Poole Ch.P., Jr. Handbook of Superconductivity

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526 Chapter 10: Thermal Properties

Fig. 10.24.

394G

h,,

~- 4

I'~OG

23t;7 G

tl~, 32SS Oe

3000 G

0 !

0 2 4 6 8

&V~ (microvolts)

Ettingshausen temperature difference vs. longitudinal potential difference for a type-II alloy

In + 40 at.% Pb in magnetic fields ranging from 39.4 to 300mT. [Solomon and Otter (1967).]

Fig. 10.25.

F-

, ...... ,, x ,,,

I ~1 I I I I I I

; i/~ B~

" ; .~ single cryslol

. : ;

)

-HX o,b ! " - i /f away

_ t ..,i

,2-,o " t& ._c--

-:7.5" : ~. ~...

:: ,5: i ~

:i " :

:'1"

70 80 90 100 410

T(K)

The transverse (Ettingshausen) temperature gradient induced in a single crystal of YBa2Cu 3

0 7

by the electric current in the presence of a magnetic field. The inset shows the probe

arrangement. [Palstra

et aL

(1990).]

D. Thermoelectric and Thermomagnetic Effects

527

Very few measurements of the Ettingshausen effect exist in either the

conventional superconductors or HTS cuprates. Figure 10.24 shows the data on

In0.6Pb0. 4 alloy (Solomon and Otter, 1967), while the behavior of the transverse

thermal gradient VyT in YBCO is shown in Fig. 10.25 (Palstra

et al.,

1990). Note

persistence of the transverse thermal gradient at temperatures some 15 K above

T c, which, according to the authors, may be due to a pronounced fluctuation

effect.

Righi-Leduc Effect

The Righi-Leduc effect is the thermal analogue of the Hall effect. A heat current

qx arising because of the applied thermal gradient

-V~T

in the presence of

magnetic field in the z-direction gives rise to a transverse temperature gradient

VyT

given by

VyT = RLBzq x,

(59)

where R L is the Righi-Leduc coefficient. Provided the carriers scatter elastically,

that is, within the regime of validity of the Wiedemann-Franz law, the Righi-

Leduc coefficient relates directly to the Hall coefficient R H as

R H = LoTR L,

(60)

where L 0 is the free-electron Lorenz number L 0 = 2.44 x 10 -8 V 2 K -2. In

analogy to the Hall effect, one defines the Righi-Leduc angle 0 as

tan 0 =

VyT/VxT.

(61)

In a normal metal the transverse thermal gradient

VyT

results from the asym-

metric scattering of the hot and cold carriers and the sign of the Righi-Leduc

effect, that is, whether the upper edge of the sample (along the y-axis) is hotter or

colder than the lower edge, is determined by the sign of the carriers.

The formalism of the Righi-Leduc effect can be applied to the mixed state

of a superconductor. However, although it is necessary for the vortices to move in

order to observe the Hall effect, the Righi-Leduc effect does not require any

motion of vortices. In fact, essentially all studies ever made of the Righi-Leduc

effect were performed with a stationary vortex lattice. The reason is simple:

Unlike an electric field, a temperature gradient can be sustained in a super-

conductor with a static vortex structure. Figure 10.26 shows the transverse

thermal gradient

VyT

as a function of magnetic field obtained on pure niobium

(Stephan and Maxfield, 1973). The behavior in the mixed state is rather dramatic

but not well understood, and the signal may even change sign as in some alloys of

conventional superconductors (Sichel and Serin, 1976).

Only two reports mention attempts to measure the Righi-Leduc effect in

HTS cuprates, both noting that the effect is very small (Galffy

et al.,

1990;

Zavaritsky

et al.,

1991). A successful use of the Righi-Leduc sample geometry to

528

Chapter 10: Thermal Properties

Fig. 10.26.

