Indian Journal of Pure & Applied Physics YoL 37, February 1999, pp. 142-149

Theory of electrical conductivity B P Bahuguna, S C Gairola", B D Indu+ & M D Tiwari""

Physics Department , H N B Garhwal University, Post Box 21, Srinagar 246 174 , Garhwal Received 14 February 1997; accepted 15 February 1999

The quantum dynamics of electrons has been investigated with the he lp of double time thermodynamic electron­Green ' s functi ons . The model Hamiltonian for the problem consists of the state of a real crystal containing phonon system (hannonic and anhannonic), electron system, localized impurities and interactions thereof. The di sorder effects include the mass and force constant changes at the impurity s ites. The life times for various scattering process have been evaluated in the new frame work . The expression for the e lectrical conductivi ty has been derived with the help of Kubo formalism . It is found that the electrical conductivity is not simply a function of electron energy, but, it depends on a large number of scattering events and other factors depending on crys tal characteristics and temperature . It has been investigated tqat the electron-phonon coupling and defects play prominent role in the electron-transport theory and hera lds the temperature dependence of electrical conductivi ty.

Inh'oduction

The electrical conductivity of metals and alloys is one of the most impOitant non-equilibrium property, which probes that how a system relaxes to its equilibrium distribution . A wealth of experimental data on electrical conductivity (or res istivity) of large number of inorganic. and organic solids have been reported during the past few decades 1-12 Some work has also been done on dis location and stacking fault-

. . 13 -18 resIstIvIty

The complete theory of electrical conductivity (or resistivity) remained an uphill task from the veTY beginning . In general, the electrical conductivity is governed by the scattering of electrons by phonons The detailed theory of scattering of electrons in a crystal which essentially limit the transport of free electrons, is in a far less satisfactory state in comparison to the theory of phonon scattering l 9

.20 The

earlier theory of electrical conductivity was based on average velocities of elect rons and Boltzmann transport equation approach on extremely simplified models for collisions between electrons and atoms using classical statistics The serious difficulties led by classical

PhYS ICS DepaJ1ment, Pauri Campus, I-I N 1:3 Garhwal University, Pauri 246 00 I , Garhwal, U P

. Physics Departmen t, Roorke.:: Un iversity, Roorke.: 247 667, U P "" Yice Chancellor, Rohilkhand Univc:rsity, Bareilly 243 006, U P

statistics were removed by Sommerfeld on the same lines USing Fermi-Dirac statistics . Instead of investigating the mechanism of lattice-electron interaction, he assumed the relaxation time as a function of electron energy only, using free electron approximation Kaveh and Witer21 modified the Boltzmann transport equation approach by including the electron-electron, electron-phonon scattering and phonon drag contribution21

. Kubo formalism was the another landmark in the field 22

, in which he presented the theory of transport coefficients and the work was \vidcly extended by several workers23-27 Despite of considerable mathematical ambiguity the results for electrical conductivity could only yield rough qualitative agreements with experiments The oversimplification of free electron model 28 and the difficulties associated with the calculation of electron scattering probability with vanous defects are formidable barriers in the progress A clue check in this line is the Bloch-T5 law, which could only envisage crude theoretical Justifications for a short period and Woods 29 reported that this law fails to give agreements with experimental values , because, T6 dependence is also found even 2,t low temperatures . T his argument was further suppOited by the work of Ekin e l al30

On the other hand, with the advent of quantum mechanics , one of thc first problems to which the solid state theoreticians turned their attention was the calculation of temperature dependent part of dc

BAHUGUNA et al.: ELECTRICAL CONDUCTIVITY 143

conductivity of pure metals, particularly at low temperature. The Boltzmann transport equation, however, is a powerful approach when it is appropriate. It is certainly not as general as the Kubo fonnalism of transport coefficients, but, difficulties arise in the evaluation of Kubo fonnula . Moreover, if one uses very abstract methods to evaluate it, one may lose sight of physical content due to the drastic approximations involved31 . Hence, one must have to deal with the problem of transport coefficients with great care.

