Thermodynamics of extended s-wave superconductor

Physica C 172 (1990) 345-351 North-Holland

Thermodynamics of extended s-wave superconductor H. C h i a n d J.P. C a r b o t t e Ph.vsi~" Department, McMaster University, tlamilton, Ontario, Canada L8S 4MI

Received 7 February 1990 Revised manuscript received 22 October 1990

We have calculated the specific heat for a model of an extended s-wave superconductor. When the coupling in the constantchannel (PI, pure isotropic) dominates over the anisotropic-channel (PA, pure anisotropic), the specific heat jump starts at the canonical value 1.43, decreases monotonically with increasingPI-PA mixing 2 ~and saturates at 1.43[4/[ 7 + 4ct3+ or4] ] ---R for ,;.~~ . Here a3 and a4 are, respectively, the average of cube and fourth power of the PA part which could be a Fermi surface harmonic. When the PA part is larger than the PI part coupling, the specific heat jump starts at the smaller value of 1.43/ot4, exhibits a minimum and then saturates to the same value R as 2 j ~oo. In both cases, PI-PA mixing decreases the jump over the pure state value.

1. Introduction The discoveries of heavy Fermion, organics and high-To oxide superconductors have triggered a renewed interest in anisotropic superconductors. Pokrovskii studied the effects of anisotropy and found that the normalized j u m p in the specific heat at T~ is smaller than 1.43 of the isotropic one. The effects of n o n m a g n e t i c impurities on the Tc of anisotropic superconductors were studied in three papers [ 2 - 4 ] . The separable model proposed by Markowitz and K a d a n o f f [2] has been widely used to study the anisotropic superconductor containing: a) n o n m a g netic impurities [2,3,9]; b) magnetic impurities [6,7]; c) the C o u l o m b repulsion [ 8 - 1 0 ] ; and d) strong coupling effects [ I 1,12]. A nonseparable model has been studied by Leaven et al. [10], Whitemore et al. [ 12 ], and P r o h a m m e r [ 13 ]. More recently, the anisotropic superconductor with s-wave and d-wave c o m p o n e n t s has been investigated for heavy fermion superconductors [ 1 4 - 1 7 ] and for high-T~ superconductors [ 18,19 ]. The thermal propertics of a d-wave or anisotropic s-wave superconductor havc been studied in refs. [ 10-28 ]. Another way to study isotropic superconductors is to use the two bands model [ 2 9 - 3 1 ] . So far, the d-wave superconductor has been studied in some detail and

has had some success in explaining some properties of the heavy fermion superconductors. In the present work, we study the t h e r m o d y n a m i c properties of an anisotropic superconductor within a nonseparable model. The plan of the paper is as follows. Section 2 gives the general formalism. In section 3 we study various limiting cases. Numerical results are given in section 4. Section 5 is a summary.

2. Formalism The anisotropic superconductor is described by the Hamiltonian

n=T.

ka

~k Ck, + t~"k,, -- Y. V k k ' C J , ~ C + _ k , C _ k . , C ~ . . . kk"

(1)

With the effective electron-electron interaction Vkk, given by t~

= Vo+ V,(Qt(fc)+Q,(f~'))+ ;~Q,(/~)Qt(Kr ' ) for I~*l and I~1
(2)

Here ~ is the single particle energy measured from

0921-4534/90/$03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland)

tl. Cht. J.P. ('arbotte / Thermodynamics of extended s-wavesuperconductor

346

the Fermi surface. The interaction (2.2) is of the extended s wave type. CZ, is the creation operator for an electron with m o m e n t u m k and spin 0/, ~o~ is the cutoff energy and ~ = k / I k l . In eq. (2), 1"o and V2 are the interactions for PI (pure isotropic) and PA (pure anisotropic) channel, respectively, and V, is the interchannel coupling given the mixing. The case I', = Vo=0 (PA) has been studied by Suzumura and Schultz [ 28 ]. Our work is a generalization of theirs. Further, the Fermi surface harmonics Q,(k') satisfies the conditions = 0 ,

(3a)

( Q ~ ( , ~ ) ) = 1,

(3b)

