Electronic heat conduction for d-wave superconductors

PHYSICA[x Physica C 234 ( 1994) 57-62

ELSEWER

Electronic heat conduction for d-wave superconductors Thomas Wiilkhausen Physihdisches Institut, Universitiit Bayreuth. 95440 Bayreuth, Germany

Received 20 June 1994; revised manuscript received 19 September 1994

Abstract The electronic thermal conductivity rc” of an isotropic s-wave and a two-dimensional anisotropic d-wave superconductor with electron-phonon interaction and impurity scattering is calculated. rce’is strongly influenced by the structure of the order parameter. In the case of an isotropic order parameter with amplitude d(O) less than A,,,( 0), an increase in the electronic thermal conductivity compared to the BCS value is found at all temperatures. A similar behavior results for an order parameter with nodes on the Fermi line. The contribution of the electrons to the total thermal conductivity is thus higher for a d-wave order parameter than for an s-wave one.

1. Introduction The measured thermal conductivity K of high-T, superconductors (HTSC’s) has been subject of interest in the last few years. K increases rapidly below T, with a maximum near TJ2, and K declines as the temperature tends to zero. This behavior is seen in single-crystal measurements of the in-plane heat conductivity K&, but absent perpendicular to the abplane. The first experiments measuring the thermal conductivity of sintered HTSC were done by Steglich et al. [ 11, Uher et al. [ 21 and Ibitzky et al. [ 31. The observed maximum could be explained within the framework of the theory of the lattice thermal conductivity by Bardeen, Rickayzen and Tewordt (BRT) [4] and the extended theory by Tewordt et al. [ 51; the latter take static defects of the lattice into account. In the BRT theory the phonons play the dominant role in the heat transport, thus providing the main contribution to the thermal conductivity, while the electrons are assumed to give negligible contributions. The electrons enter as dynamic scatterers of 0921-4534/94/$07.00 SSDIO921-4534(94)00636-9

0 1994 Elsevier Science B.V. All rights reserved

the phonons; this strongly affects the phonon lifetime and hence their mean free path. Recently, Richardson et al. [ 6 ] measured the single-crystal thermal conductivity and could explain their results on the basis of the BRT theory. Later, Salamon et al. [ 7 ] made similar measurements. They, however, assumed the phonon contribution to the thermal conductivity to be almost T independent and small and the electrons to be the dominant carriers of the heat current and that the Cooper pair condensation leads to an increase of the quasiparticle lifetime due to the reduction of the electron-electron scattering processes. Consequently, the thermal conductivity increases when the temperature is lowered from T, to TX TJ2. Then K declines with decreasing Tdue to the exponential reduction of quasiparticle excitations. The existence of two different interpretations of the same experimental results makes further theoretical investigations concerning the contributions of electrons and phonons to the total thermal conductivity necessary. In this work the Keldysh technique is used to derive the momentumand energy-dependent ki-

Th. Wdkhausen / Physica C 234 (I 994) 5 7-62

58

netic equation of an isotropic and an anisotropic weak-coupling superconductor. Then the electronic thermal conductivity is calculated, taking electronphonon interaction, electron-impurity scattering and s- and d-wave pairing into account. It should be noted that the theory of the electronic thermal conductivity of superconductors had been worked out within the framework of the Eliashberg theory by Tewordt et al. [ 8,9]. Later, Nielsen and Smith [ 10,111 used the Keldysh technique to derive an isotropic strong-coupling kinetic equation to obtain the thermal conductivity, accounting for arbitrarily large inelastic scattering rates. Pethick et al. [ 121 made calculations of the electronic thermal conductivity for an axial and a polar p-wave and an axial d-wave order parameter, taking the effect of electron-impurity scattering into account, whereas Maki et al. [ 13 ] did similar calculations for a d-wave order parameter with impurity scattering.

The self-energy be written as

2. The theory of the electronic thermal conductivity

I@,,

The electronic thermal conductivity coupling limit is written as

in the weak-

Ksel ..- - 4N(O)& cc

c FS

Y(hF, 6, t)

.

(1)

bJImp:= -inN(O)Ni,, j=R,

scattering

c dQ’I I’(.&-PF)

can

12g’(p;, e),

:s

A, K .

