! #"%$'&)(+*-,./#01 #2!3546387&/69: #2;,$=4?,.)@,.A/B2C #DFE&)$=

Contents

1 Introduction to MINIMOS 1

1.1 The New Features of MINIMOS 6 jHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 1

1.2 MINIMOS 2D Model Structure jljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 2

1.3 MINIMOS 3D Model Structure jljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 4

1.4 Transient Simulation jHjkjljHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 5

1.5 Monte Carlo Simulation of Carrier Transport jHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 8

1.6 Small-Signal AC Analysis jHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 9

1.7 Nonplanar Geometries jkjljHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 10

1.8 Three-Dimensional Simulation jHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 11

1.9 MESFET Simulation jHjkjljHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 12

1.10 Gate Charge and Capacitance Calculation jHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 13

1.11 Monitoring Program Execution jljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 14

2 The Physical Models of MINIMOS 17

2.1 Mobility Model for Silicon jHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 17

2.2 Mobility Model for Gallium Arsenide jljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 20

2.3 Generation-Recombination Modeling jljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 21

2.4 The Local Carrier Heating Model jHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 23

2.5 Modeling of Band-Structure Parameters jHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 24

2.5.1 Models for Silicon jHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 24

i

2.5.2 Models for Gallium Arsenide jHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 25

2.6 The Band-to-Band Tunneling Model jljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 26

2.7 The Dynamic Interface and Bulk Trap Model jHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 31

2.7.1 The Physical Model and Numerical Considerations jHjHjljHjkjljHjHjHjljHjkjljHjHj 32

2.7.2 Interface Traps jljHjkjHjljHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 33

2.7.3 Bulk Traps jHjHjljHjkjHjljHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 35

2.7.4 Examples jHjHjljHjkjHjljHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 37

2.8 Gate Depletion Analysis jHjkjHjljHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 40

2.8.1 The Full Two-Dimensional Model (MPOLY=1) jljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 41

2.8.2 The Approximate One-Dimensional Model (MPOLY=0) jljHjkjljHjHjHjljHjkjljHjHj 42

2.8.3 Fixed Charge and Traped Charge at the Gate/Oxide Interface jljHjHjHjljHjkjljHjHj 44

2.8.4 Examples jHjHjljHjkjHjljHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 44

2.9 Power VDMOSFET Simulation jHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 47

2.10 The Monte Carlo Transport Module jljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 50

2.10.1 Band Structure jljHjkjHjljHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 50

2.10.2 Scattering Rates jHjkjHjljHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 51

2.10.3 Impact Ionization jkjHjljHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 54

2.10.4 Material Characteristics jHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 55

2.10.5 The Monte-CarloDrift-Diffusion Coupling Technique jljHjkjljHjHjHjljHjkjljHjHj 55

2.10.6 The Monte Carlo Windows jljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 56

2.10.7 The Self-Consistent Iteration Scheme jHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 58

2.10.8 Calculation of Averages jHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 58

2.10.9 The Particle Weighting Algorithm jHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 59

2.10.10 The Self-Scattering Algorithm jHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 60

3 Reference of Input Directives 61

BIAS jkjHjljHjHjljHjkjHjljHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 62

DEVICE jljHjHjljHjkjHjljHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 64

ii

END jHjHjHjljHjkjljHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 68

FIMPLANT jHjkjljHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 71

GEOMETRY jkjljHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 73

GRID jHjHjljHjkjljHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 77

IMPLANT jljHjkjljHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 83

INTERFACE jHjkjljHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 85

MOBILITY jHjkjljHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 94

MONTE-CARLO jHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 96

OPTION jHjljHjkjljHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 103

OUTPUT jljHjkjljHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 109

PROFILE jljHjkjljHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 112

RECOMBINATION jHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 120

STEP jHjHjljHjkjljHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 125

TRANSIENT jkjljHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 128

4 Files 129

4.1 The MINIMOS 2D Files jljHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 129

4.1.1 The Input Files jkjljHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 129

One-dimensional Profiles (FILE=1-D) jHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 130

Twodimensional Profiles (FILE=2-D) jHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 131

4.1.2 The Main Ouput Files jHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 134

The Formatted Output File jHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 134

The Reference File jHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 135

The Binary Output File jljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 135

The Formatted PIF Output File jHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 138

4.1.3 Auxiliary Output Files jHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 139

Table of I-V Data jHjHjHjljHjkjljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 139

The Monte Carlo Output File jljHjHjljHjHjkjljHjHjljHjHjkjljHjHjljHjHjljkjHjHjljHj 139

iii

The Transient Output File jHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 140

The Gate Charge File jljHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 141

Poly-Gate Voltage Drop jHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 141

Charge-Pumping Output jHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 142

Band-to-Band Tunneling Currents jHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 143

4.1.4 The MINIMOS 3D Files jHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 143

The Binary Data Link File jHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 143

The Formatted Output and the Reference Files jljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 143

The Binary Output File jHjHjljHjkjHjljHjHjljHjkjljHjHjHjljHjkjljHjHjHjljHjkjljHjHj 144

Bibliography 149

Index 155

iv

1Introduction to MINIMOS

MINIMOS is a software tool for the numerical simulation of semiconductor field-effect transistors. Thefirst version was released over 10 years ago, and since then the code has undergone continuous ex-tensions and improvements, culminating in 1991s Version 5.2, which is capable of modelling siliconMOSFETs and SOI transistors as well as gallium arsenide MESFETs. The fundamental semiconductorequations wich are solved in MINIMOS numerically comprise Poissons equation and two carrier con-tinuity equations for electrons and holes. For the various coefficients such as carrier mobility, carriergeneration/recombination and impact ionization rates carefully chosen models are implemented. Thenumerical schemes of MINIMOS are designed to handle planar and non-planar device structures in bothtwo and three space dimensions.

1.1 The New Features of MINIMOS 6

MINIMOS 6 is upward-compatible to all previous versions of MINIMOS and works with any input filevalid for a previous version. The major extensions of MINIMOS 6 are listed below:

m Transient analysis.m Self-consistent inclusion of the trap-rate equations for dynamic of interface and bulk traps.m A Monte Carlo Module which couples self-consistently to the Poisson equation and replaces the

drift-diffusion approximation in critical device areas.m A band-to-band tunneling model.m Depletion analysis of poly-silicon gates.m Interface to the TCAD shell of Vista and a PIF-interface.

1

2 CHAPTER 1 INTRODUCTION TO MINIMOS

For instance, the transient facility in conjunction with the trap rate equations allow for an accuratenumerical analysis of the charge pumping-phenomenon.

The Monte Carlo Module available in this version makes MINIMOS 6 to an easy to use Monte Carlodevice simulator which describes hot electron transport and impact ionization in a fully non-local manner.

In the following some minor extensions are summarized:

Output data can be written in ASCII-PIF format (OUTPUT PIF=YES). Power Vertical DMOS transistors can be handled (OPTION LVDM=YES). P-i-n diodes can be handled (flag for p-i-n diode simulation is set if an NPP+ or PNN+ structure

is specified by appropriate profile statements). If a STEP directive is specified the resulting I-V data are written automatically to a file

(unless OUTPUT IV=NO). Gate charge can be calculated and written to a file (OPTION GCHC=YES). Coupled solution of the AC small-signal device equations for more robustness at higher

frequencies (OPTION CP=YES) A new current integration method for transient and steady-state conditions

1.2 MINIMOS 2D Model Structure

MINIMOS builds the consistent solution of the semiconductor equations in an hierarchical manner, startingwith a relatively simple model which is subsequently refined by taking into account more complicatedphysical mechanisms. This degree of sophistication is controlled by the key MODEL on the OPTIONdirective, see Page 103).

Presently the user may specify 6 distinct models. Apart from MODEL=THRES, which leads to a thresholdvoltage calculation, the models 1-D, 2-D, AVAL, HOT and MC define a sophistication hierarchy.

As an example consider the case that a user has specified a MOSFET in a MINIMOS input file, andthat the MODEL key on the OPTION directive is set to MODEL=AVAL. The following is valid only forMOSFETs, the differences in case of MESFETs are described later in this section.

MINIMOS at first tries to minimize the error norms iteratively by accounting for the simplest physicalmodel. Thus, after having made a guess for the initial solution, MINIMOS starts with the internal modeloption 1-D. In this model the MINIMOS does not solve the majority continuity equations and generationrecombination effects are neglected. The Poisson equation is solved in the full simulation domain, whereasthe minority continuity equation is just solved in the partial domain from to gateinsulator to about thehalf of the pnjunction depth. Within the deep bulk region the minorities are computed by extrapolationassuming a constant quasi fermi level. This model offers fast execution since the computational demandfor the costly solution of the carrier continuity equations is minimized. Within model 1-D MINIMOS

1.2 MINIMOS 2D MODEL STRUCTURE 3

builds its grid through an adaptive grid refinement loop. Each refinement step consists of a grid updatefollowed by a Gummel iteration loop, which terminates when all error norms lie beyond the threshold setby the key ERROR on the END directive. After completing the grid loop, that is when the mesh weightingfunction (BWF) is smaller than unity for all mesh intervals, MINIMOS switches its internal mode to2-D. Now the minority continuity equation is solved in the full simulation domain, for the majoritiesthere is still a constant quasi fermi level assumed. No grid updates are carried out. Since the deviationfrom thermal equilibrium in the deep bulk is usually small, only few Gummel iteration steps should benecessary.