(. dS~ d.",F U k

-C2(T) f

dS~. ~.~k.A3. "

(7)

J. s¥ /-'l.

Here C~(7") and C2(T) arc given by C, (T) = l o g ( 1 . 1 3 e o / T ) ,

(8)

I ('2(T) = ~ ( 3 )

(rtT)-' '

(9)

where ~(3) is the Reimann ~ function. From eq. (2). we can write A k a s Ak = J o + d l Qi (~:) -

where (...) denotes the average over the Fermi surface, Sf:, i.e., (A ( ~ ) ) = I,'r dS~"4(rK)/~v~ ,r dS~vk

(4)

(10)

Using eqs. ( 2 - 4 ) , (7) and (10), we obtain the equations for Ao and ,J~ as Ao = (,;.odo +). i.h )Ci (7") -,;.o(Ag+ 3AoA~ + ot3J~) C2(7")

where v, is the velocity of the electrons and the integration is carried out over the Fermi surface St.-. When Vo= I,'~= V2, V** given by eq. (2) reduces to the separable mode [21. To make the model more concrete, we can consider, as a possibility, an organic superconductor consisting of I D chains. A possible value for Q ( k ) is v/2 cos (~r.) and k~. is mom e n t u m perpendicular to the chains as in the work of Suzumura and Schultz [281. This function is invariant under the point group operations and so eq. (2) is an acceptable interaction. For most of this paper, however, we will not specify Q~ (/~). Only at the end will we present results for the specific model of an organic superconductor. From eq. ( 1 ), we can obtain an equation for the ordcr parametcr :J~ as follows:

1

,J~= ~ V~k J~ "l" .~. 2 . . . o),, +¢~ +a~

(5)

where m , = x T ( 2 n + l ) are the Matsubara frequencies and T is temperature. For the interaction given by eq. (2), eq. (5) can be rewritten as a~,=

f,-

dSj,. x T ~ l~,.d,~. ~- V/,.

con

2 + a1~ ),/_,.

(6)

(O,) n

For T near 7-,., wc can neglect J~ and higher order terms. Equation (6) becomes

-2~(3JgA~+3a~AoA2+0/4A3)C2(T). dl

=

(lla)

(;.Ido +22AI )Ct( T) - 2 , (Ag + 3aoA~ + a3d 3)C2(T)

-)t2(3JoA, +30/3AoJ2~ + 0 / 4 J ~ ) C 2 ( T ) .

(lib)

In the above equation, 2 , = I.;N(O). ( i = 0 . 1, 2): N(0)=

f~. dSk F

/)k

is the density of states on the Fermi surface and o~3 and 0/4 are defined as 0/3 = ( Q t 3 ( ~ ) )

,

0/4 = ((~14 ( ~ ) > .

(12a) (12b)

Now, we proceed to solve for 3o and J~ from eqs. ( 11 ). Eliminated 33 terms from eqs. ( 11 ), we get AoAo + Bodo + Co = 0

( 13 )

with Ao =32~C2J~ , Bo =Z~ +32~C20/3d~ ,

Co = ( 0/42~('2a~ - )., )a, . From eqs. (13) and (14), we have

(14)

347

I1. Chi. J.P. Carbotte / Thermodynamics of extended s- wave superconductor

With eqs. (16) and (23), eq. (29a) becomes Z~0 :

72 2, y, - 2, "

2Ao = g o J , - g , A3+O(35,)

With eqs. (25) and (26), eq. (29b) can be rewritten as

with go and g~ given by (16)

go=2J2, .

g,=

+3 2~\1 + a 3 ~-t

At =2o+2~C~

,

(29b)

(15)

,

(17) (18a)

7, 2, 2, 2r"

(29c)

Using eqs. (29c, 8, 18, 19), Tc is obtained as Tc=l.13 me

2~=22 +2~C~ ,

(18b)

2~=2~-2o22 ,

(19)

× exp[ 2° " J r A 2 - [ (,).0-~-22)2-t-4 (21--,~.0,).2) ] I/2] 2(2~ -2022) (29d)

(20)

Tc has a form similar to that of a two bands model. For T near To 3~ and d,z can be written as

Eliminating 3~ from eqs. ( 11 ), we obtain A~3~ + B , 3 , +C~ = 0

with A, = 32sC2(a 4 -a2)Z]o , Bi =a32t +a421-3a32~C2d 2,

(21)

Equations (20) and (21) give

32 =D , ( I - T / T c ) .

