3-7

A NEW ACCURATE METHOD FOR SEMICONDUCTOR DEVICE MODELLING

by V. Axelrad

Department of Electrical Engineering, Chair for Integrated Circuits,

Technical University Munich, W. Germany

Abstract

A new methodl,2 for the numerical modelling of semi­conductor devices is presented. The method is based on

the classical Fourier approach. Discretization of the differential equations is achieved through trigonometric­

series expanslons in a Galerkin procedure. The use of Fast Fourier Transform techniques renders the algorithm

computationally effective. The Fourier method assures infinite-order accurate solutions of the strongly non­

linear semiconductor equations. Excellent numerical conditioning and high efficiency of the method offer

major advantages for simulation of complex nonlinear

VLSI-devices. where high a!curacy. numerical stability and flexibility are of major concern. Properti£>s of the

method are demonstrated on application examples.

.L Introduction

Electrical behaviour of semiconductor devices is governed by a set of strongly nonlinear partial differen­

tial equations. Variation of the semiconductor functions is locally very rapid. This poses special demands to the numerical solution. Most known FEM-solutions of the device problem use the Scharfetter-Gummel approach3, based on an approximate analytic solution of the current

continuity equations in one spatial dimension. The accuracy of this approach degrades drastically with

increasing number of dimensions4. In the two-dimensional case the discretization error in currents is of merely first-order O(h -I) in the step-size h (cf. Mock'). Three­dimensional solutions for complex semiconductor devices

are still not completely successful.

In the multidimensional case, requirements of computer

resources for the solution of the discretized FEM-equa­tions rise rapidly with increasing total number of

elements M. The computer time for a direct Gaussian

elimination is at least of order O(M1.s). memory require­ments grow as O(M1.sl. An iterative solution of the large systems of linear algebraic equations is possible. How­

ever, the required number of iterations increases with M.

Numerical stability of FEM-solutions is endangered by rapidly growing loss of significant digits with increasing element densities. The quantitative measure

116

for this deterioration in a numerical method is the

condition number. Classical work on the FEM5 reports

a quadratiC increase in the condition number with the

number N es of elements per side of the domain. Linear

growth of the condition number results from the non­

uniformity of the mesh h/hm;n (average to smallest

step-size). This totals in a condition number of the

order O(N 2. hlh . ). With N = 50 .. 100 and hlh . > 102 es - Tnln es - min

for semiconductor device models. the discrete FEM-equa-

tions are singular for single precision arithmetics (6 decimal digits). Ill-conditioning impairs the effectivity of

the algorithm and limits the achievable accuracy.

The numerical Fourier methodl ,2

is based on global high-order trigonometric expansions (Fourier series) of the semiconductor functions. Coefficients of the series are calculated in a Galerkin-type procedure. The resulting numerical algorithm is infinite-order accurate under certain smoothness conditions satisfied by semi­

conductor device models. The residual error of the

solution decreases exponentially as O(e -M), where M is

the order of the Fourier polynom equal to the total number of sampJ ing points. This infinite-order accuracy

(cf. Fig. S) is guaranteed by the exponential convergence

rate of the series demonstrated in Fi g. 1 (Fourier coeffi­

cients Uv (M=25b) of the electric potential in a pn-diode

and the enveloping exponential curve).

-4 10

�8 10

-12 to

-it. \) 10 +---------�--------�----------r_--��__4 4 16

Figure 1.

64

The chosen orthonormal trigonometric basis is an

optimal one. Indeed, its elements (especially those of

high order) approach the eigenfunctions of the linearized semiconductor equations. Fourier domain equations are

therefore loosely coupled. The system matrix is diagonal­ly dominant and well-conditioned. The condition number of the equations is near unity. independent of the number of coefficients M. These properties of the equations are crucial for an efficient implementation of the ·method.

Equations for the Fourier coefficients are compiled in a general and computationally inexpensive way. Require­ments of computer resources (time and memory) for the solution of the algebraic equations rise sub-linearly with M in both the one- and multi-dimensional cases. Doubling of M doubles the number of correct decimal digits in the solution (Fig. I, Sl. High accuracy up to the exact solution (machine precision) is thus obtained at low costs.

There are significant di fferences between the numerical Fourier method and the related family of spectral methods. Spectral methods, proposed in the renowned work of Orszagb in 1971, are also based on high-order orthogonal expansions (mostly Chebyshev-polynomialsl. They enjoy considerable attention in fluid dynamics. How­ever, differential equations with nonconstant coefficients or nonlinear ones are solved in spectral methods· using explicit time-discretization only (forwards Euler), This approach is unsuitable for the semiconductor device equations. These are very stiff and demand uncondition­ally stable implicit time-discretization kf. e.g. Mock'), leading to nonlinear elliptic-type equations.

