Nov 11, 1990 - I. INTRODUCTION. URING the past two decades the numerical model-. D ing of semiconductor devices has enjoyed much at- tention of ...
IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN. VOL. 9, NO. I I . NOVEMBER 1990
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Fourier Method Modeling of Semiconductor Devices VALERY AXELRAD
results match the condition estimates given by [4] (Table 11) for the quasi-fermi solution. The discrete equations are thus not solvable in single precision arithmetics (6 decimal digits). A practically important effect of the ill-conditioning is the rapid increase of computer requirements with the number of unknowns (Fig. lO(b)). For two- and threedimensional problems the computer time is reported to be ) , M being the total number of of order O ( M ' . 5 - 2 . 5 with grid points (Fig. 10(b)). In combination with the slow algebraic improvement of the accuracy with M , high accuI. INTRODUCTION racy is practically not achievable. URING the past two decades the numerical modelOne possible source of ill-conditioning is the physical ing of semiconductor devices has enjoyed much at- problem itself. Ascher et al. [5] showed theoretically and tention of prominent scientists. The origin of the numer- numerically that stationary differential models of complex ical work in this field can be traced back to the pioneering devices containing floating isolated regions (e.g., thyrisapproach of Scharfetter and Gummel [l]. They used an tor or a bipolar transistor with a floating base) can be illapproximate analytical integral of the current-continuity conditioned. However, many standard semiconductor deequations in one spatial dimension (assuming constant vices, e.g., a single contacted p-n junction or a contacted field and vanishing generation-recombination between ad- bipolar transistor are well conditioned [ 5 ] . jacent grid points) to develop an accurate discretization An interesting fact about the physically-based ill-conscheme. This approach was first extended to two dimen- ditioning is that it does not occur in transient models of sions by Slotboom [2]. semiconductor devices. The reason is that transient charge Semiconductor device modeling is now an important storage (capacitive) effects couple the floating isolated redesign tool in the industry. However, numerical stability, gions of the device to contacted ones. accuracy, and computer requirements of presently used The physically-based ill-conditioning [5] cannot, of methods are not always satisfactory. Important classes of course, be improved by using a different numerical problems, specifically complex three-dimensional device method. All arguments here are, therefore, concerned with structures, are still beyond the reach of available algo- the standard ill-conditioning of FEM equations only, as rithms and computer codes (Mock [3]). described, e.g., by Strang and Fix [6] and verified nuA well-known drawback of classical finite element merically by Ascher et al. [ 5 ] . Comparisons of the Foumethods (FEM's) is the ill-conditioning of the (linear- rier method to the FEM are made on the basis of physiized) equations. It is increasingly important for high ele- cally well-conditioned devices (diodes and transistors) ment densities and multidimensional problems. For semi- only. According to Ascher et al. [ 5 ] , the condition numconductor models, this problem has been quantified by ber of the total Jacobian can be explained by standard Rafferty et al. [4] and more recently by an extensive in- FEM results in this case. vestigation of Ascher et al. [5], both using a one-dimenFor FEM discretizations of second-order differential sional diode as a test device. The condition number of the equations a roundoff error propagation factor ( = condiFEM-equations was estimated numerically, after an ap- tion number) of O ( h - * ) ( h is the step-size) occurs due propriate scaling. (Ascher et al. [ 5 ] suggested a scaling to the second-order numerical differentiation [5], [6]. This corresponding to a transformation from carrier densities may be easily verified by anyone having access to LINto quasi-fermi levels as solution variables.) The condition PACK [7]. For nonuniform meshes, which are unavoidnumber of the Jacobian matrix is reported to be up to lo7 able in semiconductor modeling, the condition number depending on the mesh size and nonuniformity [5]. These degrades further with hav/hmin,where hminis the smallest step size and ha, is the average step size [5], [6]. In nuManuscript received October 13. 1988; revised March I , 1989 and Aumerical models of complex devices with l/hav = 100 gust 16, 1989. This paper was recommended by Associate Editor D.Rose. (number of mesh points in one spatial dimension) and The author is with the Lehrstuhl fur lntegrerte Schaltungen, Technische hav/hmin> lo3 these two effects total to a condition numUniversitat Munchen, 8000 Munchen 2 , Germany. IEEE Log Number 90377 15. ber of the order 10'. Detailed theoretical analysis of the
Abstract-A novel high-order approach to numerical modeling of semiconductor devices is presented. The new method is a combination of the classical Fourier-series Galerkin procedure, a special matrix calculus, and fast numerical pseudospectral techniques. The proposed algorithm renders the exact solution (machine precision) of the semiconductor equations in closed form of a trigonometric polynomial. The condition number of the diagonally dominant discrete equations is near unity. As a consequence, a highly accurate solution is achieved at moderate computer costs. The method has been implemented for one- and two-dimensional device models. Properties of the new procedure are demonstrated on examples.
