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The specific heat of superconducting transition metals
Volume 19, number 2
THE
PHYSICS LETTERS
SPECIFIC
HEAT
OF
1 October 1965
SUPERCONDUCTING
TRANSITION
METALS
C. C. SUNG
Department of Physics, University of California, Berkeley, California and L. YUN LUNG SHEN
Department of Chemistry and Inorganic Materials Research Division, Lawrence Radiation Laboratory, University of California, Berkeley, California Received 1 September 1965
The e x p e r i m e n t a l d a t a on the t r a n s i t i o n m e t a l s of niobium, t a n t a l u m and v a n a d i u m show m a r k e d d e v i a t i o n f r o m the BCS t h e o r y f o r low r e d u c e d t e m p e r a t u r e s [1] ( t < 0.2, h e r e t = T / T c and T c i s the t r a n s i t i o n t e m p e r a t u r e ) *. A t w o - e n e r g y gap m o d e l that w a s p r o p o s e d by Suhl et al. [2] i s u s e d to obtain an e x p r e s s i o n f o r the s p e c i f i c h e a t of p u r e and i m p u r e s u p e r c o n d u c t o r s . We a r e a b l e to fit the d a t a to the e x p r e s s i o n and d e t e r m i n e the following p a r a m e t e r s of the t w o - g a p m o d e l : the s m a l l e n e r g y gap As, Ns/Nd, (Ns,(d) is the d e n s i t y of s t a t e s of s - ( d - ) band) and the m a g n i t u d e of i n t e r - b a n d i n t e r a c t i o n J . The H a m i l t o n i a n of a p u r e t w o - g a p s u p e r conductor
A f t e r the Bogoliubov t r a n s f o r m a t i o n , the H a m i l t o n i a n i s r e d u c e d to a s y s t e m of two t y p e s of q u a s i - p a r t i c l e s ( s - t y p e and d - t y p e ) with enerKi,e s ES kcr and Edk(7; and d i s t r i b u t i o n functions fs~a~k& E ; ( d ) = [Eks(d 2 .) + A2s(d) J]½ s(d) _ ka
+J k,~ k' (Sk+tS-k + ~dk' *d'k' t +h.c.)
exp (Ek~a~)/°'~ T ) + I
Ad(T)
r
- 2
As(T)7
n=0 ~ (-l)n K°L(n+I)--~-'J (4)
As(T)
Js - J~d(T) / Nd(JsJd - j2)
:
(1)
w h e r e e~,~ta~ is s - ( d - ) band kinetic e n e r g y m e a s u r e d fro~a~t"~e F e r m i s u r f a c e and S+ka(d+ka) Skq(dkci) a r e the c o r r e s p o n d i n g c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s . J s , J d and J a r e the c o u p l ing c o n s t a n t s of s - s , d - d and s - d bands. In eq. (1), the s u m m a t i o n s of k a r e extended o v e r v a l u e s c o r r e s p o n d i n g to the Debye cut off f r e q u e n c y O = 0.024 eV, which i s c h o s e n to be the s a m e for both bands. The n u m e r i c a l r e s u l t s of the s p e c i f i c h e a t a r e not s e n s i t i v e to w h e t h e r d i f f e r e n t cut off O a r e u s e d for d i f f e r e n t bands. * T is the product of temperature and the Boltzmann constant.
(3)
Jd - J ~'-~-~ / Ns(JsJd - j 2 )- =
= lnAs~t)
+ J d k ,~k, dk+ td-ksd-k' t +
1
The equations to d e t e r m i n e AS, Ad and T c a r e :
H=k~, EksS;aSkcr+~E kdd; dk + + J s k,k' ~ SktS-k*Sk'~Sk't + +
(2)
O~ = In Ad(T------
2 n:0 ~ (-1)nK o I ( n + l ) - -
(5)
W2(JsJ d - j 2 ) N s N d + (JsNs + JdNd)W + 1 = 0 (6) w h e r e W = In (0.88 T c / O ) and K n is the m o d i f i e d B e s s e l function of n - t h o r d e r . We n e g l e c t the i n t e r a c t i o n s between the q u a s i - p a r t i c l e s in our model, and w t r t e t o t a l e n t r o p y a s the sum of the e n t r o p i e s of s - and d - q u a s i - p a r t i c l e s . It follows that the e l e c t r o n i c s p e c i f i c heat contains two t e r m s :
Nd Ces = N s ~ V d
cd
Ns
es ~N s +N d
cs
(7)
es
where
I01
Volume 19, number 2
PHYSICS LETTERS
1October 1965
I0 e
Ces/ T_ ; Nd (2 5 2 × I~(-1)n(n+l)[K3(Ud(n)) + 3Kl(Ud(n)) ]
