Appl. sci. Res.
S e c t i o n B, V ol . 6
oo
THE
INTEGRALS
~/~(x) =
(p!)-l f ev (e + x) -1 e -~ de A N D
00
0
~ ( x ) - - - - ( p ! ) - l f e v (e+x) -2 e -~ de A N D T H E I R
TABULATION
0
b y R. B. D I N G L E *) a n d D O R E E N A R N D T D e p a r t m e n t of Physics, University of Western Australia ~f), Nedlands, Australia
a n d S. K. R O Y *) National Physical Laboratory, New Delhi, India
Summary
These integrals are fundamental to the theory of the properties of weaklydegenerate elemental semiconductors at high frequencies and/or in the presence of a steady uniform magnetic field, when conditions are such that scattering by thermal agitation or by neutral impurities predominates over ionic scattering. Their applications and properties are discussed, and various expansions and relatinships to other functions developed. The integrals are tabulated to four significant places for the arguments x = 0 (.1) 1 (.2) 2 (.5) 10 (1) 20 at half-integer spacings of the order p.
w I. Applications. (1) I t is a consequence of the B o l t z m a n n e q u a t i o n t h a t the theoretical effect of h i g h - f r e q u e n c y electric fields on t h e electrical, t h e r m a l a n d t h e r m o e l e c t r i c p r o p e r t i e s of m e t a l s a n d s e m i c o n d u c t o r s is to replace the r e l a x a t i o n t i m e v b y T
1 + io~z
T
(~OT2
1 -[-w2"c2
i
1 -[-o~2z 2 '
where ~o is the circular frequency. Moreover, the s a m e q u a n t i t y m a y be used to express instead the effect of a s t e a d y u n i f o r m m a g n e t i c field H b y p u t t i n g co = ell~me, the precessional circular f r e q u e n c y *) This work was commenced m 1954 while the authors indicated held Postdoctorate Fellowships at the National Research Council, Ottawa. t) The later work was supposted by the Research Grants Committee of the University of Western Australia.
--
144
--
SEMICONDUCTOR INTEGRALS
145
of an electronic carrier; indeed, even the simultaneous influence of high-frequency electric fields and a s t e a d y uniform magnetic field can be expressed in terms of identical types of functions of r (D i n g 1 e 3)). It follows t h a t whenever r oc e -li2, where e = carrier energy/kT, [a law valid for elemental semiconductors in which scattering b y thermal agitation or b y neutral impurities greatly predominates over ionic scattering (e.g. W i 1 s 0 n 13)1 at high frequencies and/or in a magnetic field, factors eli2/(e+x) for the real part and 1/(e + x) for the imaginary part appear in the integrals over energy involved in expressions for the theoretical properties, x being proportional to 0)2. These integrals over energy contain also : a) E n e r g y weighting factors: for quasi-free carriers (energy a: square of wave-vector) those involved are e ~12, e 512 and e 71~ b) The derivative of the Fermi-Dirac distribution function: for non-degenerate carrier systems this is proportional to e -*, and moreover for all weakly-degenerate systems it can be expanded as a series of such exponential terms. Thus, under the conditions indicated, the theoretical properties of elemental semiconductors can be expressed in terms of integrals of the type oo
~[~(x) =-
e-~ x
,
(1)
o
only integer and half-integer orders p being required for the particular case of energy bands of s t a n d a r d form. The factor (p !)-1 has been introduced into the definition of g[~ so as to permit direct interpolation between the different t a b u l a t e d orders, as well as between t a b u l a t e d arguments. For the integral itself tends to (p -- 1) ! when x --* 0 and to x - l ( p !) when x -7 co, and therefore can oscillate with p like a factorial function (cf. 0! ~ 1, ~-! = .8862, 1! = 1), t h e r e b y seriously hindering interpolation between the orders. The influences of small additional effects -- such as slight ionic scattering -- can all be expressed in terms of integrals of the t y p e co
~3,(x)
p!. !
(2)
0
These integrals also receive a t t e n t i o n in the present paper. (2) Since with the aid of the theory of partial fractions a n y
146
R. B. DINGLE, D. ARNDT AND S. K. ROY
algebraic expression involving only integral powers of ~ can be split into a series of integrals of the type 9J~(x) or of its derivatives, consideration of such basic integrals is fundamental to the discussion of a large class of more complicated integrals. (3) According to the Poisson summation formula (T i t c h m a r s h lo),
p. 60), oo
oo
Z /(n) =- 89
oo
Z f / ( n ) e~=irndn.
n=0
~' = - - o o
0
It follows that the integrals 9.I and ~8 m a y be used to facilitate summation of such functions as n~l(n + y) and n~/(n + y)~ respectively.
w
Properties. R e c u r r e n c e
Relations.
