Eindhoven University of Technology

MASTER

Implementation of physical models into the device simulator Pisces

Windgassen, Rob A.

Award date:1992

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Master thesis

Rob A. Windgassen

Implementation of physical models into thedevice simulator Pisces

EEA!439!06!1992

Eindhoven University of Technology,Department of electrical engineering, electronic devices group (EEA).

The work described has been performed at Philips Research Laboratoriesunder supervision of prof. dr. ir. F.M. Klaassen.

Summary

Nowadays designing and characterizing electronic semiconductor devices (i.e.diodes and transistors) is assisted with computer tools, such as device simulationprograms. A device simulator calculates various physical quantities such asterminal currents and also carrier densities for certain applied bias conditions.

Generally these calculations require physical models and the device structure.The models are needed to describe mobility, recombination and intrinsic carrierdensity. Into the commercially available device simulator Pisces two of thesemodels, developed at Philips Research Laboratories, have been implemented.The models of interest are a tunnel recombination model and a mobility modelthat includes several scattering mechanisms.

This report discusses the background, implementation, testing and compar­ing with measurements of these models.

The original model expressions have been reformulated to be more suitablefor use in a device simulator program. For the tunnel model an improved ap­proximation has been derived. The sensitivity of the tunnel model with respectto changes in the electric field has been investigated and it is shown that themodel is very sensitive for this parameter, making it hard to obtain very accurateresults.

It turned out to be impossible to compare results from Pisces and Trap, aone dimensional bipolar transistor simulator. Comparison with Curry, another2-D device simulator showed minor errors in both Curry and Pisces. In general,the results of Pisces and Curry agree. However, differences occur due to differentmodelling of velocity saturation at high electric fields. These differences becomesignificant for small, heavily doped devices. In larger devices, such as powertransistors, no significant differences are found.

Comparisons of simulations with measured devices show good agreement.The tunnel model seems to be suited to explain the characteristic of a gateddiode with strong fields at the semiconductor-oxide interface. With a lateralmulticollector transistor and a QUBiC transistor it is shown that the mobilitymodel improves on conventional models.

Contents

1 Introduction

2 Basic Notions2.1 Drift-diffusion based equations .2.2 Models .

2.2.1 Intrinsic carrier density and bandgap narrowing.2.2.2 Mobility ....2.2.3 Recombination

3 Device simulation3.1 Dependent Variables .3.2 Discretization .

3.2.1 Discretization of continuity equations3.2.2 Dimensions ..

3.3 Solution ~fethods . . .3.4 Supplying derivatives.3.5 Scaling.........

4 Tunnel model4.1 Introduction .4.2 Tunnel model .

4.2.1 Trap-assisted tunnelling4.2.2 Analytical approximations .4.2.3 Band-to-band tunnelling ..

4.3 Exceptions .4.3.1 Trap-assisted tunnelling, low fields4.3.2 Trap-assisted tunnelling, low 6.E / kT .4.3.3 Band-to-band tunnelling .

4.4 Model limitations .4.5 Sensitivity .

1

3

34455

6

6

679

101113

141416182124262626272728

5 Mobility5.1 Introduction .

5.2 Overview of mobility model5.2.1 Lattice Scattering5.2.2 Donor scattering ..5.2.3 Acceptor scattering.5.2.4 Electron-hole scattering5.2.5 Ultra high doping effects.

5.3 Model equations .

5.4 Determination of Pmin

5.4.1 Approximation method5.4.2 Choosing Po ..

5.5 Exceptions '"5.5.1 Expression for P5.5.2 Cluster function

5.5.3 Derivative of /lD+A+C2

5.6 Bandgap narrowing ...

6 Software implementation6.1 Introduction ..6.2 Compiling .

6.3 Numerical tests .6.4 Comparison with Trap and Curry.6.5 Robustness .. "6.6 Implementation notes

6.6.1 Module layout ....6.6.2 Activation of the models.

7 Measurements7.1 Gated diode. . ...

7.1.1 Magnitude of currents7.1.2 Resume ...

7.2 Lateral multicollector transistor .7.3 QUBiC transistor .

8 Conclusions

A Derivatives of the tunnel modelA.1 Trap-assisted tunnellingA.2 Derivatives of r, low field . .A.3 Derivatives of r, high field ..A.4 Derivatives, band-to-band tunnelling

30303132

323232

343439394141

4142

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434343444546474748

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565659

64

66666767

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B Field-effect factor r approximationsB.! Upper limit of r .B.2 Continuous gamma approximationB.3 Comparison of r approximations

B.3.! r at low fields ..B.3.2 r around x = 1 .B.3.3 r for high fields .B.3.4 r for very high fields

C Derivatives of the mobility modelC.l Parameter P .C.2 Minority scattering parameter GC.3 Electron-hole scatter parameter F .

C.4 N,c and N/Jc,ej j . . . . . . . .

C.5 "Extrinsic mobility" J.lD+A+C2

C.6 Total mobility .

D Comparing Pisces and CurryD.1 Introduction ....D.2 Velocity saturationD.3 Minority lifetimes.D.4 Bandgap .D.5 Intrinsic carrier density

E Note on accuracy with Newton and Gummel method

F Implemented modulesF.l Module tungam .F.2 Module kmob . . . .

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Chapter 1

Introduction

Since the invention of the bipolar transistor electronic devices have been evolvingrapidly and this process still continues. To enable this progress it is necessaryto understand the behaviour of electronic devices as good as possible. Thisunderstanding is obtained by knowledge of the underlying physical quantitiesthat determine device behaviour. Simply said, one wants to "look" inside thedevice and see what is happening with quantities of interest such as electric fieldsand carrier densities. These quantities are determined by a set of equations basedon semiconductor physics. Solving these equations for applied bias conditionsresults in the desired physical quantities.

Usually this is achieved by splitting a device into separate regions and obtain­ing a closed form solution based on several assumptions allowing suitable simpli­fications. An example of this met.hod can be found in popular textbooks [32, 33]where approximations for an "ideal" bipolar diode are given.

To model and understand the behaviour of modern devices correctly, thesesimplifications generally are inapplicable, furthermore, advanced insights haveresulted in more complex models restricting simplifications. Therefore accuratesolutions have to be obtained with numerical methods and a special class of com­puter tools has been evolved, the so called device simulators. This started whenGummel [36] developed a program that found a solution for a I-dimensional (1­D) bipolar transistor. Thereafter great improvements have been made and sev­eral types of device simulators exist such as dedicated bipolar [3J or MOST [39Jsimulators and general 2-D [40,41] and even 3-D simulators.

The set of equations on which most device simulators are based require phys­ical models, a doping profile and boundary conditions. The doping profile iden­tifies the distribution of acceptors and donors over the device. The boundaryconditions include the contacts with applied biases. The physical models de­scribe the intrinsic carrier density, carrier mobility and recombination. Thoughthe set of equations is well established, the physical models still evolve.

At Philips Research Laboratories recently models were developed for carriermobility and recombination rates. The mobility model incorporates different

1

mobilities for minority and majority carriers with a unified description that isbased on physical insights instead of empirical fits. The difference between mi­nority and majority carrier mobilities agrees with observations, and is importantfor bipolar devices, where the behaviour of the device is mainly determined bythe minority carriers.

The tunnel recombination model describes the effect of tunnelling on thegeneration and recombination rates in case of strong electric fields that occurin highly doped junctions of small-scale devices. The model explains why cur­rents under these circumstances can be orders of magnitude larger than withconventional models. It also correctly describes the distinctly different voltageand temperature dependence of these currents.

These models have been implemented in Pisces-2B, version 9033, a commer­cia.lly available device simulator [40] marketed by TMA. It is intended that themodifications are transferred to TMA and will be incorporated in Medici, thenext generation of Pisces. The implementation in Pisces is discussed in thispaper.

Chapter 2 introduces the physical relations that form the foundation for theoperation of most electronic devices. These relations form a set of coupled par­tial differential equations and most device simulators are based on solving theseequations. Chapter 3 briefly describes solution methods and the correspondingrequirements for model descriptions. The tunnelling and the mobility modelsare described in chapter 4 and 5, respectively. Starting from the original modeldescription by Hurkx et al. [15] of the tunnel model, expressions have been refor­mulated to be more suitable for use in device simulation. Improved approxima­tions have been derived and are given in appendix B.2. Furthermore, sensitivitywith respect to the electric field is investigated together with model limitationsstemming from local approximations. The mobility model has been adaptedto unify expressions for electron and hole mobility, enabling to implement thecalculations of the mobility of both type of carriers in a single algorithm. Im­

plementation issues and software testing are handled in chap! er 6. Simulationresults from Pisces have been extensively compared with results from anot.her

device simulat.or, Curry [41]. Comparing simulation results with measurementsis done in chapter 7. Measurements have been performed on gated diodes, alateral multicollector transistor and QUBiC transistors. For a gated diode thetunnel recombination plays a crucial role, while with the multicollector structureaccurate values of mobility can be obtained.

Finally conclusions are presented in chapter 8.

2

Chapter 2

Basic Notions

2.1 Drift-diffusion based equations

The behaviour of semiconductor devices is generally described with a set of

equations, sometimes referred to as the basic equations [29]. These equations

describe the behaviour of the electrostatic potential t/J, and the carrier densitiesnand p in a device. They consist of, firstly, the Poisson equation

(2.1)

with (. the dielectric permittivity of the semiconductor, N; and Nfj the ionizedacceptor and donor concentrations. Secondly, they include the electron and holecontinuity equations,

on-q - + V· I n = + q R,

8t

op+q - + V'. Jp = - q R.at

(2.2)

(2.3)

R denotes the net recombination rate and I n , Jp are the electron and hole currentdensities which can be written as

[kT •

I n = q n J.ln [- q v (lnnie)J + qDn 'Vn,

[kT •

Jp = q P J.lp £ + q V (In nie)j - qDp Vp.

(2.4)

(2.5)

Equations (2.4) and (2.5) are known as the drift-diffusion equations. The termswith V' (In nie) stern from a non-uniform bandgap narrowing (see below). Usuallyit is assumed that the Einstein relations

kTDnp = J.lnp-. , q

3

(2.6)

hold. After introducing the quasi-fermi potentials ¢n and ¢p by

kT ncf>n == 1/; - -In -,

q nie

kT p¢p == 1/; + - In -,

q nie

(2.7)

(2.8)

it is then possible to express the drift-diffusion equations (2.4) and (2.5) as

(2.9)

(2.10)

2.2 Models

The equations given above contain physical models for intrinsic carrier density,mobility and recombination rate. Although the foundations of the equationsare solid, the models to be used still evolve. Some models are discussed brieflywithin the framework of t.he drift-diffusion approach.

2.2.1 Intrinsic carrier density and bandgap narrowing

Experiments show that at high doping levels the bandgap tends to decrease withincreasing dopant level [6, 7J. This effect is known as bandgap narrowing, and isdenoted by AEc. The pn product in thermal equilibrium can then be writtenas

(2.11 )

where n te is the effective intrinsic carrier density, while niQ is the intrinsic carrierdensity in undoped material.

niO = JNcNv exp (-Ec/2kT) , (2.12)

Nc and IIiV denoting the effective density of states in the conduction and valencebands. The bandgap narrowing is empirically fitted [6] by

(2.13)

where N is the total impurity concentration, Va, No and Co are model parame­

ters.

4

(2.14)

2.2.2 Mobility

There exist many different models for mobility, but they usual have severalaspects in common:

1. Dependence on doping. Increasing the doping level will reduce the mobilitydue to impurity scattering.

2. Lattice scattering. At low doping levels the scattering due to lattice dom­inates.

3. Velocity saturation. The mobility is reduced at strong electric fields. Thisin fact marks the restriction of the notion mobility: the linear relationbetween electric field and current density DO longer holds at strong fields.

Sometimes additional features are taken into account, like reduction at semicon­ductor interfaces and carrier-carrier scattering mechanisms.

2.2.3 Recombination

Several recombination processes are known. 'Vell known are Shockley-Read-Hall

(SRH) [27,28] describing recombination- and generation of carriers via traps thatare located in the forbidden gap, by

2

R_ pn - n,e

SRH - ( ) ( . ,Tp n+nl +Tn p+p!J

and Auger recombination [31] which describes the process in which the energyof an electron or hole is transferred to another free electron or hole,

(2.15)

At high electric fields avalanche generation due to impact ionization will occur,described by

(2.16)

For the ionization coefficients for electrons and holes, O'n,ii and O'p,ii differentdescriptions exists, for example Sze [31J uses

where El, [I, [p and £kT are model parameters.The net recombination is obtained by adcling the different recombination

processes together

R = RSRH + RAv9 - GAV +.... (2.18)

5

Chapter 3

Device simulation

The system of three partial differential equations defined by (2.1), (2.2) and(2.3), with the dependent variables 1/;, nand p generally form the starting pointfor device simulation. Together with these differential equations boundary con­ditions must be applied. They consist of contact and insulation boundaries andalso reflective boundaries to model symmetries in a device geometry.

3.1 Dependent Variables

As stated above, the differential equations with dependent variable '1/;, nand p

are to be solved. However, sometimes it can be useful to transform the problemand use variables other than nand p. This might be done to achieve improvedr.onvergence of numerical solution methods but possibly also a to reach certainlevel of accuracy. For instance the currents calculated by the drift-diffusionexpressions (2.4) and (2.5) will suffer from numerical instability at low currentswhen the two terms are nearly equal but of opposite sign. In that case it mightbe advantageous to use rt>n and rPp as dependent variables. The currents canthen be calculated with (2.9) and (2.10) without loss of accuracy. The price tobe paid is that the continuity equatiom are exponentially nonlinear in cPn and

cPp [29J.Other sets of dependent variables also have their own advantages and draw­

back::>. Consequently, the set of dependent variabled ('I/;, n,p) is used in mostcommercial available packages, like Pisces.

3.2 Discretization

Solving the set of equations usually involves the following steps [29)

• First the device area to be simulated is partitioned into a, usually large,number of sub-areas, where an approximate solution can be obtained to a

6

certain desired level of accuracy. The partitioning of the device is describedby a mesh1, i.e. a geometrical strncture representing the sub-areas.

• The basic equations in each sub-area are approximated by algebraic ex­pressions which only depend on the variables 'l/J, nand p at discrete points,called nodes. This results in a large set of nonlinear equations. When thediscretization involves N discrete nodes, the set consists of 3 N equations(N x Poisson, 2N x continuity equations).

• Finally the large set of nonlinear equations is solved.

There are several ways to set up the mesh. Two types of meshings are oftenused, the rectangular- and the triangular mesh.

The rectangular mesh exists of lines parallel to the coordinate axes as shownin figure 3.1 b. Although the structure is straightforward, it allows not for localrefinements. For example, when a fine mesh is needed in a certain region, i.e.close to a surface or junction, the fine meshing extends to areas where this is notneeded. This implies an unnecessary increase in the number of nodes, resultingin an inefficient use of computer time and storage.

With the triangular mesh, as is used with Pisces, the mesh can be refinedlocally, as is shown in figure 3.1c. It is also possible to construct non-planarstructures. The price to be paid for the triangular meshing is the extra com-

E

c

(a)

fEDI

I

(b) (c)

Figure 3.1: Meshings: (a) example of a bipolar transistor; (b) rectangular mesh(264 nodes); (c) triangular mesh (170 nodes).

plexi ty of the geometrical structure and the possibility to introduce inaccuracieswhen the grid contains obtuse triangles [35].

3.2.1 Discretization of continuity equations

The continuity equations are often discretized with use of trial functions that takeinto account that carrier densities tend to vary rapidly, i.e. behave exponential.

1The notion grid is often used too.

7

Without special treatment this behaviour demands an extremely fine meshing.Because the continuity equations are of interest for the implementation of the

1

(a) (b)

Figure 3.2: Triangular discretization

mobility and recombination models, the discretization scheme, first suggestedby Scharfetter and Gummel [37J, will be given here for the Pisces triangularmeshing. This discretization method is based on the following assumptions:

1. The potential between mesh points varies linear with the distance, i.e. theelectric field is constant.

2. Mobility values do not change significant between mesh points.

3. Current densities arc assumed to be constant, neglecting recombinationeffects.

Using these assumptions, the differential equations (2.4) and (2.5), i.e. the drift­diffusion equations, can be solved in terms of '1/;, nand P at the nodes.