-2 4

~.,,.0 t

o Xc,~t.,.Hc,

3.OK 5.5K

" ~ ' " I ' 9 9 : =

1 i

9

..... i

H/Hc2

The transverse thermal gradient arising as a consequence of the longitudinal heat flow in the

presence of a magnetic field (Righi-Leduc effect) for pure niobium. [Stephan and Maxfield

(1973).]

study quasiparticle scattering on a pinned vortex structure and its relevance to the

issue of heat transport in HTS below T c (Krishana

et al.,

1995) is discussed in

Section C,d on thermal conductivity in a magnetic field.

g. Nernst Effect

The Nernst effect refers to generation of the transverse electric field that results

from a longitudinal thermal gradient in a perpendicular magnetic field and with

no electric current in any direction. Referring to Fig. 10.27, the Nernst coefficient

Q is defined as

Q - Ey/(VxTBz),

with j - 0. (62)

The effect is rather small in normal conductors. In contrast, the Nemst signal

becomes quite significant and is easily detected in the mixed state of super-

conductors, and this is perhaps the main reason why the Nemst effect is the most

frequently studied thermomagnetic coefficient. The Nemst effect is significant for

D. Thermoelectric and Thermomagnetic Effects

529

Fig.

10.27.

T I

U///

,. ] ....

]

~ - '/

Sample and probe configuration for measuring the Nemst effect.

its unequivocal tie with the motion of vortices. It marks the field and temperature

range where pinning is absent or very weak and vortices move freely.

Assuming pinning is absent,

Vx T

exerts the thermal force Fth-

-S, Vx T

on the vortex which moves down the thermal gradient with velocity

vr x.

The

motion is hindered by a viscous drag force f~--rlvg~,x, and the equation of

motion for the vortex is

Fth +f~ = 0. (63)

The motion of the vortex (along the x-axis) gives rise to the electric field

Ey - -%,~ B~

and, upon substitution into Eq. (63), one obtains

S4~VxT = qEy/B z.

(64)

The appearance of the transverse electric field

is an unmistakable sign of the

Nemst effect. The actual Nemst coefficient, see Eq. (62), becomes

Q -- S4~lJ 7 = pfSck/(~oBz).

(65)

The RHS of Eq. (65) follows upon substitution for the viscous drag coefficient r/

from Eq. (40) and one assumes that the viscous drag is isotropic in the

x-y

plane,

that is, the viscous drag in Eq. (63) that refers to vortex motion along the x-axis is

identical with the viscous drag in Eq. (40), which applies to the y-axis vortex

motion. Equations (64) and (65) allow determination of the transport entropy as

$4~ = Eydpo/(VxTPf ).

(66)

530

Chapter 10: Thermal Properties

For completeness, we include here the expression for the transport entropy

derived by Maki (1971) using the time-dependent Ginzburg-Landau theory,

Sc k - (~Po/4~zT)[Bc2(T)- B]L(T)/{1.16(2tC2GL -

1)+ 1} --

(~Po/T)(M)L(T).

(67)

Here (M) is the spatially averaged magnetization and

L(T)

is a numerical

function close to 1 near Tr Equation (67) is valid only near Bc2 , and in the

intermediate field range where most of the experiments are conducted it is less

reliable.

As already noted, the Nernst effect has been measured frequently in both

the conventional superconductors and in many HTS cuprates. An example of the

Nernst effect behavior in cuprates is presented in Fig. 10.28, which shows the

data of Ri

et al.

(1994) for c-axis oriented films of YBazCu307_ 6 and

BizSr2CaCu208+ x together with the respective flux-flow resistivities. Note a

significant Nernst signal extending to temperatures some 10-15K above T~,

suggesting the important role fluctuations play in these highly anisotropic

superconductors. Furthermore, we point out a much broader regime of flux

flow resistivity and a correspondingly wider temperature range of the Nernst

effect in the more anisotropic BizSrzCaCu208+ x.

Two important points need to be kept in mind when measuring the Nernst

effect in highly anisotropic superconductors: (a) The concept of rigid, tubelike

vortices needs to be replaced by that of two-dimensional pancake-like vortices

centered on individual superconducting planes and connected by Josephson

vortices. Since Josephson vortices lack the normal core, the thermal force on

such a vortex is zero and the Josephson vortices do not contribute to the Nernst

effect. (b) Even if realized, the vortex-antivortex unbinding above the Kosterlitz-

Thouless temperature is not going to contribute to the Nernst effect in spite of the

fact that it is an important resistive mechanism. Very simply, under the action of

the Lorentz force the vortex and the antivortex move in opposite directions, and,

having opposite vorticities, they will generate resistive voltage. In contrast, under

the influence of the thermal force the vortex and the antivortex both move down

the thermal gradient, and the electric fields they generate cancel each other.

With a stronger driving force in excess of the pinning force, one should be

able to extend the domain over which the thermomagnetic effects are observed.