The
obser-
vation that X
---t~+x
=1
~+x
leads immediately to the relation p~t~ + x ~ _ 1 = 1.
(3)
In applying this recurrence relation, it should be borne in mind that accuracy is preserved only if higher orders are obtained from lower orders when the argument x is small, and lower orders obtained from higher orders when the argument is large. Differentiation of (3) yields P ~ o JI- X ~i0--1 = 9~0--1"
(4)
Moreover, oo
pl ~ =
--.f e~ e-~ dI1/(e + x)] = O~x-1 + pl (91~_1 -- 9~) 0
on integrating by parts, so that ~=~1~-1--9~
if
p>0.
(5)*
*) More generally, defining oo
(e + x)q ' 0
there follows the important reduction formula 1
(~.1~_1--~--1)lfp
> 0.
SEMICONDUCTOR
147
INTEGRALS
W i t h the aid of (4) followed b y (3), this can be t r a n s f o r m e d into the somewhat more general relation ~3~ = x -1 {1 -- (p + x)9~} if p > -- 1.
(6)
D i f f e r e n t i a 1 E q u a t i o n. It follows from (6) t h a t for p~--l, x ~f'~ -- (p + x) ~f~ q- 1 ---- 0, (7) where dashes denote differentiation with respect to x. Differentiating, x~
tt
!
+ (1 -
p -
x) ~
-
~
=
0,
(8)
so t h a t ~ is a confluent h y p e r g e o m e t r i c function (J a h n k e and E m d e G), p. 275). So likewise is ~3~, since the differential coefficients of confluent h y p e r g e o m e t r i c functions are also confluent h y p e r geometric functions 11). w
Relationships
(I) T o
to other ]unctions.
lized
Ei
function.
p!~(x)
= ["e~e-*de j e~x
the
genera-
Since
oo
oo
__ x ~ [ ' u ~.e. .-.' U d u J uq- 1
0
0 oo
= x ~ e'fdx
co
oo
e-'fu
~ e - ' u d u = p! x ~ e * f x -~+1~ e - " d x
0
fe
oo
= p! e ~ f y -~+1~ e -~v dy, 1
it follows t h a t 9~(x) ~- e 9 Ei~+l(X), (9) this generalized Ei function being t r e a t e d in P 1 a c z e k s) and D i n g 1 e 4). U n f o r t u n a t e l y , this function appears as yet to have been investigated and t a b u l a t e d in detail only for integer orders and positive arguments. The following alternative derivation of the i m p o r t a n t transf o r m a t i o n (9) is of still wider application. Since ioo
1 f 2~i
J
e "u dx -
-
e + x
-,u -
e
,
--{oo
the Laplace t r a n s f o r m of ~ ( x ) is ioo
L~(u) -- 2~i
e~(x)
oo
dx=(P!)-I
e~'e-"(~+l) d e = ( u +
1)-c~+1~"
148
R . B. D I N G L E , D. A R N D T A N D S. K. R O Y
Taking the Laplace image (inverse Laplace transform) Laplace transform yields
of this
oo
~l~(x) = f e - ~ L~(u) du = e ~ Ei~+l(x ), 0
in agreement with (9). (2) T o S e r ' s i n t e g r a l , gral, and the probability
the
exponential inteintegral. B y d i r e c t corn-
co
parison, it is seen t h a t ~o(X) = f (e + x) -1 e -~ de is identical with 0
S e r's function 9) go(x), and ~l(X) ----fe(e + x) -2 e -~ d~ identical with 0
S e r ' s function 9) ~l(x). As a special case of the transformation (9), it is seen t h a t 9/0(x) = e" Eil(x ) = -- e ~ Ei (-- x),
(10)
where Ei is the well-known exponential integral (e.g. BA I2), MT 5 and 6 v)). Moreover, oo ~_l/~(x) = e ~ Eilf~(x) = x -1I~ e ~ f x -1I~ e -~ dx = (2/x) 1/~ FE(2x)~/~I, (11) X
F being the ratio of the tail area of the normal statistical curve to its bounding ordinate (BA viii)). Starting from these special cases, the functions 9/~ and ~3~ for integer and half-integer orders p can be built up b y invoking the recurrence relations (3)-(6). (3) T o t h e W h i t t a k e r function. The general definition of the W h i t t a k e r function is (eg. W h i t t a k e r and W a ts o n 1 2 ) , p. 340) co
Wk,~(x) = { ( m - k-- 89 l}-1 x 89
e- 89
~-k-89 (e + x) re+k-89 e-* de.