Then, the discretized currents flowing in the triangle along the edge fromnode 1 to 2 in figure 3.2b are

( '!h -1/J2) (1/J2 - '!d1)B tiT HI - B VT

n2

I n ,12 = Jln kT ,(3.1)/12

(3.2)Jp ,12 = Jlp kTB (¥) PI - B ( P2 ir?PI) P2

[12 '

where '1/'" ni and Pi are the values of the dependent variables at the nodes of thetriangle, B (x) is the Bernoulli function

xB(x) == () ,exp x-I

(3.3)

8

and l12 is the length of the side connecting the nodes 1 and 2, and Vr the thermalvoltage kT/q. The continuity equations are then expressed at a node (e.g. thecentral node in fig. 3.2a) as

~ [~+Jn'i(h; + hn]- A (R+ ~~) = 0,

~ [~-Jp'i(ht + hi)] - A ( R + ~~) = 0,

(3.4)

(3.5)

where the summation is over all edges that connect to a node, and h; +hi is thelength of the perpendicular bisector of edge i and A is the surface of the areaspanned by the bisectors (i.e. area enclosed by the dashed lines in fig. 3.2a).I n ,; and Jp,i are the electron and hole current densities along side i. Duringa solution iteration step, Pisces processes all triangle elements, calculates thecontribution of an element to a node (see fig. 3.2b), and adds this to a largematrix containing all contributions. It should be noted that different quantitiesare defined for different locations. For example 1/;, nand p are known at anode, while currents are calculated for edges between nodes, recombination forareas around a node (i.e. area A l in fig. 3.2b) and the electric field for the areaspanned by a triangular element. These differences tend to make things complex,and when a quantity is manipulated it should be checked for what location the

quantity has a useful meaning and what weighting factors are to be used whenquantities are added.

3.2.2 Dimensions

The discretization and the corresponding mesh determine the number of spatialdimensions in which the dependent variables 1/;, nand p are allowed to change.For a .1t1-dimensional simulator, 1vl = 1,2,3, the dependent variables can varyin M -directions.

For a one dimensional (I-D) simulator, like Trap [3], the meshing consists of asingle axis with discretized "tic marks". Only 1-D effects can be simulated withthis type of simulators, excluding all types of current spreading. The advantageof a I-D simulator is the possibility to construct a very fine meshing withoutspending much computer resources (time and storage). One must be aware thatresulting flux-like quantities, such as charge and current are expressed per unit

of area.Similarly, for a 2-D simulator, like Pisces and Curry, the mesh structure can

be represented in a x-y-plane, like the vertical bipolar transistor of figure 3.1. Intwo dimensions for example several current spreading mechanisms can be sim­ulated. The 2-D mesh usually requires more nodes compared to a I-D mesh.Consequently results are obtained at a higher computing cost, and often the

9

meshing no longer can be as fine as with a I-D simulation. Now flux-like quan­tities are expressed in units of depth. For example Pisces represents terminalcurrents in units of A/microns.

Finally with 3-D simulators, all kind of spatial effects can be studied withoutany restriction. For example parasitic capacities of crossing conductors can besimulated. A 3-D mesh consists often of very many nodes and requires verylarge computing power. This forms a restriction of this method on economicalgrounds.

3.3 Solution Methods

The solution of the large set of nonlinear equations can be done in several ways.Two well-known methods are the Newton [29] and the Gummel method [36].With the Gummel approach, the Poisson equation (2.1) and the continuity equa­tions (2.2) and (2.3) are decoupled. This means that starting from an initialguess, first the Poisson equation is solved yielding a new potential 'ljJ. Next, withthe new 7jJ, the electron continuity equation is solved yielding a new electron con­centration n. Finally the hole continuity equation is solved resulting in a newestimate of p. With the new solution of 1/;, nand p this scheme is repeated untilthe process is converged, i.e. the updates of 7jJ, nand p fall within tolerances.

With the Newton method the coupled equations are solved, i.e. with aninitial guess at once a new solution of 7/;, nand p is found2 .

Both solution methods involve the solution of a large set of nonlinear equa­tions. For a mesh with N nodes, the Gummel methods solves three times .TIiequations, while Newton at once solves 3N equations. In Pisces the set of non­linear equations is solved with a generalization of the Newton-Raphson method,which solves a single equation f(x) = O. For that case it's seen in figure 3.3 that

the update Llx is

f(x n )

Llx = Xn +l - Xn = - f'(xn

)'

Thus the equation

(3.6)

(3.7)

has to be solved. For a iV-dimensional function F(x) where both x and FareN -dimensional vectors this can be generalized to solving

(3.8)

where JXn is the Jacobian of the function F at the point Xn \ which contains thederivatives BFt/Bxnj for i,j = 1,2," .1'1. In Pisces x and F are defined by the

2In appendix E the accuracy of the methods is discussed for low biases.

10

o

Z-n

Figure 3.3: Newton-Raphson iteration for a single valued function f(x). Fromguess X n the next guess Xn+l is constructed.

dependent variables and the device equations, e.g. for the Newton method

and F = ( ~: )f p

(3.9)

(3.10)for y,x = 'Ij;,n,p

F'/J stems from the Poisson equation and Fn , Fp from the continuity equations.The Jacobian will thus contain elements of the form

aFy

oxand all these derivatives must be available. Of course it is possible to obtainthe derivatives with numerical approximation methods but this is very time con­smiling, compared with the evaluation of analytic expressions for the derivatives.As a consequence, models should not only provide a nominal quantity, but alsoprovide the derivatives with respect to the dependent variables.

3.4 Supplying derivatives

The dependencies of models on the carrier concentrations are usually straightfor­ward, and derivatives are constructed likewise. The dependence on the potential'Ij; is more complex, because models most often are not expressed explicitly interms of electric potential 'Ij;, but rather implicitly when they depend on the

electric field E.

11

In the discretized equations, the potential within a single triangular elementIS a linear function of the position, and the electric field is constant. In thatcase the electric field can be written as a linear expression of 1/J at the nodes.Splitting £ in x- and y-components, this becomes

Ex = L 1/1i' Px,i,i E nodes

£y = L 1/1i . Py,I'i E nodes

(3.11)

(3.12)

(3.13)

where Px,i and Py,i are proportionality factors depending on the geometry of thetriangle. The derivatives of Ex,y with respect to the potential at node i are then

8£x8'1j;j = Px,i,

8£y80 = Py,i'

, , (3.14)

When a model depends on the magnitude of the electric field, as is the case withthe tunnel model, the derivative of £ with respect to 'Ij; can be obtained in thefollowing manner. Starting from3

and differentiating results in

2 £ a[ = 2 E a£x 2' a£y.a.I,. x 8 I. + t y ,:) I

'1/, 1i', Ulpl

Rl'" riting yields

(3.15)

(3.16)

(3.17)a[ Ex 8Ex £y 8Ey £x £y84'i = '£ 81/Ji + '£ 81/1i = £ Px,i + E Py,i.

The derivative with respect to the potential can now be constructed with (3.17)and the chain-rule when a model supplies the derivatives with respect to theelectric field, like

8X ax a£8'1j;i = 8£ 81/J,'

where X is an arbitrary model quantity.

3 E is the magnitude of the electric field vector E, i.e. £ = Itl, and is non-negative.

12

(3.18)

3.5 Scaling

In device simulators it is common use to scale physical quantities. There aremainly two reasons to do so:

• Constant expressions like qI f and kTIq are not unnecessarily often recal­culated.

• The different orders of magnitude of several quantities is reduced, prevent­ing loss of numerical accuracy, and possible underflow or overflow.

The scaling in Pisces-2B, version 9033, is summarized in table 3.1. The columnmarked with "symbol" denotes variable names used in the program. It

Table 3.1: Pisces scaling factors

quantity units scaling factor symbol

n,p cm-3 (Nc Nv)1/4 DCSCL

V' V kTlq DKTQ

E V/cm DKTQ

R cm- 3s- 1 DKTQ· DCSCL

It cm2/Ys

oRion s-l DKTQ

oR187j; crn- 3y-1 s-1 DCSCL

(1/11)011 /8n cm3 1/DCSCL

(11 fl-)ofl-/ 8E cm/V l/DKTQ

should be noted that the mobility is not scaled. The derivatives of interest arelisted in the way as they occur in Pisces. The scaling factor for carrier densities,DCSCL, is defined in a pragmatic way and has no physical meaning. Somedevice simulators, like Curry, 5cale the densities with niO or nie' Although thishas some physical meaning, it is not practical at low temperatures. In that casethe intrinsic carrier density becomes extremely small, for example at T = lOOKnjQ is about 3 x 10- 11 cm-3. Scaling densities with niO might lead to overflow

in that case.

13

Chapter 4

Tunnel model

4.1 Introduction

During the last decades, technology has advanced in down scaling of device ge­ometries. To maintain proper functionality, the doping levels have been raisedaccordingly. This implies an increase in electric fields at junctions. As a conse­quence, it turns out that in reverse and low forward biasing of pn-junctions, thecurrents are significantly larger than is to be expected with conventional models.It is assumed that the increased currents originate from tunnel phenomena. Thetunnelling phenomena are of importance in regions where strong electric fieldsoccur, such as in base-emitter junctions of bipolar transistors with heavy dopedemitters. There it accounts for leakage currents in reverse bias, and for reducedcurrent gain at low forward bias [16].Models have been proposed [24, 25], but are not accurate and suitable to be usedwith device simulation. Recently a new model has been developed [13, 14, 15Jthat describes two types of tunnel processes, i.e. band-to-band tunnelling andtrap-assisted tunnelling.

The model is formulated in such a manner that it can be built into a devicesimulator program. This means that the recombination rate is expressed in localquantities. One should be aware of the fact that t'unnelling is a nonlocalized,i.e. spatial, phenomenon. Thus, expressing tunnelling in local quantities impliessimplifications.

In figure 4.1 the tunnel mechanisms are shown. At thermal equilibrium(fig 4.1a) there is no tunnelling and the dashed line indicates the Fermi-level.Fig 4.1b shows trap-assisted tunnelling when a moderate reverse bias is applied.Since at low fields (i.e. distance to cross is too large) it is not possible for anelectron to tunnel directly from valence to conduction band. It is possible totunnel from the valence band into a trap, and via thermal emission over a part ofthe energy and finally tunnel through the remaining potential to the conductionband. When the reverse bias is increased (fig 4.1c), the increased band-bendingreduces the tunneling distance to be crossed and the electron can tunnel directly

14

from valence to conduction band. In fig 4.1d, when a low forward bias is applied,electrons and holes recombine via the reverse process of fig 4.1b. To illustrate

-----~~------

( a)

-------

(c)

-------

v > 0

(b)

(d)

Figure 4.1: Tunnelling mechanisms in simplified energy-band diagrams of: (a) apn-junction at thermal equilibrium; (b) low reverse bias; (c) high reverse bias;(d) low forward bias.

the effects on the current characteristics, a I-D p+n-junction is simulated using

the model described in this chapter. The doping profile is shown in figure 4.2.The forward IV-characteristic (fig 4.3) including tunnelling is compared with

a simulation with conventional moads (dashed line). At low forward bias a con·siderable increase in current is observed due to trap-assisted tunnelling,The reverse characteristics, shown in figure 4.4, are simulations at low (solid line)and high temperatures (dashed line). Here three major contributions can beseen. At low bias, trap-assisted tunnelling dominates which increases with tem­perature. At medium bias, band-ta-band tunnelling dominates, which is nearlyindependent of temperature. Finally at high reverse voltage bias, avalanchegeneration due to impact ionization dominates, which decreases with temper-

15

"

le+20 dope -

.. 1..+19

"0...'"'.."'"'c~0 le+18c00

0-0...

1..+17

0.2250.2000.125 0.150 0.175distance (microns)

0.100

le+16 '--- '--- J- .L..- .l- ...L-__----'

0.075

Figure 4.2: I-D doping profile of a p+n-junction

ature. Because electric fields increase with increasing applied reverse voltagesit is seen that at strong electric fields band-to-band tunnelling dominates overtrap-assisted tunnelling.

In figure 4.5 the field dependence of the recombination mechanisms is illus­trated.

The model equations are given in the next section. A new description fortrap-assisted tunnelling is introduced that is more easy to use when derivativeshave to be constructed for a device simulator. In section 4.3 methods are pre­sented that have been developed to prevent numerical exceptions at runtime forextreme parameter values. Model limitations due to expressions based on local

quantities are discussed ill section 4.4. The influence and consequences of thestrong field dependence of the model is described in section 4.5.

4.2 Tunnel model

The two types of tunnelling account for a net tunnel recombination rate l

Rtunnel = R trap + Rbbt· (4.1)

R trap contributes for the trap-assisted tunnelling, but also includes the conven­tional Shockley-Read-Hall (SRH) recombination mechanism [27, 28], and Rbbt

IS the band-to-band tunnel induced recombination. The model accounts for

I For convenience the term recombination is used for recombination and generation. Anegative recombination denotes generation.

16

-6 tunneling ---conventioJ1el ----

-8

.,c$.. -10.."Ut7>0

.-<-12

-14

0.80.70.60.3 0.4 0.5V (volts)

0.20.1

-16 '-----'-__-'- '--__--'-__---"- L...__-'-__----'

0.0

Figure 4.3: forward IV-characteristic

-2

-4

-6

.,c

-8$...."ut7> -100

.-<

-12

T-250K ­T-370K ----.

trap-asshted

ava1anch~----------~

band-band

-14

-4-2 -3v (volts)

-1

-16 '-- --'- -'- L... -'- --'

o

Figure 4.4: reverse IV-characteristic

17

1.0E:+H

~

i 1. OE+22........

CD...'""c: 1.OE+20o.......'"c~g 1. OE+18

:

1. OE+16

band-to-band --­trap-a.sisted .....

Shockley-R....d-Ha11 ---

abadop - 2e18

n,p « nieT • 300!(

O.OE+OO 2.0£+05 4.0£+05 6.0£+058.0£+051.0£+06 1.2E+06 1.4E+06Electric Field [V/cm)

Figure 4.5: Field dependence of recombination rate for several mechanisms.

trap-assisted tunnelling in both forward and reverse biased pn-junctions, whileband-to-band tunnelling is only accounted for in reverse based junctions. Thisexcludes the correct simulation of forward biased higWy doped junctions withdegenerate valence and conduction bands, such as tunnel- or Esaki-diodes.

4.2.1 Trap-assisted tunnelling

Trap-assisted tunnel recombination is a generalization of the SRH model [27, 28]and is described by

where

ni = nie exp(+/lEtfkT) ,

(4.2)

(4.3)

(4.4)

(4.5)

!i.Et is the energy of the trap level with respect to the intrinsic level Ei. Usuallythe trap level Et is assumed to be at the intrinsic level, so that ni and PI equal

18

•

(4.6)

(4.7)

the effective intrinsic carrier density nie' The parameters Tn,p are recombinationlifetimes and r n,p are field-effect functions due to tunnelling.

In the original paper [15] r n,p was described with the parameter Kn,p thatdepends on the electric field and, implicitly, on the carrier densities. Obtainingderivatives needed for the solution algorithm lead to cumbersome expressions.In this paper a notation is introduced that eliminates the parameter Kn,p2 andthe resulting cross-dependencies. This results in much simpler, but still complexderivatives (see appendix A). The field-effect functions then become

r _ t1En,p [1 [b.En,p _ ~£o (t::.En,p)3/2 3/2]n,p - kT io exp kT u 3 £ kT u du,

where

V2mx (kT)3£0 = 2 --'------qn

Formulation (4.6) is based on a linear approximation of the spatial dependenceof the energy bands. The effective mass m x takes into account the Poole-Frenkeleffect. i.e. the effect of charged traps, resulting in a coulomb potential.

It should be noted that when £ -> 0 the field-effect factors in equation (4.6)approach zero and expression (4.2) reduces to the standard SRH formulation.Expression (4.7) can be rearranged to obtain £0 in practical units

£0 = 425881.9 /:: C~Or/2

[V jcrn]. (4.8)

For T = 300 K and m X = 0.25 rna this results in £0 = 2.12941 x 105 V jcm.The field-effect functions r n,p depend on the local electric field £ and the

energy intervals AEn,p. These energy intervals denote the height of the potentialbarrier that can be passed with tunnelling. In fig. 4.6 the energy interval is shownfor the case of an electron in a trap located at an energy level Et . In fig. 4.6athe trap level Et lies below Een , the conduction band minimum at the n-sideof a junction, while in fig. 4.6b the trap level is above Een . The conductionband minimum Een can be approximated with the local quasi-fermi level forelect.rons and then the energy interval 6.En can be expressed in the local electronconcentration, using (2.7). Applying the same methods to holes, we get

0 n> nie exp ({,ft) ,t1En

2lf3: - In n~e nl ~ n ~ nie exp (If{f) , (4.9)--=kT

k -In.!lJ... n < nl,2kT nie

2 ref [15] used the quantity K n .p , here Kn,p is eliminated, making use of ( 4.7), substituting

Kn,p = t¥ (~fTP f/2.

19

E--J

cn

(a)

~t

E---'en

(b)

Figure 4.6: Examples of energy interval 6.En : (a) trap level below Een ; (b) traplevel above Een .

