Semiconductor Device Simulation of Low-Frequency Noise Under Periodic Large-Signal Conditions

SEMICONDUCTOR DEVICE SIMULATION OF LOW-FREQUENCY NOISE UNDER PERIODIC LARGE-SIGNAL CONDITIONS

By JUAN EUSEBIO SANCHEZ

by Juan Eusebio Sanchez To those from the unincorporated area surrounding New Port Richey, Florida. ACKNOWLEDGMENTS I would like to thank my parents Drs. Juan and Ruby Sanchez, and Theresa and Larry Speer for their encouragement and support. I would also like to thank my brothers John

Sanchez and William Rockwell for their encouragement. I have been fortunate to stay in contact with many of the friends I grew up with in New Port Richey, Florida. The most “tenured” have been Christopher Tidroski and Christopher Gotwalt. Christopher Gotwalt’s enthusiasm for the PhD process reminds me why I chose graduate school in the ﬁrst place. While at the University of Florida, I have had the pleasure of meeting some incredibly unique people. Beth Chmelik, Jaime Chmelik, Jim and Sharon Chmelik, Jerome Chu, Susan Earles, Jennifer Fritz, Jennifer Laine, Aaron Lilak, Dave Maring, Carl and Aimee Miester, Heather Smith, Chad and Jodi Uhrick, Racquel White, and Chip Workman have made Gainesville a more enjoyable place to live. I would like to thank my colleagues in the Noise Research Laboratory, Fan- Chi (Frank) Hou, Lisa Kore, Jonghwan Lee, Derek Martin, and Matthew Perkins for their informative discussions. I would also like to thank my advisor, Professor Gijs Bosman, for his direction, encouragement, and careful review of this dissertation. This work was funded in part by the Semiconductor Research Corporation (SRC). Part of this work was completed at Motorola during a summer internship at Motorola in Tempe, Arizona. The methods discussed in this dissertation were implemented in the

Florida Object-Oriented Device Simulator (FLOODS), which was written by Professor Mark Law. I would like to thank Professor Law for his assistance, for helpful discussions, and for serving on my committee.

iv I would like to thank Professors Jerry Fossum, Ken O, and Loc Vu-Quoc for their suggestions and for serving as my committee members. I would also like to thank Linda Kahila for her assistance in the administrative process.

v TABLE OF CONTENTS page ACKNOWLEDGMENTS ...... iv

LIST OF TABLES ...... ix LIST OF FIGURES ...... x ABSTRACT ...... xiv CHAPTER

1 INTRODUCTION ...... 1 2 PRINCIPLES OF NOISE AND SEMICONDUCTOR DEVICE SIMULATION 5 2.1 Overview ...... 5 2.2 Fluctuation Phenomena in Semiconductor Devices ...... 6 2.2.1 Frequency Domain and Spectral Densities ...... 6 2.2.2 Types of Noise ...... 7 2.3 Newton Iteration Method ...... 9 2.4 DC Steady-State Device Simulation ...... 10 2.5 Noise Simulation of Semiconductor Devices ...... 12 2.5.1 Small-Signal AC Simulation ...... 12 2.5.2 Noise Simulation ...... 12 2.5.3 Efﬁcient Simulation of the Scalar Green’s Functions ...... 15 2.5.4 Noise Simulation Examples ...... 16 2.6 Device Simulation in Terms of Circuit Simulation Concepts ...... 17 3 HARMONIC BALANCE ...... 26

3.1 Example ...... 26 3.2 Frequency-Domain Jacobian for the Steady-State Solution ...... 28 3.3 Selection of the Number of Harmonics ...... 33 3.4 Device Simulation ...... 34 3.5 Harmonic Balance Simulation Using Iterative Methods ...... 35 3.5.1 Matrix-Vector Products ...... 35 3.5.2 Single-Tone Preconditioner ...... 37 3.6 Simulation Results ...... 38

vi 4 FREQUENCY CONVERSION OF SMALL SIGNALS ...... 49 4.1 Small-Signal Diode Example ...... 49 4.1.1 Frequency Conversion of an Applied Small-Signal Voltage .. 49 4.1.2 Frequency Conversion of Small-Signal Current Generators .. 50 4.2 Large-Signal Small-Signal Analysis ...... 51 4.3 Cyclostationary Noise Analysis ...... 55 4.4 Semiconductor Noise Analysis ...... 59 4.4.1 Frequency Conversion Matrix ...... 59 4.4.2 Noise Analysis for the Semiconductor ...... 59 4.4.3 Iterative methods for Frequency Conversion and Noise Analysis 60 4.5 Simulation Results ...... 62

5 GENERATION-RECOMBINATION NOISE IN SEMICONDUCTOR DEVICES ...... 71 5.1 Introduction ...... 71 5.2 GR Noise for the DC Steady State ...... 72 5.3 Periodic Steady-State GR Noise ...... 75 5.4 Simulation Results ...... 76 5.4.1 Resistor Simulation ...... 76 5.4.2 Resistor with 2 Trap Levels ...... 78 5.4.3 Diode Simulation with Majority Carrier Noise in Quasi-Neutral Region ...... 78 5.4.4 Diode Simulation with GR Noise in the Space Charge Region .80 6 CROSS-SPECTRAL DENSITY SIMULATION ...... 99

6.1 Introduction ...... 99 6.2 Noise Simulation Under Periodic Large-Signal Conditions ...... 99 6.2.1 The Periodic Steady State ...... 99 6.2.2 The Impedance Field Method ...... 100 6.2.3 Cyclostationary Noise Analysis ...... 102 6.3 Noise Correlation Matrix for Circuit Simulation ...... 105 6.3.1 Theory ...... 105 6.3.2 Diffusion Noise ...... 106 6.3.3 GR Noise ...... 107 6.4 Simulation Results ...... 109 6.4.1 Shot Noise ...... 109 6.4.2 GR Noise ...... 110 7 CONCLUSION ...... 129 7.1 Summary and Contributions of this Work ...... 129 7.2 Future Work ...... 129

vii APPENDIX A PERIODIC LARGE-SIGNAL STEADY STATE AND CYCLOSTATIONARY NOISE IMPLEMENTATION IN FLOODS ..... 131 B HARMONIC BALANCE AND CYCLOSTATIONARY NOISE SIMULATION MANUAL ...... 134 B.1 Introduction ...... 134 B.1.1 Harmonic Balance ...... 134 B.1.2 Noise Analysis ...... 135 B.2 Simulation Commands ...... 144 B.2.1 circuit: Specifying Stimuli ...... 144 B.2.2 hbcircuit: Extract Response at the Device Terminals . . . 145 B.2.3 hbdevice: Starting the Simulation ...... 146 B.2.4 pdbSetString Specifying the Noise Sources ...... 148 B.2.5 Accessing the Device Proﬁles ...... 148 B.2.6 Known Simulation Issues ...... 149 B.3 Examples ...... 150 B.3.1 Harmonic Balance MOSFET Simulation ...... 150 B.3.2 Diffusion Noise of a Diode ...... 151 B.3.3 Generation-Recombination Noise of a Resistor ...... 152 B.4 MOS Simulation Scripts ...... 165 B.4.1 hbdevice.ckt: Circuit Elements ...... 165 B.4.2 hbdevice.cnt: Contact Speciﬁcation ...... 165 B.4.3 hbdevice.phy: Device Physics ...... 165 B.4.4 hbdevice.test: Initial Solution ...... 166 B.4.5 hbdevice.swp-hb: Initial Solution ...... 166 B.4.6 pscripts: Prints Solution to screen ...... 167 B.4.7 sweeps.tcl: Initial Solution ...... 168 B.4.8 hbfuns.tcl: Harmonic Balance Simulaton ...... 169 B.5 GR Noise Example Script ...... 169 REFERENCES ...... 172 BIOGRAPHICAL SKETCH ...... 178

viii LIST OF TABLES Table page 2.1 Comparison of device and circuit simulation ...... 17

5.1 Trap parameters for the 5 µm resistor example ...... 78 6.1 Device simulation variables ...... 104 A.1 Components added to the simulator...... 132 A.2 Components modiﬁed in the simulator...... 133

B.1 Description of MOS simulation ﬁles ...... 151

ix LIST OF FIGURES Figure page 2.1 Overview of FLOODS ...... 18

2.2 Types of noise ...... 19 2.3 Discretized volume ...... 20 2.4 Noise contribution per unit length to the output for a 1-D diode ...... 21 2.5 The 2-D diode structure ...... 22

2.6 Comparison of simulated and theoretical results for the current noise of the 2-D diode structure ...... 23 2.7 Electron diffusion noise contributions to the contact of the 2-D diode ... 24 2.8 Hole diffusion noise contributions to the contact of the 2-D diode ..... 25

3.1 Diode circuit ...... 40 3.2 Doping profile for the 0.4 µm transistor ...... 41 µ 3.3 IDS versus VDS for the 0.4 m transistor ...... 42 µ 3.4 IDS versus VGS for the 0.4 m transistor ...... 43 3.5 The HB test circuit ...... 44 ( ) 3.6 IDS versus periods of VGS t for the circuit of Figure 3.5 ...... 45 ( ) ( ) 3.7 IDS t versus VGS t ...... 46 3.8 Fourier coefficients of the electron concentration ...... 47 3.9 Fourier coefficients of the potential ...... 48

4.1 The diode as a small-signal conductance in the presence of a small-signal voltage source ...... 63 4.2 Small-signal current response of the diode ...... 64

4.3 Small-signal representation of the diode circuit ...... 65 4.4 Small-signal representation of the mixing process ...... 66

x 4.5 Time-domain noise example for shot noise in a resistive diode ...... 67 4.6 Frequency-domain noise example for shot noise in a resistive diode .... 68 4.7 Port representation of a nonlinear device and its terminations at each sideband frequency ...... 69

4.8 Comparison of noise simulation between FLOODS, ADS, and Matlab .. 70 5.1 Example of how the superposition of Lorentzian spectra results in ﬂicker noise ...... 73

5.2 Transitions between the trap energy level and the conduction and valence band ...... 82 5.3 Resistor simulation circuit ...... 83 5.4 Upper sideband noise spectrum ...... 84 ˜ ( , ) 5.5 Comparison of Gnt 1 0 versus position ...... 85 5.6 Hole and trapped electron concentrations for a DC bias of 0.5 V ..... 86 5.7 GR Noise Spectrum for two trap levels for a bias of 0.5 V ...... 87 5.8 Comparison of the BB and USB spectrum ...... 88

5.9 Comparison of the BB and USB plateau levels for increasing AC bias ... 89 5.10 Diffusion noise, GR noise plateau, and DC current versus DC bias .... 90

5.11 Diffusion and GR noise plateaus versus increasing AC bias, + ω V0 V1 cos 0t ...... 91 + ω = , 5.12 Noise spectrum for a bias of V0 V1 cos 0t , with V1 0 100 mV ... 92 5.13 Doping proﬁle for the diode ...... 93 = . 5.14 Carrier concentrations at VDC 0 7V ...... 94 5.15 GR spectrum versus applied DC bias ...... 95

5.16 GR spectrum plateaus versus applied AC bias for ( )= . + π = 7 V t 0 7 VAC cos 2 f0t and f0 10 Hz ...... 96 ( )= . + π 5.17 BB and USB GR spectrum for V t 0 7 VAC cos 2 f0t and = 7 f0 10 Hz ...... 97 5.18 Microscopic noise source Sγ kω = Sγ kω + Sγ kω a) and scalar nt 0 n 0 p 0 Green’s functions b) in the space charge region of the diode for V (t)=0.7 + 0.1cos 2π107t ...... 98

xi 6.1 Equivalent noise current generator ...... 112 6.2 Network representation of correlated noise generators for a 1 port device . 113 6.3 Microscopic noise sources resulting in a current CSD between sidebands . 114 6.4 BJT doping proﬁle ...... 115

6.5 Base and collector shot noise of the BJT ...... 116 6.6 Test circuit for the harmonic balance and noise simulation ...... 117 6.7 Cross-spectral densities for base shot noise ...... 118 6.8 Cross-spectral densities for collector shot noise ...... 119 6.9 Real part of the correlation between the base and collector shot noise at the baseband for increasing AC bias ...... 120 6.10 Relative noise contributions of GR noise to the base current noise ..... 121 6.11 Relative noise contributions of GR noise to the collector current noise . . 122

6.12 Shot and GR current noise at the base versus IB ...... 123

6.13 Shot and GR current noise at the collector versus IC ...... 124

6.14 Electron quasi-Fermi level, Fn, and the trap energy level, ET at the surface = . for VBE 0 7V ...... 125 6.15 Upper sideband and baseband noise noise spectrums for 2 AC biases . . . 126 6.16 Upper sideband plateaus versus AC bias compared with modulated stationary noise model ...... 127 6.17 Cross spectral density to the ﬁrst USB for increasing AC bias ...... 128

B.1 Diode circuit being driven by a large-signal source ...... 139 B.2 Sample voltage and current response of the diode circuit shown in Figure B.1 ...... 140

B.3 The small-signal current response of a nonlinear device to an applied ω ω small-signal at frequency p and periodic large signal at frequency 0 . . 141 B.4 The small-signal representation of the frequency conversion of small-signals 142 B.5 Green’s Function approach for cyclostationary noise analysis ...... 143

B.6 Bias circuit for NMOS transistor ...... 154 ( ) B.7 iD t versus the period of the large-signal source ...... 155

xii ( ) ( ) B.8 iD t versus vGS t ...... 156 ( ) B.9 iD t versus time for a 200 MHz source ...... 157 B.10 Fourier coefﬁcients of the potential distribution ...... 158 B.11 Fourier coefﬁcients of the electron concentration ...... 159

B.12 Comparison of noise simulation between FLOODS, ADS, and Matlab . . 160 B.13 Comparison of noise spectrum versus AC bias when the large-signal AC source is varied between 1 kHz and 100 MHz ...... 161 B.14 Test circuit for GR noise example showing a ﬂuctuation in voltage in response to ﬂuctuation in voltage ...... 162 B.15 Baseband noise under AC bias conditions ...... 163 B.16 Upper sideband noise under AC bias conditions ...... 164

xiii Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy

SEMICONDUCTOR DEVICE SIMULATION OF LOW-FREQUENCY NOISE UNDER PERIODIC LARGE-SIGNAL CONDITIONS

By Juan Eusebio Sanchez August 2002 Chair: Gijs Bosman Major Department: Electrical and Computer Engineering Noise is deﬁned as any unintended signal that interferes with circuit operation. Although this includes spurious signals of human origin or the external environment, this investigation is limited to noise that results from microscopic ﬂuctuations within the semiconductor components of the circuit. While noise is typically seen as an analog circuits problem, it is believed that noise will become of greater concern in digital circuits as devices shrink, power supply voltages are reduced, and the number of carriers conducted by these devices is reduced.

Noise under large-signal conditions is an important consideration in the design of wireless communications circuits. It has an affect on the spectral purity of oscillators and the noise ﬁgure of mixers and power ampliﬁers. The ability to simulate the noise of semiconductor devices makes it possible to predict their performance in these types of circuits. It is then possible to have a better picture of the worst-case performance of these circuits so that overdesign or costly redesigns are not necessary. This dissertation presents the implementation and simulation of noise under periodic large-signal conditions in a partial-differential equation (PDE) based semiconductor device

xiv simulator. The harmonic balance (HB) technique is used to ﬁnd the periodic steady state of a semiconductor device. Using the impedance ﬁeld method (IFM) and cyclostationary noise sources, it is possible to simulate the self- and cross-spectral densities between sidebands of a periodic large-signal stimulus.

The ﬁrst 2-D semiconductor device simulations of diffusion and trap-assisted generation-recombination (GR) noise under periodic large-signal conditions are presented. Examples are provided that show known results for shot noise in junctions and for low- frequency noise in resistors. Additional results show the upconversion of low-frequency GR noise as microscopically stationary noise sources in resistors and as microscopically cyclostationary noise sources in diode junctions and bipolar junction transistors. It is also shown how the noise correlation matrix (NCM) for circuit simulation can be found through device simulation. It is also demonstrated that low-frequency noise due to trap-assisted GR ﬂuctuations can be modeled using the modulated stationary noise model for the purposes of circuit simulation.

xv CHAPTER 1 INTRODUCTION

Noise is an important consideration in the design of a communications circuit. The noise performance of a circuit determines the threshold for the minimum signal that can be applied at the input of the circuit and can be detected at the output. This translates into how accurately a signal can be transmitted and then be received. Although wireless communication has become a digital medium, noise is still an important consideration because analog circuitry is still required for the transmission and reception of the desired information. Noise simulation of linear circuits, such as ampliﬁers, is a fairly simple proposition. Once the DC operating point of the circuit is calculated, it is only necessary to represent the sources of noise in the circuit as current and voltage generators and to sum the contributions of each of these generators at the output. In nonlinear circuits, noise is dependent on the large-signal currents ﬂowing through the semiconductor devices. The noise is no longer time invariant and linear noise simulation is no longer adequate to characterize the noise performance of the circuit. One of the more popular methods for simulating noise in circuits under periodic large-signal conditions is the harmonic balance (HB) method with a subsequent cyclostationary noise analysis [1]. These techniques are more computationally demanding than DC circuit analysis because it is necessary to simultaneously solve for the operating point of the circuit at several harmonics of a periodic stimulus. It is also necessary to account for the cross correlations between noise current generators at each sideband frequency. These methods are used in commercial packages such as the Agilent Advanced Design System (ADS) [2]. Although HB simulation has been available in circuit simulators for the past decade [3], it has only recently been implemented for partial-differential equation (PDE)

1 2 based semiconductor device simulation [4]. In this technique, a periodic large signal is applied to an external contact of the device. From the nonlinearity of the equations describing the semiconductor device physics, the response is at the fundamental frequency of the periodic stimulus, and also at its integer multiples.1 The solution variables are then the

Fourier coefficients of the voltage, charge, and currents at each of the device terminals as well as the spatially dependent frequency components of the trap-, electron-, and hole- carrier densities and potentials. The large number of equations to be solved using the HB technique in comparison to that of standard DC simulation has made this type of simulation expensive in the computational effort and storage required. The DC simulation of semiconductor devices typically contain at least 2,000 nodes with 3 equations per node for a 2-dimensional (2-D) drift-diffusion simulation. When using direct factorization methods, the memory required grows quadratically with the number of harmonics desired [3]. It is therefore impractical to perform simulation for more than a few Fourier coefficients of the solution variables. Recent advances in the use of iterative solution methods for the HB simulation of large circuits have made the memory requirements of the harmonic balance algorithm nearly linear with the number of frequencies considered for simulation and applicable to semiconductor device simulation [5, 6, 7]. Once the steady-state solution has been attained, it is possible to simulate the small- signal mixer and noise characteristics of the semiconductor device. Small-signal phasors undergo a frequency conversion [8, 9] through the linear periodically time-varying (LPTV) system formed by linearization of the periodic steady state. These phasors appear at the sidebands of the harmonics of the periodic large-signal source. Using the impedance field

1 Quasi-periodic HB has been developed for multiple, non-harmonically related sources. This technique is not used in this work, but is described by Troyanovsky [5]. 3 method (IFM) and modeling the noise as a cyclostationary process, it is possible to calculate the noise magnitudes contributed to an external contact from within the semiconductor device being simulated [10, 11]. Although cyclostationary noise analysis has been used successfully in compact models for shot and thermal noise sources, the simulation of flicker (1/ f or 1/ f -like) and other frequency-dependent noise sources is still controversial. Several competing theories for the origin of these fluctuations exist in the literature and each has been shown to be applicable for specific cases [12]. Adequate models for low-frequency noise under periodic large- signal conditions have not been available in a theoretical or empirical manner [1, 13].

The Florida Object-Oriented Device Simulator (FLOODS) is the platform used for the course of this research. FLOODS is a PDE-based software capable of drift-diffusion and energy-balance device simulation [14, 15, 16]. The software was extended to include HB steady-state simulation and cyclostationary noise simulation for the case of diffusion (velocity ﬂuctuation) and generation-recombination (GR) noise [17]. It is believed that bulk GR processes are responsible for 1/ f -like and RTS noise in bipolar junction transistors (BJTs) [18]. In this dissertation, simulation results for the frequency conversion of trap-assisted

GR noise under periodic large-signal conditions for both resistors and junctions are presented. The basis for the method is similar to that outlined by Cappy et al. [10] and Bonani et al. [11]. However, the system of PDEs solved is extended by adding a trap- continuity equation. In our investigation we also simulate the cross-spectral densities (CSDs) between sidebands for both the case of diffusion noise and trap-assisted GR noise. This information is equivalent to the noise correlation matrix (NCM) for large-signal circuit simulation. Well known results for the case of diffusion noise [19, 20] in junctions are veriﬁed using the simulator. In addition, we show simulation results that verify the NCM entries postulated for 4 microscopically stationary noise sources [21, 22] and also show results for microscopically cyclostationary GR noise sources. The noise concepts, the Newton iteration method, semiconductor device simulation, and noise simulation using the IFM under DC conditions are presented in Chapter 2.

Chapter 3 discusses the implementation of the HB technique for semiconductor device simulation. Iterative solution techniques are presented that improve the performance of HB simulation. Simulation examples show the utility of such a tool under periodic steady-state conditions. The theory for frequency conversion of small signals and cyclostationary noise analysis using the impedance-ﬁeld method is presented in Chapter 4. Simulation results show the equivalence of microscopic semiconductor noise simulation with compact models for the case of shot noise in diodes. Chapter 5 concerns the theory of trap-assisted GR noise in semiconductor devices.

The theory is presented for a semiconductor device simulator under large-signal conditions. Device simulations are performed for the frequency conversion of low-frequency GR noise in 1-D resistors and junctions. In Chapter 6, 2-D simulation results are presented for the frequency conversion of diffusion noise and low-frequency GR noise in a BJT. By simulating the CSD between sidebands, it was possible to ﬁnd the NCM required for noise simulation in circuit simulators. Good agreement is found between the device simulation and a NCM based on the modulated stationary noise model [1]. A summary and discussion of future work are presented in Chapter 7. A description of how these techniques were implemented in FLOODS is discussed in Appendix A. Appendix B is the manual for HB and cyclostationary noise simulation in FLOODS. CHAPTER 2 PRINCIPLES OF NOISE AND SEMICONDUCTOR DEVICE SIMULATION

2.1 Overview

The simulation algorithms presented in this dissertation are implemented into the

Florida Object-Oriented Device Simulator (FLOODS) [14, 15, 17, 23]. An overview of the simulation methods available is given in Figure 2.1. This chapter presents the steady- state and small-signal simulation methods for time-invariant bias conditions (upper blocks). Block a) is a standard component of semiconductor device simulators. It finds the DC steady-state terminal currents and voltages as well as internal carrier and potential distributions. Once the DC solution has been found, it is possible to perform small-signal AC simulations (Block b)) by linearizing the steady-state system of PDEs in order to simulate the effect of small-signals applied to the device terminals. Noise simulation is a generalization of the small-signal AC case where internal microscopic fluctuations perturb the currents and voltages at the device terminals. The open arrow from Block a) to Block b) signify that a DC solution is required before small-signal AC and noise simulations can be performed. Semiconductor device simulation of noise under time-varying bias conditions is the primary focus of this work (lower blocks in Figure 2.1). Analogous to the time-invariant bias case, the periodic steady-state bias case (Block c)) is complicated by the necessity of solving for the frequency components of the semiconductor solution variables. The harmonic balance (HB) method is used and described in Chapter 3. Like small-signal AC simulation, the small-signal mixer case (Block d)) is a linearization of the steady-state solution, except now for time-varying bias conditions. In this regime of operation, applied small-signals are able to mix with the time-varying admittance of the device in order to generate components at other sidebands. Similarly, internal fluctuations

5 6 can also be frequency converted through the time-varying impedance ﬁeld. Small-signal mixer and noise simulation is described in Chapter 4. Chapter 5 presents the theory and simulation of trap-assisted GR noise under periodic large-signal conditions. Chapter 6 discusses how the cross-spectral density of noise between sidebands can be simulated.