D
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IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN. VOL Y, NO
conditioning of FEM-equations is given, e.g., in the monograph by Strang and Fix [6]. So far, Scharfetter-Gummel discretization [ 11 has been the major approach in semiconductor device modeling. It is, however, based on several important assumptions. Perhaps the most significant, is that it is basically one dimensional. Multidimensional extensions of it introduce an equivalent lumped network of one-dimensional pipes. The accuracy of this approach degrades drastically with increasing number of dimensions from one to two and three, to merely first order in the step-size O ( h l ) as shown by Mock [ 3 ] . A sufficiently accurate calculation of terminal currents requires additional work. Estimation and control of the discretization error is difficult. The numerical Fourier method [8], [9] originates in the trigonometric-series Galerkin approach. A matrix-operator calculus, developed by Axelrad [IO], and fast numerical pseudospectral techniques (Orszag [ 1 11) based on the fast Fourier transform algorithms, rendered a general and numerically effective approach suitable for the solution of strongly nonlinear semiconductor equations. The new method [8], [9] ofiers important advantages for the semiconductor device modeling. The basis functions of the trigonometric interpolation polynomial are orthogonal and smooth. They approach the eigenfunctions of the linearized equations. This assures relaxed coupling between the discretized equations. As a consequence, the equations are well conditioned. Well conditioning is of particular importance for multidimensional models, since fill-in occurring in a direct Gaussian solution of a wellconditioned linear system of equations can be neglected. This results in an asymptotically linear increase of computer time with the number of unknowns O ( M 1 ) .For the practically important medium range of values M , the increase of computer time is even slower (Fig. IO@), [ 121). The order of the trigonometric interpolation polynomial is the highest possible-equal to the number of sampling points. For all devices modeled in one and two spatial dimensions, the accuracy of the solution improves exponentially with the number of degrees of freedom (Fig. 10(a), [ 121, [14]). Machine precision (17 decimal digits, double precision) is obtained with approximately 100-200 Fourier coefficients per spatial dimension depending on the device geometry and distribution of sampling points. The discretization error is controlled by the convergence rate of the series, which is directly observable at all stages of the computation.
1 1 . NOVEMBER 1990
Fig. 1 . Regularized model of a semiconductor device: inclusion of isolators and contract-areas in the modeling domain.
The simplest modeling domain in two dimensions is a rectangle. More complex features, such as e.g., nonplanar isolating regions, trenches, etc. are described by spatially smoothly varying conductivity parameters-the intrinsic carrier density n, and mobilities p,,,p,, (Fig. 1). The most suitable boundary conditions for the application of the Fourier method are homogeneous Neumann conditions, that is, vanishing normal derivatives of the electric potential and carrier densities U , n , p at the boundary. These conditions are identically satisfied if cosineseries are used to represent U , n , p . No current flow across the boundary is possible in this case, therefore, injection of current from outer circuit components is within the modeling region. The corresponding physical picture is current-injection through contact-wires from a higher dimension (Fig. 1). Mathematically, simple linear distributed voltage-controlled current sources J,,, J,, in the current-continuity equations (2), ( 3 ) are used in the timestationary case. The resulting set of governing partial differential equations is a physically motivated extension of basic stationary versus Roosbroeck equations [3] (application to transient models via standard implicit time-discretization is straight forward): g,
= V(tVu)
E,,
=
- e
(n - p - NN)= 0
(1)
11. MATHEMATICAL MODEL
An efficient application of a global high-order method requires a regular formulation of the problem. That is, a regular-shaped domain and simple boundary conditions are of importance. Geometry features of the physical problem should be described by parameters of the differential equations rather than complicated boundary conditions and shape. A regularized formulation without loss of accuracy or generality has been developed for semiconductor models [ 141.
g,,
-V(
U
+ In n , )
V{ep,,(Vp - p . E,,)}
Ep = -V(
U
- In n ; ) .
-
R
-t J,, =
0 (3)
The current-injection model can easily be extended to account for capacitive or inductive components of the outer circuit, an advantageous property when solving
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AXELRAD: MODELING OF SEMICONDUCTOR DEVICES
transient problems. In the stationary case, a simple linear model is used: J,, =
g(z) = 0 , z = {U' n ,
C Y V ( x ) [vu- U(.)],
=
Y,(x)
PI,
g{gu, gn, g p > .
(7)
This nonlinear equation is solved incrementally with the (4) well-known Newton method
summation over n-type contacts Jp
The PDE (1)-(6) is an equation for the dependent vector variable z:
- [vx- U(.)],
:= zi
-
. g(z'),
((jg/&)-I
i
=
0,
*
, W.
(5)
(8)
In the above equations U , n , p are the electric potential, electron, and the hole densities; E , e , NN denote the dielectric constant, electron charge, and the net doping of the device. E,, E,, represent effective electric fields acting
The expression 6 g / 6 z is the Frechet- or operator-derivative of g ( z ) (for definition cf. [16]). The Frechet-derivativef 6 g / 6 z is obtained via formal differentiation of g ( z ) :
summation over p-type contacts.
VEV
f = 6g/62
=
-Vep,,nV - 6Jn 6U
Ve6,pV
e
-e
+SJP 6U
Vep,,(V
- n')/[T,(n
-SR/Sp
-6R/6n
on electrons and holes in a heterogeneous material. The space-dependent physical properties of the material are given by the dielectric constant E and the effective intrinsic carrier density n, (in general distinct for electrons and holes). These functions account for heavy doping as well as heterostructure effects and isolating layers inside the device (Fig. 1). The functions p,,, p Drepresent the electron and hole mobilities in the semiconductor. For the examples shown in this paper constant mobility was assumed. The function R represents the net recombination-generation rate in the semiconductor. For the examples presented in Section X, the Shockley-Read-Hall recombination generation model [15] is used ( T ~ rp , are electron and hole lifetimes ):
R = (n * p
+ E,,) - 6R/6n
+ n i ) + ~ , ( p+ n ; ) ] .
