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The spreading resistance of a homogeneous slab on a high-resistivity substrate: Mixed boundary value solutions
Solid-SlateEktmnics Vol. 25, No. 9, pp. 877-884,1982 Printedin Great Britain.
003a1101/821.rJ3/0 0 IS2 Peqamon Press Ltd.
THE SPREADING RESISTANCE OF A HOMOGENEOUS SLAB ON A HIGH-RESISTIVITY SUBSTRATE: MIXED BOUNDARY VALUE SOLUTIONS M. S. LEONG, S. C. CHOOand L. S. TAN Departmentof ElectricalEngineering,National Universityof Singapore,Kent Ridge, Singapore0511, Singapore (Received24 August 1981;in revisedfom 29 January 1982) Abstract-The range of applicabilityof the mixed boundaryvalue method for calculatingspreadingresistance is extended to a homogeneous slab with a disc contact source and backed by a substrateof arbitrary,but finite resistivity. Solutions are presentedin terms of the spreadingresistancecorrectionfactors and the source current density distributionsfor a slab of varyingthickness and with various high resistivity substrates.In particular,the results for a thin slab indicate that, as the substrateresistivity increases, more and more of the source currentis concentratednear the edge of the disc electrode. A comparisonis made of the source currentdensity and potentialcorrespondingto the mixed boundaryvalue methodwith those given by the uniformflux and the variableflux (power-loss)method. It is found that, except for large slab thicknesses, the source potentialdistributionsfor a slab with a high resistivitysubstrateare not strongly influencedby the particularform of the source currentdensity distributionassumed in either the uniformflux or the variable flux method. In consequence, both these two methods yield correction factors which agree quite closely with those derivedfrom the mixed boundaryvalue method.
INTRODUCTION
In two previous papers [1,2], we have used the mixed
boundary value approach to solve the spreading resistance problem of a semiconductor slab with both a uniform and a nonuniform resistivity profile. The treatment was, however, limited to structures whose substrate was perfectly conducting. As pointed out elsewhere[3], the mixed boundary value problem for structures on a perfectly insulating substrate is intractable, and the only solutions available for this sype of structures are those of the approximate methods based on assumed source current distributions. The case of a perfectly insulating substrate is of interest in spreading resistance measurements, because a perfectly insulating boundary can be used to represent a pn junction. In practice, however, it often happens that the substrate has a resistivity which, while large compared to the slab resistivity, is not infinite; and even in the case of the pn junction, a structure with a high resistivity substrate can still be used to provide a useful approximation. In this paper we shall show that the mixed boundary value approach can be applied to a slab whose substrate is of arbitrary, but finite resistivity. In particular, solutions will be presented for a homogeneous slab with substrates whose resistivities are greater than the slab resistivity by several orders of magnitude. In presenting these solutions we shall be concerned not only with the correction factors but also with the source current density distributions, since, as has been shown recently[4], a detailed study of the source current density distributions can lead to new improved methods of calculating the spreading resistance, such as the variational method. TnEPRoil~ Consider the two-layer structure shown in Fig. 1. Here the semiconductor slab of resistivity p, and thickness D 877
Fig. 1. Geometryof slab with a disc electrodeand a substrateof arbitrary,but finite resistivity.
is backed by a substrate layer of resistivity pP This second layer is very thick, so that the sink for the current can be regarded as being at z = a. Note that the radius of the source electrode is given as unity, which implies that all the geometrical dimensions given in this paper are normalized with respect to the electrode radius. We assume that the potential & and & in layer 1 and layer 2 satisfy Laplace’s equation, and that the boundary conditions are as follows: Surface
g*=v, osr<1, +o,
r>l,
z=o z=o
(1) (21
where V is the value of the potential at the source disc electrode, the reference potential being at z = m.
878
M. S. LEONG et al.
Interface of adjacent layers W,
Sneddon [5] as
2) = M,
z),
z = D
(3)
p;‘$‘(r,~)=p;~~(r,z), z=D
(13)
(4) where f(t) is the solution of the following Fredholm integral equation of the second kind:
Substrate lim &( r, z) = 0.
(5)
ZQ
By applying the above boundary conditions to Laplace’s equation, and defining the reflection coefficient k, as k
_pz-PI
’ we
A(p) = F I,’ f(t) cos (pt) dt
(6)
Pz+PI
obtain the following pair of dual integral eqns: m p-‘A(p)
dp = 1, 0 G r < 1
I0
cc
A(p
dp = 0,
r> 1
(8)
Being an even, real and continuous function, f(t) can be solved numerically by using Simpson’s rule for the integration from -1 to tl, as in the case of the perfectly conducting substrate. By way of illustration, the behaviour of f(t) for a high resistivity or insulating substrate with pZ/p, = 1999 (i.e. k, = 0.999) and D = 0.1, 1.0 and 10.0 is shown in Fig. 2. We note from the figure that the behaviour of f(t) is opposite to that for a perfectly conducting substrate, the curves in the present case being generally concave rather than convex in shape. Once f(t) is known, the correction factor FE can be obtained as follows:
I0
FE = 1 o’f(t) dt. /I
where H(p)
=
I+ k
(15)
epzDp
1- k, eC20p and J,(rp) is a zeroth order Bessel function of the first kind. The above equations are analogous to the dual integral equations found for the slab with a perfectly conducting substrate, with the kernel tanh (pD) replaced by (1 + k, em2*)/( 1 - k, e-‘*). The latter expression in fact encompasses the case of the perfectly conducting substrate, for which p2 = 0 and k, = - 1. Expanding the kernel, we have 1 t k, emzDp 1 - k, e-‘&
=1+25
“=I
k,“e-“‘@’
---J--t -I
0
(9)
f(t) -
0.2
for k,
-
p-1
0.1
tI
k(p)lJ,(pr) dp = 1, OGr
mp-‘A(p)U+
I0
(10)
=O.
r>l
,
(11)
I0
I
,
,
0
-1
m A(pV,(pr)dp
+1
,
,
,
I
I
t1
:;)IIIY,
t
, t
with k(p) = 2 “~, k,” emZnDp.