6.Ep _

kT -

o

.& I L2kT - n nie

.& -In El.2kT nie

p > nie exp (fff) ,

PI ~ P ~ nie exp (lift) ,

P < Pl·

(4.10)

Note that the trap level E t is taken into account by means of nl and PI,

allowing trap states that are not located in the middle of the energy gap3.In figures 4.7, 4.8 and 4.9 plots are shown of the p+n-junction of figure 4.2

at reverse bias (-1V). They display carrier concentrations, electric field £ andenergy intervals f::.E / kT respectively. The value along the x-axis denotes thedistance to the contact at the p+ -side of the junction (left, outside of the plots).

The electric field is of significance in the space charge (depletion) region. Inmost of that same area f::.En / kT and 6.Ep / kT are simultaneous of substantialmagnitude, i.e. the limiting value for midgap states of EG/2kT, which is about20 for Silicon at room temperature. Hence, in tms region the field-effect functionsare large, and trap-assisted-tunnelling accounts for a significant recombinationrate.

Notice, that in cases where f::.En = 6.Ep the field-effect functions r 1"1 and r p

are equal. It also must be noted, that if only one of r n or r p is considerablylarge, this will not suffice for enhanced recombination: the term with the low rin the denominator of equation (4.2) will limit the recombination rate.

At the left side ofthe plot EG is somewhat lower, which is caused by bandgapnarrowing due to the heavy p+ doping.

4.2.2 Analytical approximations

In fig. 4.10 r is plotted as a function of the electric field for two values of f::.E / kT.

It is clear that r values vary over a large range. The integral in equation (4.6)

3The implementation in Curry only allows midgap states.

20

1e+20 dope -n -,..-_.p .....

., 1e+19r::0..

"...II.....r::~ .u 1..+18 .r::0 ,u ,0' ,0.... ,,

le+17

/I

le+160.125 0.150 0.175 0.200 0.225

distance (microns)

Figure 4.7: carrier concentrations in reverse biased junction

1£+06 r-------r-------r-------r--------,

0.2250.2000.175distance (microns)

0.150

o£+oo I::======_-J... ...L-_....C::: L- .....J

0.125

8£+05

""'" f

....~.~

'"u

4£+O5l

.~.....u~....'"

2£+05

Figure 4.8: electric field in reverse biased junction

21

25 ....------"'T"-------,-------.-------..,

20 l-------:-,-------------------------------------------------

15

10

5

delta En/kT ­delta Ep/kT ----.

0.2250.2000.175distance (microns)

0.150o'------=-~-----....L------'---------'

0.125

Figure 4.9: energy intervals !:i.EIkT in reverse biased junction

L..

Figure 4.10: r versus electric field

22

(4.11)

(4.12)

can not be solved analytically but approximations exist for low and high electricfields. When the low field condition4

~ < VtlEn.1J

Eo - kT'

holds, then the approximation

r n,p = 2fi~ exp [~ (JJ 2] ,

can be used. It should be noted that this expression is independent of tl.E / kT.This is demonstrated in fig. 4.10, at low fields the two curves with differentvalues of D..E/kT coincide. Hence when the low field approximation is valid forelectrons and holes, rnand rp are equal (tunnelling occurs near the band-edgesand the magnitude of the energy interval is then irrelevant). When (4.11) isfalse, the high field approximation, a more complex form

where

(4.13)

{( E)1/2(b.E )1/4r 1 =1+(3 - ~n,p Eo kT (4.14)

(4.15)

should be used.In appendix B.3 the accuracy of the high and low field approximations are

considered in detail. Here ;35.6, a1, a2 and a3 are constants to approximate thecomplementary error function as can be found in e.g. Abramowitz [12]. and areshown in table 4.1 The expression for high field, equation (4.13) is complex, andtherefore an additional substitution is made, using the dimensionless quantityx, defined by

x == ~ (tl.En •p ) -1/2

Eo kT

The quantity x might be interpreted as a kind of normalized electric field pa­rameter. Using x simplifies the low field condition (4.11) to

x :$ 1, (4.16)

4 For convenience we speak of low field condition when (4.11) is true, although the expressionalso depends on the value of l:1EJkT, and thus might be false at relatively low electric fields.

~In ref [12] the symbol p is used instead of (3, but in this paper the symbol p is used todenote the hole concentration.

6The value (3 is adapted, in ref [12] a value of (3 =0.47047 is used, later on in appendix B.3this is reconsidered.

23

Table 4.1: Approximation parameters

parameter value

{3 0.61685

Ul 0.3480242

U2 -0.0958798

U3 0.7478556

and the high field approximation (4.13) to

jC:i. En ,p c r= ( 2 3 )fn,p = --,:;:r- va: v 7r Ultn,p+U2tn,p+a3tn,p

(6.Enp ( 2 1)]x exp ---;;T 1 - 3" . -; ,

where

(4.17)

(4.18)c 1 = 1 +13V6.En,p {..;x - ~}.

n,p kT .jx

Using the quantity x, this expression can be evaluated efficiently. This also holdsfor the associated derivatives.

:\. problem with the expressions for the low and high field approximations ltithat the transition from low to high field is not continuous. The low field approx­imation over-estimates the field effect functions rn,p near this transition region.Apart from being inaccurate, the Newton solution process can be degraded, i.e.poor or no convergence.

A way to fix this is to apply the high field approximation at somewhat lowerfields than suggested by the low field condition equations (4.11,4.16). For Siliconat T = 300 K the adapted low field condition

2 1x < -- = 0.925925···

- 30.72(4.19)

reduces most of the discontinuity. Although not elegant, it is sufficient formost cases because generally when the high field approXlluation is applied, theband-to-band tunnelling dominates over trap-assisted tunnelling. However, us­ing equation (4.19) as the condition to check whether to use the low or high fieldapproximation has one main disadvantage. It is suited when 6.£ jkT is about20, which only holds for Silicon at room temperature in the space charge region(see figure 4.9), and for not too much bandgap narrowing. At other conditions,the discontinuity will become larger.

However, this is the way the model is currently implemented. In the sameway it is implemented in Curry (release 8.6) and simulations show that Pisces

24

converges sufficiently fast. Using the same approach made it possible to compareresults of Pisces and Curry very accurately, which aided testing.

I have derived an alternative formulation for the model, that is continuousunder all conditions, it is described and evaluated in appendix B.2. This formula­tion more accurately approximates the low field regime. It also uses the original

Abramowitz parameters for the estimation of the complementary error function,resulting in a slightly better accuracy for high fields. However, both the old andnew approximations, over-estimate the field-effect functions r n,p at relative lowelectric fields, e.g. when rn,p ::::: 1. Unfortunately the new formulation has notbeen implemented yet, due to lack of time.

4.2.3 Band-to-band tunnelling

The recombination rate due to band-to-band tunnelling Rbbt in a reverse biasedjunction can be expressed [15] with

(4.20)

(4.21)

where

(E) 3/2

Ebbt = Ebbt,T==300' E G ,G,T=300

Band Ebbt,T=300 are material parameters with values, as obtained by measure·ments [15], shown in table 4.2. The parameter a depends on the type of semicon­ductor, a = 2 for direct gap and a = 5/2 for indirect gap semiconductors, likeSilicon. The prefactor D accounts for occupancy of states at the nand p-side ofa pn-junction, and can be expressed by means of the quasi-fermi energies at then- and p-side of the junction,

D= 1exp [(-E fp - q<j;) /kT] + 1

1

exp [( -Efn - q'l/J) /kT] + l'(4.~2)

This can be approximated by using the local quasi-fermi levels -q¢n and -qI/Jp,and with help of (2.7) and (2.8), expression (4.22) can be rewritten to

2D = nie - pn (4.23)

(n + nie)(P + nie)

At high reverse biases, D should be unity. Due to increased carrier densitiesthat might occur when considerable amounts of carriers are generated, the localapproximation expression (4.23) is no longer accurate. This can be fixed bysetting D to one in this case. Setting D to unity can be done when theelectron or hole current density has reached a certain fraction (e.g. 1 x 10-3 ) ofqnieV~, where v~ is the saturated drift velocity. This condition can be expressedin local quantities as

1\7I/Jn I > A nie 1\7'l/JI,n

25

(4.24)

lE25

lE20....I

"MI

!.,.0

~ lE15

lE10

LO 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0electric field (lE5 V/crn)

Figure 4.11: band-to-band tunnel originated recombination

Table 4.2: band-to-band model parameters for Silicon

parameter

£bbt.T=300

B

(J

value

1.9 X 107 Y /ern4.0 X 1014 cm-1/2y-5/2s-1

5/2

26

where

(4.25)

Formula (4.20) is to be applied in reverse mode only. This is true when j. f > O.In terms of quasi-fermi potentials this results in

(4.26)

and

(4.27)

(4.28)

4.3 Exceptions

When expressions are evaluated in a computer program, one has to be awareof special cases where numerical evaluation will fail due to overflow or divisionby zero, although algebraic expressions might produce a sensible value. Forexample it's easily seen that for values of £ and AEn,p close to zero problemsoccur in expressions like (4.15), (4.17) and (4.18). Therefore some extra care isto be taken before evaluating complex expressions. When values exceed certainlimits7

, analytical approximations of the result should be applied. In the case oftunnel-recombination the exceptions tend to happen at low values of the electricfield and energy intervals. In that case tunnel recombination is so low that itcan be neglected. When it is detected that values are below a lower bound,the appropriate actions are skipping the regular model expressions and settingtunnel recombination and associated derivatives to zero. In the next sectionsthese lower bounds are determined.

4.3.1 Trap-assisted tunnelling, low fields

Trap-assisted tunnelling is expressed with field-effect factors in the generalizedSRH formula (4.2). There the field-effect factors are added to one. So it's easyto find a practical lower limit: when r n,p ~ r min « 1 trap-assisted tunnellingcan be neglected (currently a value of 1 x 10-4 is used for r min).

Assuming rmm < 5 it's easy shown from equation (4.12) that

£ r min~ < -- => r < rmin.cO 5

Practically this means that, when £1£0 < r min/5 occurs, then r is set to zero,the regular expressions are skipped and no numerical problems arise.

7 with 'exceeding limits' also falling below a lower bound is meant here.

27

4.3.2 Trap-assisted tunnelling, low t1E / kT

Starting from the definition of r in equation (4.6) it can be shown (see appendixB.l) that

r :s; exp (1:i.ElkT) - 1

so that for values of ~E jkT ~ 1 this can be approximated by

r :s; ~EjkT

and thus

~ElkT < r min::} r < r min.

(4.29)

(4.30)

(4.31)

(4.32)

Note that the above is derived without using assumptions of either high or lowfield approximations and is valid in both cases.

4.3.3 Band-to-band tunnelling

From equation (4.20) it's not easy to obtain a lower bound that indicates wherethis type of tunnelling can be neglected. However it is necessary to have one,otherwise the exponent might underflow at low fields. So the threshold is basedon numerical grounds rather then Oll physical. In Pisces the maximal allowedargument for the exponent function is available, i.e. MAXEXP. Using this quan­tity, tunnelling is neglected when

EbbtE < Ebbt,min = MAXEXP

The actual value of MAXEXP is 80, resulting in Cbbt,min :::::: 2.4 x 105 V jcm.At T = 300 K an electric field of that magnitude results in a band-to-bandtunnelling induced recombination rate of Rbbt = 2 x 10-7 cm-3s- 1. Thus it isvalid to neglect tunnelling for electric fields that satisfy (4.32).

4.4 Model limitations

As noticed before, the model uses local quantities based on certain simplifi­cations, while tunnelling by definition is a spatial phenomenon. An essentialapproximation is the use of linearised energy bands, which makes it possible toexpress the tunnelling process in terms of electric fields (i.e. q[ = gradE). Thismight turn out to be wrong where the bands are bent strongly c,f. ref [15J, orworse, when the energy bands of the semiconductor ends at an interface. Figure4.12 shows an example of this, while in fig. 4.12a the electron can tunnel, thisis not possible in the case of fig. 4.12b where the local electric field is of thesame magnitude, but there is no state where the electron can tunnel into due tothe interface. In these cases the models might predict considerable tunnelling

28

(a) (b)

Figure 4.12: tunnelling: possible (a), impossible due to vicinity of oxide interface

at left hand side (b)

when in reality this is not possible. A way to fix this is checking whether thereexists states in the vicinity to which an electron can tunnel. This needs to bedone for every possible starting position for tunnelling processes. In fact thetunnelling process is then expressed in the quantities of interest, but it is a quitecomputational expensive approach because for processing each node, all othernodes are to be inspected8 . Therefore currently no action is performed to fixthis anomaly.

4.5 Sensitivity

Simulations show that slight variations in the electric field cause large variationsin the recombination rate. Due to this effect, small discretization errors indevice simulators are magnified and it causes significant differences in outputsfrom Pisces and Curry. To explain this behaviour, the sensitivity with respectto the electric lied has been investigated and is described next.

In general when a function f depends on quantity z the sensitivity is definedby

s = :.. 8ff 8z

and can be used to show the influence of relative changes in z on !

Do! ~ s Doz.! z

(4.33)

(4.34)

(4.35)

Applying this on the tunnel model, a<suming considerable tunnelling, i.e. r ~ 1and assuming for simplicity r n = r p = r, we get

6Rtrap 6rRtf'ap = r

8Due to the triangular mesh there is no order in internal storage that simply reflects thegeometrical order 50 that efficient short-cuts are possible.

29

and

tlf tl£r = Strap £'

where, similar to (4.33), the sensitivity is

8fEStrap = 8£ I'

resulting in

(4.36)

(4.37)

Strap =

1+ 2 (£)2~ to

al + 2a21 + 3u3t2 1312! ~EI kTalt + azt2 + a3 t3 2

X ( IX + _1 ) + (1 + £1 ~E)V J;.,(X 2 3:z: kT

low field

high field.

(4.38)

For low fields the sensitivity increases with the electric field and is has a maxi­mum when £ reaches a value of £0 JD.EfkT. Using ~E/ kT = 20, the maximalvalue is then about 14. The sensitivity for high fields is mainly determined bythe last term, that decreases with increasing electric field. So it's maximal whenx ~ 1, with a value of about 17 and decreases after that. In case of band-to-bandtunnelling the sensitivity is

(4.39)

which decreases with increasing fields. Because of the high value of [bbh thissensitivity is considerable. In the case of an electric field of about 1 x 106 Vfcmthis results in a sensitivity of about 20. Altogether we see that sensitivities rangefrom 10 to 20. This means that small variations in the electric field have a largeimpact on the resulting recombination. Suppose that due to discretization andtruncation errors an error of about 1% in the electric field occurs, then thatwill result in an error of about 20% in the recombination. Slight variations inthe doping profile will also be responsible for deviations in the electric fields,causing similar changes in the tunnel recombination. As a consequence theabsolute accuracy of the tunnel recombination and its induced currents is low.

On the other hand, if it's the purpose to find the electric field that causesa certain amount of tunnel originated recombination, this can be done quiteaccurate.

30

Chapter 5

Mobility

5.1 Introduction

In the last decade it has been shown that there is a difference between minorityand majority mobilityl. In fact it turns out that the minority mobility is sig­nificantly larger than the majority mobility in heavily doped Silicon. A model

that describes the minority and majority carrier mobilities correctly, is of greatimportance for bipolar transistors, since the collector current mainly depends onthe injected minority current into the base.

Analytical fit functions have been proposed to describe the minority carriermobility [7, 8]. However in a device simulation program it's not attractive toswitch abruptly between different models for majority mobility and minoritymobility, so a single model is desired that addresses both minority and majoritymobility. Shigyo et al. [9J presented a model based on empirical expressions formajority and minority carrier mobility as function of donor and acceptor concen­tration. However, this model is restricted to room temperature and electron-holescattering is not taken into account.

Recently a model is formulated [17], based on physical effects. The modelincludes

• Various scattering mechanisms: lattice, donor, acceptor and also electron­

hole scattering.

• Mobility degradation at ultra high doping effects, as has been found byMasetti [11], is expressed with a "cluster function".

• Temperature dependence of the various physical mechanisms, resulting in amuch stronger dependence on temperature of the minority carrier mobility.

Velocity saturation is not included in the model, usually this is done with aseparate field degradation factor (see appendix D).

1 For electrons majority mobility means the mobility in n-doped material and minority mo­bility is the mobility in p-doped material. For holes, it's just opposite.

31

In the next section the model is briefly described, for a more detailed de­scription the work of Klaassen has to be read [17, 18]. In section 5.3 the modelequations are completely given with consideration to implementation issues. Animproved, unified, algorithm for the determina.tion of the quantity Pmin (seesection 5.3) is derived in section 5.4, enabling a single algorithm for electron andhole mobility. Precautions to avoid numerical runtime exceptions are consideredin section 5.5. The relation between mobility and bandgap narrowing in bipolardevices is regarded in section 5.6.