The solid arrows in Figure 2.1 show the steps required to perform a noise simulation under periodic large-signal conditions. The DC solution provides an initial guess so that the HB simulator can converge upon a periodic steady-state solution. The small-signal mixer and noise simulation is then found through a perturbation of the HB solution. 2.2 Fluctuation Phenomena in Semiconductor Devices

2.2.1 Frequency Domain and Spectral Densities

Noise in semiconductor devices is a result of the fact that current conduction is the result of the flow of discrete charged particles, electrons. While from a macroscopic point of view that electrons are moving at an average rate in response to the conditions within the device, they are fluctuating in their velocity and position due to scattering events, recombination, or trapping and de-trapping of carriers [24]. Autocorrelation is an important quantity for noise characterization and measurement and is given [25]by , = R t1 t2 E x t1 x t2 (2.1) where E (x) is the mean of x and the autocorrelation is a measure of how long a given fluctuation exists. Under DC steady-state conditions, the autocorrelation is independent of t1 so that − = , R t2 t1 R t1 t2 (2.2)

While events in nature occur at moments in time, it is useful to observe how frequently discrete ﬂuctuation events occur. The spectral density is deﬁned as ∞ 1 − π τ S( f )= R(τ)e j2 f ∂τ (2.3) 2π −∞ 7 which has units of 1/Hz. By taking the Fourier transform of the time-domain data, it is then possible to identify the different types of noise by their characteristic shape. Figure 2.2 shows the different types of noise that may be observed at a contact of a semiconductor device. The voltage spectral density Sv is typically measured by presenting a high-impedance source to the device being tested and has units of V2/Hz. A measurement setup is described by van der Ziel [24]. Noise quantities are also referred to in terms of 2/ current spectral density Si, which has units of A Hz. More information on equivalent input current noise generators is presented in Chapter 6. While the individual noise components are shown individually in Figure 2.2,itisnot possible to isolate them because they all contribute to the noise output of the device. By varying temperature and bias, it is possible to increase the relative contributions of these noise sources so that each may be characterized more fully. 2.2.2 Types of Noise

2.2.2.1 Low-frequency noise sources

The GR and 1/ f noise components in Figure 2.2 are referred to as low-frequency noise in that their spectral density rolls off with increasing frequency. The GR noise has a Lorentzian spectrum of the form [26]

Bτ S = (2.4) i 1 +(2π f τ)2 where τ is a time-constant and B has a dependence on the current ﬂowing through the device. More details on this type of noise are presented in Chapter 5. The 1/ f noise component is typically characterized as having a current spectral density [27]of K IAF S = F (2.5) i f 8 and is associated with a superposition of Lorentzian spectra or current-density ﬂuctua- tions [12] with a microscopic spectral density of

α J2 S = H (2.6) j fn

α where H is a phenomenological parameter based on mobility ﬂuctuations, n is the number of carriers, and Jn is the current density. Because the bias dependence of this source is not known under periodic large-signal conditions, it is not considered in this work.

2.2.2.2 White noise sources

Two common noise quantities considered in semiconductor devices are thermal and shot noise shown as diffusion noise in Figure 2.2. These noise sources are also referred to as being white because they have have equal power at each frequency up into the THz region. Thermal noise for a resistor is characterized by

4kT S = (2.7) i R

2/ where Si is the current spectral density and has units of A Hz, k is Boltzmann’s constant, T is the temperature of the device, and R is the resistance in ohms. For a diode, shot noise is characterized as = Si 2qI (2.8) where q is the charge of an electron and I is the current ﬂowing through the device. While these noise sources appear to be different, with Equation 2.7 having no bias dependence and Equation 2.8 having a current dependence, they result from the same microscopic

ﬂuctuation. The spectral density of this ﬂuctuation [24]is

∂ ∂ 2 y z S = 4q Dα α (2.9) ∂x where α = n, p is the electron, hole density, respectively and Dα is the diffusivity. 9

Thermal noise in a resistor. Consider a resistor with a constant electron carrier = density of n ND, length L, and cross-sectional area A. Integrating over the device length, the current spectral density for the AC short-circuited situation is

4q2D nA S = n (2.10) i L

Using R = L/qnµA and the Einstein relation, Dn = kT µn/q, Equation 2.7 results. Shot noise in a short n+-p diode. The electron minority carrier density is given by x n2 n(x)=n(0) 1 − + i (2.11) L NA where L is the length of the quasi-neutral p-region and n(0) is the electron carrier density at the edge of the space charge region. The current spectral density at the short-circuited terminals is then

2 ( )+ ni L n 0 2 N 2 = 2 ( )∂ = 2 A ≈ 2 ( ) SiL 4q DnA n x x 4q DnAL 2q DnLAn 0 (2.12) 0 2

Using the Einstein relation, and that I = qADnn(0)/L, Equation 2.8 results. 2.3 Newton Iteration Method

Newton iteration techniques are useful for the solution of nonlinear PDEs. We begin by defining right-hand-side vector, F(X), as a function of the solution vector, X. Each row of F is an equation defined to be equal to 0 when a proper solution vector is found. Using a first-order Taylor expansion = + · − F Xk F Xk−1 J Xk−1 Xk Xk−1 (2.13) where J is the Jacobian of F and k is an index denoting the current iteration. The Jaco- bian matrix contains partial derivatives of the equations in F with respect to the solution variables [28]. We can solve for a new estimate of X as = − −1 · Xk Xk−1 J Xk−1 F Xk−1 (2.14) 10

This procedure is repeated until the solution vector has converged on a solution. Two suitable stopping criteria [29] are − < η Xk Xk−1 ∞ (2.15) and < ε F Xk ∞ (2.16) where η and ε are speciﬁed constants that reﬂect accurate convergence. The Newton iteration method has several properties useful for semiconductor device simulation. Based on a reasonable initial guess, X0, the Newton method offers quadratic convergence to a steady state solution. The Jacobian matrix is a linearization about the steady-state solution and can then be used to determine the sensitivity between solution variables. This is useful in circuit and semiconductor device simulations when a sensitivity, small-signal, or noise analysis is desired [3, 8, 30].

In actual implementation, the large-matrix sizes for HB and device simulation make inversion of J in Equation 2.14 costly in terms of storage and factorization time. Iterative and preconditioning techniques are used and are presented in Section 3.5. 2.4 DC Steady-State Device Simulation

Semiconductor device simulation in FLOODS is based on the generalized box formulation [15, 31]. The PDEs describing the semiconductor device physics are discretized using ﬁnite differences. These equations are ∇2ψ = −q − + − ε p n ND NA (2.17) ∂ n 1 = ∇ · J −U (2.18) ∂ t q n n ∂ p 1 = − ∇ · J −U (2.19) ∂ t q p p 11 and are referred to as the Poisson, electron-continuity, and hole-continuity equations, respectively. The electron- and hole-current densities are given by

Jn = −qnµn∇ψ + qDn∇n (2.20) Jp = −qpµp∇ψ − qDp∇p (2.21)

For DC simulation, the ∂/∂t terms are set equal to 0. Using Gauss’s Law, the integral form of the device equations [31] are = ∇ψ · ∂ + q − + − ∂ fψ s ε p n ND NA r (2.22) 1 f = − J · ∂s + U ∂r (2.23) n q n n 1 f = J · ∂s + U ∂r (2.24) n q p p

For the DC solution, FLOODS performs a spatial discretization of Equa- tions 2.22–2.24, and solves these equations simultaneously for all nodes in the device [32]. Figure 2.3 shows a discretized volume for device simulation. The center point is a node of the semiconductor mesh for which the equations are being formulated. Flux terms, such as current density and electric ﬁeld, are calculated along the lines connecting this node to an adjacent node and are referred to as edges. The perpendicular bisectors of each edge are used to integrate the ﬂux between the adjacent nodes. The shaded region is the approximate volume encompassing the node of interest and is used for the integration of the local quantities such as charge and recombination rate. The concept of node and edge for device simulation is directly analogous to the case of circuit simulation, except that there are three solution variables per node instead of one. Details on the discretization of the device equations can be found in Pinto [31] and Liang [32]. For derivations in future sections, the form of Equations 2.17–2.19 are used instead of the integral form of Equations 2.22–2.24 for ease of notation. 12

2.5 Noise Simulation of Semiconductor Devices

2.5.1 Small-Signal AC Simulation

The small-signal simulation of semiconductor devices is commonly implemented using sinusoidal steady-state analysis [30]. By taking the ﬁrst order Taylor expansion of the DC solution, it can be shown that a small-signal perturbation, B, in one of the contacts results in a response throughout the internal device and external contacts. Depending on the external boundary conditions, this response can be a ﬂuctuation in the terminal current or voltage. This is expressed as (J + jD)x = B (2.25) where x is the solution vector, J is the DC Jacobian, and D is a matrix with entries of 2π f along the diagonal accounting for the time-derivative terms of the carrier densities in Equations 2.18–2.19.1 This matrix is partitioned into real and imaginary parts so that       J −D  x   B    R  =  R  (2.26) DJ xI BI where the subscripts R and I denote the real and imaginary parts, respectively. By placing a perturbation in B corresponding to the row of the desired contact, it is then possible to simulate the output response of other contacts. Although not shown, external circuitry can be simulated by expanding the matrix to include the circuit equations and including the frequency dependence of these elements. 2.5.2 Noise Simulation

Noise simulation is an extension of the small-signal AC simulation. Noise simulation in FLOODS uses the impedance ﬁeld method (IFM) [23, 33, 34, 35]. The noise simulation of a semiconductor device is based on the approximation that the excitations of a noise

1 The Fourier transform of the ∂/∂t operator is j2π f , where f is the frequency being considered. 13 source within the device perturbs the steady-state solution. In practice, this approximation is valid for noise signals, because these ﬂuctuations are small enough in magnitude that the operating point of the device is not affected. Noise sources are represented by ﬂuctuations in the continuity equations (2.18–2.19) as

∂ n 1 1 = ∇ · J −U + ∇ · ξ + γ (2.27) ∂ t q n n q n n ∂ p 1 1 = − ∇ · J −U + ∇ · ξ + γ (2.28) ∂ t q p p q p p where ξα represents current density fluctuations, γα represents transition rate fluctuations, and α = n, p for fluctuations in the electron- and hole-continuity equations, respectively.

The scalar Green’s function G˜ α relates local noise sources in the continuity equations with their contribution to the potential at an external contact by

ψ˜ = G˜aγa∂r (2.29)

Using the divergence theorem [36], it is possible to define a vector Green’s function for current density fluctuations. 1 ψ˜ = G˜ ∇ · ξ ∂r (2.30) q a a ∇ · Gãξa = ∇G˜ a · ξa + G˜ a∇ · ξa (2.31) 1 1 ψ˜ = G˜ ξ · ∂s − ∇G˜ · ξ ∂r (2.32) q a a q a a

Neglecting the microscopic noise sources at the surface, the potential at the contact becomes ψ˜ = − Ga · ξa∂r (2.33) where the vector Green’s function is deﬁned as

1 G = ∇G˜ (2.34) a q a 14

Once the scalar and vector Green’s functions are known, it is possible to calculate the voltage spectral density at an external contact with = ˜ ˜ ∗ ∂ + ∗ ∂ Sv ∑ Gα Kγα ,γ Gβ r ∑ Gα K Gβ r (2.35) β ξα ,ξ α,β=n,p α,β=n,p β

where the vector and scalar Green’s functions are defined for the same contact and Kγα ,γβ is the noise source strength for transition rate fluctuations and Kξ ,ξ is the noise source α β strength for velocity fluctuation noise [34, 35]. The noise sources are spatially uncorrelated, which is why only one volume integration is required. This approximation is valid as long as the mean free path of the carriers is much smaller than the size of the discretized differential volumes. For diffusion noise, the noise source strengths are

2 K = 4q nDn ξn,ξn 2 K = 4q pDp (2.36) ξp,ξp

K = K = 0 ξn,ξp ξp,ξn where n and p are the DC carrier densities and Dn and Dp are the local diffusivities. For band-to-band and Auger recombination, the noise source is given as

= ( + ) Kγn,γn 2 R G = ( + ) (2.37) Kγp,γp 2 R G = = − ( + ) Kγn,γp Kγp,γn 2 R G where G and R are the DC generation and recombination rates between the conduction and the valence band. The case for indirect transitions, such as Shockley-Read-Hall (SRH) processes, is discussed in Chapter 5.

To find the scalar Green’s function, it is only necessary to inject sources into the rows of B corresponding to the continuity equations in Equation 2.26 and calculate the resulting potential at the desired contact. By using difference approximations, it is then possible to calculate the gradient of the scalar Green’s function field with knowledge of the scalar field at adjacent nodes in the discretized volume. The voltage spectral density of noise is 15 simulated using the AC open-circuited boundary condition. This is enforced by biasing the device using external DC current sources. When an external voltage source is used, it is possible to simulate the short-circuit current spectral density of the noise in the device. This is useful because the convergence of the DC solution is typically better when voltage bias is used and is the source commonly used for bias sweeps. 2.5.3 Efficient Simulation of the Scalar Green’s Functions

At ﬁrst inspection, it would seem necessary to do two factorizations for each discretized volume in the semiconductor in order to calculate the scalar Green’s functions. However, by using adjoint matrix methods, it is possible to calculate the scalar Green’s functions with only one matrix factorization for each output contact being simulated [36, 37].

The noise simulation problem at hand is of the form

− x = A 1b (2.38) where A is the small-signal AC matrix and x is the solution variable matrix. The b matrix is used to perturb the DC steady state and each column has a 1 in it corresponding to either the electron- or hole-continuity equation for each node in the device. Since we desire the current or voltage response at a speciﬁc device terminal, we premultiply both sides of the

T equation by ei where ei is a zero vector except for a one in the row corresponding to an external contact i. This has the effect of isolating the desired response in the solution vector. The problem is then of the form

= T = T −1 xi ei x ei A b (2.39)

By deﬁning T = T −1 yi ei A (2.40) 16 and taking the transpose of each side it is possible to solve for yi by factoring

T = A yi ei (2.41)

We then only require one matrix factorization and one vector-matrix multiplication

= T xi yi b (2.42) in order to solve for the solution at the desired contact. Besides reducing the number of matrix factorizations required to 1 per contact, it is then a simple proposition to account for when noise sources in different equations are correlated, as is the case with both direct and

−1 indirect transitions. In effect, yi is the row of A corresponding to the row of the contact and xi is the scalar transfer function for each unit source represented in b to the device terminal. 2.5.4 Noise Simulation Examples

Diffusion noise simulation was implemented in FLOODS for 1-D and 2-D structures. A 1-D 10 µm diode structure with a 0.2 µm n+ emitter and unit cross-sectional area was simulated and the spatially dependent contributions of the internal device ﬂuctuations to the external voltage spectral density is shown in Figure 2.4. As shown in the ﬁgure, the minority carriers in each region of the n+-p junction are the dominant noise contributors. A 2-D n+-p diode structure was simulated in FLOODS with the structure shown in

Figure 2.5. A plot comparing simulated and theoretical current spectral densities versus bias is shown in Figure 2.6. The theoretical current spectral density for an ideal diode (see Section 2.2.2.2)isgivenby = Si 2qI (2.43) where I is the DC current ﬂowing through the diode. The simulation was for diffusion noise only and shows that this noise source is responsible for the shot noise in the device. Diffusion noise sources are also responsible for thermal noise in resistors. The discrepancy 17 between theory and simulation results for higher bias is because Equation 2.43 does not account for the effects of high-level injection. The spatially dependent contributions to the current spectral density for the 2-D diode for electrons and holes are shown in Figures 2.7 and 2.8, respectively. The ﬁgures are consistent with the 1-D simulation in that minority-carrier diffusion noise is the dominant contributor. Figure 2.8 shows that majority carrier noise can be on the same order as minority carrier noise when the device is biased at higher voltages. 2.6 Device Simulation in Terms of Circuit Simulation Concepts

Table 2.1 compares device simulation concepts in terms of circuit simulation properties. In developing the simulation theory in Chapters 3 and 4, the concepts are presented in terms of voltage as an independent variable and the current as a response. Once the basic theory is introduced, it is expanded to where there are multiple solution variables internal to the device in response to an external bias.

Table 2.1: Comparison of device and circuit simulation

Terms Device simulation Circuit simulation Flux Current density, electric ﬁeld Current Charge Carrier density, potential Charge, voltage 18

Steady State Small−Signal

a) b) Time Invariant DC Small−Signal AC and Noise

c) d) Time Varying Harmonic Small−Signal Balance Mixer and Noise

Figure 2.1: Overview of FLOODS 19

0 10

GR diffusion -1 10 1/f Total

-2 10 /Hz) 2 (V v

S -3 10

-4 10

-5 10 0 1 2 3 4 5 10 10 10 10 10 10 f (Hz)

Figure 2.2: Types of noise 20

Figure 2.3: Discretized volume 21

10-16

10-17 / Hz cm) 2 10-18 / dx (V v 10-19 dS

10-20 0 2 4 6 8 10 x (µm)

10-16 10-17 10-18 Total 10-19 Electrons / Hz cm)

2 Holes 10-20 10-21 / dx (V

V 10-22 dS 10-23 10-24 0.0 0.2 0.4 0.6 0.8 1.0 x (µm)

Figure 2.4: Noise contribution per unit length to the output for a 1-D diode. The lower ﬁgure is a magniﬁed view of the n+-p junction. The metallurgical junction is at 0.2 µm. 22

Figure 2.5: The 2-D diode structure. The n+ region is in the upper left corner and the device dimensions are in µm. 23

10-16

10-17

10-18

10-19

10-20

10-21 / Hz) 2 -22

(A 10 i S 10-23 Simulation Theory 10-24

10-25

10-26

10-27 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

VBE (V)

Figure 2.6: Comparison of simulated and theoretical results for the current noise of the 2-D diode structure 24

−8

−10

−12

−14 0 −16 1 1 2 −5 0.8 3 x 10 0.6 4 −4 0.4 x 10 0.2 5 0 6 y x

Figure 2.7: Electron diffusion noise contributions to the contact of the 2-D diode. The device dimensions are in cm and the units of the vertical axis are ∂ /∂ ∂ log10 Si x y . 25

−8

−10

−12

−14

−16 0 −18 1 1 2 −5 0.8 3 x 10 0.6 4 −4 0.4 x 10 0.2 5 0 6 y x

Figure 2.8: Hole diffusion noise contributions to the contact of the 2-D diode. The device ∂ /∂ ∂ dimensions are in cm and the units of the vertical axis are log10 Si x y . CHAPTER 3 HARMONIC BALANCE

3.1 Example

Active devices, such as bipolar junction transistors (BJTs) and ﬁeld effect transistors (FETs) have a nonlinear relationship between terminal voltages and the resulting currents. Under large-signal bias conditions it is necessary to account for the additional current which results from the time-variation of stored charge in the device. Harmonic balance (HB) is a frequency-domain simulation method which may be used to account for these capacitive effects under periodic large-signal bias conditions.

Consider the circuit in Figure 3.1. When a periodic bias is applied by the AC voltage source, the diode current responds at DC, and integer multiples of the fundamental frequency of the source. For this circuit, the instantaneous current ﬂowing through the diode is given by ∂ i (t)=i (v (t)) + q (v (t)) (3.1) D B BE ∂t B BE ∂ qB ( ) where iB is the resistive current, ∂t is the displacement current, and vBE t is the voltage across the diode junction. The current ﬂowing through the resistor is

v (t) − v (t) i (t)= G BE (3.2) R R where the voltage at node G is given by ( )= ( )+ ( ) ω vG t VG 0 VG 1 exp j 0t (3.3)

( ) ( ) where VG 0 is the bias applied by the DC source and VG 1 is the bias applied by the AC ω / source at a frequency of 0 rad sec. Using Kirchoff’s current law (KCL), the equation

26 27 describing the circuit is then

v (t)− v (t) ∂ F (t)=0 = BE G + i (v (t)) + q(v (t)) (3.4) R B BE ∂t BE where the voltage across the diode is the only unknown. Harmonic generation occurs from the nonlinear i-v and q-v characteristic of the device [38]. Since the AC voltage is ω ( ) at a frequency of 0, it is assumed that the solution for the junction voltage is VBE k for ∈ , ω ...( − )ω k 0 1 0 K 1 0 and K is the number of positive frequency Fourier coefﬁcients ( ) of vBE t being considered, including DC. Taking the Fourier transform of this equation, the frequency-domain vector describing the circuit is   V (0)−V (0) BE G + ( )  ID 0   R   V (1)−V (1)   BE G + I (1)  =  R D  F  .  (3.5)  .    V (K−1)−V (K−1) BE G + ( − ) R ID K 1 ( ) ( ) ( ) where VBE, VG, and ID are the Fourier coefﬁcient vectors of vBE t , vG t , and iD t respectively. The frequency components of the diode current are given by

−1

∂ { }

−1 qB VBE

{ } + Ω I = i V (3.6) D B BE ∂t

−1

{} {} where is the Fourier transform operator and is the inverse Fourier transform Ω ∂/∂ Ω = , ω ,... ( − )ω operator, and is the t operator with diag 0 j 0 j K 1 0 . Since this is a set of nonlinear equations, it is desirable to solve for the frequency components of the bias across the junction using the Newton iteration method discussed in Section 2.3. Taking the 28

( ) derivatives with respect to the frequency components of vBE t , the Jacobian of F is

1 J = I+ R    ∂ ( ) ∂ ( ) ∂ ( ) ∂ ( ) ∂ ( ) ∂ ( ) IB 0 IB 0 ··· IB 0 QB 0 QB 0 ··· QB 0  ∂ ( ) ∂ ( ) ∂ ( − )   ∂ ( ) ∂ ( ) ∂ ( − )   VBE 0 VBE 1 VBE K 1   VBE 0 VBE 1 VBE K 1  ∂ ( ) . ∂ ( ) .  IB 1 .. .   QB 1 .. .   ∂ ( ) . .   ∂ ( ) . .   VBE 0  + Ω VBE 0   .   .   .   .      ∂ ( − ) ∂ ( − ) ∂ ( − ) ∂ ( − ) IB K 1 ··· IB K 1 QB K 1 ··· QB K 1 ∂ ( ) ∂ ( − ) ∂ ( ) ∂ ( − ) VBE 0 VBE K 1 VBE 0 VBE K 1 (3.7) where I is the identity matrix. In general, it is not possible to analytically calculate F and

J for arbitrary nonlinear equations. It is shown in the Section 3.2 that the elements of the Jacobian matrix can be found by taking samples of the partial derivatives in the time domain and Fourier transforming them. 3.2 Frequency-Domain Jacobian for the Steady-State Solution

We ﬁrst begin with the HB formulation for a circuit driven by a periodic bias. When nodal analysis (KCL) is used, circuits possess a current-voltage relationship at each circuit node where the voltage is the only solution variable.1 Once the technique has been demonstrated for this case, it is then straightforward to extend the technique to semiconductor device simulation where there are three solution variables (ψ,n, p). The large-signal steady-state solution of a nonlinear circuit is found using Newton iteration techniques as introduced in Section 2.3. For this purpose we require both the

Jacobian matrix and the right-hand-side vector. The time-domain right-hand-side vector

1 For the case of voltage source, its current is the solution variable when the modiﬁed nodal analysis (MNA) is used [29, 39]. For the purposes of this derivation, we shall assume that there are no voltage sources in the circuit although they are available and are used in the device simulations which appear in Section 3.6 and in subsequent chapters. 29 for circuit simulation is given [3, 29]by

∂q(v(t)) f (v(t)) = i(v(t)) + + y(t)⊗ v(t)+u(t) (3.8) ∂t where i is the resistive current, q is the instantaneous charge, y is the admittance of the linear circuit elements, ⊗ is the convolution operator, and u represents the independent current sources. Taking the Fourier transform of Equation 3.8, the frequency-domain KCL equation is

F (V (ω)) = I (V (ω))+Ω · Q(V (ω)) +Y (ω) ·V (ω)+U (ω) (3.9) where U is the vector of independent current sources, I represents the current ﬂowing through the nonlinear circuit elements, Q represents the nonlinear charge elements, and Ω is called the derivative matrix and accounts for the time-derivative operation in the frequency domain. The Jacobian is then

∂F (V) ∂I (V) ∂Q(V) J (V )= = + Ω +Y (3.10) ∂V ∂V ∂V

By taking time-domain samples of the Jacobian elements over one period of the large- signal source, the frequency-domain derivatives are found using the chain rule [8]. For a current I ﬂowing into node m at frequency index k, and a voltage V at node n at frequency index l, the frequency-domain element of the Jacobian is then

∂I (k) ∂I (k) ∂i (t) ∂v (t) m = m m n (3.11) ∂Vn(l) ∂im(t) ∂vn(t) ∂Vn(l) where ∞ jlω t vn (t)= ∑ Vn (l)e 0 (3.12) l=−∞ and T/2 1 −jkω t I (k)= i (t) e 0 dt (3.13) m T m −T/2 30

ω and 0 is the fundamental frequency of the large-signal source. The Jacobian entries [8] are then T/2 ∂ ( ) ∂ ( ) Im k 1 im t −j(k−l)ω t = e 0 dt (3.14) ∂Vn (l) T ∂vn (t) −T /2 Using the method described by Kundert [3], it is possible to explicitly form the HB Jacobian. By using the single-sided discrete Fourier transform (DFT), the HB algorithm can be efﬁciently used for computer simulation. The single-sided DFT is used since the positive- and negative-frequency frequency components are complex conjugates for a real signal. A time-domain waveform is represented in terms of its Fourier coefﬁcients in the discrete time domain using the inverse discrete Fourier transform (IDFT) [40]

K−1 j2πks/S xn (s)= ∑ Xn (k) e (3.15) k=1−K where K is the total number of frequencies considered, s is the current time sample, and S = 2K − 1 is the total number of time samples. A frequency-domain waveform can be represented by samples in the time domain using the DFT

S−1 1 −j2πks/S Xn (k)= ∑ xn (s) e (3.16) S s=0 where the Fourier coefﬁcients are decomposed into real and imaginary parts using

( )= R ( )+ I ( ) Xn k Xn k jXn k (3.17) with the superscripts R and I denoting the real and imaginary parts. Since an amplitude waveform is purely real and

ω ej t = cos(ωt)+jsin(ωt) (3.18) the DFT is then     R ( ) S−1 ( π / )  Xn k  1  cos 2 ks S  Xn (k)=  = ∑  xn (s) (3.19) I ( ) S = − ( π / ) Xn k s 0 sin 2 ks S 31

(− )= ∗( ) and since Xn k Xn k , the IDFT becomes   K−1 R ( )  Xn k  xn (s)= ∑ (2 − δ (k)) cos(2πks/S) −sin(2πks/S)   (3.20) = I ( ) k 0 Xn k where δ (k)=0 for k = 0 and δ (k)=1 for k = 0. The assumed fundamental frequency ω ,ω , ω ...( − )ω is 0 and so the frequencies considered are 0 0 2 0 K 1 0 . It is important to note that this part of the algorithm is independent of the actual fundamental frequency.

