Effects of energy gap anisotropy in pure superconductors

ANNALS

OF PHYSICS:

Effects

‘lo,

of Energy

268-295 (1966)

Gap

Anisotropy

in Pure

Superconductors*

JOHN R. CLEM~ Department

of Physics,

University

of Illinois,

Urbana,

Illinois

Various effects of anisotropy of the superconducting energy gap are theoretically considered. In order to estimate the effects of anisotropy upon the thermodynamic properties of pure, single-crystal superconductors, a factorable BCS-like model for the effective electron-electron matrix element V,,, = (1 + a=)V(l + aB,) is used. The effects of anisotropy upon the temperature dependence of the gap parameter, the critical field, and the specific heat near the critical temperature are then shown to be small and proportional to the mean-squared anisotropy (az), which is of the order of 0.02 for typical superconductors. Theoretical expressions which explicitly include the anisotropy of a general gap parameter are given for low-temperature specific heat, nuclear spin-lattice relaxation time, tunneling, surface resistance, and longitudinal ultrasonic attenuation. These processes are seen to be more sensitive to the details of the anisotropy than are the above thermodynamic properties. I. INTRODUCTION

The existence of a finite energy gap between the ground state energy and the lowest excited states of a superconducting metal plays an important role in the theory of superconductivity. The BCS theory (l)--the first successful microscopic theory of superconductivity-was the first to show that the superconducting state at finite temperatures could be described in terms of individual particle-like excitations having an energy gap. The energy of one of these quasiparticles is given by E, = [(E; + Ai)1’21, w h ere Q,is the energy, measured from the Fermi level, of a Bloch electron having wavevector p, and where AP is a quantity called the gap parameter. The minimum quasiparticle energy for a given p, attained when Q, = 0, is equal to Ap . In a pure single crystal the energy gap parameter Ap depends upon the direction of p with respect to the crystal axes and may be regarded as a basic physical property, as is, for example, the Fermi

surface,

of the metal

in question.

The

many

experimental

manifestations

* Based on a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the University of Illinois. This research was supported in part by the Alfred P. Sloan Foundation, by the U. S. Army Research Office, Durham, under contract DA-31-124-ARO(D)-114, and by the Air Force Office of Scientific Research under contract AF 735-65. of t United States Steel Foundation Fellow, 1963-65. Present address : Department Physics and Astronomy, University of Maryland, College Park, Maryland 20740. 268

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269

of the gap parameter make feasible the interesting possibility of inferring from experiment the full functional dependence of AP upon p. It is therefore of interest to examine theoretically a variety of experimental quantities and to reveal the precise role played by the angular dependence of AP in each case. In a pure superconductor the anisotropy effects may be small or large, depending upon the experimental quantity under examination. The critical field and the specific heat near the critical temperature are good examples of experimental quantities for which the effects of anisotropy are small-of the order of a few percent-even if the energy gap parameter AP should vary by as much as f25 ‘% as a function of direction. The smallness of these effects may be rmderstood by noting that the thermodynamic quantities depend upon the probability for thermal excitation of the quasiparticles. For thermal energies kT comparable with the energy gap, the probability for thermal excitation of quasiparticles is appreciable for all directions. Thus it is essentially the angular average (A,):,, of the gap parameter which determines the magnitude of t>he thermodynamic quantities. The lowest-order corrections arising from anisotropy, which are proportional to the angular average of the square of the deviation of the gap parameter from its average value, are most conveniently expressed in t*erms of the mean-squared anisotropy, defined as (a”) = ((A,

- (A,)m)“>av/(A,)tv

(1.1)

This is a small quantity for most superconductors; typically, (a’) ,z 0.02. Pokrovskii (2) theoretically obtained the effects of anisotropy upon the critical field at low temperatures and upon the specific heat jump at the critical temperature. In Section II we shall extend the calculations of the critical field and the specific heat over a wider range of temperatures by including the effect, of anisotropy upon the temperature dependence of the gap parameter. This will be accomplished by the use of a simple anisotropic model for the BCS effect,ive electron-electron interaction VP,, . The results, appropriate for weak-coupling superconductors, will be expressed in berms of the mean-squared anisotropy W. Large effects of anisotropy in pure superconductors, discussed in Section III, require the existence of selection processes, in which special groups of quasipart,icles are selected to be the dominant contributors to the experimental effect in question. Such selection processes are effective in the following experiments: low-temperature specific heat, nuclear spin-lattice relaxation time, tumreling, surface resistance, and longitudinal ultrasonic attenuation. In view of the recent experimental interest (3-8) in the low-temperature specific heat, its theoretical interpretation (2, 9, 10) will be expanded in this work. Two quantities (P(u), the anisotropy distribution function, and n(p, w),~, the Fermi surface average of the reduced effective density of states), which we feel simplify the descript,ion

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CLEM

of anisotropy effects, will be introduced. Experiments upon the nuclear spinlattice relaxation time (11-15), which have previously been interpreted as showing the effects of anisotropy (12, 14, Is), will be discussed in terms of an expression for the relaxation rate which explicitly displays these effects. Superconductive tunneling experiments have recently been used (16) to obtain information about anisotropy of the energy gap. A new expression for the tunneling current density, showing the degree of angular selection which may be expected in tunneling experiments, will be derived. Since surface resistance measurements have also shown effects of anisotropy (17), an appropriate expression for the surface impedance, based on the results of Mattis and Bardeen (18) for the extreme anomalous limit, will be presented. In addition, a discussion of the effects of anisotropy upon longitudinal ultrasonic attenuation, based on the results of previous experimental (19-23) and theoretical (24-26) investigations, will be given. II. SMALL EFFECTS OF ANISOTROPY A. THE GAP PARAMETER We wish to be able to estimate the effect of the presence of anisotropy upon the thermodynamic functions in the superconducting state. However, since these functions depend sensitively upon the gap parameter, we must first carefully examine the effects of anisotropy upon Ap , with particular attention to the dependence upon the temperature T. To do this, it is convenient to work within the framework of the BCS theory (1) . The gap parameter, Ap( T), must then be determined from the condition A = c P P’

V,,r Apl tad W%/2) 2E,s

(2.1)

where 6 = (/CT)-‘, and L is Boltzmann’s constant. Here, VP,! is a matrix element which contains the effects of the anisotropy of the electron-phonon interaction and is hence the source of the angular dependence of A2, . In the spirit of the BCS model, we shall assume that an effective phonon-induced matrix element of the form V PP’ = (1 + a,>V(l

+ a,!>

(2.2)

is appropriate for the case of a pure, single-crystal superconductor. Such a matrix element was first used by Markowitz and Kadanoff (27) to study the effects of anisotropy upon the critical temperature. The anisotropy function up, defined for quasimomenta near the Fermi surface and assumed to depend only upon the direction of p with respect to the crystal axes, is to be regarded as a fundamental property of the metal under examination, just as is the Fermi surface. The function up is defined to have zero average over the Fermi surface; using brackets

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‘27 1

( ),, to denote Fermi surface averages, we have (a& = 0. We envisage up t’o vary by perhaps SO.25 as p moves over the Fermi surface, such that the Fermi surface average of the square of the anisotropy function (a=‘),, = (a”), called the mean-squared anisotropy, has a value of about 0.02 for a typical superconductor. We note that when ap = 0 (or (a’) = 0)) our model and all the result’s thereof will reduce to those of the BCS model. Substitution of the above form for VP, into t,he gap equation (2.1 j immcdiately yields a gap parameter of the form A,(T) We note Q( 7’) as In the integrals

= edT)(l

+ apj.