0
Writingm+k-- 89
1 and m -- k -- 89-----p, i t f o l l o w s t h a t
~f~(x) = x89
e89 W_89
89 (x),
(12)
as can also be seen b y combining (9) with eq. (43) of D i n g 1 e 4). Further, writing m + k 21 -2 and m - - k - - 8 9 ~,(x) = x 89
e89 W i~+89189
).
(13)
(4) T o t h e confluent hypergeometric function. Adopting the notation of W e b b and A i r e y 1 1 ) and
SEMICONDUCTOR INTEGRALS
o
~
~
o
~
o
~
o
~
o
~
o
~
o
~
o
~
o
~
o
149
..-t.
~
\ 0
~
-
~
-
~
o
~
~
~
-
-
~
~
~
0
~
0
~
o o o o o o o o
B ~~ --~ ~ ~ -o ~ $ ~
0
~
~
o
I
o
"q"
...'-:
+
o o o o o o o o
~~ o ~ o ~ m ~ ~ ~
~
~
. . .~.
-
~R~
-
OOOOOOOOOOOOOOOOOO
~.
~
o
II
~ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
O~ ~ ~
0
~
~
~
~
7
0
~
~
0
0
~
0
~
~ O
m
0 ~
~ m
o
~
o
o
~ 0
0
o
o
~
o - - ~ o~ . . . . . o
o
o
o
0
~ ~ ~ - ~ 0
~R~ 0
0
0
0
0
o
o
+
% ~..~o 7
~ cn
0
~
0
~
~
0
~
~
~
~
~
~
~
-
~
~
~OOOOOOOOOOOOOOO
-
~
~
~
~
~-~ o ..Q~o ~o ~ ..~o~
o oO O~O O~O O O O O O O O O O O
"8
O O
,<
~
~
~
~
~
o
,'=~
o ~ OOOOOOOOOOOOOO
~o~ 9
o
o
~
~
~
~
8-~-~o-~-~
~
0~
00
ooooooo
-~o,~,~o,~
0 ~ C~I 0 C~ ~ I~. ~ ~D b'~ u~ ~:~ .~I ~
~
~
~~
~~
CW~ ~ O O O~ C,- cO ~.
~ 0"~ ~ ~ D~
~~
0
0 0 0 0
~
~
0
0
~
0 0 0 0
o
ooooo
-45 ~O uO *.O uO uD ~" 0 0 0 0 0 0 0
O 0 0 0 0 0 C
6~ O
1SO
R. B. D I N G L E , D . A R N D T A N D S. K. ROY
O - - ~ ~ O ~ O ~ O ~ O ~ O ~ O ~ O ~ O ~ O ~
\
~q
\
4~ ~
O
~
~
O
~
O
~
O
~
I
O
~F
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
GI
(N
~----~
~ ~ - - o ~
~ ~
~OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
O
~
_
_
~
~ - o ~
.....
O
~
O
~
-
-
OO O ~ ~
~
~ ~
~
~
O
o
~
~
~
o
o
~
.~.o
o o o o o o~o o o~o o oRo o o ~o o o o o o B B ~ B B B B b B B B B B B ~ B
L
~. ~ o ~ o ~ ~ ~ o ~ ~ o- ~- ~o ~~ o~ ~ -~ -o ~ o ~ ~ - - o ~ ~ o ~ o ~ ~ ~ ~
+
~
o
O
o ~~ - -o ~ ~~
- ~ o ~
~
~
-
O. ~
o ~
~0
~
~
_
O
~
~
_
~
o
o
- ~ ~
~
o
~
o
-
o
~
~ o ~ o o ~ ~ ~o ~~ _ ~ - o o o o o o o o o o o o o o o o ~ B ~ ~ ~
~ - - ~
o ~ o ~ - - o
~
0 N
-
o o o o o o o o o o o o o o o o o o o ~ ~ ~ o o o o
~~
II
~ o - ~.
m
~
~
o
~
~
-
~ -
o
~0~
e~ 0
].~.=
~m o
~
~
~
~-~ ~
o
~.~ 8 ~ o~~ o ~. ,
~
~
~
0
~ ~_ 0
0
0
0
0
._. . .