The actual fundamental frequency is only important for evaluating the linear-admittance elements and Ω. Starting with Equation 3.11 and using Equations 3.19 and 3.20, each Jacobian element is then   S−1 ( π / ) ∂Im (k) 2 − δ (l)  cos 2 ks S  ∂im (s) = ∑   cos(2πls/S) −sin(2πls/S) ∂V (l) S ∂v (s) n s=0 −sin(2πks/S) n (3.21) Using trigonometric identities this simpliﬁes to   ∂I (k) 2 − δ (l) S−1 ∂i (s)  ψ ψ  m = m  11 12  ∂ ( ) ∑ ∂ ( ) (3.22) Vn l 2S s=0 vn s ψ ψ 21 22 where ψ = ( π ( + ) / )+ ( π ( − ) / ) 11 cos 2 k l s S cos 2 k l s S ψ = −sin(2π (k + l)s/S) − sin(2π (k − l)s/S) 21 (3.23) ψ = − ( π ( + ) / )+ ( π ( − ) / ) 12 sin 2 k l s S sin 2 k l s S ψ = ( π ( − ) / ) − ( π ( + ) / ) 22 cos 2 k l s S cos 2 k l s S By deﬁning      GR (k)  1 S−1 ∂i (s)  cos(2πks/S)  ( )= mn  = m   Gmn k ∑ ∂ ( ) (3.24) I ( ) S = vn s − ( π / ) Gmn k s 0 sin 2 ks S 32 the Jacobian element is then   ∂I (k) 2 − δ (l)  GR (k + l)+GR (k − l) GI (k + l) − GI (k − l)  m =  mn mn mn mn  ∂ ( ) (3.25) Vn l 2 I ( + )+ I ( − ) R ( − ) − R ( + ) Gmn k l Gmn k l Gmn k l Gmn k l

Two important relations are (− )= ∗ ( ) Gmn k Gmn k (3.26) when k < 0 and ( )= ∗ ( − − ) Gmn k Gmn 2K 1 k (3.27) when k > K. Kundert [3] further simpliﬁes Equation 3.25 into the sum of Toeplitz and Hankel matrix elements so that

∂Im (k) 2 − δ (l) = [tmn (k − l)+hmn (k + l)] (3.28) ∂Vn(l) 2 where   R ( ) − I ( )  Gmn k Gmn k  tmn (k)=  (3.29) I ( ) R ( ) Gmn k Gmn k and   R ( ) I ( )  Gmn k Gmn k  hmn (k)=  (3.30) I ( ) − R ( ) Gmn k Gmn k are quads. The Jacobian submatrix which relates the current through the nonlinear elements at node m and the voltage at node n is then

∂Im =(Tmn + Hmn)D (3.31) ∂Vn where D = diag{1/2,1,1,...1}   ( ) (− ) ··· ( − )  tmn 0 tmn 1 tmn 1 K       tmn (1) tmn (0)  T =   (3.32) mn  . .   . ..   

tmn (K − 1) tmn (0) 33 and   ( ) ( ) ··· ( − )  hmn 0 hmn 1 hmn K 1       hmn (1) hmn (2)  H =   (3.33) mn  . .   . ..   

hmn (K − 1) hmn (2K − 2) where the rows correspond to k =(0,1,...K − 1) and the columns correspond to l = (0,1,...K − 1). The nonlinear capactive elements are done in an analogous fashion in ∂ ( ) which qm t is time sampled and Fourier transformed. ∂vn(t) Linear admittance elements, Y , are just placed along the diagonal. Each Jacobian matrix element is then given by

∂Fm (k) ∂Im (k) ∂Qm (k) = + Ω(k) +Ymn (k)δ (k − l) (3.34) ∂Vn (l) ∂Vn (l) ∂Vn (l) where    0 −kω  Ω(k)= 0  (3.35) ω k 0 0 and   ω − ω  Re Ymn k 0 Im Ymn k 0  Ymn (k)=  (3.36) ω ω Im Ymn k 0 Re Ymn k 0 Now that the large-signal Jacobian is formed, it is then possible to converge upon a steady- state solution using the Newton method as described in Section 2.3. 3.3 Selection of the Number of Harmonics

Theoretically, the number of frequencies is infinite, however the magnitudes of the Fourier coefficients decrease at higher frequencies so that their affect on the operation of the circuit is negligible. If this did not occur, the semiconductor device would draw infinite power, which is not realizable physically. ( − )ω It is necessary to choose the highest harmonic, K 1 0, so that the solution variables at this frequency have a negligible effect on the accuracy of the simulation. In a wire- ( − )ω < less communications circuit simulation, an appropriate selection would be K 1 0 fT 34 where fT is the gain-bandwidth frequency [41] and is the upper limit in which a semiconductor device can be used as an amplifier. For the HB device simulations in this work, K was selected so that the terminal currents were negligible at the highest harmonic considered.

3.4 Device Simulation

The formulation of the semiconductor HB equations is analogous to the circuit case, however, instead of there being only one equation for each node, there are three equations which are coupled with each other and to each adjacent semiconductor node. In Chapters 5 and 6, additional equations are added to investigate GR noise. We ﬁrst begin with vectors Ψ, N, and P, which are all of length 2K −1. These vectors contain the Fourier coefﬁcients for the potential ψ, electron-carrier density n, and hole- carrier density p at each node in the semiconductor mesh. These are the only data stored between each iteration and are updated as the simulator converges upon a solution. At each iteration these vectors are transformed into time-domain sampled vectors ψ, n, and p over one period of the fundamental frequency. For each time sample, the simulator generates node and edge stiff matrices. These stiff matrices are dense matrices relating variations of the semiconductor equations for a node with variations in the solution variables of another node. Using the DFT, time samples of these stiff matrices are Fourier transformed and placed in the frequency-domain Jacobian.

The Poisson, electron-continuity, and hole-continuity equations for DC simulation were deﬁned in Equations 2.17–2.19 on Page 10. In the time domain, these equations are represented as      fψ (t)   0        ∂   f (t)= f (t)  +  n  (3.37)  n  ∂t   fp (t) p 35

Taking the Fourier transform of this vector, the resulting set of equations is      Fψ   0          F (ω)= F  + Ω N  (3.38)  n    Fp P where the elements of the ﬁrst vector are calculated over one period of the large-signal source driving the device, and the second term is calculated directly in the frequency domain. The Jacobian is then   ∂FΨ ∂FΨ ∂FΨ  ∂Ψ ∂ ∂   N P   ∂F ∂F ∂F  J =  N N N  + Ω (3.39)  ∂Ψ ∂N ∂P  ∂ ∂ ∂ FP FP FP ∂Ψ ∂N ∂P where each submatrix relates frequency components of the equations to the frequency components of the solution vectors. The presence of external sources and linear passive elements are treated as boundary conditions for the device simulation [16]. The resulting matrix is partitioned into  

 Jsc A  J =   (3.40) BJckt where Jsc is the Jacobian for the semiconductor device mesh, Jckt is the Jacobian for the linear circuit elements, and A and B couple the external circuit elements with the semiconductor device mesh. 3.5 Harmonic Balance Simulation Using Iterative Methods

3.5.1 Matrix-Vector Products

Using direct methods to solve the frequency domain are expensive in computational and memory requirements. Forming the frequency-domain Jacobian explicitly and inverting the matrix can be a tremendous computational burden for more than a few harmonics for a 2-D semiconductor simulation. State of the art HB for both circuit and semiconductor 36 device simulation has led to the use of iterative methods such as Krylov subspace methods as an iteration inside of an outer Newton loop [5, 42]. These algorithms iteratively attempt to minimize functions of the form r = Ax − b (3.41) where r is the residual vector and x is the update. The iterative routine returns r and x, given A and b. The iterative routine finishes when r has been reduced to within a specified tolerance or gives up. The outer loop continues by updating the solution variables and formulating a new A and b. A significant advantage of using this method is that the inverse of the A matrix is not explicitly formed. For the HB system of equations, the Jacobian matrix is divided into time-independent and time-dependent parts and is given by

= + Ω J Jt J f (3.42)

where J f is the part of the Jacobian which accounts for the capacitive terms in the conti- Ω ∂/∂ nuity equations, accounts for the t operation in the frequency domain, and Jt is the resistive part of the Jacobian. Applying the chain rule [3],

= Γ Γ−1 Jt jt (3.43)

Γ where jt is a matrix with the time samples of the Jacobian of the nonlinear elements, and is the Fourier transform operator and Γ−1 is the inverse Fourier transform operator. Simi- larly = Γ Γ−1 J f j f (3.44)

On each iteration of the inner loop, the residual is then = + Ω ∆ − r Jt J f X F (3.45) 37 and is evaluated as matrix-vector products, for example, ∆ = Γ Γ−1∆ Jt X jt X (3.46) so that the operations are performed from right to left and no matrix-matrix operations are required. The computational expense of forming the Jacobian is thus decreased using matrix- vector products. In effect, the Fourier transform operations are being taken of a vector, while using the direct method takes the DFT of the elements being entered into the frequency-domain Jacobian. This is signiﬁcant for device simulation, where there are multiple edges referring to the same node and the DFT is performed for each edge stiff matrix and one node stiff matrix. An additional savings in computational expense is possible when the fast Fourier transform (FFT) is used instead of the DFT. The order of operations ∆ ( 2) ( ( )) for forming Jt X is reduced from O N to O N log2 N [40]. 3.5.2 Single-Tone Preconditioner

In order to speed up the iterative solution process, a preconditioner which is much less costly to invert than the original Jacobian is used. The residual which is then solved is

− r = P 1 (Ax − b) (3.47) where P is the preconditioner. A desirable property of the preconditioner is that it is based on knowledge of the problem being solved and reduces the number of iterations required to minimize the residual. Without the preconditioner, it is often not possible to solve the system of nonlinear equations at all. The basis of the single-tone preconditioner is that the strongest coupling between the equations is with other equations at the same frequency. In fact, if there were no large signals driving the HB system of equations, the frequency- domain Jacobian would be block diagonal and only same-frequency coupling would exist between the equations. 38

This preconditioner is referred to as the low-distortion Jacobian [5] and is found by using only the diagonal elements of Equations 3.32 and 3.33.IfJ is ordered so that all equations at the same frequency are grouped together, the matrix becomes block diagonal. The inverse of this matrix is then simply the inverse of K sparse matrices. A dramatic reduction in storage and factorization time is realized making it possible to simulate additional Fourier coefﬁcients of the solution variables within the memory constraints of the computer workstation being used. Further discussion of this preconditioner for HB can be found in Troyanovsky [5] and Feldmann et al. [6]. 3.6 Simulation Results

HB simulations were performed on a 0.4 µm NMOS transistor. The doping profile is shown in Figure 3.2. The Darwish mobility model [43] was implemented in FLOODS to properly account for the surface mobility in the channel of the device. A comparison between simulation and measurement for IDS versus VDS and IDS versus VGS is shown in Figures 3.3 and 3.4. Although there was not an exact match for the terminal currents, the trends for the device are reasonable for the purposes of demonstration of the HB algorithm. = = . Steady-state simulations were performed for the transistor with the VGS VT 0 5V. ( ) The simulation circuit is shown in Figure 3.5. Figure 3.6 shows IDS t for a 1 kHz and 178 Mhz source. The low-frequency source is the resistive case where the charge storage in the device is minimal. Figure 3.7 shows hysteresis which occurs in IDS versus VGS at higher frequencies. In this figure, the resistive (low frequency) case corresponds to the i-v characteristic of the DC sweep shown in Figure 3.4. A 100 MHz simulation was performed on the same device. Figure 3.8 shows the first 3 Fourier coefficients of the electron concentration in the device. Of particular interest are the first and second harmonics of the electron concentration shown in Figures 3.8 b) and 3.8 c). The depletion regions bordering the bulk and source and drain regions have a ridge which is associated with the modulation of the width of the depletion regions. The 39 variation of the electron density is observed along the channel and a minimum is observed at the junction of the channel and the drain which corresponds to pinchoff. The harmonics of the potential distribution are shown in Figure 3.9. It can be seen in Figure 3.9 b) that the response in the polysilicon gate tracks the applied sinusoidal source as well as in the channel. A potential variation is observed in Figure 3.9 c) in the channel near the source at twice the fundamental frequency. The potential near the drain is fixed at pinch off. 40

iR G B

i + D − E

Figure 3.1: Diode circuit 41

−4 x 10 0

20.5 0.2

20 0.4

19.5 0.6

19 0.8

18.5 1

depth (cm) 18 1.2

17.5 1.4

17 1.6

16.5 1.8

16 2 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 −5 position (cm) x 10

Figure 3.2: Doping proﬁle for the 0.4 µm transistor. The units of the doping are − log10 ND NA . 42

600

500

400 m) µ 300 A/ µ Sim: Vb=0.0 V, Vg=2.5 V

Id ( 200 Sim: Vb=0.0 V, Vg = 3.3 V Msr: Vb=0 V, Vg=2.5 V Msr: Vb=0 V, Vg=3.3 V 100

0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Vd (V)

µ Figure 3.3: IDS versus VDS for the 0.4 m transistor. Actual measurements are shown in solid lines while the dashed lines show simulation results. 43

60 m) µ 40 A/ µ Id (

20 Sim: Vb=0.0V, Vd=0.1V Msr: Vb=0.0V, Vd=0.1V

0 0.0 1.0 2.0 3.0 4.0 Vg (V)

µ Figure 3.4: IDS versus VGS for the 0.4 m transistor. Actual measurements are shown in solid lines while the dashed lines show simulation results. 44

VGS(1)

VGS(0) VDD

= . ( )= . Figure 3.5: The HB test circuit. For the simulation VDD 1 0V,VGS 0 0 5 V and ( )= VGS 1 200 mV . 45

−7 x 10 14 f=1 kHz f=178 MHz

8 m) µ 6 (A/ D I

−2 0 0.5 1 1.5 2 2.5 3 periods of the fundamental

( ) Figure 3.6: IDS versus periods of VGS t for the circuit of Figure 3.5. The simulation was performed for a 1 kHz and a 178 Mhz applied signal. 46

−7 x 10 14 f=1 kHz f=178 MHz

8 m) µ 6 (A/ D I

−2 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 V (V) GS

( ) ( ) Figure 3.7: IDS t versus VGS t 47

0 0

−5 10 x 10

15 −2 −1 −4 −3 −6 −5 −7 −5 −8 x 10 depth (cm) 20 −10 −9 position (cm)

0 0

−5 10 x 10

15 −2 −1 −4 −3 −6 −5 −7 −5 −8 x 10 depth (cm) 20 −10 −9 position (cm)

0 0

−5 10 x 10

15 −2 −1 −4 −3 −6 −5 −7 −5 −8 x 10 depth (cm) 20 −10 −9 position (cm)

Figure 3.8: Fourier coefﬁcients of the electron concentration at a) f = 0 MHz, b) f = 100 MHz, and c) f = 200 MHz. The vertical scale is a logarithmic scale. 48

1.2

0.8

0.6

0.4

0.2

0 0

−5 10 x 10

15 −2 −1 −4 −3 −6 −5 −7 −5 −8 x 10 depth (cm) 20 −10 −9 position (cm)

0.1

0.08

0.06

0.04

0.02

0 0

−5 10 x 10

15 −2 −1 −4 −3 −6 −5 −7 −5 −8 x 10 depth (cm) 20 −10 −9 position (cm)

−3 x 10

0 0

−5 10 x 10

15 −2 −1 −4 −3 −6 −5 −7 −5 −8 x 10 depth (cm) 20 −10 −9 position (cm)

Figure 3.9: Fourier coefﬁcients of the potential at a) f = 0 MHz, b) f = 100 MHz, and c) f = 200 MHz. CHAPTER 4 FREQUENCY CONVERSION OF SMALL SIGNALS

4.1 Small-Signal Diode Example

4.1.1 Frequency Conversion of an Applied Small-Signal Voltage

Consider the diode circuit shown in Figure 3.1 on Page 40. The equation describing the current ﬂowing through a purely resistive diode [27]is ( ) vBE t ( ( )) = nV − iB vBE t IS e T 1 (4.1)

where IS is the saturation current, VT is the thermal voltage, and n is the ideality factor. The instantaneous small-signal conductance is then

∂ ( ) ( ) ( )= iB t = iB t gBE t ∂ (4.2) vBE nVT which can be expressed as the sum of its frequency components as ∞ ( )= ( )+ ( ) ω gBE t GBE 0 2 ∑ GBE k cos k 0t (4.3) k=1 ( )= ω when the large-signal voltage source supplies vG t VG cos 0t V to the circuit. The time-varying conductance is what causes the frequency conversion of small-signal voltages into currents at other frequencies [9, 38]. Figure 4.1 shows the small-signal representation of the diode circuit (Figure 3.1) in the presence of an applied small-signal voltage source and a series resistance of 0 ohms. This source is represented as

( )= (ω ) (ω ) vSS t VSS p cos pt (4.4)

49 50 where ωp is the baseband (BB) frequency. Multiplying this phasor by the instantaneous conductance (Equation 4.3), the resulting current response is ∞ ∞ ( )= (ω ) · ( ) ω + ω + ( ) ω − ω iSS t VSS p ∑ GBE k cos k 0 p t ∑ GBE k cos k 0 p t k=0 k=1 (4.5) where the first term represents the current flowing through the diode at the baseband and the upper-sidebands, and the second term represents the currents flowing at the lower-sideband frequencies. Figure 4.2 is a pictorial example of the resulting small-signal currents which results at the lower sidebands (LSBs) and upper sidebands (USBs) of the large-signal harmonics. 4.1.2 Frequency Conversion of Small-Signal Current Generators

Figure 4.3 shows the small-signal representation of the circuit in Figure 3.1 with a small-signal current source added in parallel with the resistor and the time-varying conductance. Once the steady-state solution for the circuit is known, a frequency conversion matrix is formed using Equation 4.3 so that the phasors of the small-signal voltage can be found. For example, if 3 frequency components of the diode current are signiﬁcant, the conversion matrix is then   ( )+ ( ) ( )  GBE 0 GS GBE 1 GBE 2       G (1) G (0)+G G (1) G (2)   BE BE S BE BE    Y =  G (2) G (1) G (0)+G G (1) G (2)   BE BE BE S BE BE     ( ) ( ) ( )+ ( )   GBE 2 GBE 1 GBE 0 GS GBE 1  ( ) ( ) ( )+ GBE 2 GBE 1 GBE 0 GS (4.6) 51 where Gs = 1/Rs. The small-signal voltages at the junction of the diode are then found using     ω − ω  VBE 2 0 p   0           V ω − ωp   0   BE 0      −    V (ω )  = Y 1  I (ω )  (4.7)  BE p   ss p       ω + ω     VBE 0 p   0  ω + ω VBE 2 0 p 0 where Iss (ωp) is the small-signal current injected to the circuit at the baseband. In Sec- tion 4.2, a general approach for small-signal analysis in the presence of a periodic large signal is described.