( 2.3 J

that A,( 2’) might vary by f25 % about its average value (A,(T)),,. = p moves around the Fermi surface. following treatment we shall often convert sums over wavevrct’or p to via (3.4 1

We remark that for nonspherical Fermi surfaces the density of Bloch states of one spin in energy at t’he Fermi surface is actually anisotropic and depends upon the position of the quasimomentum p on the Fermi surface. Therefore, this effect should be included if the procedure of converting the sum over wavevectors to an integral over Bloch energies plus an average over the Fermi surface is to he correct,. However, since this anisotropy here plays a role secondary to t.hat of t,he angular dependence of V,,t , we have suppressed it by replacing the density of states by a corresponding isotropic average N (0). The quantity EO(T) remains to be determined from a gap equation which may be written in the form

1 = N(O)v ((1 + ap)2 / de-,tanh2;$,/2)',, , il" . The limit,s for the integral are found from the cutoff at) E, = wD , :I typical phonon energy. (We work in units for which fi = 1.) By examination of Eq. (2.5) we are able to determine in the weak-coupling limit (wD >> LT,) the effects of the anisotropy upon (A,(O)),, = Q(O), upon 111~ critical temperature T, , and upon the temperature dependence of the gap parameter as expressed as A,( T)/A,(O) = cO(T)/eO(0) vs. T/Tc . We shall soon see that the modifications to the corresponding BCS model result#s are small and are proportional to (a”). First, we confine our attention to the solution of Eq. (2.5) at T = 0. Since the hyperbolic tangent is then unity, we may easily perform the ep integration. We next, expand, in powers of the anisotropy function up, the resulting quantiby

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CLEM

within the brackets. We then perform the Fermi surface average, noting that (l)m = 1, (a& = 0, and (a;>,v = (a’), but neglecting terms of order (a,3),, . Finally, after rearranging terms, we find in the weak-coupling limit ( wD>> ~(0)) : Q(O) = 2WD [1+(&--g)

@)]exp[--&I.

(2.6)

This result differs from the corresponding BCS isotropic-gap result only by virtue of the term proportional to (a”}. Next, in order to obtain a convenient expression which will enable us to find the temperature dependence of co(T) and the critical temperature T, (where Q(T) + 0), we subtract the finite temperature gap equation (2.5) from its zero temperature version. In the weak-coupling limit we may then write with little error

.I-(&( T))\ s,dtp E,(T) /av ' [ 1 ((l+Up)2

-(Cl + up)2)8v In $$J =

(2.7)

where f( U) = (1 + exp pw)-’ is the Fermi function. The steps used to obtain Eq. (2.7) and many of the succeeding results follow those first used by Muhlschlegel (as), who obtained the thermodynamic functions using the BCS isotropic-gap model in the weak-coupling limit. However, since we are considering the case for which the gap parameter is anisotropic, the important modification here is that a Fermi surface average must be taken after the energy integrals have been performed. By examination of the behavior of Eq. (2.7) as CO(T) tends to zero (28) we find -((l

+ ap)2)&y In

[

$$J

2 c =

1

= ((1 + a,)‘ln

(1 + op))av,

(2.8)

where yB = ec = 1.78107 . . . and C is Euler’s constant. Inserting numerical values and performing the Fermi surface average after expanding in powers of up, we find, through order (a’),

(2.9) which differs from the corresponding BCS result only through the (a’) term. Combining (2.9) with (2.6), we obtain for the critical temperature kT, = 1.134wo[1

+&I

exp[l

- kV],

(2.10)

a result which was first obtained by Markowitz and Iiadanoff (27). The correction to the corresponding BCS result is rather small. For example, with (u*> NN 0.02

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273

SUPERCONDUCTORS

and l/N(O) V M 4, the presence of anisotropy gives rise to an 8 % increase in the critical temperature. Finally, in order to find the effect of the presence of anisotropy upon t#he temperature dependence of the energy gap parameter, we direct our attention to the behavior of A,(T)/A,(O) = Q(T)/Q,(O) as a function of t = T/Tc , t.he reduced temperature. We shall now derive a power series expansion of E~(T)/Q(O) in powers of (1 - t) which adequately describes t’his behavior. For such a calculation it is convenient t,o make use of a function A (r), int,roduced by Riuhlschlegel (28) :

A(x)

= -f

II

G?UIn {l + exp [-P(U’

+ r)“‘]J + 5 In (yB xl”) [

- i

1

1’2.11)

+ i.

This function is regular as x -+ 0, and its derivatives at x = 0 can be expressed in terms of Riemann zeta functions. The quantity In [Q( T)/e,(O)], as well as all the thermodynamic funct)ions, can be expressed in terms of A (x) and a few of its derivatives. If we define X

then, combining

(2.7)

p = MWd~1

+ up)‘,

(2.12j

and (2.8) and making use of (2.11)) we obtain

((1+ ~P>“A’bP>>m. + Int = 07 ((1+ a,>%v where A’(x) = dA(z)/clx. Equation cients of the power series expansion

[ 1 CO(T) ’

__ EOKO

= Co(1 -

(2.13)

t)[1 + C,(l

(2.13)

may be used to obtain

-

t) + C2(1 -

1)’ + ...I

the coeffi-

(2.14,

in the following manner: First, we expand A’(x,) about x1, = 0 in powers of :cP ; t.he known derivatives A” (0)) A”‘(O), A”“(O), . . * , appear as coefficients. In turn, making use of the expressions (2.12) and (2.14)) we expand x, , wherever it appears, in powers of (1 - t). W e next express t,he left-hand side of Eq. (2.13) as a power series in (1 - t), then find Co , Cl , Ca , . . . by solving t#he equations which result when we equate to zero the coefficient of each power of (1 - t). The resulting expressions for CO , Cl , Cz , . . . involve Fermi surface averages of various known functions of a, . We expand t,hese functions in powers of ap , then perform the Fermi surface averages, keeping only terms through order (a”). Finally, inserting numerical values for the Riemann zeta funct,ions,

274

CLEM

we obtain

[ 1 co(T) do>

2

= 3.0160(1 - 2(a2))(1 X [l -

- t)

(0.8190 - 2.724(a2))( + (0.0425 -

1.752(a2)( 1 -

Taking the square root, we find the following convenient temperature dependence of the energy gap parameter: A,(T) AdO)

= -co( T) = 1.7367(1 EO@) X [l -

(a2))( 1 (0.4095 -

(2.15)

1 - t) t)” + -. .I. expression

for the

t)“’ 1.362(a2))( 1 - t)

(2.16)

(0.0626 + 0.318(a2))( 1 - t)” + . . .I.

For (a”) = 0 this power series expansion accurately reproduces Muhlschlegel’s (28) numerical calculations of E~( T)/eo(0) vs. t for the BCS model, provided t > 0.3. The error is less than 0.1% for t > 0.4 and less than 0.5 % for t = 0.3, for which Q( T)/co(0) = 0.997. (The remarkable accuracy of this power series over such a wide temperature range has also been noted by Ferrell (29).) For t < 0.3,we have eo(T)/co(0)NN1. Thus, in the plot of CO(T)/eo(O)vs. t in Fig. 1 the power series expansion for ($) = 0 is indistinguishable from Muhlschlegel’s results for t greater than 0.3, whereas eo(T)/eo(O)is indistinguishable from unity for t less than 0.3. For typical values of (a”), the coefficients of the various powers of (1 - E) in Eq. (2.16) are altered in such a way as to cause the curve of EO( T)/eo(O)vs. t to lie slightly below that of the BCS isotropic-gap model. This behavior is illustrated in Fig. 1 for ($) = 0.04. We thus observe that the correction to the temperature dependence of the gap parameter arising from anisotropy is small and of the order of (u’). With the above knowledge of the effects of anisotropy upon A,(O), T, , and A,( T)/A,(O) vs. t we are now able to proceed with a calculation of the thermodynamic functions in the superconducting state. In particular, we wish to treat the critical field and, near the critical temperature, the specific heat. B. THE