(6)
The set of (1)-(6) constitutes a general physically motivated regularized model of a semiconductor device. The physical consistency of the model is certainly comparable to conventional mixed Dirichlet-Neumann formulations. In addition, outer circuit components are naturally integrated in the calculation (Fig. 1). 111. METHODOF SOLUTION The Newton iteration is the best known procedure to solve nonlinear systems of equations. The Fourier method differs from FEM's in the kind of the discretization basis and correspondingly discretized equations. This section describes the general structure of the method. It introduces the Frechet-derivative of the governing equations and briefly discusses discretization approaches.
Vecc,(V - E p ) - S R / S p
With R according to (6):
6R/6n
[p'~,+ , pni(T,, +
=
SR/SP =
+ .",I/
T ~ )
[(n+
ni)Tp
[ n2Tp
+ nni (T,,+ T ~ +) n27
[ < n+
ni)Tp
+ (P +
+ (P +
ni)Tn12
ni)Tn12*
For the resistive current injection models
6 J n / 6 u , 6 J p / 6 u read: 6 J , , / & ~=
-
C
y,(.),
summation over n-type contacts
SJ,/Su =
-C
Y,(x), summation over p-type contacts.
(11)
Each iteration (9) implies a solution of a coupled system of linear differential equations for the vector update Az:
f(z')
*
A 2 = -g(z').
(12)
Since the solution of (12) cannot be performed analytically, we envisage a numerical procedure. Numerical calculation demands that the dependent variable z and the nonlinear function g ( z ) be approximated using finite sets of numbers. The Frechet-derivativef( z ' ) transforms then to a matrix of corresponding dimension (Jacobian matrix) and the coupled system of differential equations transforms to a linear algebraic system of equations. Obviously, there are various possibilities to approximate a continuous function using a discrete set of numbers. Any such representation is a kind of interpolation
IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN, VOL. 9. NO. I I . NOVEMBER 1990
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with the discrete numbers determining the interpolation coefficients. At this point the Fourier method and FEM's separate. In the FEM, dependent functions are represented locally, usually with low-order piecewise polynomials. Correspondingly, the discretization error is of low order in the step-size h . The basis of the Fourier method is the global trigonometric polynomial. Its component functions are orthogonal and smooth. Each of the coefficients represents its specific part of the entire function. The discretization error (truncation error of the series) is determined by the properties of the interpolated function only. Smoothness of the natural semiconductor functions U , n , p discussed in Section IX results in a rate of convergence faster than any, however high, power function of the coefficient number. In other words, the series coefficients decrease exponentially (Figs. 2 and 6). The orthogonality of the basis functions and rapid exponential decay of high frequency coefficients explain the observed weak coupling between discrete variables (Fourier coefficients) of the same function (e.g., U). Of course, coupling between the PDE, which can lead to physically based ill-conditioning for certain devices [ 5 ] , is unaffected.
In vector notation (vectors and matrices boldfaced):
G ( Z ) = 0.
(16)
An incremental solution of the coupled system of nonlinear Galerkin equations (16) leads to the transformed or discretized equation corresponding to the incremental differential equation (12):
F . A Z = -G
(17)
where A Z denotes the vector of cosine coefficients of the unknown's increment A z , G is the vector of cosine coefficients of the residual g , and finally F is the Jacobian matrix corresponding to the Frechet-derivative f (9). The Jacobian matrix F is a three by three array of matrix operators, each corresponding to an appropriate element of the Frechet-derivative array f(9). The differential operators f k l are interpreted as an application of several basic operators in sequence, i.e., a noncommuting product of basic operators. The corresponding matrix operators Fklare assembled as a noncommuting product of basic matrix operators. Merely two distinct types of basic operators are required: the differentiation operator and the convolution (multiplication of series) operator. These operators are derived in the next section.
METHOD IV. NUMERICAL FOURIER
V . BASICOPERATORS ON FOURIER SERIES
In the generalized Fourier method [8], [9] linearized differential equations (12) are solved using trigonometric series decompositions for dependent variables. This paper deals with basic features of the method and the underlying theory. The multidimensional theory must remain outside the scope of this paper. However, since such problems are the main aim of the method, the author could not refrain from presenting some multidimensional results (Section X). For the state function z = { U , n , p } global cosine-series decompositions in the domain x E [0, L ] are used according to Section 11:
A. Multiplication by a Constant
The simplest possible operation on a Fourier series is multiplication by a constant. Since the Fourier transform is a linear operation, the coefficients of the series are simply multiplied by this constant. The matrix operator [ k ] of the multiplication by a constant k is a diagonal matrix with constant values k on its main diagonal (this is a special case of the convolution operator (27) presented further):
k
- z(x) = c k [ k ] , , = k,
Z , cos ( v ~ x / L ) for v
=
0, 1, 2,
*
-
(18)
00.