(12)
The solution to eqns (10)-(12) is then given by
-1
0
+1
Fig. 2. Behaviour of the function f(t) for a slab with an insulating substrate: h/p, = 1999 (k, = 0.999). The normalised thickness D =O.l, 1.0 and 10.0.
The spreadingresistanceof a homogeneousslab on a high-resistivitysubstrate
al9
Table 1. Correctionfactors as a function of normalisedthickness D for p~/pl=19,199and 1999,correspondingto k, = 0.9,0.99 and 0.999, respectively pk D P - 1% 7.
1
P = 1990) 2
p = 1999P1 *
0.01
15.32
79.03
202.1
0.03
12.09
44.37
90.05
0.05
10.33
32.42
60.54
0.07
9.138
26.03
46.36
0.10
7.900
20.43
34.79
0.30
4.614
9.267
14.13
0.50
3.481
6.341
9.265
0.70
2.885
4.949
7.039
1.00
2.380
3.834
5.299
2.00
1.720
2.451
3.184
3.00
1.485
1.973
2.461
4.00
1.365
1.731
2.097
5.00
1.292
1.585
1.877
10.00
1.146
1.292
1.440
50.00
1.029
1.059
1.088
The correction factors are shown in Table 1 for values of thickness D ranging from 0.01 to 50.0 at pZ/p, = 19 (t = 0.9), p2/p, = 199 (k, = 0.99) and pZ/p, = 1999 (kr = 0.999). It is observed that, as expected, the correction factor increases as the substrate becomes more insulating. It is also noticed that as the slab thickness increases, the correction factor decreases asymptotically to unity. This is consistent with our physical intuition, for, as the substrate recedes from the source, its influence on current flow in the slab decreases and the situation approaches that of an infinitely thick slab. Neverthless, even with a thickness of 10, the correction factor is quite different from unity, showing that it is still necessary to apply corrections to the measured spreading resistance for such large thickness. COMPARISON WITH AF’PROXlMATEMElMOlW3J
It is of interest to compare the solutions obtained from the mixed boundary value or exact method with those given by the approximate methods[3] based on assumed source current distributions. In making such a comparison, the quantities of interest are, besides the correction factors, (a) the current density distributions corresponding to the constant-potential boundary condition in the exact method and (b) the potential distributions corresponding to the two source current density distributions assumed in the approximate methods, viz. the uniform current distribution and the variable current distribution given by the classical solution for the infinitely thick slab.
(a) Current density in the exact method The source current density distribution is related to f(t) in eqn (14) as follows [31: J(r,O)=p;‘V~J&p)dp~~~(t)cos@t)dt. (16) In Ref. [3], J(r, 0) was computed from the above equation after f(t) was expressed as a Fourier series in the interval (0,l). The use of a Fourier expansion for f(t), however, tends to give rise to slight oscillations in J(r, 0), especially near the edge of the disc contact. These oscillations are well known in Fourier analysis as Gibbs phenomenon. The oscillations can be minimised by taking a large number of terms in the Fourier expansion for f(t) and by using various smoothing functions (e.g. Lanczos’ or Fejer’s method) to accelerate the convergence. However, since f(t) is an even, smooth and wellbehaved function a computationally more efficient method is to approximate it by an even-degree polynomial
with bo, b2, b4,. . . determined by a least-square fit of f(t). The number of terms required to achieve a given accuracy depends on the substrate resistivity and on the slab thickness. However, it was found that even in the
880
M.S.LEONGdol.
worst case, by taking the first five terms in the polynomial, the approximation obtained does not deviate by more than l/4% from f(t) at any point within the range (-l,+l). By using the polynomial approximation up to the b,t* term then, it can be shown (see Appendix) that the current density distribution under the source electrode is given by J(r,0)=~[c,(1-r3-“‘-c!~(1-r3”’ 1 t 3
1 C,( 1 - r2)‘” - jj C,( 1 - r2)5/z
t j& C,( 1 - ?)“‘I
(17)
where the coefficients C,, . . . , C, are related tG b,,, . . . , b, through eqn (A4) in Appendix. From eqn (17), it is clear that the present procedure for computing J(r,O) is simpler and more direct than the method earlier proposed, as it does not involve a numerical integration as in eqn (3) of Ref. [3]. (b) Potential distributions in the approximate methods Uniform flux method [3]: For an assumed source current distribution which is uniform, the potential distribution over the source region is easily shown to be given by: Mr, 0) = $6
H(p) y
J&r) dp
(18)
where I is the total current entering the source electrode, and .J1@)is a first order Bessel function of the first kind. Variable flux method[3]: For an assumed source current distribution of the classical form, it can likewise be shown that t$,(r, 0) = $
lrn H(p) y
J&r) dp.