In the present paper, the theory of electrical conductivity has been investigated with the help of quantum dynamical approach . The one electron Green 's function is obtained via a new Hamiltonian which includes the effects of electrons, phonons, their inter actions and that of randomly distributed point impurities . This Green ' s function obtained with the help of least approximations so that the physical content and generality would be maintained, is then used in Kubo fonnula to obtain the expression for electrical conductivity. This theory, investigated on the ground of many particle system, reveals that the electrical conductivity is extremely complicated phenomenon and cannot be explained with the help of simple models as discussed earlier.

2 Formulation of Problem

Following Kubo fonnalism the electrical conductivity tensor aap(eo) can be defined in tenns of current-current correlation functions as 22.23

00 p ~ ~

aap(eo)= J I dt dA. < Ja (q,t) J p (q,t + ihA.) e -iwt o 0

(I)

where

Ja(q,t) = L qa PaQ·q(~) b; o(t) bQq (t) . (2a)

Pa

Q. q (1) = (e / 2m,) f d 3 r e' ii r

[\jf~(;)Va \jfQ (r) - \jfQ(;).V a \jJ~(;)J (2b)

where q and Q = q + k are the electron states

associated with some unperturbed Hamiltonian chosen

~

as the basis for perturbation, k is phonon wave

vector,a stands for spin and b 0q (J (b q (J ) are the creation (annihilation) operators for the electrons and \jf are the wave functions of respective states . For simplicity we shall condense the indices such that qa=q and Qa= Q. More details about the derivation and symbols are available in the references else where32 Substitution of Eq(2a) in ~q . (l) develops a correlation function of the fonn :

which can be simplified by using the usual decoupling scheme33 to give the two particle correlation function as

00

= Ideo N(eo) ePhOl e - ,wt lim - 00 ('-+0"-

[Gqq . (E + iE')

- Gqq . (E - iE') ]

.(3)

where

N(co) = (e Ph Ol + lr1and Cqq , (E ± iE') is the

Fourier transfonn of the electron - Green 's function :

with a small quantity £' --; 0 +. Substitution of Eqs

(3) and (4) in Eq. (I) and evaluation of I and A. integrals in the Cesaro and Abel limits23 yields the electrical conductivity expression in the fonn

[Cqq, (E + iE') - Cqq.(E - iE')]

[Cq.q (E + iE') - Cq.q (E -iE') ]

where n (E) is Bose function and

Pup(q ,q') = Pa Q. q (q) Pl' (q)

(5)

. . (6)

144 INDIAN] PURE APPL PHYS VOL 37, FEBRUARY 1999

Eq (5) reveals that most of the physics of the problem rests in the configuration of Green's function Gqq· which, in tum, makes its evaluation essential.

3 Defect Induced Electron-Phonon Anharmonic Hamiltonian

To deal with the problem of electrical conductivity in a more realistic way we consider the many body problem which can be investigated by taking the Hamiltonian (almost complete in present context) in the fonn :

H = Hop+Hoe + Hep +HA + HD

where

phonon Hamiltonian)

. (7)

(8a)

Hoe = L qa q b;a bqa (Electron Hamdtonian) (8b)

He p = g Lq a b;a bqa Bk (Electron-phonon

Hami Itonian)

(Anhannonic Hamiltonian)

and

(8c)

(8d)

HD = L k,k, [D(kJ ,k2 ) Ak , Ak, - C (kJ ,k2 ) Bk , Bk,]

(Defect Hamiltonian)

where

Ak =ak + a:k = A: and Bk

=ak -a:k = - B~kak (a;) and bqo (b;o)

(8e)

are the phonon and electron destruction (creation) operators respectively with phonon wave vector

k (== k ) of branch index j and electron wave vector J

and g stand for the elcctron

energy, phonon cnergy and electron-phonon coupling cocfficient,respcctively . Vs (k , k2 .. .... . ks ), C (k , k2 )and D(k , k 2 )

represent the anhannonie eouplmg coeffiCients, mass change and force constant change parameters

. I 33 34 respect ive y . .

4 Electron Green's Function via Disordered Electron-Phonon Anharmonic Hamiltonian

Let us consider the evaluation of electron Green 's function Gqq.(t, (' )using equation of motion technique of quantum dynamics33

-38

. Differentiating Eq .(4) twice, with respect to time argument (via Hamiltonian given in Eq . (7) we obtain the following Fourier transfonned electron Green 's function :

27t (E 2_ E~)Cqq . (E) =(E+ Eq)Eqq.