5.2 Overview of mobility model

The mobility model is set up in such a way that in case of majority mobil­ity in uncompensll.ted ma.terial it conforms with the expression suggested by

Masetti [ll]

/1max - /1min

/1 = /l-min + 1 + (NIN )"'1Tef,l

11-1 (5.1)

where N is the impurity concentration and Ji.=in, Ji.max, /-Ll, NTe!,i and Qi aremodel parameters. In figure 5.1 the mobility is plotted with this relation forelectrons and holes. The third term at the right hand side of equation (5.1)

Masetti J n ­Caughy-Thomas I n -,_.­Masetti, p .....Caughy-Thomas, p _.. _.-

1600

1400

1200

-;:> 1000.....'"e.£

BOO..,.,.........,.,600.&J

0II

400

200

--~'--'--" --~~-_._~._~-.._- .....

.................,

..................__ _-=---_._._ ..-;.,.-,,..,,..,, , ,oL--'-.....L...---'-...J_~.....L-~~_............u-~~=S~~;;;J

1E+14 1E+15 1E+16 1£+17 1£+18 1£+19 1E+20 1E+21 1E+22impurity concentration (cm-3)

Figure 5.1: Majority mobility versus impurity concentration

15 of significance only at very high doping levels. Ignoring this term leads tothe Caughy-Thomas expression [10] for majority mobility, which is plotted in

figure 5.1 too.

32

5.2.1 Lattice Scattering

For simplicity we regard the case of electron mobility in this section, hole mobilityis fully analogous, just the role of acceptors and donors has to be exchanged.

Lattice scattering is obtained by the low concentration limit of equation (5.1)

I-ln,lat = I-lmax'

5.2.2 Donor scattering

(5.2)

Donor scattering is derived in such a manner that for majority carriers it willresult in the Masetti expression (5.1), but it also includes the effect of screening

P-n,D(ND, n+p) = Ji-~ax (Nref,n)o<nJi-ma:r - flmin N D

+ }lminflmax (nN+DP).flmax - Ji-min

(5.3)

(5.4)

5.2.3 Acceptor scattering

Acceptor scattering resembles donor scattering, however acceptors represent re­pulsive potentials while donors are attractive potentials for electrons. This canbe formulated with

r _ Ji-n,D(ND = IVA, n+p)Ji-n,A(fl. A, n+p) - G(P)

where G(P) is the ratio between collision cross sections of attractive (donor)and repulsive (acceptor) potentials. The partial-wave method has been used toobtain the cross-sections for linear-momentum relaxation [17J.

G(P) = crT,repcr r,atlT

(5.5 )

and P = 4k2r~ where k is the wave vector and ra is the Debye screening length.The G-function is plotted for several temperatures in figure 5.2.

5.2.4 Electron-hole scattering

Electron-hole scattering is described in the same manner, noticing that holescan be regarded as "moving donors" and thus

Ji-n,p(n,p) = fln,D(ND = p, n + p) F(P), (5.6)

where the mobility ratio F(?) accounts for the difference in scattering betweenthe moving holes and the donors that have infinite mass, and is calculated us­ing the Brooks-Herring approximation. A plot of the F-function is shown infigure 5.3.

33

1.2n---~--r--~----,---~~...---~_'_""",--__-"

1.0

p

Figure 5.2: G versus P at several temperatures

8. 0 rr-----,---~----,---..--.,.._----.-...,....---~__.

5.0I

4.0 ~.

rnl/m2-l.00 ­ml/m2-1. 33ml/m2-0.67·····7.0

6.0

3.0

2.0

1.0

".".........

'"

...,.., ,

- ..=.~..""'"'::._._------:1

10000100010010O. 0 ~---'-~-'---~-"'-'---~~-'----~~........._-~---'-'

0.1p

Figure 5.3: F-function versus P for several rnl/rn2 ratios

34

5.2.5 Ultra high doping effects

Ultra high doping effects, represented by the third term of the Masetti equa­

tion (5.1) are modelled as a clustering of impurities. This results in an impurityof Z electronic charges, with a concentration of N /Z, where N is the impurityconcentration. Because collision cross-sections are proportional to Z2, (Le. see[17)) in formula's like (5.3) the quantity fit has to be replaced with Z x N.

A plot of the cluster function Z versus the impurity concentration is shownin figure 5.4

6.0

5.5

5.0

4.5...0., 4.0III""'.. 3.5...," 3.0"...u

2.5

2.0

1.5

1.0

Donors ­ACCQptors _•.•.

lE~20 lE+21 lE~22

impurity concentration (cm-3)

Figure 5.4: Cluster function Z versus impurity concentration, for acceptors (B)and donors (As, P)

The resulting minority mobilities for electrons and holes are shown in fig­ures 5.5 and 5.6.

5.3 Model equations

The next section provides expressions that are valid for electron and for hole mo­bilities. Therefore a notation of indices is introduced with the following mean­ing. In case of electron mobility the index 1 is used to refer to electron relatedquantities, such as effective mass of an electron, electron concentration, donorconcentration. The index 2 then refers to hole related quantities. In case ofhole mobility, the indices are swapped. The quantity Ci is used to denote carrierconcentrations. Some examples are shown in table 5.1. Although this addedlevel of abstraction makes things a little harder, it enables the possibility to get

35

1400 r--~~"""T--~-.,....--~~r--~~""""--~~-r---~~

1200

1000

800

EOO

400 I

200

--- minority carrieremajority carriers

1E+20 1E+211£+17 1E+18 lE+19impurity concentration (cm-3)

1E+16

o '-_--'_.........__..J_~.J___~_L...__ __'_.........__..J_~.........__- ...--_--~-=-

1E+15

Figure 5.5: Comparison of electron minority and majority carriers mobility

minority carriersmajority carriers

500

450

400

" '"I~N

0 300

>..~ 250.....~.!l

200 I""0•150 l•....

0.s:

"'fSO

0lE+15 lE+lE 1E+17 1£+18 lE+19

impurity concentration (cm-3)1E+20

-.._- ..

1E+21

Figure 5.6; Comparison of hole minority and majority carriers mobility

36

(5.7)

Table 5.1: Example of used notation

symbol electron mobility hole mobility

Jl. Jln JLp

C1 n p

C2 P n

N1 ND NAN2 NA NDml me mh

m2 mh me

single unified expressions for holes and electrons, that can be used efficientlJ' inprogram code.

The clustering of the impurities is described by

ZI(NI) = 1 + 1 2'

CI + (NJJf,l )

Here CI and Nre!,J are model parameters with values listed in table 5.2. Thecluster function should be applied to the (non-clustered) impurity concentrationNIa with

(5.8)

(5.9)

Note that at low impurity levels ZI --' 1 and clustering has no effect.

Table 5.2: Cluster parameters for donors (As, P) and acceptors (B) in Silicon

Parameter Value

NreJ,D 4.0 X 1020 cm-3

CD 0.21

Nre!,A 7.2 X 1020 cm-3

CA 0.50

In case of weak screening, i.e. when n+p --+ 0, the Brooks-Herring parameterP --+ 00. This problem is circumvented by replacing P by a weighted mean basedon the Brooks-Herring and the Conwell-Weisskopf approach

(few iBH )-1

P(NSCl n + p) = Pcw(NJc

) + PEH (n + p) ,

37

where few and fBH are weighting parameters listed in table 5.3, and

N se = N A + N D + C2.

Table 5.3: Weight factors

(5.10)

Parameter

fBH

few

Value

3.828

2.459

For Silicon the Brooks-Herring approximation results in

P ( ) 1.36 X 1020 (m1 ) ( T ) 2BH n+p = - - ,

n + p mo 300(5.11)

where ffi1 and mo represent the effective mass of the carriers and the electronfree mass respectively. The Conwell-\Veisskopf approach results in

(5.12)

(5.13)

To make sure that only two-body scattering is taken into account with thenearest-scatterer, the impact parameter has to be limited to half the averageseparation distance, thereby using lYse instead of loll in (5.12).

Now the G function, denoting the ratio of repulsive and attractive collisioncross-sections, is

$1 85

G(P) = 1 - [ P (!llilL)S4]S3 + (p (!!ll.300)37J'SG'82 + m 1 300 mo T

The function G reaches a minimum at values of P between 0.1 and 1 (see fig­ure 5.2). This minimum Pm;n stems from the fact that collision cross-sectionfor an attractive potential reaches a maximum as a function of the temperature.This maximum is related to the Rarnsauer effect [18] which is not incorporated inthe Brooks-Herring and the Conwell- \Veisskopf approximations. Therefore, thiseffect is to be eliminated from expression (5.13). This can be done effectively by

replacing G(P) by G(Pmin ) for values of P smaller than Pmin'In table 5.4 the parameters 81 ... 57 are listed. For the F -function, used

to describe the electron-hole scattering, the next relation holds

(5.14)

where ffil and m2 have the meaning as is listed in table 5.1. The parametersTl ••. T6 are listed in table 5.4.

38

Table 5.4: G- and F-function parameters

Parameter Value Parameter Value

T1 . 0.7643 51 0.89233

r2 2.2999 82 0.41372

T3 6.5502 83 0.19778

T4 2.3670 54 0.28227

r5 -0.8552 55 0.005978

r6 0.6478 56 1.80628

87 0.72169

Following the same approach for the collision cross-sections as with theConwell-Weisskopf expression for Pew (5.12) to ensure that only two-body scat­tering is taken into account for the collision cross-sections, it is possible to obtaina relative simple expression for the mobility due to donor, acceptor and electron­hole scattering. After introducing the quantity

N.c,ejJ = N 1 + G(P)Nz + F~~) (5.15)

this results in

N.c (lv'ref)Cr (n+ p )IlD+A-f- C2 = Ji,v-j\f T + Ilc -N '

.c,ejJ l.c Jc,efJ(5.16)

where

(5.17)Jiz ( T )30-15mal'

Ill\' = -Ilmax - i-LTTlin 300

and

(3~0)O.5

IlminJimazIlc =

fLmax - Ilmin(5.18)

The lattice scattering is described with

(300)8

Illat! = Ilmax T (5.19)

The total mobility can be constructed using Mathiessen's rule

-1 -1 -1Iltot = Jl'at + Jl D+A+C2' (5.20)

Values of parameters are listed in tables 5.5 and 5.6.

39

Table 5.5: mobility parameters

Parameter

f)

rn

Electrons

2.285

1.000

Holes

2.247

1.258 rna

Table 5.6: mobility parameters, dopant dependent

Parameter As (n) P (n) B (p)

J-Lmin 52.2 86.5 44.9 cm2jVs

J-Lmaa: 1417.0 1414.0 470.5 cm2jVs

Nref 9.68 x 1016 9.2 x 1016 2.23 X 1017 cm-3

a 0.68 0.711 0.719

5.4 Determination of Pmin

5.4.1 Approximation method

Expression (5.13) is only valid for P ~ Pmin, where Pmin is the value of P forwhich (5.13) is minimal. Originally an approximation for Pmin was suggested byKlaassen [20J but the latter appears to have two disadvantages, caused by thedependence of G(P) on the carrier effective mass:

• Two separate formulations were needed, one for electrons and one for holes .

• Parameter values, including effective masses, were substituted inline, mak­ing it impossible to use other values when that appears to be necessary.

An alternative approximation for Pm in will be given next. This new approxima­tion does not suffer from these drawbacks. At the minimum

dG =0dP

holds. Differentiating (5.13) leads to

~~ = 81 83 (S2 + Pt"4 )-"3- 1 t"4 - 85 86 (PC"7) -"0-1 C"7 = 0,

where

rna Tt=--.

m1 300

40

(5.21)

(5.22)

(5.23)

(5.24)

After some rearrangements, this can be written as

(

8 8 ) • ~ 1 '3 +1~t34-!6n 6 p2=(82+Pt!4)2'6+1,$5 8 6

At the left hand side of (5.24) we see a true quadratic expression in P, but atthe right hand side (RHS) this is not the case. It should be noted that theexponent at the RHS 2~: t~ :::::l 0.854, 80 that the RHS is nearly linear in P. Agood approximation of the right hand side can be made with a second orderTaylor series around a certain Po. Denoting the RHS with Irh3 and the Taylorseries approximation with irh3 we get

( ) ()I l::..p2 II (

Irh! Po + LlP ~ Irh! Po + LlPlrh3(Po) + -Z-frh. Po)

= irh.(PO + LlP). (5.25)

Substituting irh. in place of frh3 in expression (5.24) leads to a true quadraticexpression

(5.26)

where

q = 2$3 + 1.$6 + 1

The above equation can be solved simply, resulting in

(5.27)

(5.28)

(5.29)

(5.30)

(5.31)

and thus Pmin simply is found from

Pmin = Po + LlP. (5.33)

Now the solution with the plus-sign must be used (the minus-sign results innegative, unphysical, P-values). Equation (5.32) is a rather complex one, andit would be economical to find a simpler expression for l::..P. Fortunately this is

41

possible. In the root of (5.32) the term (2C/Po - 1I)2 dominates and the rootcan be approximated using a well-known first order Taylor series expansion

v'a + b ~ va (1 + fa) if a» b.

After applying (5.34) to equation (5.32) the much simpler form

!:l.P ~ fo - Cz P62CrPo - II

(5.34)

(5.35)

results, and it can be noted that the second order derivative of f.rh~ no longer isnecessary.

5.4.2 Choosing Po

The best results are obtained when Po is close to Pmin' Inspection of figure 2 ofref [17], the curves shown in fig (5.2) and several numerical observations lead toan empirical guess

Po = 0.15 + 0.15 t (5.36)

that gives good results. In numerical calculations over the temperature rangefrom 100 K up to 500 K, Pmin is evaluated up to five digits accurately with theabove mentioned method, three to four digits more than the earlier suggestedapproach (20J. It should be noted that, since the slope of G near P = Pmmis very low, small errors in Pmin result into much smaller errors in G(Pmin)'In the case of the described met.hod, G(Pmin) has been obtained up to 8 to 9digits accurate, while the original suggested approach is still up to 3 to 4 digitsaccurate. The choice of Po turns out to be not very critical, setting Po = a leadsto results that are not as good as obtained with (5.36) but they are still at leastas good as the original method.

Resuming, it can be stated that a more flexible and unified expression forPmin is found, that enables the model to be expressed in single algorithm forelectrons and holes. This is achieved without loss of accuracy.

5.5 Exceptions

Similar to section 4.3 cases with extreme parameter values are considered here,that cause numerical problems when no care is taken.

5.5.1 Expression for P

In the Brooks-Herring expression (5.11) a division occurs with the sum of thecarrier densities in the denominator. This might result in a division exceptionwhen the carrier concentrations are very low. Similar problems arise with theConwell-Weisskopf approximation (5.12) when N 3C becomes very low. This can

be remedied by using the inverted quantities 1/PBE and 1/Pew.

42

5.5.2 Cluster function

When .TVr ~ 0 evaluation of expression (5.7) results in a division exception. Thiscan simply be remedied by multiplying the numerator and denominator of (5.7)with .TV}, resulting in

(5.37)

5.5.3 Derivative of IlD+A+C2

The derivative of IlD+A+C2' i.e. expre55ion (C.10) li5ted in appendix C, containsa division by n + p. To prevent problems, the Pisces variable denoting thesmallest safely usable real, MINDBL, is added to this term. Currently MINDBLis 1 x 10-38 .

5.6 Bandgap narrowing

Bandgap narrowing is usually modelled with a formula like equation (2.13),

and the model parameters Vo, No and Co are obtained empirically. It's noteasy to measure Lhe apparent bandgap directly. Most often collector satura­

tion currents are obtained with special test structures, in which case p n~c isdetermined. Depending on the used mobility, the bandgap narrowing can becalculated. The original Slotboom and de Graaff [6J parameters were obtainedby using the majority mobility. When one takes in account that collector cur­rent mainly depends on the minority current, and uses the Klaassen [17J mobilitymodel, other bandgap narrowing parameters result [19], see table 5.7. A resultof applying the new mobility model is that the bandgap narrowing is consideredto be independent of temperature.

It should be stressed that the bandgap and mobility model are tightly cou­pled. Using old bandgap parameters and the new mobility model, or vice versa,result in inconsistent calculations.

Table 5.7: Bandgap narrowing parameters for old (no difference between major­

ity jminority mobility) and new mobility models

parameter old new

Vo 9.0 6.92 mV

No 1.0 x 1017 1.3 X 1017 cm-3

Co 0.5 0.5

43

Chapter 6

Software implementation

6.1 Introduction

A device simulator program is a complex piece of software, for example the origi­nal Pisces-2B, version 9033, contained 490 fortran modules. Modification of sucha program must be done with care to retain a level of reliability. Apart from theprogram being large, the expressions it contains are complex, for example see theexpressions listed for the derivatives in appendices A and C. To prevent unnec­essary errors testing has been carried out. Several tests that were performed aredescribed below. Beside testing, some decisions about implementation detailsare discussed.