3
m
z(x) =
C
Z,
x E [0,L ]
cos ( v a x / L ) ,
(13)
v=O
z
=
iL
L o z(x)
cos ( v a x / L ) d r ,
v = 0 , 1, 2,
- *
3
0
(14)
3
The series (13) satisfies identically the specified homogeneous Neumann boundary conditions (Section 11). Coefficients of the series 2 , are determined by Galerkin equations, i.e., requirement that the residual of the PDE g ( x ) be orthogonal to the cosine basis functions:
G = L S'g(x) o v
=
*
cos ( v ~ x / Ldr )
0 , 1, 2, . . .
3
03.
=
0, (15)
B. D@erentiation Differentiation of series is nearly as simple as multiplication by a constant. The matrix operator [ d l d x ] = A is a diagonal matrix: d -
dx
z(x)
A,,
=
d c 2, cos ( v a x / L ) dr
=
C
=
va/L,
- v a / ~*
Z , sin ( v a x / ~ )
for v = 0, 1, 2, 3 ,
..
7
03.
(19) The vector of sine coefficients of ( d / d x ) z ( x ) denoted by Z' is the product of the matrix operator A by the vector of cosine coefficients of z(x):
Z'
=
-A
*
Z.
(20)
AXELRAD: MODELING OF SEMICONDUCTOR DEVICES
1229
The series of higher order derivatives Z x are obtained by repeated application of the operator A . Changing series types for uneven order derivatives as well as changing signs must not be overlooked. Matrix operators [ k ] and A are diagonal operators. They do not couple coefficients with different indexes.
The complex Fourier coefficients C, are related to sineand cosine-coefficients denoted by CS,, C: according to [17] as
C. Multiplication of Series (Convolution) A further type of operator is the convolution matrix. This is a nondiagonal matrix, responsible for coupling of coefficients with different indexes in the matrix equation. The purpose of this operator is to obtain series coefficients of a product of two functions using their individual Fourier coefficients. This operator is derived here via the convolution theorem (e.g., [17]) for complex Fourier series. The result is then specialized for products of sine- and cosine-series. An equivalent approach [ 101 uses well-known expressions for products of trigonometric functions like sin ( a ) cos (0)= 0.5 (sin ( a - 0) + sin (CY + 0)). The complex Fourier coefficients C, of the product a ( x ) b ( x ) = c ( x )are calculated in terms of the Fourier coefficients A,, B , of the multiplicands
Inserting the relations (24) into the sum (23) and rearranging terms renders the following expressions for sineand cosine-coefficients:
+m
a(.)
*
C,, = 0.5 . ( C :
i
C.;)
f i , C : = C Y ,CS,= CY,. (24)
+m
C' = [ a ] :
*
-m
-m
7
=
v
B'
*
-CO
A substitution of the summation index ders
*
Cs= [ a ] : B'
+m +m -m
i
Following [ 101 these sums are written as matrix by vector products
c A , e f U x c B,e"" = c AYB,ei(vffl)X.
b(x) =
=
-
+ p ren-
+ [a]: . Bs + [ a ] : B'. *
(26)
Notation of [lo] is adopted: the superscript "c" or "s" indicates the type of the series A and the subscript '' + " or ''-" indicates the type of B ( " " for cosine series; for sine-series). The expressions for [ a ] correspond to the sums of (25). Their main parts 7,p = 1, 2, 3, * * , 00 read:
+
' ' - 1 9
/
+a
\
+m
The vector of Fourier coefficients C is thus determined by the convolution sum:
[ a ] : r , = 0.5
+m
C, =
c
p = -m
AT-,
*
=
[a]* B
[ a ] , = Ar-,,
for
7,p =
-03,
*
(Af~-pI
*
(+sign
(21 1
B,.
The convolution (21) is obviously equivalent to a matrix by vector product. This matrix [ a ] is the complex convolution operator matrix:
C
[allr,= 0.5
-
*
-2, 1 , 0 , 1 , 2 ,
* "
, 00.
(22) To derive the matrix operators for real sine- and cosineseries the sum (21) is rearranged. Its two branches for positive and negative p are united into a sum over positive indexes p only, as required by sine- and cosine-series: m
(7 - p )
*
ATr-pl
+ AS,,,). (27)
In special cases where A and B are pure sine- or cosineseries, the matrix expression (26) simplifies to a single matrix by vector product. As an example, for the A , B pure cosine-series, the cosine series C is the product [a]', B'. The matrix operator [a]: is a sum of a ToeA f r - , l and a Hankel matrix 0.5 * plitz matrix 0.5 A : + ? (except for the zero column p = 0). The Toeplitz matnx dominates for high row indexes since the elements of the Hankel matrix decrease rapidly with an increase of the row index corresponding to the convergence rate of the series A'. A 4 by 4 part of the matrix operator [a]:
-
,
+
-
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The third example is the element h3 = 6 g n / 6 p . This derivative describes the coupling between carrier densities n, p . For Shockley-Read-Hall generation-recombination (6) the Frechet-derivative f23 = - S R / S p is given in (10) (other models are easily included). Since SR/Sp is a mere convolution operator, the corresponding Jacobian matrix element is:
reads :
f23
= [n27p
+Y
[(n+ F23
These two basic operations-the differentiation and convolution (multipliction of series)-are sufficient to assemble the Jacobian matrix F. VI. MATRIXEQUATIONS Using the described two basic operator types, the Jacobian matrix F of the matrix equation (17) is composed by simple one-to-one substitution. This procedure is shown on some examples.