0
0
0
0
0
~
~
0
0
~ ~ _ o 5 ~ ~ -o ~- * ~~ 0
0
~
0
0
0
0
O
0
0
0
~
0
0
0
0
~
0
0
~ o~
,
~
~
O~
~ O ~ ~ ~ O O O O O O O O O O O O O O O O O O O O O O O O O O
~
~
O
~
~
~
_
~
_
~
O
~
m
~
~
~
~OOOOOOOOOOOOOOOOOOOOOOOOOO
d~ I
o z
151
SEMICONDUCTOR INTEGRALS
Jahnke function"
and
Erode6)
for
the
confluent
hypergeometric
xM" + (7 -- x) M' -- ~ M = 0, M ( a , y , x ) ~-- 1 + - - x + 7
]
m ~ + 1 x2 + . . . . 77+
12!
l
(14)
'
it follows from the differential equation (8) that, except when p is an integer, ~(x)--
aM(l,
1 - - p, x) + bx ~ M ( i + p ,
1 -}-p, x),
where the coefficients a and b are independent of x but m a y depend upon p. Perhaps the simplest m e t h o d of determining a and b is to examine the behaviour at the limits x = 0 and x - ~ oo. For it is oo
clear from the original definition ~l~(x) = / P ! ) - i f ~ ( e + x) -1 e -~ de 0
t h a t ~ ( 0 ) ---- p-l, yielding a = p-1 since M(a, 7, 0) == 1 b y (14). At the other limit, x - + 0% the fact that the dominant term in M(a,y,x) is 6) 11) {(y _ 1 ) ! / ( a - - 1)l} e ~ x "-~ shows that the dominant term in 9~(x) for x - + oo must be {a ( - - p ) l + b} x ~ e ~. Since the integral obviously does not increase exponentially with x, b=--a(--p)!------a/(p!) sin a p (e.g. J a h n k e and E r o d e 6 ) , p. 11). Thus ~t ~(x) ~ p-1 M ( 1,1 -- p, x) -- {~/(p l) sin a p} x ~ M( 1 + p, 1 + p,x). (15) B y (14), the second M in (I 5) is precisely equal to e x. The differentiation formula 11) M'(~, 7, x) = (s/W) i ( ~ .4- 1, 7 -+- 1, x) then gives ~ : ( x ) = - - ~l':(x) - tr~
pl s i n a p
1 p ( p --1)
Mr2, 2 -- p, x) +
{x ~ M(2 + p, 2 + p, x) + px ~-1 M(1 + p, 1 + p, x)}. (t6)
B y (14), the last two M's in (16) are each equal to e x. In view of the failure of the expressions for 9J and ~ in terms of confluent hypergeometric functions when p happens to be an integer, their expansions will be determined b y a more w 4. E x p a n s i o n s .
152
R. B. D I N G L E , D. A R N D T A N D S. K. R O Y
powerful method.
Since ( J a h n k e
and E m d e ~ ) ,
pp. 20, 11)
co
f x u-1 (e + x) -1 dx = e u-1 ( - - u ) ! (u - - 1)! = ~e~-l/sin ~u, 0
the Mellin t r a n s f o r m ( T i t c h m a r s h l ~ of gt(x) is oo
9Jr(u) =
DoetschS))
~(u)
oo
x u-t 9~(x) dx
-
-
e l~
p! sin ~ u j - - -
0
e -* de
=
p! sin ~u
0
According to the inversion formula e+ioo
9.1(x) =-
~(u)
x -~ d u ,
r
the ascending series for ~[(x) is equal to the sum to the residues of 93~(u) x -u lying in the complex u-plane to the le[t of a line drawn parallel to the i m a g i n a r y axis and passing t h r o u g h u = c, while the descending series is equal to minus the sum of the residues at all poles lying to the right of this line (e.g. D i n g 1 e ~)). W h e n p is not an integer, the poles at u = -- n, w h e r e n = 0,1, 2 . . . . . due to the vanishing of sin ~u are all single and yield the contribution co y. ( p - - n - 1)!x n ~0
p Tcos ~n
to t h e ascending series, while the poles at u = -- (p + n), where again n = 0, 1, 2 . . . . . due to the infinities of (p + u -- 1)l yield the f u r t h e r contribution ~ -- ~ = o p ! s i n ~ ( p + n )
(-- 1) n x ~+~ _ n!
nx ~
plsin~p
oo x,~
X---n=0n!