Finally, MINIMOS executes in AVAL mode. The Poisson equation and both carrier continuity equationsare solved in the full domain. The recombination rate is computed at every Gummel step, the impact-ionization and tunneling generation rate, however, only after a sufficient decrease in the error norms. Thisleads to an avalanche subcycle. MINIMOS completes its execution when the last carrier generation updatedoes not increase the error norms over the threshold set by ERROR.

For a MESFET MINIMOS proceeds slightly different. In the 1-D mode both carrier continuityequations are solved, however generation/recombination is neglected. Then in 2-D mode the gener-ation/recombination mechanisms, impact ionization excluded, are switched on. Finally, the AVAL modefor the MESFET has the same meaning as the AVAL mode for the MOSFET.

In model HOT another cycle is superimposed on the previous model AVAL. To account for carrier heatingeffects local carrier temperatures are introduced. After the completion of each AVAL cycle a carriertemperature update is performed. MINIMOS in HOT mode terminates when the error norms in bothcarrier temperatures have decreased beyond the error measure TCERR, specified within the END directive.

In modelMCMINIMOS generates an initial solution for the subsequent Monte Carlo iteration by processingthe internal models 1-D through AVAL. The resulting distribution of the electrostatic potential serves asinput for the Monte Carlo simulation. As output the distributions of electron mobility, electron temperatureand ionization coefficient are obtained in critical device regions. With the updated mobility and temperaturedistributions a gummel iteration is performed. After its termination an avalanche subcycle is imposed toaccount for the altered ionization coefficients. This procedure yields the potential for the next Monte Carloiteration. All the above described steps which are needed to obtain the new potential from the old onecan be considered as one Monte Carlo Poisson iteration step. MINIMOS terminates after a user-definednumber of such iteration steps has been executed.

In the following we summarize the model hierarchy of MINIMOS. A model can be specified by theMODEL key on the OPTION directive.

4 CHAPTER 1 INTRODUCTION TO MINIMOS

MODEL Description1-D Bulk-MOSFET: Poisson, minorities, partial domain, no nkoqp

SOI-MOSFET : Poisson, minorities, full domain, no nkoqpMESFET : Poisson, both carriers, full domain, no nkoqp

THRES Loop for threshold voltage calculation by model 1-D2-D MOSFET : Poisson, minorities, full domain, no nJoqp

MESFET : Poisson, both carriers, full domain, nkoqp , no impact ionizationAVAL MOSFET, MESFET: Full equation set, full domain, nJoqp including impact ionization.HOT Local carrier tempearture model in addition to model AVAL.MC Full set of equations with electron temperature, electron mobility and ionization

coefficient from a Monte Carlo simulation.

1.3 MINIMOS 3D Model Structure

MINIMOS 3D relies on a similar model structure as MINIMOS 2D. With the exception ofMODEL=THRES,MINIMOS 3D can execute in the same modes as MINIMOS 2D. In other words, the key MODEL on theOPTION directive directs the computation of the three-dimensional solution in an equivalent way. Howeveran additional parameter, M3MODE is required to control MINIMOS 3D execution. The combination ofboth parameters allows the setting of a variety of three-dimensional models. At first it is assumed thatM3MODE=1 was set on the OPTION directive. M3MODE=1 leads MINIMOS 3D to build the mesh in z-direction together with a coarse three-dimensional solution by solely solving the Poisson equation in threedimensions. The carrier concentrations are calculated from their quasi fermi level, assuming vanishingcurrent densities in the z-direction. The solution of the previously solved two-dimensional problem,corresponding to the setting of the MODEL key, is taken as a Dirichlet boundary condition in the firstx-y-plane. This first x-y-plane is assumed to be the center of symmetry with respect to the z-direction.This model offers comparatively fast execution since no solution of the carrier continuity equations iscarried out.

After completion of the model M3MODE=1 MINIMOS 3D either terminates execution or continues withM3MODE=2, if this was specified on the OPTION directive. With M3MODE=2 the grid adaptation isdeactivated. The carrier continuity equations are solved according to the setting of the MODEL key on theOPTION directive. Thus, if MODEL=1-D or MODEL=2-Dwas specified within the MINIMOS input deckthe minority carrier continuity equation is solved and no avalanche and hot carrier effects are regarded. Inall other cases, namely MODEL=AVAL or MODEL=HOT, both carrier continuity are solved and avalancheand hot carrier effects are taken into account analogously to the two-dimensional case. Generally, the firstx-yplane is treated as a Neumann boundary, thus indicating that the device is modelled symmetricallywith respect to the first x-yplane.

The second character in the M3MODE specification allows the use of a previous three-dimensional solutionof the same example as initial solution, which can be helpful and CPU time saving during testing,development etc. For example consider the case that MINIMOS 3D has been run with the settingof M3MODE=1. To obtain a solution with the setting M3MODE=2 change the setting of M3MODE to

1.4 TRANSIENT SIMULATION 5

M3MODE=2O, rename the binary outputfile of the previous MINIMOS 3D run appropriately and restartMINIMOS 2D. See Chapter 4. MINIMOS 2D is terminated after having loaded the changed common-block data. Now invoke MINIMOS 3D. If MINIMOS 3D finds a file with the old three-dimensionalsolution of M3MODE=1 it continues immediately with M3MODE=2. See also the key M3MODE on theoption directive Page 103 to obtain further information.

Model Option Key Summary (MINIMOS 3D)

1N Poisson only, full domain, no n:r;p1O Same as 1N, starting with prev. initial solution2N Equation set according to MODEL key2O Same as 2N, starting with prev. initial solution

1.4 Transient Simulation

An efficient algorithm for solving the time-dependent semiconductor equations has been built into MIN-IMOS. The dynamics equations for arbitrary distributed interface and bulk traps are consistently solvedwith the basic semiconductor equations (see Section 2.7).

The solution of the time-dependent semiconductor equations in MINIMOS employs a hybrid algorithmusing a fast decoupled solution of the semiconductor equations [1]. The algorithm proceeds in twomain steps: At first the successive procedure devised by Mock computes an initial solution [2]. Thisinitial solution phase is indicated by a minus sign preceeding the Poisson solver iteration count in thesimulation reference file. In the subsequent correction steps Gummels algorithm is applied to compute theselfconsistent final solution. By this approach the stability can be achieved at very short time steps, whilethe Poisson equation is fulfilled with desired (controlled) accuracy independent of the actual time step.The algorithm has been tested on a wide range of MOSFETs under a manifold of operating conditionsincluding biasing and switching times, ranged from very short (1sAt ) to very long (1 t ) and has proven tobe both, fast and stable.

After each time step a summary of terminal currents is printed in the reference file. These currents aresplit up into the electron (ele), hole (hol) and displacement (dis) current, each of these components beingprinted on a separate line. The last line of this current information record prints the sum of electron, holeand displacement current, i.e. the total electrodynamic current for each individual contact. The last entryin this line is the sum of all terminal currents. Due to the conservation law of charge this must be a smallnumber, which depends on the solution error of the semiconductor equations. After the last time stepaverage values of all terminal currents across the total simulation time are printed in the reference file.For the calculation of the terminal currents in transient conditions we employed an accurate weightingfunctions-based technique.

6 CHAPTER 1 INTRODUCTION TO MINIMOS

The evolution of the terminal voltages and currents are written to file unit NUMTRA.

Specification of the terminal voltages

Each transient simulation initially requires the computation of a steady-state solution. The terminal biasfor this steady-state is specified on the BIAS directive (see BIAS directive Page 62). After convergenceMINIMOS will start a transient analysis, if a TRANSIENT directive (see Page 128) is found in theMINIMOS input deck. The TRANSIENT directive is used to specify all required input parameters tobuild waveforms for voltage-controlled contacts.

By default the voltage waveform for an individual contact is independent from the waveforms on othercontacts. The default voltage waveform for each contact is a constant voltage equals to the steady statebias. To specify a different waveform the waveform-switch Wi is used, where i stands for B (bulk contact),D (drain contact), G (gate contact) or S (source contact). Wi must be an integer out of u 1,2,3 v , with 1 beingthe default value. Wi=1 denotes a constant voltage on the contact i. The two further choices, Wi= u 2,3 v aredescribed in detail below.

Wi=2 denotes a periodic trapezoidal voltage with the starting point at t=0 located at the beginning ofthe falling edge of the trapeze (see Fig. 1.1). To specify parameters for this waveform the Tij and Uijparameters are used, where i denotes the contact as outlined above and j enumerates the parameters of theindividual waveform. All T parameters are specified in seconds, all U parameters in volts.

The waveform parameters for wAxgy 2 are the following: Ti1 is the period length of the trapeze pulse ( z ),Ti2 is the fall time ( {}| ), Ti3 the lower-plateau time ( {}~ ), Ti4 the rise time ( {' ) and Ti5 is the simulation endtime. Note that the upper-plateau time ( {} ) of the waveform needs not be specified. If Ti5 is larger thenTi1, more than one period will be simulated. The parameter Ui1 is by default equal to the correspondingvoltage given in the bias directive and needs not be specified. Ui2 specifies the amplitude of the trapezoidalwaveform. The given value for the amplitude may be positive or negative. If a negative value is specifiedfor Ui2, the waveform is mirrored with respect to the voltage axis of the initial steady-state bias. In thiscase the specifications falling edge and rising edge interchange their meaning and the transient analysisstarts at {6y 0 with a rising edge. Clearly, Ti2 denotes the rising edge and Ti3 denotes the falling edgewhen x 2 0.

The parameters Ui3 and Ui4 control the time-step lengths when a rising or falling voltage ramp is steppedthrough during a transient simulation. Specifically, Ui3 denotes the voltage increment to be used for thecontact i for the falling edge (if x 2 0, otherwise the rising edge), Ui4 denotes the voltage increment tobe used for the contact i for the rising edge (if x 2 0, otherwise the falling edge).