4.2 Large-Signal Small-Signal Analysis

Large-signal small-signal analysis [8] assumes that there are 2 signals in the system, a periodic large signal and a small signal which does not affect the steady-state operation of the circuit. A conversion matrix relates a small-signal perturbation at one node and sideband frequency to a response at another node and sideband. This matrix is a linearization about the steady state and is referred to as a linear periodically time-varying (LPTV) network. A procedure similar to generating the Jacobian for a steady-state solution can be used for the assembly of the conversion matrix. Figure 4.4 shows the small-signal representation which is used in the literature [8, 19, 20] and is the one used here. The horizontal axis includes both positive and negative frequencies with the center dashed line representing 0 Hz. The dashed lines represent the frequency components of the large-signals in the system. The solid lines represent the small-signal phasors which are a distance of ωp/2π Hz from each harmonic and accounted ω + ω for in the simulation. Being real signals, each sideband at p k 0 has a complex conju- − ω + ω gate at p k 0 , where k is the index of the sideband being considered. These complex conjugate phasors are left out of the ﬁgure for ease of visualization and are accounted ω + ω for in the simulation by scaling the phasor at p k 0 by a factor of 2. 52

Positive frequencies represent USBs and negative frequencies represent the LSBs. Using this representation, it is seen that all of the small-signal frequencies are related by ω integer multiples of the fundamental frequency 0 of the applied stimulus. Since each large-signal harmonic has both USB and LSB, the conversion matrix has twice the dimension of the HB Jacobian. For the small-signal conversion analysis, we start with the frequency domain KCL equation (Equation 3.9 on Page 29) Taking a ﬁrst-order Taylor expansion the resulting small-signal KCL equation is

∂I (V) ∂Q(V) δI = δV + Ω δV +Y δV (4.8) ∂V ∂V where δI is the vector of small-signal currents at each sideband and δV is the vector of small-signal voltages. The small-signal conversion analysis is based on the model that a small-signal voltage ω + ω phasor at node n and frequency p l 0 (ω + ω ) δ ( )=δ ω + ω j p l 0 t vn t Vn p l 0 e (4.9)

ω + ω is related to the small-signal current injected at node m and frequency p k 0 (ω + ω ) δ ( )=δ ω + ω j p k 0 t im t Im p k 0 e (4.10)

For a nonlinear device, these terms are related through ∂ δi (t)= g (t)+ c (t) δv (t) (4.11) m mn ∂t mn n where ∞ ∂Im (k) ( − )ω ( )= j k l 0t gmn t ∑ ∂ ( ) e (4.12) k−l=−∞ Vn l is the instantaneous conductance and

∞ ∂qmn (t) ∂Qm (k) ( − )ω ( )= = j k l 0t cmn t ∂ ( ) ∑ ∂ ( ) e (4.13) vn t k−l=−∞ Vn l 53 is the small-signal capacitance of the device. It can be shown that by applying Equa- tions 4.9–4.13 that ∂Im(k) ∂Qm(k) δIm ωp + kω = + Ω ωp + kω +Ymn ωp + kω δ (k − l) 0 ∂Vn(l) 0 ∂Vn(l) 0 ×δ ω + ω Vn p l 0 (4.14) where    Re Ymn ωp + kω −Im Ymn ωp + kω  ω + ω =  0 0  Ymn p k 0 (4.15) ω + ω ω + ω Im Ymn p k 0 Re Ymn p k 0 represents the admittance of the linear elements present in the circuit and the derivative matrix is given by    0 − ωp + kω  Ω ω + ω =  0  p k 0 (4.16) ω + ω p k 0 0 and accounts for the ∂/∂t terms. Using Equation 3.14 on Page 30, the conversion matrix elements are evaluated in a similar means as the Jacobian matrix entries. The single-sided DFT method for the HB Jacobian elements does not properly account for the mixing of a baseband phasor so a double-sided DFT is used for the small-signal conversion matrix. Transforming Equa- tion 3.14 on Page 30 into the discrete time domain

∂ ( ) S−1 ∂ ( ) Im k = 1 im s ( ( π ( − ) / ) − ( π ( − ) / )) ∂ ( ) ∑ ∂ ( ) cos 2 k l s S jsin 2 k l s S (4.17) Vn l S s=0 vn s

∂ ( ) and Qm k is found in a similar fashion. Splitting the phasors into real and imaginary parts, ∂Vn(l) the current due to resistive mixing is then     R R  δI ωp + kω  ∂I (k)  δV ωp + lω   m 0  = m  n 0  ∂ ( ) (4.18) δ I ω + ω Vn l δ I ω + ω Im p k 0 Vn p l 0 54 the quad representation of the element for the conversion matrix is then   ∂I (k)  GR (k − l) −GI (k − l)  m =  mn mn  = ( − ) ∂ ( ) tmn k l (4.19) Vn l I ( − ) R ( − ) Gmn k l Gmn k l where Gmn is evaluated using Equation 3.24 on Page 31. The block submatrix of the nonlinear conductance elements is then   ( ) (− ) ··· ( − )  tmn 0 tmn 1 tmn 2 2K      ∂I  tmn (1) tmn (0)  m = T =   (4.20) ∂ mn  . .  Vn  . ..   

tmn (2K − 2) tmn (0) where the rows correspond to k =(1 − K,2 − K,...K − 1) and the columns correspond to l =(1 − K,2 − K,...K − 1). The same procedure is carried out for the capacitive terms as well. From the Nyquist Theorem [40], it is necessary to sample a waveform at least twice within the period of the highest frequency considered. While this condition is met for the single-sided Jacobian formulation, it is not met for the elements in Equation 4.17 when |k − l| > K. It is assumed that these terms are negligible and this results in a banded conversion matrix. This is a valid assumption considering that the instantaneous admittance of a nonlinear device is often based on the large-signal currents flowing through the device [44]. For example, the Fourier coefficients for the conductance terms in a resistive diode is ∂ ( ) I (k − l)ω Im k = 0 ∂ ( ) (4.21) Vn l nVT ω th where I k 0 is the k Fourier coefficient of the current flowing through the device. Selec- tion of K is based on the approximation that higher harmonics of current and voltage are negligible (see Section 3.2). 55

Once the conversion matrix is formed, the relationship between a voltage perturbation and a current source is δI = YδV (4.22) where Y is the assembled frequency conversion matrix, δV is the vector of voltage perturbations, and δI is the vector of current sources at all the nodes in the circuit. The voltage response to a small-signal current source is then

− δV = Y 1δI (4.23)

For noise simulation, the columns of δI are correlated noise generators for each sideband frequency and this is discussed in Section 4.3. 4.3 Cyclostationary Noise Analysis

The cyclostationary noise analysis is based on the modulated stationary noise model [13]. The basis of this model is that a noise process may be modeled as

n(t)=h(t)x(t) (4.24) where h(t) is the an instantaneous function of bias and x(t) is a white noise source with x(t)=0 and an autocorrelation of Rx,x (t,t + τ)=δ (τ) This model is valid when the autocorrelation time of the noise processes being modulated is much shorter then the period of the large-signal stimulus. In semiconductor devices, this is valid up into the THz region of operation. This formulation is based on that of Dragone [19] for the case of shot noise in diodes, but it is easily extended for the case of microscopic GR noise sources in Chapter 5. The instantaneous current spectral density of the shot noise in a diode is

( )= ( ) Si t 2qI t (4.25) where q is the electron charge, and I (t) is the large-signal current ﬂowing through the 2/ device and Si has units of A Hz. 56

A typical shot noise waveform is shown in Figure 4.5 a) for the case of a DC bias. Under large-signal conditions, the diode current depicted in Figure 4.5 b) modulates this process and the resulting noise waveform is depicted in Figure 4.5 c) and is what would typically be observed on an oscilloscope. Figure 4.5 d) shows the instantaneous noise power given by Equation 4.25. The large-signal current causes a frequency conversion of the noise phasors which occur at each sideband frequency. Figure 4.6 a) shows noise phasors at the BB, 1st LSB and 1st USB. The Fourier coefﬁcients of the large-signal diode currents are shown in Fig- ure 4.6 b). The noise phasors are modulated by the instantaneous current so that correlated cross-spectral density components appear at the other sidebands in Figure 4.6 c). The noise correlation matrix (NCM) contains the cross-spectral densities between sidebands and is used to calculate the total noise contribution of a semiconductor device. The derivation of the NCM begins with the instantaneous noise amplitude versus time (Figure 4.5 c))[19]. Since we are only considering the harmonically related components which exist at ωp/2π Hz away from each large-signal harmonic, the time-domain representation of the noise amplitude contributed by these harmonically related phasors is given by ∞ (ω + ω ) δ ( )= ω + ω j p k 0 t n t ∑ N p k 0 e (4.26) k=−∞ This equation is recast into the form

δn(t)=h(t)x(t) (4.27) where h(t) models the large-signal modulation and is given by ∞ ω ( )= ω jk 0t h t ∑ H k 0 e (4.28) k=−∞ 57 and x(t) is a white noise source (Figure 4.5 a)) whose harmonically related components are ∞ 1/2 j(ω +sω )t jϕ t x(t)= ∑ ∆ f e p 0 e s (4.29) s=−∞ where ϕs denotes the statistically independent random phase angle at each sideband and ∆ ω + ω f is the bandwidth centered around p s 0. The instantaneous spectral density is given by ∞ ω ( )= ω jk 0t S t ∑ S k 0 e (4.30) k=−∞ and is related to h(t) through h(t)= S(t) (4.31)

It can be shown using Equations 4.28, 4.30, and 4.31 that ∞ ω = ω ( − )ω S r 0 ∑ H k 0 H r k 0 (4.32) k=−∞ and by substituting Equations 4.29 and 4.28 into Equation 4.27 that ∞ ∞ / (ω + ω ) ϕ δ ( )=∆ 1 2 ( − )ω j p k 0 t j s n t f ∑ ∑ H k s 0 e e (4.33) s=−∞ k=−∞

By comparison of this expression with Equation 4.26, the noise amplitude phasors are given by ∞ / ϕ ω + ω = ∆ 1 2 ( − )ω j s N p k 0 f ∑ H k s 0 e (4.34) s=−∞ and using Equation 4.32 and that    = ϕ − ϕ 1 r s j s j r = e e  (4.35)  0 r = s

ω + ω ω + ω the cross-spectral density between p k 0 and p l 0 is then ∗ N ωp + kω N ωp + lω 0 0 = S (k − l)ω (4.36) ∆ f 0 58 so that the noise correlation between sidebands is based on harmonics of the instantaneous spectral density. These elements are then positioned into the NCM. Each column represents the spectral density at a sideband and its cross-spectral density at the other sideband frequencies. The correlated sideband components for a noise phasor originating at the baseband is demonstrated in Figure 4.6 c) for the diode example. Equation 4.22 is then used to couple the noise current fluctuations to a desired node in the circuit. In order to account for the correlation between phasors originating at different sideband frequencies, it is necessary to sum up the noise sources quadratically. The resulting voltage density is then. δVδVH = Z δNδNH ZH (4.37) where the angular brackets denote a statistical average, H denotes the Hermitian conjugate, Z = Y−1, and the NCM is given by δNδNH . The voltage noise spectral density which is observed at electrode i at a frequency of ω + ω p k 0 is then S = Z BZH (4.38) vi,k i,k i,k where Zi,k is the row of Z corresponding to the electrode and δNδNH B = (4.39) ∆ f is the NCM. For shot noise in a diode, the terms which go into the the NCM [20] are = ( − )ω Bk,l 2qI k l 0 (4.40) ( − )ω ( − )ω where I k l 0 is the steady-state current flowing at k l 0. In order for this equation to be valid, it is necessary that I (t) > 0 in Equation 4.25 so that the time-varying current spectral density is always a positive quantity. If the current does not meet this condition, it would be necessary to use the Fourier coefficients of |I (t)| instead. 59

4.4 Semiconductor Noise Analysis

4.4.1 Frequency Conversion Matrix

The frequency conversion matrix for the semiconductor is analogous to the one for circuit simulation discussed in Section 4.2. Like periodic steady-state simulation, small- signal device simulation is more complicated than circuit simulation because there are additional equations per node. Simulation of the conversion of external small-signals is straightforward. Since the circuit and device equations are coupled, it is only necessary to perturb the equation representing the external source and observe the response. This type of simulation is useful in determining how the small-signal transconductance of three- terminal devices are affected by the presence of a large-signal carrier. 4.4.2 Noise Analysis for the Semiconductor

The frequency conversion matrix is used to simulate the transfer of noise ﬂuctuations in the semiconductor mesh to voltage or current noise at the contacts of the device. The method described is analogous to the case of DC steady-state noise simulation, except for the correlation of noise phasors at different sideband frequencies.

Since a modulated noise source which originates at a speciﬁc sideband frequency has correlated components at all the other frequency sidebands, it is necessary to inject these frequency components simultaneously. This process must be repeated for each noise source at each sideband and each position within the device. Each source which was once represented as a column vector for the DC steady state becomes a submatrix of the NCM where each column represents the noise spectral density at each sideband and its cross- spectral density at other sidebands. The total spectral density due to microscopic diffusion noise sources is found using [10, 11] ∞ , ∗ x x ω + kω ,ω + lω = G x (k,u) · K (u,v) · G x (l,v) ∂r (4.41) Sdiff p 0 p 0 ∑ ∑ α ξ ,ξ β r α=n,p u,v=−∞ α α 60

x where Gα (k,u) is the spatially dependent vector Green’s function which couples velocity ∂ ω + ω ﬂuctuations for each differential volume r within the device at p u 0 to electrode x at

ωp +kω . The noise source strength for velocity fluctuations K (u,v) is the correlation 0 ξα ,ξα ω + ω ω + ω α = of microsopic velocity fluctuations between p u 0 and p v 0 in the conduction ( n) or valence (α = p) band. The noise source strength tensor used in Equation 4.41 for velocity fluctuations is 2 K (u,v)=4q Dα α (u − v)ω (4.42) ξα ,ξα 0 α ω th where Dα is the diffusivity and k 0 is the k harmonic of the carrier density. It is important to note that the presence of an external load can have a significant effect on the frequency conversion of signals. This is demonstrated using the port representation for noise correlation in a 2-terminal device and is shown in Figure 4.7. At each sideband ω + ω δ frequency, p k 0, there exists a current generator Nk whose cross-spectral density with noise generators at other sidebands is given by Equation 4.36. The nonlinear device draws δ a current Ik from each sideband which is determined by the parallel combination of its ( ) instantaneous admittance y t and its termination Yk. If the termination is a short-circuit, the noise current cannot be frequency converted to other frequencies via the time-varying admittance. Short-circuit terminations are used in Chapters 5 and 6 to find the current spectral density of GR noise at each sideband due to microscopic noise source modulation and independent of small-signal mixing. 4.4.3 Iterative methods for Frequency Conversion and Noise Analysis

Similar to the linear noise simulation case, it is possible to take advantage of the matrix method in Section 2.5.3 which reduces the number of factorizations required to one per contact at a speciﬁed sideband [34, 37].

Since each dimension of the frequency conversion matrix is twice that of the steady- state Jacobian, the computational cost of the inverting the matrix is greatly increased over a HB simulation. The noise analysis is adapted for use with Krylov subspace methods 61 through the use of matrix-vector products. The conversion matrix is divided into time- independent and time-dependent parts given by

= + Ω Y Yt Y f (4.43)

Ω where Y f accounts for the time-dependent terms in the continuity equations, is the fre- ∂/∂ quency domain t operator, and Yt accounts for the time-independent terms of the semiconductor device equations. The matrix Yt is now formed [3] from

= Γ Γ−1 Yt yt (4.44)

Γ where yt is the time-sample matrix for frequency conversion and is the complex Fourier transform operator and Γ−1 is the complex inverse Fourier transform operator. For the ∂/∂t terms, = Γ Γ−1 Y f y f (4.45)

The transfer of a conversion matrix for a small signal is then = Γ Γ−1 + ΩΓ Γ−1 ∆ − 0 yt y f Xi N (4.46) where N is a vector of correlated noise generators and ∆X is the response. It would then be necessary to repeat this for every noise phasor originating at every node in the device. As discussed for the DC steady state in Section 2.5.3 on Page 15, the number of matrix factorizations can be reduced to one by minimizing T = Γ Γ−1 + ΩΓ Γ−1 ˜ − r yt y f G ex,k (4.47)

˜ where G is the vector of scalar Green’s functions and ex,k is an elementary vector with a ω + ω 1 in the row corresponding to the desired contact x at frequency p k 0. This equation is then recast into the form of matrix-vector operations by applying the transpose operator to each individual matrix. The vector G˜ consists of the scalar Green’s functions for all the device nodes at all frequencies to the external contact at frequency k. The vector Green’s 62 functions are found from

x 1 x Gα (k,u)= ∇G˜ α (k,u) (4.48) q and the diffusion noise is found using Equation 4.41. Although not explicitly shown in this section, the single-tone preconditioner described in Section 3.5.2 is applied to solve the system of equations so that only same-frequency conversion is considered. The inversion of this preconditioner is then equivalent to inverting 2K − 1 small-signal AC matrices. 4.5 Simulation Results

The diffusion noise simulation of the 1-D diode structure demonstrated in Sec- tion 2.5.4 was repeated for the case of the periodic steady state. The circuit of Figure 3.1 was used to simulate the voltage noise of the diode at the base contact. Extracting the values of IS and nVT in Equation 4.1 on Page 49 from one DC bias point, HB noise simulation was performed using Matlab [45] and the Agilent Advanced Design System (ADS) [2]. Good agreement between FLOODS, ADS, and Matlab were found and the results are shown in Figure 4.8. Results for the conversion of low-frequency GR noise are shown in Chapters 5 and 6. 63

Vss g(t)

Figure 4.1: The diode as a small-signal conductance in the presence of a small-signal voltage source 64

Large Signal

Small Signal

ω ω ω ω 0 p 0 2 0 3 0

Figure 4.2: Small-signal current response of the diode 65

Rs Iss g(t)

Figure 4.3: Small-signal representation of the diode circuit 66

Large Signal

Small Signal

ω − ω ω − ω ω ω + ω ω + ω p 2 0 p 0 p p 0 p 2 0

Figure 4.4: Small-signal representation of the mixing process 67

4 3 2 1 0 −1 −2 −3 −4 0 0.5 1 1.5 2 2.5 3 3.5 4

0.02 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 0 0.5 1 1.5 2 2.5 3 3.5 4

−1

−2

−3 0 0.5 1 1.5 2 2.5 3 3.5 4

8 7 6 5 4 3 2 1 0 0 0.5 1 1.5 2 2.5 3 3.5 4

Figure 4.5: Time-domain noise example for shot noise in a resistive diode. Figures are shown with arbitrary units. a) White noise source; b) Instantaneous diode current; c) Noise amplitude as a function of periods of the large-signal source; d) Instantaneous noise power. 68

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 −5 −4 −3 −2 −1 0 1 2 3 4 5

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 −6 −4 −2 0 2 4 6

Figure 4.6: Frequency-domain noise example for shot noise in a resistive diode. Figures are shown with arbitrary units. a) Harmonically related noise phasors for a white noise source; b) Fourier coefﬁcients of the diode current; c) Correlated noise spectral densities from the modulation of the noise phasors in a). 69

ω − ω - p 2 0 # δI −2- δ Y−2 N−2 "! y(t)

ω − ω - p 0 # δI −1- δ Y−1 N−1 "! y(t)

ωp - # δI 0 - Nonlinear Device δ Y0 N0 "! y(t)

ω + ω - p 0 # δI 1 - δ Y1 N1 "! y(t)

ω + ω - p 2 0 # δI 2 - δ Y2 N2 "! y(t)

Figure 4.7: Port representation of a nonlinear device and its terminations at each sideband frequency 70

−13 x 10 1.7 Matlab FLOODS ADS 1.6

1.5

1.4 /Hz) 2 m

µ 1.3

2 (V v S 1.2

1.1

0.9 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 V (1)(V) BE

Figure 4.8: Comparison of noise simulation between FLOODS, ADS, and Matlab. The x axis is the magnitude of the fundamental frequency of the large-signal voltage applied to the base of the diode when the AC source was swept from 0 to 200 mV. CHAPTER 5 GENERATION-RECOMBINATION NOISE IN SEMICONDUCTOR DEVICES

5.1 Introduction

In compact circuit models, ﬂicker noise is modeled as an equivalent input noise source with a current spectral density of the form

KIα S = (5.1) i f β where K, α, and β are experimentally determined constants, I is the current ﬂowing through the device, and f is the frequency at which the noise is being observed. When

β ≈ 1, this is referred to a 1/ f or 1/ f -like noise. In general, such noise is called low- frequency noise if it dominates at low frequencies and becomes negligible at high frequencies. Because of the frequency dependence of this type of noise, and the fact that α and β can be non-integer numbers, it is not clear how low-frequency noise should be modeled under periodic, large-signal conditions. It is sometimes assumed that ﬂicker noise processes are stationary [46, 47] so that only DC component of the current is used, even when time-varying bias conditions exist. Circuit simulators, such as Agilent Advanced Design System (ADS) [2], currently use this model.

It is not known if the modulated stationary noise model described in Section 4.3 is applicable to nonlinear noise simulation of circuits. The primary difficulty is that the long- term autocorrelations of a low-frequency noise process seem to preclude the use of this model. Recently, researchers [1] have begun to apply this type of cyclostationary noise analysis to flicker noise by treating the frequency-dependent term in Equation 5.1 as a stationary filter multiplied by the time-varying bias-dependent term. To date, experimental and theoretical justification for the application of this model has been lacking [13, 22].

71 72

The McWhorter theory [24] states that the superposition of many Lorentzian spectra with continuously distributed time constants results in 1/ f noise. Each Lorentzian is of the form N τ S ∝ t (5.2) i 1 +(ωτ)2 where Nt is the trap density, ω is the frequency being observed, and τ is the characteristic time of the trap level. These Lorentzian spectra arise from the trapping and de-trapping of carriers in defect centers in a semiconductor device [48]. Figure 5.1 shows a pictorial example of how the superposition of these spectra results in 1/ f -like noise. If the low-frequency noise in a device is the result of McWhorter 1/ f -like noise, it is possible to simulate it in a device simulator under periodic large-signal conditions since the microscopic generation-recombination (GR) processes which produce the Lorentzian spectra are white and the modulated stationary noise model can be used. In this chapter, the theory for GR noise simulation under cyclostationary conditions is presented. Simulation results showing the upconversion of low-frequency GR noise are presented for 1-D resistors and junctions. In Chapter 6, GR noise simulations of a 2-D BJT are presented as well as a methodology to extract correlated equivalent current-noise generators from device simulations.

5.2 GR Noise for the DC Steady State

In this section, we continue with the noise simulation method of Section 2.5, extended for the case of trap-assisted GR noise. The transitions between a trap level in the forbidden band and the conduction or valence band are shown in Figure 5.2. For each trap level, it is necessary to add a continuity equation [49, 50]

∂n t = R − G + G − R + γ (5.3) ∂t n n p p nt

γ where nt is a ﬂuctuation in the transition rate, and Rα and Gα are the recombination and generation rates for electrons in the conduction band (α = n) and holes in the valence band (α = p). It is assumed that these transitions are Shockley-Read-Hall processes [51]. 73 SpectralVoltage Density (a.u.)

100 101 102 103 Frequency (Hz)

Figure 5.1: Example of how the superposition of Lorentzian spectra results in ﬂicker noise 74

This equation assumes that transitions occur directly between each trap level and the conduction and valence bands without interaction with any other trap levels which may be present. This is because, if inter-trap transitions were allowed, a Lorentzian frequency spectrum would occur, instead of a 1/ f -like spectrum [12]. The Poisson, electron- continuity, and hole-continuity equations are then modiﬁed accordingly ∇2ψ = −q − + − + − ε p n ND NA Nt nt (5.4) ∂n 1 = ∇ · J + G − R + γ (5.5) ∂t q n n n n ∂ p 1 = − ∇ · J + G − R + γ (5.6) ∂t q p p p p where Nt is the density of donor-like traps and nt is the density of trapped electrons. γ = γ − γ Accounting for the fact that nt p n, the noise source strengths [52] are then

Kγ ,γ = −Kγ ,γ = 2(Rn + Gn) n n n nt

Kγ ,γ = Kγ ,γ = 2(Rp + Gp) p p p nt (5.7) = = Kγn,γp Kγp,γn 0

Kγ ,γ = 2(Rn + Gn + Rp + Gp) nt nt The noise contributions of the rate ﬂuctuations to this trap-energy level to voltage spectral density of a contact is then (see Section 2.5.2) ∗ = ˜ α γ ,γ ˜ ∂ Sv ∑ G K α β Gβ r (5.8) α,β= , , n p nt For the DC steady state, the time-derivative term in Equation 5.3 is 0 and it is possible ψ to solve exactly for nt in terms of the other solution variables ( , n, p). It is then only necessary to solve Equations 5.4–5.6 for the DC solution, no matter how many trap levels are present. The resulting DC Jacobian is then reduced from a 4N by 4N matrix to a 3N by 3N matrix where N is the number of nodes being considered in the simulation. ψ Equation 5.3 is linearized to ﬁnd the variation of nt as a function of changes in , n, p, 75

γ γ γ γ and nt . By modifying n and p and adding a noise source ψ to Equation 5.4, the noise γ contribution of nt can be accounted for by a 3N by 3N system of equations [50, 53]. 5.3 Periodic Steady-State GR Noise

For the periodic steady state, it is not possible to solve for nt in terms of the other ∂ /∂ = solution variables since nt t 0. For each trap energy level, a continuity equation (5.3) must be solved in addition to Equations 5.4–5.6. The resulting number of equations is then ( − ) + ( − ) + 2K 1 3 Ntraps Nnodes for the steady-state simulation and 4K 2 3 Ntraps Nnodes for the cyclostationary noise simulation. It is apparent that the number of equations to be solved can quickly become prohibitive based on the large memory requirements of having multiple trap levels. In addition to the correlated noise components in the transitions between the trap levels and the conduction and valence bands, there are also correlated frequency components which occur as a result of the modulation of the trap- and band-carrier densities. Similar for the case of shot noise in Section 4.3, the autocorrelation of the microscopic transition- rate ﬂuctuations has an instantaneous dependence on bias and is given [52]by

( , + τ)= ( ( )+ ( )) δ (τ) Rγα ,γα t t 2 Gα t Rα t (5.9)

When a semiconductor device is driven by a periodic stimulus with a fundamental fre- ω quency of 0, the nonlinearity of the semiconductor equations can cause the instantaneous { ,ω , ω ,...} GR rates to respond at 0 0 2 0 . The fluctuations present in the semiconductor interact with the time-varying impedance field to generate components at sidebands of the large-signal bias and is depicted in Figure 4.4 on Page 66 where ωp is the baseband frequency. The magnitudes of the GR fluctuations may be then instantaneously modulated by the macroscopic carrier densities and potential which results in correlated components at each sideband and must be accounted for in the simulation. By analogy with shot noise in a diode [19] and extension to the case of correlated fluctuations, the noise source strengths 76 are then Kγ ,γ (k,l)=−Kγ ,γ (k,l)=Sγ (k − l)ω (5.10) n n n nt n 0 Kγ ,γ (k,l)=Kγ ,γ (k,l)=Sγ (k − l)ω (5.11) p p p nt p 0 ( , )= ( , )= Kγn,γp k l Kγp,γn k l 0 (5.12)

Kγ ,γ (k,l)=Kγ ,γ (k,l)+Kγ ,γ (k,l) (5.13) nt nt n n p p where ∞ − jkω t Sγ kω = 2 Rγ ,γ (t,t)e 0 ∂t (5.14) α 0 −∞ α α is the kth harmonic of the instantaneous spectral density of the transition-rate ﬂuctuations ∗ given by Equation 5.9 and Kγ ,γ = K . α β γβ ,γα For trap-assisted GR ﬂuctuations, the spectral density between at an external circuit ω + ω node x at p k 0 is ∞ x,x x x ∗ ω + ω ,ω + ω = ˜ ( , ) · γ ,γ ( , ) · ˜ ( , ) ∂ Sgr p k 0 p l 0 ∑ ∑ Gα k u K α β u v Gβ l v r r α,β= , , , =−∞ n p nt u v (5.15)

x where G˜α (k,u) is the scalar Green’s function which couples GR fluctuations for each dis- α = α = α = cretized volume from the electron- ( n), hole- ( p), or trap- ( nt) continuity ω + ω ω + ω equation at frequency p u 0 to a response at x at frequency p k 0. The method for finding the scalar Green’s functions are discussed in Section 4.4.3. In the simulation results in this chapter and in Chapter 6, modified nodal analysis [39] is used so that x can represent current through a voltage source. 5.4 Simulation Results

5.4.1 Resistor Simulation

Simulation setup. A cyclostationary noise simulation was performed for { ,ω , ω } µ = 0 0 2 0 on a 2 m Si resistor using the circuit of Figure 5.3 with Rs 10Rm and 77

= . ω 1 Vi 1 01cos 0t V. The resistor Rm was modeled in FLOODS with one trap level and = 13/ 3 = −7 3/ = −8 3/ = 15/ 3 − = Nt 10 cm , cn 10 cm sec, cp 10 cm sec, ND 10 cm , and EC ET 0.28 eV.

