Poole Ch.P., Jr. Handbook of Superconductivity

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496 Chapter 10: Thermal Properties

where here Ces is the electronic term immediately below T c. This discontinuity

has been observed, as shown in Fig. 10.2 and by the data presented in Table 10.1.

The BCS theory predicts that far below T c the specific heat depends exponentially

on the temperature

Ces(T) ~ a exp(-A/k B T),

(17)

where Eg

2A is the energy gap in the superconducting density of states.

When a magnetic field is present there is still a discontinuity in the specific

heat, but in addition there is a field-dependent latent heat given by

L(~) - (Sn - Ss)rc(~),

(18)

so the transition is now first-order, where Tc(B ) is the transition temperature in the

presence of the field. If the assumption is made that the critical field

Be(T)

has a

parabolic dependence on the temperature,

Bc(T ) - Bc(0)[1 -

(T/Tc)2],

(19)

then this expression can be inverted to give the magnetic field dependence of the

critical temperature,

Tc(B ) - T~[1

B/Bc(O)] 1/2,

(20)

where T c is the transition temperature in the absence of the field, and of course

B < Bc(O ) for the material to remain superconducting.

Fig. 10.2.

Aluminum

coo

ms1

ing

0.5 1 I .5 2

T {K}

Specific heat of elemental aluminum in the normal and superconducting states showing the

jump predicted by the BCS theory. [Phillips (1959); see Crow and Ong (1990), p. 225.]

C. Thermal Conductivity 497

References for Section B

J. E. Crow and N.-P. Ong, in

High Temperature Superconductivity

(J. W. Lynn, Ed.), Chapter 7.

Springer-Verlag, Berlin, 1990.

A. Junod, in

Physical Properties of High Temperature Superconductors

(D. M. Ginsberg, Ed.), Vol. 2,

Chapter 2. World Scientific, Singapore, 1990.

N. E. Phillips,

Phys. Rev.

114, 676 (1959).

C. P. Poole, Jr., H. A. Farach, and R. J. Creswick,

Superconductivity.

Academic Press, New York, 1995.

S. V. Vonsovsky, Yu. A. Izumov, and E. Z. Kumaev,

Superconductivity in Transition Metals.

Springer,

New York, 1982.

Thermal Conductivity

Ctirad Uher

a. Introduction

Among the important properties of high-temperature superconductors (HTS) is

their ability to conduct heat. There is not only an obvious technological interest in

how efficiently and by what means the heat flows in these solids, but also a deep

theoretical desire to understand the electronic and vibrational properties of these

materials. In HTS, such information is especially valuable due to the fact that

traditional galvanomagnetic probes such as resistivity, Hall effect, and thermo-

power are inoperative in the (now) wide temperature range below T c. Detailed

account of heat transport in HTS can be found in Jezowski and Klamut (1990)

and Uher (1990, 1992).

For any isotropic solid, the thermal gradient V T imposed across the sample

results in the rate of flow of heat across a unit cross-section perpendicular to the

direction of heat flow governed by Fourier's law,

U = -tcVT. (21)

Here ~c is the thermal conductivity and the negative sign implies that the heat

flows down the thermal gradient, that is, from the warmer to the cooler end of the

sample.

Measurements of the thermal conductivity can be accomplished by various

experimental arrangements depending on the structural form of a sample,

temperature interval of interest, and the desired accuracy of the data. One is

mostly interested in the behavior of the thermal conductivity at and below room

temperature and, for samples of bulk form, the most convenient and by far the

most frequently used setup is the

longitudinal steady-state technique

illustrated in

Fig. 10.3. One end of a sample with a uniform cross-sectional area A is attached

to a cold tip of the cryostat while a small heater (typically a metal-film resistor) is

498

Chapter 10: Thermal Properties

fastened to the other end of the sample. Electrical power Q dissipated in the heater

provides a heat flow. The resulting temperature difference AT = T h - T c between

two points separated by a distance L is measured by a pair of thermometers or a

differential thermocouple. The thermal conductivity ~: is then calculated from

= QL/(A AT). (22)

For not too large AT, the value of tc obtained from Eq. (22) will be that

corresponding to the mean temperature between the thermometers. For very short

samples, it is frequently preferable to use the so-called

two-heater~one thermo-

meter technique,

wherein a thermometer is attached to the free end of the sample

while the power is switched between two heaters spaced along the length of the

sample. Since heaters can be made very small (e.g., by evaporating resistive

strips), they can be easily accommodated on short samples. Thermal conductivity

of samples in the form of thin films can be determined by the so-called 3-09

method (Cahill, 1990). Experimental techniques of measuring thermal conduc-

tivity are discussed in several monographs and review articles, for example,

Berman (1976) and Uher (1996).