With wxy 3 three-level voltage pulses can be constructed (see Fig. 1.2). The following parametersspecify all required values: Ti1 is the waveform period (z ), Ti2 the fall time prime ( {=

|

), Ti3 the mid-levelduration ( {' ), Ti4 the fall time double prime ( {

|

), Ti5 the rise time ( {= ), Ti6 the top-level duration ( {

) andTi7 the end of the simulation. Specification of the low-level duration ( {}~ ) is not necessary. The waveformis periodic with the period length Ti1. Ui1 equals the steady-state bias and needs not be specified, Ui2is the waveform total amplitude and can be positive or negative. In the negative case the waveform ismirrored with respect to the axis placed on the top level. Both falling edges change to rising edges, the

1.4 TRANSIENT SIMULATION 7

Figure 1.1: Trapezoidal waveform

rising edge changes to the falling edge and the simulation starts at the first rising edge ( { |

). Ui3 specifiesthe mid-level voltage, which must range within [Ui1,Ui1-Ui2].

The parameters Ui4, Ui5 and Ui6 control the time step lengths for the three voltage ramps: Ui4 is the rampstep voltage increment of fall time prime, Ui5 is the ramp step voltage increment of fall time double primeand Ui6 is the ramp step voltage increment of the rising voltage ramp. For a negative amplitude Ui2 theseparameters change their meaning accordingly.

Figure 1.2: Three-level waveform

If no voltage waveform is specified on any contact, all terminal biases remain constant within the wholesimulation time. In this case the obligatory key TG1 determines the first and the obligatory key TG2 thesecond time step. The following time steps are expanded using a multiplication factor of the ratio of thefirst two time steps. The transient simulation is stopped, when the end time given by the obligatory keyTG3 is exceeded. This time-stepping mode is prospective to be employed in some simulation modes infuture.

Different waveforms can be applied on different terminals simultaneously. Each waveform must becompletely described, in other words, all parameters must be given for each particular waveform, inde-pendently of other waveforms and of the end time. End time of the simulation must be the same for all

8 CHAPTER 1 INTRODUCTION TO MINIMOS

waveforms. The waveform parameters for each specified contact will be checked for completeness andphysical consistency. End time can be any positive number, also less than the period. Therefore, one canfocus on a part of the specified waveform when analyzing only e.g. turn-on and turn-off of device.The actual time stepping is controlled by waveforms on all contacts simultaneously, fulfilling the require-ments specified by user for the particular waveforms.

1.5 Monte Carlo Simulation of Carrier Transport

Todays semiconductor devices are characterized by large electric fields in conjunction with steep gradientsof the electric field and the carrier concentrations. As these variations may occur over distances comparableto the carriers mean free path the widely used drift-diffusion approximation is losing validity. Simulationof charge transport based on a Monte Carlo particle model is physically much more accurate under suchconditions.

Currently, in MINIMOS a particle model is implemented which describes the motion of electrons insilicon. The band-model of silicon accounts for the six equivalent minima near the Xpoints. Eachof these valleys is assumed non-parabolic and anisotropic. Models for phonon scattering, scattering ationized impurities and at rough surfaces, as well as impact ionization, are described later in this manualin more detail.

MINIMOS utilizes a hybrid approach which combines the Monte Carlo and the drift-diffusion modelsfor the analysis of a single device. In critical device regions, position-dependent coefficients such asmobility, carrier temperature and ionization coefficient are extracted from a Monte Carlo simulation.These coefficients, after being extended analytically over the rest of the simulation domain, are then usedin a generalized drift-diffusion current equation to simulate the overall device. The above mentionedcoefficients are sometimes also referred to as Monte-Carlodrift-diffusion coupling coefficients. Thetheoretical foundation of the hybrid approach is given by the Boltzmann transport equation [3] [4].

In MINIMOS self-consistency of the carrier distributions with the electrostatic potential is achieved byiteratively updating the MonteCarlodriftdiffusion coupling coefficients. Each step of the iterationscheme requires one Monte Carlo simulation and one self-consistent solution of the semiconductorequations (see also Section 1.2).

Currently the Monte Carlo module in MINIMOS deals with electrons only. Therefore, holes are treatedby the drift-diffusion approximation even in model MC.

The analytic band model of MINIMOS allows the equations of motion to be integrated analytically. Thetime of a carriers free flight is determined by an optimized self-scattering algorithm.

To deal properly with scarcely populated regions in both real and energy space a trajectory multiplicationscheme has been implemented. For this purpose the simulation domain can be divided in up to fourdifferent regions, in each of which a unique weight is assigned to the carriers. The positions of theboundaries of these regions and the corresponding weights are determined automatically by MINIMOSIn energy space a carrier trajectory is multiplied if the energy threshold for impact ionization is exceeded.

1.6 SMALL-SIGNAL AC ANALYSIS 9

During a Monte Carlo simulation statistical data, sometimes also referred to as raw-data, are recordedon an auxiliary grid which is independent of the MINIMOS grid. After a Monte Carlo simulationthe raw-data are assigned to the grid points of the MINIMOS grid by means of a convolution method.The parameters for the weighting function used for this step can either be specified in the input file ordetermined automatically by MINIMOS.

1.6 Small-Signal AC Analysis

To perform AC analysis with MINIMOS an AC bias has to be specified within the BIAS directive. Thisbias voltage is defined by its real part VR and its imaginary part VI. If one of these keys is missing a zerovalue is used for it as default.The bias is applied to the gate electrode by default. Any of the 4 terminals can be specified by assigningthe key JCAP on the BIAS directive one of the following five symbolic values: G (Gate), S (Source), D(Drain), B (Bulk). Specifying A (all terminals) leads to the application of AC bias to all terminals oneafter the other.The absolute value of the bias is not significant due to the linearity of the problem, in other words, the resultof the simulation does not depend on the absolute value of VR and VI. However the iteration convergencemay be influenced by both the absolute value as well as the angle of the complex input voltage. Thefrequency of the applied AC bias can be set by the key OMEGAwithin the OPTION directive. The iterationof the AC-analysis can be controlled by the keys CAPERR and NAC on the END directive. The assignedvalue of CAPERR is the desired final error of the AC-analysis and, if specified, the value of the key NAClimits the number of AC-analysis iterations.The AC-analysis is preceded by a normal run of MINIMOS 2D. AC-analysis is restricted to planar two-dimensional devices only. MINIMOS runs according to the degree of sophistication specified by theMODEL key on the OPTION directive. The specified model influences the result and the convergenceproperties the AC-analysis. It is recommended to use MODEL=AVAL for the AC-analysis, see also Section1.2. With MODEL=THRES no AC-analysis is performed.

The result of the AC-analysis is one row of the admittance matrix, if just one terminal bias was specifiedby the JCAP key, or the whole 4 4 matrix if JCAP=A was declared. The admittance matrix data areprinted (together with the other output data) in the formatted output file. The left item is the real part inSiemens ( ), the right item is the capacity in Farad ( ).

If convergence problems arise with the default settings one should invoke the coupled solution algorithmwith CP=YES on the the OPTION directive. To obtain accurate results it is recommended to explicitlyspecify the length of the source and drain terminals by the LSOURCE and LDRAIN keys on the OPTIONdirective.

10 CHAPTER 1 INTRODUCTION TO MINIMOS

AC Analysis Key Summary

Key Description DirectiveJCAP ACbias terminal(s) BIASVI Imaginary part of applied voltage BIASVR Real part of applied voltage BIASCAPERR Final Error ENDNAC Iteration limit ENDCP switch to coupled solution method OPTIONLDRAIN Drain contact length OPTIONLSOURC Source contact length OPTIONOMEGA ACbias angular frequency OPTION

1.7 Nonplanar Geometries

MINIMOS is capable of simulating nonplanar structures. This is true for both the twodimensional andthe threedimensional version. The nonplanarity is given by the oxide body. To model the nonplanar oxidebody a set of parameters is provided within the GEOMETRY directive (see Page 73). These parameters maybe divided into length specifications of straight lines and lengths of transitions. All length specificationsare to be given in . The horizontal lengths of the transitions are measured from 10% to 90% of theabsolute values of the vertical distances of the slopes. The other horizontal length specifications aremeasured from the 50% values of the transitions. Two drawings of nonplanar crossections are insertedto support the spatial imagination of the geometry. Fig. 1.3 is a crossectional view through the centralxy plane, the center of symmetry of the threedimensional transistor at 1yr 2 , where W denotes thechannel width. Fig. 1.4 is a crossection through the yz plane at half gate length Gy ~2 .

MINIMOS 2D automatically switches to nonplanar mode if the GEOMETRY directive is present withinthe input deck. MINIMOS 3D calculates always nonplanarly in the zdirection. It runs nonplanar in allspatial directions only if the GEOMETRY directive is given on the input deck.

The nonplanarity is handled internally by a point characteristics and a box integration concept. Due to thenecessity of resolving the nonplanarities sufficiently accurate there will generally be increased mesh sizesand thus higher computer resources will be required. This is particularly true for full twodimensionaland threedimensional nonplanar simulation.

1.8 THREE-DIMENSIONAL SIMULATION 11

+TINS/2

Y

XOXIDEF/2

DCGRDG2GRD

SGAP DGAP

DRGRDR2GRDSRGRD R1GRD

D2GAP

D1GAP

G1CGRD

S2GAP

SCGRD

L

DDREOX

TINS

S1GAP

SDREOX

RECESS

Figure 1.3: Geometry parameters in the xy plane.