(19)
The only diierence between eqns (18) and (19) and the corresponding solutions for the slab with the perfectly conducting substrate131 lies in the replacement of tanh(pD) by H(p). This is also true of the solutions for the correction factors. REWLTSANDDIsCUSSlON
In order to provide a uniform basis for comparing the potential and current density distributions under the source electrode calculated by the various methods, we set the current I in the approximate methods to unity; and since, in the exact method, the current I entering the source electrode is given by 4p;‘VJ,‘f(t)dt, we obtain I = 1 in this case by making V = p,/4 j’,’f(t) dt. By using the expressions given in eqns (17)-(19) and with the above values for V and I, the current density tResults for Schumannand Gardner’smethod have not been included,because the methodis inherentlyinconsistent[3].
and potential distributions were computed for values of slab thickness D ranging from 5.0 to 0.1. The substrateto-slab resistivity ratio pJp, used was 1999, corresponding to a reflection coefficient of 0.999. The results are shown in Figs. 3-6. Several interesting points can be observed from this set of figures. First of all, consider the current density distributions shown in parts (a) of the figures. It is observed that for a thin slab (D = 0.10, for example) most of the current is concentrated near the edge of the disc source contact so that the exact current density distribution is quite unlike either of the distributions assumed in the approximate methods. This current crowding effect can be explained physically by the tendency of the insulating substrate to repel the current carrying charges towards the edge of the disc contact, an effect which is to be contrasted with the attractive affinity of a conducting substrate [3]. Turning our attention now to parts (b) of the figures, we observe that the potential distributions under the source contact are not strongly influence by the assumed current density distributions. The variable flux method, in general, gives rise to a more nearly uniform potential distribution under the source contact than does the uniform flux method. Nevertheless, except for large slab thickness (e.g. D = 5.0), the source potential distribution as computed by the uniform flux method is not too different from the exact distribution. For instance, the maximum difference in potential between the two methods for D = 1.0 is less than 6%. The correction factors FE calculated from the mixed boundary value method are shown in Table 2, together with those calculated from the approximate methods-t We observe from the table that the variable flux method (power-loss definition for the spreading resistance) gives a correction factor Fp which is quite close to FE of the exact method throughout the range of thickness considered. Nevertheless, the correction factor Fu computed by the uniform flux method is not too different from FE either, the maximum deviation being only 7.2% at D = 0.01. The behaviour of both Fp and Fu is thus consistent with that observed for the source potentials. In particular, the good agreement of Fp and Fu with FE is seen to be a consequence of the insensitivity of the source potentials to the assumed source current density distributions. In order to investigate the phenomenon of current crowding further, we have obtained results for a slab with a thickness 6xed at D = 0.1 but with pJp, varying from 19 to 199,999. The results for the current density and potential distributions are shown in Fig. 7(a) and (b). Reference to Fig. 7(a) shows that increasing the substrate resistivity causes more of the current to be pushed to the edge of the disc electrode. However, by the time that p2/p, has exceeded a value of 1999, the current crowding effect has essentially saturated, so that further increase in substrate resistivity produces relatively little change in the shape of the current density distribution. Similarly, from a careful examination of Fig. 7(b), it can be seen that as p2/p1 increases beyond 1999, there is not much change in the percentage deviations of the variable- and
The spreading resistance of a homogeneous slab on a high-resistivity substrate 4
1
0.9 -
0. j?..
Io=s.ol
0.6 -
0.7
----.-
Q 0.6-
kG Q2
0.5 -
10r5.0
0.8 I.,
.
2 Y
881
EXACT VARIABLE UNIFORM
-
FLUX FLUX
------. -
]
EXACT V4fflABlE UNIFOW
FLUX FLUX
o.c-
0.2
a1
0.1
t
OO I
0.2
0.6
0.c
0.6
1.0
OO
r
02
0.b
0.6 c
0.6
1.0
Fig. 3. The source current density I(r,O) and potential +(r,O) distributions calculated by using the exact and approximate methods for a slab with pJpl = 1999(5 = 0.999).D = 5.0.
0.9 3.!
lo.-1
0.8
I 0.7 0 i Y
0.6
-----.-
EXACT VARIABLE UNIFORM
I
3.1 --------
FLUX FLUX
EXACT VARIABLE UNIFORU
FLUX FLUX
2.5
1.0
0.2
44
0.6 r Ia1
R’I
ti0
) lb)
Fk. 4. The source current density I(r,O) and potential &r,O) distributions calculated by using the exact and approximate methods for a slab with h/p, = 1999(&,= 0.999).D = 1.0.
P
M. S. LEONGet al.
882
1 D=O.S 1 0.6 .,
-------.-
;; i <
EXACT VARIABLE UNIFORU
FLUX FLUX
3.0 ..
0.7..
I I I
06..
:
ic
2.5.: ---- .-.-_-_____
I I
i?i ci 0.s..
___--~:_y<
-=Z
0 ;
2.0..
Q )3 Q s Q
G i= 2 c” 2
0.4.. -____.---.-.-. a3..
---------
1.5 .,
EXACT VARIABLE UNIFORN
FLUX FLUX
0.8
1.0
1.0..
00
0,2
0.4
06
0.8
1.0
OO
0.2
0. c
0.6 r
r 101
lb1
Fig. 5. The source current density J(r, 0) and potential &r,O) distributions calculated by using the exact and approximate methods for a slab with h/p, = 1999(k, = 0.999). D = 0.5.
Ll7:O.l
1 4.5
-----.-
EXACT VARIABLE UNlFaPN
cc FLUX FLUX
3.5
lo=ar --------
1
EXACT VARIABLE UNIFORM
FLUX FLUX
---.-_ ___--_-_____Y-_.._
d
0.2
0.4
0.6 P
0.8
1.0
Fig. 6. The source current density J(r,O) and potential $(r,O) distributions calculated by using the exact and approximate methods for a slab with h/9, = 1999(k, = 0.999). D = 0.1.
.
The spreading resistance of a homogeneous slab on a high-resistivity substrate ,
----.-
Ia’.
