Poole Ch.P., Jr. Handbook of Superconductivity
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486
Chapter
9:
Characteristic Parameters
et al.
(1997); McDougall
et al.
(1995); and Hoyt and Mota (1976). Values from
transport are known for many elements and appear consistent with other values
when available. It would be very interesting to have reliable values for ferro-
magnetic elements. However, one expects that there should be quite different
values of 2 for the up and the down spin species, but two values (21, and 2r
cannot be extracted independently from transport measurements. Therefore, we
must wait for theoretical values and measurements by spin-sensitive techniques.
As an example, Gd is ferromagnetic with a Curie temperature of 292K.
Photoemission below this temperature has seen a surface state carrying the
majority (?) spin, and its 2 value is listed.
Table 9.11 lists most of the intermetallic compounds where there is reliable
information (Dynes, 1972; Kihlstrom
et al.,
1985; Kwo and Geballe, 1981;
Rudman and Beasley, 1984; Schulz
et al.,
1992; Glassford and Chelikowsky,
1994; Rathnayaka
et al.,
1997; Pickett and Singh, 1994; Kim
et al.,
1995; Singh,
1996; Michor
et al.,
1996; Singh, and Pickett, 1995). Evaluating 2tr for
compounds is often risky. There are two main difficulties: (1) the measured
p(T)
is sometimes sample dependent, especially in oxide materials where
polycrystalline bulk, single crystal, and thin film samples give different results;
Table 9.10.
Values of 2 for non-superconducting crystalline elements.
Metal 2~r '~LDA
'~other
Li 0.35 0.45-0.51 b
Na 0.14
K 0.11
Rb 0.15
Cs 0.16
Cu 0.13 0.14 a
Ag 0.12
Au 0.15
Mg 0.20 g
Ca 0.05
Ba 0.27
Sc 0.51 g
Y 0.62 g
Pd 0.47 0.35 a
Pt 0.66
Gd
0.24 c
0.14 4- 0.02 e
0.2Y
0.6 h
aUnless otherwise noted, from Allen (1987b). bLiu and Quong
(1996). cSurface quantum well state (Carlsson
et al.,
1997).
d Savrasov and Savrasov (1996). e Surface state on Cu(111)
(McDougall
et al.,
1995). fExtrapolation from superconducting
alloys (Hoyt and Mota, 1976). gSanbom
et al.
(1989). hSurface
state with quantum numbers 5d(z 2, ?) on Gd(0001), P. D.
Johnson, private communication.
G. Electron-Phonon Coupling Constants 487
it is hard to know the "true"
p(T),
but usually safe to assume that the smallest
values are the best; (2) it is important to be sure that the transport mean free path
is at least 10 A; otherwise, wavevector k is not a good quantum number, Bloch-
Boltzmann theory is not applicable, and 2tr is ill-defined. In Eq. (35), the group
velocity
hvk, x = Oe/Ok x
is only well defined when wavevector k is a good
quantum number. The superconducting 2, however, is still well defined from
Eq. (35) provided the quantum numbers k, k' are reinterpreted as labels for the
exact eigenstates of the disordered material.
RuO 2 is an interesting case where the transport value of 2 suggests the
possible occurrence of superconductivity at not too low a temperature, and this
may not have been tested below helium temperature. The layered intermetallic
borocarbides received a great deal of attention. In the case of LuNizBzC,
McMillan and transport values agree, which suggests (as does other evidence)
conventional superconductivity. For the related material La3NizBzN3_6, transport
and McMillan values badly disagree. My guess is that the transport value may
change with better data, but usually the transport value goes down in time as
samples improve, rather than up in time. An alternate possibility is the occurrence
of unconventional superconductivity. A similar situation holds for cuprates,
where transport values of 2 of order 1 are found using LDA bands, whereas to
account for
T c
near 100 K, one needs 2 of order 3 or higher. There are clear signs
Table
9.11.
Values of 2 for crystalline compounds and ordered intermetallics.