CRITICAL

FIELD

The critical field H, for a bulk specimen of unit volume is given by the difference between the normal and superconducting free energies. We shall assume that the lattice contribution to the free energy in the normal state is the same as that in the superconducting state. The calculation of the superconducting state free energy, taking into account anisotropy of the energy gap parameter,

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SUPERCONDUCTORS --

r--.

pi.---

r

_.__,

- -~

4

pi I.Oi t o.ej-: 0.6

For T/T. %(T) = G(O)

t

2

0.3

1.7367( Ir *[I - (0.4095-1.362<1 - (0.0626+0.318<0~>)

= 0.04’

0.4'r

0

I 0.1

I

0.2

I

0.3

I

_ 0.6

I

0.4

0.5 T/ T,

FIG. 1. The temperature dependence of the energy gap parameter: Ap(2’)/Ap(0) = Q,(!P)/E~(O) versus T/To . The curve with (a”) = 0.04 illustrates t,he small corrections, of t hr order of the mean-squared anisotropy (a*), which arise from the anisotropy of the energy gap parameter.

follows closely that of the BCS theory weak-coupling limit:

HC = ; N(O)(A,“),, 8s

and yields the following

result, in the

- f ?r2N(0)/fi2 (“.I7

The critical

field at zero temperature,

HO 87r

1

Ho , is simply given by

= f N(O)~c30N

1 +

(2).

To obtain the effects of anisotropy upon the t’emperature dependence of If,. , we begin by confining our attention to temperatures such tha.t t 5 0.3. For suc*h a case, as we saw earlier, we have A,(T) .X A,(O). Further, the third term on the right-hand side of Eq. (2.17) is negligibly small in comparison wit,h t,hc first, two terms. Combining Eqs. (2.17), (2.18), and i2.9), we then have h s He/H, = (1 - 2k’t2’)“2,

(“.l!I’)

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WhtP.3

K = 2Try(T,/H,)2

= 1.057(1 + 2(a2))

(2.20)

and

y = g7r2N(o)k2.

(2.21)

The present model thus yields h = 1 -

1.057(1 + 2(a”>>t” -

0.559(1 + 4(a2))t4 + *. * .

(2.22)

The critical field can be measured so accurately that h is sometimes plotted in terms of its deviation from the parabola (1 - t”) . In terms of the deviation function D (1) (SO), the above result becomes

D(t)

= h - (1 - t”) = -(0.057

+ 2.11($)>t”

- 0.559(1 + 4(u2))t4 + * * * .

(2.23)

For temperatures such that 1 is between 0.3 and 1 the temperature dependence of the critical field is more difficult to obtain, because of the involved nature of the Fermi surface averages over the anisotropy in the superconducting state free energy. Progress can be made, however, by expanding about the critical temperature in powers of (1 - t), as was done for Q( T)/EO(O). To do this, we first observe that we may express Eq. (2.17) in terms of the auxiliary function A (x) in the following manner:

Hc = 3 i?(x, A’&) ___ 8ayT,2 2

- A(z,)),,

.

We next expand A (z,) and A’(z,) in powers of 2, ; the known derivatives, AN(O), A”‘(O), ... , appear as coefficients. With the help of Eqs. (2.12) and (2.15) we in turn expand xp , wherever it appears, in powers of (1 - t). The coefficients of the various powers of (1 - t) involve Fermi surface averages of functions of ap . We expand these functions in powers of ap and then perform the Fermi surface averages, keeping only terms through order (a*). Finally, inserting numerical values, we obtain

HZ -

8?ryT,2

0.71306( 1 -

4(a2))( 1 - t)‘[l -

Taking

-

(0.5460 - ~l.816(a2))(1

(0.1153 + 0.422(c?))(

- t)

(2.25)

1 - t)’ + . . .I.

the squre root and making use of Eq. (2.20), we obtain

h = 1.7367(1 -

(a”>)(1

-

t) [I -

(0.2730 - 0.908(&) (0.0949 - 0.037(u2))(1

(1 - t) - t)” + - - *I.

(2.26)

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SUPERCONDUCTORS

If we express the above result in terms of i’, we observe that the deviation tion may be written as D(t)

= -0.1317(1

+ 6.60(a2))(1

- t”) + 0.0986(1 + 3(&(1

+ 0.0287(1

+ 615(a”))(l

- t”)”

- I’)“+

...

func-

(2.27) .

To determine the validity of the power series expansions (2.22) and (2.26 ), we may compare the numerical values obtained from theve expressions when (a’) = 0 with the corresponding BCS isotropic-gap model results computed by Muhlschlegel (28). We find that the expansion (2.22) with (a’) = 0 gives good agreement with Muhlschlegel’s results for t 5 0.3, while t,he expansion (2.26) with (a”) = 0 gives good agreement for t 2 0.7. Neither expansion should he trusted when 0.3 < t < 0.7. The above expressions show that in the weak-coupling limit the anisotropy corrections to the temperature dependence of the critical field given by tlm HCX isotropic-gap model are small and of the order of (a’), the mean-squared anisotropy of the energy gap. For typical values of (a’> the effect of anisotropy is t.o cause the curve of h vs. t to lie slightly below that of the BCS model. This behavior is shown in Fig. 2, where the value (a”) = 0.04 has been used in the expansions (2.22) and (2.26) to illustrat’e this effect. Figure 3 rc-eshib& t’hese

-

0.4

For

For

T/Tc L 0.7

!k

= 1.7367(1-)(l-+)*

HO

0.2

L 01

T/T, S 0.3

*[ I-0.2730(1-3.326:0*‘)(1 -0.0949(1-0.388

I 0.1

I 0.2

I 0.3

)

I 0.4

-+ , -3 t

Tc r1 I I 0.5 0.6

-

I 0.7

0.8

0.9

1.0

T/ r, temperature dependence of anisotropy are demonstrated by the using (a3 = 0.04. FIG.

2. The

of the critical field: H,/Ho versus curves obtained from the power

T/T, series

. The effects expansions,

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results in terms of the deviation function D(t) . The expansions (2.23) and (2.27) in their appropriate ranges of validity, have been used. The presence of anisotropy, its effects now magnified by the nature of the deviation function, causes the curve of D(t) vs. t” to lie somewhat below that of the BCS isotropic-gap model. The accuracy of critical field measurements may make it possible to observe the small effects of anisotropy in experiments designed to determine the effect of impurity doping upon the critical field curves for weak-coupling superconductors. Though certain strong-coupling effects complicate the analysis, we would expect the gross features of the electron-phonon coupling to be essentially unchanged for small impurity concentrations. On the other hand, the addition of nonmagnetic impurities should significantly alter the anisotropy effects, causing the terms of order (a”) to disappear (as in the impurity doping effect upon the critical temperature) and thereby producing an upward shift in the critical field curves. C. THE SPECIFIC HEAT

NEAR THE CRITICAL

TEMPERATURE

The electronic contribution to the specific heat in the superconducting state C, exhibits some rather dramatic effects of the anisotropy at low temperatures. However, we shall postpone a discussion of these effects until Section III and confine our attention for the moment to the behavior for temperatures near the critical temperature. t’=

(T/Tel*

0.1317(1+6.60~02~)(l-t2) +0.0986(l+3~a2~)(l-tZ)z +0.@287(1

+ 6.15

)

(1-t’)’

-0.04-

-o.os-

-0.06

For t’~0.l D(t) = -0.057 -0.559 I

(I+ 369)t’ (It 4 <02>)t’ I

I

I

I

I

I

I

I

I

FIG. 3. The deviation function, D(l) - H,/Ho - (1 - P), versus P. (t = T/T..) The effects of anisotropy are demonstrated by the curves obtained from the power series expan sions, using (a”) = 0.04.