(3Oh)

Using eqs. (16-18), (23-28). we obtain Do and D~

Do = T¢ ~

~

(22) 1

with

x2(x2+ 1 )

= C o x4+6x2+4a3x+a4

(23)

f~ =72/71 ,

f,=

(30a)

as

C I ~-~ -- ( C~'42r + ~321 -- Ot'42 s C2 d 2 )z~0 .

d, =foAo -.~ 3i] + O(d~)

d~] =Do( 1- T / T c ) ,

y, ) , ~ . + 3

(a.-o~)y,

D,=L

I

(25)

72 =afi.l + a 4 2 r .

(26)

Substituting eq. (22) into (15), we obtain

a~ _

x2+

,

(31b)

1

= CO x 4 + 6 x 2 + 4 a 3 x + a 4

where x=2°/2, =2,/2 o .

fog,,- 1 f i g , +gof, "

(31a)

dA~ d T r,.

(24)

Yl ----'O~32t "~-O~421 ,

,

(27)

Similarly, substituting eq. (15) into eq. (22), we have

(32)

I n the above equations, the superscript "'0" means that the quantity is calculated at To. The specific heat of the system is calculated from

C~=2 ~ E,

Of(Ek)

,~ = fogo-I fog, +go3f~ •

(28)

Therefore, the transition temperature is given by fogo = l .

(29a)

- -

with

l"

-:ze~

vk \

2- ~ / 0 E ~

(33)

H. Chi, J.P Carbotte / Thermodynamics of extended s-wave superconductor

348

E~ = ( ~ +zl~),/2

(34)

dx\TTc]=O using eqs. (3), (4) and (33), C~ for T-,7~. is calculated as

,.,

:,'i. L\

where y=2rt2N(O)/3 and the normalized specific heat j u m p at T,. is obtained from eqs. ( 3 5 ) , ( 3 1 ) as C.. ~_¢'n AC (x2+ I )2 C, ~'~ - ;.,~,. - 1.43 x 4 + 6 x 2 + 4 a 3 x + c ~ , ,

(41a)

or 0x -- =0.

(41b)

O),,

From eqs. (4 i a) and ( 36 ), we have 2X 3 "k- 30~3X2m -- ( 3 - - O t 4 ) X m - - a 3 = 0 '

if or3 = 0, eq. (42) becomes

"

Xm = ( ( 3 - o t 4 ) / 2 ) '/2 (X2+I)

= 1.43

(36)

From eq. (43), one notes that xm< 1 as a 4 > 1. Further, Xm does not exist for a 4 > 3. The m i n i m u m AC/ yT,. becomes \/(}~,.) AC

3. Q u a l i t a t i v e f e a t u r e s a n d l i m i t i n g c a s e s

= 1.43 ot-----~ 95 -- a 4 " if =

From eqs. (30, 31, 36), we note that Do, D, and AC/yT¢ decrease as or3 and a4 increase. From eq. ( 36 ), we note that the normalized specific heat j u m p at T¢ is always less than the BCS value 1.43 as a 3 x > 0 and a 4 > 1. a3x>O is obvious and a 4 > l has been noted before [ l ]. We can prove it by looking at the following inequality >0.

(37)

Using eq. (4), this becomes (38)

It is also noted that AC/TT~ depends on 20, :.~ and 22 through x which does not depend on a3 and a4. We can rewrite eq. ( 32 ) by using eq. (18a) as follows: 1

l ( k o - , ~ 2 ) + [ ( , ~ , , - a 2. / , 2 -~-A' ^:l, J, / ' - , J-

(39)

Let us depart for a moment to consider a PA state with the symmetry (organic) Q~ =,¢/2 cos(/~,,). For the case of a Fermi surface [28], we have a 3 = 0 , a 4 = 1.5, Ym=,V/~/2 and (AC/yT~) ....... =0.667. If a3 # 0, we can solve for x,, from eq. (42) and then calculate the j u m p from eq. (36). Now we look at several limiting cases. ( 1 ) Pl-case, i.e. 2. = 2 2 = 0 . From eqs. (39), we have x = o o . Then eqs. (29 d) and (36) become

AC

yT~

02,\77.]=~\.,~¢]~=0, AC/yT~ has an extreme when

i=0, 1,2.