-_---_ __---
0.8
EXACT VARIABLE UNiFO(?U --
FLUX FLUX
---_
-_zz;m_
16,
p&y
= 199999
14.. -----_ -----------_----_~
--u_
12
19999
10.. __L-_Z_,_~, 1999
8,.
2,-_--zxz-_;y----
---=--
OO
0.2
0.4
IQ)
0.6 r
0.8
19
1.0
lb1
Fig. 7. Effect of substrate resistivity on source current density I(r, 0) and potential ~$(r,0) distributions., D = 0.1.
Table 2. Correction factors calculated as a function of normalised thickness D by using the mixed boundary value method (FE), the variable flux method (FP), and the uniform flux method (FU). pJp, = 1999, corresponding to k,= 0.999
D
SSE Vol. 2J. No. 9-D
FE
PP
Fu
0.01
202.1
208.4
216.7
0.03
90.05
92.12
94.96
0.05
60.54
61.71
63.42
0.07
46.36
47.13
48.35
0.10
34.79
35.28
36.13
0.30
14.13
14.20
14.47
0.50
9.265
9.283
9.450
0.70
7.039
7.045
7.171
1.00
5.299
5.300
5.401
2.00
3.184
3.184
3.268
3.00
2.461
2.461
2.543
4.00
2.097
2.097
2.178
5.00
1.877
1.877
1.956
10.00
1.440
1.440
1.517
M. S. LEONGet al.
884
the uniform-flux potential distributions from the exact distribution. One would therefore expect the correction factors to display a saturation type of behaviour also, and this is borne out by Table 3, which shows the change in the percentage deviations of FP and I$, from FE slowing down considerably for p2/p, > 1999. SUMMARY ANLl coNcLusIoNs
In this paper we have extended the range of applicability of the mixed boundary value method to a slab backed by a substrate of arbitrary but finite resistivity. From the solutions obtained for the source current density for thin slabs, we have found that, the more insulating the substrate becomes, the greater is the current density near the edge of the disc electrode. We have also found that, except for large slab thickness, the potential distributions under the source contact for a slab with an insulating substrate are in general not strongly influenced by the particular form of the source current distribution assumed
in either
the variable
or the uniform
flux
method-in contrast to the situation for a slab with a perfectly conducting substrate. Both these two methods thus yield correction factors which agree quite closely with those given by the mixed boundary value method. For the calculation of the source current density distribution, a new method has been proposed which is simpler and more direct than the one previously reported for a slab backed by a perfectly conducting substrate. With this method, it has been found that the source current density distribution can be closely approximated by a polynomial of the form:
REWltENCEs 1. M. S. Leong, S. C. Choo and K. H. Tay, Solid-St. Electron. 19,397 (1976). 2. M. S. Leong,S. C. Choo and L. S. Tan, SOWSI. E/e&on. 22, 527 (1979). 3. M. S. Leong, S. C. Choo and L. S. Tan, Solid-St. Electron. 21, 933 (1978). 4. S. C. Choo, M. S. Leong and L. S. Tan, Solid-St. Electron. 24, 557 (1981). 5. I. N. Sneddon, Mixed Boundary Value Problems in Potential Theory, pp. 106-110.North-Holland, Amsterdam (1966). 6. Y. L. Luke, Integrals of BesselFunctions, p. 325. McGrawHill, New York (1%2).
APFXNDJX Current density distribufion under the source contact The current density distribution under the disc source contact
is given by eqn (16) as
J(r,O)=p;‘V~J&p)dp~$j(0cos(pt)dt. Let f(f) he approximated by the polynomial bOt b2tZt b.,t4t b6t6t b#.
=~~‘(b,,+b2~2tb4f4tb6~6tbs~*)cos(pt)dt =-2P ?r J( ; > GP-“*J&p) + C~P-“*J~,Z(P)-
p,
FP- FE F
E (%)
C~P-~‘*JS,AP)
c4P-7’2J,,2(P)
(0
c5P-9’2J9,2(P)l
C,=b0tbztb4tb6tbs, C2=2b,t4b4t6b6t8b,, C, = 8b4t 24b6t 48b,, C4=48b6t 192bs
E
the first kind and order v/2 where v is an odd integer. Therefore, the current density distribution OD
J(r,O)=Fp;lV
7.90
2.8
10.1
199
20.43
2.2
6.2
1999
34.79
1.4
3.9
19999
49.42
1.0
2.7
199999
64.07
0.8
2.1
I0 4
5 J&Wlp”2J~~2W >
-
c2P-“2J3,2(P)
-
C~P-“~J~,~(P)+
+
c3P-3’2J5,2(P)
C,~-“~J9,&01 dp.
(As)
From Luke[6], we have
(2)
19
(A4)
In eqn (A3) and subsequent eqns, J&p) is a Bessel function of
Fu- Fl
F
and
Cs = 384b8.
Table 3. Correction factors calculated as a function of substrateto-slab resistivity ratios, dp,, by using the various methods. The results for FP and FU are expressed in terms of percentage deviations from FE
FE
-
where
This suggests that, in the variational method of spreading resistance calculation, a source function based on such a polynomial may be used in place of the previously-proposed linear combination of the uniform flux and variable flux distributions[41. In a separate paper, we shall show that this can indeed be done, resulting in improved accuracy in the solutions for the potential and current density distributions.
$
W)
Then,
+
J(r,0)=K,(1-rZ)-“2tK,(ltr~1’2tK,(l-r2)3’2t~~~
(Al)
I0
0) J.+,(at)J,(bt)t*-‘dt (a* _
= 0, a
b*)y-‘b’
=2y-Pav+lr(v_pt,).
a>b
646)
withRe(vtl)>Re(&>-1. Applying this formula to eqn (As), we finally obtain the expression for the current density distriiution as given by eqn
(17).