The Frechet-derivativef, defined in (9), is a three-bythree matrix of nine differential operatorsfkl, k, I = 1, 2, 3. Each one is transformed to a corresponding Jacobian matrix Fkl independently. The two simplest elements off are Frechet-derivatives of the left-hand side of (1) g , ( u , n , p ) with respect to n and p . Obviously, these derivatives are constants f i 2 3 &,/tin = -e andf13 = 6 g u / 6 p = e . Corresponding elements of the Jacobian matrix F a r e constant matrices (18): h2 =
-e
FI2= [-e]. (29) The second example provides the Frechet-derivative f,I 3 6 g U / 6 u = V E V .This differential operator is the application in sequence of a differentiation, multiplication, and again differentiation operator. The corresponding element FIIof the Jacobian matrix is (notice the minus sign produced by the second-order differentiation): f,l
= VEV
Fll
=
-A[E]CA.
(30)
The convolution operator [ E ] Y represents the multiplication of a cosine by a sine series (27), since E is represented by a cosine series ( E is nonzero) and in (17) E is multiplied by V ( u ' + ' - U ' ) , which is a sine series.
( 7 n
ni)7p
+ .p) + n f 4 / + ( P + ni)7n12
= [&I:.
(31)
To construct the matrix F23, point values off,, are calculated first, the discrete cosine transform 3, { f& ] is obtained via the fast Fourier transform (cf. Section IX) and with these cosine coefficients the [ ]', matrix operator (27) is constructed (plus-type since h3is applied to the cosine series p I - p ). Use of numerical algorithms for direct harmonic analysis (fast Fourier transforms) makes the method applicable and effective for arbitrary nonlinear terms in the equations. In this straightforward manner the total Jacobian matrix F is assembled. The result reads with Y" = [E Y"]',, Y p = [C Y * ] % cf. , (11): +
'
The matrix operators (17) and (26) have been derived for infinite series, therefore, the Jacobian matrix F is an infinite-dimensional matrix. For practical calculation the infinite series are truncated and the Jacobian matrix F becomes a square matrix of finite dimensions 3 M x 3M, where M is the number of cosine coefficients for each of U, n , p .
Only a small part of the matrix F is actually relevant for the numerical solution of the discrete Newton equation (17). To assure sufficiently accurate calculation of the Newton updates, a small number of elements grouped along the main diagonal is considered. The implemented data structure of Fo is a band matrix with decreasing bandwidth for high row numbers (Fig. 2, inset). Elements of a row, which are smaller than a 0.001 part of the main diagonal element, are dropped. An estimate for the necessary bandwidth is obtained using the expressions for the Jacobian F,, of the thermal equilibrium equation (special case of PDE (1)-(3)):
F,,
=
- A [ E ] ~-A [ e ( . + p ) ] : .
(33)
The accuracy of the iterative Newton's solution is solely determined by the residual G ( Z ), therefore, no loss of accuracy results from the truncation of the Jacobian matrix. Calculation of G ( Z ) is described in the next section.
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AXELRAD: MODELING OF SEMICONDUCTOR DEVICES
IOL8 [cm-
loL4
IO’O
IOb
IO2 I
I
I6
256
64
Fig. 2. Fourier coefficients of electrons and holes in a p+n-diode on a loglog scale (255 sine coefficients). Convergence rate is exponential. Inser: Structure of the Jacobian matrix.
VII. EVALUATION OF THE VECTORRESIDUAL Semiconductor equations must be solved very accurately for physical reasons (cancellation of drift and diffusion currents). The right-hand side of the discrete equation (17), i.e., the cosine transform of the residual of the PDE must be calculated carefully. Its accuracy determines the accuracy of the iterative solution. To achieve high accuracy, all differentiations are performed on series coefficients using the matrix operator A ( 19), whereas nonlinear operations (e.g., multiplications) are performed on point values of dependent functions. This so-called pseudospectral strategy [ 111 is one of the key steps to an effective application of the Fourier method to strongly nonlinear semiconductor equations. The first step in calculating G is the evaluation of the functions U , n , p, U’ = du/a!x, n’ = dit/&, p f = dp/a!x, and E,,, E,, using the series coefficients U,N, P ( %,Isdenotes the cosine- /sine-transform and 3i,sits inverse ): U =
5,1{u}
U’ =
5s-1{AU}
E,, =
-U’
- (In ni)‘.