Fig. 10.3.

Experimental setup to measure thermal conductivity using the longitudinal steady-state

method.

C. Thermal Conductivity

499

b. Normal State

Heat in a solid is carried by two distinct entities, free carriers, ~%, and quantized

lattice vibrations called phonons, top. The total thermal conductivity, to, is then

K -- K e -[-" Kp.

(23)

The relative magnitude of the two contributions in Eq. (23) can be used to classify

solids in a way analogous to the magnitude of the electrical resistivity, Fig. 10.4.

Each heat-conducting channel (carriers and phonons) is subject to relaxation

mechanisms (scattering) that ensure the stationary nature of the heat-conducting

process. Charge carriers are scattered by phonons yielding the thermal resistivity

contribution

We_p,

by defects yielding We_d, and by other charge carriers, We_ e.

Phonons can scatter on free carriers, Wp_e, on defects, Wp_ d and in interactions

with other phonons, Wp_p. According to Matthiessen's rule, the scattering

processes within each channel are additive, yielding

We - 1/• e

- We_ p -[- We_ d -'[-

We_ e (24)

Wp -- 1/tCp - Wp_ e + Wp_ d + Wp_p. (25)

Depending on the carrier density, the density of defects, and the temperature

range, the magnitude and the temperature dependence of each term in Eqs. (24)

and (25) can be evaluated (see, for example, Klemens, 1965). For instance, a

Fig. 10.4.

\ /

METAL

SEMICONDUCTOR

]~e = K~ A K~p __~ KZ e

K:p small

Main contributions to the thermal conductivity of various solids.

500

Chapter

10:

Thermal Properties

normal metal (ic

Ice) can be modeled by the following expression with the

temperature dependence

W e = 1/tc e -aT

2 + b/T,

(26)

where the first term on the RHS stands for the cartier scattering by phonons and

the second term represents the interaction of carriers with static lattice defects.

Carrier-cartier interaction is usually small and is neglected in Eq. (26).

Assuming

elastic scattering,

the thermal conductivity associated with free

carriers is related to the electrical resistivity via the

Wiedemann-Franz law,

Ice -- LoT~P,

(27)

h(~B) 2~2 0 -8

VZK-2

where L 0 --5- -- - 2.44 x 1 is the Lorenz number.

Althoug pure metals are very good heat conductors, the highest values of

the thermal conductivity are achieved in covalently bonded insulators with high

Debye temperatures such as diamond and boron nitride. The thermal conductivity

of isotopically pure diamond is so high that it is extremely difficult to determine

accurately. Table 10.2 gives room-temperature values of the thermal conductivity

of several selected materials.

Conventional superconductors are metals or alloys, and for T > T c their

thermal conductivity has a substantially metallic character with a small contribu-

tion due to phonons that increases (as a percentage of the total thermal

conductivity) as the metal becomes less pure and more disordered. HTS materials

have a significantly reduced carrier density in comparison with typical metals and,

Table 10.2.

Thermal conductivity of several selected materials at a room tempera-

ture.

Material to(300 K) in W/m-K Reference

Diamond 2200 1

Silver 420 1

Copper 390 1

Gold 320 1

Aluminum 240 1

Brass 120 1

MgO 60 1

Niobium 51 1

Lead 35 1

Constantan (55 Cu 45 Ni) 23 1

Stainless Steel 14 1

YBa2Cu307 (ab-plane) ,~ 10 2

Pyrex 1 1

Plexiglass 0.2 1

Styrofoam 0.04 3

References: 1, Childs

et al.

(1973); 2, Uher (1992); 3, Waite (1955).

C. Thermal Conductivity SO1

as a consequence, phonons are the dominant heat conducting channel. For

instance, in bulk sintered cuprates phonons account for 90-95% of the total

thermal conductivity, and even in the best single crystal available phonons carry

at least half of all heat in the normal state.

c. Superconducting State

The lowering of the temperature below the superconducting transition tempera-

ture T~ and the ensuing formation of the Cooper condensate lead to a sharp

change in the electromagnetic and kinetic response of a material, and a

consequent drastic modification of its heat flow pattern. Two properties of the

condensate provide the overriding influence:

1. Cooper pairs carry no entropy

2. Cooper pairs do not scatter phonons

The first condition implies that the electronic thermal conductivity vanishes

rapidly below T c. In fact, the decrease is approximately exponential and has been

justified by Bardeen, Rickayzen, and Tewordt (1959) in the so-called BRT

Theory, and by Geilikman and Kresin (1958).

