On superconductivity in degenerate semiconductors

Volume27A, number 4

PHYSICS LETTERS

ON S U P E R C O N D U C T I V I T Y

IN DEGENERATE

1 July 1968

SEMICONDUCTORS

J. A P P E L

Gulf General Atomic Incorporated, John Jay Hopkins Laboratory for Pure and Applied Science, San Diego, California, USA Received 20 May 1968

For the Bogoliubov kernel describing the attractive interaction between electrons due to the exchange of acoustic phonons, the frequency factor ~ in the gap formula A = ~ exp (-l/k) is found as a function of the electron concentration n or of the ratio EF/O) D (E F = Fermi energy, wD = Debye energy).

In the BCS theory of s u p e r c o n d u c t i v i t y [1] the p a i r i n g of e l e c t r o n s in s i n g i e t spin states o c c u r s in a thin shell a r o u n d the F e r m i s u r f a c e . The t h i c k n e s s of this shell i s of the o r d e r of the Debye energy O~D, which in m e t a l s i s much s m a l l e r than the F e r m i energy E F. In e x p e r i m e n t a l studies of d e g e n e r a t e polar s e m i c o n d u c t o r s such as GeTe, SnTe, I n T e , and SrTiO3..x , superconductivity has been found for r a t h e r s m a l l e l e c t r o n c o n c e n t r a t i o n s where the r a t i o E F/o~ D i s of the o r d e r of magnitude 1 or s m a l l e r . The q u e s t i o n a r i s e s : How does the energy gap Ao depend on the e l e c t r o n c o n c e n t r a t i o n n or on E F / ~ D ? F o r a given compound, an a c c u r a t e evaluation of Ao or of the t r a n s i t i o n t e m p e r a t u r e T c r e q u i r e s the knowledge of the effective i n t e r a c t i o n between two conduction e l e c t r o n s a s a function of energy and m o m e n t u m t r a n s f e r and a s a function of the e l e c t r o n c o n c e n t r a t i o n . An a c c u r a t e evaluation of this i n t e r a c t i o n i s a difficult task for s e v e r a l r e a s o n s . F o r example, the n e g l e c t of e l e c t r o n - p h o n o n v e r t e x c o r r e c t i o n s a s s o c i a t e d with the r a n d o m - p h a s e a p p r o x i m a t i o n for the e l e c t r o n - p h o n o n s y s t e m , has not been studied for EF/OJ D <~1. B e c a u s e of such difficulties, it a p p e a r s worthwhile to find for d e g e n e r a t e s e m i c o n d u c t o r s a s i m p l e and qualitative r e l a t i o n of the BCS f o r m ,

,% = • exp (-I/~.),

(1)

between energy gap A o and e l e c t r o n c o n c e n t r a t i o n n. We a d d r e s s o u r s e l v e s h e r e to the calculation of ~(n) u n d e r the a s s u m p t i o n that i n t r a v a l l e y s c a t t e r i n g of acoustic phonons m e d i a t e s an a t t r a c t i v e i n t e r action r e s p o n s i b l e for superconductivity. The Coulomb r e p u l s i o n i s ignored. It i s s m a l l in the highly polar s e m i c o n d u c t o r s w h e r e s u p e r c o n d u c t i v i t y has been observed. To find ~(EF/O~D) , it i s convenient to s t a r t f r o m the energy gap equation that Bogoliubov [3] has d e r i v e d with his c o m p e n s a t i o n method:

,,(k) =

k')

4~ o

[A2(~,) +~2(k,)]½ '

(2)

where e(k) = E(k) - E F = (1/2m*)(k 2 -k2F), and V(k,k') i s the i n t e r a c t i o n k e r n e l (m* = band m a s s ) . The energy gap 4 o = 4(k F) is found in the BCS f o r m (1), with ~) = "~(EF/WD), by solving the i n t e g r a l equation with Bogoliubov's analytic method. T h i s method s t a r t s f r o m the o b s e r v a t i o n that the condensation a m plitude has a n a r r o w peak at E F so that it is useful to r e w r i t e eq. (2) in the f o r m