(34)
With these functions (carrier densities and their derivatives calculated analogously) cosine coefficients of the residual G = 5,{ g } are calculated: G, = -A
*
G,, = -A G,,
=
-A
S,{EU‘}- 3.{ e ( n 3.{ p,,(n’
*
+n -
TS{p P ( p ’ - p
*
-p
- NN))
E,,)} - %,{R
+J
~ }
VIII. COORDINATE TRANSFORMATION A characteristic property of the semiconductor state variables U , n , p is their extremely intensive variation in small critical areas of the devices (e.g., inversion layers or space-charge zones). In typical devices most of the solution’s variation is confined to a small part of the device. Fig. 3(a) presents as an example a MOS-capacitor under strong inversion conditions. The thin needle-shaped inversion layer (electron density n ( x ) ) is recognized at the Si-Oxide boundary. According to (1) this electric charge distribution, together with a rapid change of the dielectric constant from csi = 11.9 to cOxide = 3.9, causes fast variation of the electric field E = - d u / & at the Si-Oxide interface. Obviously, such a device is not effectively described using equidistant sampling as required by the fast Fourier transform. To achieve the necessary sampling density in the inversion layer, which is merely some l o w 3pm wide, the overall number of sampling points would be enormous. The solution to this problem is a coordinate transformation, stretching critical zones of the modeling domain and compressing parts with slow variation of the state variables. Fig. 3(b) shows the results of a possible coordinate transformation applied to the device of Fig. 3(a). The needle-shaped inversion charge of Fig. 3(a) transforms to a smooth bell-like function in the new coordinate 4. The modeling of this MOS capacitor is based on the assumption that the effective bandgap increases smoothly from 1.12 eV for Si to 9 eV for Si02 over the distance of approximately two atomic layers of silicon ( = 1 nm). This assumption reflects the impossibility of abrupt changes of physically observable quantities; it complies with the nonlocal nature of the quantum mechanical energy-band model. The inversion charge density n ( 5 ) does not, therefore, change abruptly on the extended scale 4 in Fig. 3(b) from silicon to oxide. Nevertheless, the variation of the inversion charge n ( x ) is still very fast on the real world scale (Fig. 3(a)). The Fourier method is applied to the coordinate transformed problem. The transformed equations are obtained as follows. The coordinate transformation is defined by
4
=
4w.
The independent variable x is substituted by the new independent variable 4 ( x ) . This transforms differential operators to
E,,)} - S , { R - J,,}.
(35) The use of direct harmonic analysis approximated with the discrete Fourier transform FFT’s provides accurate transforms of the residuals of the PDE in a simple and effective way, not being restricted to any particular type of nonlinearities in the PDE.
(36)
(37) The convolution operator (27) remains unaffected. The differentiation operator (19) transforms to:
[i]
=
-
[CY]‘A.
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e.g., in Fig. 3) since a ( x ) is sampled densely where it is large. It is well represented using a comparatively small number of cosine coefficients (typically 3-5). The operator matrix [ a ] is thus a narrow-band matrix.
0.50
-1.00
Inversion Layer 0.00
0.13
0.25
0.38
0.50
X [pml (a)
2.00
1
’ I
I
\
\
M
0.25
0.98
I
0.50
t [(lml (b) Fig. 3. Coordinate transformation. Electric potential, electron, and hole concentrations in a MOS-capacitor in strong inversion. (a) U , n, p in x-coordinates,,real world scale. (b) U , n, p in E-coordinates, critical zone enlarged.
Equations (37) and (38) mean that nonequidistant sampling requires a multiplication by a( 4 ) ( 1/stepsize, i.e., the point density) when differentiating. This result corresponds to the well-known difference expressions in the classical methods. Consider a simple example of a coordinate transform. After choosing a set of sampling points x, the new coordinate values are defined by 4, := v. The differentiation can be approximated by the central difference:
-
=
(Y”+l
a” *
- Yu-I)/(~,+l
=
U ( . )
Inversion Layer 0.13
dY/h =
!
\
-1.00 0.00
IX. TRIGONOMETRIC POLYNOMIAL A. The Discrete Transform In a practical application of the Fourier method, the infinite series must be truncated to workable size. The resulting trigonometric polynomial is an approximation to the series. The approximation error is the truncation e or of the series, the part of the series not included in the polynomial. The convergence rate of the series is, th refore, of crucial importance for any practical applic tion of the Fourier method. For natural semiconductor state variables U , n , p, a simple estimate for the conver ence rate is obtained, assuring exponential decrease of th truncation error with an increasing number of polynomi coefficients. This estimate is confirmed by numerical results (Figs. 2, 6, and 10(a)). The series (13) is truncated to the polynomial
- &-I)
( Y ” + I - Y,-I)
a, := l / ( x , + l - X , - I ) . (39) Equation (39) is merely a crude approximation. In the present implementation of the Fourier method the functions defining the coordinate transform a ( 4 ), x ( 4 ) are calculated iteratively. The point density function a is either controlled automatically by, e.g., the charge density or chosen by hand. The mapping function a ( 4 ) thus obtained is a very smooth one even in cases when a (x)changes rapidly (as,
c U”
”=O
(40)
cos ( v n x / L ) .
+
To calculate M 1 coefficients of an interpolation polynomial M 1 samples of the interpolated function are necessary and sufficient. For equidistributed sampling points x, the integral expression (14) is approximated by the sum
+
-
U, =
M
1 ~
M
+1
c
u(x,)
p=o
I
- cos (v.x,/L),
+ 1)
L. (41) These two expressions (40) and (41) are discrete cosine
x,
= p/(M
*
transform (DCT) equations. Written in matrix form, they read: U =
c-‘
U,
*
U =
c . U.