The effect of the second condition is more subtle. Provided that the mean

free path of phonons at T > T c is limited by scattering on change carriers, on

passing into the superconducting state the phonon thermal conductivity will rise

because the number of normal carriers (more precisely, the density of quasi-

particle excitations) rapidly decreases. The competition between the rapidly

diminishing ~c e on the one hand and the increasing/s

the other will determine

the overall dependence of the total thermal conductivity of any given super-

conductor. Because of the dominant contribution of charge carriers, a vast

majority of conventional superconductors show a decrease in the ratio of the

thermal conductivities in the superconducting and normal states,

KS(T)/~cn(T),

for

T < T c. In the case of elemental superconductors, the thermal conductivity may

be reduced by 2-3 orders of magnitude in comparison to its normal-state value.

On the other hand, in a few alloys, sufficiently disordered so that tce is relatively

small and

represents a significant fraction of the total thermal conductivity,

and, at the same time, when the phonon-carrier interaction in the normal state is

significant, the entry of such material into the superconducting domain may be

accomplished by a rise in the thermal conductivity, Fig. 10.5. Such a rise is the

consequence of an enhanced mean-free path of phonons as progressively fewer

normal carriers are available to scatter them below To. Of course, at very low

temperatures, all conventional superconductors will behave as ordinary dielectric

solids, because phonons are the only entity that can carry heat.

Phonons carry the bulk of the heat in the normal state of HTS, and it is

reasonable to assume that phonons also play a prominent role at temperatures

502

Chapter

10:

Thermal Properties

Fig. 10.5.

"5 ..........

0"2

.....

0.1

0"05

0"02

"6 0-01

.Iz

0-005

O-OOZ

oC~

NbC

~X J

0 X

X /

Ox x

X lX

I I

.......

I I

mo 2o ~ "

Temperature (K)

Thermal conductivity of niobium carbide. Open circles refer to a superconducting

NbC0.96

with T c -~ 10 K. Crosses represent a nonsuperconducting NbC0.76. [After Radosevich and

Williams (1969).]

below Tr The behavior of Xp being limited by carrier scattering was originally

treated in the BRT theory and supplemented by Tewordt and Wolkhausen (1989)

to account for the relevant scattering processes and the anisotropy of the phonon-

carrier interaction as appropriate to HTS cuprates. The in-plane phonon thermal

conductivity becomes, Peacor

et al.

(1991 a)

OD/r 1

Kp,ab(T ) --

(kB/2n2v)(kB/h)3T 3 J

dx x4eX/(e x -

1) 2

Id( 3/2(1

- (2)~-(T, x, (),

0 0

(28)

where the overall scattering rate is given by

~--I(T,

x, ~) - B Jr- Opt4X 4 -+-

Osft2x 2 -'l-

gtxg(x,

y)(1

- ~2)3/2 _1_

eT4x 2.

(29)

Coefficients B, Dp, Dsf, E, and U in Eq. (29), in turn, refer to phonon scattering

by boundaries, point defects, sheetlike faults, charge carriers, and other phonons.

The function

g(x,y)

is the ratio of the phonon-carrier scattering times in the

normal and superconducting states, and its exact form depends on the super-

C. Thermal Conductivity 503

conducting energy gap. The theory can accommodate the strong coupling limit

for both the s-wave and d-wave pairing mechanisms, and provides a plausible

explanation for the characteristic rise and the peak in the thermal conductivity of

HTS below T c, Fig. 10.6. The peak has been observed in samples of all major

families of HTS except for BaKBiO and NdCeCuO structures. While excellent

fits with perfectly reasonable parameters exist for both sintered and single crystal

data (e.g., Peacor

et al.

1991 a), phonons may not be the sole entity responsible for

the rising thermal conductivity and the peak below To.

Microwave surface resistance studies (Bonn

et al.,

1992) and ultrafast laser

pump-probe studies of cartier relaxation (Chwalek

et al.,

1990) have clearly

shown that the relaxation time of quasiparticles in HTS is dramatically enhanced

below T~. Such an unusually long quasiparticle lifetime provides and altemative

explanation for the peak in the thermal conductivity (Yu

et al.,

1992). In this case

the fit to the experimental data is made with the aid of the formula for the cartier

Fig. 10.6.

e untwinned

-- @

:t% o

O0 000 0 0 0 ~

twinned

qlZl~ D

intered

1 I J_l l I.I Jl , i I.i ~ 1 i , ,

0 5O 100 150 2O0

T(K)

Thermal conductivity of sintered, twinned, and untwinned (a-direction) samples of

YBa2Cu307_ ~. [After Uher

et al.