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The specific heat in the superconducting state, following the BCS theory hut including anisotropy of the energy gap, may be expressed in the weak-coupling limit as

cs= 2NONP

J!!,2

(2.28)

Since we are interested in the behavior near l’, , it is again convenient, to obtain a power series expansion for C, in powers of ( 1 - t) . For this purpose it is helpful to express c, in terms of the auxiliary function A (.r) as

c, “fTc- t - ; t(zp)av+ 3t(z,A’bp) - A(x,))av+ ;4 t (xP2$&rP a”, (x2!)) which may be obtained from Eq. (2.28) with the help of Eq. (2.13). By expantling the righthand side of the above equation in powers of i 1 - 1) in a manner similar to t,hat used for the critical field, we find c, __ = 2.4261(1 y ‘I’,.

- 2.351(a2))

- 4.7621(1 + 1.3498( 1 -

- 4.792(&(1 12.432(a2))11

- t)

( 2.30 i

- t)” + . . .

In order to determine the range of temperatures for which this result is valid we may compare numerical values of C,/rT, obtained with (a”) = 0 with the corresponding values given by Muhlschlegel’s numerical computations (28). Wc find t#hat the above power series expansion gives good agreement with the exact results of t’he BCS isotropic-gap weak coupling model, provided t > 0.7. The effect,s of anisotropy upon t,he specific heat near the critical temperatluro are again seen to be small and of the order of the mean-squared anisotropy (a”). We note that the presence of anisotropy reduces the magnitude of t,he specific: heat jump C, - C,, evaluated at T, and decreases the slope of the curve of (7, vs. T just’ below the critical temperature. These effects are illustrated in Fig. 4, whcrc the power series expansion (2.30), with (u’) = 0 and (a’) = 0.04, has been ~isetl to obtain the curves near T, , We emphasize that all the above calculations arc valid only for weak-coupling superconductors, of which there are few examples in nature. Although a direct, comparison with experiment cannot easily be made at present,, our results should be regarded as giving a good estimate of the magnitude and Dhe direction of eff rcts which may be expected to arise from the presence of anisotropy. For the t,hermodynamic functions we have considered in this section it is of interest to note that strong-coupling effects and anisotropy cff ects are genernlly in opposite directions. For example, strong-coupling eff cot’s produce uwa~l shift,s in (1) the value of 2(A,(O)),,/kT,, (2) the curve of A,(T)/A,(O) vs. I, (3) the curve of h vs. t, and (4) the specific heat jump between the superconduct -

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2.4

-

2.0

-

1.6 c* YTC 1.2 0.8-

0.4

Of 0

0.2

0.4

0.6

0.8

T c

1.0

FIG. 4. The temperature dependence of the specific heat near the critical temperature: C./-yT, versus T/T, . The power series expansion (2.30) for C,/rT, in the absence of anisotropy ((aZ) = 0) is shown for T/To > 0.5. The effects of anisotropy are demonstrated by the dashed curve, which is obtained from the power series expansion with (a”) = 0.04. ing and normal state at the critical temperature. However, as we have seen in this section, the presence of anisotropy tends to produce downward shifts in all these quantities. Thus, since both strong-coupling effects and anisotropy effects are likely to be present in most pure superconductors, the two effects compete in such a way as to make their separation difficult if only thermodynamic data is examined. III.

MORE

SENSITIVE

PROBES

OF THE

ANISOTROPY

A. THE SELECTIONPROCESS In the preceding section we discusseda number of quantities upon which the anisotropy of the superconducting energy gap has little effect-even rather large amounts of anisotropy gave rise to small corrections of the order of (a”). Within the context of the BCS theory the physical interpretation of the smallnessof these effects follows from the fact that the thermodynamic quantities depend upon the probability for thermal excitation of quasiparticles over the energy gap. At temperatures near the critical temperature, excitations for all directions of p are thermally created with roughly the same weight. Summing the excitations

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IN SUPERCONDUCTORS

281

from all parts of the Fermi surface essentially averages the features of the gap parameter, and only the gross features such as its average value and the meansquared anisotropy survive. Thus, the thermodynamic quantities are generally not sensitive probes of the anisotropy. On the other hand, the low-temperature specific heat in the superconducting state is an exception t,o this rule in that it samplesthe features of the anisotropy in those directions p for which AI, takes on its smallest values. In this c*asethere is a thermal selection of the participating quasiparticles; at low temperatures only those quasiparticle states sufficiently populated by thermal excitation---those for which the energy gap 2A, is small-are able to contribute to the specifit heat. In a similar manner, the nuclear spin-lattice relaxation time at low temperatures depends upon thermally selected excitations, and hence it, also probes the anisotropy in directions for which Ap has its smallest values. Both of these quantit,ies, therefore, are able to sample certain details of the anisotropic gap parameter by selecting special quasiparticles to be the dominant contributors 1.0 the experimentally observed effect. A processof selection is effective also in other experimental quantities, though the selection may not be of thermal origin. For example, since the tunneling current in superconductive tunneling arises chiefly from electrons which are incident normally upon the insulating junction barrier, the tunneling characteristics of pure, single-crystal superconductors reflect the nature of the gap parameter Ap appropriate to directions p approximately perpendicular to the junction barrier. Another method of selection arises from the constraint imposed by the simultaneous conservation of energy E, and momentum p. For example, only those quasiparticles which satisfy both conservation laws are able to contribut,r t,o surface resistance or to longitudinal ultrasonic attenuation. In each of the experiments in which a selection process is effective, the sellsitivity to tbe angular dependence of the gap parameter provides the means for probing someof the details of the anisotropy. We shall now turn our attention t,o the calculation of a few of these experimental quantities. We remark that these resuhs will hold for general A* and will not depend upon the anisotropy 1nodc4 (2.2) used in the previous section. B. LOW-TEMPERATURE

SPECIFIC

HEAT

We wish to examine the effects of anisotropy upon the low-temperature behavior of the electronic specific heat C, in the superconducting state. WC shall confine our attention to temperatures low enough that the temperature dependence of the gap parameter may be neglected. After a change of variables, Ii’,q. (2.28) may then be written in the form c, = 2N(0)kf12 --r* dM$(w)Il s

- fCw)l(n(p, w)),, .

( 3.1 )

252

CLEM

We have introduced

the anisotropic,

reduced, effective density

n(p, LI) = Re [a/(~”

of states

- AP2)“‘],

(3.2)

where the sign of the square root is chosen to obey (0” - APz)“’ + w as 1w ( -+ 00, so that n(p, U) is an even function of w. We note that although n(p, LO) has inverse square root singularities at w = &A,, as in the BCS theory, the Fermi surface average (n(p, a)),, , in general, does not. Instead, it has structure which depends upon the distribution of values of AP over the Fermi surface. We now define the anisotropy function by means of the relation up = (A,(O)

- @dO))av>/(A,(O>)av

.

(3.3)

It is also convenient in this point to introduce an anisotropy distribution P(a), which describes the distribution of the various values of up as p moves over the Fermi surface. We define P(a) da to be the fraction of the Fermi surface for which up has a value in the interval a to a + da. Just as the anisotropy function up is to be regarded as a fundamental property of the metal under examination, so should the anisotropy distribution function P(a), which is derived from up . The introduction of this distribution function allows us to convert Fermi surface averages of functions of the anisotropy function up into one-dimensional integrals over a. The Fermi surface average of the reduced, anisotropic, effective density of states may thus be written as (n(p, CO)),, =

daP(a)

Re l4~”

-

@,>%1

+

a)21”21.