Although it is easy to increase the Lorentz force by increasing the transport

current, the small heaters typically used in the thermal force-driven experiments

do not supply enough power to create the large thermal gradients needed to

overcome strong pinning forces. With the use of focused laser light one may

overcome this limitation, as shown by Lengfellner

et al.

(1990) and, more

recently, by Zeuner

et aL (1994),

who were able to explore all three regimes of

flux motion: thermally assisted flux flow, flux creep, and viscous flux flow.

Table 10.4 gives values of the transport entropy collected from various

experiments on conventional superconductors and HTS cuprates. In general, the

Fig. 10.28.

[o) 150

E 100

::1.

o..

0 i_.=

Cb)

:~2

I;>,

u.J ='" F

9 H= 0T

9 H= 1 I"

9 H= 2T

9 H= 3T

9 H= 4T

O H= 6T

O H= 8T

'~ H= 10 T

a H= 12T

"'i I!

Y1302Cu307- 6

80 90 100 '

.... i ....

i' ' "'"'1

....

u_ qT

,JJ

110

(e) ~50

100

~ 50

(d) 5

..-.4

1--.2

J"l

50 70 90 110

0 L

0 I

70 Bo 90 100 110 30 so 70 90 ~o

T (K) T (K)

The Nemst effect data and the associated flux-flow resistivities for c-axis oriented films of YBa2Cu307_ 6 and Bi2Sr2CaCu208. [Ri

et al.

(1994).]

532 Chapter 10: Thermal Properties

S0[J/m-K ] T[K]

Table 10.4.

Experimental values of the transport entropy S o .

B[T] Material Reference

1.6 x 10 -12 3.0 0.1 In+40% Pb

6.0 • 10 -12 1.6 0.15 Pb+4.56% In

4.4 x 10 -17 87 0.54 YBCO, c-axis oriented film

3.0 x 10 -14 87 4 YBCO, c-axis oriented film

4.3 x 10 -16 77 4 YBCO, polycrystalline slab

1.1 • -16 87 4 YBCO, polycrystalline film

2.6 x 10 -15 87 4 YBCO, epitaxial film

5.0 x 10 -16 87 5 YBCO, epitaxial film

6.5

• 10 -13

87 4 YBCO, epitaxial film

1.0 • 10 -15 87 4 BSCCO, 2212 single crystal

3.7 xl0 -13 81 4 BSCCO, 2212 c-axis film

6.0 x 10 -16 87 5 BSCCO, 2223 polycrystal

9.5 x 10 -13 87 4 TBCCO, epitaxial film

3.0 x 10 -16 80 1 TBCCO, 2212 single crystal

1.0 • 10 -16 87 5 TBCCO, 2212 polycrystal

1.0 x 10 -16 70 5 TBCCO, 2223 polycryst, film

1.6 x 10 -15 34 3 LSCO, bulk polycrystal

1.6 • -14 18 0.5 NCCO, c-axis oriented film

5.0 x 10

-16

16.8 1.5 NCCO, epitaxial film

Solomon and Otter (1967)

Vidal (1973)

Zeh

et al.

(1990)

Hagen

et aL

(1990)

Kober

et al.

(1991)

Kober

et aL

(1991)

Kober

et al.

(1991)

Hohn

et al.

(1991)

Ri et al.

(1994)

Zavaritsky

et aL

(1991)

et aL

(1994)

Dascoulidou

et al.

(1992)

Hagen

et al.

(1991)

Logvenov

et aL

(1992)

Dascoulidou

et al.

(1992)

Zeuner

et al.

(1994)

Hohn

et al.

(1994)

Haensel

et aL

(1995)

Jiang

et aL

(1995)

Table 10.5.

Transport entropy obtained from the slope of the transport energy.

-dU4~/dT

[J/m-K] B[T] Material Reference

8.9 x 10

-13

4 YBCO, epitaxial film

6.6 x 10 -13 3 YBCO, single crystal

4.0 x 10 -13 3 YBCO, single crystal

2.2 x 10 -13 3 YBCO, epitaxial film

1.4 x 10 -14 3.5 YBCO, epitax film, 6 < 7

6.7 x 10-14 3 YBCO, single crystal

2.0 x 10 -13 4 YBCO, epitaxial film

2.5 x 10 -13 4 YBCO, epitaxial film

2.5 x 10 -13 1 YBCO, thin film

1.2 x 10 -12 4 YBCO, melt grown

4.8 x l0 -17 5 BSCCO, 2223 bulk polycr.

