Theory of specific heats of dilute alloys

J. Phys.

Chem. Solids

THEORY

Pergamon

Printed in Great Britain,

Press 1964. Vol. 25, pp. 1435-1439.

OF SPECIFIC

HEATS

OF DILUTE

ALLOYS

A. D. BOARDMAN Department

of Pure and Applied Physics, Royal College of Advanced Technology,

Salford

N. H. MARCH Department

of Physics, The University, (Received

23 April

Sheffield

1964)

Abstract-Two results which appear to have relevance in the interpretation of specific heats of dilute alloys are derived. The first concerns the density of states at the Fermi level, calculated using Bloch waves in the unperturbed matrix, and treating the impurity potential to first order. The result is exact in band theory to this order. Secondly, it is pointed out that the temperaturedependent self-consistent perturbing potential should be used in the evaluation of the alloy specific heat. The magnitude of the effect is calculated for plane waves, and multiplies the usual first order result by a factor of 4/3. The effect of band structure and temperature together will evidently have to be estimated before detailed comparison with experiment can be made, but this necessitates accurate knowledge of the Dirac density matrix for the pure metal and this information is not at present available.

1. INTRODUCTION

OUR interpretation of the specific heats of dilute alloys such as Cu-Zn and Cu-Ge is still very incomplete. Following earlier work by FRIEDEL, who derived the change in specific heat to first-order for free electrons, HOUGHTON(~)and the present writers(s) worked out the theory for plane wave scattering to second-order in the impurity potential. Our own calculations then revealed that, provided in this latter case the potential is calculated self-consistently to second-order, then the firstorder result for the change in specific heat is largely regained. In addition to the assumption of plane waves these earlier arguments assumed that the perturbing potential was independent of temperature. We shall show in this note how both these assumptions can be removed. However, we can only estimate the second effect quantitatively for plane waves. We have made some tentative estimates of the effect of Bloch waves for Cu. but have found that the lack of knowledge of the Dirac density matrix in this case precludes useful quantitative work. We stress that when this knowledge becomes available, our theory should provide a basis for quantitative

calculations for Cu-Zn alloys and similar systems. We expect the theory to fail for strong perturbations, and also when the impurity has a complex electronic structure such as an incomplete d shell, as, for example in Cu-Pd. 2. DENSITY

OF STATES BLOCH

AT FERMI WAVES

LEVEL

FOR

We use the density matrix theory of MARCH and MURRAY (3a, b, c), which starts out from the canonical density matrix C(rr@) defined in terms of one-particle wave functions $4 and corresponding energy levels Ei as C(r, r0,P)

= 2 #:(r)&(r0)exp(-BE& t

(2.1)

This satisfies the Bloch equation H(r)C(r,

r0, B) = - $

(2.2)

with initial condition C(rro0) = 8(r-ro), where the one-particle Hamiltonian H(r) is such that

1435

1436

A.

D.

BOARDMAN

If V is the impurity potential, and CO is the canonical density matrix corresponding to I’ = 0, then (2.1) and (2.2) may be written in integral equation form as

C(r, ro,B) = Co(r, ro,B)- j drlV(rd i 4% 0

x Co@,r, B-&>C(rl, ro,PI).

(2.4)

To first order in V, we can replace C in the integral in (2.4) by CO and then we have

/4%

C(r, ro,B) = GO-,ro,B)-- 1 drlV(rl)

0

x C0(r, rl, B-Bl)C0(rl,

r0, PI).

out, if we go on to the diagonal in (2.5) by putting ro = r and assume that V(q) is slowly varying and can therefore be replaced by V(r), then we can use the result that drlCo(r, rl, B-Bl)Cs(rt,

From (2.8) the density of states is readily obtained by differentiating with respect to E, and hence we find

N(E) = No(E)-

j drV(r)sr,

r, PI) = COP, r, Bl)

as is easily verified from the definition (2.1) using the orthonormality of the wave functions. This leads to FRIEDEL’S@) generalized first order result. However, we emphasize here that to obtain the density of states, we can proceed from the partition function, which is the integral of C(r, r, 8) through the volume, without making any assumptions about the spatial variation of V(r). Thus, putting rs = r in (2.5) and integrating we find

Ef

We can rewrite (2.9), using the identity

using (2.8).

dr~(r)BG(r, r, B)

Transforming

(2.7)

to the particle density

p(rE) in the manner of MARCH and MURRAY@*) we find j drp(r, E) = j drpo(r, E)

-

I’

dr V(r)-f$r,

E).

(2.8)

azpo

1 =-----' Ells

aE2

l @PO 2E31s aE

(2.11)

AZ].