Discussion. Figure 5.4 shows the GR noise spectrum of the upper sidebands (USBs) ω when 0 is varied. The 1 MHz spectrum is larger than the others since lower sideband noise has folded over the 0 frequency axis and is superimposed over the USB spectrum. The ratio = = of the f0 100 MHz, 10 GHz to the f0 0 Hz spectrum is 4, validating the model discussed by Lorteije and Hoppenbrouwers [54] in which bias-independent resistance fluctuations are coupled out to the contact by the time-varying current. This is represented by the ∆R and ∆V components shown in Figure 5.3. Figure 5.5 compares the magnitude of the spatially dependent scalar Green’s function ˜ ( , ) Gnt 1 0 for each input frequency. This function couples the low-frequency fluctuations + at 1 Hz from the trap level to the USB at f0 1 Hz. The noise sources in the electron- and hole-continuity equations did not significantly contribute to the noise. Since the simulation assumed an infinite recombination velocity at the contacts, fluctuations in nt close to the γ ( , ) contacts have less of an affect on the total noise. The nt 1 0 for the 10 GHz AC bias is slightly distorted and is attributed to the capacitive effects within the resistor. From a compact modeling perspective, these simulation results would not have been predicted if the stationary low-frequency noise model presented by Gray and Meyer [47] was used. The resistance fluctuation model of Lorteije and Hoppenbrouwers [54] predicts how upper and lower sideband noise is generated when an AC current is passed through a noisy linear resistor. This model is directly compatible with the modulated stationary noise model, which is presented in Section 6.3 on Page 105.

1 ω Being a linear resistance, no signiﬁcant response was expected or observed at 2 0. 78

Table 5.1: Trap parameters for the 5 µm resistor example

Level 1 Level 2 Units Type acceptor donor · 11 · 11 / 3 Nt 8 10 5 10 cm −9 −8 3 cp 9 · 10 2.3 · 10 cm /sec 3 cn 0 0 cm /sec − . . ET EV 0 4 0 32 eV

5.4.2 Resistor with 2 Trap Levels { ,ω ,..., ω } µ Simulation setup. A simulation was performed for 0 0 4 0 on a 5 mSi p+/p/p+ resistor to which an AC bias was applied directly. The resistor was modeled in FLOODS with two trap levels with the parameters in Table 5.1. Figure 5.6 shows the hole and trap carrier densities for a DC bias of 0.5 V. The p+ doping at the contacts causes spillover of holes into the p region so that the position of the most active traps varies with bias and temperature. Figure 5.7 shows the resulting current density spectra for the same DC bias with the contributions of each trap separated. Under a bias of 0.1 + 0.1cos 2π107t the BB and USB spectra are placed on the same plot shown in Figure 5.8. To be able to compare the shape of the USB with the BB, f0 was subtracted from the USB spectrum. The relative magnitude of USB to BB GR noise versus increasing AC bias is shown in Figure 5.9 and a

DC bias of 1.0 V. The USB noise approaches that of the AC case for increasing bias.

Discussion. While stationary resistance ﬂuctuation models [54] are appropriate for a homogeneous resistor [17], the resistor and example shown here is more complicated since the noise sources and impedance ﬁeld are both spatially and bias dependent. The exponential dependence of carrier and trap densities on potential allows for USB levels to approach the BB levels in both Figure 5.9. The slight difference in shape between the USB and BB spectra in Figure 5.8 may also be due to this nonlinear nature.

5.4.3 Diode Simulation with Majority Carrier Noise in Quasi-Neutral Region + µ = 18/ 3 Simulation setup. Ap -n diode with a 0.1 m emitter doped with NA 10 cm = 15/ 3 and a base doped with ND 10 cm was simulated with traps distributed across the 79

= 14/ 3 = −11 3/ = − = . semiconductor with NT 10 cm , cn 10 cm sec, cp 0 and Ec ET 0 26 eV for a donor-like trap. This results in a signiﬁcant trapping noise in the quasi-neutral region of the device under low- and high-level injection at the location where the quasi-Fermi level for electrons coincides with the trap level. The current spectral density versus DC bias is shown in Figure 5.10 + ω A simulation was performed with a bias of V0 V1 cos 0t for a 1-D device with 751 = . grid points and for 5 harmonics, including DC. The bias was chosen with V0 0 7 V and 7 V1 was swept from 0 to 200 mV at 10 Hz. The relatively low number of harmonics used in the simulation was possible since the bias was applied directly to the semiconductor device which reduced the amount of distortion induced by the higher frequency currents in the presence of a termination. Figure 5.11 shows the GR noise plateaus at the BB and USB as well as the diffusion noise level for increasing AC bias. Figure 5.12 shows the BB and USB spectra for an AC bias of 0 and 100 mV.

Discussion. The junction case is a more complicated situation than the resistor. Not only are the scalar Green’s functions for electron, hole, and traps spatially and bias dependent, but the magnitude and position of the microscopic ﬂuctuations are as well. For a DC bias, the position of the trap with the most signiﬁcant noise is located where the electron quasi-Fermi level crosses the trap energy level. Under AC bias conditions, the position in the device where this occurs is dependent on the instantaneous bias.

Another important effect is that the baseband GR noise rises above the shot noise floor of the diode for increasing AC bias when the sidebands are short-circuited. While the DC voltage bias is constant, the DC current increases due to the nonlinear i-v characteristic of the device. As shown in Figure 5.10, the GR noise increases faster than the shot-noise floor. In addition, the GR transitions rates are proportional to the DC carrier concentrations which have a similar bias dependence. In Figure 5.11, the USB GR noise plateau approaches the BB level. This demonstrates that the BB noise may be modulated by the AC currents flowing through the device, as was 80 discussed for the linear resistor (Section 5.4.1). This will be further investigated for the next simulation result (Section 5.4.4). 5.4.4 Diode Simulation with GR Noise in the Space Charge Region { ,ω ,..., ω } µ Simulation setup. A simulation was performed for 0 0 5 0 on a 3 m ω = π 7 diode example using a doping and trap profile as shown in Figure 5.13 and 0 2 10 Hz. = 17/ 3 The acceptor-like traps were located in the space charge region with Nt,peak 10 cm , = −12 3/ = −11 3/ − = . cn 10 cm sec, cp 10 cm sec, and EC ET 0 55 eV. This is verified as shown in Figure 5.14. Since the modified nodal analysis was used to formulate the equations of circuit elements, it is possible to simulate the current noise density flowing through a voltage source directly applied to device contact. Figure 5.15 shows how the GR noise plateau, diffusion noise, and DC current increases under increasing DC bias. Figure 5.16 shows how the GR = . noise plateau increases under increasing AC bias (0-190 mV) at VDC 0 7 V. The resulting = noise spectrum for VAC 100 mV is shown for the baseband (BB) and USB in Figure 5.17. Figures 5.15 and 5.16 consider the diffusion noise level separate from the GR components while Figure 5.17 considers the total noise for each sideband. It was found that the dominant noise contribution was from the Langevin term in the trap-continuity equation. The harmonics of the microscopic noise sources and the scalar Green’s function which couple them to the noise current are shown in Figure 5.18.

Discussion. It can be shown by analytically applying the admittance matrix method [20] that it is not possible for a stationary low-frequency equivalent current noise generator to upconvert when a diode is AC short circuited. Since the diode simulations were performed under this condition, Figures 5.16 and 5.17 show that a compact noise model for flicker noise would require a cyclostationary model and thus contradicts the arguments for a stationary noise model such as that in Gray and Meyer [47]. While the stationary resistance fluctuation model of Lorteije [54] is appropriate for the resistor, the diode is a much more complicated situation since the noise sources and 81 impedance field are both spatially and bias dependent. Any compact model for the diode is then specific to the region of the device which is contributing the noise. It should be noted that Equation 5.15 differs from that in Bonani et al. [11] which considers band-to-band transitions and in Cappy et al. [10] which uses equivalent current- density fluctuations. The equivalent model would also be appropriate for mobility fluctuations [12] and the vector Green’s function would then be used to simulate the contributions to an external contact. This model would be appropriate where the macroscopic quantities 2 are invariant to applied bias and the local noise spectral density shows a J dependence, such as the poly-mono Si interface in the emitter of a bipolar junction transistor [18].

A cyclostationary noise analysis may be required for sources in the space charge region of a junction, such as end-of-range defects in ion-implanted p+ shallow junctions [55]. In Figure 5.18 a) it appears that the noise sources are cyclostationary and the ﬁrst 3 frequency components are of the same order of magnitude for this AC bias condition.

The scalar Green’s functions in Figure 5.18 b) mainly couples the BB components to the output. This supports the equivalent current-density ﬂuctuation model [10] in which only the DC component of the microscopic current ﬂuctuations are considered. An interpreta- tion of this is that while the GR transition rates may vary instantaneously with applied bias, the trap is not able to respond as quickly. These simulations were only performed for AC short-circuited conditions and further investigation is necessary to determine whether this is always the case. Regardless of whether or not the microscopic noise sources are stationary, a cyclostationary noise model would always be required for circuit simulation due to the time-varying nature of the Green’s functions. 82

EC Rn = cnn(Nt − nt) Gn = ennt

Rp = cp pnt Gp = ep (Nt − nt)

Figure 5.2: Transitions between the trap energy level and the conduction and valence band. The constants cn,cp and en,ep are the capture and emission coefﬁcients for electrons and holes. 83

Rs V+∆V

Rm Vi

∆R

Figure 5.3: Resistor simulation circuit 84

-14 10

-15 10

-16 10

-17

/Hz) 10 2

(V -18 10V S f0 = 0 Hz -19 6 10 f0 = 10 8 f = 10 -20 0 10 10 f0 = 10 -21 10 0 1 2 3 4 5 6 7 8 9 10 10 10 10 10 10 10 10 10 10

f-f0 (Hz)

Figure 5.4: Upper sideband noise spectrum 85

-20 3×10

-20 2×10 (1,0) t n G

-20 f = 0 Hz 1×10 0 6 f0 = 10 8 f0 = 10 10 f0 = 10 0 -5 -4 -4 -4 0 5×10 1×10 2×10 2×10 x (cm) ˜ ( , ) Figure 5.5: Comparison of Gnt 1 0 versus position 86

N 16 A 10 Holes

) Trap 1 3 Trap 2 14 10

12 10 concentration (/cm

10 10 012345 x (µm)

Figure 5.6: Hole and trapped electron concentrations for a DC bias of 0.5 V 87

-24 10 Diffusion Trap 1 -25 Trap 2 10 Total /Hz)

2 -26 10 (A i S

-27 10

-28 10 3 4 5 6 7 8 10 10 10 10 10 10 Frequency (Hz)

Figure 5.7: GR Noise Spectrum for two trap levels for a bias of 0.5 V 88

-25 10 /Hz)

2 -26 10 (A i S

-27 10 BB USB Diffusion -28 10 3 4 5 6 10 10 10 10 Frequency (Hz)

Figure 5.8: Comparison of the BB and USB spectrum 89

-25 10 /Hz)

2 -26 10 (A i S

-27 10 BB USB Diffusion -28 10 0.2 0.4 0.6 0.8 1 AC Bias (V)

Figure 5.9: Comparison of the BB and USB plateau levels for increasing AC bias 90

-18 -4 10 10

-8 10 -24 10 I -12 DC 10 /Hz) 2

-30 -16 I(A)

(A S 10i gr 10 S Sdiff -20 10 -36 10 -24 10

0 0.2 0.4 0.6 0.8 DC bias (V)

Figure 5.10: Diffusion noise, GR noise plateau, and DC current versus DC bias 91

-20 10

Sgr BB -21 10 Sgr USB

Sdiff -22 10 /Hz) 2 (A i -23 S 10

-24 10

-25 10 0.00 0.05 0.10 0.15 0.20

V1 (V) + ω Figure 5.11: Diffusion and GR noise plateaus versus increasing AC bias, V0 V1 cos 0t 92

-21 10

SGR BB (V1 = 0 V)

SGR BB (V1 = 100 mV) -22 S USB (V = 100 mV) 10 GR 1 /Hz)

2 -23 10 (A i S

-24 10

-25 10 1 2 3 4 5 6 7 10 10 10 10 10 10 10 f (Hz), f-f0 for upper sideband + ω = , Figure 5.12: Noise spectrum for a bias of V0 V1 cos 0t , with V1 0 100 mV 93

19 10

18 NA

) 10 3

17 10 NT

16 10

Concentration ( #/cm 15 10 ND

14 10 -5 -4 -4 -4 -4 0.0 5.0×10 1.0×10 1.5×10 2.0×10 2.5×10 x (cm)

Figure 5.13: Doping proﬁle for the diode 94

19 10 Doping 18 p

) 10 n 3 t n

17 10

16 10

Concentration ( #/cm 15 10

14 10 -5 -4 -4 -4 -4 0.0 5.0×10 1.0×10 1.5×10 2.0×10 2.5×10 x (cm)

= . Figure 5.14: Carrier concentrations at VDC 0 7V 95

-22 -3 10 10

-23 -4 10 10

-24 -5 10 10 /Hz)

2 -25 -6 10 10 I (A) (A I S -26 -7 10 10

SGR @ 1Hz -27 -8 10 Sdiff 10 IDC -28 -9 10 0.5 0.7 0.910

VDC (V)

Figure 5.15: GR spectrum versus applied DC bias 96

-4 10

-23 10

-5 10 /Hz) 2

-24 I (A) (A 10I S

I DC -6 10 Sdiff @ 1 Hz S @ 1 Hz -25 GR 10 SGR @ (f0 + 1) Hz 0.00 0.05 0.10 0.15 0.20

VAC (V)

Figure 5.16: GR spectrum plateaus versus applied AC bias for ( )= . + π = 7 V t 0 7 VAC cos 2 f0t and f0 10 Hz 97

-23 10 /Hz) 2 (A I S

-24 BB V = 0 V 10 AC BB for VAC = 0.1 V USB for VAC = 0.1 V

0 1 2 3 4 5 6 7 10 10 10 10 10 10 10 10 f (Hz) for BB f-f0 (Hz) for USB ( )= . + π = 7 Figure 5.17: BB and USB GR spectrum for V t 0 7 VAC cos 2 f0t and f0 10 Hz 98

21 4×10 S (0) G (1,0) -23 nt nt 10 S (1) G (1,1) nt nt S (2) G (1,2) 21 n n -24 3×10 t t 10

-25 21 10 2×10

-26 10 21 1×10

-27 10

0 -5 -5 -5 0 5×10 0 0.0 5.0×10 x (cm) x (cm)

a) b) Figure 5.18: Microscopic noise source Sγ kω = Sγ kω + Sγ kω a) and scalar nt 0 n 0 p 0 Green’s functions b) in the space charge region of the diode for V (t)=0.7 + 0.1cos 2π107t . CHAPTER 6 CROSS-SPECTRAL DENSITY SIMULATION

6.1 Introduction

In this chapter, simulation results are presented for the frequency conversion of both diffusion noise and trap-assisted GR noise under periodic large-signal conditions for a bipolar junction transistor (BJT). In our investigation we also simulate the current cross-spectral densities (CSDs) between sidebands for both the case of diffusion noise and trap-assisted GR noise. This information is equivalent to the noise correlation matrix (NCM) for large- signal circuit simulation. Well known results for the case of diffusion noise [19, 20] in junctions are demonstrated using the simulator. In addition, we show simulation results which verify the NCM entries postulated for modulated stationary noise sources [13, 21, 22] when applied to microscopically cyclostationary GR noise sources. 6.2 Noise Simulation Under Periodic Large-Signal Conditions

6.2.1 The Periodic Steady State ω When a periodic bias with a fundamental frequency of 0 is applied to a semiconductor, the exponential relationship between potential and electron- and hole-carrier density results in higher order harmonics of these quantities being generated. The harmonic balance (HB) method is used to simulate the periodic steady-state solution of the semiconductor device. This is a frequency-domain technique in which the Fourier coefficients of the solution variables are found [3]. While time-domain methods such as the transient or shooting method could be used to find the periodic steady state [56], the advantage of the HB method is that the procedure to find the linearized equations used to converge upon a solution can also be used for finding the small-signal conversion matrix required for a noise analysis.

99 100

Since it is necessary to solve for all frequency components for each equation at each node in the semiconductor mesh simultaneously, the resulting matrix is often too large to solve using direct methods. Recent advances in the use of iterative methods for HB circuit simulation have been applied to semiconductor simulation so that it is a tractable problem for modern computer workstations [4]. 6.2.2 The Impedance Field Method

By linearizing the frequency-domain equations, a linear periodically time- varying (LPTV) network is formed which models how fluctuations within the semiconductor are frequency converted to the sidebands of the large-signal currents flowing through the ω + ω device. As depicted in Figure 4.4 on Page 66, a phasor located at a frequency of p k 0 is frequency converted via the harmonics of the time-varying impedance field to generate components at each of the other sidebands. The Shockley-Read-Hall (SRH) recombination model [51] (depicted in Figure 5.2 on Page 82) was used to describe band transitions and noise. Table 6.1 defines the constants used in the SRH equations. For the presence of a trap level, the trap-continuity equation is then ∂n t = R − G − R + G + γ (6.1) ∂t n n p p nt and the electron- and hole-continuity equations are

∂n 1 = ∇ · J + G − R + γ (6.2) ∂t q n n n n

∂ p 1 = − ∇ · J + G − R + γ (6.3) ∂t q p p p p and Poisson’s equation is stated as ∇2ψ = −q − + − + − ε p n ND NA Nt nt (6.4) 101

γ γ γ where Nt is the density of donor-like traps, and n, p, and nt are the Langevin terms corresponding to ﬂuctuations in the steady-state GR transition rates in the electron-, hole-, and trap-continuity equations, respectively.

By placing a source γα in each linearized PDE describing the semiconductor device, it is possible to couple perturbations in the microscopic GR rates in the semiconductor with a voltage or current response at the device terminals. For trap-assisted GR ﬂuctuations, the ω + ω ω + ω CSD between a response at external circuit node x at p k 0 and node y at p l 0 is ∞ x,y x y ∗ ω + ω ,ω + ω = ˜ ( , ) · γ ,γ ( , ) · ˜ ( , ) ∂ Sgr p k 0 p l 0 ∑ ∑ Gα k u K α β u v Gβ l v r (6.5) r α,β= , , , =−∞ n p nt u v

x where G˜α (k,u) is the scalar Green’s function which couples GR fluctuations for each dis- α = α = α = cretized volume from the electron- ( n), hole- ( p), or trap- ( nt) continuity ω + ω ω + ω equation at frequency p u 0 to a response at x at frequency p k 0. Using modified nodal analysis, x and y can also represent current through a voltage source. The noise ( , ) source strength, Kγα ,γβ u v , is the correlation of transition-rate fluctuations between equa- α β ω + ω ω + ω tions and at sideband frequencies p u 0 and p v 0, respectively, and is defined in Section 6.2.3.

Diffusion noise is found using [10, 11] ∞ , ∗ Sx y ω + kω ,ω + lω = G x (k,u) · K (u,v) · G y (l,v) ∂r (6.6) diff p 0 p 0 ∑ ∑ α ξ ,ξ β r α=n,p u,v=−∞ α α where

x 1 x Gα (k,u)= ∇G˜ α (k,u) (6.7) q is the vector Green’s function and K (u,v) is the noise source strength tensor for veloc- ξα ,ξα ity ﬂuctuations. The total CSD is then given by x,y ω + ω ,ω + ω = x,y ω + ω ,ω + ω + x,y ω + ω ,ω + ω Stotal p k 0 p l 0 Sgr p k 0 p l 0 Sdiff p k 0 p l 0 (6.8) 102

6.2.3 Cyclostationary Noise Analysis

6.2.3.1 Modulated stationary noise model

The modulated stationary noise model [13] is used to calculate the instantaneous bias dependence of the microscopic noise sources. This model has been used to describe the modulation of shot noise in junction diodes [19, 20]. The time-varying spectral density of a noise process, n(t), may be modeled as

Sn (t)=Cf(t)Rx,x (t,t) (6.9) where C is a constant, f (t) is the modulating function and is an instantaneous function of the solution variables, and Rx,x (t,t + τ)=δ (τ) is the autocorrelation of a stationary white noise source ∞ (ω + ω ) ϕ ( )= α ω + ω j p k 0 t j kt x t ∑ p k 0 e e (6.10) k=−∞ α ω + ω = ϕ where p k 0 1 and k is a random phase angle. It can be shown that the CSD ω + ω ω + ω between noise phasors at p k 0 and p l 0 is then [19] ω + ω ,ω + ω = ( − )ω Sn p k 0 p l 0 CF k l 0 (6.11) ω ( ) ω ϕ where F k 0 is the frequency component of f t at k 0. Since k is randomly distributed across frequency, the correlation exists due to the modulating function, f (t). In order to properly account for the total noise, it is necessary to account for the contribution of both the self- and cross-spectral components to the output response of the semiconductor device through the time-varying Green’s functions as described by Equations 6.5 and 6.6.