(4)

Nimp denotes the concentration of impurities in the system, and V represents the electron-impurity interaction in momentum space on the Fermi surface. The following abbreviations are used for the retarded (R), advanced (A) and Keldysh (K) propagators: g~:=g~(p~, e), g’-‘:=gj(p;, D”:=D’(p;

-pF, E’--t)

@':=g'R-g'A,

E) ,

c’), $:=&j(Pr, ,

gD':=D'R_D'A,

j=R,A,K.

(5)

With the self-energies as above, Eqs. (2), (3) and (4)) the collision integral of the corresponding generalized Boltzmann equation takes the form e) =Tr(j?ft

[ (BR-eA)F+

-6K]}.

(6)

When the self-energies are inserted according to the expressions given above, Eq. (6), the collision integral can be written as the sum of Z,, and limp. The operators P2A are the Shelankov projectors [ 141 defined in terms of the retarded and advanced Green’s functions:

ch2W/2) IA/

ps := 1 kg” and p”, := 1~2~ .

E is the quasiparticle energy, A the order parameter, pF the Fermi momentum and t = T/T,. N( 0) denotes the density of states at the Fermi surface for a single spin in the normal state, Y describes the deviation of the quasiclassical distribution function of the electronic system from its equilibrium value, and the integral JFs dQ denotes the Fermi surface integration which reduces to a contour integral in the twodimensional case, normalized with respect to 2~. To derive the kinetic equation determining the deviation function Y one starts with the contributions to the self-energy due to the electron-phonon interaction [lo]: ^%A := a~( 0)

dSZ’{g’R,AD’K+g’KD’A,R} ,

dc’

O,P

_K OeP :=

term due to impurity

s

$N(O)

dQ’{glKDfKs FS

@‘6D’}

.

gK(pF, E):=&(F+PR+BA

2” is given by

+F_BRPA+).

F+ := 1-2f*

are the unknown tions, linearly approximated by

.f+ (PF, ~)=fO+f’(l f-

(8)

distribution

-f")h.VW~~,

6))

func-

(9)

(pF,c)=l-Ifo+fo(l-fo) X~F.VT~(-PF,

(10)

-~11 ,

where f” is the Fermi function. With these approximations, the kinetic equation takes the form

(2)

Tr{SA_PR+iUF.VF+}=Zep(PF,E)+Ii,p(PF,~).

s FS

de’ s

The Keldysh Green’s function

(7)

(3)

The retarded written as

and advanced

phonon

propagators

(11) are

Th. Wiilkhausen / Physica C 234 (I 994) 5 7-62

DR.*&,

E):= E

I&,J’

j dwp~_~;~;

,

(12)

where 1 denotes the band index, p the polarization index, g$“* the electron-phonon vertex, w the phonon frequency and pnr(pF, o) the phonon spectral density. Inserting the retarded and advanced phonon propagators and evaluating the traces, the kinetic equation is given by

59

priate to intermediate coupling, and 0~ denotes the Debye frequency set to wn = 15T,. Then the kinetic equation can be rewritten as

-.4(r)

1 (23)

+[52(0+53(01~=0>

with t = T/T,, &= = c/A, and (AZ-B’)

3

= s de’V(e, e’)cu2F(e’-e) Z,(r):=

x I dQ{WZG, e’)d-

W.P,, e)a} ,

(13)

Fs

s dQ’ 9 dc’ Fs IV1 y(Z,b, Aor’)

with &:=

(AZ-D2)

( >

(,‘2-D’2)

.@:=fi’-DD’

l+

g

,

,

(15)

&,

C+iD:=

J&,

(24)

(14)

.Z2(<):= j dS2’ 3 dr FS

A+iB:=

,

(16)

x

IV1

%(rl-O(C-W)

+~S(r’+o(4?+@@‘) 7

u(t, d):= A:=A(p,,

ch(W2)

ch(/?e’/Z)sh(P(e’--e)/2) t), A’:=A(p;,

(17)

’

(25)

E’) etc., /I= l/T and A the

1

order parameter. The weak-coupling kinetic equation is obtained by replacing the functions (A’- D’), d and ~8by their weak-coupling limits:

(A2-D2)+@(l~l - IAl), ~~~~I~I-l~l~~~l~‘I-l~‘I

(18) 1,

The weak-coupling order parameter A and the gap enhancement factor x are defined in the following way (t=T/T,,xkO):

andAo(t):=xA,,,(t)

e(t) :=

a2W08

(26)

(27)

(19)

(20)

A:=Ao(t)$(bF)

Aox7t’Imp ’

.