Metal T 2tu n 2tr 2Me M
c
In2Bi 5.6
Bi2T1
6.4
T17Sb 2 5.2
V3Si 17.1
Nb 3 A1 18.5
Nb3Sn 17.8
Nb3Ge ~20
NbN 16.0
NbO 1.4
ReO 3 < 0.02
RuO 2 < 4.2
CoSi 2 1.22
Pd2Si
LuNi 2
B 2 C 16.1
La3NizBzN3_ 6 12.25
1.40 a
1.63 a
1.43 a
0.89 b
1.7 r
1.75 a
1.7 b
1.46 b
0.51 e 0.41 e
0.35 r
0.54-0.1 g
0.44 f
o.15-o.2o r
0.9?, 0.8 i 1.0 h
0.2~ 0.86 /
aDynes (1972). 6Kihlstrom
et al.
(1985).CKwo
and Geballe
(1981). dWolf
et al.
(1980);
Rudman and
Beasley (1984). eSchulz
et
al.
(1992).fAllen
and Schulz (1993). gGlassford and Chelikowsky
(1994).
bUsing data from Rathnayaka
et al. (1997)
and theory from Pickett and Singh
(1994) and Kim
et al.
(1995). iSingh (1996).
JUsing data from Michor
et al.
(1996)
and theory from Singh and Pickett
(1995).
488 Chapter
9:
Characteristic Parameters
of unconventional behavior, which in my mind invalidates the transport analysis
based on Bloch-Boltzmann theory. Another interesting intermetallic, Sr2RuO4, is
also not listed. Nice-looking resistivity measurements are now being published,
but do not accord well with the Bloch-Griineisen formula, which is another sign
of unconventional behavior. It is not clear whether the concept of an electron-
phonon 2 value can be retained in such cases. In most materials where the
confusion is large, I have not tabulated any 2 values, but La3Ni2B2N3_ 6 is
included as a sign of the potential hazards.
Acknowledgements
I thank R. C. Dynes and W. E. Pickett for help with the manuscript. This
work was supported in part by NSF Grant No. DMR-9725037.
References for Section G
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(1986).
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and G. Dresselhaus, Phys. Rev. Lett. 64, 2172 (1990).
A. Carlsson, B. Hellsing, S.-A. Lindgren, and L. Walldrn, Phys. Rev. B 56, 1593 (1997).
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R. C. Dynes, Phys. Rev. B 2, 644 (1970).
R. C. Dynes, Solid State Commun. 10, 615 (1972).
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K. M. Glassford and J. R. Chelikowsky, Phys. Rev. B49, 7107 (1994).
G. Grimvall, The Electron-Phonon Interactim in Metals, North-Holland, Amsterdam, 1981 (Vol. XVI
of Selected Topics in Solid State Physics, edited by E. P. Wohlfurth.)
J. J. Hopfield, Phys. Rev. 186, 443 (1969)
J. J. Hopfield, Proceedings of the Rochester Conference on d- and f-Band Superconductors
(D.H. Douglass, Ed.) p. 354. AIP Conference Proceedings, 1972.
R. E Hoyt and A. C. Mota, Solid State Commun. 18, 139 (1976).
W. N. Hubin and D. M. Ginsberg, Phys. Rev. 188, 716 (1969).
K. E. Kihlstrom, R. W. Simon, and S. A. Wolf, Physica 135B, 198 (1985).
H. Kim, C. Hwang, and J. Ihm, Phys. Rev. B 52, 4592 (1995).
K. Knorr and N. Barth, Solid State Commun. 8, 1085 (1970).
J. Kwo and T. H. Geballe, Phys. Rev. B 23, 3230 (1981).
A. Y. Liu and A. A. Quong, Phys. Rev. B 53, R7575 (1996).
L. E Lou and W. J. Tomasch, Phys. Rev. Lett. 29, 858 (1972).
B. A. McDougall, T. Balasubramanian, and E. Jensen, Phys. Rev. B 51, 13891 (1995).
G. Electron-Phonon Coupling Constants 489
W. L. McMillan,
Phys. Rev.
167, 331 (1968).
W. L. McMillan and J. M. Rowell,
Phys. Rev. Lett.
19, 108 (1965).
W. L. McMillan and J. M. Rowell, in
Superconductivity
(R. D. Parks, Ed., Vol. 1, p. 561. Marcel Dekker,
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H. Michor, R. Krendelsberger, G. Hilscher, E. Bauer, C. Dusek, R. Hauser, L. Naber, D. Werner,
P. Rogl, and H. W. Zandbergen,
Phys. Rev. B
54, 9408 (1996).
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72, 3702 (1994).