(3.4)

It is clear from this expression that the structure of the anisotropy distribution function P(a) is partially reflected in the energy dependence of (n(p, w)),, . For example, if we assume, for simplicity, a rectangular model for P(a) : P(a)

=

uIn,x)-1,

Chin t

<

U <

amax

otherwise,

where a., ax = -CL,, in = (3(~‘))“~, then (n(p, w)),, has the form shown in Fig. 5. Returning to Eq. (3.1), we see that the specific heat at low temperatures is given by an integral over all energy of the Fermi surface average of the reduced, effective density of states, weighted by the temperature-dependent factor w’f( 1 - f) . At very low temperatures this factor tends to w2 exp ( -pw), which gives much greater weight to those values of (n(p, w)),, for which w is small than to those values for which w is large. Thus, the quasiparticles for which the energy gap is small become the dominant contributors to the electronic specific heat at low temperatures. The low-temperature specific heat will then appear experimentally to bs characterized by an effective gap parameter which is smaller

2S8

GAP ANISOTROPY IN SUPERCONDUCTORS

2.0-

05-

0

02

0.4

0.6

0.0

I .o

1.2

14

I6

I .0

w/~~ FIG.

5. The

(n(p,w)h. with

(a’)

Fermi

, versus

surface ~/(4~(0)),,

average of the reduced, , for the rectangular

anisotropic, anisotropy

effective distrihution

density of st~Aes, function (3.5)

= 0.03.

than (A,(O)),, and which decreases with temperature.

In certain cases t,his may at t,he low temperature end of IL give rise to a so-called “upward curvature” curve of log (C,/rT,) vs. To/T. For example, such behavior is seen in Fig. 6, which has been obtained by inserting the rectangular distribution function iS.T, 1 into Eqs. (,3.4) and (3.1) and by assuming /3(A,(O)> C. NUCLEAR

SPIN-LATTICE

= 1.764(1 - :2/(u’))(T,/T). RELAXATION

( 3.6 i

TIME

Let us now consider the nuclear spin-lattice relaxation time T, and examine t,he effects of anisotropy. The experimental quantity of interest is the r’tltcl R = l/T1 at which the system of nuclear spins comes into equilibrium wit,h th(h lattice. In a met,al the dominant mechanism for this process arises from t’hr coupling of the nuclear spins to the spins of the conduction electrons, which arc assumed to have the temperature T of the lattice. The inter:tct,ion Hamiltonian which describes t,his process is (31) H8~ = F3 Ye Yn C

z

C Si.Ij s(ri - Rj ), i

( 3.7 1

284

CLEM

0.01 cs yT, 0.001 -

3

4

5

6

T,

7

6

9

FIG. 6. The low-temperature specific heat: C,/yT, versus T,/T, for various values of the mean-squared anisotropy (aa), cakulated using a rectangular model for the anisotropy distribution function.

where ye and yn are the gyromagnetic ratios, Si and Ij are the dimensionless spin operators, and q and Rj are the position vectors of the ith electron and the jth nucleus, respectively. (We use units in which fi = 1.) Using matrix elements of Hs, between initial and final states of the systems of nuclear spins and electrons, the nuclear spin-lattice relaxation rate may be calculated with the aid of the Golden Rule. The derivation of this rate for the superconducting state within the context of the BCS theory was first given by Hebel and Slichter (11). When we extend this calculation to include anisotropy of the energy gap parameter A, , we obtain

Rs= k3 = ‘$ re”rn” 1~(o)\~~N(o)]~ l: ciof(w)[l - f(w)] (&

_““,c)l,2

>\”1 /-

(3.8)

’

where u(O) is the value at a nuclear site of the Bloch wave function of a typical

GAP

ANISOTROPY

IN

SUPERCONDUCTORS

2&j

electron at the Fermi surface. We have neglected the splitting of the nuclear energy levels, since this splitting is usually extremely small compared with the relevant energies, AP or kT. We note that when AP = 0, the above expression reduces t,o the form appropriate for the normal state i I1 ) :

R, = $

= ‘$

Y: rn” 1 u(0)14 [N(0)12 kT.

III

If AP is assumed to be isotropic, the resulting singularity in the reducbed, effcctive density of states, n(p, 0) = Re [~/(a” - A,“)““] (as well as, of course, in the related quantity Re [A,/ (0” - A,*)““]), causes the int,egral over w to be logarithmically divergent. This difficulty was recognized by Hebel and Slicht)er (11)) who, in order to obtain theoretical agreement with their experimental measurements of T1, in superconducting aluminum, introduced an empirical ‘%~vel broadening” which was designed to smear out the singularity in the density of states. For Al an empirical smearing of the density of states over approximately Ho the gap width was necessary to obtain agreement with experiment. However, as we have remarked, AP is expected to be anisotropic in pure superconductors. Since it is the Fermi surface average (n(p, w))%” of the effectJive, anisotropic, density of states which enters the expression (3.8) for R, , there is t,hus no divergence difficulty. We interpret the so-called “level broadening” as just the effect of the Fermi surface average of n(p, a). As we have seen earlier, (n(p, w)),, has the appearance of being “smeared.” A similar interpret,at,ion has been given by previous authors (12, 14, 15). Measurements of other experimental quantities indicate that the anisotropy function up in aluminum varies by at least fO.l as p moves about t’he Fermi surface. The surface resistance experiments of Biondi, Carfunkel, and Thompson (17) indicate such behavior, and experiments determining the effect of impurit#y doping upon the critical temperature have been interpreted by Markowitz and Kadanoff (27) as revealing a mean-squared anisotropy (a’) = 0.011 for aluminum. This amount of anisotropy would introduce a “smearing” of the anisotropic densit,y of states to the degree observed in the nuclear spin-lattice rclax:ttion experiments. Further evidence that the ‘(level broadening” arises from anisotropy is the effect, of the addition of impurities. The resulting decrease of the effect,ive “level broadening” is easily interpreted as arising from the “washing out” of t,he anisotropy by t,he impurity scattering (15). We remark, in passing, that the anisot,ropy model (2.2) yields :I gap paramctcr for which the anisotropy function a3 =

[A,(T) - @,CT)MI@,(?‘))av

ie independent of temperature. This behavior is in agreement with t’ure dependence observed experimentally (14, 15).

(3.10) the tempcra-

256

CLEM

The presence of the Fermi functions f(u) in Eq. (3.8) gives rise to a thermal selection of the participating quasiparticles, just as in the case of the lowtemperature specific heat. The behavior at low temperatures is thus dominated by the excitations for which the gap parameter is small. The expression (3.8) also holds when the energy dependence of the gap parameter is important, provided AP is then replaced by the complex gap parameter A(p, w). This result may be derived with Green’s function techniques, starting from the approach used by Kadanoff (32) for the normal state, but employing the Gor’kov factorization (33) which is appropriate for the superconducting state. The properties of the matrix Green’s function for the superconducting state (34-V) may then be used to obtain the appropriate form for R, . (For an isotropic gap the Fermi surface averages are unnecessary, and Eq. (3.8) then reduces to the result given by Fibich (38) for strong-coupling superconductors.) D. TUNNELING Superconductive tunneling is one of the most direct probes of the anisotropy of the energy gap parameter. The idealized experimental arrangement is essentially the following: A single crystal of some superconducting metal A, having gap parameter APA, is prepared. Upon some flat surface of the crystal, having unit normal ri pointing in some known direction with respect to the crystal axes, a thin, uniform layer (~20 & thick) of insulating material is deposited or allowed to form. Upon this layer is deposited a thick layer of metal B, which, if it is superconducting, is characterized by the gap parameter APB. A voltage V is applied to the resulting sandwich, and the current I, which arises from the quantum mechanical tunneling of the electrons through the barrier, is measured. The characteristics, curves of 1 vs. V, give information about the gap parameters APA and APB. The derivation of the first-order tunneling current density j from the microscopic theory (35, 36, 39-44)) when extended to include anisotropy of the gap parameter, yields j