(21)

The phonon spectral function is assumed to be a Debye spectrum:

where a common factor 2Jw/ch( (Ao/2T)<) is extracted in Eq. (23), and qS:=+(@;). The isotropic expression can be derived by taking $- 1. The integrals .Z,and J2 are obtained from the electron-phonon collision operator Zepand .Z3from the electron-impurity collision operator Zi,, which is assumed to be independent of&. 3. The anisotropic electronic thermal conductivity

2

a2F(w):=A

(> E

8(w,-co))

(22)

where the coupling constant was set to A= 1, appro-

Fig. 1 shows the normal-state electronic thermal conductivity normalized with respect to its value at Tc: ic$$c&. The parameter c:= ri,/2r& describes

Th. Wdkhausen

60

Electronic

/ Physica C 234 (I 994) 5 7-62

Thermal Conductivity

ElectronicThermal Conductivity

( c = T,,,/ 2s”II’V’ ) 2’

I

( x=WA,,(t) )

I

I

8.0

2.0

0.0 )

0.2

0.4

0.6 0.8 t = T/Tc

1 .o

1.2

1.4

Fig. 1. The ratio of the electronic thermal conductivity in the normal state is shown normalized with respect to its value at T,.

the ratio of the lifetimes of the quasiparticles due to inelastic and impurity scattering, and 1/ ZinTcz 0.1. The upper long-dashed curve describes the electronic thermal conductivity without impurity scattering (c=O). In this case the number of phonons will decrease as the temperature is lowered causing the electronic thermal conductivity to increase and to diverge as l/T’. If impurity scattering is taken into account, the respective curves display a maximum near TJ3 which becomes more and more suppressed and shifted towards higher temperatures as the scattering rate is increased. The dot-dashed curve represents the dirtylimit result (c=co) where the temperature dependence of ufi jr&c shows a linear behavior. Fig. 2 depicts the electronic thermal conductivity in the superconducting-state normalized with respect to its value at T. of the gap C' IC$/K~' N,Tc. The influence

0.0

0.2

0.4

0.6

0.8

1.o

t=T/Tc Fig. 2. The ratio of the electronic thermal conductivity of the superconducting state (s-wave pairing) to its value at T= T, is shown depending on the gap enhancement factor x=A/As,,.

enhancement factor x:=d,, (1) /dBcS( t) (2 1) is shown, and impurity scattering is not taken into account (c=O). A decrease of x from 1 to 0.8 leads to an increase in K:/K&,, and the maximum is shifted towards lower temperatures. In the limit of vanishing x the corresponding curve describes the normal-state electronic thermal conductivity, complying with the clean-limit result in Fig. 1 (long dashed curve). If x increases, the opposite behavior is observed: the thermal conductivity decreases, and the maximum is shifted to higher temperatures. For x> 2, this maximum is suppressed and K$/K~'N,T, is an increasing function of temperature. The derivative (d/ dt) [lc$/&Tc] at T, (t=T/T,) is negative for 05~~ 1 and decreasing with decreasing x. For x2 1 this derivative changes sign and increases for higher

Th. Wdkhausen

/ Physica C 234 (I 994) 5 7-62

values of x. Therefore, the thermal conductivity changes its slope at T, with varying x. The reason for the increase in K$/K&.~ when x decreases at a given temperature is connected to the smaller amplitude of the order parameter and thus to the smaller energy gap, because in this case the number of excited quasiparticles is higher, causing the electronic thermal conductivity to increase. Fig. 3 shows the electronic thermal conductivity of the superconducting s- and d-wave states normalized with respect to their values at Tc: K$/K&-,. For the d wave pairing the thermal conductivity shows a stronger dependence on the impurity scattering parameter c, than for s-wave pairing. The total thermal conductivity of phonons and electrons has higher electronic contributions in the d-wave than in the swave pairing state. This is due to the nodes of the d-