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Phys. Rev.
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Phys. Rev.
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Phys. Rev. B
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This Page Intentionally Left Blank
Chapter lO
Thermal Properties
Ctirad Uher
Physics Department, University of Michigan, Ann Arbor, Michigan
Charles R Poole, Jr.
Department of Physics and Institute of Superconductivity,
University of South Carolina, Columbia, South Carolina
Alan B. Kaiser
Physics Department, Victoria University, Wellington, New Zealand
A. Introduction 492
B. Specific Heat 492
a. Meaning of Specific Heat 492
b. Electronic Specific Heat 493
c. Phonon Specific Heat 493
d. Discontinuity at T C 495
References for Section B 497
C. Thermal Conductivity 497
a. Introduction 497
b. Normal State 499
c. Superconducting State 501
d. Effect of a Magnetic Field 504
References for Section C 509
D. Thermoelectric and Thermomagnetic Effects
a. Introduction 510
b. Lorentz and Thermal Forces 512
c. Seebeck Effect 514
ISBN: 0-12-561460-8
$3O.OO
510
HANDBOOK OF SUPERCONDUCTIVITY
Copyright 9 2000 by Academic Press.
All rights of reproduction in any form reserved.
491
492 Chapter 10: Thermal Properties
A
d. Peltier Effect 521
e. Ettingshausen Effect 525
f. Righi-Leduc Effect 527
g. Nernst Effect 528
References for Section D 533
Introduction
The present chapter deals with various thermal type properties of superconductors.
It begins with the specific heat, which has been extensively studied for many
superconductors. These data are of interest because the BCS theory makes explicit
predictions about the behavior of the specific heat in the neighborhood of the
transition temperature To. This chapter also contains more extensive sections on
the thermal conductivity and various thermoelectric and thermomagnetic effects.
Specific Heat
a. Meaning of Specific Heat
The specific heat C of a material is the change in internal energy U brought about
by a change in temperature,
dU
C- dT'
(1)
where the usual units for C are J/mol-K. We do not differentiate between the specific
heat at constant volume, which is defined by Eq. (1), and that at constant pressure,
which is the change in enthalpy brought about by a temperature change, because for
solids the two are almost indistinguishable. Ordinarily the specific heat is measured
as the heat input
dQ
that raises the temperature by a unit amount
dT:
dQ- C dT.
(2)
In metals C is mainly the sum of an electronic term C e due to the conduction
electrons and a phonon contribution Cph due to lattice vibrations:
C= C e +Cph. (3)
The electronic term is only appreciable at low temperatures and changes
dramatically at the superconducting transition, whereas the phonon contribution
dominates at room temperature and is largely undisturbed by the transition at T c.
We will investigate these two contributions to the specific heat in the next two
sections.
B. Specific Heat 493
b. Electronic Specific Heat
In normal metals C contains a contribution from the conduction electrons Ce that
increases linearly with temperature:
C e = 7T (4)
The so-called Sommerfeld constant 7,
7 = (rc2/3)D(Ev) kg,
(5)
provides an experimental estimate of the density of states D(EF) at the Fermi
level. Table 10.1 lists values of 7 in the normal state for a number of super-
conductors. For many metallic elements 7 is close to the free electron value 7o,
70 =
rc2R/2TF = 4.93R/Tv,
(6)
where R =
NAk B
is the gas constant and T F is the Fermi temperature. The fact that
T F appears in the denominator of Eq. (6) makes it clear from Eq. (4) that C e << R
for all temperatures of experimental interest. The ratio of the effective mass m* to
the free electron mass m is
m*/m -
7/70 -
7TF/1 rcZR.
(7)
The free electron approximation also provides an expression for the ratio of 7o to
the (Pauli) conduction electron susceptibility Ze,
70/Ze -- 1 (7~kB/I.I,B)2 (8)
c. Phonon Specific Heat
The Debye model for the phonon contribution
Cph to
the specific heat assumes
that the sound velocity v is isotropic and independent of the frequency, and that
there is a maximum vibrational frequency co D called the Debye frequency with an
associated Debye temperature 0 D,
]'1co D = kBO D,
(9)
which gives for the phonon density of states per unit volume Dph(co )
3o92
Dph(co ) --
dn/dco --
2rczv3 . (10)
Standard solid-state physics texts derive the following integral for the vibrational
or phonon specific heat:
Ii D x4 eX dx
Cph --
9R(T/OD) 3
( e~ _ 1) 2 , (11)
494 Chapter 10: Thermal Properties
Table 10.1
Specific heat and related data for some selected superconductors. Data are from Table 4.1 of Poole
et
aL
(1995); see also Junod (1990) and Vonsovsky
et aL
(1982).