(2.12)

as

N(E) = No(E)-

jdrV(r)

x [Ells-&s)+

Finally, at the Fermi level, making use of (2.10), (2.12) becomes

N(Ef) = No(Ef)-

x

s

. (2.10)

Z = j dr(p-ppo)B, = - j drV(r)(g)

r, B> = j drG(r, r, B> -

(2.9)

We thought it worthwhile to record in Appendix 1 a different derivation of (2.9) using the full perturbation expression of MARCH and MURRAY@~) for the Dirac density matrix directly. This derivation also suggests an alternative form for (2.9) when we evaluate it at the Fermi surface E = Ef. The answer then comes out in terms of the charge normalization integral which, for an impurity having Z valence electrons additional to those of the matrix atom, is given by

(2.6)

j drC(r,

E).

(2.5)

As MARCH and MuaaAv(sb) pointed

1’

and N. H. MARCH

J drF’(r)E+/2

[-&(&gg

f.

c2e1 3,

It is perhaps worth noting that in Appendix 1 it is shown that the charge normalization term Z/2Ef arises very naturally, and in fact amounts to summing part of the Dirac density matrix to infinite order in V. However, the remainder, leading to the second term on the right-hand side of (2.13), is summed, of course, only to order V. Equation (2.13) may eventually prove a more favourable form than (2.12) for calculation, as it will be noted that the second term on the

THEORY

OF

SPECIFIC

HEATS

right-hand side of (2.13) is identically zero for free electrons. Unfortunately, the basic information required for evaluation of (2.9) or (2.13) is not at present available, the band structure calculations of SEGALL(~) and BURDICK@) giving the eigenvahres but not the wave functions for Cu. We shall therefore content ourselves with the remark that since, for Cu-based alloys, the specific heat results for attractive impurities such as Zn and Ge accord reasonably with a change in the density of states at the Fermi level of 2/2Ef, the other term will presumably be found to be small, when an accurate density function is available for Cu. 3. TEMPEIWTUFUZ DEPENDENCE OF V(r) Having obtained a formula valid for the scattering of Bloch waves to first order in the impurity potential, we turn to our second point; namely that it is strictly necessary to calculate the specific heat more carefully than is usually done. The customary procedure is to assume that C, is directly proportional to the density of states at the Fermi surface, and the potential V(r) is taken to have a form appropriate to T = 0. It is shown in detail in Appendix II that V(r) is temperature dependent to 0( Tz), so that the above assumption will not in general be valid. We must, instead, determine the free energy of the perturbed non-interacting electron gas and use the appropriate thermodynamic relations to obtain CV. The free-energy of an electron gas, as was pointed out by Peierls and used, among others, by SONDHEIMERand WILSON,(‘) is given by co

F-NEF

= 2

s

OF

DILUTE

ALLOYS

1437

namely AF, is m

AF=

-2

ss

r

dEdr-$V(r)pa(r,

E).

0

The evaluation of this, using Bloch waves, requires a detailed knowledge of po(r, E) which is not available at present. The effect of temperature dependence of V(r) will therefore be estimated for plane waves. Thus, to O(Tz), the change in the free energy is AE = &*

p(O)*(2E~)~‘~fl+

f(z)‘]

(3.4)

where p(O) is the zeroth Fourier component of V(r). The specific heat of the electron gas is given by

V

and the entropy S by

s=

-(g+-(-.f&.~

inwhichthelast term vanishes since (BF/~EF)T = 0. When V(r) is independent of temperature we recover the usual result that the change in specific heat ACv is

&‘kzT Z ACv = -.3 ~EF(O)

(3.5)

where EF(O) is the value of the Fermi dEP(E);

(3.1)

0

where N is the number of electrons per unit volume, EF is the temperature dependent Fermi energy, f(E) is the usual Fermi-Dirac distribution function and P(E) is related to C(r, r, /I) as follows: j drC(r, r, /3) = ,19 7 dE exp(-fiE)P(E).

It can now be shown, with the aid of equation (2.7), that the change, to O(V), in the free-energy,

energy at

T = 0. However, if V(r) is temperature dependent can be seen, using equation (A2.8) that AF=

- y(l+

f($r).

it

(3.6)

Therefore,

(3.2)

0

(3.3)

--

-

(3.7)

The ratio of two results is 413 so that a non-trivial correction arises from this effect.

A.

1438

D.

BOARDMAN

4. CONCLUSION

and

N.

H.