+ « Fkq (I) ; bq. (I') »E

.(10)

where « ... » , stand for Fourier transformation and

Fkq «() =2gcq b; - g26 Qq b; B; Bk

+gckAkb; +6g L k,k ,V3 (k , , k 2 -k)

Ak , Ak , b; + 8g

L V4(k"k2,k3,-k )Ak , Ak, Ak,b; k I . k 2.k 1

+ 4g t D(k" - k)Ak, b;

(II)

The Green 's function « Fkq(t);b q·, (I ' ) » can be evaluated with the help of equation of motion method Double differentiation of it with respect to t' via Hamiltonian given in Eq . (7) yields thirty SIX higher order Green 's functions , but out of them only eight Greeri ' s functions contribute significantly and the rest either vanish identically or have negligibly small contributions . Substituting the systematically evaluated remaining Green 's functions into Eq.(lO), the equation of motion for G qq' ( t, t) can be written in the fonn of Dyson 's equation:

Gqq. (E) = Co(:::;) + Go(£) P(kqq' ,£) Go (E)

=Go (£) + Co (£) TC (kqq',E) Cqq. (E)

(12)

where

(13)

and the self-energy function is obtainable in the fonn

7r(kqq' ,£) = P(kqq',c) [1 + Go (£) P (kqq',£) ] -'

. ( 14)

}

BAHUGUNA et al.: ELECTRICAL CONDUCTIVIlY 145

with

P ( kqq , , f,) = 21t (f, - f, q ) (f, + f, q r I «Fkq (I); b.·(I'»>E

and under a reasonable approximation

.. . ( 15)

G". (E) -= (cSqq' / 2X )[E - Eq -(27tr' P(kqq' ,E) r ... (16)

The response function P(kqq',& ) can be obtained from the estimation of Green's functions involved

in«Fkq(t) ;bq• ( t' in the form :

P (kqQ, f,)= g2 NQ {(a' / NQ ) (D_q - D+q)

+ ro;q [X,(k, +q)D_k -XI (k,-q)D+ k ]

+72 L IV3 (k, +k2 , -k) 1211, .

k,.k,

+Sa,(XI (a 2 ,+q ) D_a 2 - X, (a 2 ,-q ) D+a) ]

+96 I IV4 (k,+k2 ,k3,-k) 12 k, .k, .k)

T'l 2[SIlI (X, W, ,+q)D_p, X,(J3J -q)D +p,

+ 3S~2(X, (J32,+q ) D_Il, - X, (J32,-q) D+p,) ]

,'. 64 rr L 1 D(k, ,-k) 12 .,

.. . (17)

In Eq. ( 17) the various symbols used are derivable in the form :

a'= aJ2Eq(Eq + 1) .. . ( lSa) .

a=6!n. +2&,€.n.+46 2 n.+72 L 1V,(k"k, ,-k)12 2nk nk q *1 11 1 1

... (ISb)

" =< Ak Bk > ; nk = < Bk Bk >

... (lSc)

R,(k,q) = 4f,q {f,;qX2 (k, -q)

+72 L IV3(k" k2,-k )12

11 ,[So., X3(a, ,q) k I . k2

112 [Sill X2 (J3"q)+3SIl , X 2(J32,q)]

+ 641t L I D(k,,- k) 12 X2 (k, ,q)} k,

R2 (k ,q) = a g / NQ + 2X3 (k,q)

... (lSd)

+72 L IV3(k" k2, - k )12

11,[So., X 3(a, ,q) k ,. k2

+ 96 L IV4 (k"k2 ,k3,-k) 12

112[SIl, X l (J3 " q) K,. k, . k}

+ 3Sp X3 W2, q)] + 64rr I 1 D(k, ,- k) 12 X3 (k , ,q) , k ,

. (lSe)

. (lSf)

.. . (lSg)

X 2 (i ,q)= '~ ; (f, ~ -f,! r2; i =k, a 1,a 2 ,J3 J> J3 2

X3 (i,q)= (&;2 + &:) &,-' (&;2 - &: r'; i = k, a" a2 , A ,fJ 2

.. . (lSh)