×
f
dAd(t) 1 + 4 d ( 1 / T ) TAd(T) ud(n) = (n+l)
+
to c
Kl(Ud(n))} I(J I \
Ad(T) T
a =0.5 As=0.14 a=0.5 As=0.18 a =0.5
As=0.16
and y = ~ n 2 ( N s + N d) × (Boltzmann constant) 2 . By exchanging the label (s) and (d) in eq. (85 we obtain Cdes. Which e n e r g y gap, As or Ad, is l a r g e r is d e t e r m i n e d as follows: since Nd/(Ns+Nd) 1 and Ns/(Ns+Nd) << 1 [3], Ad has to be a s signed as the l a r g e r gap so that at high r e d u c e d t e m p e r a t u r e eq. (7) gives Ces ~ Cdes which fits the e x p e r i m e n t a l data [4]. NsCSes/(Ns+Nd) b e c o m e s d o m i n a n t in the low t e m p e r a t u r e r e g i o n w h e r e T << Ad. Let x(T) = ~s(0)/As(T) and a = -[Ad(0)/As(0) ] × × j/(NsN d . j2). Equation (4) b e c o m e s oO
2 ~0(-1)n
Ko[us(n) ]
= In x(T) - a(1-x)
(8)
dAs (T)
d(I/T)
1 T~s(T) -
2Us(0) ~ (-1) n (n+l) n=O
Kl(Us(n)) (9)
oO
1 + ax - 2Us(0) n=~0(-1)n=(n+l)
Kl(Us(n) 5
P a r a m e t e r s used in d e t e r m i n i n g CSes a r e Ns/Nd, a , and /', . Equation (85 is solved n u m e r i c a l l y for x a n d S u b s t i t u t e d in eq. (9) to obtain CSes . A ( 0 ) is r e l a t e d to the l i m i t i n g slope of the e l e c t r o n i c specific heat at a low t e m p e r a t u r e limit. T h e b e s t f i t is given by A s = 0.16 Tc, a = 0.5 and Ns/N d = 1.5 × 10 -2 as shown in fig. 1. We note that the p a r a m e t e r to m e a s u r e the s t r e n g t h of i n t e r b1a n d coupling J( YsYd )2 /Ys \2 --
JsNsJdNd
1
-
0.06.
F r o m the p a r a m e t e r s As, Ad and Ns/N d, which a r e d e t e r m i n e d by fitting the data and eqs. (4-65 we c a n c a l c u l a t e Jd, Js and J. However, b e c a u s e of the s m a l l v a l u e s of Ns/N d and As/A d and the l a r g e e x p e r i m e n t a l u n c e r t a i n t y of T c and Ad we a r e unable to obtain r e l i a b l e v a l u e s of Jd, Js and J. The above model gives a r e a s o n a b l e specific heat of pure niobium over the whole m e a s u r e d t e m p e r a t u r e range. Even though detailed band
102
\
(J 1(~3 ~
tO4
•
~
*A
"" 8,25e_l,51Tc/T
..
\ I (~5
i 5
~, i 10
ira5
m~/m
L 20
m 25
30
Fig. 1. The calculated Ces for various value of AS(0) .
and
=
162
s t r u c t u r e c a l c u l a t i o n s a r e lacking in niobium, we get Ns/N d f r o m specific heat along. The d i s c r e pancy between e x p e r i m e n t a l value and c a l c u l a t i o n may be due to the fact that the p u r e s a m p l e is not 100% pure as it is a s s u m e d in the calculation. No effort was made to fit t a n t a l u m data b e c a u s e they a r e not available over a wide enough r e d u c e d t e m p e r a t u r e range. We c a l c u l a t e the effect of i m p u r i t i e s , a s s u m ing they a r e n o n - m a g n e t i c . The r e s u l t s cannot fit the e x p e r i m e n t a l data. It s e e m s that the e l e c tronic s t r u c t u r e of the i m p u r i t i e s [5] in t r a n s i t i o n m e t a l s has to be taken into account to explain the e x p e r i m e n t a l data. It is a p l e a s u r e to thank Dr. P. Flude for h e l p ful d i s c u s s i o n s and Dr. N. E. P h i l l i p s for his i n t e r e s t s in this work. This work was done u n d e r the a u s p i c e s of the U. S. Atomic E n e r g y C o m m i s s i o n .
References 1. L.Y.L.Shen, N.M.Senozan and N.E.Phillips, Phys. Rev. Letters 14 (1965) 1025. 2. H. Suhl, B.T.Matthias and L. R. Walker, Phys. Rev. Letters 3 (1959) 552. 3. J.M. Ziman, Electrons and pnonons (Oxford Press, 1960). 4. H.A. Leupold and H. A. Boorse, Phys. Rev. Letters 134 (1964) A1322. 5. A.M. Clogston, Phys.Rev. 116 (1964) A1417.