6.2.3.2 GR noise sources

If the fundamental source of the 1/ f -like ﬂuctuations in a semiconductor can be attributed to GR events, it is possible to apply the IFM and the modulated stationary noise model. The GR transition-rate ﬂuctuations between the conduction or valence band and the trap level are white processes so that Equation 6.11 applies. For one trap level, the 103 time-derivative term in Equation 6.1 results in a Lorentzian frequency spectrum when coupled to the device contact. For a proper distribution in energy or position for the traps, the low-frequency noise can produce a 1/ f spectrum [12]. Being a shot process, the microscopic GR noise sources have an autocorrelation of [52] ( , + τ)= ( ( )+ ( )) δ (τ) Rγα ,γα t t 2 Gα t Rα t (6.12) where Gα (t) and Rα (t) are the instantaneous GR rates between the conduction band (α = n) or valence band (α = p) and the trap level. Using Equation 6.11, the noise source strengths for use in Equation 6.5 are then Kγ ,γ (u,v)=−Kγ ,γ (u,v)=Sγ (u − v)ω (6.13) n n n nt n 0 Kγ ,γ (u,v)=Kγ ,γ (u,v)=Sγ (u − v)ω (6.14) p p p nt p 0

Kγ ,γ (u,v)=Kγ ,γ (u,v)+Kγ ,γ (u,v) (6.15) nt nt n n p p ( , )= ( , )= Kγn,γp u v Kγp,γn u v 0 (6.16) where ∞ −jkω t Sγ kω = 2 Rγ ,γ (t,t)e 0 ∂t (6.17) α 0 −∞ α α is the kth harmonic of the instantaneous power of the transition-rate ﬂuctuations given in ( , )= ( , )∗ Equation 6.12 and Kγα ,γβ u v Kγβ ,γα u v . In the literature, there are methods where this 4 equation system of PDEs is reduced to

3 by deﬁning the trap density in terms of ψ, n, and p [50]. Under large-signal conditions, such a method could only be used for a fast trap such as the virtual recombination center used to model recombination currents in the space charge region of a junction device. The ∂/∂t term from the trap level then becomes negligible, except at high frequencies. The trapped electron density nt is then instantaneously dependent on the other solution variables 104

Table 6.1: Device simulation variables ψ Potential /cm3 n Electron carrier density /cm3 p Hole carrier density /cm3 = / 3 ni Intrinsic carrier density, ni n1 p1 cm / 3 n1 n when quasi-Fermi level crosses trap cm / 3 p1 p when quasi-Fermi level crosses trap cm / 3 nt Trapped electron density cm / 3 Nt Trap-density cm 3/ cn Capture coefficient for electrons, enn1 cm sec 3/ cp Capture coefficient for holes, ep p1 cm sec 6 en,ep Emission coefficient for electrons, holes cm /sec / 3 Gn Electron generation rate, ennt 1 cm sec ( − ) / 3 Rn Electron recombination rate, cnn Nt nt 1 cm sec / 3 Gp Hole recombination rate, cp pnt 1 cm sec ( − ) / 3 Rp Hole generation rate, ep Nt nt 1 cm sec and net recombination rate is then 2 cncpN np− n U = R − G = R − G = t i (6.18) SRH n n p p + + + cp p p1 cn n n1 and

( , )= ( , )= ( − ) Kγn,γn k r Kγp,γp k r 2USRH k r (6.19) ( , )= ( , )= Kγn,γp k r Kγp,γn k r 0 (6.20) and no trap-continuity equation need be considered. Since the ∂/∂t term is not considered, there is no roll-off at high frequencies. Since the primary interest of this investigation was the low-frequency noise, we did not consider this noise source in Section 6.4.

6.2.3.3 Diffusion noise sources

The noise source strength tensor used in Equation 6.6 for velocity ﬂuctuations is 2 K (u,v)=4q Dα α (u − v)ω (6.21) ξα ,ξα 0 105

th where Dα is the diffusivity and α (k) is the k harmonic of the carrier density. For the purposes of our investigation, we use the Einstein relationship

kT µα Dα = (6.22) q to find the noise source from the low-field mobility. For 2-D and 3-D simulation it is appropriate to use a diffusivity tensor to account for crystallographic direction [11] and frequency components of the diffusivity/carrier-density product to account for the electric field dependence of the diffusivity [10]. Since our focus is the 1/ f -like low-frequency noise, no attempt was made to account for the diffusivity outside of equilibrium. Reason- able results can be attained using this model, however, a more accurate investigation would require an approach similar to that of Jungemann et al. [57]. 6.3 Noise Correlation Matrix for Circuit Simulation

6.3.1 Theory

For DC steady-state noise analysis in circuits, an equivalent current-noise generator is used to model the noise of the semiconductor devices in a circuit at the small-signal frequency being considered. The current generator for a 2-terminal junction device is shown in Figure 6.1. Since the noise is instantaneously dependent on bias, a NCM is required to account for the cross-correlation between the current phasors at each sideband under periodic large-signal conditions. When a device is placed in a circuit under periodic steady-state conditions, the response is − δV = Y 1δI (6.23) where δI is a vector of correlated current phasors at the harmonically related sideband frequencies, Y is the admittance matrix formed by linearizing the nodal circuit equations, and δV is the voltage response at each sideband for every node in the circuit [58]. Using mod- iﬁed nodal analysis [39], the current response through each voltage source is also found. 106

ω + ω The resulting noise spectral density at node i and frequency p k 0 is then † S , ω + kω ,ω + kω = Z SZ (6.24) vi vi p 0 p 0 i,k i,k

−1 ω + ω where Zi,k is the row of Y corresponding to output node i at frequency p k 0,† denotes the transpose conjugate, and S = δI δI† is the NCM where each column is a vector of correlated noise components. Figure 6.2 shows a network representation of how noise current generators at each sideband frequency contribute to the total noise spectral density for a 1 port device. A port ω + ω ( , ) representing sideband p l 0 has a noise generator, S l l , which represents the current spectral density at that sideband, independent of the small-signal conversion through the transfer impedance matrix, Z. Each of these noise generators may be partly correlated, as described in Section 6.2.3.1. Figure 6.3 illustrates how the microscopic noise sources result in correlated sideband noise currents under AC short-circuit conditions for an arbitrary semiconductor device. The short-circuit current noise simulated at each sideband can only be due to the noise generator at that frequency. Equation 6.24 is then used to account for how these correlated components contribute to the total noise spectral density at each sideband.

Using a semiconductor device simulator, Equation 6.5 can be used to ﬁnd the the CSDs for low-frequency GR noise sources for use in circuit simulation. The resulting NCM entry is then ( , )= x,y ω + ω ,ω + ω Sx,y k l Sgr p k 0 p l 0 (6.25)

In Section 6.4, simulation results are presented which demonstrate how Equation 6.5 is used to generate the NCM entries for low-frequency GR noise sources. 6.3.2 Diffusion Noise

For a junction diode, the shot noise is instantaneously dependent on bias so that the current spectral density is ( )= ( ) Si t 2qi t (6.26) 107

( ) ( ) 2/ where i t is the large-signal current and Si t has units of A Hz. Using Equation 6.11, ω + ω the NCM entries which couple the noise generator at p k 0 with the noise generator at ω + ω p l 0 [20]is ( , )= ( , )= ( − )ω Sx,x k l Sy,y k l 2qI k l 0 (6.27)

( , )= ∗ ( , )=− ( − )ω Sx,y k l Sy,x k l 2qI k l 0 (6.28) ( − )ω ( − )th where I k l 0 is the k l harmonic of the large-signal current ﬂowing through the diode, and x and y are the nodes to which the current generator is attached. 6.3.3 GR Noise

The GR transition rates may be invariant (stationary) or variant (cyclostationary) with respect to an applied bias. Regardless of whether or not the microscopic noise sources are bias dependent, a compact model would require a NCM. This is since harmonics of the periodic scalar Green’s functions could modulated bias independent GR fluctuations into correlated current density fluctuations at the device terminals. For a linear resistance, low-frequency GR fluctuations may be viewed as bias independent (stationary) resistivity fluctuations which are modulated by the large-signal currents flowing through the device so that correlated current components exist at the LSBs and USBs. If we consider these resistivity fluctuations as being a filtered white noise source, the modulated stationary model [22] applies and Equation 6.11 becomes ∞ ω + ω ,ω + ω = ( − )ω α ω + ω 2 ∗ ( − )ω Sn p k 0 p l 0 C ∑ H k u 0 p u 0 H l u 0 (6.29) u=−∞ where ∞ − ω ω = ( ) jv 0t H u 0 ∑ f t e (6.30) v=−∞ 108

The NCM entries for a stationary low-frequency noise source with quadratic current- dependence are then ∞ ( , )= ( , )= ( − )ω ω + ω ( − )ω ∗ Sx,x k l Sy,y k l ∑ I k u 0 c p u 0 I l u 0 (6.31) u=−∞ ω + ω = α ω + ω 2 where c p u 0 p u 0 . For a single trap

C c ωp + uω = (6.32) 0 + ω + ω 2 τ2 1 p u 0 and for stationary ﬂicker noise ω + ω = C c p u 0 ω + ω (6.33) p u 0 ω + ω ≈ | | > ω ω + ω For RF and microwave applications, c p k 0 0 for k 0 and 0 0 p 0 0 so that ( , )= ( , )= ω (ω ) ω ∗ Sx,x k l Sy,y k l I k 0 c p I l 0 (6.34)

In Sanchez et al. [17], simulation results are presented demonstrating the appropri- ateness of this model for trap-assisted GR noise in linear resistors. When the microscopic noise sources are functions of the instantaneous bias, it is not known if it is possible to use an expression like Equation 6.31 to describe the equivalent input current noise generator.

While the modulated stationary noise model is appropriate for microscopic GR ﬂuctua- tions resulting in low-frequency noise, the frequency dependent nature of low-frequency noise in general makes it unclear whether this model is always applicable. In the literature, the modulated stationary noise model has been applied in general to low-frequency noise [13, 22], although it is not known if it is appropriate for all types of low-frequency noise. Commercial circuit simulators, such as Agilent Advanced design system (ADS), treat low-frequency noise as being dependent on the DC component of the current ﬂowing through the device [2]. However, such a model does not predict the presence of low- frequency noise around the upper and lower sidebands under AC short-circuited conditions, 109 as was observed for a linear resistor and a p+-n junction [17]. In Section 6.4 we present simulation results which compare Equation 6.34 with a semiconductor noise simulation under periodic large-signal conditions. 6.4 Simulation Results

6.4.1 Shot Noise

A silicon n+/p/n BJT was simulated in FLOODS with the dimensions and doping proﬁle shown in Figure 6.4. The emitter is located in the upper-left corner and was modeled

5 with a surface recombination velocity of Sp = 10 cm/sec. The collector contact is along the bottom and the base contact is located in the upper right hand side. The device had an emitter depth of 70 nm and a base width of 70 nm. The diffusion noise for increasing = . DC bias on the base was simulated for VCE 2 0 V. The simulation results for base and collector shot noise versus VBE are shown in Figure 6.5. These results agreed well with the shot noise expressions of S = 2qI and S = 2qIc. ib b ic = . For the HB simulation, the circuit shown in Figure 6.6 was used with vdc 0 7V, = . = − π 6 vce 2 0 V and vac VAC cos 2 10 t V. A noise simulation was performed for each sideband for increasing AC bias. The steady state simulation was performed with 5 frequencies, including DC. This results in a conversion matrix relating 9 sideband frequencies. Figure 6.7 shows the current CSDs for the base current between the 1st upper side- ( , ) band (USB) and other sidebands denoted by Si 1 k where k denotes the sideband at fre- ω + ω quency p k 0. Excellent agreement was found between the simulated data and NCM entry for shot noise given by Equation 6.27. Collector shot noise for increasing AC bias is shown in Figure 6.8. The simulation begins to deviate from Equation 6.27 for higher AC bias. This corresponds to the devia- tion between the simulation and analytical shot-noise model for the DC results shown in Figure 6.5 and is attributed to high-level injection effects. 110

Figure 6.9 shows the correlation between the base and collector shot noise at ω p which was calculated as ( , ) Si ,i 0 0 c = b c (6.35) S (0,0)S (0,0) ib ic This indicates that the equivalent current noise generators for the base and collector can be treated independently, even under large-signal AC conditions. 6.4.2 GR Noise

−12 3 A BJT device was simulated with a electron capture coefﬁcient of cn = 10 cm /sec − = . and an energy level of EC ET 0 31eV. Similar trap parameters were reported for a polysilicon-emitter BJT in the literature [26] with the traps ascribed to the emitter region at = / 3 or near the polysilicon-monosilicon interface. A constant trap density of Nt 1 cm was simulated for the device in order to ﬁnd the relative contributions of the device regions. The distributed contributions to the short-circuit base current noise and collector current = . noise for VBE 0 7V are shown in Figures 6.10 and 6.11, respectively. For this trap energy level, the dominant noise contribution to the base noise is between the base and the emitter near the surface. A possible source of traps for this region would be the interface between the silicon and the oxide spacer between the base and emitter contacts [18]. Noise was then simulated for traps at the surface between the base and emitter contact = · 9/ 2 1 with Nt 8 10 cm . Figures 6.12 and 6.13 shows both the GR and shot noise versus current at the base and collector, respectively. Figure 6.14 shows the electron quasi-Fermi = . level for VBE 0 7V at the surface of the device. It shows that near this bias condition, the electron quasi-Fermi-level is close to the trap-level over the entire interface between the contacts. At lower biases a quadratic dependence is observed and is attributed to trapping noise in the space charge region of the device. As bias is increased, the space charge region

1 The surface density is a volume density distributed over a depth of 1.5 nm at the surface. 111 narrows, and the position where the quasi-Fermi level for electrons crosses the trap level moves into the quasi-neutral region and the quadratic current dependence is lost. At much higher biases, high-level injection of electrons causes the location of this crossing to move toward the base contact so the area of active traps is reduced and the noise rolls off.

Using the same bias conditions as in section 6.4.1 the GR noise was simulated versus frequency for increasing AC bias. In Figure 6.15 the USB and baseband spectra are shown for an AC bias of 0, 50, and 100 mV. It is apparent that the corner frequency, fc ≈ 50 Hz, is invariant to the AC bias. The USB plateaus versus AC bias are shown in Figure 6.16. The baseband level was fit to the model of Equations 6.34 and 6.32 with C = 1.21 · 10−9 /Hz and using the DC component of the base current for increasing AC bias. Using this value of C, the linear GR noise model was applied to each of the sidebands in Figure 6.16 and good agreement was found. Figure 6.17 shows the current CSD between the 1st upper sideband and several other sidebands. Again the data was fit to and good agreement was found between the simulation and the model of Equation 6.34. The GR noise shown in the simulations had a near quadratic dependence on current. The good agreement for the stationary fluctuation model derived as Equation 6.31 and the simulations was surprising since the trap density was a function of instantaneous bias and the current dependence shown in Figure 6.12 did not have an exact exponent of 2. This variation is attributed to the spatial position of active traps versus bias. In Cappy et al. [10], the authors model low-frequency GR and flicker noise as being microscopically stationary and consider equivalent current noise sources. They speculate that since the bias is varying at a frequency much larger than the corner frequency of the noise, that the trap is not able to respond. In the literature, Equation 6.31 is typically used for cyclostationary analysis in noise simulators without experimental or theoretical verification. The simulations presented in this chapter provide evidence that such a model is applicable to low-frequency noise due to trap-assisted GR noise. 112

Figure 6.1: Equivalent noise current generator 113

ω − ω p 2 0

(− ) ( , ) ZL 2 S -2 -2 Z (k,-2)

ω − ω p 0

(− ) ( , ) ZL 1 S -1 -1 Z (k,-1)

ωp Nonlinear Device ( ) ( , ) ZL 0 S 0 0 Z (k,0)

ω + ω p 0

( ) ( , ) ZL 1 S +1 +1 Z (k,+1)

ω + ω p 2 0

( ) ( , ) ZL 2 S +2 +2 Z (k,+2)

Figure 6.2: Network representation of correlated noise generators for a 1 port device. By short circuiting the terminations at each frequency, it is possible to ﬁnd the cross-correlation between the noise generators at each sideband. 114

S(k,l)

∗ Gα (l,v)

Gα (k,u)

K (u,v) ξα,ξα

Figure 6.3: Microscopic noise sources resulting in a current CSD between sidebands k and l. The total CSD is found by summing over u and v. 115

−4 x 10 0

19.5

19 0.5

18.5

18 depth (cm)

17.5 1

16.5

1.5 16 0 1 2 3 4 5 −5 position (cm) x 10 − Figure 6.4: BJT doping proﬁle. The greyscale units are log10 ND NA . 116

-21 10

-22 10 S ib -23 10 S ic -24 10 Shot noise model

-25 10

-26 10 /Hz)

2 -27 10 (A i

S -28 10

-29 10

-30 10

-31 10

-32 10

-33 10 0.4 0.5 0.6 0.7 0.8 0.9 1

VBE (V)

Figure 6.5: Base and collector shot noise of the BJT 117

vac

+ + vdc − − vce

Figure 6.6: Test circuit for the harmonic balance and noise simulation 118

-27 3×10 Shot noise model

Si (1,1) -27 2×10 Si (1,0)

Si(1,-1) S (1,-2) -27 i 1×10 Si(1,-3) /Hz) 2 0 (A i S

-27 -1×10

-27 -2×10

-27 -3×10 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11

VAC

Figure 6.7: Cross-spectral densities for base shot noise. The dotted lines use the model described by Equation 6.27. 119

-25 3×10

-25 2×10

-25 1×10 /Hz) 2 0 (A i

S Shot noise model S (1,1) -25 i -1×10 Si (1,0)

Si(1,-1) -25 S (1,-2) -2×10 i Si(1,-3)

-25 -3×10 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11

VAC

Figure 6.8: Cross-spectral densities for collector shot noise. The dotted lines use the model described by Equation 6.27. 120

0.008 c (0,0) ibic

0.007

0.006

0.005

Re{c} 0.004

0.003

0.002

0.001

0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

VAC (V)

Figure 6.9: Real part of the correlation between the base and collector shot noise at the baseband for increasing AC bias. The imaginary part was negligible. 121

−4 x 10 0 −36

−38

−40

−42 0.5 −44

−46

depth (cm) −48

1 −50

−52

−54

−56

1.5 0 1 2 3 4 5 −5 position (cm) x 10

Figure 6.10: Relative noise contributions of GR noise to the base current noise. The greyscale units are log ∂S /∂r. 10 ib 122

−4 x 10 0

−34

−36

−38

0.5 −40

−42

−44 depth (cm)

−46 1 −48

−50

−52

−54 1.5 0 1 2 3 4 5 −5 position (cm) x 10

Figure 6.11: Relative noise contributions of GR noise to the collector current noise. The greyscale units are log ∂S /∂r. 10 ic 123

-22 10

-23 10 Shot GR

-24 Shot noise model 10

-25 10

-26 10 /Hz) 2 (A

i -27

S 10

-28 10

-29 10

-30 10

-31 10 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 10 10 10 10 10 10 10 10 10 10

IB (A)

Figure 6.12: Shot and GR current noise at the base versus IB 124

-20 10

-21 10 Shot GR -22 10 Shot noise model

-23 10

-24 10

-25 10 /Hz) 2 -26 (A i 10 S -27 10

-28 10

-29 10

-30 10

-31 10 -12 -8 -4 10 10 10

IC (A)

Figure 6.13: Shot and GR current noise at the collector versus IC 125

0.5

-0.5 Energy (eV)

EV -1

ET Fn -1.5 0 0.1 0.2 0.3 0.4 0.5

Position (µm)

Figure 6.14: Electron quasi-Fermi level, Fn, and the trap energy level, ET at the surface = . for VBE 0 7V 126

-25 10 Si(0,0) 0 mV

Si(0,0) 50 mV

Si(1,1) 50 mV

Si(0,0) 100 mV

Si(1,1) 100 mV

-26 10 /Hz) 2 (A i S

-27 10

1 2 3 10 10 10 f (Hz)

Figure 6.15: Upper sideband and baseband noise noise spectrums for 2 AC biases 127

-25 10

-26 10 /Hz) 2 (A i -27 S 10 Si(0,0)

Si(1,1)

Si(2,2) S (3,3) -28 i 10 GR model

Si Shot

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11

VAC (V)

Figure 6.16: Upper sideband plateaus versus AC bias compared with modulated stationary noise model 128

-25 1×10

-26 8×10 GR Model Si (1,1) × -26 6 10 Si (1,0) S (1,-1) -26 i 4×10 Si(1,-2) -26 S (1,-3) 2×10 i /Hz) 2 0 (A i

S -26 -2×10

-26 -4×10

-26 -6×10

-26 -8×10

-25 -1×10 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11

VAC

Figure 6.17: Cross spectral density to the ﬁrst USB for increasing AC bias CHAPTER 7 CONCLUSION

7.1 Summary and Contributions of this Work

While the device simulation of cyclostationary noise was reported in the literature, it was only capable of 1-D diffusion and band-to-band generation-recombination noise simulation [11, 59]. The implementation presented in this dissertation is unique in that it is capable of 2-D device simulation and able to solve for the GR noise of an arbitrary number of trap levels [17, 60, 61]. This allows for simulating 1/ f -like noise as the sum of Lorentzian spectra.

This implementation in FLOODS facilitated the 1-D and 2-D simulation results presented in this work. In Chapter 5, results demonstrating the upconversion of low-frequency noise in resistors and diode juctions are presented. These simulation results are the first to be reported for the case of trap-assisted GR noise in a PDE-based semiconductor device simulator. In Chapter 6, results are presented which support the theory that surface traps between the base and emitter can have a significant contribution on the low-frequency noise in BJTs under both DC and periodic large-signal steady-state conditions. A methodology has been developed for finding noise correlation matrices for compact circuit simulation using cross- spectral density device simulation. Evidence supporting the modulated stationary noise model for low-frequency noise is also given. The results presented in this chapter are the first 2-D, 3-terminal semiconductor device noise simulations to be reported for periodic large-signal conditions.

7.2 Future Work

While this dissertation has presented simulation results for how low-frequency due to bulk and interface GR traps are upconverted in frequency, it does not model how other

129 130 types of low-frequency noise may be modulated. This includes low-frequency noise due to tunneling in the mono/polysilicon interface of poly-emitter bipolar junction transistors [18] or to mobility fluctuations [62]. In MOS devices, it is typically believed that oxide-trapping noise is the dominant contribution to the low-frequency noise spectrum. This would require a simulation approach such as that of Hou [63] extended to large-signal simulation. Currently, experimental evidence for the modulation of bulk GR noise does not exist for bipolar devices in the literature and such information would provide additional verification. For RF applications, load-pull measurements and simulations are performed by varying the impedance seen by the device under test (DUT). By varying the impedance presented to the input and output terminals it is possible to simulate the optimal impedance for minimum noise figure, and maximum gain under large-signal conditions. Load-pull device simulations have been performed [64], but not accounting for noise performance. Such simulations would provide an additional tradeoff for a device engineer when optimizing a device structure. APPENDIX A PERIODIC LARGE-SIGNAL STEADY STATE AND CYCLOSTATIONARY NOISE IMPLEMENTATION IN FLOODS The FLOODS simulator is an object-oriented platform written using the C++ pro- gramming language [65]. In order to implement the harmonic balance and noise simulation routines, it was only necessary to create the appropriate objects specific to the algorithm being used. The added code was able to use the same functions and classes as the DC and small-signal AC simulators. Using operator overloading, it was possible to replace code so that it is more specific to the simulation problem. For example, the iterative solver takes the matrix object as an argument. When it requires a matrix-vector multiplication, the routine calls the matrix object’s multiply routine to perform the operation. For the harmonic balance, this required the use of DFT and IDFT operations discussed in Section 3.5 on Page 35. Table A.1 lists the components added to the FLOODS simulator. In addition, it was necessary to modify the components listed in Table A.2 in order to interface the code with the rest of the simulator. The manual for use of the simulator is included as Appendix B.

131 132

Table A.1: Components added to the simulator.

Filename Class Provided Function HBDevSolv.cc HBDevSolver Form the time-sample Jacobian matrix, RHS vector, and block-diagonal preconditioner for the HB steady state. HBNoiseSolv.cc HBNoiseSolver Form the time-sample conversion matrix and block-diagonal preconditioner for noise and set the noise output sideband contact HBPrecond.cc HBPreconditioner Factor the steady-state and noise preconditioner HBmatrix.cc HBMatrix Loads the Jacobian, noise conversion, and preconditioner matrices. Also performs DFT and IDFT operations for use in iterative solver. hbcircuit.cc N/A Retrieves circuit solution variables to shell. hbdevice.cc N/A Command to start simulation and set parameters. 133

Table A.2: Components modiﬁed in the simulator.

