Poole Ch.P., Jr. Handbook of Superconductivity

Подождите немного. Документ загружается.

36 Chapter

Properties of the Normal Metallic State

Fig. 2.4.

kzltetrogonat axis

Geometric relationship between adjacent Brillouin zones of

La2CuO4.

Note that the

k x and ky

axes of this figure and those of Fig. 2.3 are rotated by 45 ~ relative to each other. [Kulkarni

et al.,

1991.]

Fermi Surface

Conduction electrons follow Fermi-Dirac statistics with the FD distribution

function,

f(E)

exp[(E -

#)/k B

T] + 1' (17)

where the chemical potential/~ is related to the Fermi temperature T F by

# ,~ EF = kB T F

(18)

and T F is typically about 105 K. Equation (17) is plotted in Fig. 2.5a for T = 0

and in Fig. 2.5b for T > 0. We see from the figures that

f(E)

is 1 for energies

below E F, it is zero above E F, and it assumes intermediate values only in a narrow

region of width k B T near the Fermi energy E F.

The electron kinetic energy

E K - hZkZ/Zm - (h2/Zm)(k 2 + ~ + k 2)

(19)

is quantized in k-space, and at 0 K these k-space levels are doubly occupied by

electrons of opposite spin up to the Fermi surface at E = EF,

E F -- hZkZ/Zm,

(20)

as indicated in the figure. Partial occupancy occurs in the narrow region of width

k B T at the Fermi surface shown in Fig. 2.5b, and the levels are empty for higher

E. Fermi Surface 37

Fig. 2.5.

f(E)

(a)

l"1"=0 I

EF E

f(E) kBT

1 ->,t k(-

E F E

(b)

Fermi-Dirac distribution function

f(E)

for electrons (a) at 7' = 0 K, and (b) above 0 K for the

condition 7' << 7"f. [Poole

et al.,

1995, p. 9.]

energies. For the isotropic case of a spherical Fermi surface, the electron density

n at this surface is

1 2)3/2

n -- k3/3rc 2 - -~2(2mEF/h .

(21)

The derivative

dn/dE

with E F replaced by E provides the density of states

D(E)

per unit volume,

d 1 2)3/2E1/2

1/2

D(E) -- -d-E n(E) - ~-~(2m/h - D(Ev)[E/EF] .

(22)

With the aid of Eqs. (18) and (20),

D(E)

at the Fermi level can be written in two

equivalent ways:

_ I 3n/2kB TF

D(EF)

(23)

mkF / h 2 rc 2"

The electron density n and the total electron energy E T can be expressed in

terms of integrals over the density of states multiplied by

f(E)

from Eq. (17):

n - ID(E)f(E) dE

(24)

E T - f D(E) f(E) E dE.

(25)

The energy dependence of

D(E)f(E)

in these integrands is given in Fig. 2.6a for

T = 0 and in Fig. 2.6b for T > 0.

38 Chapter 2: Properties of the Normal Metallic State

Fig. 2.6.

(a)

EF E

tll

Ill

ksr

,<_

Iv,01

(b)

Energy dependence of the occupation by electrons of a free electron energy band (a) at 0 K and

(b) for T > 0 K for the condition T << T F. [Poole

et al.,

1995, p. 10.]

The kinetic energy near an energy gap may be expressed in terms of an

effective mass m*,

where

Ek -- h2k2/2m *,

(26)

m* h 2

a2Ek

dk2

(27)

and the density of states, which is proportional to the effective mass m*,

D(g) - m*kF/h2g 2,

(28)

becomes large near the gap, as indicated in Fig. 2.7.

E Electromagnetic Fields

Fig. 2.7.

D(E)

L..J

Energy dependence of the density of states

D(E)

in the presence of an energy gap.

Electromagnetic Fields

The electromagnetic fields obey Maxwell's two homogeneous equations [Jackson,

1998; Lorrain

et al.,

1988],

V. B = 0 (29)

V x E + aB/at

= 0, (30)

as well as the corresponding inhomogeneous ones,

V.D=p (31)

V x H = J + 0D/0t, (32)

where p and J, respectively, are the free charge density and the free current

density. They are called "free" because they do not arise from the reaction of the

medium to applied fields, charges, or currents. The constitutive relations between

the B, H fields and the E, D fields, respectively, in a medium characterized by a

permeability # and dielectric constant e, are as follows:

B : #H : #o(H + M)

(33)

D = eE = eoE + P. (34)

These, of course, are SI formulas. When cgs units are used we have #o = Co = 1

in free space, and the factor 4n is added in front of M and P in Eqs. (33) and

34). Otherwise, for the SI case #o and e o have the values given in Table 1.1.

At the interface of two media of respective permeabilities #' and #" in

contact with each other, the B', H' fields in one medium are related to B", H" in

the other medium through the following two boundary conditions:

40 Chapter 2: Properties of the Normal Metallic State

Fig. 2.8.

B' H'

t H' B'

H" B"

Boundary conditions for the components of the B and H magnetic field vectors perpendicular

to and parallel to the interface between regions that have different values of the perme-

ability. The figure is drawn for the case #" = 2#'. [Poole et al., 1995, p. 15.]

