved numerically provided that appropriate initial and 20 boundary conditions are given. In fact, conventional knowledge of deriv- ing the quasi-static charge density is sufficient to solve for these conditions [1]. They are: (3) Initial condition: The dc drain cun'ent I DC can be found in any conventional do charge- sheet model. ¢s(y,0) can then be solved using the following implicit expression: [DC - y = P(y,0) - P(0,0) (2.13a) where P01 ,0) can be expressed as the following 1 l P (11.0) = [Va(0) - VFB + F] (ISOLO) — 39301.0) .11 2 l 2 - ?x[ ¢s(nro) - VB(O) - F] 2:. B Solving (2.13a), (2.13b) requires an iterative procedure. By substituting + i [¢s(n90) _ VB(O) - fi] 2 . (2.13b) ¢s(y,0) into equation (2.9n), the initial charge density distribution Qn(y,0) can be solved. 21 (b) Boundary condition: By using a QS model [1,6], mm) and o,(1,:) are found by solving the following implicit equation %{3 [no.0 - 24>}: - Vo) ]} = [VG(t) - VFB - ¢s(yrt)]2 - 12[¢S()’,t) - V30) - '1' [3 with V(0) = V5 and W1) = VD for ¢s(0,t) and ¢S(1,t) respectively. By itera- ] (2.14) tively solving (2.14), one obtains ¢S(O,t) and ¢s(1,t) which allows the boundary charge densities Qn(0,t) and Qn(1,t) to be determined. 2.3 Formulation of the Compact Model In this section, an approximation is made for the behavior of the non- quasi-static charge distribution. Consider equation (2.11), one can make the approximation that + I z 8. Nae.» — V30) - 1/B Parameter 6 takes into account the body factor and short channel effect 1 (2.15) [10]. This simplification is commonly used in strong inversion dc models since it improves the efficiency of the models [11,12]. In a transient analysis, though 5 varies with time, one can still make the approximation that 22 Q,.(y.t) x 2‘\l¢s(y,t) - V30) - up The justification is based on the following reasoning: one.» 5 l l ——‘= —. 2.16 + I3 + I3 ( ) 1+ 1) In the strong inversion region: typically in the strong inversion region, 8 is in the range from 1 to 2. The approximation is reasonable. 2) In the weak inversion region: since Q” is quite small, QnO’J) 1 < — . 5 B Thus, the accuracy of 8 is of little importance and (2.16) is justified. Using (2.16), (2.11) can be simplified to be i H 2.0.» + L J 32.0,» ] ay 6 B a) 2 = fi—%Qn(y,t), (2.17) One efficient method of obtaining an approximate solution is the weighted residual method which extracts an ordinary differential equation from equation (2.17) [3]. The use of the weighted residual method requires that an approximate expression be assumed for Qn(y,t). To formulate the approximate expres- sion, consider a MOSFET operating in the linear region with a time vary- ing signal being applied to the gate as in Fig. 5a. The non-quasi-static charge distribution for a rising and falling signal can be analyzed by the 10—- V,, . (Volts) '— 2 l l l l 0 , l 2 3 4 Mus} (a) 10 \ ‘9‘ 8-— ‘3 ‘ Norm 6—1\2 Charge Density 1 2— ‘° 0 l l l l 0 0.2 0.4 0.6 0.8 Norm. Chan. Length (y) (C) 1 23 Norm. Charge Density (Qn Oh! )) lO—\_ 2" I I l l 0 1 2 3 4 t(ns) (b) 10 to 8—-t\ 6—K 41 ,3 \ 2.— 0 l I I I 0 0.2 0.4 0.6 0.8 1 Norm. Chan. Lcngtiro) (d) Fig. 5. Normalized time varying charge densities for a MOSFET with VFB = - 0.8 volt, width = 10 um, oxide thick- ness = 0.1 um and doping = 1015 cm‘3. (a) The rising vol- tage ramp. (b) The falling voltage ramp. (c) Normalized charge density distributions when the voltage ramp of (a) is applied. to = 0.9 ns, t1 = 1.2 ns, t2 = 1,6 as, :3 = 20 ns, VD = 1.0 volt. (d) Normalized charge density distributions when the voltage ramp of (b) is applied. Time instants and VD are the same as (c). 24 exact numerical model described in Section 2.2 and a typical result is shown in Fig. 5b and Fig. 5c. It is reasonable that the first order accuracy can be maintained if one were to approximate the non-quasi-static charge distribution by a quadratic function, i.e. 2.0.» = a(t)y + B(t)(1-y)+ B(txf—y) (2.18) where B(t) is an undetermined function, and B(t) = Qn(0,t) a0) = 2.04) represent the boundary conditions. The error, or residual, introduced by using the approximation can thus be defined as: R = _a_ H on.» + _1_ ] 82.0.0 ]_ L_2 35.0.» 