The numerical Fourier method!,2 extends previous work in several respects. Arbitrarily nonlinear differential equations are handled using Newton's qUasilinearization. Equations for the Fourier coefficients are compiled in a straight-forward and efficient matrix-operator form7 via the Frechet-derivative of the governing equations. Solu­tions of strongly nonlinear elliptic-type equations are thus obtained. The method is applicable to a wide class of differential equations, offering maximum accuracy, well-conditioning and effectivity. It is a competitive alter­native to the FEM in semiconductor device modelling.

I, Mathematical Model The electrical behaviour of semiconductor devices is

governed by a set of nonlinear partial differential equa­tions first given by v. Roosbroeck8:

gu" 'lEY'u-( n - p - N ) =0 (1) gn" Y'l'n[Y'n - n Y'(u+ln nj))- R - In - cln/elt = 0 (2)

gp"- V'l'p[Y'p + P Y'(u-ln nj)) + R -Jp + clp/dt = 0 (3) The classical equations8 are presented here in an ex-

tended form, accounting for space-dependent intrinsic carrier density nj (e.g. heterostructures or isolators) and distributed current sources J ,J . All other symbols have their standard meaning (;f. �.g. Sze9),

A device is connected to the outer world via ohmic contacts. In the contact areas current is injected. The magnitude of this current is controlled by the difference

117

between a fixed outer potential V and the potential u()i) of the contact area. This leads to the following voltage­controlled current-source model for the electron current­injection J (hole currents represented analogously):

� n ....,. --'7 In(x) = z: G�(x)' [V� - u(x)) (4) Where G�(iJ is the eqUivalent conductance of the

contact area and V� is the voltage applied to the n-type contact i. summation is over all n-type contacts.

G" 1

Figure 2.

vn 2

?> n-type j=O

A requirement of vanishing currents across the boundary (homogeneous Neumann boundary conditions for u, n, p) completes the mathematical model.

lrregu lar device geometries are accounted for by the inclusion of space-dependent ni (small in isolators) and contacts arbitrarily placed inside the modelling domain, which may be chosen to be rectangular without loss of generality (Fig. 21. Unlike the usual idealized ohmic contact models, a finite contact resistance is included. This can be of advantage in transient models.

The extended eqs. (1-3) provide a general and physically consistent device-model for the simulation of irregularly shaped devices within simple boundaries.

;!, Numerical Fourier Method The numerical Fourier method provides approximate

solutions of differential equations using high-order trigonometric expansions (Fourier series) in a Galerkin approach. A general and flexible solution of nonlinear equations is obtained by qUasilinearization (Newton's method) prior to the trigonometric expansion. The Fourier method is applied repeatedly to the linear nonconstant-coefficient differential equations, which originate in the iterations of Newton's process.

With the vector notation z={u n p} T, g={g g g }T <Cf. eqs. (1-3»), the i+l-th Newton's iterate of the �ol�tion z reads:

f=6g/oz (5) Where f is the so-called operator- or Frechet-derivative

of g(z). For the three-equations system eqs. (1-3), f is a three-by-three array of differential operators (statio­nary case assumed Ci/elt=O, En� Y'(u+ln nj), Ep= Y'(u-ln nj»

� V,V

f E -'71' n'7 +IGn n ,

-'7l-'pp'7 +IG�

-1 liR '71-',r-En)- �n

iR 8n " J sp

-'711('7+E )+� p p p

(I.)

In a Galerkin-type solution of the differential equa­tions (\-3) a finite set of functions "'v' each of which satisfies the boundary conditions, is used to approximate the solution z = I Zv"'v' Coefficients Zv of this poly nom are determined by the requirement that the residuum of the equation g(z) be orthogonal to each "'v:

Jg ' '''v dx = O v=O . .. ,M (7)

A suitable orthonormal set of basis functions "'v' matching specified homogeneous Neumann boundary conditions (Sect. 2), is spanned by cosines (xj € [0, lj]):

"'v,v2v3 = cos(v,rr/L

,x,l' . . 'cos("3rr/L3x3) (8)

With this choice of "'v' the Galerkin coefficients of z are its cosine coefficients (an alternative appl'Oach based on sine-series is possible1.2) , They constitute a matrix

of coefficients Z. The orthogonality condition eq. (7) amounts to vanishing cosine coefficients of g(z), The system of differential equations (1-3) transforms to the Fourier coefficient equation:

G(Zl = 0 (9) The solution of this equation Z is determined by the

transformed eq. (5): Zi+1 := Zi _ F-1 (Zi). G(Zi) (to) The cosine-co<,fficients of the residuum G(Z) are

calculated accurately, evaluating linear operations (differentiation) on Fourier coefficients and nonlinear operations (multiplication, transcendental functions) on

spatial values of unknowns. Transition from Fourier coefficients to spatial values and vice v<'rsa is performed effectively via the FFTlO. This approach has been proposed by Orszag"- Evaluation of G(Z) for semiconductor device equations is described in detail by Axelrad2. Jacobian Matrix

The transformed Frechet-derivative F, i,e. the Jacobian matrix of the discrete equations, is of central importance for the method. In the iteration eq. (10) most of the numerical work is spent to solve the linear algebraic system of equations F<lZ=-G. The structure of the Jacobian matrix and its mathematical properties are dpcisive for the success of the solution.

The use of orthonormal trigonometric series offers

significant advantages in comparison to e.g. FEM shape functions or Chebyshev polynomials. A formulation for the matrix elements can be given. which is both general and simple to evaluate . Differential operators (dominating for higher spatial frequencies) are diagonal. This guaran-

1 18

tees diagonal dominance and well-conditioning of the Jacobian, allowing omission of small far-off-diagonal elements as well as the fill-in in the Gaussian elimination.

Elements of the Frechet-derivative array f eq. (6) are compound differential operators, which are built up using two basic operators: differentiation and multiplication by a function. The Jacobian matrix F is compiled using two corresponding types of matrix-operators described by Axelrad?

The differentiation operator-matrix is diagonal, with linearly increasing elements for higher indices (its nonzero elements are Avv =!rr/L·v for one spatial dimension). Of course, series' type (cosine/sine) and/or

signs change when this operator is applied.

The transform of a product of two functions c=a'b is the convolution of their transforms C=A'B. The mUltiplication operator2 is obtained writing the convo­lution as a matrix-by-vector product C=[al B. For complex series a(x) = L A eivx this operator has the struc-v ture of a Toeplitz matrix: [alV[1= AV-l-" For real trigono-metric functions cosine/sine, their symmetry properties lead to Toeplitz-Hankel matrices denoted by [al:�� .

The superscript denotes the series-type of a, subscript "+/-" is for cosine/sine-type of the factor b respectively (cf. Axelrad7). The Jacobian matrix F for the one­dimensional stationary semiconductor equations reads2 (Gn=[LG� l� , third row omitted for space reasons): tAlElCA ' -I i

A[l-' nlcA+Gn'-AlI-' lCA-AlI-' E lS_[�lC

n - n - n n + on + -- ------------+---- - ----- -

Discrete Equations The matrix F is diagonally-dominant (especially for

higher indices) and well-conditioned. It is therefore suffi­cient to consider only a small part of its elements for the solution of the system of discrete equations (to).

256

16 x 16

�

Figure 3. Figure 3 displays the locations of the significant

elements of the block-structured Jacobian F for the two­dimensional case (16x16 coefficients).

In a direct (Gaussian) solution of eq. (10) fill-in is

negligible, leading to a merely linear rise of computer time with the order of equations M. Actually the rise

is even weaker since the bandwidth of the matrix F decreases for higher indices (Fig. 3).

� Properties of the Method

The numerical Fourier method is characterized by very

high accuracy, well-conditioning and flexibility in the handling of nonlinearities and multidimensional problems. Its properties are demonstrated here using a one-dimen­sional bipolar transistor model and a two-dimensional

off-state diode model.

Semiconductor functions u, n, p are infinitely differen­tiable. Their Fourier series converge faster than algebrai­

cally (exponentially) under certain periodicity conditions at the boundaries (vanishing odd-order derivatives for

cosine-series). This is assured if all geometry functions

(NN' e, nil have vanishing odd-order derivatives at the

boundaries, a condition which can be satisfied without

loss of generality.

An example is given by Figs. 4,5 presenting a one­

dimensional model of a bipolar pnp-transistor in the

active region liBE=0.25 Y, liCE=1 Y. Fig. 4 displays the

electric potential and the conductance functions G(x}

(cf. eq. (4», whereas Fig. 5 shows the current densities

in the device. Non-uniform sampling (indicated by vertical lines at sampling points) is realized using a coordinate

transformation technique2. Contacted areas in the

emitter. base and collector are defined by large (Gma,,) values of the equivalent conductance G(x) (eq. (4». Voltage-controlled current-injection (Sect. 2) occurs in

these regions (Fig. Sl. Homogeneous Neumann boundary conditions for u, n, p (vanishing normal derivatives at

the boundary) result in vanishing current density across the boundary (Figs. 2,5).