(42)
With the symmetric and orthogonal cosine transform matrix and its inverse
c,’
= cos
(
up _ _
M L )
) :
c,, = M 1+ 1 cos ( v p M + 1 ~
)
*
(43)
The vector of cosine coefficients U is obtained as a product of the transform matrix C by the vector of sampling values U. The practical evaluation of the cosine transform and its inverse is carried out effectively via the fast Fourier transform reducing the number of arithmetic operations from O ( M 2 ) for a direct evaluation of (42) down to O ( M log2 M ) .
B. Truncation Error of the Series-Regularity
of the Solution. The error of replacing the series by the polynomial is approximately the tail of the series (there is an additional
/
AXELRAD: MODELING OF SEMICONDUCTOR DEVICES
1233
error of the same order in the retained coefficients U,,v = 0, 1, * * , M )
-
m
dl
=
c
u=M+I
m
u,cos (v.rrx/L) I
c
v=M+I
p,(.(44)
The discretization error 03 crucially depends on the convergence rate of the series U,,which is determined by the global regularity of the solution. In this important question we must distinguish between two basically different things. Physical (quantum mechanical) reality and functions representing it (e.g., electric potential, current densities, etc.). These are smooth in all derivatives (C”). Solutions of mathematical equations modeling the reality. Adequate approximations to real physics a) by mathematical models are smooth. It is, of course, possible to use other, e.g., piecewise approximations with corresponding penalty in the accuracy. All relevant parameters of the semiconductor PDE (1)(6) represent physical quantities. In the Fourier method, they are modeled by smooth C mfunctions. Most important is, of course, the net doping density NN, a smooth physical quantity produced by diffusion processes. It is a given function modeled by a short trigonometric polynomial. Further coefficients of the semiconductor equationspn, pp, R , €-are modeled by smooth functions of the solution u , n , p and geometry/material parameters NN, Eg. Homogeneous Neumann boundary conditions that are used in the Fourier method are C” (except for the corners, which do not seem to restrict the regularity of the solution for practical semiconductor devices, Figs. 9 and 10). The theory of elliptic partial differential equations guarantees infinite smoothness of the solution U , n, p E C” since all coefficients, right-hand sides, and boundary conditions of the PDE belong to C”-the class of infinitely differentiable functions (Gilbarg, Trudinger [2 11). According to a well-known convergence theorem for Fourier series [ 171, if the A-order derivative of a periodic function u ( x ) = u ( x + 0 2 L ) exists for allx, the series coefficients of this function converge at least as fast as v (notation of (14)):
-’
Iu,~
< c U-’, forA 03. (45) For cosine-series representation of U , periodicity requirements amount.to vanishing odd-order derivatives at the boundary points x = 0, L (or boundary lines/surfaces in the multidimensional case). This condition can be satisfied for semiconductor devices without loss of generality (cf. Figs. 3, 5, 7(a), 9(a)). With the estimate (45) for the convergence rate of the series, an estimate for the truncation error is obtained: -+
The truncation error thus decreases faster than any power function of the polynomial order M , i.e., exponentially:
dl
=
O(ePM).
(47)
The numerical results in Figs. 2 and 6 closely follow this estimate. For large v , the series coefficients N , , P , approach the limiting exponential function 10I8 . exp ( - v * const) with const = 0.2. For different coordinate transformations (Section VIII) and device structures this constant varies, but the exponential shape of the curves remains unchanged.
X . APPLICATION EXAMPLES The generalized Fourier method has been implemented in C on a UNIX workstation. Several one- and two-dimensional devices including a MOS-capacitor, various p-n-junctions, pin-diodes, and bipolar transistors have been successfully modeled. The calculation of the Jacobian matrix and the solution of the discretized system of equations have been carried out using single precision arithmetics (6 decimal digits). This reflects well conditioning of the method for all modeled devices. No damping was required to achieve convergence of the iterative solution. The accuracy improves exponentially with increasing M , doubling the number of correct digits in the solution when M is doubled, whereas the computer time and memory requirements rise slowly with the number of Fourier coefficients M . This behavior is of particular importance for the multidimensional case (Fig. lo@)). Two examples of application are presented. The onedimensional model of the bipolar n-p-n transistor is based on the sine-series approach suitable for Dirichlet boundary conditions [8], [9]. The two-dimensional model of a diode uses a cosine-series approach [ 141 described in this paper. The n-p-n transistor is a 4-pm device with a base width of approximately 0.8 pm. The doping concentration of the device is shown in Fig. 4 on a log scale. The current injection in the base was modeled similar to Cook [19]. The potential distribution in the device for bias conditions: UBE = 0.52 V , UcE = 1.04 V is presented in Fig. 5 . The nonequidistant sampling is indicated by vertical lines drawn at sampling points. Dense sampling especially at the high-doped base-emitter p-n junction is clearly seen (a total of 256 points). This concentration assures exponential convergence rate of the Fourier series (Fig. 6), corresponding to the estimate (47). The shape of plots in Fig. 6 resembles closely the results of diode simulation Fig. 2. Modeling results of the transistor in the active region UcE > UBE (base-emitter junction forward-biased, basecollector junction reverse-biased) are presented in detail.