,,(k) = Q(~,k F)

o

dk'

, +

(e'2+A'2) ~

o

[Q(k,k')-O(k, kF) ] (e,~.÷,,,~.)½,

(3)

where Q(k, k') = (1/4~2)V(k, k')k'2A(k'). The i n t e g r a n d of the second t e r m v a n i s h e s at k F and t h e r e f o r e , u n d e r the s q u a r e root, A' = A(k') may be set equal to z e r o provided A o << E F. After s o m e a n a l y s i s , one a r r i v e s at an i n t e g r a l equation whose f i r s t i t e r a t i o n y i e l d s the solution in f o r m of eq. (1) with

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Volume27A, number 4

~=44~-EFexp

PHYSICS LETTERS

1July1968

Ik'-kFJ 1 kF .)elk' ,

l l ; d [V(kF,k')V(k',kF) k'2 (k''kF)~ -~

~

m*kF

V2

~-jlog(2

(4)

and = N(o)r

t

1

'

2kF < qD '

{qD/2kF }2 '

2kF > qD "

H e r e N(0) = k F m * / 2 ~ 2 and E F i s t h e F e r m i e n e r gy. T h i s r e s u l t f o r A (k) b e c o m e s an exact s o l u tion of eq. (4) a s V ~ 0 [4]. F o r the Bogoliubov i n t e r a c t i o n k e r n e l , the f a c t o r ~ depends only on the r a t i o V(k, k F ) / V and thus i s a l m o s t i n d e p e n dent of the coupling p a r a m e t e r . Umklapp s c a t t e r ing a f f e c t s X, not 0. It i s s e e n f r o m fig. 1 that ~ / W a c ( 2 k F ) v e r s u s n i s n e a r l y constant, except for l a r g e - e l e c t r o n c o n c e n t r a t i o n s , w h e r e 2k F > F o r k_ < k F < ½qD, one h a s ~ ~ 1qD" > . 5 Wac(2~F) oc nZ/S. T h i s p r o p o r t i o n a l i t y a r i s e s b e c a u s e the d i s p e r s i o n of V(kF, k) o c c u r s n e a r k F and b e c a u s e the p h a s e s p a c e that counts f o r the effective i n t e r a c t i o n i s p r o p o r t i o n a l to the phonon f r e q u e n c y of the wave v e c t o r at which the d i s p e r s i o n t a k e s p l a c e (k ~ kF).

(s)

2.0

\ 1.0

qo = 108 CM

3

k 0 = 2m~'c = 5 X I O 5 CM-I 0.5

i

I0 -z

I

jO 16

iO -I

J

jO 17

i0 0

I

i018

EF l ~ 0 l 1019

iO t

I

1020

i i I0 2

L

i0 21

I

fO22

n (CM -3}

It i s a p l e a s u r e to thank P r o f e s s o r W. Kohn for s e v e r a l v a l u a b l e d i s c u s s i o n s .

Fig. 1. The pre-exponential frequency factor ~/0Jac(2kF) as function of EF/0J D (or of n).

References 1. J.Bardeen, L.N. Cooper and J . R . Schrieffer, Phys. Rev. 108 (1957) 1175. 2. L.V. Keldysh, Usp. Fiz. Nauk 86 (1965) 327; Soviet Phys. Solid State 8 (1965) 496; J.Appel, Phys. Rev. Letters 17 (1966) 1045; H. Schuster, Diplomarbeit (unpublished), Erlangen (1968). 3. N.N. Bogoliubov, Zh. Exp. i Teor. Fiz. 34 (1958) 58; Soviet Phys. JETP 7 (1958) 41. 4. A.Quattropani and C. P. Enz, Phys. Letters 26A (1967) 100.

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