1.8 Three-Dimensional Simulation

Three-dimensional simulation of MOSFETs is necessary to accurately account for the influence of thetransition from the gate oxide to the field oxide, the so-called birds beak. It has been shown in theliterature that those effects are not negligible for channel widths of roughly less than 5 .A three-dimensional cartesian coordinate system is introduced. The x-direction points into the maincurrent flow direction, the y-direction from the surface into the bulk, same as in MINIMOS 2D. Thez-direction points into the channel accordingly. The coordinate system origin is assumed to be at theintersection between the gate-oxide to field-oxide transition and the source-sided gate edge. Thus, thesymmetry plane (x-y plane) has the z-coordinate r 2 .The transition from gate-oxide to field oxide introduces a nonplanarity in z-direction. The form of thisnonplanarity can be modelled by the keys BEAKL and OXIDEF within the GEOMETRY directive. BEAKLspecifies the length of the birds beak, OXIDEF the thickness of the field oxide. Default values for thoseparameters are supplied as outlined in Chapter 3. since the specification of the GEOMETRY directive isoptional. Thus, with no GEOMETRY directive present, MINIMOS 3D calculates fully planar in x- andy-direction and nonplanar in zdirection, otherwise nonplanar in all three dimensions. See Section 1.7 fora detailed explanation of nonplanar simulation. The layered model structure MINIMOS uses to obtain athree-dimensional solution of the semiconductor equations is explained in Section 1.2.Three-dimensional simulation is comparatively memory and CPU time expensive due to the large ranks

12 CHAPTER 1 INTRODUCTION TO MINIMOS

Z

Y

W/2 W/4

OXIDEF

TINS

RECESS

OXIDEF/2-TINS/2

Figure 1.4: Geometry parameters in the yz plane.

of the arising linear systems of equations. A high performance can be achieved by running MINIMOS 3Don vector- or vector-concurrent computers.MINIMOS 3D expects a binary linkfile with all required data as input. This linkfile is produced by aprevious run of MINIMOS 2D with the M3MODE key specified in the input deck. The various settingsof this key are explained in Section 1.2. For information of the MINIMOS 3D input and output files seeChapter 4.

1.9 MESFET Simulation

MINIMOS is capable of simulating MESFETs on either silicon or GaAs substrates. In the program aSchottky contact model is included which accounts for a current dependent surface recombination velocityand a field dependent barrier height [5], [6].

The oxide thickness given by the key TINS on the DEVICE directive acts as the switch from MOSFETto MESFET simulation. Setting TINS to zero forces the program to use a Schottky contact at the gate.In this case the GATE key on the DEVICE directive is interpreted as the barrier height of the Schottkycontact, which has to be supplied by the user.

The specific differences for GaAs versus silicon as substrate materials and the facilities to specify variousMESFET geometries are explained in detail in the sectionsDEVICE,GEOMETRY,IMPLANT,MOBILITY,

1.10 GATE CHARGE AND CAPACITANCE CALCULATION 13

PROFILE and RECOMBINATION. For the foundation of the mobility model for GaAs in MINIMOS seeSection 2.2. Both, steady-state and transient simulation, self-consistently including transient rate equationsfor (deep) bulk traps, can be normally performed with MESFETs, Section 2.7.

1.10 Gate Charge and Capacitance Calculation

MINIMOS offers a possibility to obtain the total gate charge H by calculating the flux of the electricfield through a contour in the oxide around the gate. The contour can be seen in Fig. 2.4 and Fig. 2.5(dot-dashed curve). In particular, the charge assigned to the source-sided and drain-sided edges, to thebottom of the gate, as well as the total charge are calculated and written to both the reference file and thefile unit NUMGC1 (see file cmsysx.inc).The quasi-static gate-capacitance is defined as

^y

H

1 j 1

where H is the total charge in the gate. Performing the simulation at two gate biases

r

o 2 and

o 2 the quasi-static gate capacitance may be calculated by

ZZ

k

o 2 Lr!H

r

Zo 2

1 j 2

The differentiation should be done externally after a MINIMOS run. Proper results can be obtainedassuming

i 25 . The value of

can be specified on the STEP directive by the VG key.Note that the grid must be frozen (OPTION GRIDFREEZE=YES). Comparing the numerical capacitancewith the analytical result, assuming long-channel devices with uniformly doped bulk, we found that thissimple technique to calculate

is very accurate. The numerical error is small, whereas an influence ofthe discretization error is quite reduced by using the same grid for both bias points. In Section 2.8 theaccuracy of this technique is demonstrated.

Keys for Gate Charge Calculation

Key Description DirectiveGCHC Invoke calculation of total gate charge OPTIONGRidf freeze the discretization mesh OPTIONVG for each gate bias UG two DC-calculations are performed

at UG-VG/2 and UG+VG/2 STEP

The calculation of the gate charge is possible only for planar device structures.

14 CHAPTER 1 INTRODUCTION TO MINIMOS

1.11 Monitoring Program Execution

To survey the MINIMOS execution the reference file (see also Chapter 4) collects important runtime data.The reference file offers the user information concerning:

m Input deck, MINIMOS 2D only.The entire input deck is included.

m Important numerical constants, MINIMOS 2D only.m Grid updates.m Gummel iterations.m Virtual memory statistics.m CPU time consumption.

The displayed data are grouped according to the active model 1-D, 2-D, THRES, AVAL or HOT forMINIMOS 2D and 1, 3 for MINIMOS 3D. Grid information is given selectively when grid adaption iscarried out i.e. when MODEL=1-D and/or M3MODE=1.A grid adaption record contains the following information:

m The mesh coordinate, x, y or z.m The grid index at which a meshline is inserted.m The grid coordinate at which a meshline should be inserted.m The mesh spacing at this position.m The value of the grid weighting function at the inserted meshline.

A Gummel iteration record contains the following information:

m The consumed CPU time since MINIMOS start in t .m The iteration count per Gummel loop.m The error norm for the majorities (LMAJ).m The error norm for the minorities (LMIN).m The error norm for the space charge (LF).m The error norm for the potential distribution (LPSI).m The error norms for the carrier temperatures if the HOT model is active.m The inner iteration count for the Poisson equation (PO).

1.11 MONITORING PROGRAM EXECUTION 15

m The inner iteration count for the majorities (MA), MINIMOS 3D only.m The inner iteration count for the minorities (MI), MINIMOS 3D only.m The drain current in .m Either the bulk current for MOSFETs in if the model AVAL or HOT is active, or the gate current

for MESFETs in for all models.

Note: If the Gummel iteration count is preceded by a minus sign, MINIMOS tries to accelerate conver-gence.When MODEL=HOT is active a carrier temperature subiteration is performed. At each carrier temperatureupdate an informational record is issued, which contains:

m The hot subiteration iteration count, 1 \ item.m The carrier temperature error norm for the majorities, 2 a item.m The carrier temperature error norm for the minorities, 3 item.

MINIMOS controls convergence by the values of certain error norms. A Gummel iteration or a HOTsubiteration loop is terminated if all required error norms are smaller than the threshold set by the keyERROR on the END directive. All error norms are dimensionless real numbers.

16 CHAPTER 1 INTRODUCTION TO MINIMOS

2The Physical Models of MINIMOS

2.1 Mobility Model for Silicon

In this subsection the expressions representing the MINIMOS mobility model for silicon are summarized.A more detailed discussion can be found in [7] [8].

The temperature dependence of lattice mobility in pure silicon is modeled by simple power laws.

~

y 1430cm2

Vs z

300K

2

2 j 1

~

y 460cm2

Vs z

300K

2 18

2 j 2

To account for mobility reduction due to ionized impurity scattering the formulae of Caughey and Thomas[9] is used in conjunction with temperature dependent coefficients.

~_

S

yC

W

_

~

_

r

8

S

1 !

8

2 j 3

8

y 80cm2

Vs z

300K

0 45

z 200 K

2 j 4

8

y 80cm2

Vs 200K300K

0 45

z

200K

0 15

z 200 K

2 j 5

8

y 45cm2

Vs z

300K

0 45

z 200 K

2 j 6

17

18 CHAPTER 2 THE PHYSICAL MODELS OF MINIMOS

W

y 45cm2

Vs 5200K300K #

0 45

z

200K

0 15

z 200 K

2 j 7

|

y 1 j 12

1017cm

3

z

300K

3 2

2 j 8

|

y 2 j 23

1017cm

3

z

300K

3 2

2 j 9

_

y 0 j 72

z

300K

0 065

2 j 10

Surface scattering is modeled by the following empirical expression:

~[N

_

y

|

_

~[

_

r;

}

|

S

1 rP

1 b

_

o>

|

S

\

2 j 11

|

y 638 cm2

Vs z

300K 1 19

2 j 12

|

y 240 cm2

Vs z

300K 1 09

2 j 13

Zy

2

exp

r

o

|

2

1 exp

r 2

o

}

|

2

2 j 14

The pressing forces

and in Equation 2.11 equal to the magnitude of the normal field strength at theinterface, if the carriers are attracted by the interface, otherwise they are zero.

Deviations from the ohmic low-field mobility are given for both electrons and holes by

~[NS

_

y

2

~_N

S

1 1

2 q\ N}

Z

1

2 j 15

y

grad Pr 1

grad (5

)

2 j 16

y

grad ^ 1s

grad 5

sAS

2 j 17

2.1 MOBILITY MODEL FOR SILICON 19

Here _

represent the driving forces for electrons and holes and

_

are the carrier temperaturevoltages.

The saturation velocities are assumed temperature dependent.