3y 6 B 3y tin 3; ' By assuming the weighting function to be uniform over the channel, weighted residual method requires that, 1 IRdy=O 0 i.e. N J , a HEW) 1] 65.0.0] _ 22 82.0.0 '_..p_a 25 Substituting equation (2.18) into (2.19) results in 2.312..- b. _ 2 ‘ ' dt _ 6&2[(a [3) +B(r)(a+B+28¢7-)]+3(ct+[3)(2.20) =fl a0) . B(t) .B(t) ) where (I),- 5 l/B and (it, B are the time derivatives of a(t) and B(t). B(t) in equation (2.20) can now be solved using numerical methods for ordinary differential equations provided that suitable initial and boun- dary conditions are given. For the initial condition B(O), note that prior to the starting of the transient, all derivative terms in (2.20) must be zero, thus - _ (a - B)2 3(0) _ (a + B + 25%). (2.20a) For the boundary condition, one can use the method described in Sec- tion 2.2 which requires solving an implicit equation by iterative pro- cedure. This method, however, is time consuming particularly in the weak inversion operation region. In the following section, a simple and efficient procedure is presented which avoids the iterative procedure. 2.4 Boundary Conditions for the Compact Model The conventional method of solving for the carrier densities at both ends of the channel is to iteratively find the surface pontential near the source and drain of the MOSFET. In this section, however, Qn(0,t) and 26 Qn(1,t) are solved by analytically summing the weak inversion and strong inversion charges to get the total charge density. The charge summing method is more suitable for CAD applications. One may formulate this procedure by the following three sections. 2.4.1 Strong Inversion Charge Density In strong inversion, the surface potential can be written as 9,0,!) = 2¢B + V(y) where V(y) is the voltage at point y in the channel, and $3 is defined as Therefore, at the source and drain boundaries, the surface potentials are ¢s(ort) = 2¢B and ¢s(19t) = 2¢B + VD“)- Substitute these two equations into (2.9n), gives normalized charge densi- ties of B,(t) = Q..(O.t) = V00) - V7130) (221a) and as(t) = Qn(1,t) = VG(t) — VTa(t) (2.21b) where Vmo) a VF,3 + 2¢B + 2.42453 — VB(t)-'1/B (2.21o) 27 and Vmo) a VFB + 2o, + VDo) + 1mg - V300) - up. (2.21a) Equations (2.21a) and (2.21b) give the charge densities for the source and drain ends in the strong inversion region. 2.4.2 Weak Inversion Charge Density When either end of the channel is in the weak inversion region, the charge density is calculated by considering a MOSFET biased in the subthreshold region. Since Qn(y,t) is very small, equation (2.6) can be simplified as ' Zun 3Q" I , =- . 0’) B ay Integrating with respect to y, one obtains 2B,. 1,, = — LB Q..0=L> - Q..(y=0)]. Since the source end is biased more negatively than the drain end, Qn(y=0) > Que/=1» one obtains, 214,. ID = L13 Qn(y=0). (2.22) The subthreshold current is known to be proportional to eXBI’: [7], where x stands for the nonideality factor resulting from surface state charge [3]. xB¢ From (2.22), the charge density is also proportional to e ’ , hence the 28 normalized subthreshold charge density may be written as , V - V Q "(y=o) E NAS e[ XB( G T)] where Q’n(y=0) stands for the charge density and N AS stands for the charge density at the threshold voltage. Both Q’n(y=0) and N A5 are nor- malized by Cox- The charge density can thus be expressed as Q,,(y=0) a NAS e[ 1W“ ' VT” C0,, . (2.23a) N AS and x can be found by fitting (2.22) to the transistor’s I-V characteris- tics. For example, a typical transistor I-V curve is plotted in Fig. 6, (V61, [01) and (V02. 