=

eNA

(O)NB(O)

(4rAr)F’

f: ch [f(m) - f(w’)l (3.11)

. s dQ s dQ’ I T(p, P')]' na(p,~h(p',

~'1,

where o’ = o + ev. Here, nA(p, W) and nB(p’, w’) are the reduced, effective, anisotropic densities of states for metals A and B. (In the normal state n(p, w) = 1.) The tunneling matrix element T(p, p’) is the probability amplitude that an electron with momentum p on the Fermi surface of metal A will tunnel through the barrier, emerging with momentum p’ on the Fermi surface of metal B. For simplicity these Fermi surfaces are taken to be spherical; the angular integrals

GAP

ANISOTROPY

IN

2Si

SUPERCONl~~CTOIZS

are over all elements of solid angle &(dQ’) within which the moment~um p(p’ 1 may lie as it moves around the Fermi surface of metal A( 13). Here, N,(O) :mtl NB(0) are t’he densities of Bloch states of one spin per unit, energy interval in metals A and B, and AI is the area of the insulating barrier. In order to proceed, we need further information about the tunneling matris element T(p, p’). Harrison (44) has used the WIiB met)hod to c~alculate such :L matrix element for the case of specular transmission of the electrons through the barrier. His result for 1 T(p, p’) 1’ is proportional to the exponential qu:mtit,y exp (- q), where, if the barrier potential is taken t’o hc :L c*onst:mt OVIY t IN. barrier thickness d, v may be written as ?j = 242m/fiy%?

+ ,uey,

( 3.1‘2 1

where 0 is bhe angle bet’ween 6 = p/p and ti. Here, since the nM3l-barrier work function + is typically one electron volt and (1 is approximately 20 K, wc have 2~2(2n14,)“~/fiw 20. Thus the quantit’y t = cxp [-22tZ(2r,&)1’S/fi.] < < < 1. Let us further expand the square root’ in (3.12). We may then writ)c exp C-0) Fz t exp (-{e’),

(3.13)

where, in terms of the chemical potential p and a quantJit,y cx = ( 2,lr+ j’if::fi,d, w(’ have r = 214fi2a. Since { is usually quite large, typically 20 to 100, exp i - 7) is strongly dependent upon the direction of ?j with respect, t#ot#heunit, normal ii of the junction barrier (directed outward from metal A), and is largest for 1; perpendicular to the barrier. This factor reflects the fact that the electrons which move approximately normal to the barrier are the dominant contributors lo thrb tunneling current. The assumption of specular transmission requires essentially t,hat an c4cctrou moving in a certain direction before it tunnels t’hrough the harrier emergesfrom the barrier moving in approximately the same direction. Further, since t,he rclcvant momenta are approximately normal t,o the barrier, the desired tunneling makix element may thus be written in the form

a{A&s(@ - 1;‘)

I UP> P’Y = (2n)2N*(0)Ne(O) exp (-s-e”).

:x14 1

The tunneling current density then becomes

To account for the fact that electrons moving away from t.he harrier cannot, tunnel, the integral over solid angles is restricted to those dir&ions of fi for which 6 < a/2. Here, j(P, V) = e44*)-’

/ dw[f(w)

- f(~0’)ln,(p,

W) k,(P, a’),

( 3.16 1

288

CLEM

where w’ = w + eV, is the potential contribution to the tunneling current arising from electrons moving in the direction $. The actual contribution from these electrons is reduced by the factor exp ( -[ti2), as is seen from Eq. (3.15). Because of the term exp ( -J??‘), the tunneling current depends upon the values of APA and APB for directions fi clustered tightly about the unit normal fi perpendicular to the junction barrier. The angular size of this “cone” depends upon the value of f; the larger { is, the smaller the size of the “cone.” A measure of this dependence is given by the quantity (3.17) which is roughly the fraction of the tunneling current carried by electrons for which the angle 8 between $ and A is less than B. . We see that eo2 = - f’ In (1 - fo) for small eo. For a barrier thickness of 20 8, using @ = 1 eV and fi = 10 eV, this result tells us that approximately 50 % of the tunneling current is carried by electrons for which e is less than 4.7’, and 95 % of the current is carried by electrons for which 0 is less than 10”. In the case that n,(p, w) and nB(p, w’) are slowly varying functions of direction compared with the variation of the exponentia function examined above, they may be replaced by their values at J? = ri. The remaining solid angular integral in Eq. (3.15) may then be performed, using the condition that { >> 1. The tunneling current then becomes j = eat(47r)-2

s

dw[f(w)

- fb’)hA(ri,

whB(fi,

d,

(3.18)

which depends only upon the values of the gap parameters APA and A,* for the direction $ = fi, perpendicular to the junction barrier. Near threshold the tunneling characteristics for such a situation should then look very much like those derived using the BCS density of states, with the two appropriate values of the gap parameter AnA and AnB. (We shall not be concerned with the fine structure at higher energies which srises from the phonon spectrum.) However, in single crystals it is presumably very often the case that the effective densities of states nA(p, w) and n,(p, w) vary as a function of direction at least as rapidly as does the function exp (-ye”). We must then use the full expression (3.15) for the tunneling current density. The tunneling characteristics are then a superposition of the characteristics of the contributions j(p, V), each of which enters with a weight specified by the function exp ( -co”). Complicated characteristics of this origin have been found by Zavaritskii (16)) who has observed the tunneling current through a thin tin oxide barrier between single crystal tin, having gap parameter ApA and “dirty” tin, having isotropic gap parameter APB = AB. He has found that the gap parameter in single-crystal

GAP

ANISOTROPY

IN

299

SUPERCONDUCTORS

tin is a strongly varying function of the direction of p, varying by as much as f2.5 % as p moves over the Fermi surface. He has also made some progress in correlating the value of APA with the posit,ion of p on the rather complicated Fermi surface of tin. E.

SURFACE

RESISTANCE

As seen in recent experimental investigations of aluminum by Biondi, Garfunkel, and Thompson (17), measurements of the surface resistsnce in singlccryst,al superconductors show the influence of anisotropy of the energy gap. In order to see how this comes about, we shall I~OW briefly discuss the phenomenon of surface impedance. Let us consider a large single crystal of a superconducting metal, whose surf:tc:o is cleaved in a plane of high symmetry. We apply, parallel to the surface, :k microwave-frequency electric field, pointed in a direction of symmetry in such a way that the current induced in the crystal flows in the same direction. The surface impedance is defined as

where E(0) is the value of the electric field at the surface and.?(<) is the cnrrcnt density in the simple a distance E from the surface. Here, R is called t’hr surl’accl resistance, and X, the surface reactance. The properties of the metal in question are contained in the response function K( 4, w), which relates the Fourier components of t’he current density j(q) :mt,l the vector potential A(q) via the relation j(q)

= - (cIW~(q,

(,X20)

4A(q).

Here, the wavevector q is perpendicular to the surface whereas j and A arc parallel to the surface. For random scattering at the surface of the metal, the surface impedance may be expressed in berms of K ( q, w) as (’1, 45)

We shall be interested only in the extreme anomalous limit, in which K( q, w) is proportional to (l/n) for those values of 4 which are most important, in the integrand of Eq. (3.21). In the normal state we then have (18) Kn(q,~)

=

-

47rzwcr,( c2

q,w)

=

-

37riw h2(o)ffF

( 3 .I“‘2 ) q

where X,(O) = (47rnoe”/mcz)-1’2 is the London penetration depth at, zero ternperature, no is the number of conduction electrons per unit volume, and vp is

290

CLEM

the Fermi velocity. z

The surface n

impedance

= 2?rw &i -C?(3J(

In the superconducting

then becomes

4XL2(0)VF 1’3 1 - i 43 2 >.

state we have (18,

K,(q, w) = -4Tiwu;;qy where us/u” independent

(3.21)

(3.23)

46)

w, = -

0

37riw 4xL2(o)vF q

;

is complex and dependent upon the angular of q. The surface impedance becomes

(3.24)

frequency

w, but is (3.25)

2, = Zn(u,/up3.