\N

y 1 j 45

107 cms S

tanh

155Kz

2 j 18

\N

y 9 j 05

106 cms

tanh

312Kz

2 j 19

Several parameters in the MINIMOS mobility model can be weighted. The weight coefficients which canbe specified on the MOBILITY directive have the following meaning:

Key SymbolLattice and Impurity Scattering

ML weight for ~

and ~ in Equation 2.1 and Equation 2.2, respectively.MI weight for

|

and

|

_

in Equation 2.3.Surface Scattering

MC weight for

and in Equation 2.11.MR weight for |

and } | in Equation 2.11.MS weight for

|

and

|

in Equation 2.14.MT weight for |

and | in Equation 2.11.High Field Mobility

MV weight for '

and ' in Equation 2.15.MB weight for the exponents

and in Equation 2.15.

All coefficients multiply the coresponding parameters, except MS, which divides the default

|

_

. Thoseweights should only be used for fine tuning, since the default mobility parameters in MINIMOS are chosencarefully and agree rather well to experiments. The default parameters in MINIMOS are:

20 CHAPTER 2 THE PHYSICAL MODELS OF MINIMOS

~

= 1430 2 o>t ~ = 460 2 o>kt

W

= 80 2 o>t 8 = 45 2 o>kt

|

= 1 j 12

1017

3

} |

= 2 j 23

1017

3

= 0 j 72 = 0 j 72

|

= 638 2 o>t | = 240 2 o>kt

} |

= 7 j 0

105 Boq | = 2 j 7

105 Boq

= 1 j 69 = 1 j 00

} |

= 10

10

7

|

= 10

10

7

'

= 1 j 45

107 oat ' = 9 j 05

106 oat

= 2 = 1

All constants refer to a simulation temperature of 300 , except \N_

that obey the law given by Equation2.18 and Equation 2.19

2.2 Mobility Model for Gallium Arsenide

In this subsection the expressions representing the MINIMOS mobility model for gallium arsenide aresummarized. A more detailed discussion can be found in [10] [11].

The temperature dependence of lattice mobility in pure gallium arsenide is modeled by simple power laws.

~

_

yC

0 _

5

z

300K #

2 j 20

Impurity scattering is modeleled by the Caughey and Thomas [9] expression Equation 2.3. For galliumarsenide the asymptotic mobilities at high impurity concentrations, 8

and 8 , can be weighted. Thecoresponding key is MB key on the MOBILITY directive. The default value of MB is zero in case of galliumarsenide, so that the influence of W

_

is suppressed. Note that the MB key acts differently for galliumarsenide and silicon (see key description Page 94).The high field mobility for electrons is given by

~_

y

~[

1 FA

~[

N

1

1 FA

~[

N

2 j 21

'

y 6

106 cms 8

1

~_

104 cm2Vs

r

~_

5 j 477

106 cm2Vs

2

2 j 22

:y 10

2 0 j 6

exp ~[

103 cm2Vsr 2

exp 7 r ~_

285 j 72 cm2Vs

#

1

2 j 23

2.3 GENERATION-RECOMBINATION MODELING 21

{Zy 4 1280

sinh

1

~_

250 cm2Vs

2 j 24

For holes the high field mobility reads

~_

y

~_

1 q

N

2 j 25

'

y 1 j 5

107 cms

2 j 26

and represent the driving forces for electrons and holes according to Equation 2.16 and Equation2.17, respectively.

For gallium arsenide several parameters can be weighted in the mobility model. The weights which canbe specified on the MOBILITY directive have the following meaning:

Key SymbolMB weight for 8

and W in Equation 2.3 (default MB=0).MI weight for

|

and

}

|

_

in Equation 2.3.ML weight for ~

and ~ in Equation 2.1 and Equation 2.2, respectively.MV weight for '

and ' in Equation 2.21 and Equation 2.25, respecitively.

All weights multiply the corresponding parameters. The default settings of the various parameters in thegallium arsenide mobility model are:

0

= 8000 2 o>Jt 0 = 380 2 o>Jt

= 1 = 2

8

= 1500 2 o>Jt 8 = 50 2 o>Jt

~

= 8000 2 o>Jt ~ = 380 2 o>Jt

|

= 1017

3

|

= 3 j 232

1017

3

= 0 j 5 = 0 j 4956

2.3 Generation-Recombination Modeling

In this section we present models for the standard generation-recombination mechanisms which areaccounted for by MINIMOS in the AVAL mode. Dynamic generation-recombination via interface and

22 CHAPTER 2 THE PHYSICAL MODELS OF MINIMOS

bulk traps is described below in Section 2.7. A description of generation-recombination due to interbandtunneling as it is modeled in MINIMOS can be found in Section 2.6.

The parameters in the following expressions can be set on the RECOMBINATION directive.

Shockley-Read-Hall recombination rate:

p

S

y

sr

2_

( ) [

(s ) 2 j 27

where , s and are the local concentrations of electrons, holes and the intrinsic concentration, respec-tively. The lifetimes TN and TP can be specified by the user.

Auger recombination rate:

plJy

sr

2

s

2 j 28

with CN and CP being the Auger coefficients. Both, p S and p

are independent of the bulk-trapspecification and are superposed on the generation-recombination calculated for bulk traps.

Surface recombination at the gate-oxide/bulk interface:

p

|

y

sr

2

s%

2 j 29

with SN and SP being the surface recombination velocity for electrons and holes, respectively at theinterface between substrate and gate insulator. This surface recombination is independent of terminal biasand interface trap density. It is superposed to the surface recombination rate calculated for interface trapsas explained in the previous section. If

y 0 or

y 0 is specified this surface generation-recombinationvanishes. Note, default values are not 0.

Surface recombination at the silicon-film/bulk-insulator interface in SOI devices:

pH

|

y

sr

2

s

2 j 30

with VN and VP being the surface recombination velocity for electrons and holes, respectively. Thissurface recombination is independent of terminal bias and interface trap density. It is superposed tothe surface recombination rate calculated for interface traps placed at the back interface. This surfacegeneration-recombination can be suppressed by

y 0.

2.4 THE LOCAL CARRIER HEATING MODEL 23

Impact Ionization

MINIMOS has been enhanced to include a depth dependent ionization model that has been verified againstn-channel substrate current measurements. The ionization coefficients are given by

Zy:

exp r

Z

2 j 31

Zy:

exp r

2 j 32

with

denoting the local electric field and

the depth from the Si-SiO2 interface. The existence of adepth dependent factor

_

was suggested by Slotboom [12] and is also used in a number of luckyelectron models of impact ionization [13].The depth dependence of

_

is given by the expressions

gy

1

exp r

+

c

2 j 33

Zy

5

1

exp r

[

j

2 j 34

The form of the depth dependence is similar to that used in the above references. The effect of thedepth dependence is to increase the

S

factor near the surface, which is similar to decreasing themean free path near the surface. Default settings for the electron coefficients have been derived fromexperimental measurements of the substrate current over a number of n-channel devices with channellength in micrometer and 3 o 4 micrometer range. Hole parameters have not been calibrated against localexperimental data and rather use the so-called dark space coefficients from MINIMOS 4. Since noexperimental data were available for checking a depth dependent ionization model for holes, the defaultis to apply the electron depth dependence to holes analogously.

All the above described generation-recombination mechanisms are active in steady-state as well as intransient conditions.

2.4 The Local Carrier Heating Model

In MINIMOS enhanced drift-diffusion equations are implemented which allows for elevated carriertemperatures. The carrier temperatures are obtained from a local approximation of the energy balanceequations [14].

S

y

z

S

y

5

0 23

S

\N

_

2

1

~[N)

_

r

1

~[N

_

2 j 35

The energy relaxation times

and are constant with respect to the carrier temperatures. They aremodeled by

24 CHAPTER 2 THE PHYSICAL MODELS OF MINIMOS

y

32 a5 0

~_N

'

2

2 j 36

y

32 ) 0

~[N

'

2

2 j 37

The energy relaxation time weights UN and UP have a default value of 0 j 8 and can be specified on theOPTION directive.

2.5 Modeling of Band-Structure Parameters

In this section the models for the effective masses of electrons and holes, band gap and intrinsic concen-tration in silicon and gallium arsenide are presented.

2.5.1 Models for Silicon

The temperature dependence of the effective masses is modeled by

yC

1 j 045 4 j 5

10

4

z

CZ

2 j 38

yC

0 j 523 1 j 4

10

3

z

r 1 j 48

10

6

z

2

2 j 39

where z is the absolute temperature in . denotes free-electron mass in vacuum.

The intrinsic concentration, if not explicitly specified by the INTRINSIC key on the OPTION directive,is modeled by

y

exp

r

E 2

z

2 j 40

where E is the energy bandgap for silicon E

r E according to

E y 1 j 1700 1 j 059

10

5

z

C

r 6 j 05

10

7

z

C

2

z 170

2 j 41

E Uy 1 j 1785 r 9 j 025

10

5

z

r 3 j 05

10

7

z

2

z 170

2 j 42

with E being obtained in q .

2.5 MODELING OF BAND-STRUCTURE PARAMETERS 25

The relationship between the density of states product

and the effective masses is given by

y 2 j 51

1019

z

300K

2 3 2

3j

2 j 43

2.5.2 Models for Gallium Arsenide

The band gaps between the valence band and the lowest three conduction bands are given by

E y 1 j 519 r 5 j 405

10

4

z6oq^

2

z6oq^ 204 2 j 44

EL y 1 j 815 r 6 j 050

10

4

z6oq^

2

z6oq^ 204 2 j 45

EX y 1 j 981 r 4 j 600

10

4

z#oq

2

z6oq 204 q 2 j 46

The electron effective mass is modeled by

yC

3 2

L

3 2exp

r

L

z

X

3 2exp

r

X

z

2 3

2 j 47

with y 0 j 067

2 j 48

L y 0 j 55

2 j 49

X y 0 j 85

2 j 50

L y EL r E

2 j 51

X y EX r E

2 j 52

The hole effective mass is modeled by

y

3 2

!