1234
IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN, VOL. 9, NO. l l , NOVEMBER 1990 L O I-
10
-7
-.. 4 -.
N N(xl
-
1
I
LO
LO
10
I
*. 2.00
3.00
4.00
0.00
2.00
1.00
3.00
X [pl
4.00
x Iuml
(a)
Fig. 4. Bipolar n-p-n transistor doping concentration. 10
LO
-
jn diff
\
00
:
-O.SO)___-0.00
1.00
2-00
3.00
..00
X [pml Fig. 5 . Potential distribution in the transistor for Us, = 0.52 V, U,, 1.04 V.
10 e
=
LO -2
LO
10-0
LO
0
1.00
2.00
3.00
I. 00
10
X [wml 5.3 0.00
2.00
4.00
6.00
00
log* f
Fig. 6. Fourier coefficients U,,N u , P , on log-log scale.
Fig. 7Ca) shows the charge carrier concentrations in the device on a log scale. A high global accuracy of the solution is observed-values ranging from lo” down to 10” for electrons and from 10l6down to lo6 for holes are accurate. This accuracy is necessary to calculate the current densities. Fig. 7(b) and (c) show the drift and diffusion current components together with the total electron and hole currents. Drift and diffusion currents are large and vary intensively inside the device (broken lines). Their resultants, the total electron and hole currents, are smaller by several orders of magnitude. Even for this relatively large collector current of 0.5 A/cm2, still approximately
-[ t
-0.30
jn
~
:
:
2.00
3.00
- 0 . EO 0.00
1.00
1.00
X [pml
(d) Fig. 7. Bipolar transistor in active regime UeE = 0.52 V , LICE = 1.04 V. (a) Charge camer concentrations n , p (log scale). (b) Drift, diffusion, and total current densities of electrons (log scale). (c) Drift, diffusion, and total current densities of holes (log scale). (d) Hole, electron, and total current densities (linear scale).
AXELRAD: MODELING OF SEMICONDUCTOR DEVICES
5 decimal digits in electron current (Fig. 7(b) and 10 decimal digits in hole current (Fig. 7(c)) are cancelled. Fig. 7(d) presents the electron and hole currents on a linear + .scale. The hole-current injection in the p-doped base is observed as a change in hole current and total current in the base, whereas the electron current density remains virt I/ tually unchanged throughout the device (recombinationgeneration effects are not significant for these bias conditions). The device was modeled for various bias conditions. Results are compiled for the following figures: Fig. 8(a) shows the collector, emitter, and base currents for a fixed UCE [VI base-emitter voltage (UBE = 0.52 V) versus collectoremitter voltage ( U,, from 0 to 1 V). Saturation of the collector current in the active region is observed. Fig. 8(b) displays a family of collector currents versus collector-emitter voltage for several base-emitter voltages, demonstrating exponential control of the collector current by the base-emitter voltage. Fig. 8(c) shows the collector, emitter, and base-current densities on a log scale versus base-emitter voltage U,, for fixed U,-, = 1.04 V. All currents rise exponentially over several orders of magnitude ( 10-5-103 A/cm2) with increasing base-emitter voltage. Finally Fig. 8(d) presents the static common emitter current gain Po = Z c / Z b as a function of the base-emitter UCE [VI voltage U B E . Nonideal effects degrading the current gain are observed. Recombination in the base-emitter junction contributing to the base current (Sah-Noyce-Shockley current [20]) reduces the current gain for low UBE.High injection effects-injected carriers effectively increase the base doping-reduce the current gain for high U B E (Webster effect [20]). The two-dimensional example [12] is a 0.7-pm by 0.7pm diode with a peak doping concentration of 10l6 ~ m - ~ . Fig. 9(a) and (b) present the electron concentration and the current density in the device at 5 U, forward bias. Contact regions in the corners of the device are clearly 0.om recognized: current injection is according to the current UBE [VI sources model (4) and (5); thermal equilibrium n p = (C) n? in contact areas is achieved by locally reduced electron and hole lifetimes (cf. (6)). The accuracy of the two-dimensional solution is presented in Fig. 10(a) [12]. It improves again exponentially with the number M of Fourier coefficients as in the onedimensional case [14]. This infinite order accuracy is a consequence of the exponential convergence of the series in one and two spatial dimensions [14], [18]. Diagonal dominance and well-conditioning of the equations result in a slow rise of the computer time for the approximate Gaussian decomposition of the Jacobian (Section VI) with increasing number of equations M (Fig. 0.so UBE [VI 10(b) [12]). For practically relevant values of M( 162 12g2) the increase of the computer time with increasing M is slower than linear. A comparison with the numeri- Fig. 8. Current-voltage characteristics of the n-p-n transistor. (a) Collector-, emitter- and base-current densities versus U,-,for U,, = 0.52 V. cally verified O ( M 2 ) rise in computer time for the solu(b) Collector current density versus U,, of various LIB,. (c) Collector-, tion of finite difference equations is given in Fig. 10(b). emitter-, and base-current densities versus U,, for U,, = 1.04 V. (d) Both the accuracy and computer time curves of Fig. 10(a) Static common-emitter current gain Po = j