4.2 x 10 -12 4 TBCCO, 2212 epitax film

8.3 x 10 -16 1 TBCCO, 2212 single crystal

1.4 x 10 -14 3 LSCO, granular bulk

2.4 x 10 -t3 0.5 NCCO film

Hagen

et aL

(1990)

Palstra

et al.

(1990)

Hao

et aL

(1991)

Kober

et al.

(1991)

Huebener

et al.

(1991)

Oussena

et al.

(1992)

et al.

(1993)

et aL

(1994)

Haensel

et aL

(1995)

Sasaki et al. (1996)

Dascoulidou

et aL

(1992)

Hagen

et aL

(1991)

Logvenov

et al.

(1992)

Hohn

et al.

(1994)

Haensel

et al.

(1995)

D. Thermoelectric and Thermomagnetic Effects 533

transport entropy of conventional superconductors is 2-3 orders of magnitude

larger than the transport entropy in HTS materials. Among the cuprates, the data

frequently differ by a wide margin, and it is important to understand why such

large discrepancies may arise. First of all, the transport entropy is sensitive to the

temperature and magnetic field, and because the superconducting transition

temperature may differ by several degrees even among the same family of

cuprates, a direct comparison of the respective transport entropies at a particular

fixed temperature is not very revealing or meaningful. It is more practical to make

use of the transport energy per unit vortex length,

U4) - TS4),

and specifically its

temperature derivative,

dU4)/dT,

at a fixed value of the magnetic field. This

approach tends to suppress the influence of different Tc's among the family of

cuprates. The values of the transport entropy obtained by this procedure are

collected in Table 10.5.

References for Section D

J. L. Cohn, E. E Skelton, S. A. Wolf, and J. Z. Liu,

Phys. Rev. B

45 13140 (1992).

A. Dascoulidou, M. Galffy, C. Hohn, N. Knauf, and A. Freimuth,

Physica C

201,202 (1992).

A. Freimuth, in

Selected Topics in Superconductivity, Frontiers of Solid State Science,

Vol. 1 (L. C.

Gupta and M. S. Multani, Eds.). World Scientific, Singapore, 1992.

M. Galffy, A. Freimuth, and U. Murek,

Phys. Rev. B

41, 11029 (1990).

H. Ghamlouch and M. Aubin M,

Physica C,

269, 163 (1996).

V. L. Ginzburg,

Sov. Phys. Usp.

34, 101 (1991).

H. J. Goldsmid,

Electronic Refrigeration.

Pion Limited, 1986.

H. Haensel, A. Beck, E Gollnik, R. Gross, R. P. Huebener, and K. Knorr,

Physica C

244, 389 (1995).

S. J. Hagen, C. J. Lobb, R. L. Greene, M. G. Forrester, and J. Talvacchio,

Phys. Rev. B

42, 6777 (1990).

S. J. Hagen, C. J. Lobb, R. L. Greene, and M. Eddy,

Physica C

185, 1769 (1991).

Z. Hao, J. R. Clem, M. W. McElfresh, L. Civale, A. P. Malozemoff, and E Holtzberg,

Phys. Rev. B

2844 (1991).

C. Hohn, M. Galffy, A. Dascoulidou, A., Freirnuth, H. Soltner, and U. Poppe, Z.

Phys. B--Condensed

Matter

85, 161 (1991).

C. Hohn, M. Galffy, and A. Freimuth,

Phys. Rev. B

50, 15875 (1994).

R. E Huebener,

Magnetic Flux Structures in Superconductors.

Springer-Verlag, Berlin, 1979.

R. P. Huebener, A. V. Ustinov, and V. K. Kaplunenko,

Phys. Rev. B

42, 4831 (1990).

R. P. Huebener, E Kober, H.-C. Ri, K. Knorr, C. C. Tsuei, C. C. Chi, and M. R. Scheuermann,

Physica

C 181, 345 (1991).

X. Jiang, W. Jiang, S. N. Mao, R. L. Greene, T. Venkatesan, and C. J. Lobb,

Physica C

254, 175 (1995).

A. B. Kaiser,

Phys. Rev. B

35, 4677 (1987).

A. B. Kaiser and C. Uher, in

Studies of High Temperature Superconductors,

Vol. 7 (A. V. Narlikar, Ed.),

p. 353. Nova Science, New York, 1991.