3 2"!$#

2 3

2 j 53

with

"

y 0 j 076

2 j 54

!

y 0 j 50

2 j 55

The density of states of the conduction and the valence band is calculated by

y 8 j 63

1013

z

3 2

1 r 1 j 93

10

4

z

C

r 4 j 19

10

8

z

C

2 21

exp

r

L

zG

44

exp

r

X

z

3

26 CHAPTER 2 THE PHYSICAL MODELS OF MINIMOS

y 2 2 %

z

& 2

3 2

3

2 j 56

Using these formulations the intrinsic carrier concentration will be determined by

y

'(

exp

r

E2

z%

j

2 j 57

2.6 The Band-to-Band Tunneling Model

Interband tunneling has recently become increasingly important for Si technology. It is a source of leakagein memory cells ( [15, 16]) and MOSFETs ( [17, 18]). This effect can produce a degradation in static( [19, 20]) and an enhanced degradation in transient bias conditions ( [21]). It is prospective to be usedto design programable memories with a high injection efficiency ( [22]), as a monitor for hot-carrierdegradation ( [23, 24, 25]) and to develop devices whose operation is based on this effect ( [26, 27]).Tunneling is a sensitive function of the electric field, whereas the field depends on several parameters suchas oxide thickness, dopant concentration, trap density, geometry and terminal biases. As a consequence,tunneling in MOSFETs can only be properly modeled by a numerical approach [28, 29, 30, 31].

We have developed a two-dimensional numerical model of the band-to-band tunneling. Two approacheshave been implemented in MINIMOS:

One-dimensional model, in which the tunneling path is searched from a particular point in the bulk indirection perpendicular to the oxide/bulk interface, as often applied in the literature, e.g. [17, 32].

Two-dimensional model, where we search from a particular point in the bulk discretization mesh, whichrepresents the starting point, in two dimensions for the nearest endpoint whose potential difference islarger than the characteristic tunneling band gap (says E ). The tunneling path found in this way is normalto the corresponding equipotential lines assuming sufficiently fine local grid. Since the tunneling length)

is shorter than the curvature of the equipotential lines, the tunneling between the starting point and theendpoint may be treated like a planar one-dimensional problem.

The first integer number in the key MTUN in the RECOMB directive specifies which of these two approachesis used, as explained bellow.

To calculate the generation rate associated with an individual path, models for direct and indirect tunnelingas function of the starting field

, an average field along the tunneling path

y E o*)

and the fieldvariable from

to

(endfield) are implemented. Which field-model is applied in the calculation canbe specified by the third integer number in the key MTUN.

The calculated generation rates of electron and holes are separated in the position space. They are coupledwith the continuity equations via the generation term in a selfconsistent manner after filtering. The totalcharge in device is strictly conserved. Filtering is used for smoothing the distributions only, to allow anefficient and automatic grid adaption in the critical areas. Standard deviations for filtering associated with and

directions can be scaled by the keys SXT and SYT, respectively in the GRID directive. The default

2.6 THE BAND-TO-BAND TUNNELING MODEL 27

values of the standard deviations are chosen carefully, being appropriate for the grid adaption when thetunneling is localized in the gate/drain overlap region. Therefore, it is not recommended to change thesekeys when analyzing the standard gate-induced-drain-leakage (GIDL) problem ( [17, 18, 23, 25]). Itis, however, necessary to increase SXT up to 5 when tunneling takes place along the whole MOSFETchannel, as in the BBISHE injection [22].Since the tunneling rate is strongly dependent on the electric field, the tunneling analysis is done on thegrid which is MXT times finer for the -mesh and MYT times finer for the

-mesh than the MINIMOSmesh for solving the basic semiconductor equations in the bulk. Increasing MXT and MYT quite reducethe discretization error in the terminal currents, but can prolong the computation time. The default valuesof 2 should represent a proper choice.

Terminal currents are calculated by a very accurate technique based on the local concentration-dependentweighting functions (a similar method is proposed in [33]).

After an individual bias point is calculated, some quantities characteristic for the band-to-band tunnelinganalysis are printed in the reference file. These cover the number of active starting points in the finediscretization mesh, the actual fine mesh for the analysis in the bulk, the minimal and an average tunnelingdistance, the maximum electric field involved in the calculation of the tunneling rates (either startingor an average field, dependent on the specification) and an average electric field across all paths found.If the variable-field model is specified, the field variation along the shortest tunneling distance and anaverage field variation across all paths are printed, as well. In addition, the total tunneling generationrates of electrons and holes in the whole device are printed; they should be equal. Particularly, the carriersgenerated in the source-half ( ,+-o 2,

0) and the drain-half of device are presented. Note thatthe tunneling can occur in both junctions at some bias conditions (sufficiently large bulk reverse bias).After the tunneling rates the total avalanche generation rate in device, as well as the sum of the both, totalavalanche and tunneling rates are printed ( [29]). These rates should be compared with the electron andhole components of the terminal currents, printed in the current-information record. Note that additionalgeneration-recombination processes also contribute to the terminal currents.

Physical model for the tunneling rate

Direct tunneling is modeled by the Kane-Keldysh expression [34, 35, 36]

y

.-0/

2

1 2

18

%

& 2

E1 2 2

exp r % 1 2

E3 22

&

2143

2 j 58

The values of the physical parameters are given below. This model is derived for the internal-field emissionin an infinite material, assuming a constant field and two-band interacting over 5

5

6

perturbation [35].Because of the later, it is convenient for narrow-gap semiconductors, but not for silicon. Regarding tosilicon MOS devices, the second restriction is the assumption of an infinite medium. In MOSFETs,the tunneling occurs very close to the interface, where the movement becomes quantized in the normaldirection. Third important restriction is a variation of the electric field along the individual tunnelingpaths, an effect normally taking place in devices.

28 CHAPTER 2 THE PHYSICAL MODELS OF MINIMOS

In Equation 2.58

stands either for an average field E o* )

,

)

being the tunneling length, or for the fieldat the starting point

. For the gap E the user can choose both, indirect and direct silicon band gap, inspite of the former being physically unreasonable. TLIF and TEXF are fitting parameters with a value 1as default.

We observed significant variations of the electric field along the tunneling paths, from the starting field

to the endfield

.

o

ranges from 1 j 5 to 5 for GIDL problem in common cases. This ratio can alsobe smaller than 1, as for tunneling towards the interface in the BBISHE injection in n-channel devices andfor the GIDL problem in p-channel devices. To analyze the impact of the field variation on the tunnelingcurrent, we develop a model for linearly variable field:

)

8y

r

r

)

o

)

. It is based on WKBJand the two-band 5

5

6 dispersion relation. Assuming that

is independent of the energy associated withthe momentum perpendicular to the tunneling direction, after several approximations we arrive at [31]

|y

2-7/

118

%

2

& 2

E1 2

1 3 8

2

o 16 exp r%

2

E3 2

&

1 3 8

2

o 32

2143

2 j 59 with

Jy:9

2

2

o 2 and8

Jy

2

r

2

o

2

2

. In the derivation of Equation 2.59, fieldvariations less then four times are assumed. The WKBJ part was multiplied with % 2 o 9 1 j 1 in thetransmission coefficient in a non-constant field (cf. [36]). Because of the later, this model reducesexactly to Equation 2.58 in a constant field case. For moderate field variations

| becomes close to theconstant-field model

which employs an average field

y E o*)

y

o 2, Fig. 2.1.

Indirect phonon-assisted tunneling employs the model by Keldysh derived for a constant field [36, 37]

!

y

.-0/

5 2";

7 4

2< =

@

1 @

BA

2 j 60

with phonon-absorption and emission related factors:

y exp r 4 2

1 2

3

&

E 6r &4C

3 2

2143

2 j 61

-y exp r 4 2

1 2

3

&

E W &4C

3 2

2143

2 j 62 @

being Bose-Einstein population@

y

1

exp

&C

z

r 1 2 j 63

and z the absolute temperature.

If the indirect tunneling with the variable-field model is specified, the Equation 2.60 is used assuming anaverage field

y

o 2. After extensive comparisons between Equation 2.58 and Equation 2.59we found an average-field approach to be satisfactory for direct tunneling problems if the field variationsare not too large. We may expect an average-field approach together with Equation 2.60 Equation 2.62may be sufficient for engineering use regarding to phonon-assisted tunneling, since other effects are soand so not accounted for, as the realistic band structure, manifoldness of the phonon branches and the

2.6 THE BAND-TO-BAND TUNNELING MODEL 29

interface-vicinity effect.An anisotropy in silicon has not been accounted for in the models implemented. The anisotropy effectsseem to be small at higher fields which are of interest [38].

Note that the starting-field model and the 1D approach for the tunneling direction are relevant for theinvestigation purposes only.

List of the physical parameters:

elementary charge; y 1 j 602

10

19

E for indirect band gap of silicon see Section 2.5;for direct tunneling band gap in silicon we assume E

y 3 j 42 q

&

reduced Planks constant (Diracs constant); & y & o 2 %y 1 j 0544

10

34 Dt

Boltzmanns constant;

y 1 j 3804

10

23 Doq

free-electron mass in vacuum; #y 9 j 108

10

31 2E

reduced mass in tunneling process; #y 1 o

1 o

1;

y 0 j 26 ,

yC

F! ; y 0 j 1 , is used for both direct and indirect tunneling.

y 0 j 588 The effective masses in silicon are:

y 0 j 98 , conduction band electron longitudinal mass;

y 0 j 19 , conduction band electron transversal mass;

F!

y 0 j 16 , valence band light hole mass;

!"!

y 0 j 49 , valence band heavy hole mass.For a possible impact of the field on the hole mass see discussion in [38].