Filename Class Provided Functions Added circuit.cc N/A Added commands to tcl interface for HB sources. CircuitNode.cc CircuitNode HB and noise solution variables to store on external circuit nodes. Also update these variables. CktElem.cc CktElem Evaluate HB and noise circuit element parameters. Contact.cc Contact Maintain HB and noise solution variables. DevControl.cc DevController Control HB steady-state, small-signal mixer, noise, and cross-spectral density noise simulation iterate.cc GMRes Added transpose solve for doing noise conversion matrix factorization. Solution.cc Solution Store Solution Variables APPENDIX B HARMONIC BALANCE AND CYCLOSTATIONARY NOISE SIMULATION MANUAL B.1 Introduction

B.1.1 Harmonic Balance

Harmonic balance (HB) simulation [3, 5] has been implemented in FLOODS for the periodic steady state. A large sinusoidal signal is applied to the semiconductor device through an external source. Figure B.1 shows an example of a diode being driven by a periodic stimulus. For a nonlinear device driven by a periodic bias, the current response occurs at integer multiples of the fundamental frequency of the source. The HB algorithm then converges on a solution by enforcing Kirchoff’s Current Law (KCL) at each frequency for each node. Figure B.2 shows an example solution for the diode circuit. The Fourier coefficients (ω) of the base emitter voltage, VBE , is the desired solution set. Using iterative techniques, (ω)= (ω) a solution is found when the ID IR as shown in Figure B.1. Since the resistor is (ω) (ω) (ω) linear, it is possible to calculate IR as a function of VG and VBE . It is not possible (ω) (ω) to find ID directly from VBE since it is a nonlinear function. Instead it is necessary ( ) ( ) to calculate iD t from vBE t and then taking the discrete Fourier transform (DFT) to find (ω) ID . ω Since this source is a periodic stimulus with a fundamental frequency, 0. The non- ω linear response of the device occurs at integer multiples of 0. The solution variables for a HB device simulation are Ψ, N, and P which are the Fourier coefficient vectors of the potential, electron density, and hole density distribution throughout the device. In addition the frequency components of the terminal currents and voltages are solved. Linear circuit

134 135 elements are available (R, L, C) which allow for enforcing frequency dependent boundary conditions for each harmonic being considered in the device simulation. The number of frequencies considered, including DC, is a user selected parameter, K. The number of frequencies required is based on the magnitude of the periodic source and the nonlinearity of the device being considered. The number of equations which are considered is 2K −1 times that of a DC simulation. The direct factorization of the Jacobian matrix formed from these equations quickly becomes prohibitive in computation and storage required. For this reason, FLOODS uses iterative methods to solve the HB system of equations. The GMRES algorithm is used to converge upon a steady state solution. More information on these iterative methods can be found in Feldmann et al. [6] and Saad [42]. The memory requirements for an HB simulation are then linear with K, and the number of nodes in the semiconductor mesh. FLOODS has been rewritten to allow the solution of PDEs in addition to the standard drift diffusion equations. For example, it is now possible to solve the trap continuity equation with the trap carrier density, nt as a solution variable. The noise analysis now allows the specification of number and velocity fluctuation noise under periodic large-signal conditions. Number fluctuations correspond to the vari- ance in the generation and recombination rates between the conduction and valence bands and the trap level in the bulk of the semiconductor and commonly referred to as GR noise. Velocity fluctuations correspond to the thermal noise observed in resistors and the shot noise observed in junctions. B.1.2 Noise Analysis

B.1.2.1 Frequency conversion of applied external signals

The frequency conversion of small-signals is simulated using the small-signal large- signal approximation [8]. Small-signals applied to the device are frequency converted by the time varying small-signal admittance of the semiconductor device being simulated.

This is useful when a mixer simulation is desired with a large signal driving the circuit and 136 a small signal which is not able to change the steady-state operation of the semiconductor device. The response to the applied small signal is then at the frequency at which it was applied and each sideband. Figure B.3 shows an example of the small-signal current response of a nonlinear device when a small-signal voltage is applied to the device terminal at the baseband frequency ωp. The dashed lines are the frequency components of the large-signal current response. The solid lines represent the small-signal current response at each sideband. Figure B.4 is the small-signal representation used in FLOODS. The positive frequency axis contains the baseband frequency ωp and the upper sideband frequencies. The negative frequencies represent the lower sideband frequencies. Using this representation, it is shown ω that each sideband frequency is related through integer multiples of 0. For example, a ω ω + ω small-signal voltage applied at frequency p is related to a current response at p 0 by ω the ﬁrst harmonic ( 0) of the small-signal conductance at frequency and the response at ω − ω ω p 2 0 by the second harmonic (2 0) of the small-signal conductance [9]. B.1.2.2 Cyclostationary noise analysis

The noise analysis for the periodic steady state uses the Green’s function approach [66, 67] to couple the internal ﬂuctuations of the carrier densities to a noise response at an external contact. Unlike the DC steady state, the concept of an open circuit voltage or short circuit current noise no longer applies since the termination at each sideband frequency has an affect on the measured noise magnitude. In addition, the instantaneous bias of the device modulates the noise sources within each differential volume of the semiconductor device so that correlated noise components exist at each sideband simultaneously [19].

Diffusion Noise. Figure B.5 demonstrates the case for diffusion noise. The vector Green’s function, Gk,l is the transfer function of a ﬂuctuation in the current density in a ω + ω discretized volume at a frequency of p l 0 to an external contact of the device at an 137

ω + ω ω + ω frequency of p k 0. The total voltage spectral density observed at p k 0 is then ∗ = ∗ ∗ ∂ VkVk ∑∑Gk,l SlSm Gk,m r (B.1) r l m where the spectral density of electron current density ﬂuctuations is given by H = 2 ∆ Sl Sm 4q DNl−m f (B.2)

where Nl−m is the frequency component of the periodic steady electron concentration at ( − )ω l m 0. For the case of hole current density fluctuations, the spectral density of the noise fluctuations is then H = 2 ∆ Sl Sm 4q DPl−m f (B.3) where Pl−m is the frequency component of the periodic steady hole concentration at ( − )ω l m 0. Bulk Generation Recombination Noise. The case for bulk generation recombination noise is handled similarly to the case of diffusion noise. The total noise power at ω + ω frequency p k 0 is then ∗ ∗ α (i) β V V = ∑ ∑∑G , K (l,m)G ∂r (B.4) k k k l γα ,γβ k,m α,β= , , r n p nt l m ˜α α = where Gk,l is the the scalar Green’s function which couples fluctuations in the electron ( α = α = ω + ω n), hole ( p), and trap ( nt) continuity equations at frequency p l 0 to an external ω + ω contact at p k 0. Fluctuations in the bands and trap levels are correlated since a loss of an electron in one level corresponds to an addition in another level [24]. It is then necessary to account for the correlations using a correlation matrix, K, where 138

γ ,γ ( , )=− γ ,γ ( , )=( − )+ ( − ) K n n l m K n t l m 2Rn l m 2Gn l m (B.5)

γ ,γ ( , )= γ ,γ ( , )=( − )+ ( − ) K p p l m K p t l m 2Rp l m 2Gp l m (B.6) ( , )= ( , )= Kγn,γp l m Kγp,γn l m 0 (B.7)

γ ,γ ( , )= γ ,γ ( , )+ γ ,γ ( , ) K t t l m K n n l m K p p l m (B.8)

th with Gα (l − m) and Rα (l − m) is the (l − m) frequency component of the large-signal generation and recombination rates of equation α, respectively. 139

iR G B

i + D − E

Figure B.1: Diode circuit being driven by a large-signal source 140

( ) (ω) vBE t VBE

0 0.5 1 1.5 2 2.5 3 -1 0 1 2 3 4 5 6

0 1 2 3 -1 0 1 2 3 4 5 6 ( ) (ω) iD t ID

Figure B.2: Sample voltage and current response of the diode circuit shown in Figure B.1 141

Large Signal

Small Signal

ω ω ω ω 0 p 0 2 0 3 0

Figure B.3: The small-signal current response of a nonlinear device to an applied ω ω small-signal at frequency p and periodic large signal at frequency 0 142

Large Signal

Small Signal

ω − ω ω − ω ω ω + ω ω + ω p 2 0 p 0 p p 0 p 2 0

Figure B.4: The small-signal representation of the frequency conversion of small-signals 143

Gk ,l

2 Sl Vk G Sm Gk ,m

Figure B.5: Green’s Function approach for cyclostationary noise analysis 144

B.2 Simulation Commands

B.2.1 circuit: Specifying Stimuli

B.2.1.1 Example usage circuit add name=Vss from=Vn1 to=Vn2 voltage eq = {Vn1+$Vss} ac.r= {$V1} ac.i= {0} hb.r = {$V2} hb.i= {0}

The circuit add command appends a new circuit element to the network connected to the semiconductor device. The voltage option adds a equation for which the solution is a branch current using modified nodal analysis (MNA). This is useful for specifying a voltage source or an inductor element. The eq option specifies the steady-state equation which is used for DC and small-signal AC analysis. Any variables prefixed with a $ are tcl variables and are used to specify constants. It should be noted that the tcl variables should be surround by curly braces, {}, so that they are not evaluated immediately and can be varied for sweeps.

B.2.1.2 Harmonic balance

The hb.r and the hb.i options specify the magnitude of the periodic steady equation such that the time-varying source is of the form ( )= ω + ω v t hb.rcos 0t hb.isin 0t (B.9) and the boundary conditions of eq are enforced for each moment in time.

B.2.1.3 Small-signal mixer

The ac.r and the ac.i options specify the magnitude of a small-signal phasor of the form

j(p+kω t) v(t)=(ac.r + jac.i)e 0 (B.10) where p and k denote the sideband as shown in Figure B.3. The sideband frequency is then speciﬁed when the simulation starts using the hbdevice command.

B.2.1.4 Noise simulation

It is only necessary to be sure that the periodic stimuli (hb.r, hb.i) is set correctly. 145

B.2.2 hbcircuit: Extract Response at the Device Terminals

B.2.2.1 Harmonic balance set freq [ hbcircuit freq ] set xr [ hbcircuit value name=cur_Vgs real ] set xi [ hbcircuit value name=G imag ] set xm [ hbcircuit value name=G magnitude ss ] set xp [ hbcircuit value name=G phase ]

The hbcircuit value command returns a space, ‘ ’, delimited list which contains the desired form (real|imag|magnitude|phase) of the Fourier coefficients of the node voltage specified with name. The current going through a voltage circuit element can be specified by prefixing its name with cur . A list of frequencies of the Fourier coefficients is given using the freq modifier. It should be noted that the frequency components are the positive frequency form.

B.2.2.2 Small-signal mixer

By appending ss to the hbcircuit command, the phasor response for the small- signal mixer case is returned as shown in the example above. Negative frequencies returned correspond to lower sideband frequencies as described in Section B.1.2. The complex conjugate of negative frequency phasors is required to convert them to positive frequency components.

B.2.2.3 Noise simulation set sf [ hbcircuit value name=cur_Vgs noisefreq ] set sd [ hbcircuit value name=cur_Vgs Sdiff ] set sg [ hbcircuit value name=cur_Vgs Sgr ]

The noise magnitude for the diffusion noise and GR noise can be extracted for the noise frequency last speciﬁed using the Sdiff and the Sgr option. The noisefreq option can be used to extract the noise frequency of the noise result. 146

B.2.3 hbdevice: Starting the Simulation

B.2.3.1 Harmonic balance hbdevice hb= fund=

hb This sets the number of frequencies to consider, including DC. This parameter should be set so that the harmonic content of the solution variables are contained within the simulation. This is a function on the magnitude of the external source and the boundary conditions enforced on the device. fund This speciﬁes the fundamental frequency of the large-signal source. The HB simulation requires an initial guess for the solution variables and so the DC simulation is used to provide initial values. If it is desired to use higher input power levels, it is possible to aid convergence by sweeping the source up to the desired magnitude. An error is returned if the simulation does not converge and the solution variables are restored from any previous simulation . A simulation with a 1 MHz fundamental frequency and 4 frequency components (0-3 MHz) is speciﬁed using. device hb=4 fund=1e6

Reducing the number of frequencies considered reduces the speed and memory requirements of the simulation, but leads to a less accurate solution if the response of the highest harmonic considered is not negligible small.

B.2.3.2 Mixer simulation device hb=4 fund=1e6 freq=1 preuse

By specifying freq, a small-signal mixer simulation is started. The location of this source is speciﬁed by placing values for ac.r and ac.i in the appropriate circuit add state- ment. Inverting the preconditioner is the most expensive computational burden. If frequencies being considered for mixer or noise simulations are close to each other or the same relative frequency to a different sideband, it is possible to use the preuse boolean to reuse the same preconditioner as the previous noise simulation. 147

B.2.3.3 Noise simulation

device hb= freq= noise= device hb=4 fund=1e6 device hb=4 noise=base freq=1e3

Usage

hb This specifies the number of frequency components to consider in the small-signal simulation. freq This sets the frequency for the small signal source specified using the circuit add command. For the noise simulation, this specifies the frequency being simulated. noise This specifies the name of the node at which the noise is being simulated. If the noise current is desired, the name of a voltage source prefixed with cur (eg. cur Vg) should be specified. The fundamental frequency is not specified since the solution variables from the last HB simulation depend on this frequency. Once a steady-state solution has been found, it is possible to perform either mixer

or diffusion noise simulation, depending on whether the noise option is speciﬁed. The following commands perform a noise simulation where a fundamental exists at 1 MHz and the noise is being calculated at 1 kHz. 148

B.2.4 pdbSetString Specifying the Noise Sources pdbSetString Si Elec Noise.V "4.0 * $Vt * $mun * Elec" pdbSetString Si Hole Noise.V "4.0 * $Vt * $mup * Hole" pdbSetString Si Elec Noise.N_Elec "2.0 * ($Gn + $Rn)" pdbSetString Si Hole Noise.N_Hole "2.0 * ($Gp + $Rp)" pdbSetString Si nt Noise.N_nt "2.0 * ($Gn + $Rn + $Gp + $Rp)" pdbSetString Si Elec Noise.N_nt "- 2.0 * ($Gn + $Rn)" pdbSetString Si Hole Noise.N_nt "- 2.0 * ($Gp + $Rp)" pdbSetString Si nt Noise.N_Elec "- 2.0 * ($Gn + $Rn)" pdbSetString Si nt Noise.N_Hole "- 2.0 * ($Gp + $Rp)"

Noise Sources can be specified in the same fashion as for the DC steady state. The first argument of the pdbSetString command is the material being considered. The next argument is for the PDE being considered. The Noise.V specifies a velocity fluctuation term is being considered. A vector Green’s function is used to integrate these noise sources. The Noise.N syntax specifies a number fluctuation term where the suffix specifies the PDE for which a cross-correlation is given. The Noise.N statements implement Equa- tions B.5–B.8. It is important to note that these equations are samples over 1 period of the stimulus driving the source. The velocity noise as shown above takes the Fourier transform of the (mobility)-(carrier density) product so that the effects of a field dependent mobility can be considered. B.2.5 Accessing the Device Profiles

The positive frequency coefficients of the solution variables are stored on the field server of the simulator. The potential distributions are available as DevPsi Rx and DevPsi Ix where R is the real part, I is the imaginary part and the desired frequency component (0 − (K − 1)) replaces x. The electron and hole distributions are stored as Elec Rx, Elec Ix, Hole Rx, and Hole Ix. To save the real part of the potential dis- ω tribution at frequency 0 into the file DevPsi R1.dat, select z=DevPsi_R1 print.data o=DevPsi_R1.dat 149

B.2.6 Known Simulation Issues

1. The parameters of the solver operations for the HB and noise simulation are hard wired into the code. All of the simulations presented in Section B.3 worked without needing to modify the parameters as they are set now. The GMRES iterative method is used for steady state, mixer simulation, and noise simulation. A restart value of 10 is set for the GMRES solver and it has been found to afford better convergence than when a lower value is specified. 2. The simulator has better performance since the individual blocks of the block diagonal matrix are now factored individually. This means that the machine can swap pieces of the inverted preconditioner as necessary to perform larger simulations. 3. Harmonic balance simulation is limited to a maximum of 32 frequencies while the mixer and noise simulation are limited to 16. It is likely that the large memory requirements for 2-D HB simulation restrict the number of frequencies for simulation further. The matrix factorization package currently used for the HB simulation does not support dynamic memory allocation and it is difficult to determine a priori the memory required for inverting the preconditioner matrix so the memory allocation is overestimated and can lead to problems when the simulation problem is large. 4. Circuit solutions are not stored permanently when the structure file is written out, so it is not possible to restart an HB simulation from a saved steady state condition. It is also necessary to do a DC simulation after creating a new circuit network. 5. The individual particle currents as integrated at the contacts are not currently available. 150

B.3 Examples

B.3.1 Harmonic Balance MOSFET Simulation

A HB simulation for a 0.4 µm 2-D NMOS device were performed using the circuit shown in Figure B.3.3. Scripts for performing the simulation are presented in

Appendix B.4 as described in Table B.1. The structure file is omitted, but is referred to as hbdevice.geo in the scripts. The script in Section B.4.5 is used to simulate the 0.4µm device. In this example, the peak magnitude of the sinusoidal source was swept from 0 to 500 mV with a DC voltage of 0.5 V applied at the gate and a 1.0 V bias at the drain. The appendhbdatarow function is included in Section B.4.8 and uses the circuit command as described in Section B.2.1 to extract circuit solution data and append it to a file. The simulations shown used 5 frequency components and required 1400 MB of memory. ( ) Figure B.7 shows a comparison of iD t when the frequency of the source is 0 and 200 MHz, respectively. The peak magnitude of the sinusoid is 100 mV. Figure B.8 shows a ( ) ( ) comparison of iD t versus vGS t for the same conditions. It demonstrates the of FLOODS to capture the non quasi-static effects of the drift-diffusion equations [68]. ( ) Figure B.9 shows iD t for a 200 MHz sinusoidal source where is varied from 100 to 400 mV. The threshold voltage of the transistor is about 0.5 V so the transistor is being switched between cutoff and saturation. Only 5 frequency components (0, 200, 400, 800 MHz) were specified for this simulation and it is apparent that at higher AC bias a ripple is apparent when the transistor is off. This indicates that not enough frequency components are specified. While this simulation gives a good approximation to the correct waveform, it would be necessary to add more frequency components to get a more accurate result. It is important to note that if the frequency of the AC source was increased, the capacitances of the device would short these higher harmonics and a more accurate result would occur for the same number of frequencies specified. 151

Table B.1: Description of MOS simulation ﬁles

Filename Purpose Section hbdevice.ckt Circuit Speciﬁcation B.4.1 hbdevice.cnt Contact Speciﬁcation B.4.2 hbdevice.phy Device Physics B.4.3 hbdevice.test Simulation B.4.4 hbdevice.swp-hb Performs Sweep B.4.5 pscripts Prints Solution to Screen B.4.6 sweeps.tcl Automates Simulation Sweep B.4.7 hbfuns.tcl Automates HB Simulation Sweep B.4.8

An useful feature HB device simulation is the ability to plot the Fourier coefficients of the potential and carrier distributions within the device simulated. The first 3 Fourier coefficients for the potential distribution for the MOS device biased as in Figure B.3.3 is shown in Figure B.10. The electron distributions are shown in Figure B.11. B.3.2 Diffusion Noise of a Diode

The diffusion noise simulation of the 1-D diode structure is shown for the case of the periodic steady state. The circuit in Figure B.1 was used to simulate the voltage noise of the diode at the base contact. The AC source was swept from 0 to 200 mV which resulted in an 0 to 80 mV fundamental frequency component across the base-emitter junction. The diode equation for a resistive diode [27]isgivenby ( ) ( )= · vBE t − iD t IS exp 1 (B.11) nVT where IS is the saturation current, VT is the thermal voltage, and n is the ideality factor.

Extracting the values of IS and nVT in Equation B.11 for one DC bias point, HB noise simulation was performed using a Matlab script and the Agilent Advanced Design Sys- tem (ADS). Both of these simulations used the macroscopic current noise density of

S (t)=2qI (t) (B.12) iD D and the cyclostationary noise analysis described by Held and Kerr [20]. The same simulation was done with ﬂoods using the microscopic approach described in Section B.1.2. 152

Good agreement between FLOODS, ADS, and Matlab were found and the spectral density of the diffusion noise is shown in Figure B.12. This shows the validity of the microscopic noise simulation approach. ω The frequency dependence of the diode is investigated by varying VG 0 from 0 to 300 mV for a fundamental frequency of 1 kHz, 1 MHz, and 100 MHz. The resulting noise spectrum at 1 Hz is shown in Figure B.13. At low frequencies the large-signal source is able to drive the circuit and cause the noise to decrease with AC bias. At 100 MHz, the diode capacitances short the source so that the noise becomes nearly stationary. B.3.3 Generation-Recombination Noise of a Resistor

When a resistor is biased with a DC supply, the low-frequency noise has the form [24] of S ( f ) S ( f ) α i = v = (B.13) I2 V 2 f where α is a constant dependent on trap properties of the device and f is the frequency being observed. This 1/ f noise is referred to as a resistance ﬂuctation because of the I2 dependence on the current. This α parameter is a constant independent of bias for a homogeneous linear resistor. For a sinusoidal large-signal bias, it can be shown that [54]

S (∆ f ) α v = (B.14) 2 ∆ Vp 4 f = π ∆ when V Vp cos 2 f0t is applied to Equation B.13. The symbol f denotes a frequency − of f f0 where f0 is the frequency of the source driving the circuit and f is the frequency being observed. The signiﬁcance of this is that when a sinusoidal bias is applied, an image of the low-frequency noise appears around the sidebands of the oscillator frequency and the baseband noise disappear. Using the bulk GR simulation feature of the simulator, it is possible to show the frequency conversion of GR noise in resistors. The circuit is shown in

Figure B.14 where Vi is the source driving the circuit and the noise spectrum is measured across Rm. 153

= Figure B.15 demonstrates the noise spectrum observed near DC when Vi πω = Vg cos 2 0t . The baseband noise when Vi Vg (DC bias) is superimposed on the ﬁgure to demonstrate the reduction in noise. The noise observed at the baseband is much less than the case when a DC only bias is applied. The only exception is demonstrated where = 6 f0 10 Hz and the lower sideband of the noise spectrum extends into the baseband. In Figure B.16 the upper sideband noise simulation data is 1/4 the magnitude of the = 6 baseband data for the DC bias condition. Again, the exception is for f0 10 Hz The script used to generate the data for Figures B.15 and B.16 is shown in Appendix B.5. This example for GR noise in resistors demonstrates a case for when the microscopic

fluctuations are invariant to instantaneous bias, but that the transfer functions of these fluctuations to an external contact are being modulated. This situation is only true for a homogeneous resistor. For the case of a junction, the microscopic fluctuations and their transfer functions are both being modulated in time. 154

VGS(1)

VGS(0) VDD

Figure B.6: Bias circuit for NMOS transistor 155

−8 x 10 10 f =0 Hz 0 f =200 MHz 0

4 m) µ (t) (A/ D i 2

−2

−4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Periods of the fundamental