146 INDIAN] PURE APPL PHYS. VOL 37, FEBRUARY 1999

C UI = ck i ± Ek 2

a2

... (l8i)

±Ek 3

... (l8j)

S =nk ± n k ; S =1 ± nk, nk, +nk ,nk, ±nk, nk, 0., ' '13, U2 P2

... (l8k)

... (181)

... (l8m)

where &; describes the renormalization energy. The

response function P(kqq I , &) describes the shifts f.. ep

(kqq I , &) and widths r.p(kqq I , &) for electron and phonon energies as

I P(kqq' ,E) = 271: P(kqq' ,E +i8)

= !1e/kqq',E)- j r ep (kqq' ,E); 8~o t

(19)

The shift can be obtail1led from the principal value of P(kqq I , &) and the width is obtainable in form •

r.p (kqq',E)

== NQ g 2 {NQI U'[8(E - Eq ) -8(E - Eq) ]+ ffi~q }

+ 72 I k ,. k, 2

8 (E - Eo.,) - XI (a l ,-q) 8 (E + Eo., »

+ S o. ,(X , (a 2 ,+q) 8 (E - Eo.)

- X 2 (a 2 ,- q) 8(E + E0.2» ]

+96 I IV4 (kl , k2, k3,-k)I\1 2 [S13, (XI )(~I, + q) k ,.k, . k,

8(£ - £p)-X, Cf3 ,,-q) 8(£ + Ep))

+3Sp, (X, (13 2 ,+ q)

8(£ -£13, )-- XI 03 2 ,-q) 8(E + E13,))]

+6471: L I ~k"k)12[X, (k,,+q)8 (E- Ek ) Y. I I

- X, (k, ,-q"/J (E + Eji)]+R(k ,q)8(E + £q) }

-. (20)

where

R(k,q) = R,(k,q)+ R2 (k,q) .(21)

The last term in Eq . (20) is obtained under a reasonable algebraic manipulation. Th,e use of Eq. (19) in Eq.(l6) yields.

C;q. (E ) = [ ] 21t E -Eq +i r ep + ( kqq' ,E )

. (22)

where Eq is the perturbed mode energy given by

. (23)

5 Electrical Conductivity

Substituting Eq . (22) in Eq .(5) ,the expression for electrical conductivity can be obtained in the form .

... (24)

This conductivity integral cannot be evaluated exactly. Under a reasonable and widely accepted approximation known as Breit-Wigner approximation, let us peak the expression in the vicinity of & = "'i q to

get .

Gap (E) = L PaP (q, q ') (nE- I)

q

eP<ql 2 tanh (P £~ /2) r,-~ (kqQ,E )

.. (25)

The width r ep (kqQ, E) is the measure of the life

times of various scattering events and is related to the relaxation time t by the result.

BAHUGUNA et af.: ELECTRICAL CONDUCTIVITY 147

... (26)

6 Discussion and Conclusions

The present formulation of electrical conductivity reveals that the electrical conductivity tensor decisively depends on electron-phonon scattering line

width fep (kqQ,E) . None of the previous workers

has described the electron-phonon scattering as comprehensively as in the present paper, because, it describes the problem of scattering of electrons in various phonon fields, namely : localized phonon fields, cubic and quartic anharmonic phonon fields, etc. In the limit of no electron decay (fep -->o ) the Breit-

Wigner function

f e p (kqQ, e) / [( e - E: q ) 2 + fe p (kqQ, e ) ]

becomes approximately equal to 7r t5 (e - E: q) which

shows that a ap (e) becomes finite and independent of

f e p (kqQ, e ) .