6.2 Compiling

The version of Pisces as it exists at TMA, is written in ratfor, this is a higherlevel computer language based on fortran. After processing a ratfor programwith a ratfor pre-processor a fortran source program results. The fortran codegenerated by TMA, is currently used at Philips Research Laboratories.

In the fortran computer language it is not necessary to specify explicitly thetype of a variable. The type of an unspecified (untyped) variable is determinedby the first letter of its name. It is strongly recommended to avoid this feature,because simple typing errors result in generation of "ghost" variables and er­roneous program behaviour. The used HP FORTRAN 77 compiler fortunatelyhas the option to check and warn for untyped variables. All modified and newlyadded modules were checked with this option.

Beside this option, another option was used that warned against the use ofconstructs that are not available in standard ANSI FORTRANI.

lOften compilers contain extensions on the standard language definition. Using these ex­tensions results in programs that are not easily to transfer to another computer system withanother compiler that doesn't recognize these specific extensions.

44

6.3 Numerical tests

The used analytic expressions are very complex, this means that

• Errors easily occur, for example caused by mistyping.

• Occurred errors are very hard to detect. It is nearly impossible just to lookat a formula or some lines of fortran code and see whether it is correct ornot, when the length of the total expression by far extends the length of aline.

Confronted with these facts, it is necessary to perform some validity tests. Partlythis is done by testing analytical expressions, that are complex, against numericalapproximations that are relatively simple2 .

The analytic expressions for the tunnelling field-effect factor r, the minimumof the mobility G-ratio function and the derivatives of the recombination andmobility expressions have been checked against numerical approximations. Thederivatives are simply checked by approximations based on difference quotients,using small intervals. This is not done for the final expressions of the derivativesonly, like op/on, but also for several sub-expressions, like BO/on. This has beendone to prevent that possible errors, causing no significant deviation in the finalexpressions under test conditions, would not be detected.

The integral that defines the field effect factor r, equation (4.6), is approx­imated by a midpoint sum where the summation intervals are generated adap­tively [38].

The applied methods resulted in the detection of a couple of mistakes in thederivatives, showing the profit of this method.

The errors mainly originated in the cumbersome derivatives based on theoriginal description of the trap-assisted tunnelling with the Kn,p parameter [15J.This initiated the description as given in section 4.2.1 that led to simpler deriva­tives. Using the new expressions for the derivatives solved the problems and theresulting Pisces version shows good convergence. That the original descriptionresulted in tricky derivatives is illustrated by the fact that the Curry implemen­tation, which is based on the original description, also contained errors (up toversion 8.6) in the same derivatives. As a consequence Curry showed poor or noconvergence.

Because only a limited number of small errors have been found after extensivetesting, the code is assumed to be quite reliable.

The midpoint sum method is also used to analyze the weak spots in theanalytical approximation for the field-effect factor r, resulting in the improvedapproximation that is discussed in appendix B.2.

2These numerical approximations are not used under normal program operation becausethey spend much more computing time. For the tests as described above, this is not crucial.

45

..

6.4 Comparison with Trap and Curry

Beside numerical testing, the results of the Pisces implementation have beencompared with the results of Trap [3] and Curry [41]. The base contact in Trapis modelled as a special type of contact: the quasi-fermi potential of the majoritycarrier is set equal to the applied base-bias while the electrostatic potential andthe quasi-fermi level of the minority carriers are not set to a fixed value. Thistype of contact does not exist for Pisces. It has been tried to use a Schottkycontact for the base, setting the recombination velocity for the majority carriersto a high value. However it failed totally to achieve similar solutions at highinjection, and at low injection significant differences showed too. Therefore itwas not possible to compare Pisces with Trap.

Next, comparisons with Curry are described. Running equivalent simulationswith Pisces and Curry must result in nearly3 the same results. Testing of themodels has been performed on test structures, for simulation purpose only. Thetest structures included:

• Homogeneous doped, rectangular silicon samples. Useful for checking pn­

product. Because Curry and Pisces both output the electron and holeterminal currents, it is possible to obtain and compare the minority andmajority mobility for this simple case.

• One dimensional short and long pn-diodes. The diode is considered shortwhen the distance bet.ween junction and contacts is small compared to thediffusion length, for long diodes the opposite holds.

• Large lowly doped transistor, with low fields in the junctions, and thus nosignificant velocity saturation,

• Small heavily doped transistor, with high fields in the base-emitter junc­tion, causing significant velocity saturation.

'When a difference in the output occurred, the source of that difference hadto be located. This was performed by hand calculations for several nodes of themesh.

Although no real program errors in the implemented models were found,other crucial bugs or features showed. The most important ones are listed here:

• Differences in settable parameters like bandgap and bandgap narrowingwere found, caused by different model descriptions in Curry and Pisces.This forced me to set up a parameter set for Pisces so that simulationconditions are equivalent to Curry. This parameter set is described inappendix D.

3 Small differences are allowed due to round-off errors and different discretization schemes.Usually a difference of less than 1 percent is considered acceptable.

46

..

• A bug showed in the Pisces bandgap narrowing code. Bandgap narrowingwas calculated as a function of the Det dope, IND - N A I instead of the totaldope ND + NA. This caused different effective intrinsic carrier densities nie

in heavily doped junctions, like the base-emitter junction in a small-scaledevice.

• Pisces doesn't calculate the velocity saturation correct if the current den­sity becomes very low, about J.L x 1.7 x 10- 19 A/rnicron2 at room temper­ature. As a result, it happened that Pisces and Curry calculated the samesolution of 'l/J, n and p, displayed equal total terminal currents, while thedisplayed minority carrier terminal currents, that were very low, clearlydiffered.

• Plotting or printing recombination with the tunnel model resulted in erro­neous output in Curry due to an error in the postprocessing routines.

• Differences occur due to different models for velocity saturation. In Piscesthis is a function of the electric field, while in Curry this is a function ofthe quasi-fermi levels (see appendix D). The differences become significantin heavily doped junctions that are forward biased or when ohmic contactsare within several diffusion lengths from a junction. In the latter case

strong recombination occurs at the contact, causing strong bending of thequasi-fermi level of the minority carrier, implying considerable saturationeffects with Curry. The electrostatic potential is not changing that muchand no significant saturation occurs with Pisces.

• Due to the high sensitivity of the tunnel recombination rate with respect tothe electric field it is hard to obtain nearly the same results, when tunnelrecombination contributes significantly to the solution. Small differences inthe electric field of about 0.3%, caused by different discretization schemes,are amplified up to differences of about 6% in the resulting recombination.Knowing that this is inherent to the tunnel model, this has to be accepted.

6.5 Robustness

The implemented expressions must be robust against extreme values. Apart fromthe actioDs described in section 4.3, care is taken that no problems occur whennon-physical values are presented. This occurs when a solution is constructed:before the final iteration, the values are not correct. In Pisces this means thatcarrier concentrations can have a zero value. All new code is set up to take tillspossibility into account.

47

1

6.6 Implementation notes

6.6.1 Module layout

PISCES

"SOLVR

1\MN

EPAEN~!,\SOLVSS SOLVT

I X~ASSl\ffiJ RECOMB _ .. tungarn

/\G"\TNEFDSID DMOBL........ Jonab

ASSMB

Figure 6.1: Part of Pisces calling tree

The part of the pisces calling tree of interest for the model routines is shown infigure 6.1. The main module pisces calls the solution module, solvr when asolution statement is processed. Depending on steady state- or transient analysisone of the modules solvss or solvt is called. Similar, depending on the solutionmethod the gummel or newton module is called. Both methods require theassembling of matrices that represent the set of equations to be solved. Theassembling of matrices is performed by the module assmb. This module callsfor every element of the mesh the module elemn that supplies the values of thebasic equations and their derivatives for an element. Elemn calls several modelroutines, e.g. recomb for the recombination. The current along a. side is obtainedby calling assmbj that uses the mobility models that are provided by dmobl.

In the original Pisces version the calculation of Shockley-Read-Hall andAuger recombination was implemented in the module recomb. This moduleis called during the solution phase of Pisces, as is illustrated in figure 6.1, butalso when output is generated during post-processing phase, because the recom­bination is not stored, but recalculated with use of the stored solution (1/J, n,p).It was therefore advantageous to expand the existing recombination module

1. Trap-assisted tunnelling is closely related to conventional SRH-recombination.

2. Using the existing recombination module, no a.dditional calls have to beadded for the existing output statements, a.nd no new user output com­mands have to be added.

The parameter list of the recomb subroutine is expanded, caused by the Dumberof parameters needed by the tunnel model. These additional parameters require

48

changes in the calling routines.The algorithms to compute the field-effect functions r nand r p are quite

similar and have been unified into the single subroutine tungam. This routinecalculates the appropriate r depending on the parameters provided.

The mobility model, although much more complex compared with the exist­ing models, did not require major changes in the module structure or parameters.The model is coded in the subroutine kmob that is called by dmobl that takescare of the other mobility models.

6.6.2 Activation of the models

Extending the Pisces user command language is not easy. Although the key­words for statements and corresponding parameters are stored in a text file thatcan easily be changed, the number of keywords is protected via a coded binaryfile. Furthermore each keyword is associated with a number used to identifythat keyword. Adding new keywords demands the addition of correspondingnumbers. This however is tricky, because at TMA new features are added too,making it hard to use the numbers for identifying the keywords without interfer­ing with their new additions. To enable some setting of models and parametersa temporarily ad hoc method (a "quick & dirty" method) is implemented. Whenthe implementation is transferred to TMA, it must be adapted, so that the newmodels are available with the normal Pisces command language.

Currently, in the quick & dirty method the new models are activated withthe TITLE command. When the title string contains a special format, it isinterpreted to activate the appropriate models.

The format of the title string is

TITLE "\#### rest of title string"

where #### represent four decimal digits. The meaning of the digits is positiondependent, from left to right the meaning is:

. 1. Low field mobility parameter.

• 0 = Conventional Pisces (ANALYTIC, CONMOB, ... )

• 1 = Masetti with carrier-carrier scattering like Curry (QUAFERM)

• 2 = Klaassen's multi scatter model

2. Tunnel model parameter

• a= Conventional Pisces (no tunnelling)

• 1 = tunnel model, incorporating trap-assisted and band-to-band tun­nelling

3. Band-to-band proportionality factor B

49

• 0 = B equals default value, i.e. 4 x 1014

• 1 = B equals zero, i.e. switch off band-to-band tunnelling

4. Calculation of field-effect factors rn, rp

• 3 = Calculate both r nand r p

• 2 == Calculate only r n, set r p to zero

• 1 = Calculate only rp, set rn to zero

• 0 == Calculate none, set both rnand r p to zero

The Curry-alike mobility model has been implemented to facilitate testingand comparing Pisces and Curry results.

Examples of title commands to activate implemented models

TITLE "\0000 conventional Pisces"TITLE "\1103 both trap assisted and band-band tunnelling"TITLE "\1113 only trap assisted tunnelling"TITLE "\1100 only band-band tunnelling"TITLE "\2000 advanced mobility"

50

Chapter 7

Measurements

In this chapter the implemented models are used to simulate fabricated devicesand to compare the results with measurements. Included are results from

• A gated diode structure. The behaviour of the reverse leakage currents ofsuch a device is quite adequately modelled with the tunnel model.

• A lateral multicollector bipolar transistor. With this device it is possible toinvestigate the transport of minority carriers in the base region accurately.It is therefore used as case for the implemented mobility model.

• QUBiC bipolar transistors. With these devices, the mobility model isregarded for collector currents.

7.1 Gated diode

The operation of a gated diode is first described for a simple case, shown infigure 7.1. The device consists of a homogeneous doped p-type bulk, a n+ -type

lp

...L

I\{;

Figure 7.1: Example of a gated diode structure.

contact region and a gate located on top of an oxide layer. It is assumed that the

51

bulk is connected to ground potential and the n+-region to a positive voltage,e.g. 1V. Thus the junction is reverse biased and only the gate voltage will bevaried.

According to Grove [30] the behaviour of the reverse current has to be con­

sidered for three cases:

1. Accumulation. For low gate voltages the surface under the gate will be ac­cumulated, and the current originates from generation within the depletionregion of the metallurgical pn-junction.

2. Depletion. At moderate biases, the surface is depleted, and recombination­generation centers at the surface will contribute to the reverse current,showing a significant increase in current.

3. Inversion. When the surface under the gate is inverted, i.e. for high gatebiases, generation in the depletion region in the bulk below the gate, butno longer at the surface, contributes to the current. The current will beless than with the depleted surface, but due to the depleted bulk it is largerthan in case of accumulation.

Grove assumes that the contribution of the surface depletion is constant, regard­less of the gate bias, and only the contribution of the bulk generation increases,due to the growth of the depletion region until the surface layer is inverted.This behaviour is shown in figure 7.2, which shows a characteristic "hump" inthe IV-curve when the surface is depleted. It should be noted that the humpcontains a (nearly) fiat plateau (the small increase is due to the growth of thebulk depletion region).

IK

(lin)

Accumulation Depletion---... II ..

In_ersioo

L.... V p'"

Figure 7.2: Reverse current of a gated diode with an uniform doped bulk, ac­cording to Grove [30].

Next the measured device is considered. It is similar to the gated diodedescribed above, but it contains an additional p+-region, see figure 7.3. This

52

1p

/'i

Figure 7.3: Structure of measured gated diode.

structure is used to investigate leakage currents in CCD-image sensors. Followingthe arguments of Grove, expecting a "hump" in the IV-curve when the surfaceis depleted, now two humps are expected. The humps then correspond to thebiases at which the p and p+ surface regions are depleted respectively. Results ofmeasurements on several devices have been provided by H.L. Peek and three ofthese devices will be discussed. Properties of the devices are listed in table 7.1.

Table 7.1: Properties of measured gated diode structures

property value

n+·dope 1.4 x 1018 cm-3

bulk p-dope 3 x 1015 cm- 3

oxide thickness 100 nm

area p-region 1.85 x 10-3 "cm-

area p+-region 0.35 x 10-3 cm2

device #1 p+ -dope 7.0 x 1017 cm-3

device #2 p+-dope 2.8 x 1017 cm-3

device #3 p""-dope 1.4 x 1017 cm- 3

In figure 7.4 results a.re plotted for the three devices. Although the currentsare noisy for the lowest values (it is very difficult to measure these small currents!)indeed two humps can be identified. Note that at increasing doping levels ofthe p+-regions the humps shift to higher gate voltages. This agrees with theassumption that the humps denote a depleted surface: the increase of dopinglevels require an increase in ga.te bias to reach deep depletion and inversion.However calculations show that the surface is depleted already before the large

53

7.4 9.5 It6

VG (VOll)\,1 ~.J

,

..,,

······ \'.- .-'..,'...... \, ".,' -\)..~.

13.7 15.8 17.9 20.0

Figure 7.4: :Vfeasured reverse currents for devices #1, #2 and #3.

hump really shows, e.g. for device #1 the surface is depleted at about 4V, whileat about 8V the surface is at weak inversion (i.e. surface concentrations areabout n,e)' Thus the "hump" occurs mainly in the deep depletion regime.

~lore observations can not be explained with Grove's arguments. Clearly thehump related to the depletion/weak inversion of the p- surface region is muchlarger then the other hump, although the corresponding surface is much smaller.The shape of the large hump remarkedly differs from what Grove expects, itclearly contains not a. nearly fiat plateau (note that a logarithmic scale is usedalong the vertical axis!).

Because the electric fields at the semiconductor surface are quite strong, e.g.about 3 x 105 V/ cm when a. ga.te bias is applied of IOV. it is expected thattra.p-assisted tunnelling contributes to the generation of carriers at :he surface.Simulations to investigate this ?ossibility have been performed for device # l.

The device is simulated with the implemented tunnel model and. for comparison.with conventional Shockley-Read-Hall recombination. It is assumed that thesurfa.ce states, resulting in the hump. can be described with a thin surface layerof 50A, with a short minority lifetime, e.g. 5 x 10-8 sec.

Results of this simulation are shown in figure 7.5. The simulation shows thatwith conventional SRH recombination (dashed line) the hump related to the p­

region is a rather small one, and also note that this hump has a fiat plateau. Bothobservations disagree with the measured data. However, the simulation resultsfor the tunnel model (solid line) result in a big hump, without a. fiat plateau.

54

10- 13

~ 10- 14c0u'E"- 10- 15 -~

C / ------~ /

:0 10- 16 /

u /

,- -10- 17

0 4 8 12 16 20Gate Bios (V)

Figure 7.5: Simulation of device #1, with tunnel model (solid curve) and con­ventional SRH (dashed curve).

Thus, due to the high electric fields at the surface, the trap-assisted tunnellingcontributes considerable and the hump is amplified at higher gate voltages. Allthree devices have been simulated with the tunnel model and results are plottedin figure 7.6. The resemblance with the measurements is striking.