( ) Figure B.7: iD t versus the period of the large-signal source 156

−8 x 10 10 f =0 Hz 0 f =200 MHz 0

4 m) µ (t) (A/ D i 2

−2

−4 0.4 0.45 0.5 0.55 0.6 0.65 v (t) GS

( ) ( ) Figure B.8: iD t versus vGS t 157

−6 x 10 7 0.1 mV 0.2 mV 0.3 mV 6 0.4 mV

4 m) µ 3 (t) (A/ D i

−1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −8 t (sec) x 10

( ) Figure B.9: iD t versus time for a 200 MHz source 158

1.2

0.8

0.6

0.4

0.2

0 0

−5 10 x 10

15 −2 −1 −4 −3 −6 −5 −7 −5 −8 x 10 depth (cm) 20 −10 −9 position (cm)

a) f=0 MHz

0.1

0.08

0.06

0.04

0.02

0 0

−5 10 x 10

15 −2 −1 −4 −3 −6 −5 −7 −5 −8 x 10 depth (cm) 20 −10 −9 position (cm)

b) f=100 MHz

−3 x 10

0 0

−5 10 x 10

15 −2 −1 −4 −3 −6 −5 −7 −5 −8 x 10 depth (cm) 20 −10 −9 position (cm)

c) f=200 MHz

Figure B.10: Fourier coefﬁcients of the potential distribution 159

0 0

−5 10 x 10

15 −2 −1 −4 −3 −6 −5 −7 −5 −8 x 10 depth (cm) 20 −10 −9 position (cm)

a) f=0 MHz

0 0

−5 10 x 10

15 −2 −1 −4 −3 −6 −5 −7 −5 −8 x 10 depth (cm) 20 −10 −9 position (cm)

b) f=100 MHz

0 0

−5 10 x 10

15 −2 −1 −4 −3 −6 −5 −7 −5 −8 x 10 depth (cm) 20 −10 −9 position (cm)

c) f=200 MHz

Figure B.11: Fourier coefﬁcients of the electron concentration. The vertical scale is a logarithmic scale. 160

−13 x 10 1.7 Matlab FLOODS ADS 1.6

1.5

1.4 /Hz) 2 m

µ 1.3

2 (V v S 1.2

1.1

0.9 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 V (1)(V) BE

Figure B.12: Comparison of noise simulation between FLOODS, ADS, and Matlab. The x axis is the magnitude of the fundamental frequency component of the large-signal voltage resulting at the emitter of the diode. 161

−13 x 10 2 1 kHz 1 MHz 1.8 100 MHz

1.6

1.4 /Hz 2 1.2 m µ

2 ) V 0 1 ω ( V S

0.8

0.6

0.4

0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 V (ω ) V G 0

Figure B.13: Comparison of noise spectrum versus AC bias when the large-signal AC source is varied between 1 kHz and 100 MHz 162

Rs V+∆V

Rm Vi

∆R

Figure B.14: Test circuit for GR noise example showing a ﬂuctuation in voltage in response to ﬂuctuation in voltage 163

−12 10

−13 10

−14 10

−15 10

/Hz) −16 2 10 (V v

S −17 10

−18 10 DC f = 106 Hz 0 −19 f = 108 Hz 10 0 f Hz = 1010 Hz 0 −20 10 0 2 4 6 8 10 10 10 10 10 f (Hz)

Figure B.15: Baseband noise under AC bias conditions 164

−12 10

−13 10

−14 10

−15 10

/Hz) −16 2 10 (V v

S −17 10

−18 10 DC f = 106 Hz 0 −19 f = 108 Hz 10 0 f Hz = 1010 Hz 0 −20 10 0 2 4 6 8 10 10 10 10 10 f−f (Hz) 0

Figure B.16: Upper sideband noise under AC bias conditions 165

B.4 MOS Simulation Scripts

B.4.1 hbdevice.ckt: Circuit Elements circuit clear circuit add name=Vgs from=G voltage eq = {G-$Vg} hb.r = {$Vhb} hb.i = {0} circuit add name=Vbs from=B voltage eq = {B-$Vb} circuit add name=Vds from=D voltage eq = {D-$Vd} circuit add name=Vss from=S voltage eq = {S-$Vs}

B.4.2 hbdevice.cnt: Contact Speciﬁcation ################################################################ # Definition of contact regions for device # ################################################################

# # Set the thickness for contact boxes # set eps 0.001

# # Define geometrical locations for contact boxes # contact name=D silicon ylo=-0.25 yhi=-2.7755575615629e-17 xlo=[expr {0-$eps}] xhi=[expr {0+$eps}] contact name=S silicon ylo=-1.072 yhi=-0.822 xlo=[expr {0-$eps}] xhi=[expr {0+$eps}] contact name=G silicon ylo=-0.762 yhi=-0.31 xlo=-0.029 xhi=-0.027 contact name=B silicon xlo=[expr {2.0-$eps}] xhi=[expr {2.0+$eps}]

B.4.3 hbdevice.phy: Device Physics

# Physics: # virtual trap center using srh # constant mobility solution name=Potential nosolve solution add name=DevPsi solve damp negative solution add name=Elec solve !negative solution add name=Hole solve !negative # solution add name=nt solve !negative set T 300 set k 1.38066e-23 set q 1.60218e-19 set Vt [expr {$k*$T/$q}] set ni 1.4e10 set mun 200 set mup 100 set cn 1.0e-7 set cp 1.0e-7 set n1 $ni set p1 $ni set Nt 1.0e10 set taun [expr {1.0 / ($cn*$Nt)}] set taup [expr {1.0 / ($cp*$Nt)}] set U "( $taup*(Elec+$n1) + $taun*(Hole+$p1) )" set U "(Elec*Hole - $ni*$ni) / $U" set esi [expr 11.8 * 8.85418e-14] set eps [expr $esi / $q] set eqnP "$eps * grad(DevPsi) + Doping - Elec + Hole" set eqnE "ddt(Elec) - $mun * $Vt * sgrad(Elec, DevPsi/$Vt) + $U" set eqnH "ddt(Hole) - $mup * $Vt * sgrad(Hole, -DevPsi/$Vt) + $U" pdbSetDouble Silicon DevPsi DampValue 0.005 pdbSetDouble Silicon DevPsi Abs.Error 1.0e-9 pdbSetString Silicon DevPsi Equation $eqnP pdbSetString Silicon Elec Equation $eqnE pdbSetDouble Silicon Elec Abs.Error 1.0e-5 pdbSetString Silicon Hole Equation $eqnH pdbSetDouble Silicon Hole Abs.Error 1.0e-5 166

pdbSetString Silicon Elec Noise.V "4.0 * $Vt * $mun * Elec" pdbSetString Silicon Hole Noise.V "4.0 * $Vt * $mup * Hole"

#G pdbSetBoolean G Elec Fixed 1 pdbSetBoolean G Hole Fixed 1 pdbSetBoolean G DevPsi Fixed 1 pdbSetString G Elec Equation "Doping - Elec + Hole" pdbSetString G Hole Equation "DevPsi + $Vt*log(Hole/$ni) - G" pdbSetString G DevPsi Equation "DevPsi - $Vt*log(Elec/$ni) - G" pdbSetString G Equation "$q * (Flux_Hole - Flux_Elec)"

#S pdbSetBoolean S Elec Fixed 1 pdbSetBoolean S Hole Fixed 1 pdbSetBoolean S DevPsi Fixed 1 pdbSetString S Elec Equation "Doping - Elec + Hole" pdbSetString S Hole Equation "DevPsi + $Vt*log(Hole/$ni) - S" pdbSetString S DevPsi Equation "DevPsi - $Vt*log(Elec/$ni) - S" pdbSetString S Equation "$q * (Flux_Hole - Flux_Elec)"

#D pdbSetBoolean D Elec Fixed 1 pdbSetBoolean D Hole Fixed 1 pdbSetBoolean D DevPsi Fixed 1 pdbSetString D Elec Equation "Doping - Elec + Hole" pdbSetString D Hole Equation "DevPsi + $Vt*log(Hole/$ni) - D" pdbSetString D DevPsi Equation "DevPsi - $Vt*log(Elec/$ni) - D" pdbSetString D Equation "$q * (Flux_Hole - Flux_Elec)"

#B pdbSetBoolean B Elec Fixed 1 pdbSetBoolean B Hole Fixed 1 pdbSetBoolean B DevPsi Fixed 1 pdbSetString B Elec Equation "Doping - Elec + Hole" pdbSetString B Hole Equation "DevPsi + $Vt*log(Hole/$ni) - B" pdbSetString B DevPsi Equation "DevPsi - $Vt*log(Elec/$ni) - B" pdbSetString B Equation "$q * (Flux_Hole - Flux_Elec)"

# Oxide set eso [expr 3.9 * 8.85418e-14] set epo [expr $eso / $q] set eqnO "$epo * grad(DevPsi)" pdbSetDouble Oxide DevPsi DampValue 0.005 pdbSetDouble Oxide DevPsi Abs.Error 1.0e-9 pdbSetString Oxide DevPsi Equation $eqnO set eqnsurf "DevPsi_Oxide - DevPsi_Silicon" pdbSetBoolean Oxide_Silicon DevPsi FixedOxide 1 pdbSetBoolean Oxide_Silicon DevPsi FluxConstant 1 pdbSetString Oxide_Silicon DevPsi Equation_Oxide $eqnsurf

B.4.4 hbdevice.test: Initial Solution

# make sure the circuit elements are added before contacts math device dim=2 umf gmres row scale #math device dim=2 superlu none row scale #source hbdevice.geo struct inf=test.str source hbdevice.phy source hbdevice.ckt source hbdevice.cnt sel z=$ni*exp(DevPsi/$Vt) name=Elec sel z=$ni*exp(-DevPsi/$Vt) name=Hole

# Use preconfigured scripts for printing circuit values source pscripts source sweeps.tcl set Vd 0.0 set Vg 0.00 set Vb 0.0 set Vs 0.0 device

B.4.5 hbdevice.swp-hb: Initial Solution init inf=hbdevice.vg0.5vd1.0 #init inf=hbbias.str source pscripts source sweeps.tcl set PHYFILE hbdevice.phy set CKTFILE hbdevice.ckt 167

set CNTFILE hbdevice.cnt set MCAEXT hb source $PHYFILE source $CKTFILE source $CNTFILE set Vg 0.5 set Vd 1.0 set Vs 0.0 set Vb 0.0 set Vhb 0.0 device source hbfuns.tcl set fund 2e8 hbdevice hb=5 fund=$fund hbsweep Vhb 0 1.0 0.1 hbsweep$fund.dat phb exit

B.4.6 pscripts: Prints Solution to screen

#################### proc pac {} { puts "\nAC Solution\n\tReal\t\tImaginary" foreach i [circuit nodes] { set x [ circuit value real name=$i ] set y [ circuit value imag name=$i ] puts [ format "%s\t%1.5e\t%1.5e" $i $x $y ] } } proc pac2 {} { puts "\nAC Solution\n\tMagnitude\t\tPhase" foreach i [circuit nodes] { set x [ circuit value mag name=$i ] set y [ circuit value phase name=$i ] puts [ format "%s\t%1.5e\t%1.5e" $i $x $y ] } } proc pdc {} { puts "DC Solution" foreach i [circuit nodes] { set x [ circuit value name=$i ] puts [ format "%s\t%1.5e" $i $x ] } } proc phb {} { puts "HB Solution" set f [ hbcircuit freq] puts [ format "freq\t%s" $f ] foreach i [circuit nodes] { set xr [ hbcircuit value name=$i real ] set xi [ hbcircuit value name=$i imag ] puts [ format "%s\t%s" $i $xr ] puts [ format "\t%s" $xi ] } } proc phb2 {} { puts "HB Solution" set f [ hbcircuit freq] puts [ format "freq\t%s" $f ] foreach i [circuit nodes] { set xr [ hbcircuit value name=$i mag ] set xi [ hbcircuit value name=$i pha ] puts [ format "%s\t%s" $i $xr ] puts [ format "\t%s" $xi ] } } proc phbac {} { puts "HB Solution" set f [ hbcircuit ss freq] puts [ format "freq\t%s" $f ] foreach i [circuit nodes] { set xr [ hbcircuit value name=$i real ss ] set xi [ hbcircuit value name=$i imag ss ] puts [ format "%s\t%s" $i $xr ] puts [ format "\t%s" $xi ] } } proc phbac2 {} { puts "HB Solution" set f [ hbcircuit ss freq] puts [ format "freq\t%s" $f ] foreach i [circuit nodes] { set xr [ hbcircuit value name=$i mag ss] 168

set xi [ hbcircuit value name=$i pha ss] puts [ format "%s\t%s" $i $xr ] puts [ format "\t%s" $xi ] } } proc mydump {ddir} { catch {exec mkdir $ddir} cd $ddir set f [select] foreach i $f { if { ![string compare $i EdgeCouple] } continue # puts $i select z= $i print.data outf=$i.dat } cd .. }

B.4.7 sweeps.tcl: Initial Solution

#tcl procedures for doing sweeps # requires pscripts proc Ramp1 { varname initv finalv maxstep body} { global Vg Vd Vb Vs upvar $varname v set v $initv device uplevel $body while { [expr abs($initv) ] < [expr abs($finalv) ] } { set initv [ expr $initv + $maxstep ] if { [expr abs($initv)] > [expr abs($finalv)] } { set initv $finalv } set v $initv device uplevel $body } } proc linsweep { varname initv finalv maxstep filename text1 text2 body } { global Vg Vd Vb Vs Vhb upvar $varname v puts "writing to $filename" set ofile [ open $filename w ] set ns " " set msg "MC-ASCII text {$text1} {$text2} \n\n record $ns Vd $ns Id $ns Vg $ns Ig $ns Vs $ns Is $ns Vb $ns Ib\n" set un " V \\8m\\5A/\\8m\\5m" append msg " None $un $un $un $un\n" set ns " " append msg " 1.0 $ns 1.0 $ns 1.0 $ns 1.0 $ns 1.0 $ns 1.0 $ns 1.0 $ns 1.0 $ns 1.0" puts $ofile $msg flush $ofile

set rcd 100

set v $initv while { [expr abs($initv) ] < [expr abs($finalv) + 1.0e-4 ] } { set ret [ catch { device } ] if { $ret } { set initv [ expr $initv - $maxstep ] set v $initv set maxstep [ expr $maxstep * 3.0 / 4.0 ] if { $maxstep < 0.01 } { puts " Too small of a step!!!!!" break; } puts [format "Non-Convergence!! Reducing Step to %1.7e" $maxstep] } else { #device #device #device uplevel $body set msg " $rcd" set foo [circuit name=D value] append msg [ format " %1.7e" $foo ] set foo [expr [circuit name=Cur_Vds value] * 1e6 ] append msg [ format " %1.7e" $foo ] set foo [circuit name=G value] append msg [ format " %1.7e" $foo ] set foo [expr [circuit name=Cur_Vgs value] * 1e6 ] append msg [ format " %1.7e" $foo ] set foo [circuit name=S value] append msg [ format " %1.7e" $foo ] set foo [expr [circuit name=Cur_Vss value] * 1e6 ] append msg [ format " %1.7e" $foo ] set foo [circuit name=B value] append msg [ format " %1.7e" $foo ] set foo [expr [circuit name=Cur_Vbs value] * 1e6 ] append msg [ format " %1.7e" $foo ] 169

puts $ofile $msg flush $ofile }

set initv [ expr $initv + $maxstep ] set v $initv # if { [expr abs($initv)] > [expr abs($finalv)] } { # set initv $finalv #} incr rcd } close $ofile set v $finalv }

B.4.8 hbfuns.tcl: Harmonic Balance Simulaton

# This procedure takes a list of electrodes, and a list of ckt elements and puts all the data to a file # This file is opened for append with a comment ( which can be a independent variable ) # The file is created if it doesn’t exist # Each circuit node and voltage source is printed out proc appendhbdatarow { Comment Filename} { set f [open $Filename a ] puts $f [format "var\t%s" $Comment ]

# get the frequency set freq [join [ hbcircuit freq] \t] puts $f [ format "freq\treal\t%s" $freq ]

foreach i [circuit nodes] { set xr [ hbcircuit value name=$i real ] puts $f [format "%s\t%s\treal\t%s" $Comment $i $xr] set xi [ hbcircuit value name=$i imag ] puts $f [format "%s\t%s\timag\t%s" $Comment $i $xi] } flush $f close $f } proc hbsweep { varname initv finalv maxstep filename body } { global Vg Vd Vb Vs Vhb upvar $varname v puts "writing to $filename"

set v $initv while { [expr abs($initv) ] < [expr abs($finalv) + 1.0e-4 ] } { set ret [ catch { hbdevice } ] if { $ret } { set initv [ expr $initv - $maxstep ] set v $initv set maxstep [ expr $maxstep * 3.0 / 4.0 ] if { $maxstep < 0.01 } { puts " Too small of a step!!!!!" break; } puts [format "Non-Convergence!! Reducing Step to %1.7e" $maxstep] } else { appendhbdatarow $varname $filename uplevel $body }

set initv [ expr $initv + $maxstep ] set v $initv } set v $finalv }

B.5 GR Noise Example Script solution name=Potential nosolve solution add name=DevPsi solve damp negative solution add name=Elec solve !negative solution add name=Hole solve !negative solution add name=nt solve !negative set T 300 set k 1.38066e-23 set q 1.60218e-19 set Vt [expr {$k*$T/$q}] set Nv 1.04e19 set Nc 2.8e19 set Eg 1.12 set ni 1.4e10 set mun 1450 set mup 500 set dEc -0.28 set n1 [expr ($Nc*exp($dEc/$Vt))] set p1 [expr ($ni*$ni/$n1)] set cn 1.0e-8 170

set cp 1.0e-7 set Nt 1.0e13 set Rn "$cn * Elec * ($Nt - nt)" set Gn "$cn * $n1 * nt" set Rp "$cp * Hole * nt" set Gp "$cp * $p1 * ($Nt - nt)" set esi [expr 11.8 * 8.85418e-14] set eps [expr $esi / $q] set eqnP "$eps * grad(DevPsi) + Doping - Elec + Hole - nt" set eqnE "ddt(Elec) - $mun * $Vt * sgrad(Elec, DevPsi/$Vt) + $Rn - $Gn" set eqnH "ddt(Hole) - $mup * $Vt * sgrad(Hole, -DevPsi/$Vt) + $Rp - $Gp" set eqnt "ddt(nt) - ($Rn - $Gn) + ($Rp - $Gp)" pdbSetDouble Si DevPsi DampValue 0.025 pdbSetDouble Si DevPsi Abs.Error 1.0e-9 pdbSetString Si DevPsi Equation $eqnP pdbSetString Si Elec Equation $eqnE pdbSetDouble Si Elec Abs.Error 1.0e1 pdbSetString Si Hole Equation $eqnH pdbSetDouble Si Hole Abs.Error 1.0e1 pdbSetString Si nt Equation $eqnt pdbSetDouble Si nt Abs.Error 1.0e1 pdbSetString Si Elec Noise.V "4.0 * $Vt * $mun * Elec" pdbSetString Si Hole Noise.V "4.0 * $Vt * $mup * Hole" pdbSetString Si Elec Noise.N_Elec "2.0 * ($Gn + $Rn)" pdbSetString Si Hole Noise.N_Hole "2.0 * ($Gp + $Rp)" pdbSetString Si nt Noise.N_nt "2.0 * ($Gn + $Rn + $Gp + $Rp)" pdbSetString Si Elec Noise.N_nt "- 2.0 * ($Gn + $Rn)" pdbSetString Si Hole Noise.N_nt "2.0 * ($Gp + $Rp)" pdbSetString Si nt Noise.N_Elec "- 2.0 * ($Gn + $Rn)" pdbSetString Si nt Noise.N_Hole "2.0 * ($Gp + $Rp)" pdbSetBoolean top Elec Fixed 1 pdbSetBoolean top Hole Fixed 1 pdbSetBoolean top DevPsi Fixed 1 pdbSetBoolean top nt Fixed 1 pdbSetString top Elec Equation "Doping - Elec + Hole - nt" pdbSetString top Hole Equation "DevPsi + $Vt*log(Hole/$ni) - top" pdbSetString top DevPsi Equation "DevPsi - $Vt*log(Elec/$ni) - top" pdbSetString top nt Equation "Elec*$Nt/(Elec+$n1) - nt" pdbSetString top Equation "1.619e-19 * (Flux_Hole - Flux_Elec)" pdbSetBoolean bot Elec Fixed 1 pdbSetBoolean bot Hole Fixed 1 pdbSetBoolean bot DevPsi Fixed 1 pdbSetBoolean bot nt Fixed 1 pdbSetString bot Hole Equation "Doping - Elec + Hole - nt" pdbSetString bot Elec Equation "DevPsi - $Vt*log(Elec/$ni) - bot" pdbSetString bot DevPsi Equation "DevPsi + $Vt*log(Hole/$ni) - bot" pdbSetString bot nt Equation "$p1*$Nt/(Hole+$p1) - nt" pdbSetString bot Equation "1.619e-19 * (Flux_Hole - Flux_Elec)" circuit clear circuit add name=Vss from=Vn1 voltage eq = {Vn1+$Vss} ac.r= {$V1} ac.i= {0} hb.r= {$V2} hb.i= {0} circuit add name=Rs from=Vn1 to=top eq = {(top - Vn1)/857459} circuit add name=Gnd from=bot voltage eq=bot line x loc=0.0 spac=0.02 tag=top line x loc=2.0 spac=0.02 tag=bot region xlo=top xhi=bot silicon init contact name=top silicon xlo=-0.1 xhi=0.1 contact name=bot silicon xlo=1.91 xhi=2.1 sel z=1.0e15 name=Doping # sel z=1.0e19*exp(-x*x/(0.2*0.2))-1.0e16 name=Doping sel z=0.5*(Doping+sqrt(Doping*Doping+4.0e20))/$ni name=arg sel z=$Vt*log(arg) name=DevPsi sel z=$ni*exp(DevPsi/$Vt) name=Elec sel z=$ni*exp(-DevPsi/$Vt) name=Hole

# Use preconfigured scripts for printing circuit values source pscripts set Vss 10.10 set V1 1 set V2 10.10 device device device pdc set conv 1.0e8 set fstep [expr pow(10,0.2)] set Vss 1e-20 set fu 1e9 device 171

device device device pdc for {set fu 1.0e6} {$fu <= 1e10} {set fu [expr $fu * 100]} { set foo [format "gr_aconly_%1.0e" $fu] set out [open $foo.dat w] puts $out [format "Vdc=%1.4e Vac=%1.4e fund=%1.4e" $Vss $V2 $fu] puts $out "%freq\tSgrbb\tSgrlsb\tSgrusb" for {set freq 1.0} {$freq < [expr $fu]} {set freq [expr $freq * $fstep]} { hbdevice fund=$fu hb=3 hbdevice phb

set foo [expr $freq+$fu] hbdevice noise=top freq=$foo set Sgrusb [expr {$conv*abs([circuit value Sgr name=top])}]

hbdevice noise=top freq=$freq preuse set Sgrbb [expr {$conv*abs([circuit value Sgr name=top])}]

set foo [expr $freq-$fu] hbdevice noise=top freq=$foo preuse set Sgrlsb [expr {$conv*abs([circuit value Sgr name=top])}]

puts $out [format "%1.7e\t%1.7e\t%1.7e\t%1.7e" $freq $Sgrbb $Sgrlsb $Sgrusb] flush $out } close $out } exit REFERENCES [1] K. Mayaram, D. C. Lee, S. Moinian, D. A. Rich, and J. Roychowdhury, “Computer- aided circuit analysis tools for RFIC simulation: Algorithms, features, and limita- tions,” IEEE Transactions on Circuits and Systems – II, vol. 47, pp. 274–286, Apr. 2000. 1, 3, 4, 71

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[65] S. B. Lippman, C++ Primer. Reading, MA: Addison Wesley, 2 ed., 1991. 131 [66] F. Bonani, S. D. Guerrieri, G. Ghione, and M. Pirola, “A new approach to the physics- based noise analysis of semiconductor devices operating in large signal, (quasi) periodic regime,” in IEDM Technical Digest, pp. 321–324, 1997. 136 [67] F. Danneville, G. Dambrine, and A. Cappy, “Generalization of the impedance ﬁeld method for noise modelling of devices under nonlinear operation,” in 14th Interna- tional Conference on Noise in Physical Systems and 1/f Fluctuations, pp. 140–143, 1997. 136 [68] Y. Tsividis, Operation and Modeling of the MOS Transistor. Boston: McGraw-Hill, 2 ed., 1999. 150 BIOGRAPHICAL SKETCH Juan Eusebio Sanchez graduated cum laude with a Bachelor of Science in electrical engineering from the University of South Florida in May 1995. He has been attending the

University of Florida since the Fall of 1995 and received a Master of Science in electrical and computer engineering in August 1997. Since then he has been pursuing a PhD in the same discipline. His research interests have focused on noise phenomena in semiconductor devices under periodic large-signal conditions.

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