THE
PHYSICS LETTERS
SPECIFIC
HEAT
OF
1 October 1965
SUPERCONDUCTING
TRANSITION
METALS
C. C. SUNG
Department of Physics, University of California, Berkeley, California and L. YUN LUNG SHEN
Department of Chemistry and Inorganic Materials Research Division, Lawrence Radiation Laboratory, University of California, Berkeley, California Received 1 September 1965
The e x p e r i m e n t a l d a t a on the t r a n s i t i o n m e t a l s of niobium, t a n t a l u m and v a n a d i u m show m a r k e d d e v i a t i o n f r o m the BCS t h e o r y f o r low r e d u c e d t e m p e r a t u r e s [1] ( t < 0.2, h e r e t = T / T c and T c i s the t r a n s i t i o n t e m p e r a t u r e ) *. A t w o - e n e r g y gap m o d e l that w a s p r o p o s e d by Suhl et al. [2] i s u s e d to obtain an e x p r e s s i o n f o r the s p e c i f i c h e a t of p u r e and i m p u r e s u p e r c o n d u c t o r s . We a r e a b l e to fit the d a t a to the e x p r e s s i o n and d e t e r m i n e the following p a r a m e t e r s of the t w o - g a p m o d e l : the s m a l l e n e r g y gap As, Ns/Nd, (Ns,(d) is the d e n s i t y of s t a t e s of s - ( d - ) band) and the m a g n i t u d e of i n t e r - b a n d i n t e r a c t i o n J . The H a m i l t o n i a n of a p u r e t w o - g a p s u p e r conductor
A f t e r the Bogoliubov t r a n s f o r m a t i o n , the H a m i l t o n i a n i s r e d u c e d to a s y s t e m of two t y p e s of q u a s i - p a r t i c l e s ( s - t y p e and d - t y p e ) with enerKi,e s ES kcr and Edk(7; and d i s t r i b u t i o n functions fs~a~k& E ; ( d ) = [Eks(d 2 .) + A2s(d) J]½ s(d) _ ka
+J k,~ k' (Sk+tS-k + ~dk' *d'k' t +h.c.)
exp (Ek~a~)/°'~ T ) + I
Ad(T)
r
- 2
As(T)7
n=0 ~ (-l)n K°L(n+I)--~-'J (4)
As(T)
Js - J~d(T) / Nd(JsJd - j2)
:
(1)
w h e r e e~,~ta~ is s - ( d - ) band kinetic e n e r g y m e a s u r e d fro~a~t"~e F e r m i s u r f a c e and S+ka(d+ka) Skq(dkci) a r e the c o r r e s p o n d i n g c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s . J s , J d and J a r e the c o u p l ing c o n s t a n t s of s - s , d - d and s - d bands. In eq. (1), the s u m m a t i o n s of k a r e extended o v e r v a l u e s c o r r e s p o n d i n g to the Debye cut off f r e q u e n c y O = 0.024 eV, which i s c h o s e n to be the s a m e for both bands. The n u m e r i c a l r e s u l t s of the s p e c i f i c h e a t a r e not s e n s i t i v e to w h e t h e r d i f f e r e n t cut off O a r e u s e d for d i f f e r e n t bands. * T is the product of temperature and the Boltzmann constant.
(3)
Jd - J ~'-~-~ / Ns(JsJd - j 2 )- =
= lnAs~t)
+ J d k ,~k, dk+ td-ksd-k' t +
1
The equations to d e t e r m i n e AS, Ad and T c a r e :
H=k~, EksS;aSkcr+~E kdd; dk + + J s k,k' ~ SktS-k*Sk'~Sk't + +
(2)
O~ = In Ad(T------
2 n:0 ~ (-1)nK o I ( n + l ) - -
(5)
W2(JsJ d - j 2 ) N s N d + (JsNs + JdNd)W + 1 = 0 (6) w h e r e W = In (0.88 T c / O ) and K n is the m o d i f i e d B e s s e l function of n - t h o r d e r . We n e g l e c t the i n t e r a c t i o n s between the q u a s i - p a r t i c l e s in our model, and w t r t e t o t a l e n t r o p y a s the sum of the e n t r o p i e s of s - and d - q u a s i - p a r t i c l e s . It follows that the e l e c t r o n i c s p e c i f i c heat contains two t e r m s :
Nd Ces = N s ~ V d
cd
Ns
es ~N s +N d
cs
(7)
es
where
I01
Volume 19, number 2
PHYSICS LETTERS
1October 1965
I0 e
Ces/ T_ ; Nd (2 5 2 × I~(-1)n(n+l)[K3(Ud(n)) + 3Kl(Ud(n)) ]
×
f
dAd(t) 1 + 4 d ( 1 / T ) TAd(T) ud(n) = (n+l)
+
to c