003a1101/821.rJ3/0 0 IS2 Peqamon Press Ltd.
THE SPREADING RESISTANCE OF A HOMOGENEOUS SLAB ON A HIGH-RESISTIVITY SUBSTRATE: MIXED BOUNDARY VALUE SOLUTIONS M. S. LEONG, S. C. CHOOand L. S. TAN Departmentof ElectricalEngineering,National Universityof Singapore,Kent Ridge, Singapore0511, Singapore (Received24 August 1981;in revisedfom 29 January 1982) Abstract-The range of applicabilityof the mixed boundaryvalue method for calculatingspreadingresistance is extended to a homogeneous slab with a disc contact source and backed by a substrateof arbitrary,but finite resistivity. Solutions are presentedin terms of the spreadingresistancecorrectionfactors and the source current density distributionsfor a slab of varyingthickness and with various high resistivity substrates.In particular,the results for a thin slab indicate that, as the substrateresistivity increases, more and more of the source currentis concentratednear the edge of the disc electrode. A comparisonis made of the source currentdensity and potentialcorrespondingto the mixed boundaryvalue methodwith those given by the uniformflux and the variableflux (power-loss)method. It is found that, except for large slab thicknesses, the source potentialdistributionsfor a slab with a high resistivitysubstrateare not strongly influencedby the particularform of the source currentdensity distributionassumed in either the uniformflux or the variable flux method. In consequence, both these two methods yield correction factors which agree quite closely with those derivedfrom the mixed boundaryvalue method.
INTRODUCTION
In two previous papers [1,2], we have used the mixed
boundary value approach to solve the spreading resistance problem of a semiconductor slab with both a uniform and a nonuniform resistivity profile. The treatment was, however, limited to structures whose substrate was perfectly conducting. As pointed out elsewhere[3], the mixed boundary value problem for structures on a perfectly insulating substrate is intractable, and the only solutions available for this sype of structures are those of the approximate methods based on assumed source current distributions. The case of a perfectly insulating substrate is of interest in spreading resistance measurements, because a perfectly insulating boundary can be used to represent a pn junction. In practice, however, it often happens that the substrate has a resistivity which, while large compared to the slab resistivity, is not infinite; and even in the case of the pn junction, a structure with a high resistivity substrate can still be used to provide a useful approximation. In this paper we shall show that the mixed boundary value approach can be applied to a slab whose substrate is of arbitrary, but finite resistivity. In particular, solutions will be presented for a homogeneous slab with substrates whose resistivities are greater than the slab resistivity by several orders of magnitude. In presenting these solutions we shall be concerned not only with the correction factors but also with the source current density distributions, since, as has been shown recently[4], a detailed study of the source current density distributions can lead to new improved methods of calculating the spreading resistance, such as the variational method. TnEPRoil~ Consider the two-layer structure shown in Fig. 1. Here the semiconductor slab of resistivity p, and thickness D 877
Fig. 1. Geometryof slab with a disc electrodeand a substrateof arbitrary,but finite resistivity.
is backed by a substrate layer of resistivity pP This second layer is very thick, so that the sink for the current can be regarded as being at z = a. Note that the radius of the source electrode is given as unity, which implies that all the geometrical dimensions given in this paper are normalized with respect to the electrode radius. We assume that the potential & and & in layer 1 and layer 2 satisfy Laplace’s equation, and that the boundary conditions are as follows: Surface
g*=v, osr<1, +o,
r>l,
z=o z=o
(1) (21
where V is the value of the potential at the source disc electrode, the reference potential being at z = m.
878
M. S. LEONG et al.
Interface of adjacent layers W,
Sneddon [5] as
2) = M,
z),
z = D
(3)
p;‘$‘(r,~)=p;~~(r,z), z=D
(13)
(4) where f(t) is the solution of the following Fredholm integral equation of the second kind:
Substrate lim &( r, z) = 0.
(5)
ZQ
By applying the above boundary conditions to Laplace’s equation, and defining the reflection coefficient k, as k
_pz-PI
’ we
A(p) = F I,’ f(t) cos (pt) dt
(6)
Pz+PI
obtain the following pair of dual integral eqns: m p-‘A(p)
dp = 1, 0 G r < 1
I0
cc
A(p
dp = 0,
r> 1
(8)
Being an even, real and continuous function, f(t) can be solved numerically by using Simpson’s rule for the integration from -1 to tl, as in the case of the perfectly conducting substrate. By way of illustration, the behaviour of f(t) for a high resistivity or insulating substrate with pZ/p, = 1999 (i.e. k, = 0.999) and D = 0.1, 1.0 and 10.0 is shown in Fig. 2. We note from the figure that the behaviour of f(t) is opposite to that for a perfectly conducting substrate, the curves in the present case being generally concave rather than convex in shape. Once f(t) is known, the correction factor FE can be obtained as follows:
I0
FE = 1 o’f(t) dt. /I
where H(p)
=
I+ k
(15)
epzDp
1- k, eC20p and J,(rp) is a zeroth order Bessel function of the first kind. The above equations are analogous to the dual integral equations found for the slab with a perfectly conducting substrate, with the kernel tanh (pD) replaced by (1 + k, em2*)/( 1 - k, e-‘*). The latter expression in fact encompasses the case of the perfectly conducting substrate, for which p2 = 0 and k, = - 1. Expanding the kernel, we have 1 t k, emzDp 1 - k, e-‘&
=1+25
“=I
k,“e-“‘@’
---J--t -I
0
(9)
f(t) -
0.2
for k,
-
p-1
0.1
tI
k(p)lJ,(pr) dp = 1, OGr
mp-‘A(p)U+
I0
(10)
=O.
r>l
,
(11)
I0
I
,
,
0
-1
m A(pV,(pr)dp
+1
,
,
,
I
I
t1
:;)IIIY,
t
, t
with k(p) = 2 “~, k,” emZnDp.