102) are any two points below the threshold voltage VT. From equation (2.22) and (2.23a), N AS and x can be solved as l = In I /I 2.231) X B(VGI _ V02) ( 01 02) ( ) and AS -_. inflam- ch .—. Mew” - V62) . (223a) ZunCox Zuncox Equation (2.23a) expresses the boundary charge density in the weak inver- sion region as nsfia) = NAS emeGQ) " V730» where N AS, x, VTB are given in (2.23b), (2.230), and (2.21c) respectively. To suppress the exponential increase of the charge density above the thres- hold region, an upper limit of this weak inversion charge was proposed by 29 1 _. 10‘2 - Drain Current 10—4 _ (V02 v [D 2) Var . 101) 10“ - l l I l l 2 3 4 5 0 1V1- Vgs (volts) Fig. 6. Typical I-V characteristics for a MOSFET. The currents for gate voltages less than the threshold voltage exhibit exponential dependence on the gate voltage. 30 Antognetti and coworkers [5]. The upper limit was "1 = 7| N AS where 11 is the suitable fitting parameter. The weak inversion charge den- sity can be expressed by a continuous, smooth function given by awn) = fl . (2.24a) "SB '1' nx In a sirniliar manner, the boundary condition at the drain end can be derived as "sanx t = — 2.241) a,.() "m + "x ( ) where um = NAS exB(Va(t) - Vro(t)) . 2.4.3 Total Charge Density For the transition between strong and weak inversion regions, the charge density in the strong inversion region and in the weak inversion region can be summed together to obtain the overall charge density as Bm=m+m an» and 0t(t) = as + aw (2.25b) where BS, BW, as, aw are give in equations (2.21) and (2.24) respectively. In Fig. 7, the strong inversion charge density, the weak inversion charge density, the charge density calculated from the conventional iterative 31 1 _ Norrn.10”2 " : Charge :' Density :- Q" 10"4 - - —: Strong lav. Charge Wuklnv.Charge ,- 10"6 —I I :Surnrnad Charge 6 l l l I 0 l 2 3 4 5 Vgs (volts) (a) 1 _ 2 Nor-mm- "‘ Charge Density 9" 10-4 -— — : Compact Model 10-6 — I: Exact Nam. Model I l l l 0 1 2 3 4 5 Vgs (volts) 0)) Fig. 7. (a) Illustration of the charge-summing method. The MOSFET parameters for the charge-summing method. are: doping = 1015 cm‘3, flat band voltage = 0.3 volt, oxide thickness = 0.1 um, N45 = 0.1, x = 0.7, T] = 0.25. (b) Com— parision between the I-V curves generated by the charge- sumrning and the exact numerical solution. 32 method and the charge density calculated from equation (2.25) are plotted to show the smoothness and accuracy of equation (2.25). 2.5 NQS Drain And Source Currents The transient drain and source currents for the NQS model developed in Section 2.3 are derived in this section. Integrating the normalized continuity equation (2.2n), one obtains 2 10 o- 1(0.t)- - fi——J Qnaz.» dc (2.26) After one more integration, equation (2.26) becomes L21 I 10 t)dy- 1(0 t) = If' I J Q,.(c MW (227) n 0 0 The integration on the right hand side of equation (2.27) can be simplified by integration-by-parts to be 1y 1 I l QACMCdy = I (l-y)Q,.(y.t)dy (2.28) 0 0 0 Substituting equations (2.28). (2.10), (2.16) together, one obtains the expression for the source transient current to be 150) = - [(0, :) QnO’J)+13Qn(y,t)d L_2d = l B ay d“ From (2.26) and (2.29), the expression for" the transient drain current is flame (y Ody (2.29) 33 2 ID(:)=1<12)=1(0:) + 11—; Q,(c 0d: ,. .t) a ,. ,t) 2 =l[Q(5y +115]qu dy-L—IanOtMy (230) The first term and second term of (2.29) and (2.30) may be identified as the dc current and transient current components respectively. By substitut- ing the charge density Q" given by (2.18) into (2.29), one obtains the expression of the transient source current of the compact model as _ fl “Lil _ _; Ian-[‘1 25 +¢1(0t- 13)] “n [60” :13 123] Substituting B of (2.20) into the above equation gives Is(t) = [123ng + M0 - [3)] —%[] _ _ _ 2 261[(a [3) + B(oc + B + 25%)] . (2.31) The transient drain current can be similarly derived to be 34 ’00) = - [%E + 91(0‘ " 5)] — 218- [(a - (3)2 + B(a + B + 28%)] . (232) Since the