In figure 7.7 measured currents are shown for device #1 at temperatures of2SoC and 8SoC. With conventional SRH recombination, the generation rate isproportional to the intrinsic carrier density nie in case of depletion. Increasingthe temperature would then lead to an increase of both humps with the same fac­tor, retaining their shape. However, the measurements show that the little humpincreases somewhat more than the large hump does. This weak temperature de­pendence of the large hump agrees with the theory of trap-assisted tunnelling.The simulation results shown in figure 7.8 agree with the measurements.

7.1.1 Magnitude of currents

It should be noted that the magnitude of the simulated currents is incorrect.The measured structure has an area of 2.2 x 10-3 cm- 2

1 while the simulateddevice has a width of 2 x 10-3 em. Therefore the simulated currents, expressedin A/micron, must be multiplied with 1.1 cm = 11000 micron. This resultsin simulated currents being about 100 times too large. This stems from the

used value of minority lifetimes and the assumption that the traps are locatedexactly in the centre of the energy gap. For example when traps are used thatare located at about 0.11 eV from the intrinsic level, the simulated currents have

55

8 12 16 20Gote Bios (V)

4

\\

I

I

I\

II

I\

10 - 1 7 L......---'-'-~........J_..L.. ....1.!_.- - -....L-~.~-L...-.-.....J.

o

'0-"1C 10- 14

~u'E? 10- 15

Figure 7.6: Simulation of devices #1 (solid line), #2 (dashed line) and #3(dotted line) with tunnel model.

10"

<l's ',:

, ..---------.,\•.

'.

::: ~...

.... _.-".--

10'"-lO II ~.2 5.3 7.4 9~ Ill; 13.7 :5.8 17.9 20.0

VG (VOLT)

Figure 7.7: Measured device #1 at different temperatures, i.e. 2SoC (solid curve)and 8SoC (dotted curve).

56

-,--,\

\\

4 8 12 16Gote Bios (V)

20

Figure 7.8: Simulated device #1 with tunnel model at different temperatures,i.e. 250C (solid curve) and 85DC (dotted curve).

about the same magnitude as the measured currents. The location of the traplevel, and the related temperature dependence of the recombination rate, needsmore investigation.

7.1.2 Resume

It is shown that Grove's descriptions, predictiug a constant surface recombina­tion rate, although commonly used, are not adequate in describing the reversecurrent of a gated diode with a heavily doped p+ region and strong electric fieldsat the semiconductor-oxide interface. Instead it is found that in this case the de­vice behaviour is mainly determined by contributions of trap-assisted tunnelling.

7.2 Lateral multicollector transistor

With a lateral npn multicollector transistor, shown in figure 7.9, it is possible toobtain the parameters that determine the collector currents, i.e. noDn and Ln ,

from a single device [21, 22] accurately. It should be noted that the base consistsof the uniformly doped bulk region. The sma.ll spacing between the collectorsminimize the disturbing effects of surface recombination.

The collector currents strongly depend on f-Ln and nie' These parameters aredetermined by the mobility and the bandgap narrowing model. Because the testconditions are well-defined, this device is suited to investigate the performanceof these models.

57

base

I e~ittercollectors

~

p-substrate

Figure 7.9: Lateral multicollector n-p-n bipolar transistor

The device is biased with a negative emitter voltage, while the base andthe collector contacts are held at ground potential. The measured currents,provided by F. van Rijs [23], are plotted versus the base-emitter voltage infigure 7.10 Simulations have been performed with conventional1 and the new

1.00.4 0.6 0.8Vbe (v)

10- 11

1 0 - 13 '---~'--""--L-.J--'-------'-_..L-~~0.2

10- 1 r-~-r----'-'-r---r--r----"-,

10-31

Figure 7.10: Gummel plot of emitter and collector currents

implemented mobility model (see chapter 5), both used with a correspondingbandgap narrowing model. The minority lifetime Tn, that depends on tbe wafer

1 A model with equal mobility for minority and majority carriers, e.g.QUAFERM [41).

Curry's

58

processing, is set to the value obtained by van Rijs [23]23. Other parameterswere set to standard values (see appendix D). Specific parameters for the devicestructure are listed in table 7.2. The measured collector currents are

Table 7.2: Simulation parameters for lateral multicollector npn transistor

parameter value

substrate dope 7.07 x 1018 cm-3

emitter dope 5 x 1020 cm-3

collector dope 5 x 1020 cm-3

Tn 25.22 ns

Temperature 296.2 K

collector length 200 micron

emitter length 245 micron

10- 10Mo.J/t; codector structure n n

I

r = O.60V

10- 11 ~

~c 10- 12 1-

0 Iv L'E , 3/......~ 10-' '--

C ~~:J 10- 14 >--u

10-15~" """ 1

,~

~ '11

10- 16 / I2 3 4 5 6

collector nr

Figure 7.11: Collector currents at VEE = 0.6V for: measured data (symbols),new model (connected by solid line) and conventional model (connected bydashed line).

compared with the simulated currents for a certain applied emitter bias. The

2The measured lifetime incorporates all recombina.tion processes. Therefore only the SRHrecombination model must be used a.nd Auger recombination must be switched off.

3The minority lifetime has been obtained independent from the mobility [211

59

Ybe = O.75Y

,""

"

coflector nr

Figure 7.12: Collector currents at VEE = O.75V for: measured data (sym­boIs), new model (connected by solid line) and conventional model (connectedby dashed line).

results are plotted in figures 7.11 and 7.12 for base-emitter voltages of O.60V andO.75V respectively.

The implemented mobility model agrees well with the measurements, whilethe collector currents with the old model drop off too fast. It should be notedthat tuning of parameters of the mobilit.y model might improve the conventionalmodel, but that will invalidate the majority carrier mobility!

7.3 QUBiC transistor

For QUBiC, a fabrication process for integrated circuits, the collector currenthas been investigated by using measurement data provided by E.P.M. Bladt. ASIMS doping profile for a bipolar npn transistor is shown in figure 7.13. Thebase sheet resistance is determined by the doping profile of the base region.Measurements of the base sheet resistance for several processed wafers showconsiderable variations. Thus, the doping profile for the different wafers is notexactly the same. Therefore, simulation results have been compared with wafersthat show similar base sheet resistances.

In figure 7.14 the measured current densities at VEE = O.6V, VeB = l.OVhave been plotted versus the base sheet resistance, for the processed wafersE14010-8 and E15118-7. Results of simulations4 with old models for the carrier

40nly I-D Trap simulations have been performed because no 2-D dUIJing profile was

60

OJC­oo

0.10 0.20 0 ..30X [um)

0.40 0.50

Figure 7.13: SIMS doping profile of a QUBiC npn transistor

1.6

1.4

xN

0 1.2 new model, 300K <)

'-~ old model JOOK +

new model 299K D

~ old model 299K x

.,; 1.0 msmt: E14010-8 100M100 ..<)

I msmt: E15118-7 lOOM100 •§ D

..,0.8~

u.,0.6 !

800075005000 5500 6000 6500 7000ba~e sheet re~i8tAnCe (ohm)

4500

O. 4 '--__.L-__--'-__--'-__~_____' .L.....___..!____'

4000

Figure 7.14: Simulated and measured collector current densities at VSE = O.6V.Errorbars with the measurements denote the variation in base sheet resistanceof a wafer.

61

mobility and the bandgap narrowing5 and the new models from chapter 5 areshown for T = 300K and T = 299K in the same figure. A change of 1K resemblesthe accuracy of the control system that had to keep the temperature at T = 300Kduring the measurements. The simulated currents are considerable larger thanthe measured currents. This holds for the simulations with the old and thenew models. However, with the new models the difference has been reduced to'" 50%, while with the old models the difference is '" 100%. The simulated dataat T = 299K show that, in case of a somewhat lower temperature during themeasurements, the differences between simulated and measured currents becomesmaller.

Next it is investigated whether the still considerable difference between mea­surements and the simulation results with the new models can be explained.Possible reasons for the difference might be

• Variations in the doping profile

• Variations in the temperature

In fig. 7.14 it is seen from the simulations that a. variation of 1K in temperatureresults in a change in current of,...., 10%. Thus the scattering of the measureddata points of wafer E14010-8 might be caused by a variation in temperature.However, it is clear that the uncertainty in temperature cannot fully explain thedifference between measured and simulated currents.

Now let us consider the doping profile. Although the base sheet resistanceof the transistors from wafer E14010-8 and the simulated values are the same,it is still possible that the doping profiles are different. The global shape andthe peak value of the base profile might be the same, resulting in equal basesheet resistance, but possibly significant differences exist near the junctions. Asignificant difference in the base-emitter junction would cause a different collectorcurrent.

The doping profile near a junction can be checked with the profile used forsimulation by comparing measured and simulated depletion capacitances [31J.Unfortunately, there are no results of capacitance measurements available forthe QUBiC devices6 .

It has been tried to eliminate the effect of differences in the doping near thebase-emitter junction, which is described next. Due to different base-emitterjunctions, different Ic-VBE characteristics will result, mainly caused by thereverse-Early effect. Like the (forward) Early effect, it results from a changein the effective base width, in this case caused by changes in the width of the

available.'I.e. models that use the equal minority and majority carrier mobility with the associated

bandgap narrowing model.6 Due to problems with the measurement equipment the capacitances could not be measured

when this report was written.

62

(7.1)

base-emitter depletion layer. For a considerable collector current, i.e. whenIn(Jc) is proportional to VEE, the collector current density for VeE = constantcan then be described by

Jc = Jc,oexp (~::),

where Je,O is the collector saturation current density and m, a non-ideality factor,which stems from the reverse-Early effect. Usually m will be close to UUlty. Withtwo measured or simulated bias points and expression (7.1) values for m and Je,ocan be obtained by

and

J = J (Jcl) VBE;~EVEEIc,a cl J

c2(7.2)

(7.3)

where (VEEl, Jed and (VBE2, Jc2) denote the two bias points. For biases ofOAV ::; VEE::; O.6V and VCB = l.OV the non-ideality factor m and the collectorsaturation current le,a have been obtained for the measured and the simulatedtransistors. The results are plotted in figure 7.15.

1.020 ..A

aimul, new, 300J( 0

eimul. old, 300K ...eimul, new, 2991< 0simul, old, 2991< )(

melle, E14010-e ..melle, E15118-? •

)(

1. 015

,."" f..A

e

••

I1.005

•1.000

0.4 0.6

o

0.8 1~cO 11e-l0 A/cm2)

1.2

...

1.4

Figure 7.15: Non-ideality factor versus collector saturation currents for measured

and simulated transistors.

Although the measured m values vary significantly, part of the differencescan be explained by temperature differences of about 1K, as can be seen from

63

expression (7.3) for T = 300K. Therefore it is not clear whether the differencesmainly stem from temperature or from profile variations. Although the measured

6.0

+eooo75005000 5500 6000 6500 7000

ba.e sheet resistance (ohm)4500

4. 0 L.-_---'L.-__L.-__L.-__L.......__.1...-__.1...-__..i.-_~

4000

Figure 7.16: Simulated and measured collector saturation current densities. Er­rorbars with the measurements denote the variation in base sheet resistance.

Jc,O values are close to the simulated values with the new models (see fig. 7.16),i.e. the difference is '" 30%, no further conclusions can be drawn on this data.

It is therefore concluded that with the new models the difference betweenmeasured and simulated currents is at least reduced from"'" 100% to "oJ 50%.

To obtain a more accurate evaluation of the models, it is necessary to

• have sufficient knowledge of the doping profile of the device. Apart fromSIMS measurements, the profiles of the devices should be checked withcapacitance measurements .

• perform measurements at a well-known and constant temperature, I.e.with an accuracy of about 0.1K.

64

Chapter 8

Conclusions

Although the implementation of physical models into a device simulator appearsto be straightforward several notes can be made about the implementation andthe implemented models.

• Pisces produces with the implemented models the same output as Curry,provided that the different model for velocity saturation is not crucial.

• The implemented models behave well, i.e. simulations converge as rapidlyas Curry and no problems occur with numerical exceptions like "divisionby zero".

• To achieve a reliable device simulator, it's necessary to perform extensivetesting, and even then you cannot be sure whether all possible errors havebeen detected.

• The formulation of the trap-assisted tunnelling contains a discontinuitythat can be removed with the suggested description in appendix B.2. Thisformulation is also somewhat more accurate for high electric fields. How­ever, both the old and new formulation, over-estimate the field-effect func­tion r at relative low electric fields.

• Tunnelling is a non-local phenomenon and in the implementation simpli­fied expressions are used that expresses the resulting recombination in localquantities. It should be investigated whether the errors introduced due tothis simplification require special treatment in the vicinity of semiconduc­tor surfaces where the approximation clearly breaks down. This is mainlythe case with band-to-band tunnelling.

• The mobility model is formulated in a way that it can be implementedeasily in a single algorithm that calculates both electron and hole mobili­ties. The used methods allow for a flexible change of parameters like theeffective masses of carriers.

65

• The tunnel model is very sensitive to small changes in the electric field.This makes it hard to test. For simulations the electric field must be knownaccurately, requiring a fine mesh in high field regions. It also implies thatsmall variations in doping profiles of a device might result in considerablevariations in the tunnel originated recombination rate.

The evaluation of the models, performed by comparing measurement andsimulation results lead to the following observations:

• Comparison with measurements show that the tunnel model can be usedto explain leakage currents in gated diodes.

• Comparison with measurements of a lateral multicollector transistor and aQUBiC transistor show that the mobility model improves on conventionalmodels that assume equal mobility for minority and majority carriers.

66

Appendix A

Derivatives of the tunnelmodel

This section contains some tedious formulas, the expressions tend to be longand complex, but they are necessary for the implementation of the model. Forbrevity the subscripts nand P are dropped in cases where no confusion can arise.

A.I Trap-assisted tunnelling

The derivatives with respect to [, nand P are given below. Because r dependson [ and the carrier concentrations, several subexpressions have to be evaluated

(A.I)

(A.2)

(A.3)

oRtrap _ oRtrap I + oR/rap orn a~:- C:;E 'an an r==cs! orn Ow an

oRtrap _ oRtrap I + oR/rap ofp a~:- C:;E •op op r==cst arp 0 kT ap

According to (4.9) and (4.10) the derivatives w.r.t. nand p of I::i.E/kT are

oD.E/kT = { -~an 0

oD.E/kT = { -~op a

7"1,1 ~ 7"1, ~ nie exp (IT/f)elsewhere

PI ~ P ~ nie exp (2tf )elsewhere

(A.4)

(A.5)

And from (4.2) can be obtained that

oRtrap _ Rtrap T: (P + PI)

ern - r;(n+nl)+r:(p+Pl) l+fn '

67

(A.6)

8Rtrap _ RtTap r; (n + nd

8fp 1";(n+nl)+r:(P+Pl) l+fp

8Rtrap I _ P - RtTa.p r;an r==c31 - r; (n + nl) + 1"~ (P + PI)'

8RtTa.pI n- Rtrap r;ap r==C3t = r; (n + nl) + 1"~ (P + PI)'

where

• 1"n pr = I

n,p 1 + r n,p

(A.7)

(A.S)

(A.9)

(A.10)

Note that equations (A.8) and (A.9) alrea.dy had to be calculated with f = 0for conventional SRH recombination.

A.2 Derivatives of r, low field

Because f is independent of b..E / kT

af- = 0 and8n

af = 08p . (A.H)

Derivative with respect to E

8f f ( 2 (E )2)a£ = £ 1 + 3 £0 .

A.3 Derivatives of r, high field

(A.12)

Here we get the real complex expressions, which are broken into practical parts.

First

and

next,

8r 8r I 8f 8toboE = 8/).E + at 8 boE

itT itT I=c.. t itT

of ar I ar ataE = 8£ t=c..t + at 8£'

af _ al + 2 a2 t + 3 a3 t2 fat - al t + a2 t 2 + a3 t3 '

~I =r{~(tlE)-l +1-~}[)6.E 4 kT x '

kT t==C3t

68

(A.13)

(A.14)

(A.I5)

(A.16)

(A.I7)

(A. IS)

(A.19)

A.4 Derivatives, hand-to-hand tunnelling

Luckily some expressions turn out to be simple! From equa.tion (4.20) it's easilyshown that

and

BRbbt = Rbbl {u + Cbbt }

BE £ E(A.20)

where

v = n or P, (A.21)

and

aD fi-ie

an = - (n+nie)2(A.22)

(A.23)

Of course, when D is set to unity a.ccording to (4.24), the derivatives of D areto be set to zero.

69

Appendix B

Field-effect factor rapproximations

B.l Upper limit of rUsing (4.6) expression (4.29) will be proved next. For brevity the subscripts n

and p are dropped and the substitution z = u t1E / kT is made

rAE/kT

[ 2 [0 ]r = Jo

exp z - 3""[ z3/2 dz.