[Vl

0.0 0.0

-0.5 u(x)

-1.0

-1.5 0.5 1.0 1.5 2.0 [[lml

Figurp 4.

119

0.5 �---------------,

[-.A..2l cm

0.25

0.0

( j .,

0.5 1.0

Figure S. 1.5 2.0 [[lml

Fig. 6 displays the residual error in the ca1culatpd electric potential for the one-dimensional bipolar tran­sistor model as a function of the series' order M. Accu­

racy of the solution improves faster than algebraically

(exponentially) with an increase in M. Close relation to the exponential convergence rate of the series (Fig. t) is evident. The series' convergence gives a reliable esti­mate for the accuracy of the numerical solution.

10°

10-3

10 -.

-9 10

to -12

10 -15

8 16 32

M 64 128

Figure 6. 256

No one-dimensional assumptions or approximations

(e.g. of Scharfetter-Gummel type) have been made to

achieve the accuracy indicated in Fig. 6. Implementation

of the method in several spatial dimensions is straight­

forward. The Jacobian matrix is again diagonally

dominant and well-conditioned (Fig. 3), A direct Gaussian

solution of the discrete multidimensional equations is

particularly effective since fill-in is insignificant.

Figure 7 presents the electric potential and its Fourier

coefficients for a two-dimensional diode (off-state). Exponential series' convergence is again observed. It

closely resembles the one-dimensional case kf. Fig. 1, log-log scale).

Figure 7. This convergence rate of the series assures exponen­

tial decrease of the approximation error with increasing order of the trigonometric series (cf. Fig. h).

� Conclusions

The Fourier approach renders a highly accurate, well­conditioned and flexible method,,2 for semiconductor device modelling. The mt'thod has been implemented for

one and two spatial dimensions and applied to several semiconductor device-structures including diodes, tran­

sistors. MOS-capacitors and th)ristors. Results indicate competitive features of the method in comparison to

the classical FEM.

il High accuracy requirements of semiconductor

models are matched by the infinite-order accuracy of the Fourier method.

iD Optimal numeriral stability is guaranteed b) low condition numbers 0(1) independent of the numb"r of equations.

iii) High numel'ical efficiency due to slow, sub-linear Increase of the computer costs and exponential decrease of the solution's prrnr with increasing number

of discrete variables.

Significant advantages for the applications are expected in particular for complex multidimensional

numerical models of sub-micron semiconductor devices.

For the solution of two-dimensional stationary and transient models of complex devices a computer code is now under construction. Extensive comparison of the method to the FEM is being carried out.

Acknowledgement

Thp author is indebted to Prof. Dr, I.Ruge for support

of this work. to Prof. Dr. E.L Axelrad, H.Thurner, S.Bamberg and S.Eckart for many discussions and to

L.BluC"her for reading the manuscript.

120

References

[1] V. Axelrad, Halbleiter-Modellierung mit der Fourier­

Methode, Dissertation TU Munchen, 1987 [2] V. Axelrad, Fourier Approach to Semiconductor Device

Modelling, accepted for publication in the Int, Journal of Numerical Modelling

[3] D. Scharfetter, H.K. Gummel, Large Signal Analysis

of a Silicon Read Diode OScillator, IEEE Trans. Electron Devices, Vol. ED-lb, 1969

[4] M.S.Mock, Basic Theory of Stationary Numerical Models, in Advances in CAD for VLSI, Vol. I, W.LEngl, editor, North-Holland 1986 [S] G.Strang. G.J.Fix, An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs N.] .. 1973

[6] S.A.Orszag, Numerical Simulation of Incompressible

Flows Within Simple Boundaries. I. Galerkin (Spectral)

Representations, Studies in Applied Mathematics, Vol. L, No. 4. Dec. 1971

[7] E.L Axelrad, Theory of Flexible Shells, North-Holland Amsterdam, 1987 [8] W.V. v. Roosbroeck, Theory of Flow of Electrons and

Holes in Germanium and Other Semiconductors. Bell Syst. Techn. J.. Vol. 24, pp. 5&0-607, 1950 [9] S.M. Sze, Physics of Semiconductor Devices, John

Wiley & Sons 1981

[to] J.W.Cooley, J.W.Tukey, An Algorithm for Machine

Calcu lation of Complex Fourier Series, Math. Compu­tation Vol. 19, pp. 297-301, April 1%5