~x ~ e ~
p!sin~p "
Thus, w h e n p is not a n integer, g/,(x) =
1-- p - ~
+ ( P - - 1 ) ( p - - 2 ) -- " ' "
p! sin ~p
(17)
W h e n p is a n integer, the poles at - - u -- 0, 1, 2 . . . . (p -- 1) are due only to the zeros of sin ~u, and are therefore single. The poles at - - u = p, p + 1. . . . . are however double, owing to the simultaneous infinities of the factor (p + u -- 1)! in the n u m e r a t o r . As
SEMICONDUCTOR
153
INTEGRALS
shown in D i n g 1 e 4), pp. 405-6, the ascending expansion is then*)
1{
x
~ ( x ) -----p- 1
x~
P-- 1 +(p-
(--x) ~-i
1) ( p - - 2 ) - - " " " -~ ( - p ~ . T J +
+ (--x)~ ~ x ' { ~ ( t ) - - l n x } , P! ,=0 t!
(18)
whereT(t) : d l n ( t ! ) / d t (Jahnke and E m d e 6 ) , p. 18). Independently of whether p is an integer or not, the poles at u = n, where n = 1, 2, 3 . . . . . due to the vanishing of sin nu are all single and yield the asymptotic descending series
9~(x):--~ (p-Fn-1)!x-n,,=1 p! cos nn
1 x
{ 1 p~-I ,
Jr
(#+ 1)x 2(#+2)
x
}.(19) --""
This expression can also be obtained directly from the original integral by expanding (e + x) -1 in falling powers of x, and then integrating term by term. The descending series (19) is unfortunately only applicable when x ))) p. A far more useful series is obtained by invoking the transformation (9), 9/~_1(x) : e ~ Ei~(x), and making use of the expansion of Ei~(x) developed by B 1a n c h 1) (see also D i n g 1 e 4)), giving
1 {1 +
9/~-1(x) = p + x
+
_p Jr p(p -- 2x) (p + x) (p + x) 4 +
p(p2 _ 8px + 6x 2) p(p3 _ 22p x + 58px (P+x) 6 + (p+x) s
--
24x3) ./ -"
J,
(20)
an expression convenient when both p and x are quite large. The same type of series can also be obtained from the original integral by expanding (e + x) -1 = {(e -- p) + (p + x)}-1 in failing powers of (p + x), integrating term by term, and finally regrouping the terms. Expansions for ~ can be obtained by differentiation of those for ~[~, and need not be reproduced here. R e c e i v e d 12th April, 1956.
*) T h e r e is a m i s p r i n t in eq. (27) of D i n g i e i). The l a s t factor in t h e d e n o m i n a t o r of the first s u m m a t i o n s h o u l d r e a d ( n - r 1). Appl. sei. Res. B 6
154
SEMICONDUCTOR INTEGRALS
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8)
9) 10) 11) 12) 13)
Blanch, G., 1946. Appendix A to P l a c z e k 1946. British Association Mathematical Tables, Cambridge University Press. Dingle, R . B . , P h y s l c a 2 2 ( 1 9 5 6 ) 701. Dingle, R . B . , A p p l . sci. Res. B 4 (1955) 401. D o e t s c h, G., Theorle und Anwendung der Laplace Transformation, Berlin, 1937. J ahnke, E. and F. E r o d e , Tables of Functions, Dover, New York. t945. Mathematical Tables, U.S. Government Printing Office, Washington 25, D.C. P 1 a c z e k, G., The Functions En(X), MT-I National Research Council of Canada, Division of Atomic Energy, 1946. (Reprinted in Applied Math. Series 37, pp. 57-111, U.S. Gov. Printing Office, 1954). S e r, M. J., Bulletin des Sciences Math6matiques 6 2 (1938) 171. T i t e h m a r s h, E.C., Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford, 1937. W e b b , H . A . a n d J . R. A l r e y , Phil. Mag. 36 (1918)129. W h i t t a k e r, E . T . and G. N. Watson, A Course of Modern Analysis, 4th edition, Cambridge University Press, 1946 W i 1 s o n, A. H., The Theory of Metals, 2nd edition, Cambridge University Press, 1953, pp. 264-269.