(40)

(45)

- 1.43.

( 2 ) PA-case, i.e. ,i.o=)-, = 0. Then we have x = 0, Tc and the j u m p become Tc = 1.13 o9¢ e x p ( - 1/).2) AC

From this equation one notes that x > 0 and x and then AC/TT¢ depends on the difference 2 o - 2 2 . It is also noted that x > 1 if;to>3.2 and x < I if;to<22. As AC/;,T,.< 1.43, let us look for a possible minimum. From

(44)

~ m

7",. = 1.13 ¢o,, exp( - 1/,,1.o) ,

( Q~ ) =or4 > 2 Z - Z 2< i .

.v=~.

(43)

2

(x2+ l )2+4x2+4a3x+oca-- 1

((Q~-Z)2)

(42)

7T¢

1

- 1.43--.

(3)

(46) (47)

Og 4

Decoupled PI and PA case superconductor,

2, = 0. Then T¢= 1.13 o9¢ e x p ( - i/2ma~),

(48)

AC _~'1.43 7T~-(l.43/a4

(49)

for for

20>2: 20<22

Here ;tma, is the larger one of 2o and )-2.

H. (.'hi, J.P. Carbotte / Thermodynamics o f extended s-wave superconductor

(4) When the coupling strength of PI and PAchannel are equal, ;to=k2. Then x = 1 and Tc = 1.13 co~.exp z~C

ko+tl

' ' ' '

I ' ' ' '

349

I ' ' '

I ' ' '

I'

'

as=0.0, .a4=2.14

........ AI=0.04

1.5

'

'

- -

'

'

~t=0.0

- - - X,=O.08

4

),7~ - 1.43 7+4o~3 +o~4 '

'

ko=0.16

- - -

~=0.12

.....

~u=0.16

(51)

which is less than 1.43/~4 for most PA states. It is worth noting that, in this specific case, Z~C/TT~ is independent of).1 although T~ does depend on kl. In this case, A,] has the same slopes as A2, at "F~ and this slope does not depend on ;t,. With the help of the above discussion, and in summary, we can understand the quantitative dependence o f AC/77% on the coupling constant ;to, ,i 1, and 22. We need only consider the dependence of AC/TT¢ on ;t, and ;t: (or 20) since AC/TT¢ depends only on ;to-).2. Let us first consider its dependence on ;tl. If ;to > k 2, from eq. ( 39 ) one sees that x decreases when 2, increases and x > I. Therefore AC/,/Tc decreases monotonically from 1.43 as ;t i increases but is never less than AC/TT~I,-=I=4/[7+4x3+ot4]. If ;to<;t~, then x increases from zero as ;t i increases from zero. In some cases, we can have a m i n i m u m value AC/ 7T~I ..... between AC/}'TcI~,=o = 1.43/o:4 and AC/ 7Tcla, .~ = 1.43{4/[7+4o~3+ot4]}, so that AC/TT,. decreases as k, increase for k, < (;t,)m and increases for ;t i > (;tl),~.

4. Numerical results In this section, we give numerical results for the normalized specific heat j u m p at T¢ given by eq. (36). We study its dependence on three coupling parameters ;to (Pl-channel), ,;-2 (PA-channel) and ;t~ (coupling between the two channels). To make the weak coupling theory meaningful, we have used, in the numerical work, the constraint Tc/toc < 0.1. Also, to be concrete, we will use the model o f an organic superconductor introduced by Suzumura and Schulz [28] with Q , ( / ~ ) = x / 2 c o s (,~.). In fig. 1, we show the normalized j u m p AC/yTc as a function of PAchannel coupling 22 for a fixed value o f Pl-channel coupling ;t o = 0.16 and several values o f interchannel coupling ;t,, namely ; t l = 0 . 0 (solid curve), 0.04 (short dashed), 0.08 (intermediate dashed), 0.12

'~

1.0

"''\..