re
Material (K)
19o 7 A D(EF)
(K) (mJ/mol-K 2) (Cs-Cn)/TT c (mJ/mol-K 4) (states/eV)
Cd 0.55 252 0.67 1.36
A1 1.2 423 1.36 1.45
Sn, white 3.72 196 1.78 1.60
Pb 7.19 102 3.14 2.71
Nb 9.26 277 7.66 1.93
Zr0.7Ni0. 3 2.3 203 4.04 ~1.65
V3Ge A15 8.2 7.0
V3Si A15 17.1 17.0
Nb3Sn A15 18.0 13.0
HfV 2 Laves 9.2 187 21.7
(Hfo.5Zr0.5)V 2 Laves 10.1 197 28.3
ZrV 2 Laves 8.5 219 16.5
PbMo6S 8 Chevrel 12.6 79.0
PbMo6Se 8 Chevrel 3.8 28.0
SnMo6S 8
Chevrel 11.8 105.0
YMo6S 7 Chevrel 6.3 34.0
UPt 3 Hvy ferm 0.46 460.0 ~0.9
UCdll Hvy ferm 5. 200 290.0
URu2Si 2 Hvy ferm 1.1 31.0 0.42
CeRu2Si2 Hvy ferm ~0.8 340.0 3.5
(TMTSF)2C104 Org 1.2 213 10.5 1.67
K-(ET)2Cu(NCS)20rg 9.3 34.0
BaPbl_xBixO3 Perovskite 10.0 0.6 8.0
(Lao.9255ro.o75)2 CuO 4 37.0 360 4.5 2.0
YBa 2 Cu 3 07 92.0 410 20-35 1.3-2
Bi2Sr2CaCu208 95.0 250 0-10
T12Ba2CaCu208 110.0 260 <6.8 >6
T12Ba2Ca2Cu3Olo 125.0 280 ~58 ~ 1.4
0.23
1525
115
11.4
1.55
2.0
2.3
3.0
1.9
9.3
0.24
1-3
where x D - 0D/T, and this expression is plotted in Fig. 10.1 using experimental
data from Cu and Pb. The molar phonon specific heat has the following limiting
behaviors:
Cph --
(12rc4 /5)R(T/OD) 3 - 234R(T/OD) 3,
Cph -- 3R, T >> 0 o.
T << 0 D (12a)
(12b)
The figure shows that at T c the element Pb and LaSrCuO are in the T 3 region,
while YBaCuO is significantly above it.
B. Specific Heat 495
Fig. 10.1.
3R ......... |
2R
=o
" YkCuO
oO
R-
,:3i
~j~ .... Pb
0 l
o 0.5
, ,,,
w--
oPb
= Cu
9 SC at Tr
...... ]
1 .o 1.5
T/eu
Debye model (solid curve) of the phonon specific heat with the T 3 low temperature
approximation (dashed curve) indicated. Experimental data for the elements Cu and Pb are
given. The specific heats of the three superconductors Pb, (Lao.925Sro.075)CuO4, and
YBa2Cu307_ 6 at their respective transition temperatures are indicated [From Poole
et al.
(1995 p.14).]
If
Cexp/T
is plotted vs T 2 at low temperature,
Cexp/T -- 7 q- AT2,
(13)
the slope gives the phonon part
A -- 234R/03
and the intercept at the origin,
T - 0, gives the electronic coefficient 7. Sometimes other terms also contribute.
d. Discontinuity at T~
The phase transition from the normal to the superconducting state in the absence
of an applied magnetic field is second order, so the Gibbs free energy and its
temperature derivative are continuous:
Gs(T~)
= Gn(Tc) (14)
dG___2s = dGn .
(15)
dT dT
There is no latent heat, but there is a discontinuity in the specific heat, and the
BCS theory predicts a dimensionless jump in the normalized electronic specific
heat at T~ given by
(Ces-
7Tc)/TT ~
= 1.43, (16)