The ratio r = RJR, of the surface resistance in the superconducting state to that in the normal state, a quantity often measured experimentally, is thus r = Re [( 1 - i&)

(a,/a,)-1’3].

(3.26)

With u3 = usI + iua2, we note that (3.27) r E 1 - 3-1’2 (zy’”

(z)

,

(3.28)

for 2 << 1.

We are interested in the effects of anisotropy upon the superconducting kernel K8(q, w) and hence us(q, w) in the extreme anomalous limit:

LJ/VFP<< 1,

4o)lvFq

<< 1,

cl/P, x

ql << 1,

1,

state (3.29)

where pF is the Fermi wavenumber and 1 is the electronic mean free path. When anisotropy of the energy gap parameter AP is included in Mattis and Bardeen’s (18) treatment of the anomalous skin effect, the following expression is obtained: (3.30) where

ll dw’ Mw’) - f(w” )I n

&(p, w) = w-l .Re

(w””

:

&2)1/Z

X

[ Re ( cw,,-@;nI,2) AP (w”2

_

&?)1/2

>I

(3.31)

’

and w” = w + w’. The angular integration extends over all elements of solid angle do within which the direction ?; = p/p may lie as p moves over the Fermi surface, here assumed to be spherical. We note that Re [w/(w2 - AP2)1’2] is an

GAP ANISOTROPY

IN SUPERCONDUCTORS

291

even function and Re [A,/( o2 - AP2)1’2]is an odd function of w. Under the conditions of the extreme anomalous limit, the quasiparticles which are effective, as we seefrom the presence of the delta function 6( fi. 4)) are those which have wavevectors p very nearly parallel to the surface. From the presence of the term (6.6)’ we see that somewhat greater weight is given to those excitations for which p has an appreciable component in the direction &of the electric field. We note, in passing, that for an isotropic gap parameter USI -= un

lTl?T-l

s

dn(p+?)‘s($~j

= 51.

( x:x 1

The imaginary part of us is related to the real part by means of KramersKronig relations; thus us2(o)/ un depends upon the values of gal(~‘)/an for all values of the argument w’. Although excitations from all parts of t,he Fermi surface then play a role in uS2(w), the most influential values of Ap are st,ill those for which fi.4 NN0. We thus have to good approximation us2 --.-=a un

-1

ds2(fi.a)*s(~+t2(p, s

4,

( 3.33 )

where

and mN= w + w’. The expressions (3.30) and (3.33) may be regarded as quantitative statements of Biondi, Garfunkel, and Thompson’s speculations (17) (based on Pippard’s “ineffectiveness concept” (47)) concerning t,he excitations expected to play a role in the anomalous limit. At zero temperature we see that ~,~/a,, is equal to zero when the angular frequency o is less than twice the minimum value of Ap on the “effective zone” of the Fermi surface. For larger frequencies there are contributions from these excit ations; thus u,Ju, and r, as seen from Eq. (3.27), become nonzero. As the frequency is further increased, the quasiparticles having larger values of Ap are able to participate; thus u~I/u,, and r increase, but at a rate which depends upon the distribution of values of Ap around the “effective zone.” This distribution is therefore revealed from the details in the experimentally obtained curves of 1’ vs. w (17). F. LONGITUDINAL

ULTRASONIC

ATTENUATION

The effect of anisotropy of the superconducting energy gap upon the longitudinal ultrasonic attenuation, first observed in single-crystal tin by Morse, Olsen,

292

CLEM

and Gavenda (19)) has been observed in various metals by numerous investigators (ZO-WS). Although this effect has been discussed by previous authors (24-d%), we wish to emphasize its simple interpretation in terms of the selection process. We consider the idealized experimental arrangement in which a beam of pure longitudinal phonons of wavevector q and angular frequency C+ = sq (where s is the sound velocity) propagates in a symmetry direction within a pure, singlecrystal metal specimen. The attenuation of this beam, characterized by an attenuation coefficient a, arises from the absorption of the phonons by the conduction electrons at the Fermi surface or, in the case of the superconducting state, by the corresponding thermally excited quasiparticles. The calculation of the attenuation coefficient can be carried out most simply by means of a Golden Rule (Born approximation) treatment (2.4, 48). In the calculation for the normal state there occur processes in which an electron of wavevector p and Bloch energy eP at the Fermi surface absorbs a phonon of wavevector q and energy wpand scatters into the state p + q having energy Q,-+~ = Q, + uq . From this relation, assuming a free electron Fermi surface and using the typical experimental values oq = sq E 10’ see? and s M 5 X lo5 cm/see, we find that the angle 0 between ?j = p/p and 4 = q/q obeys the relation cos 0 = fi.4 = S/V, - q/2p, NN 5 X 10m3<< 1. The only electrons able to participate in the attenuation are those which have wavevectors p very nearly perpendicular to q, thus defining an “effective zone” of the Fermi surface. For the superconducting state the same argument, with the Bloch energies cp replaced by the quasiparticle energies E, = ( cp2 + AP2)1’2, may be applied. In this case the angle between $ and 4 depends upon the quasiparticle energy: cos8=1).B=~{[I+2(~)+($]I12-(~)}-&

(3.35)

For Q, >> AP the quasiparticles behave as normal particles; hence cos 19<< 1 and 8 m ?r/2, as before. However, for smaller values of cP, the angle 0 decreases. When Q, = 0 (E, = A,,), the angle B is given by cos 0 = (s/z+) (1 + ~A,/w,)“~ q/2p, ; with oq z 10’ see-’ and AP = 1012 see-‘, we then have cos 0 E 10-l. Thus, in the superconducting state the only excitations selected to contribute to the attenuation are those having wavevectors p confined to a thin, ring-like “effective zone” of the Fermi surface. The directions ?j of these wavevectors are typically within a few degrees of those appropriate to the normal state effective zone. For this reason the ratio of the attenuation coefficient in the superconducting state to that in the normal state may be written in the form (3.36)

GAP

ANISOTROPY

IN

293

SUPERCONDUCTORS

The delta function a(?j a4) within the solid angular integration over the directions of fi evaluates the Fermi function f( AP) for wavevectors p on the effective zone. To obtain Eq. (3.36)) as in the BCS theory (1)) the matrix element M,, I for scattering from the state p to the state p’ = p + q is assumed to be the same in both normal and superconducting states and to be independent of the direction of p with respect to the crystal axes (aside from a momentum-conserving delta-function). If some directional dependence of this matrix element. is assumed via the relation M,,l = M,6,r,,+, , then

This result is essentially the same as that obtained somewhat different method. VI.