(12)
The solution to eqns (10)-(12) is then given by
-1
0
+1
Fig. 2. Behaviour of the function f(t) for a slab with an insulating substrate: h/p, = 1999 (k, = 0.999). The normalised thickness D =O.l, 1.0 and 10.0.
The spreadingresistanceof a homogeneousslab on a high-resistivitysubstrate
al9
Table 1. Correctionfactors as a function of normalisedthickness D for p~/pl=19,199and 1999,correspondingto k, = 0.9,0.99 and 0.999, respectively pk D P - 1% 7.
1
P = 1990) 2
p = 1999P1 *
0.01
15.32
79.03
202.1
0.03
12.09
44.37
90.05
0.05
10.33
32.42
60.54
0.07
9.138
26.03
46.36
0.10
7.900
20.43
34.79
0.30
4.614
9.267
14.13
0.50
3.481
6.341
9.265
0.70
2.885
4.949
7.039
1.00
2.380
3.834
5.299
2.00
1.720
2.451
3.184
3.00
1.485
1.973
2.461
4.00
1.365
1.731
2.097
5.00
1.292
1.585
1.877
10.00
1.146
1.292
1.440
50.00
1.029
1.059
1.088
The correction factors are shown in Table 1 for values of thickness D ranging from 0.01 to 50.0 at pZ/p, = 19 (t = 0.9), p2/p, = 199 (k, = 0.99) and pZ/p, = 1999 (kr = 0.999). It is observed that, as expected, the correction factor increases as the substrate becomes more insulating. It is also noticed that as the slab thickness increases, the correction factor decreases asymptotically to unity. This is consistent with our physical intuition, for, as the substrate recedes from the source, its influence on current flow in the slab decreases and the situation approaches that of an infinitely thick slab. Neverthless, even with a thickness of 10, the correction factor is quite different from unity, showing that it is still necessary to apply corrections to the measured spreading resistance for such large thickness. COMPARISON WITH AF’PROXlMATEMElMOlW3J
It is of interest to compare the solutions obtained from the mixed boundary value or exact method with those given by the approximate methods[3] based on assumed source current distributions. In making such a comparison, the quantities of interest are, besides the correction factors, (a) the current density distributions corresponding to the constant-potential boundary condition in the exact method and (b) the potential distributions corresponding to the two source current density distributions assumed in the approximate methods, viz. the uniform current distribution and the variable current distribution given by the classical solution for the infinitely thick slab.
(a) Current density in the exact method The source current density distribution is related to f(t) in eqn (14) as follows [31: J(r,O)=p;‘V~J&p)dp~~~(t)cos@t)dt. (16) In Ref. [3], J(r, 0) was computed from the above equation after f(t) was expressed as a Fourier series in the interval (0,l). The use of a Fourier expansion for f(t), however, tends to give rise to slight oscillations in J(r, 0), especially near the edge of the disc contact. These oscillations are well known in Fourier analysis as Gibbs phenomenon. The oscillations can be minimised by taking a large number of terms in the Fourier expansion for f(t) and by using various smoothing functions (e.g. Lanczos’ or Fejer’s method) to accelerate the convergence. However, since f(t) is an even, smooth and wellbehaved function a computationally more efficient method is to approximate it by an even-degree polynomial
with bo, b2, b4,. . . determined by a least-square fit of f(t). The number of terms required to achieve a given accuracy depends on the substrate resistivity and on the slab thickness. However, it was found that even in the
880
M.S.LEONGdol.
worst case, by taking the first five terms in the polynomial, the approximation obtained does not deviate by more than l/4% from f(t) at any point within the range (-l,+l). By using the polynomial approximation up to the b,t* term then, it can be shown (see Appendix) that the current density distribution under the source electrode is given by J(r,0)=~[c,(1-r3-“‘-c!~(1-r3”’ 1 t 3
1 C,( 1 - r2)‘” - jj C,( 1 - r2)5/z
t j& C,( 1 - ?)“‘I
(17)
where the coefficients C,, . . . , C, are related tG b,,, . . . , b, through eqn (A4) in Appendix. From eqn (17), it is clear that the present procedure for computing J(r,O) is simpler and more direct than the method earlier proposed, as it does not involve a numerical integration as in eqn (3) of Ref. [3]. (b) Potential distributions in the approximate methods Uniform flux method [3]: For an assumed source current distribution which is uniform, the potential distribution over the source region is easily shown to be given by: Mr, 0) = $6
H(p) y
J&r) dp
(18)
where I is the total current entering the source electrode, and .J1@)is a first order Bessel function of the first kind. Variable flux method[3]: For an assumed source current distribution of the classical form, it can likewise be shown that t$,(r, 0) = $
lrn H(p) y
J&r) dp.