MARCH

where the nth term is

To first order in the localized potential, an exact band theory formula has been derived for the change in the density of states at the Fermi surface in a dilute alloy. Experimental results suggest that the correction term to the usual free-electron theory can be small, even when we have a very nonfree electron like density of states as in Cu. Rigid band arguments based on the density of states are clearly seen to be inadequate, the energy derivative of the diagonal element of the Dirac density matrix being required. Finally, we show that the usual first order specific heat, for plane wave scattering by the impurities, should be increased by a factor of four thirds when the correct screening potential at elevated temperatures is employed.

s1fJ# ( -dr17(*) )

pn(r, E) =

m-l

n+l

I

“pp4

APPENDIX Partial

summation

The electronic infinite series

n+l

m!sm with E = k2/2, + VL(T). ~G(T) nth term of density so that over all space,

ro = r,Sm = Ir,--r~~ll, VT(T) = v(p) is the periodic potential and pan(r) is the the unperturbed, periodic, electronic from (A1.2) we obtain, after integrating

f(r, E) =

can be expanded

2 fn(r, E)

?Z=O

v(r)1

- drm[VL(r)+

-drmV~(rm)

-

277

jo(klr-rnl+k

)I

f Sm> m=2

(Al .3)

fisrn m=2 By writing (Al .3) as

Apn(r, E)dr

=f$zFn(k).

(Al .4)

?a=1T

which can be regarded as defining F,(k), density of states in the alloy is

we find that the

1 N(E)

of perturbation series density

(A1.2)

X

Acknowledgements-One of us (N.H.M.) wishes to thank Dr. V. HEINE for a valuable discussion in which the interest in calculating the screened potential at T # 0 first came up. We wish also to acknowledge that this work was performed under partial contractual support from the U.S. Army through its European Research Office.

REFERENCES 1. HOUGHTONA., J. Phys. Chem. Solids 20,289 (1961). 2. MARCH N. H. and BOARDMANA. D., J. Phys. Sot. Japan, 18, Suppl. 11,80 (1963). 3. MARCH N. H. and MURRAY A. M. (a) Phys. Rew. 120, 830 (1960); (b) Proc. Roy. Sot., Lond. A261, 119 (1961); (c) PYOC. Roy. Sot., Lond. A266, 559 (1962). 4. FRIEDEL J., Advanc. Phys. (Phil. Mag. Suppl.) 3, 44 (1954). 5. SEGALL B., Phys. Rev. 125, 109 (1962). 6. BURDICK G. A., Phys. Rev. 129, 138 (1963). 7. SONDHEIMERE. H. and WILSON A. H., Proc. Roy. Sot., Lond. A210, 173 (1951). 8. MARCH N. H. and MURRAY A. M., Proc. Phys. Sot., Lond. 79, 1001 (1962). 9. GASKELL T. and MARCH N. H., Phys. Lett.7, 169 (1963).

i

into the

(AH)

=

NO(E)+

f$

&

/

b(r,

E)

dr

n=o

+c

(Al .5)

THEORY Charge normalization

zo

OF

SPECIFIC

HEATS

requires that

j &(r,

E) dr =

(A1.6)

2

OF

DILUTE

where we have considered an attractive centre of excess valency Z and f’(E) is the energy derivative of the Fermi-Dirac distribution function

1 f(E)

where Z is the excess valency of the impurity. Although the latter summation can easily be achieved, the third term in (A1.5) can only be summed to O(V) (though with no restriction on Vs(r)). Thus, after lengthy, but straightforward, manipulation of the integrals involved we obtain

1439

ALLOYS

=

1 +w{(E-EF)/kT)’

In order to cast (A2.1) into a more convenient form, we consider the following integral. cc

I = -

dEEl’zf’(E)

s

(k2-8E)ln

k-d8E

d(8E)k

Ik+2/8E

I) ’

(iQ.4)

By partial integration this readily reduces to w =-

drV(r)El~~(---&f$).

(A1.7)

I = -2/(S)

0

p(k) now become

The Fourier components The density of states, in the alloy, evaluated at the Fermi level, becomes

N(EF) = NO&F)+

(A2.5)

jdEf(E);lnl;+;;;l.

-4772 v(k)

(A2.6)

= al

&F

kz-

?- j dEf pO

This form is valid for an arbitrary temperature and is identical to the form derived by GA.%ELL and MARCH@) in an investigation of ionic screening in liquid metals.

(A1.8) The term Z/2& has thus arisen quite naturally as a partial sum of the series expansion for N(E) to infinite order in V. APPENDIX

The quantity of interest here is p(O) and this, from (A2.6), is - 1/(2)&z P(0) = co

s

(A2.7)

f (E&G2.

2

The localized potential due to an impurity of excess valency 2 in an electron gas at an elevated temperature T, can be obtained from Poisson’s equation

VW(r) = -bAp(r)+hZS(r)

0

The denominator, in the degenerate panded in powers of T so that

(A2.1)

where Ap(r) is the displaced electronic charge density, which is, as shown by MARCH and MURRAY@)

WX

= &&(I+

where EF is the temperature

case, can be ex-

z(z)‘) dependent

W.8) Fermi-level.

(A2.2) The Fourier components

of the localized potential are

v(k)

-4T.Z

=

42 k2- 7

s O”

0

6

kz-SE 2+ d(8E)kln

I

k- d8E k+2/8E

II

(-42.3)