Now, let us consider the term f e p (kqQ, e) in

more details . Eq.(20) exhibits that the electron-phonon coupling coefficient g appears throughout in all the constituent terms which has never been observed in any of the earlier formulations and underlines the importance of coupling phenomena of electrons with phonons . Remarkably, the dectron-phonon line width f e p (kqQ, e) described here is solely different

from earlier ones, because it is a sum of a large number of possible scattering events, which of course occur in the crystal and are accorded in the quantum dynamics of electrons . Let us rewrite f e p (kqQ, e) In

the following form , for better understanding :

f ep (kqQ , £)=I,o +lpo+I ),P + I- .,p + I ' D

.. (27)

r. o = a' g2 [S(£ - £q) +(R(k ,q) -1) S(£- £q) ]

(28a)

rpo= NQ g 20)~q

[ (ck + c q r 2 &(c - ck) - (ck + c q r 2 &(c +ck) ]

.. (28b)

r3•p = 72N Q g2

L I V3(k l ,k2 ,_k)12

111 {SaJ(E k + Ekl + Eq r 2

Kl *) I

x 8 (E - Ek , - Ek)

- (Ek, -Ek1 - Ekqf2 8 (E +Ek , +Ek) .J

+ Sn,[(Ek , - Ekl - Eqf2 8 (E - Ek , - Ek )

- (E - E - E )-2 k, kl q

. .. (28c)

r •. p =96NQq2

k, I ,k,iv• (k"k2> k) -k)i\1 2 {Sp, [( Ek , + Ek , + EE, + Eqr2

and

-( '& k - '& k - '& k - eq ) - 2 , 2 )

[(E k,+ Eqf2 8(E

8 (E + Ek)' ]

... (28d)

... (28e)

In expressIOns Eq.28e f eo provides the

contribution to the energy width at electron energies £=

± £q This contribution has the magnitude g2a / 2eq (eq

+ I) for & (£- £q) peak and l a R(k,q)- 1/2 £q (c'q +l) for & (£-£q) and both are temperature dependent via a

and R(k.q) . The temperature dependence of f eo at

energy £=£q roughly appears to be of the form f eo

-AIT+A2 r + A3 r . where Ai are some p~rameters Obviously, the study of a and R(k.q) reveals that the cubic and quartic anharmonicities alongwith defects contribute significantly to the electron energy at £=£q .

148 INDIAN J PURE APPL PHYS. VOL 37, FEBRUARY 1999

r po describes the electron-phonon energy exchange at

c= ± Gk with its temperature dependence via N Q which is given by :

N Q =- 2fN(&) 1m G (& + i &) d& ... (29) QQ'

The cubic and quartic anharmonic fields exchange the phonon energies with the electrons with the energy equivalences g =±(Ek ± £k ) and :s =±(Ekl ± Ek ± Ek )n

1 2 2 J

ces with strong temperature dependence N Q Sa; and N Q Si3, ' respectively. Hence, the terms

r3ep and r4 ep describe the role of electron in the

cubic and quartic anharmonic phonon fields. The variation of temperature dependent terms

N Q Sa; and N Q Sp; appears to be :

N Q S OJ ::= ei3E Q

[1+ 2 n~1 (e - nDE

12/2 ± enPEk

, /2 ) ]

low temperature limit ... (30a)

dependence via ~p (kqq' , G) [Eq.(23)].These contributions are found maximum around

g = Ek ± Ek and g = Ek ± Ek ± Ek I 2 I 2 3

The term r eD describes the contribution to electrical conductivity due to the force constant changes due to the substitutional impurities . This exhibits the temperature dependence via NQ and is highly sensitive to force constant changes and electron­phonon coupling constant.

It emerges from present study that the electrical conductivity of crystalline solids containing isotopic impurities can be studied with the help of defect induced electron-phonon anharmonic Hamiltonian . The concept that the electrical conductivity is derivable by simply considering the electron Hamiltonian is not well justified. It is also inferred that the electrical conductivity is not merely a function of electron energies but it depends on a large number of factors , namely; Fermi energy, nature of Fermi surface, phonon energies in the form of electron and phonon energies and their combination bands, temperature, defect concentration and electron-phonon coupling coefficient.

References

high temperature limit

NQSfJ' ::= efJeQ{I ± fJ 2

... (30b) Kos J F, Can} Phys , 51 (1973) 1602.