(19)
The only diierence between eqns (18) and (19) and the corresponding solutions for the slab with the perfectly conducting substrate131 lies in the replacement of tanh(pD) by H(p). This is also true of the solutions for the correction factors. REWLTSANDDIsCUSSlON
In order to provide a uniform basis for comparing the potential and current density distributions under the source electrode calculated by the various methods, we set the current I in the approximate methods to unity; and since, in the exact method, the current I entering the source electrode is given by 4p;‘VJ,‘f(t)dt, we obtain I = 1 in this case by making V = p,/4 j’,’f(t) dt. By using the expressions given in eqns (17)-(19) and with the above values for V and I, the current density tResults for Schumannand Gardner’smethod have not been included,because the methodis inherentlyinconsistent[3].
and potential distributions were computed for values of slab thickness D ranging from 5.0 to 0.1. The substrateto-slab resistivity ratio pJp, used was 1999, corresponding to a reflection coefficient of 0.999. The results are shown in Figs. 3-6. Several interesting points can be observed from this set of figures. First of all, consider the current density distributions shown in parts (a) of the figures. It is observed that for a thin slab (D = 0.10, for example) most of the current is concentrated near the edge of the disc source contact so that the exact current density distribution is quite unlike either of the distributions assumed in the approximate methods. This current crowding effect can be explained physically by the tendency of the insulating substrate to repel the current carrying charges towards the edge of the disc contact, an effect which is to be contrasted with the attractive affinity of a conducting substrate [3]. Turning our attention now to parts (b) of the figures, we observe that the potential distributions under the source contact are not strongly influence by the assumed current density distributions. The variable flux method, in general, gives rise to a more nearly uniform potential distribution under the source contact than does the uniform flux method. Nevertheless, except for large slab thickness (e.g. D = 5.0), the source potential distribution as computed by the uniform flux method is not too different from the exact distribution. For instance, the maximum difference in potential between the two methods for D = 1.0 is less than 6%. The correction factors FE calculated from the mixed boundary value method are shown in Table 2, together with those calculated from the approximate methods-t We observe from the table that the variable flux method (power-loss definition for the spreading resistance) gives a correction factor Fp which is quite close to FE of the exact method throughout the range of thickness considered. Nevertheless, the correction factor Fu computed by the uniform flux method is not too different from FE either, the maximum deviation being only 7.2% at D = 0.01. The behaviour of both Fp and Fu is thus consistent with that observed for the source potentials. In particular, the good agreement of Fp and Fu with FE is seen to be a consequence of the insensitivity of the source potentials to the assumed source current density distributions. In order to investigate the phenomenon of current crowding further, we have obtained results for a slab with a thickness 6xed at D = 0.1 but with pJp, varying from 19 to 199,999. The results for the current density and potential distributions are shown in Fig. 7(a) and (b). Reference to Fig. 7(a) shows that increasing the substrate resistivity causes more of the current to be pushed to the edge of the disc electrode. However, by the time that p2/p, has exceeded a value of 1999, the current crowding effect has essentially saturated, so that further increase in substrate resistivity produces relatively little change in the shape of the current density distribution. Similarly, from a careful examination of Fig. 7(b), it can be seen that as p2/p1 increases beyond 1999, there is not much change in the percentage deviations of the variable- and
The spreading resistance of a homogeneous slab on a high-resistivity substrate 4
1
0.9 -
0. j?..
Io=s.ol
0.6 -
0.7
----.-
Q 0.6-
kG Q2
0.5 -
10r5.0
0.8 I.,
.
2 Y
881
EXACT VARIABLE UNIFORM
-
FLUX FLUX
------. -
]
EXACT V4fflABlE UNIFOW
FLUX FLUX
o.c-
0.2
a1
0.1
t
OO I
0.2
0.6
0.c
0.6
1.0
OO
r
02
0.b
0.6 c
0.6
1.0
Fig. 3. The source current density I(r,O) and potential +(r,O) distributions calculated by using the exact and approximate methods for a slab with pJpl = 1999(5 = 0.999).D = 5.0.
0.9 3.!
lo.-1
0.8
I 0.7 0 i Y
0.6
-----.-
EXACT VARIABLE UNIFORM
I
3.1 --------
FLUX FLUX
EXACT VARIABLE UNIFORU
FLUX FLUX
2.5
1.0
0.2
44
0.6 r Ia1
R’I
ti0
) lb)
Fk. 4. The source current density I(r,O) and potential &r,O) distributions calculated by using the exact and approximate methods for a slab with h/p, = 1999(&,= 0.999).D = 1.0.
P
M. S. LEONGet al.
882
1 D=O.S 1 0.6 .,
-------.-
;; i <
EXACT VARIABLE UNIFORU
FLUX FLUX
3.0 ..
0.7..
I I I
06..
:
ic
2.5.: ---- .-.-_-_____
I I
i?i ci 0.s..
___--~:_y<
-=Z
0 ;
2.0..
Q )3 Q s Q
G i= 2 c” 2
0.4.. -____.---.-.-. a3..
---------
1.5 .,
EXACT VARIABLE UNIFORN
FLUX FLUX
0.8
1.0
1.0..
00
0,2
0.4
06
0.8
1.0
OO
0.2
0. c
0.6 r
r 101
lb1
Fig. 5. The source current density J(r, 0) and potential &r,O) distributions calculated by using the exact and approximate methods for a slab with h/p, = 1999(k, = 0.999). D = 0.5.
Ll7:O.l
1 4.5
-----.-
EXACT VARIABLE UNlFaPN
cc FLUX FLUX
3.5
lo=ar --------
1
EXACT VARIABLE UNIFORM
FLUX FLUX
---.-_ ___--_-_____Y-_.._
d
0.2
0.4
0.6 P
0.8
1.0
Fig. 6. The source current density J(r,O) and potential $(r,O) distributions calculated by using the exact and approximate methods for a slab with h/9, = 1999(k, = 0.999). D = 0.1.
.
The spreading resistance of a homogeneous slab on a high-resistivity substrate ,
----.-
Ia’.
-_---_ __---
0.8
EXACT VARIABLE UNiFO(?U --
FLUX FLUX
---_
-_zz;m_
16,
p&y
= 199999
14.. -----_ -----------_----_~
--u_
12
19999
10.. __L-_Z_,_~, 1999
8,.