~

,... "..

'..

-'~~~:-z== 0.5

, 0.0

,

,

,

1 , 0.1

,

,

,

1 . . . . 0.2

I 0.3

,

~ ,

,

I 0.4

,

,

,

, 0.5

Am

Fig. I. Specific heat j u m p as a function of coupling ,~2 in the PAchannel for a fixed value of Pl-channel coupling ko=0.16. When the interchannel coupling 2 j =0, the specific heat jump is a constant equal to 1.43 for 20>22 and 0.953 for 22<20 and correspond, respectively, to the PI and PA case. When ).~ ~0, the two channels are coupled and the specific heat jump is always less than the corresponding uncoupled case.

(long dashed) and 0. i 6 (short intermediate dashed curve). For the solid curve k, = 0 . 0 and we are dealing with a Pl-superconductor for 20>22 and a PA superconductor for ;t2>ko. Thus, the solid line is equal to a constant 1.43 for ;t2<0.16=;to and another constant 0.953 for 2 2 > 0 . 1 6 = 2 o with a sharp discontinuous behaviour at 22 = 20. The other curves show no such discontinuities but, initially, decrease smoothly as k: increases. When 22 =;to, all curves pass through the same value, namely 0.673, after which they display a m i n i m u m o f 0.667 before rising gradually towards the solid curve value of 0.953 as 22 becomes larger then 20 and k,. For small 2, values, the approach to 0.953 is faster than for large values o f In fig. 2, we show results for the normalized specific heat j u m p AC/yTc as a function ofinterchannel coupling 2t for PI channel coupling 2 2 = 0 . 1 6 and various values o f Pl-channel coupling ;to, namely 2 0 = 0 . 0 (solid curve), 0.04 (short dashed curve), 0.08 (intermediate dashed curve) and 0.12 (long dashed curve). Note that all these values are smaller than ;t2. In this instance, all curves start at the pure

tt. ('hi, J.P. ('arbotte / Thermodynamics of extended s-wave superconductor

350

0.8

. . . .

I ' ' ' '

I ' ' ' '

I'

' ' ' 1

- hz=O.16

........

as=O.O,

ct4=2.14

-

_

__

'

''

'

'

Xo=O.O0 Xo=O.04 _ Xo=O.O8 _

1.5

ho=O.12

0.7

¢.) <1

i

<1

I'

' ' ' I '

' ' ' I '

Xz=O.18

' ' ' I ' - -

ct~=0.0,

o4=2.14

........

h°=0'20

-

h o = 0 . 2 4

-

- -

.

.

.

:1

\

.

.

'

'

,

,

ho=O.16

-

-

.

x~=o.28

x~=o.32

1.0

~,,,.~-'~,___ --_-..--...............

0.6

' ' '

Li',. ,\',, \ \. ......... Z.2..-:..-...-:...

0.5

' 0.0

'

'

,

I 0.1

,

,

,

,

I

,

,

0.2

,

,

I 0.3

,

i,,1|1,, 0.4

0.5 0.5

, 0.0

,

,

,

I 0.1

,

,

,

,

I

,

,

PA value o f 0.953 and d r o p as the interchannel coupling is increased. Each curve shows a m i n i m u m value o f 0.667 before tending asymptotically for ).~ - - , ~ towards the saturated value o f 0.673. We note than in all cases the switching on o f the interchannel coupling reduces the j u m p over its ideal pure PA value. Also, the initial decrease with increasing 2. is more rapid for finite ;to than when ,;.0=0.0. A similar situation holds when )t2 is less than )to as shown in fig. 3. Here the j u m p is shown as a function o f interchannel coupling ;t, for one value o f PA-channel coupling )t 2 = 0.16 and several values o f Pl-channel coupling, namely 20 = 0.16 (solid curve ), 0.20 (short dashed curve), 0.24 ( i n t e r m e d i a t e dashed curve), 0.28 (long dashed curve), and 0.32 (short-intermcdiate dashed curve). All values o f 2o are larger than ).2=0.16 except for).o=)t2 (solid curve). In this case, AC/TTc is i n d e p e n d e n t o f 2, and simply equal to 0.673. In all other cases, the j u m p starts at the canonical value o f !.43 for no interchannel coupling and then drops m o n o t o n i c a l l y as )t, increases but always remains above 0.673 which is achieved only in the a s y m p t o t i c limit. As ;to increases, the d r o p towards the saturated value is less rapid as expected