&k, 6\'[ 1- fJ (Ck, + Ck2 ) r [1 - P (Ek2 + EkJ] -1

[1- P (Ek' + EkJ r } low temperature limit .. (3 Oc)

::=4(fJe;" &k2

e;,J-1 &Q (1- fJ&Q)

high temperature limit ... (30d)

Eq. (30) reveals that the contribution of anharmonicities to electrical conductivity cannot be ignored in either of low or high temperature. Further, the presence of perturbed mode energies CI in above equations also enters' the implicit temperature

2 Gugan D, Proc R Soc, A325 (1971) 223 .

3 Bass J, Adv Phys, 21 (1972) 431.

4 Cimberle M R, Babel G & Rizzuto C, Adv Phys, 23 ( 1974) 639.

5

6

7

Pratt W P, Lee C W, Hearle M L, Heinen V, Bass J, Rowlands J A & Schroeder P A, Physico, 108B (198 1) 863 .

Barnard B ~ & Caplin A D, Commn Phys, 2 ( 1977) 223.

Uher C, Khoshnevisan M, Pratt W P Jr & Bass J, } Low Temp Phys, 36 ( 1979) 539.

8 Sinvani M, Greenfield A J, Bergmann A, Kaveh M & Wiser N,} Phys F.Metal Phys, II (1981) 149; ibid, II (198 1) L73 .

9 Sudharshanan R & Clayman B P, Phys Status Solidi (b), 128 (1985) 329.

10 Levy B, Sinvani M & Greenfield A J, Phys Rev Lett , 43 (1979) 1822.

II Rowlands J A, Duvvury C & Woods S B, Phys Rev Lett, 40 ( 1978) 1201.

12 Van Kempen H, Lass J S, Ribot J H J M & Wyder P, Phys Rev Lett, 37(1976) 1574 .

13 Kagan Yu & Zhemov A P, Sov Phys }ETP, 33 (i 97 1) 990.

14 Engguist H L & Grimvall G, Phys Rev B, 21 ( 1980) 2072.

,

BAHUGUNA et af.: ELECTRlCAL CONDUCTIVITY 149

15 Tremblay A M, Patton B, Martin P C & Maldague P F, Phys Rev A. 19 (1979) f721 .

16 Brown R A, Can J Phys, 60 (1982) 766.

17 Gantmakher VF, Gasparov V A, Kulesko G I & Matveev V N, Sov Phys JETP, 36 (1973) 925 .

18 Rowlands J A & Woods S B, J Phys F, 5 (1975) LIOO ; ibid, 8 (1978) 1929.

19 Grimvall G, The electron-phonon interaction in metals (North Holland, Amsterdam), 1981, Ch 4& 8.

20 Rosenberg H M, Low temperature solid state physics (Clarendon Press, Oxford), 1965, Ch4 .

21 Kaveh M & Wiser N, Phys Rev Lett, 26 (1971) 635; ibid 29 (1972) 1374; Phys Rev B, 9 (1974) 4042.

22 Kubo R, J Phys Soc Japan , 12 (1957) 570.

23 Fetter A L & Walecka J D, Quantum Theory of Many Particle Systems (McGraw Hill, New York), 1971 .

24 Velicky B, Kirkpatrick S & Ehrenreich H, Phys Rev, 175 (1968)747.

25 Velicky B, Phys Rev, 184 (1969) 6 14.

26

27

28

29

30

31

32

33

34

35

36

37

38

Hoshino K & Watabe M, J Phys C:Solid State Phys. 8 (1975) 4 193 .

Gallinar J P, Phys Rev B, 17 (1978) 807.

Ashcroft N W & Mermin N D, Solid State Physics (Holt , Rinehart & Winston, New York, 1976).

Woods S B, Can J Phys, 34 (1956) 223 .

Ekin J W & Maxfield B W, Phys Rev B, 4 (1971) 4215 .

Peierls R E, Transport Phenomena, eds J Ehlers, K Hepp & H A Weidenmiller (Springer, Berlin), 1974, p. 2.

Mahan G D, Many particle physics (Plenum Press, New York) 1971 , p. 192 .

Indu B D, Int J Mod Phys B,4 (1990) 1379.

Painuli C P, Bahuguna B P, Indu B D& riwari M D, Int J Theor Phys , 31 (1992) 81.

Kstura S & Horiguchi T, J Phys Soc Japan , 25 (1968) 60.

Mavroyannis C & Pathak K N, Phys Rev, 182 (1969) 872.

Sharma P K & Bahadur R, Phys Rev B, 12 (1975)1522 .

Sahu D N & Sharma P K, Phys Rev B.28 (1983) 3200.