E Kober, H.-C. Ri, R. Gross, D. Foelle, R. P. Huebener, and A. Gupta,

Phys. Rev. B

44, 11951 (1991).

K. Krishana, J. M. Harris, and N. P. Ong,

Phys. Rev. Lett.

75, 3529 (1995).

H. Lengfellner, A. Schnellbogl, J. Betz, W. Prettl, and K. E Renk,

Phys. Rev. B

42, 6264 (1990).

G. Yu. Logvenov, M. Hartmann, and R. P. Huebener,

Phys. Rev. B

46, 11102 (1992).

K. Maki,

Physica

55, 124 (1971).

E. A. Meilikhov and R. M. Farzetdinova,

J. Supercond.

7, 897 (1994).

S. D. Obertelli, J. R. Cooper, and J. L. Tallon,

Phys. Rev. B

46, 14928 1992).

M. Oussena, R., Gagnon, Y. Wang, and M. Aubin,

Phys. Rev. B

46, 528 (1992).

T. T. M. Palstra, B. Batlogg, L. E Schneemeyer, and J. V. Waszczak,

Phys. Rev. Lett.

64, 3090 (1990).

H.-C. Ri, A. Kober, A. Beck, L. Alff, R. Gross, and R. E Huebener,

Phys. Rev. B

47, 12312 (1993).

534

Chapter 10: Thermal Properties

H.-C. Ri, R. Gross, E Gollnik, A. Beck, R. P. Huebener, E Wagner, and H. Adrian,

Phys. Rev. B

50,

3312 (1994).

T. Sasaki, M. Sawamura, S., Awaji, K. Watanabe, N. Kobayashi, K. Kimura, K. Miyamoto, and M.

Hashimoto,

Jpn. J. Appl. Phys.

35, 82 (1996).

M. Sera, S. Shamoto, and M. Sato,

Solid State Commun.

68, 649 (1988).

E. K. Sichel and B. Serin,

J. Low Temp. Phys.

24, 145 (1976).

P. R. Solomon and E A. Otter,

Phys. Rev.

164, 608 (1967).

C. H. Stephan and B. W. Maxfield,

J. Low Temp. Phys.

10, 185 (1973).

C. K. Subramaniam, C. V. N. Rao, A. B. Kaiser, H. J. Trodahl, A. Mowdsley, N. E. Flower, and J. L.

Tallon,

Supercond. Sci. Technol.

7, 30 (1994).

C. K. Subramaniam, H. J. Trodahl, A. B. Kaiser, and B. J. Ruck,

Phys. Rev. B

51, 3116 (1995).

Uher,

J. Appl. Phys.

62, 4636 (1987).

E Vidal,

Phys.

Rev. B 8, 1982 (1973).

Z. Z. Wang and N. P. Ong,

Phys. Rev. B

38, 7160 (1988).

N. V. Zavaritsky, A. V. Samoilov, and A. A. Yurgens,

Physica

C 180, 417 (1991).

M. Zeh, H.-C, Ri, E Kober, R. P. Huebener, J. Fischer, R. Gross, H. Muller, T. Sermet, A. V. Ustinov, and

H.-G. Wener,

Physica C

167, 6 (1990).

S. Zeuner, W Prettl, K. E Renk, and H. Lengfellner,

Phys. Rev. B

49, 9080 (1994).

Chapter 11

Electrical Properties

Charles R Poole, Jr.

Department of Physics and Institute of Superconductivity~

University of

South

Carolina, Columbia,

South

Carolina

A. Introduction 535

B. Hall Effect 536

C. Tunneling 537

D. Josephson Effect 541

E. Magnetic Field and Size Effects 544

E Superconducting QUantum Interference Device (SQUID)

References 546

545

Introduction

This chapter deals with some of the electrical properties of superconducting

materials. One of the most important of these is the critical current density Jc, and

measured values are tabulated for many superconductors in Section E of Chap. 9.

It was mentioned there that experimentally determined critical current densities

seldom come close to theoretically predicted limits such as the depairing value.

Researchers around the world are presently spending a great deal of time and

effort trying to increase attainable critical currents in wires and tapes. We now

proceed to examine some specific electrical properties such as the Hall effect,

tunneling, Josephson junctions, and Superconducting QUantum Interference

Devices, commonly referred to as SQUIDs.

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535