;

number of valence band electrons per unit cell, taking place in tunneling;

;

y 2 .< =

1 leadsto a lower mobility and viceversa.

Type: Default: 1Range: 0 2 MC 2

MI Impurity Scattering WeightThis key specifies the weight for the critical concentration in the mobility model due to impurity scattering.MI > 1 leads to a higher mobility and viceversa.

Type: Default: 1Range: 10 3 MI 106

ML Lattice Mobility WeightThis key specifies the weight for the zero field lattice mobility. ML > 1 leads to a higher lattice mobilityand viceversa.

Type: Default: 1Range: 0 5 ML 2

MOBILITY 95

MQ Weight for the Coulomb Scattering due to Interface StatesThe total number of surface charged-sites determines the mobility reduction. All traps and fixed chargesfrom the INTERFACE directive contribute to the scattering. MQ > 1 leads to a lower mobility andviceversa. If this key is omitted, no mobility reduction due to this type of scattering will be taken intoaccount.

Type: Default: 0Range: 0 MQ 103

MR Weight for the Mobility at the SurfaceThis key specifies the weight for the surface mobility at the zero normal field (max. mobility at thesurface). MR > 1 leads to a higher mobility at the surface and viceversa.

Type: Default: 1Range: 0 5 MR 2

MS Surface Scattering Distance WeightThis key specifies the weight of the characteristic distance for the transition from normalfield dependentsurface mobility to bulk mobility. For MS > 1 this distance is decreased and viceversa.

Type: Default: 1Range: 0 1 MS 10

MT Critical Normal Field WeightThis key specifies the weight for the critical normal field in the surface mobility model. MT > 1 leads to ahigher mobility and viceversa. For galliumarsenide substrates MT weights the critical field in driftfieldrelation.

Type: Default: 1Range: 0 1 MT 102

MV Saturation Velocity WeightThis key specifies the weight of the saturation velocity. MV > 1 leads to a higher saturation velocity andviceversa.

Type: Default: 1Range: 0 5 MV 103

96 CHAPTER 3 REFERENCE OF INPUT DIRECTIVES

Directive MONTE-CARLO

AUXS Source Sided Extension of Auxiliary GridType: Default: 0 1 10 4Units: Range: 0 AUXS 0 5 10 4

AUXD Drain Sided Extension of Auxiliary GridType: Default: 0 2 10 4Units: Range: 0 AUXD 0 5 10 4

BDR Back Diffusion Ratio at Drain ContactThis key specifies the back diffusion ratio at the absorbing drain contact. A carrier is reflected at thecontact BDR/(1-BDR) times before it is absorbed. BDR=0 means that all arriving carriers are absorbed,BDR=1 means that all arriving carriers are reflected.

Type: Default: 0 5Units: 1Range: 0 BDR 0 999

CNTMIN Threshold for Statistical SignificanceTo identity an average at a given mesh point as statistical reliable more than a specified minimum numberof scattering events have to contribute to this average. This minimum number is obtained by multiplyingCNTMIN with the total number of simulated scattering events.

Type: Default: 10 5Units: 1Range: 0 CNTMIN 1

DTK ) , *,+ 1 - 2 - 3 - 4 - 5 - 6 Coupling Constants of the Intervalley PhononsWith these keys six phonon coupling constants can be specified. The ponons with ordering numbers 1 2 3are g-type phonons (the number of final valley .0/ equals to 1), those with ordering numbers 4 5 6 aref-type phonons ( .0/ 4 ' . The program accounts for six intervalley phonon scattering mechanims in anycase, even if some of them should be physically disabled by setting DTK

0.Type: Default: 0 5 108

0 8 10811 0 1080 3 1082 0 1082 0 108

Units: 919Range: 0 DTK 1010

MONTE-CARLO 97

DEFPOT Acoustic Deformation PotentialType: Default: 9 0Units: Range: 0 DEFPOT 102

DEPSS Depth to Which Surface Scattering is ActiveType: Default: 25 10 7Units: Range: 0 DEPSS 0 1 10 4

DFMIN Threshold Parameter for Mobility WindowType: Default: 2000Units: 19Range: 0 DFMIN 105

DLSS Surface Roughness ParameterThis key specifies the product of correlation length times standard deviation of surface roughness.

Type: Default: 3 10 15Units: 2Range: 0 DLSS 10 12

DOVL Drain Sided Overlap of Monte Carlo WindowType: Default: 0 3 10 4Units: Range: 0 DOVL 2 10 4

E1SELF Energy Parameter for the First Self-Scattering IntervalType: Default: 0 1Units: Range: 10 2 E1SELF 1.0

E2SELF Energy Parameter for the Second Self-Scattering IntervalType: Default: 5 0Units: Range: 1 E2SELF 10

EMULT Trajectory Multiplication Factor in Energy DomainType: Default: 10Units: 1Range: 1 EMULT 104

98 CHAPTER 3 REFERENCE OF INPUT DIRECTIVES

EPH ) , *(+ 1 - 2 - 3 - 4 - 5 - 6 Energies of the Intervalley phononsWith these keys the energies of the six intervalley phonons can be specified. See also the description ofthe DTK keys.

Type: Default: 12 1 10 3

18 5 10 362 0 10 319 0 10 347 4 10 359 0 10 3

Units: 9Range: 10 3 EPH 0 5

EREF Reference Field for Surface Roughness ScatteringType: Default: 105Units: 29Range: 1 EREF 106

EXII Exponent in Keldysh FormulaType: Default: 2Units: 1Range: 0 2 EXII 5 0

EXSS Exponent in Surface Roughness Scattering ModelThis key specifies the power law for the normal-field dependence in the surface roughness scatteringmodel.

Type: Default: 2Units: 1Range: 0 1 EXSS 10

MCMOD Monte Carlo Simulation ModeThis key has the following meaning:

MCMOD Description1 Perform Monte Carlo Device Simulation (default)2 Calculate material characteristics under uniform conditions3 Print debye-length versus doping to output file4 Print scattering rates versus energy to output file5 Print self-scattering rate to output file

In MCMOD=2 material characteristics are calculated for a range of electric field strengths and for a rangeof doping levels. These ranges can be specified on the STEP directive.

Type: Default: 1Range: 1 MCMOD 5

MONTE-CARLO 99

MCPIT Number of Monte-CarloPoisson IterationsThis key specifies the number of self-consistent iterations to be executed. The program terminates whenthe specified number of iterations are done. The convergence history is printed to the reference file.

Type: Default: 1Range: 1 MCPIT 20

MLONG Longitudinal Effective Electron-MassThis key specifies the ratio of the longitudinal electron-mass to the free electron-mass.

Type: Default: 0 9163Units: 1Range: 10 2 MLONG 10

MTRAN Transversal Effective Electron-MassThis key specifies the ratio of the transversal electron-mass to the free electron-mass.

Type: Default: 0 1905Units: 1Range: 10 2 MLONG 10

NONPAR Non-Parabolicity FactorThis key specifies the non-parabolicity factor of the conduction band of silicon.

Type: Default: 0 5Units: 1Range: 0 NONPAR 10

NSB Number of Particle Split BoundariesThis key specifies how often the trajectory multiplication algorithm is applied in real space. Up to threeboundaries can be specified. That means that the simulation domain is divided into one common domainand three rare domains. The positions of the multilication boundaries in real space and the correspondingmultiplication factors are automatically chosen by MINIMOS. In energy space a carrier trajectory ismultiplied if the energy threshold for impact ionization is exceeded.

Type: Default: 3Range: 0 NSB 3

NUMB Number of Scattering Events per Single MC-Poisson IterationType: Default: 20 106, MCMOD=1

106, MCMOD=2Range: 105 NUMB 5 108

100 CHAPTER 3 REFERENCE OF INPUT DIRECTIVES

NUM ) , * =1,2,3,4,5 Number of Scattering Events for a Specific MC-Poisson IterationFor the lowest five self-consistent iterations the number of scattering events can be specified individually.The key NUM overwrites the key NUMB for the -th iteration. With these keys one can reduce the numberof scattering-events for the very first iterations, when the solution is far away from self-consistency. Inthe final iterations that determine the final results the noise can be reduced by using a higher number ofscattering events, that might be specified by the NUMB key.

Type: Default: NUMBRange: 105 NUM / 108

RHO Semiconductor Mass DensityThis key specifies the mass density of silicon

Type: Default: 23 28 10 4Units: 43 3Range: 10 4 RHO 10 2

SAV ) , *(+ 1 - 2 - 3 - 4 - 5 Save Results after a Specific MC-Poisson IterationWhith this keys up to five iteration numbers can be specified, after which certain distributed quantities areto be written to file unit NUMUAI.

Type: Default: Range: 1 SAV MCPIT

SEED Initial Seed of the Pseudo Random Number GeneratorType: Default: 78Range: 1 SEED 168

SOVL Source Sided Overlap of Monte Carlo WindowType: Default: 0 3 10 4Units: Range: 0 SOVL 2 10 4

TRACE Trace Execution of MC-ModulIf TRACE=YES is specified addtitional information is written to the reference file. A message is writtenif a particle is launced or absorbed, and if a particle is split into light ones or light particles are gatheredto a heavy one.

Type: Default: NO

UTMIN Threshold Parameter for Temperature WindowType: Default: 1 05Units: 1Range: 1 UTMIN 10

MONTE-CARLO 101

VSOUND Sound Velocity in the SemiconductorType: Default: 90 33 104Units: 5Range: 104 VSOUND 106

WBH Weight of Brooks-Herring Scattering RateThis key weights the Brooks-Herring scattering rate in Ridleys formula for ionized impurity scattering.