SUMMARY

AND

by Pokrovslcii

($?!j) by n

DISCUSSION

In this paper we have attempted to present a basic picture of the various eff rcts of anisotropy of the superconducting energy gap. We first discussed certain quantities which are relatively insensitive to the details of anisotropy; the anisotropy effects are then of the order of the mean-squared anisotropy (a’). WC bhen examined various experimental quantities which, by means of certain selection processes, are sensitive to the details of the anisotropy function up or of the anisotropy distribution function P(a). The new results, interpretations, and simplifications given should contribute to a better underst.anding of the effect’s of anisotropy. Much interesting work concerning anisotropy remains to be done. The expcrimental techniques discussed above may be used to probe the features of Ap for various metals and to obtain the particular functional dependence upon p in each case, just as various experimental techniques are now being used to obtain Fermi surfaces. In fact, the appropriate way t.o display the wavevector dependence of Ap is in terms of contours of constant A drawn upon the Fermi surface. Excellent progress is already being made along these lines for the case of tin (IS). The high degree of selection available in superconductive tunneling makes it a particularly valuable experimental technique for such investigations. We not,e that for the interpretation of certain of the above experimental quantities, it will be necessary to include in the present theory the departure from sphericity of the Fermi surface. Knowledge of the wavevector dependence of AP obtained from experiment should provide theoretical information about the electron-phonon interaction, since the anisotropy of the gap parameter originates from the anisotropy of t,he

294

CLEM

electron-phonon interaction. Further, theoretical investigations of this connection might complement band structure calculations of the Fermi surface, since the anisotropy of the electron-phonon interaction in turn arises partly from the wavevector dependence of those single-particle Bloch states at the Fermi surface which appear in matrix elements of the electron-ion potential. ACKNOWLEDGMENTS The author wishes to express his gratitude to Professor Leo P. Kadanoff for his guidance in this research and to Professor John Bardeen for his helpful advice. He also wishes to thank Professor D. M. Ginsberg, Professor Dillon E. Mapother, Wesley N. Mathews, Sang Boo Nam, John Shier, and David C. Montgomery for useful discussions and suggestions. RECEIVED:

March

29, 1966 REFERENCES

1. J. BARDEEN, L. N. COOPER, AND J. R. SCHRIEFFER, Phys. Rev. 108,1175 (1957). 2. V. L. POKROVSKII, Zh. Eksperim. i Teoret. Fiz. 40, 641 (1961) [English transl.: Soviet Phys.-JEW 13, 447 (1961)]. 3. R. GEISER AND B. B. GOODMAN, Pkys. Letters 6, 30 (1963). 4. H. A. LEUPOLD AND H. A. BOORSE, Phys. Rev. 134, Al322 (1964); ibid. 137, AB3 (Erratum) (1965). 5. H. R. O’NEAL AND N. E. PRTLLIPS, Phys. Rev. 137, A748 (1965). 6. N. E. PHILLIPS, Phys. Rev. 134, A385 (1964). 7. L. Y. L. SHEN, N. M. SENOZAN, AND N. E. PHILLIPS, Phys. Rev. Letters 14, 1025 (1965). 8. B. J. C. VAN DER HOEVEN, JR. AND P. H. KEESOM, Phys. Rev. 184, A1320 (1964); ibid. 136, A631 (1964); ibid. 137, A103 (1965). 3. L. N. COOPER, Phys. Rev. Letters 3, 17 (1959). f0. B. T. GETLIKMAN AND V. Z. KRESIN, Fiz. Tver. Tela 6, 3549 (1963) [English transl.: Soviet Phys.-Solid State 6, 2605 (1964)]. if. L. C. HEBEL AND C. P. SLICHTER, Phys. Rev. 107, 901 (1957); ibid. 113, 1504 (1959). 12. L. C. HEBEL, Phys. Rev. 116, 79 (1959). 13. A. G. REDFIELD, Phys. Rev. Letters 3, 85 (1959). 14. Y. MASUDA AND A. G. REDFIELD, Phys. Rev. 126, 159 (1962). 16. Y. MASUDA, Phys. Rev. 126, 1271 (1962); I.B.M. J. Res. Develop. 6, 24 (1962). 16. N. V. ZAVARITSKII, Zh. Experim. i Teoret. Fiz. 46, 1839 (1963) [English transl.: Soviet Phys.-JETP 18,126O (1964)]; ibid. 48,837 (1965) [English transl.: Soviet Phys.-JETP 21, 557 (1965)]. 17. M. A. BIONDI, M. P. GARFUNHEL, AND W. A. THOMPSON, Phys. Rev. 136, A1471 (1964). 18. D. C. MATTIS AND J. BARDEEN, Phys. Rev. 111, 412 (1958). 19. R. W. MORSE, T. OLSEN, AND J. D. GAVENDA, Phys. Rev. Letters 3, 15 (1959); ibid. 4, 193 (Erratum) (1959). 20. P. A. BEZUGLYI, A. A. GALKIN, AND A. P. KOROLYUK, Zh. Eksperim. i Teoret. Fiz. 39, 7 (1960) [English transl.: Soviet Phys.-JETP 12, 4 (1961)]. $1. H. V. BOHM AND N. H. HORWITZ, in Proc. Eighth Intern. Conf. Low Ttimp. Phys., R. 0. Davies, ed. p. 191. Butterworths, Washington, 1963. 22. E. R. DOBBS AND J. M. PERZ, Rev. Mod. Phys. 36,257 (1964). 23. G. A. SAUNDERS AND A. W. LAWSON, Phys. Rev. 136, A1161 (1964). 24. A. R. MACKINTOSH, in “Phonons and Phonon Interactions,” T. A. Bak, ed., p. 181. Benjamin, New York, 1964.

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26. V. L. P~KRO~SKII, Zh. Ezperim. i Teorel. Fiz. 40, 898 (1961) [English transl.: Soviet Phys.-JETP 13, 628 (1961)]. 26. I. A. PRIVOROTSKII, Zh. Eksperim. i Teoret. Fiz. 42, 450 (1962) [English transl.: Soviet Phys.--JETP 16, 315 (1962)l. 27. D. MARKO\VITZ AND L. P. KADANOFF, Phys. Rev. 131, 563 (1963). 28. B. M~HLSCHLEGEL, 2. Physik 166, 313 (1959). 29. R. A. FEKRELL, 2. Physik 182, 1 (1964). 30. 1). E. MAPOTHER, Phys. Rev. 126,202l (1962). 31. C. P. SLIGHTER, “Principles of Magnetic Resonance,” p. 91. Harper and Row, New York,1963. 32. L. P. KADANOFF, Phys. Rev. 132, 2073 (1963). 33. L. P. GOR’KOV, Zh. Eksperim. i Teoret. Fiz. 34,735 (1958) [English transl.: Soviet Php--JETP 7, 505 (1958)]. Mechanics.” Benjamin, New 34. L. P. KADANOFF AND 0. A. BAYM, “Q uan t urn fitatistical York,1962. 36. L. P. KADANOFF, in Proc. 1963 RavelEo Spring School Phys. (t.0 be published). 36. J. R. SCHRIEFFER, “Theory of Superconductivity.” Benjamin, New York, 1964. 37. A. A. ABRIKOSOV, L. P. GOR’BOV, AND I. E. DZYALOSHIXSKI, “Methods of Quantum Field Theory in Statistical Physics.” Prentice-Hall, Englewood Cliffs, New Jersey, 1963. 38. M. FIBICH, Phys. Rev. Letters 14, 561 (1965). 39. J. R. SCHRIEFFER, D. J. SCALAPINO, AND J. W. WILKINS, Phys. Rev. Letters 10, 336 (1963). 40. J. BARDEEN, Phys. Rev. Letters 6,57 (1961); ibid. 9, 147 (1962). 41. M. H. COHEN, L. M. FALICOV, AND J. C. PHILLIPS, Phys. Rev. Letters 8, 316 (1962). .@. R. E. PRANOE, Phys. Rev. 131, 1083 (1963). 43. J. W. WILKINS, Ph.D. thesis, University of Illinois, 1963 (unpublished). #. W. A. HARRISON, Phys. Rev. l23, 85 (1961). 45. R. B. DINGLE, Physica 19, 311 (1953). @. P. B. MILLER, Phys. Rev. 118, 928 (1960). 47. A. B. PIPPARD, Advanc. Electron. Electron Phys. 6, 1 (1954). 48. R. W. MORSE, in “Progress in Cryogenics,” K. Mendelssohn, ed., Vol. I, p. 219. Academic Press, New York, 1959.