first term and second term of (2.31) and (2.32) may be identified as the dc current and transient current components respectively [3]. (2.31) and (2.32) may be rewritten as Is(t) = - I 0C0) — ITR(t) (2.33a) [0(t) = 10d!) " ITR(t) , (2.331)) where [DC and [TR are defined as 2 _ 2 10d!) = PT“ + (p103 - ot) (2.33c) and 1 2 [m0 = E [(a - [3) + B(a + B + 26%)] . (2.33d) 2.6 Further Improvements on the Compact Model Comparisons were made between the compact model and the exact numerical model for the different operation regions. For example, the transient simulation described in Fig. 8 was performed on a MOSFET in the linear region. The transient currents of the compact model, the exact numerical model and the Q8 model are shown in Fig. 8(b). The charge distributions at several instants during the simulation are also shown in Fig. 8(c). It is seen that the compact model exhibits the non-quasi-static characteristics and that the quadratic approximation of the NQS charge 35 16 12—4 8 .— Normalized Drain 4 -— Current O—CD 3 4 -4_J -8 D :NQS Num Model - :NQS Compact Model FET operating in the strong inversion region. (VD = 1 volt, channel width = 10 um, v“, = 0.2 volt, doping = 1015 cm-3, oxide thickness = 0.1 pm, V,,, = 1.17 volt) (a) The voltage ramp applied to the gate is a (2-10) volt ramp in 1 ns. (b) Comparision between the transient drain and source currents generated by the above models. (c) Normalized charge den- sity distributions at various times of simulation: to = 1.0 ns, Norm. Chan. Length (C) Fig. 8. Comparision between the NQS compact model, the NQS numerical solution and the Q8 model for :1 M08- t1 = 1.5 ns, t3 = 2.0 ns, (4 = 3.0 ns. 3.0 36 distribution (2.18) is in good agreement with the exact numerical model under strong inversion operation. When the compact model is extended to the weak inversion operating region for fast operating speeds, it suffers some discrepancy that requires further changes to the model. To understand the discrepancy, consider the MOSFET operating as described in Fig. 9(a). The MOSFET is initially biased in the subthreshold region. A rising voltage ramp is then applied on the gate to turn the MOSFET from subthreshold to saturation region. From the charge distribution of the exact numerical model shown in Fig. 9(b), it is observed that the inversion charge injected from the source end needs some time to diffuse to the drain end. Therefore, a certain amount of delay time is expected before the transient drain current begins to increase as in Fig. 9(c). Fig. 9(d) shows the simulation results of the compact model for the same operating conditions. The quadratic approximation of the charge dis- tribution is inadequate in this situation since it predicts that the drain end is immediately affected after the ramp is applied. This inadequacy causes the erroneous jitter as shown in Fig. 9(e). The same discrepancy is observed for falling voltage ramps as shown in Fig. 10. In order to overcome this inaccuracy, the following algorithm is 37 3 3 ‘ . 2 ....................... 5 -— Norm. Normalized 4 — Charge 1 _ Drain l E ' V‘: 3 -— DenSity cunent -— - - . . . - .4;- ............... (Volts) 2 — 5 : l 0 0 : 3 0 I : 0-5 1.0 2.0 3.0 (a) (b) (c) 3 3 _ _ 2 _. Norm. . Normalized Charge 1 Drain 1 _ Density —l\ Current 0 0 — . . l l l l I 0.5 1.0 2.0 3.0 Norm. Chan. Length Time (ns) ((1) (8) Fig. 9. Comparision between the NQS compact model and the NQS numerical solution for a MOSFET Operating in weak inversion and saturation region. (VD = 5 volt, the other physical quantities for the MOSFET’s are the same as in Fig. 8) (a) The voltage ramp applied to the gate. (b) Numerical solution of the normalized charge density distributions at 1 ns, 1.5 ns, 2.0 ns and 3.0 ns. (c) The transient drain current generated by the NQS numerical solution. (d) The normalized charge density distribution generated by the quadratic approx- imation of the NQS compact model at time instants same as (c). (e) The current generated by the uncorrected compact model is represented by the solid line and the current gen- erated by the correcting scheme is represented by the dotted CUI’VC. (Volts) 38 3 3 . 