The components of B normal to the interface are continuous across the

boundary

B',

(35)

The components of H tangential to the interface are continuous across the

boundary

H i

HII, (36)

as shown in Fig. 2.8. For a surface current density Ksurf at the interface the

second condition (36) becomes

n x (H'- HI( ) - Ksurf,

(37)

where the unit vector n is defined in the figure, and Ksurf, which has the units

A/m, flows perpendicular to the field direction. For the analogous electric case

the normal components of D and the tangential components of E are continuous

across an interface, and the condition on D is modified in accordance with Eq.

(31) when there are surface charges present.

The Lorentz force F on a charge q moving at the velocity v is expressed in

terms of the macroscopically measured magnetic and electric fields B and E,

respectively:

F - q(E + v • B). (38)

The field B is sometimes called the magnetic flux density because

B- ~/A

where 9 is the magnetic flux and A is the cross-sectional area perpendicular to

B. The magnetic flux inside Type II superconductors is quantized with the value

t~ o - h/2e -

2.0678 x 10 -15 T m 2, where the unit tesla meter squared is called a

weber.

References

N. W. Ashcroff and N. D. Mermin, Solid State Physics, Saunders, Philadelphia, 1976.

T. A. Friedmann, M. W. Rabin, J. Giapintzakis, J. P. Rice, and D. M. Ginzberg, Phys. Rev. B 42, 6217

(1990).

References 41

J. D. Jackson,

Classical Electrodynamics,

3nd ed., Wiley, New York, 1998.

C. Kittel,

Introduction to Solid State Physics,

7th Ed., Wiley, New York, 1996.

H. Krakauer, W. E. Pickett, and R. E. Cohen,

J. Supercond.

1, 111 (1988).

A. D. Kulkarni, E W. De Wette, J. Prade, U. Schr6der, and W Kress,

Phys. Rev. B

43, 5451 (1991).

P. Lorrain, D. P. Corson, and E Lorrain,

Electromagnetic Fields and Waves,

W. H. Freeman and

Company, New York, 1988.

S. Martin, A. T. Fiory, R. M. Fleming, L. E Schneemeyer, and J. V. Waszczak,

Phys. Rev. Lett.

60, 2194

(1988).

W. E. Pickett,

Rev. Mod. Phys.

61, 433 (1989).

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

Superconductivity,

Academic Press, New York, 1995.

C. P. Poole, Jr.,

The Physics Handbook,

Wiley, New York, 1998.

This Page Intentionally Left Blank

Chapter 3

Superconducting State

Charles

Poole, Jr.

Department of Physics and Institute of Superconductivity,

University of South Carolina, Colombia, South Carolina

A. Introduction 43

B. Persistent Current and Surface Resistance 44

C. Fields Inside Superconductors 45

D. Temperature Dependencies 47

E. Critical Magnetic Field Slope 51

Introduction

This chapter will provide a short introduction to the nature of the superconducting

state, with an emphasis on its two salient properties, namely, zero resistance and

perfect diamagnetism. The emphasis will be on Type I superconductors, with

Type II mentioned briefly at the end. Much of the material in this chapter was

elaborated on in our earlier work (Poole

et al.,

1995).

ISBN: 0-12-561460-8 HANDBOOK OF SUPERCONDUCTIVITY

All fights of reproduction in any form reserved.

44 Chapter 3: Superconducting State

Persistent Current and Surface Resistance

Even though superconductors have, by nature, zero dc resistance, it is still of

interest to see how close they come to zero. Ideally an electric current established

in a loop of superconducting wire will persist forever. An upper limit on the

resistivity p is set by the duration of persistent current flow. For a loop of radius

r = 15cm and wire radius a--1.5mm the ratio of the inductance L =

#or[ln(8r/a) - 2] to the resistance R = 2rp/a 2 provides the time constant

= L/R, and we find

p~ ~ 6.6 • 10 -1~ f~ cm sec. (1)

Since the current flows undiminished for well over a year,

z >> 3.2 x 10 7 sec,

(2)

the resistivity is far less than the limiting value

p << 2.1

• 10 -17

~ cm.

(3)

A special case to consider is current flow along a film of thickness d. For the

square region of surface with dimensions a • a shown in Fig. 3.1, the current

encounters the resistance R s -- pa/A, where A = ad, to give

n s = p/d. (4)

This resistance p/d is called the sheet resistance, or the resistance per square,

because it is for a square section of film independent of the length of the side a. It

is analagous to the surface resistance R s -- p/6 of a metal with the skin depth

6. There is a quantum of resistance h/4e 2 with the value

h/4e 2 -- 6.45 kf~, (5)

which is one-fourth of the Hall effect resistance R H = h/e 2. When the sheet

resistance in the normal state just above Tr exceeds this value (5), the metal does

not become superconducting, as illustrated in Fig. 3.2 for bismuth. The condition

R s < h/4e 2 (6)

must be satisfied for the material to superconduct.

Fig. 3.1

f.1.- j

Geometrical arrangement and current flow direction for sheet resistance measurement. [From

Poole et al. (1995), p. 32.]

C. Fields inside Superconductor 45

Fig. 3.2

Temperature dependence of the sheet resistance of films of Bi deposited on Ge as a function of

the film thickness in the range from 4.36 A to 74.27 A. [From Haviland

et al.

(1989).]

Fields inside Superconductor

More quantitatively, when an idealized Type I superconducting cylinder of radius

R much greater than the penetration depth 2-

(m/~onse2) 1/2

is placed in an

external field Bap p --B o = ~oHo directed along its axis; then demagnetizing or

shielding current Jsh is induced to flow in circles around the outside regions of the

cylinder. Far inside the cylinder the H field balances the magnetization,