Electronic

Thermal Conductivity

( s-wave and d-wave pairing

61

wave order parameter on the Fermi line: if PF points into these directions, A@,) vanishes, and the dominant contributions to the Fermi line integral and thus to thermal conductivity emerge. In the clean limit (c=O) they even diverge like l/T* as Tvanishes. It is the impurity scattering which makes them remain finite, but they are still very large compared to those which result if d z dscs is assumed. The similarity of the normal-state thermal conductivity to that in the superconducting state with d-wave pairing is clearly visible (see Figs. 1 and 3 ). K:’ reflects the temperature dependence of K: if the impurity scattering is less effective. For higher ratios of c the K$' (d wave) exhibits more and more the character of K$ (s wave). The anisotropy due to the dwave pairing is washed out by the increasing impurity-scattering rate. The same behavior is observed in theoretical results by Klemm et al. [ 15 ] concerning the electric conductivity.

)

I’

---

-

4.0

c c c c c c c

= 0, d-wave = 0.01, d-wave = 0.1, d-wave = 1 .O, d-wave = 0.01, s-wave = 0.1, s-wave = 1.0, s-wave

4. Concluding

t" z ;iy . xyw 2.0

c

0.0

0.2

0.6 0.4 t = T/TC

0.8

1.o

Fig. 3. The ratios of the electronic thermal conductivity of the superconducting state for s- and d-wave pairing are shown depending on the strength of impurity scattering c=q,/2r&.

remarks

The electronic thermal conductivity K: depends sensitively on the amplitude of the superconducting order parameter. This explains the higher values of of for the d-wave in contrast to the s-wave pairing state. Consequently the contributions of the electrons to the total thermal conductivity is much higher for d-wave pairing, but impurity scattering lowers this difference. It should be noted that the measured sudden increase in the total thermal conductivity cannot be explained within this electronic weak-coupling model. Finally, it should be mentioned that the use of an isotropic Debye model for the phonons is of limited applicability to high-T, superconductors if one takes strong-coupling effects into account where the cr*F shows structure. Therefore, the weak-coupling calculations should be extended such that strong-coupling effects are taken into account (e.g. use of a Debye spectrum from measurements). The quasiparticle bandstructure has been neglected here and this may be another critical point. But to obtain first the effect of the order parameter anisotropy one should neglect the influence of the bandstructure.

62

Th. Wdkhausen / Physica C 234 (1994) 5 7-62

Acknowledgements I would like to thank C.T. Rieck, D. Walker and P. Esquinazi for many valuable discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG).

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[4] J. Bardeen, G. Rickayzen and L. Tewordt, Phys. Rev. 113 (1959) 982. [ 5 ] L. Tewordt and Th. WSlkhausen, Solid State Commun. 70 (1989) 839. [ 61 R.A. Richardson, F. Nori, S.D. Peacor and C. Uher, Phys. Rev. B44 (1991) 9508. [ 71 M.B. Salamon, J.P. Lu, R.C. Yu and W.C. Lee, Phys. Rev. Lett. 69 (1992) 1431. [8] L. Tewordt, Phys. Rev. 128 (1962) 12. [9] L. Tewordt, Phys. Rev. 129 (1963) 657. [lo] J.B. Nielsen and H. Smith, Phys. Rev. B 82 (1984) 283 1. [ 111 J.B. Nielsen and H. Smith, Phys. Rev. Lett. 49 ( 1982) 689. [ 121 B. Arfi, H. Bahlouli and C.J. Pethick, Phys. Rev. B 39 (1987) 8959. [ 13 ] H. Won and K. Maki, Phys. Rev. B 49 ( 1993) 1397. [ 141 L. Shelankov, Zh. Eksp. Teor. Fiz. 78 (1980) 2359. [ 151 R.A. Klemm, K. Scharnberg, C.T. Rieck and D. Walker, Z. Phys. B 72 (1988) 139.