,

I 0.3

,

,

,

,

I

,

0.4

, 0.5

ht

At

Fig. 2. Specific heat j u m p as a function of interchannel coupling strength ;.j for fixed strength of the PA-channel coupling 3.2 = 0.16 and various values of PI-channel coupling 3.o with 3.0<3. 2. The specific heat j u m p starts at the PA-channel value of 0.953 and decreases with increasing 3. ,, shows a m i n i m u m of 0.667 before increasing again and saturation at large ~., to 0.673.

,

0.2

Fig. 3. Specific heat j u m p as a function of intcrchannel coupling

strength 2, for fixed value of d-wave channel coupling 3.2= 0.16 and various values of PI-channel coupling 3.o with 2o>-3.2. For 3.o= 3.2. there is no dependence of the jump on 3.~ and it is equal to 0.673. For the other cases, the jump starts at the canonical value of 1.43 and drops monotonically with increasing ;. ~. since, in this case, the Pl-channel is more robust.

5. Summary and conclusions In this paper, we have calculated the specific heat for a coupled PI and PA-channel superconductor. If the coupling strength in the PI and PA-channels are d e n o t e d respectively by 2o and 22 and the P I - P A coupling by 2,, several specific cases need to be considered. For 2o> ).2, we find that the normalized specific heat j u m p at To AC/TTc starts at 1.43 for).~ = 0 and decreases monotonically with increasing). ~. The decrease is less rapid as the discrepancy between ),2 and 2o is increased. In the limit )., --,oo, AC/yTc tends towards 1 . 4 3 { 4 / [ 7 + 4 a 3 + a 4 ] } whatever the associated value o f ) t , might be. Here a3 and a4 are the third and fourth m o m e n t o f the Fermi surface harmonic describing the PA-channel. For )to=)t2, that is equal coupling PI and PA-channel AC/yT¢= i.43{4/ [ 7 + 4 a 3 + a 4 ] } and is independent o f the P I - P A coupling 2 ~. Finally, for ). 2 > ;to, the specific heat j u m p at Tc starts at the reduced value 1.43/ot4 then decreases as )t, increases, it then shows a m i n i m u m

H. Chi, J. I~ Carbotte / Thermodynamics of extended s-wave superconductor

value before t e n d i n g t o w a r d s 1 . 4 3 { 4 / [ 7 + 4 a 3 + o ~ 4 ] } as ;t ~--,~. F o r the specific case o f an o r g a n i c superc o n d u c t o r with Q(/~)=x/-'5 cos(/~,,), a 3 = 0 and a 4 = 1 . 5 , a n d the m i n i m u m v a l u e o f the j u m p is 1.43 [ ( 5 - a4 ) / (9 - ol4) ] = 0.667 w h i c h o c c u r s for X = X / ~ 3--a4)/2-----0.866 where x = (,7to-22) /

2k, + x/i--t- [ (.;.o -k2 )/22, ]2 Acknowledgements T h i s research was s u p p o r t e d in part by the N a t u r a l Sciences and E n g i n e e r i n g R e s e a r c h C o u n c i l o f C a n ada ( N S E R C ) . J.P. C a r b o t t e is also s u p p o r t e d by the C a n a d i a n I n s t i t u t e for A d v a n c e d R e s e a r c h ( C I A R ) . O n e o f us ( H . C . ) w o u l d like to t h a n k M. P r o h a m m e r , P. W i l l i a m s a n d E. N i c o l for useful discussions.

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