2,-_--zxz-_;y----
---=--
OO
0.2
0.4
IQ)
0.6 r
0.8
19
1.0
lb1
Fig. 7. Effect of substrate resistivity on source current density I(r, 0) and potential ~$(r,0) distributions., D = 0.1.
Table 2. Correction factors calculated as a function of normalised thickness D by using the mixed boundary value method (FE), the variable flux method (FP), and the uniform flux method (FU). pJp, = 1999, corresponding to k,= 0.999
D
SSE Vol. 2J. No. 9-D
FE
PP
Fu
0.01
202.1
208.4
216.7
0.03
90.05
92.12
94.96
0.05
60.54
61.71
63.42
0.07
46.36
47.13
48.35
0.10
34.79
35.28
36.13
0.30
14.13
14.20
14.47
0.50
9.265
9.283
9.450
0.70
7.039
7.045
7.171
1.00
5.299
5.300
5.401
2.00
3.184
3.184
3.268
3.00
2.461
2.461
2.543
4.00
2.097
2.097
2.178
5.00
1.877
1.877
1.956
10.00
1.440
1.440
1.517
M. S. LEONGet al.
884
the uniform-flux potential distributions from the exact distribution. One would therefore expect the correction factors to display a saturation type of behaviour also, and this is borne out by Table 3, which shows the change in the percentage deviations of FP and I$, from FE slowing down considerably for p2/p, > 1999. SUMMARY ANLl coNcLusIoNs
In this paper we have extended the range of applicability of the mixed boundary value method to a slab backed by a substrate of arbitrary but finite resistivity. From the solutions obtained for the source current density for thin slabs, we have found that, the more insulating the substrate becomes, the greater is the current density near the edge of the disc electrode. We have also found that, except for large slab thickness, the potential distributions under the source contact for a slab with an insulating substrate are in general not strongly influenced by the particular form of the source current distribution assumed
in either
the variable
or the uniform
flux
method-in contrast to the situation for a slab with a perfectly conducting substrate. Both these two methods thus yield correction factors which agree quite closely with those given by the mixed boundary value method. For the calculation of the source current density distribution, a new method has been proposed which is simpler and more direct than the one previously reported for a slab backed by a perfectly conducting substrate. With this method, it has been found that the source current density distribution can be closely approximated by a polynomial of the form:
REWltENCEs 1. M. S. Leong, S. C. Choo and K. H. Tay, Solid-St. Electron. 19,397 (1976). 2. M. S. Leong,S. C. Choo and L. S. Tan, SOWSI. E/e&on. 22, 527 (1979). 3. M. S. Leong, S. C. Choo and L. S. Tan, Solid-St. Electron. 21, 933 (1978). 4. S. C. Choo, M. S. Leong and L. S. Tan, Solid-St. Electron. 24, 557 (1981). 5. I. N. Sneddon, Mixed Boundary Value Problems in Potential Theory, pp. 106-110.North-Holland, Amsterdam (1966). 6. Y. L. Luke, Integrals of BesselFunctions, p. 325. McGrawHill, New York (1%2).
APFXNDJX Current density distribufion under the source contact The current density distribution under the disc source contact
is given by eqn (16) as
J(r,O)=p;‘V~J&p)dp~$j(0cos(pt)dt. Let f(f) he approximated by the polynomial bOt b2tZt b.,t4t b6t6t b#.
=~~‘(b,,+b2~2tb4f4tb6~6tbs~*)cos(pt)dt =-2P ?r J( ; > GP-“*J&p) + C~P-“*J~,Z(P)-
p,
FP- FE F
E (%)
C~P-~‘*JS,AP)
c4P-7’2J,,2(P)
(0
c5P-9’2J9,2(P)l
C,=b0tbztb4tb6tbs, C2=2b,t4b4t6b6t8b,, C, = 8b4t 24b6t 48b,, C4=48b6t 192bs
E
the first kind and order v/2 where v is an odd integer. Therefore, the current density distribution OD
J(r,O)=Fp;lV
7.90
2.8
10.1
199
20.43
2.2
6.2
1999
34.79
1.4
3.9
19999
49.42
1.0
2.7
199999
64.07
0.8
2.1
I0 4
5 J&Wlp”2J~~2W >
-
c2P-“2J3,2(P)
-
C~P-“~J~,~(P)+
+
c3P-3’2J5,2(P)
C,~-“~J9,&01 dp.
(As)
From Luke[6], we have
(2)
19
(A4)
In eqn (A3) and subsequent eqns, J&p) is a Bessel function of
Fu- Fl
F
and
Cs = 384b8.
Table 3. Correction factors calculated as a function of substrateto-slab resistivity ratios, dp,, by using the various methods. The results for FP and FU are expressed in terms of percentage deviations from FE
FE
-
where
This suggests that, in the variational method of spreading resistance calculation, a source function based on such a polynomial may be used in place of the previously-proposed linear combination of the uniform flux and variable flux distributions[41. In a separate paper, we shall show that this can indeed be done, resulting in improved accuracy in the solutions for the potential and current density distributions.
$
W)
Then,
+
J(r,0)=K,(1-rZ)-“2tK,(ltr~1’2tK,(l-r2)3’2t~~~
(Al)
I0
0) J.+,(at)J,(bt)t*-‘dt (a* _
= 0, a
b*)y-‘b’
=2y-Pav+lr(v_pt,).
a>b
646)
withRe(vtl)>Re(&>-1. Applying this formula to eqn (As), we finally obtain the expression for the current density distriiution as given by eqn
(17).