Kl(Ud(n))} I(J I \
Ad(T) T
a =0.5 As=0.14 a=0.5 As=0.18 a =0.5
As=0.16
and y = ~ n 2 ( N s + N d) × (Boltzmann constant) 2 . By exchanging the label (s) and (d) in eq. (85 we obtain Cdes. Which e n e r g y gap, As or Ad, is l a r g e r is d e t e r m i n e d as follows: since Nd/(Ns+Nd) 1 and Ns/(Ns+Nd) << 1 [3], Ad has to be a s signed as the l a r g e r gap so that at high r e d u c e d t e m p e r a t u r e eq. (7) gives Ces ~ Cdes which fits the e x p e r i m e n t a l data [4]. NsCSes/(Ns+Nd) b e c o m e s d o m i n a n t in the low t e m p e r a t u r e r e g i o n w h e r e T << Ad. Let x(T) = ~s(0)/As(T) and a = -[Ad(0)/As(0) ] × × j/(NsN d . j2). Equation (4) b e c o m e s oO
2 ~0(-1)n
Ko[us(n) ]
= In x(T) - a(1-x)
(8)
dAs (T)
d(I/T)
1 T~s(T) -
2Us(0) ~ (-1) n (n+l) n=O
Kl(Us(n)) (9)
oO
1 + ax - 2Us(0) n=~0(-1)n=(n+l)
Kl(Us(n) 5
P a r a m e t e r s used in d e t e r m i n i n g CSes a r e Ns/Nd, a , and /', . Equation (85 is solved n u m e r i c a l l y for x a n d S u b s t i t u t e d in eq. (9) to obtain CSes . A ( 0 ) is r e l a t e d to the l i m i t i n g slope of the e l e c t r o n i c specific heat at a low t e m p e r a t u r e limit. T h e b e s t f i t is given by A s = 0.16 Tc, a = 0.5 and Ns/N d = 1.5 × 10 -2 as shown in fig. 1. We note that the p a r a m e t e r to m e a s u r e the s t r e n g t h of i n t e r b1a n d coupling J( YsYd )2 /Ys \2 --
JsNsJdNd
1
-
0.06.
F r o m the p a r a m e t e r s As, Ad and Ns/N d, which a r e d e t e r m i n e d by fitting the data and eqs. (4-65 we c a n c a l c u l a t e Jd, Js and J. However, b e c a u s e of the s m a l l v a l u e s of Ns/N d and As/A d and the l a r g e e x p e r i m e n t a l u n c e r t a i n t y of T c and Ad we a r e unable to obtain r e l i a b l e v a l u e s of Jd, Js and J. The above model gives a r e a s o n a b l e specific heat of pure niobium over the whole m e a s u r e d t e m p e r a t u r e range. Even though detailed band
102
\
(J 1(~3 ~
tO4
•
~
*A
"" 8,25e_l,51Tc/T
..
\ I (~5
i 5
~, i 10
ira5
m~/m
L 20
m 25
30
Fig. 1. The calculated Ces for various value of AS(0) .
and
=
162
s t r u c t u r e c a l c u l a t i o n s a r e lacking in niobium, we get Ns/N d f r o m specific heat along. The d i s c r e pancy between e x p e r i m e n t a l value and c a l c u l a t i o n may be due to the fact that the p u r e s a m p l e is not 100% pure as it is a s s u m e d in the calculation. No effort was made to fit t a n t a l u m data b e c a u s e they a r e not available over a wide enough r e d u c e d t e m p e r a t u r e range. We c a l c u l a t e the effect of i m p u r i t i e s , a s s u m ing they a r e n o n - m a g n e t i c . The r e s u l t s cannot fit the e x p e r i m e n t a l data. It s e e m s that the e l e c tronic s t r u c t u r e of the i m p u r i t i e s [5] in t r a n s i t i o n m e t a l s has to be taken into account to explain the e x p e r i m e n t a l data. It is a p l e a s u r e to thank Dr. P. Flude for h e l p ful d i s c u s s i o n s and Dr. N. E. P h i l l i p s for his i n t e r e s t s in this work. This work was done u n d e r the a u s p i c e s of the U. S. Atomic E n e r g y C o m m i s s i o n .
References 1. L.Y.L.Shen, N.M.Senozan and N.E.Phillips, Phys. Rev. Letters 14 (1965) 1025. 2. H. Suhl, B.T.Matthias and L. R. Walker, Phys. Rev. Letters 3 (1959) 552. 3. J.M. Ziman, Electrons and pnonons (Oxford Press, 1960). 4. H.A. Leupold and H. A. Boorse, Phys. Rev. Letters 134 (1964) A1322. 5. A.M. Clogston, Phys.Rev. 116 (1964) A1417.