5 —- Norm 2 4 Normalized2 — 4 Charge Drain l 3 " \ Density 1 - Current 1 _ 0 — i o l:__l__l__J . ° : : 0 l 2 3 1 I | I l , 0,5 1.0 2.0 3.0 ‘ (ns) Norm. Chan. Length Time ("5) (a) (b) (c) 3 3 .. . 2 - Norm. Normalized Charge Drain Density1 .. \ Current 1 d 0 — 0 -- Ill ii 0.5 1.0 2.0 3.0 Norm. Chan. Length Time (ns) ((1) (6) Fig. 10. Comparision between the NQS compact model and the NQS numerical solution for a MOSFET operating in saturation region. (VD = 5 volt, the other physical quantities for the MOSFET’s are the same as in Fig. 9.) (a) The voltage ramp applied to the gate. (b) Numerical solution of the nor- malized charge density distributions at 1 ns, 1.1 ns, 2.0 ns and 3.0 ns. (c) The transient drain current generated by the NQS numerical solution. (d) The normalized charge density distribution generated by the quadratic approximation of the NQS compact model at time instants same as (c). (e) The current generated by the uncorrected compact model is represented by the solid line and the current generated by the correcting scheme is represented by the dotted line. 39 implemented in the compact model: 1) IF (IDC increasing) AND (ITR > 0 ) AND ( Drain end pinched-off ), THEN: I D is allowed to increase, ELSE, hold at it’s previous value. 2) IF (IDC decreasing) AND (1" < 0 ) AND ( Drain end pinched-off ), THEN: I D is allowed to decrease, ELSE, hold at it’s previous value. This algorithm has been tested for all operation regions. The corrected I D of this modified compact model is shown in Fig. 9(e) and Fig. 10(e) with the dotted line. It can be seen that not only the erroneous jitter is prevented but it also includes the delay for the conduction of the carriers from the source to the drain. CHAPTER 3 Implementation of the Compact Device Model into Circuit Simulator To investigate the significance of non-quasi-static effects in various MOSFET circuit applications, the NQS compact device model was imple- mented into a circuit simulator called TABLET (TABLE model based Transient simulator) [4]. MOSFET devices in the TABLET simulation program are represented by the model shown in Fig. 11. The MOSFET behavior is solved by evaluating the terminal charges and the drain to source dc current. The NQS effects are included in the four terminal charges: (1) Q00), the drain charge, (2) Q50), the source charge, (3) Q60), the gate charge and (4) QB(t), the substrate charge. The formula- tion of the compact model for implementation in the TABLET simulation program expresses the four terminal charges in terms of the MOSFET device parameters (sizes, dopings, flat band voltage, gate oxide thickness, etc.) and the circuit parameters (biases on the terminals, time derivatives of the biases, etc.). The expressions are developed in Section 3.1. The modifications to the original TABLET simulation program and the changes to the original user’s manual are summarized in Section 3.2. 40 41 Gate lde(t) m m 6 Substrate Fig. 11. Representation of the NQS compact model for the MOSFET device. 42 3.1 NQS Terminal Charges In this section, the expressions for QD(t), Q50), Q00), and Q30) are derived. Q B(t) and Q50) are obtained by partitioning the total surface sheet 1 charge, QN(t) = IQn(yJ)dt, according to the following definition [1,13]: 0 1 tht) = Jyenoady (3.1) 0 1 . Qs(t) = I(l-y)Q,,(y.t)dy . (3.2) 0 By using (3.1) and (3.2), (2.29) and (2.30) can be written as de 150) -— - Ipc(t) + T—th) (3.3) and de ID(t)= Inca) - T—Qbm (3.4) By integrating (3.3) and (3.4), the source and drain terminal charges may be expressed as Q=s(t) QS(O) + —”-j( 15(2) + 10cc) )dt (3.5) and 90(0- - (20(0) + —j( IDc