Type: Default: 1Units: 1Range: 0 WBH 102

WBII Weight of the Prefactor in the Keldysh FormulaType: Default: 3 9 10 2Units: 1Range: 0 WBII 103

WDIST Weight of the Average Distance Between IonsIn Ridleys model for ionized impurity scattering the average distance between ions in conjunction withthe actual carrier velocity determines the upper limit for the scattering rate. This key allows to weight thisdistance which is obtained from the local impurity concentration by 6

WDIST &

2 87:9

'

1 ; 3.

Type: Default: 1Units: 1Range: 10 2 WDIST 102

WEII Weight of The Engergy Threshold for Impact IonizationType: Default: 1Units: 1Range: 10 2 WEII 102

WINFREEZE Freeze Window for MC-Transport ParametersIf this key is set to YES the windows in which the analytical transport parameters are replaced by theMC generated ones do not change between succsessive MC-Poisson iterations. Otherwise the criteria forwindow boundary placement are evaluated after each MC-Poisson iteration.

Type: Default: NO

WSCREEN Weight of the Screening LengthWith this key the influence of the screening length on the ionized impurity scattering rate can be weighted.

Type: Default: 1Units: 1Range: 10 2 WSCREEN 102

102 CHAPTER 3 REFERENCE OF INPUT DIRECTIVES

XWIDTH Lateral Width of Assignment FunctionType: Default: Minimum of

OPTION 103

Directive OPTION

The OPTION directive serves as a directory for general purpose keys. Some physical device parametersare collected herein as well as parameters to control MINIMOS execution. This directive can be omitted.No key of this directive is required.

ACCELERATION Try to Accelerate ConvergenceThis key may be used to suppress the use of an acceleration algorithm in the iteration process.

Type: Default: YES

CCLP Critical Carrier Concentration for Determining the Capture BoundaryThis key specifies the critical electron or hole concentration for finding the x-coordinate which deter-mines the area for the total electron and hole capture at the specified terminal bias (charge-pumpingthreshold and flat-band voltage). The output is written in the file *.LATPROF. The result is in a format&

=> ?A@B ?1 ?ACD FEHG

@

FEHG

C

#< IEJG

@

FE5K

@

FE5K

C

#< LE5K

@

NMPO

' , where EHG@

, EJGC

, EQK@

and E5KC

are the co-ordinates at the drain and source side for electrons and holes, respectively. < is the gate length (LGATE).If the coordinate does not exist the output is the total left or right coordinate of the simulation domain.Remember that the source-sided gate edge is the origin of the MINIMOS x-coordinate axis. Note that thecalculation assumes DC condition.

Type: Default: Units: 3Range: 105 CCLP 1020

CP Flag for Coupled Solution of AC EquationsType: Default: NO

CURRENT Drain Current for Threshold CalculationThis key is active only for threshold voltage calculation, i.e. ifMODEL=THRES is specified on theOPTIONdirective. In that case the CURRENT key determines the drain current for threshold condition.

Type: Default: 10 7 RS for nchannel devices

10 7 RS for pchannel devicesUnits: TRange: 2 10 3 CURRENT 2 10 3

EA Acceptor Activation EnergyThis key specifies the difference between the impurity level of acceptors and the valence band edge(ionization energy). If EA is not specified, total ionization of acceptors is assumed.

Type: Default: Total IonizationUnits: Range: 0 001 EA 0 1

Typical: 0 045 for boron in silicon

104 CHAPTER 3 REFERENCE OF INPUT DIRECTIVES

ED Donor Activation EnergyThis key specifies the difference between the conduction band edge and the impurity level of donors(ionization energy). If ED is not specified total ionization of donors is assumed.

Type: Default: Total IonizationUnits: 9Range: 0 001 ED 0 1

Typical: 0 039 for antimony in silicon0 045 for phosphorus in silicon0 054 for arsenic in silicon

GCHCALC Calculation of the Total Gate-ChargeIf GCHCALC is specified, the total charge in the gate will be calculated by evaluating the electric field fluxthrough a contour around the gate. The results are the total charge U

, the source-sided charge U DVWV ,the bottom charge U (X and the drain-sided charge U (YZV . The result is written to file unit NUMCG1 in theform

&

=> ?A@B ?2 ?AC( U[

U[DV\V8 U (X U (YZV '

.

Type: Default: NO

GRIDFREEZE Freeze GridThis key is active only if any of the step parameters has been chosen on the STEP directive. This key maybe used to determine whether grid modifications should be performed between voltage steps or not.

Type: Default: NO

IDEALITY Ideality Factor for Gate DiodeThis key specifies an ideality factor for the gate diode of MESFETs. It affects the gate current in theforward biased region. This factor is defined by

]

9

@

1

^

&`_

]Qa

'

^

; cb 3

This key is only effective in case of MESFET simulation. If this key is omitted an ideal gate diode will beassumed.

Type: Default: 1Units: Range: 1 ID 5

INTRINSIC Intrinsic ConcentrationThis key specifies the intrinsic concentration at the simulation temperature. If this key is omitted, abuiltin model for the temperature dependence of the intrinsic concentration will be used.

Type: Default: Builtin ModelUnits: 3Range: 10 20 INTRINSIC 1015

OPTION 105

LCPUMP Charge Pumping Relevant Data on OutputIf LCP is specified, all quantities necessary to perform postprocessing for the charge-pumping analysiswill be written on output after transient simulation. Independent of LCP all specified interface and bulktraps are always taken into account in the selfconsistent calculation.

Type: Default: NO

LDRAIN Drain Contact LengthThis key specifies the length of the drain contact in xdirection

Type: Default: Units: Range: 10 6 LDRAIN 0 1

LSOURCE Source Contact LengthThis key specifies the length of the source contact in xdirection.

Type: Default: Units: Range: 10 6 LSOURCE 0 1

LVDMOSFET Device is Interpreted as Vertical DMOS TransistorIf this key is specified, doping has to be read in from a two-dimensional doping file. Furthermore, thelength of the source contact (see OPTION directive) and the depth of the simulation domain must be given(see DEVICE directive).

Type: Default: NO

MODEL MINIMOS 2D Model OptionThis key specifies which mode of calculation MINIMOS 2D will perform. For a detailed explanation ofthe five modes see Section 1.2. Only the first character of the specified character string is significant.

Type: Default: 2-DRange: 1-D

THRES2-DAVALHOT

MPOLY Model for the Gate Depletion AnalysisThis key is active only in the gate depletion analysis.

Type: Default: 1Range: 0 one-dimensional model

1 full two-dimensional solution in the poly-gate area

106 CHAPTER 3 REFERENCE OF INPUT DIRECTIVES

M3MODE MINIMOS 3D Model OptionThis key is required if MINIMOS 3D is to be invoked. It specifies the mode of calculation for MINI-MOS 3D. For a detailed explanation of the four modes see section Section 1.2. Only the first character ofthe specified string is obligatory. The second character N - new, O - old, is optional. Default is N.

Type: Default: Range: 1N

1O2N2O

OMEGA ACAnalysis FrequencyThis key specifies the angular frequency for which ACanalysis should be carried out.

Type: Default: 104 2 7Units: rad d 1Range: 1 OMEGA 6 3 1010

PDOP Activated Doping in the Poly-GateThis key specifies the activated impurity concentration in the poly-gate area. The type of impurity isdetermined by the GATE directive. This key activates the gate depletion analysis. Thickness of thepoly-gate layer is given with the GCTHE key.

Type: Default: Units: 3Range: 1014 PDOP 1021

PHYSCK Check Physical ErrorThis key specifies whether physical errors should be processed. Physical errors are out-of-rangeerrors of any numerical input value as well as sign errors within the BIAS directive. Another meaningof this key is to decide whether execution should be terminated if any iteration seems to diverge (e.g.threshold voltage infinite due to punch-through).If physical errors are detected by MINIMOS with PHYSCK=NO, a warning message will be issued. Ifthis key is omitted, physical errors will be processed. For instance, if a positive bulksource voltageshould be specified for an n-channel device PHYSCK=NO has to be given. Since any guarantee of regularexecution is lost with PHYSCK=NO, use of this statement is not recommended.

Type: Default: YES

OPTION 107

RBULK Effective Bulk ResistorThis key specifies an effective bulk resistor. When this key is given, the effective substrate bias is calculatedas efhgfNe

M4i

Zj CNkXlV . For UB see Page 62. As a large value of RBULK can affect the computation time inan undesirable manner, it is recommended to determine the effective substrate bias by the UB key directly,provided that j CmkXlV is known. This is not applicable if the STEP directive is given. This key is active onlyif MODEL=AVAL or MODEL=HOT is specified.

Type: Default: 0Units: Range: 0 RBULK 105

RDRAIN Effective Drain ResistorThis key specifies an effective drain resistor. When this key is given, the effective drain bias is calculatedas enGogPn4g

qpsr

j8@ . For UD see Page 62. As a large value of RDRAIN can affect the computation timein an undesirable manner it is recommended to determine the effective drain bias by the UD key directly,provided that j @ is known. This is not applicable if the STEP directive is given.

Type: Default: 0Units: Range: 0 RDRAIN 100

RSOURCE Effective Source ResistorThis key specifies an effective source resistor. When this key is given, the effective source bias is calculatedas eqtugqt4veg

xw

sj8C . For US see description of the BIAS directive. As a large value of RSOURCE canaffect the computation time in an undesirable manner it is recommended to determine the effective sourcebias by the US key directly, provided that j8C is known. Thi