(1994).]

504

Chapter

10:

Thermal Properties

contribution in the superconducting state derived by Kadanoff and Martin (1961)

and independently by Tewordt (1962),

Kse --

1/(2kB T2m)

j d3p z4/r)

sech2(Ep/2kB T) ~-

~/F.

(30)

Here Ep- (ep 2 + A~) 1/2, where ep is the normal-state dispersion and Ap the

superconducting gap. The scattering rate F is taken as F ,~

(T/Tc)n+ wi,

implying a power law dependence augmented by a constant term wi that stands

for the residual scattering rate due to impurities. Yu

et al.

(1992) find that a d-

wave pairing state fits best, and the relaxation rate varies as the fourth power of

temperature.

The difficulty in making an unambiguous choice between the two compet-

ing interpretations rests in the fact that the charge carriers and phonons contribute

roughly equally (in single crystals) to the heat transport in the normal state, and

the physical processes that give rise to enhancements in either tOp(T) or tCe(T )

below T c have rather strong temperature dependences that lead to peaks in the

thermal conductivity at virtually the same temperatures.

At very low temperatures,

TIT c

<< 10 -2, any anomalous behavior should

cease and the only mode of heat transport should be via phonons, with grains and

boundaries being the dominant scatterers, that is, tc ~ T 3. Although this is the

case for insulating cuprates such as YBa2Cu306 (Cohn

et al.,

1988), the

superconducting YBCO (Gottwick

et al.,

1987) and LaSrCuO (Uher and

Cohn, 1988) show a T-linear limiting dependence, Fig. 10.7, which is well

approximated by

~c(T) = aT + bT 3.

(31)

Although various mechanisms were proposed to account for the linear term, the

consensus converges on the presence of a small number (10-15%) of uncon-

densed carriers, and this provides support for the d-wave (nodes) symmetry of the

superconducting state. BiSrCaCuO, on the other hand, displays a T2-dependence

below 2 K regardless of its structural form (Peacor and Uher, 1989; Zhu

et aL,

1989; Sparn

et aL,

1989). A correlation seems to exist between the T-linear

limiting behavior and the magnitude of the 7-term in the specific heat.

HTS cuprates possess considerable structural anisotropy, which is reflected

in the behavior of the thermal conductivity. Figure 10.8 provides the temperature

dependence of the anisotropy ratio between the in-plane and the c-axis thermal

conductivities, tCab/~C c, for several cuprates.

d. Effect of a Magnetic Field

An external magnetic field B > Bcl drives a superconductor into the mixed state

characterized by the presence of Abrikosov vortices of core radius ~ that contain

bound excitations not too different from normal electrons. Close to Tc the heat is

C. Thermal Conductivity 505

Fig. 10.7.

i0 0

10-1

10-2

10-3

10-4

' ' ' ' ''''1

"" ' ' ' ' ' ''I ' ' .....

10-5

..... , ,t ......

0.01 0.1 1

9215

x~p

x d

~ 9 o

.. /

~rn + 0 /

. ~ ~

9 ,+

, L 1

T (K)

Thermal conductivity of several samples of sintered

YBa2Cu307_ 6 at

very low temperatures.

The slopes for linear (T) and cubic

(T 3)

temperature dependences are indicated. [After Uher

(1992).]

carried by unbound excitations (uncondensed electrons), and they scatter on

vortices, resulting in the thermal resistance

We(B)-

We(O)[1 -t-(lea/dPo)B]

(32)

where I e stands for mean free path (mfp) of quasiparticles and a is the effective

vortex cross-section. There is no general theory to cover the entire range of

magnetic fields, but beyond a certain field strength the scattering weakens as the

vortices start to overlap. At the same time, tunneling between vortices drives the

thermal conductivity toward its normal-state value tcn as B--+ Bc2. How the

thermal conductivity actually attains its normal-state value depends critically on

how large is I e in relation to the coherence length ~ (Caroli and Cyrot, 1965;

Maki, 1967). Unbound quasiparticles may also undergo the Andreev (1964)

process, where an incoming electron-like quasiparticle scatters on the modula-

tion of the order parameter A(r) and transforms into an outgoing hole-like