Now z, [ and [0 are all positive, consequently

rAE/kT

r :5 Jo

exp(z) dz = exp (b.ElkT) - l.

which proves (4.29).

B.2 Continuous gamma approximation

(B.l)

(B.2)

An expression for the low-field approximation is derived analogous to the originalone [15] but without neglecting a substantial term. It's possible to write (4.6)as

r = ai 1exp(j(u))du,

where

(B.3)

a = t1EjkT and (B.4)

Tills integral can not be solved analytically. The contribution to the integral ismaximal near the maximum of feu). For a good approximation of the integral

iO

the function f (u) will be approximated in this region with a second order Taylorseries expansion j (u)

- 1 2 1 a ( )2f(u) :::::: f(u) = -ax - -2 u - x 2.

3 4x(B.5)

(B.6)

(B.7)

As stated, near the maximum of f (u) the contribution to the integral is maxi­

mal. The original approach therefore assumed that the other contributions wereneglectable and so the lower and upper limit of the integral were set to -00 and+00. When the maximum is near u = 1 this results in large errors, i.e. r isover-estimated for the highest fields of the low field approximation. To avoidthis, only the lower limit is set to -00, while the upper limit is left unchanged,i.e. 1. Using j(u) instead of f(u) in expression (B.3) and substituting

w = ~ (u - x2)

results in

(1 ) f~(J--x)

r = 2xja exp 3"ax2 J-: Z exp( _w2) dw.

This is the integral of a gaussian, which results in expressions with the comple­

mentary error function. Using x = (£/£o)(l/JO.) leads to

(B.8)

Note that

1. For very low fields, x L 0 and l/x -+ +00, thus the complementary error

function erfe in (B.8) approaches zero, and expression (4.12) results.

2. When x = 1, i.e. on the transition from the 10w- to high field approxi­mation, the complementary error function erfc yields 1 and thus expres­sion (B.8) is fifty percent smaller than (4.12).

3. The high field approximation is derived in an equivalent way, resulting insmooth transition at x = 1 for all ~E / kT values. 1

Using an approximation for the erfc function [12], r results in

r = vrr~ {exp [~ (:J 2]

- 9 (t) exp [- ~ (%J 2 - (a~r+ 2a] } , (B.9)

IThis can be checked by substituting x:::: 1 in equations (4.17) and (B.B) and using ~: =(

1 [ )2x rr; .

71

where

(B.10)

and

(B.ll)

where f3 = 0.47047 and aI, a2 and a3 are constants shown in table 4.1.

B.3 Comparison of r approximations

The original [15] and the proposed (section B.2) r -approximation are comparedwith results of numerical integration of equation (4.6). Because r varies over awide range, several area's are inspected.

B.3.! r at low fields

0.300.250.20

numeric ­appro" ••••.

0.15x

,,

"/.//

..""/

..........'"...., .. ' ..

".'............

0.100.05

2

1

ol.....-'...__.L.-__..J........__-'-__--l.__-J__~

0.00

Figure B.1: low field r for ~E/kT = 20

In fig. B.1 r is plotted versus the normalized quantity x for ~E/kT = 20. ~\Jte

that for low fields tlE/kT doesn't really matter (see fig. 4.10) except that thex-axis is more or less stretched when using the x quantity instead of the electricfield. The solid curve results from numerical integration and the dashed linestands for the low field approximation. For the latter it doesn't matter whetherthe original or new approximation is used, for x « 1 they are the same. Inthe plot it's clear that the approximation over-estimates r. In approximating

72

equation (4.6) the lower limit is set to -00 assuming that the contributions forz below 0 can be neglected. However this is not true at very low fields.

B.3.2 r around x = 1

..... ",/.. ' ,.

..... ~..", .'

1. 05

,.i

i .,.'::.I /

;'i

/

1.000.95

,.'

numeric -

,~.

,~.

f'

J/,.>.

approx old ..... ...,.approx ne'" _.- - .,-:/....,/

.' ""

0.90

10000

9000

8000

7000

~ 6000

~~() 5000

4000

3000

2000

10000.80 0.85

x

Figure B.2: low field r for !:::.E/kT = 20

In figures B.2 and B.3 r is plotted around x = 1 for two values of !:::.E/kT.The discontinuity at x = 0.925··· is clearly seen in the curve with the oldapproximation, and significant deviation from the numerically calculated values.Also it's seen that the new approxima.tion has no discontinuity at all.

B.3.3 r for high fields

Now we get in the range x > 1, and the high field approximation is of interest.Here use is made of an approximation for the complementary error function erfc,I.e.

(B.I2)

where

(B.I3)1

t= .l+,6z

In the original approximation [15J an adapted value of f3 is used, while for thealternative approximation of section B.2 the original value of Abramowitz [12] isused. In figures BA and B.5 it's clearly seen that the alternative approximationwith the original ,6 matches best the numerical obtained curve.

73

32

numeric30 Ilpprox old

approx new

28

26

24

E 22........... ;.:;.-_.-:-:.-: ..-: ..... :--:::.~

"20 _.- ,.-

18

16

14

120.80 0.85 0.90 0.95 LOO 1.05

x

Figure B.3: low field r for AE/kT = 5

numeric ­approx old .....approx new _._.-

4.5e.07

~4.0e+07

3.5e+07

f3.0.+07

., 2.5e+07

~.," 2.0.+07

1. 5.+07

1.Oe+07

5.0.+06

0.0.+00LO 1.5 2.0 2.5 3.0

X

3.5 4.0 4.5 5.0

Figure B.4: high field r for AEIkT = 20

74

100 ,---...,.....---,.--..,....----.,..--,.---...,.....---,.-------,

90

80

70

60

50

40

numeric ­approx old .....approx new -'--

,.'....'."/"

"/

".--

_.-

20

1 0 '-_~_ __l.__.....-_--'-__'-_->-_ __l._-----J

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0x

Figure B.5: high field r for L::..E/kT = 5

B.3.4 r for very high fields

In figures B.6 and B.7 the case for very high fields is shown. Here we see theopposite of the previous section, for v~ri high electric fields the approximationwith adapted f3 performs better. !\ow it's the question what is the best ap­proximation. To answer that the magnitude of electric field in these cases isconsidered. In figures B.4 and B.5 it's seen that the original f3 suffices up tox = 5. With equation (4.15)

roE£=x - Eo

kT(B.14)

it can be seen that for L::..E/ kT is respectively 5 and 20, this is equivalent withfields up to 2.2 x 106 and 4.4 x 106 V jem. When fields get much larger thanthat (!), the adapted /3 proofs to be useful. Practically fields won't be that high,because breakdown fields are much lower [31].

Therefore it's recommended to use the original Abramowitz parameters incombination with the low field approximation as stated above in expression (B.B).When using (4.12) as low field approximation one must be aware that the 10w­to-high field transition at x = 0.925925 is obtained with the adapted {3.

75

6.0e+08

5.08+08

,4.0e+08 , ,

, ,

.. /'!

E 3.0.+08IICl

2.0e+08

1. Oe+08

.- .

numeric -app:cox old .approx new -_.-

O. Oe+OO '--_~ "-__--' .....I- J.-__~__--'

20.0 40.0 60.0x

80.0 100.0 120.0 140.0

Figure B.6: very high field r for AE j kT = 20

180 ,--.,----,.------,.---...,....---,.---.,------,

numeric ­approx old .....approx new - _.-

gO

90

70 L..-_--l.-__---' -'-__-l. ..L-__-'-__--'

20.0 40.0 60.0x

80.0 100.0 120.0 140.0

Figure B.7: very high field r for AEjkT = 5

76

Appendix C

Derivatives of the mobilitymodel

In Pisces the derivatives of mobility with respect to the carrier concentrations nand p and to the electric field £ must be known to set up the Jacobian used inthe solution methods. Fortunately the field dependence is treated as a separatefield degradation factor (see appendix D) and the code for that part also takesinto account the derivatives with respect to the electric field. The derivativeswith respect to the carrier concentrations are given below.

General expressions are provided that hold for both electron and hole mo­

bilities. This is established by using the (extra) quantities C1 and C2 that denotecarrier densities. In case of electron mobility Cl equals the electron concentrationand C2 equals the hole concentra.tion. In case of hole mobility the meaning of q

and C2 are swapped. In the same way mi and N, denote the effective masses andimpurity concentration respectively. Expressing the derivatives not with respectto n or p but with respect to co, the expressions simultaneously hold for hoth.

C.l Parameter P

oP _ p2 {f ,p-2 oPcw + f p-2 aPBH }~ - cn cw ~ BH BH !:lvC; VCi vC;

and

8N,.c = {O if i = 1oc, 1 if i = 2

i7

(C.l)

(C.2 )

(C.3)

(CA)

C.2 Minority scattering parameter G

Using t

ma Tt=-­

m 300(C.5)

we get

oG = {S3 t64 81 0 _ ~ 85

oP 82 + Pt64 (52 + Pt'4 )J3 P (pt67 r6

if P < Pmin

jf P 2: Pmin(C.6)

C.3 Electron-hole scatter parameter F

From equation (5.14) it's easily shown that

of r -1 TIT4 - T2 + (TITS - T3)~- = T6P 0 2

oP ( pT6 + T4 + T5 fffi-)

From equation (5.10) it's seen that

8N6C = {o if i = 1OCi 1 if i = 2

and from equation (5.15)

C.5 "Extrinsic mobility" J.LD+A+C2

Using expression (5.16) we get to

(C.7)

(C.B)

(C.9)

(C.lO)

C.6 Total mobility

Combining previous expressions, we arrive at

o/-Ltot ( /-Ltot ) 2 allN1 + N2 +C2

8;: = IlN l +NZ+C2 OC;

78

(C.Il)

Appendix D

Comparing Pisces and Curry

D.l Introduction

During implementation of the tunnel and mobility models it was useful to com­pare Pisces results with those from Curry and Trap. Therefore it's necessaryto tune all relevant model parameters. Many parameters just can be set appro­priate, some, like the doping dependence of the minority lifetimes, are modelledthe same but are expressed with differently algebraic equations and a simpletransformation suffices. Other parameters, like the intrinsic carrier density, aremodelled quite different. In the next sections the differenl.:es are inspected forthe following items:

• Velocity saturation.

• Doping dependence of minority lifetimes.

• Bandgap.

• Intrinsic carrier density.

D.2 Velocity saturation

When electric fields are sufficiently high the drift velocity of carriers is no longerproportional with the electric field [31]. Usually this phenomenon is modelledas a mobility degradation factor D with

J.1.(£) = J.1.(E = 0) . D (D.l)

In Pisces, with the FLDMOB and EJ.MOBIL models, D is a function of theelectric field,

(f3)1/f3fL(O)£1I

D(£II) = 1 + [~]

79

(D.2)

with [II the field component parallel with the current flow. If current flow andelectric field have opposite direction, then [II is set to zero

([.j )

[II = MAX J'O (D.3)

However in Curry, using the QUAFERM model, D is a function of the gradientof the quasi-fermi potentials

(D.4)

(D.5)

This is inherently different, except in cases where transport of carriers stemstotally from the electric field and no diffusion occurs. The best that can be doneis to set the parameters v~(lt and f3 in Pisces equal to Curry. Pisces default usesa saturation velocity of

~at 2.4 x 107

v = 1 + O.8exp (ot'o)'while Curry uses

(D.6)

where viat, vzat

, f31 and f32 are model parameters (e.g. see ref [41]). For Piscesto deviate from (D.5) the satura.tion velocity can be set to a specified value,without any temperature dependence. The same holds for the parameter /3, inPisces this is constant, while in Curry a temperature dependent expression isused

(D.7)

To match Pisces parameters with Curry, they should be evaluated for each tem­perature of simulation. For T = 300 K the Curry parameters are shown intable D.l.

Table D.1: velocity saturation coefficients at T = 300 K [41]

parameter va.lue

v~at 1.0705 x 107 cmlsn

v 3at 8.3447 x 106 cmlsp

f3n 1.3298

f3p 1.2130

80

D.3 Minority lifetimes

In Pisces the minority lifetimes of the Shockley-Read-Hall recombination in (withthe CONSRH model), are described by

1 1 ( N tot )-= -- l+-r- •r TO,pi~c 1\ .rh

(D.8)

Here N tot is the total impurity concentration NA + N D while TO.pi~c and N.rh aremodel parameters. Curry (with the SRH, STANDARD and TUNNEL recombi­nation models) uses

1 1 r (300)2- = -- (l+ro C.rh Ntot ) T 'T rO,cur

(D.9)

where C~rh is a model parameter. Note that Curry includes a temperaturedependence while Pisces doesn't I, To transform C.rh and TO,cur into the Piscesparameters N.rh and TO,pi3C use

lI.' 1l'f .rh =----

rO,cur C.rh

and

(T )2

~ . -r -O,puc - O,eur 300

Parameters at T = 300 K are shown in table D.2.

(D.lO)

(D.ll)

It should be noted

Table D.2: minority lifetime coefficients at T = 300 K

parameter value

TnO 4 x 10-5 s

TpO 2 x 10-5 5

lll.rh,n 1.25 x 1017 cm- 3

NJrh.p 6.25 X 1016 cm-3

that SRH minority lifetimes depend on the wafer processing and therefore arenot easy to model in general.

1 Recently Kla.assen [18J suggested a. slightly different temperature dependence of the minor­ity lifetime.

81

D.4 Bandgap

(D.12)

There are different empirical expressions m use to describe the temperaturedependence of the bandgap energy. One of them is an expression obtained byThurmond [1]

Q'T2

EG=EGo---T+j3'

where EGO = 1.17 eV, a =4.73 x 10-4 eV/K2 and j3 = 636K. With this formulaa good approximation is possible over a wide temperature range, as can be seenin figure D.l. Another possibility, like Green [5], is to express the bandgap for

1.18 ...---r---r---.---.---.---.---r---r---r-----.

1.16

1.14

1.12

1.10

1. 08 L

eq Thurmond ­eq Gre.n -_ ...

cla~a Bl udau 0claea Groon +

1. 06 '--_"-_'---_'------.J'------.Jl....-_'--_'--_'--_'----'o 50 100 150 200 250 300 350 400 450 500

eamporaturo (It)

Figure D.l: Bandgap versus temperature

several ranges with formula's of the form

EG = A + BT +CT2 (D.13)

and parameters A, Band C as in table D.3. Note that for the temperaturerange of 250 - 415 K the parameter C is zero, and the bandgap is approximatedby a linear temperature dependence.

Pisces is based on the Thurmond expression, equation (D.12), but quantitiesare expressed in room temperature parameters,

(3002 T2)

Eo = EG300 + a 300 +;3 - T + i3 .

82

(D.14)

Table D.3: Bandgap Coefficients [5]

T range (K)

0-190

150 - 300

250 - 415

A

1.1700

1.1785

1.206

B

1.059 X 10-5

-9.025 X 10-5

-2.73 X 10-4

C

-6.05 X 10-7

-3.05 X 10-7

o

Curry and Trap have no explicit bandgap model and are not fully consistentat this point. For the intrinsic carrier density they are based on expression likeGreen (D.13) with the parameters for the temperature range 250 - 415 K. Forthe tunnel model the bandgap described with the Thurmond expression (D.12)is implicitly used, and parameters cannot be changed.

To match Pisces and Curry EG300 should be set to the value listed in ta­ble D.4.

D.5 Intrinsic carrier density

A well known expression for Hi is

n; = Nc Nvexp(-EG/kT) (D.15)

(D.16)

where N c and Nv both are proportional to r 3/ 2 • In Pisces this is expressed as

n~ = NC300 NV300 (3~0) 3 exp(-EG/kT)

where NC300 and NV300 are the values of Nc and Nv at T = 300K.An alternative way to express the intrinsic carrier density is handled next. In

section DA we have seen that in the temperature range 250 - 415 K the bandgapdepends linearly on the temperature. In that range the intrinsic carrier densitycan be expressed with [4]

n; = Co T 3 exp (-gA/kT) (D.17)

with A the parameter from table D.3. This behaviour was observed by Putleyand Mitchell [4], with the same value for A and Co = 9.61 x 1032 . In this wayCurry models the intrinsic carrier density. One should be aware that A is not thebandgap, but a model parameter from equation D.13. Unfortunately in Currythe parameter A is called Vgo, causing possible confusion. Note that (D.17) isonly valid in a limited temperature range, and Curry thus will be inaccurate at

low temperatures.Because the bandgap is modelled implicitly, and in a different way, into the

tunnel model of Curry, one should be aware that altering the parameter Vgo

83

has effect on the intrinsic carrier density, but it has no influence at all on theimplicitly used bandgap for the tunnel model.

To set Pisces parameters, the bandgap at T = 300 K is calculated withequation (D.13) and then NC300 and NV300 are set to obtain the same intrinsiccarrier density. It should be noted that the intrinsic carrier density is verysensitive to changes in EG 2 , and therefore EG300 is expressed in sufficient numberof digits3 . In figure D.2 the intrinsic carrier is plotted versus the temperature

lE+l0

;;;E

lE+OO~....

>.......~

~ 1E-l0•"~•,~~

"• 1E-20\)

\),~

~

~....~... lE-30c:

'001

11:-40

50.0 100.0 150.0 200.0 250.0temperature (K)

300.0 350.0 400.0

Figure D.2: Intrinsic carrier density versus temperature.

for the formulations used in Pisces and Curry. As can be seen, the differentexpressions are in agreement, only for low temperatures small differences can beobserved. Figure D.3 displays the relative difference in niQ between Curry andPisces expressed in percents. This results in a more detailed view, but it is seenthat the differences are less than one percent for temperatures above 280 K.

2 At T = 300 K the exponent in (D.15) is about -40, and a change of 1 percent in bandgapenergy, will cause about 40 percent change in n~.

3To make comparison between Pisces and Curry output as easy as possible, it is necessarythat the numerical difference is sma.ll as possible. This is achieved by setting parameters likebandgap with many significant digits, a.lthough in rea.lity this level of accuracy can not be met.

84

2r---...,....-__r--,.----r--~-_r--r__-...,....-__r-___,

o

-2

-4

-6

o -8o~

-10 '----'--'-_---.J.__-'--_--'-__.L-_-'-__'--_-'-_---1_--J

200 220 240 260 280 300 320 340 360 380 400t"",p"%,,ture (Xl

Figure D.3: Relative difference in intrinsic carrier densities of Curry and Piscesversus temperature.

Table D.4: Bandgap parameters

parameter value

EG300 1.1241 eV

Cl' 4.73 x 10-4 eVjK2

f3 636 K

NC300 *NV300 (3.3037 x 1019 )2 cm-6T-3

85

Appendix E

Note on accuracy withNewton and Gummel method

Several observations show that with the Gummel method the currents are moreaccurately computed at low biases than with the Newton method. This can beseen by displaying the right-hand-side (RHS) of the solved Poisson and conti­nuity equations (e.g. see equation (3.8)). Tills is possible in Pisces with theMETHOD RHSNORM command. When the solutions have been converged(this can be checked with the METHOD LIMIT command) it turns out that theRHS with Newton is much larger compared to the RHS with Gummel. For ex­ample the results of a simpIe 1-D p +n-diode at a -1V bias are shown in table E.1.

Table E.1: RHS values after convergence for a p+n diode at -1V bias

RHS

Gummel

Newton

F,p (C/micron)

1 X 10- 27

5 X 10-29

Pn (A/micron)

8 X 10-22

3 X 10- 16

Fp (A/micron)

6 x 10-20

2 x 10- 15

Mark that the RHS values ofthe Newton solutions of the continuity equationsare five to six orders of magnitude larger than with Gummel. As a consequencethe terminal currents with Gummel can be computed accurate for much lowerbiases than with the Newton method. Plots of current densities in the exam­ple device show a neat straight line for the total current with Gummel, whilewith Newton this quantity is very noisy in the heavily doped p+ region. It isassumed that these effects stem from a relatively poor conditioned matrix withthe Newton method. It should be investigated whether this holds in general. Ifthis is the case, it is advantageous to finish a Newton iteration process with a"post-processing" step that consists of one or two iteration steps that solve the

86

decoupled continuity equations (i.e. to "smooth" the Newton solution). Thesesteps are similar to those performed with the Gummel method.

87

Appendix F

Implemented modules

The implemented modules tungam and kmob described in section 6.6.1 axe listedbelow.

F.l Module tungam

C23456789012345678901234567890123456789012345678901234567890123456789012C 1 234 5 6 7

SUBROUTINE TUNGAM (INODE,DC1,TRAPC,EM,GAMMA,DGDC,DGDE)INTEGER INODEDOUBLE PRECISION DC1,TRAPC,EM,GAMMA,DGDC,DGDE

c

cc NAME:c FUNCTION:cc TYPE:c CALL:

TUNG AMComputes field effect factor GAMMA for trapassisted tunnellingSUBROUTINECALL TUNGAM(INODE.DC1.TRAPC,EM,GA~~A,DGDC.DGDE)

cc PARAMETER INPUT:c INGDE INTEGER node numberc DCl DOUBLE (l!DCSCL) * carrier concentration (p or n)c TRAPC DOUBLE (1!DCSCL) * Nie • exp(+/-ETRAP!kT)c EM DOUBLE magnitude of electric field E (V!cm)c PARAMETER OUTPUT:c GAMMA DOUBLE field effect factor

88

ec

c

c

DGDCDGDE

DOUBLEDOUBLE

(DCSCL) * d(GAMMA)/d(Carrier cone)d(GAMMA)/d(E)

c-----------------------------------------------------------------------c COMMON BLOCKS:

include 'tmpeom.inc'include 'seteo.inc'include 'emacop.inc'

c----------------------------------------------------------------------c local variables:

c----------------------------------------------------------------------double prec1s10n al.a2,a3,beta,fac23,x072,SqrtPi,Gmindouble precision dWkT,ER,WgHlf,x,SqrtX,btt,dGdtdouble precision SqrtW, Tnp, fTnp, dfTnp, dtdE,dtdW,dGdWinteger ireglogical lZrdW

parameter (Gmin;le-4)parameter (al;O.3480242, a2;-O.0958798, a3;O.7478556)parameter (beta=O.61685, fac23=O.66666666667, x072=O.9259259)parameter (SqrtPi=1.772453851)

c SqrtPi SQRT(pi)c beta, al,c a2, a3 coefficients for approximation function of "erfc"c Gmin when gamma about less than Gmin, no calculations need toc be performed. and gamma is set to zero,c thus preventing under/overflow and waste of computing timec ER reduced el. field EM/Eref, where Eref is ref Electric field.c dWkT delta W (energy) in units of kTc WgHlf Wgap/2 in units of kTc SqrtW SQRT(d~kT)

c x (EM/Eref)/DSQRT(delta W/kT)c SqrtX SQRT(x)c x072 (2/3)/0.72, transition value low -> high field approximationc lZrdW delta Wis at limit value, derivatives w.r.t. Wmust be zeroed

C-----------------------------------------------------------------------C start of tungam

C-----------------------------------------------------------------------c

89

GAMMA = 0.0DGDC = 0.0DGDE = 0.0dGdW = 0.0

c

c obtain half of the local bandgap energy. takingc possible bandgap narrowing effects into account

ireg = itype(inode)WgHlf = O.S*(lgdnc(ireg)+lgdnv(ireg»-lgcnieCinode)

c

lZrdW = .false.if (del .It. trapc) then

dWkT = WgHlf - dlogCtrapc/cnie(inode»lZrdW = .true.

elsedWkT = WgHlf - dlog(OC1/enie(inode»

e Note that the next test includes the case dWkT <= 0if CdWkT .It. Gmin) then

RETURNendif

endif

ER = EM !CDBLE(tterefCireg» * (temp/300.0dO)**1.5)x = er / dsqrt(dWkT)

c calculate GAMMA .if (x .It. x072) then

c first approximation: low fieldif (er .gt. O.2.Gmin) then

GAMMA = 2 * SqrtPi • ER * dexp(ER*ER/3.0)DGDE = GAMMA*(l + fac23*ER*ER)/EM

endifelse

c second approximation: high fieldSqrtX = dsqrt(x)SqrtW = dsqrt(dWkT)tnp = 1.0/C1.0 + beta * SqrtW * (SqrtX - 1.0/SqrtX»fTnp = Tnp * (al + Tnp * Ca2 + Tnp * a3»dfTnp = al + Tnp * (2 * a2 + Tnp * 3 * a3)

c

gamma = SqrtW*SqrtX*SqrtPi*fTnp*dexp(dWkT*(1.0-fac23/x»cc derivatives w.r.t. electric field

90

btt = beta*tnp*tnpdtde = -O.5*(btt/em)*SqrtW*(SqrtX + 1.O/SqrtX)dGdt = gamma * dfTnp/fTnpDGDE = (g~a/em).(0.5 + fac23*dWkT/x) + dGdt * dtdE

c

if (.not. lZrdW) thenc derivatives w.r.t. carrier concentrations (i.e. w.r.t. delta W)

dtdw = -0.2S*(btt/SqrtW)*(SqrtX - 3.0/SqrtX)dGdW = gamma*(O.25/dWkT + 1.0 - 1.O/x) + dGdt * dtdWDGDC = -dGdW/DCl

endifendif

returnend

91

mobility for the sideDCSCL * (l/MU) * d(MU)/d(NSIDE)DCSCL * (l/MU) * d(MU)/d(PSIDE)

kmobComputes low field mobility according to D.B.M. Klaassen's modelMobility is computed along a side connecting two nodesof a triangleSUBROUTINECALL KMOB(lpcont,nside,pside,ireg,nl,n2,mu,dmudcn,dmudcp)

F.2 Module kmob

C23456789012345678901234567890123456789012345678901234567890123456789012C 1 2 3 4 5 6 7

subroutine kmob(lpcont,nside,pside,ireg,nl,n2,mu,dmudcn,dmudcp)logical lpcontdouble precision nside,pside,mu,dmudcn,dmudcpinteger ireg,n1,n2

C

C

C NAME:C FUNCTION:C

CC TYPE;C CALL:C

C PARAMETER INPUT:C LPCONT LOGICAL True (false) for hole (electron) mobilityC NSIDE DOUBLE electron concentration for the side (#/cm~3)

C PSIDE DOUBLE hole concentration for the side (#/cm-3)C I REG INTEGER region numberC Ni, N2 INTEGER numbers of the two nodesC PARAMETER OUTPUT:C MU DOUBLEC DMUDCN DOUBLEC DMUDCP DOUBLEC

include 'blank. inc ,include 'emacop.inc'include 'mobco.inc'include 'seteo.inc'

c constants for F and G function (see also data stmt)real fr1,fr2,fr3,fr4,fr5,fr6real gsl,gs2,gs3,gs4,gs5,gs6,gs7

c effective massesreal melee, mhole

c constants for clustering functionsreal cD, NrefD, cA, NrefA

c weighting factors Conwell-Weisskopf, Brooks-Herringreal feW, fBH, pwrCW

92

c temperature dependence of lattice scatteringreal tetan,tetap

double precision mumax,mumin,alpha, nref,mulatdouble precision cl,c2,Na,Nd,dmudcl,dmudc2double precision Nsc,Nsceff,dNefdl,dNefd2double precision mUl,mu2,mub,mua,mux,dtmpdouble precision Nle,N2e,Gc,Fcdouble precision dGdcl,dGdc2,dFdcl,dFdc2double precision ctotdouble precision Td300double precision ml, m2, t,Pc,dPdcl,dPdc2,dGdP,dFdPdouble precision fPbhi, fPcwidouble precision tgs4,denoml,terml,term2double precision pO, Prnin, Cl,fOpO,f1pO,pwrdouble precision pfr6, mlrn2

parameter (frl=O.7643, fr2= 2.2999, fr3=6.5502)parameter (fr4=2.3670, fr5=-O.8552, fr6=O.6478)parameter (gsl=O.89233, gs2=O.41372, gs3=O.19778, gs4=O.28227)parameter (gs5=O.005978, gs6=1.80618, gs7=O.72169)parameter (fCW=2.459, fBH=3.828, pwrCW=-O.666666667)

data cD/O.21/, NrefD/4.0e20/, cA/O.50/, NrefA/7.2e20/data melee/i.OOOI, mhole/l.2581

data tetan/2.285!. tetap/2.247!

Td300 = temp/300.0dtmp = (r1(n1)+rl(n2»*dcscl

c due to rounding errors negative Nd, Na might occur, this isc to be prevented

Na drnaxl(O.25*(tconc(nl)+tconc(n2)-dtmp), mindbl)Nd = dmaxl(O.25*(tconc(nl)+tconc(n2)+dtmp), mindbl)

cc apply cluster functions for the dopingc

dtmp = (Nd!NrefD)**2Nd = Nd*(1.0 + dtmp/(cD*dtrnp + 1»dtmp = (Na!NrefA)**2Na = Na*(l.O + dtmp!(cA*dtmp + 1»

cc map electron/hole quantities onto the right parametersc i.e. carrier type 1 and 2.

93

c

if (lpcont) thenmumax : mupmx(ireg)murnin : mupmn(ireg)alpha: alphp(ireg)Nref =nrefp(ireg)mulat = mumax/(Td300**tetan)Nle = NaN2e = Ndml = mholem2 = meleecl = psidec2 = nside

else= murunx(ireg)= munmn(ireg)= alphn(ireg)= nrefn(ireg)= mumax/(Td300**tetap)Nd

Na

mumaxmuminalphaNrefmulatNle :

N2e =m1 = meleem2 = mholecl = nsidec2 = pside

endifmu1 = mumax*mumax/(mumax-mumin)*(Td300**C3*alpha-l.5»mu2 = mumin*murnax/(mumax-mumin)/sqrt(Td300)Nsc = Nle + N2e + c2ctot = cl + c2

c

c again watch out for underflow due to n=p=Oc

if (ctot .It. mindbl) thenctot = mindbl

enditcc compute Pc = P(ctot,Nsc,ml,temp)c

dtmp = Td300*Td300fPbhi = fBR * ctot/(1.36d20*ml*dtmp)fPcwi = feW /(3.97d13*dtmp*(Nsc**pwrCW»Pc = l/(fPcwi + fPbhi)

94

dPcicl = -fPbhi*Pc*Pc/ctotdPdc2 = dPdcl + pwrCW*fPcwi*Pc*Pc/Nsc

c

c compute Fe = F(Pc,ml,m2)c

pfr6 =Pc**fr6mlm2 =ml/m2Fc = (frl * pir6 + fr2 + fr3 * mlm2)/(pfr6 + fr4 + frS * mlm2)dFdp = fr6 * (pfr6/Pc) * «frl*fr4-fr2) + (frl*fr5-fr3)*mlm2)/

+ «pfr6 + fr4 + fr5*m1m2)**2)cc compute Gc = G(Pc,m1,temp)c

t = (1.0/m1)*Td300tgs4 = t**gs4pwr = 2.0*(g53 + 1.0)/(gs6 + 1.0)Cl = C(gsl*gs3/Cgs5*gs6» • t**(gs4-gs6*gs7) )**(2.0/Cgs6+1.0»

cc determine Pmin, using several taylor series approximations around pOc

pO = 0.15 + 0.15 * tfOpO = (gs2 + pO*tgs4)**pwrf1pO =fOpO*pwr*tgs4 / (gs2 + pO*tgs4)Pmin = pO + «fOpO - Cl*pO*pO)!(2.0*Cl*pO-f1pO»

c note: Pc can be modified, because F(Pc) is already computed,c and we don't need Pc furthermore after computing Gc

if (Pc .It. Pmin) thenPc = Pmin

endifdenom1 = (gs2 + Pc*tgs4)terml = gsi/(denoml**gs3)term2 = gs5/«Pc*(t**(-gs7»)**gs6)if (Pc .ge. Pmin) then

dGdp = (gs3 * tgs4/denom1)* term1 - (gs6/Pc)*term2else

dGdp = 0.0endifGc = i.OdO - term1 + term2

cc combine partial derivativesc

dGdc1 = dGdP * dPdc1dGdc2 = dGdP • dPdc2

95

dFdci = dFdP * dPdcldFdc2 = dFdP * dPdc2

c

c compute effective scatter concentration Nsceffc

Nsceff = Nie + N2e * Gc + c2/FcdNefdl = N2e * dGdcl - dFdcl * c2/(Fc*Fc)dNefd2 = N2e * dGdc2 + (1 - c2*dFdc2/Fc)/Fc

c

c and combine things into mobilityc

mua = mu1 * (Nsc/Nsceff) * (Nref/Nsc)**alphamub = mu2 * ctot / Nsceff

mux = mua + mubdmudci = mua * (- dNefd1/Nsceff) +

+ mub * (1.0/ctot - dNefdl/Nsceff)dmudc2 = mua * «1.0-alpha)/Nsc - dNefd2/Nsceff) +

+ mub * (1.0/ctot - dNefd2/Nsceff)mu = 1.o/(1.O/mux + 1.O/mulat)

c

c combine derivatives and apply scaling factorsc

dtmp =dcscl*mu/(mux*mux)dmudcl = dtmp * dmudcldmudc2 = dtmp * dmudc2

c

c map carrier type 1 and 2 back to electron/holesc

if (lpcont) thendmudcp = dmudcldmudcn = dmudc2

elsedmudcn = dmudcldmudcp = dmudc2

endifcc .. Done.

returnend

96

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