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Volume 99A, number 8 PHYSICS LETTERS 19 December 1983 COMMENTS ON THE W1DOM HALL EFFECT CURRENT FORMULA J. HAJDU and Ute GUMMICH Instimt J~ir Theoretische Ph)'sik, Universitdt zu Kdln, D-5000 Cologne 41, West Germany Received 12 September 1983 Revised manuscript received 18 October 1983 Recently Widom has derived a formula lor the Hall current. Ihe aim of the following comments is to claril} the re- lation between Widom's formula and the usual one. Widom [1] defines the Hall current by j = curl M, (1) where M is the total magnetization of the system con- sidered per unit volume (respectively unit area). Assum- ing that M depends on the space coordinates only via the chemical potential j = E x a, (2) a = e(OM/O~)T,B = e(On/OB)T,~. (3) Here E = grad ~'/e is the electric field, B is the magnetic field, -e (e > 0) is the electron charge and n is the elec- tron density. In (3) a Maxwell relation has been utiliz- ed which holds for the total magnetization. The con- ductivity corresponding to (2) and (3) is Oxy : - o),x = e (On/OB) T. ~ , (4) when B = (0, 0, B) is chosen. In order to see the relation between (4) and the "usual" Hall conductivity which is in the most simple cases Oxy = - en/B , (5) we consider a system of electrons interacting with ran- domly distributed impurities. Assuming, in the thermo- dynamic limit, macroscopic rotational and translational invariance, the impurity averaged equilibrium distribu- tion function (J(H)) I will be diagonal in the Landau representation Is), c~ = (n, ky, kz), and independent of the quantum number k v selected by the gauge A = (0, 8x, 0). 396 0.03 n = Vol 1 Tr (f(H)) I _ eB ,~[~<c~l(,t(H)>ii~> ' (6) 2nLz kz (using the usual Landau counting of degenerate states). Thus, (On/OB)T,~ = ,z/B + (OML /O~)T,B , (7) where M L is the average of the magnetic moment over the Landau states, M L = Vo1-1 ~ (oeI(f(H)(-OH/OB)) I I&). O~ (8) Eq. (7) can also be obtained from the relation M = -S2/B + ML, (9) (pointed out by Zyryanov [2], cf. also refs. [3,4] ), where gZ is the thermodynamic potential. Thus, in order to obtain (5) from (1) we have to change the sign and to subtract curl M L. To see the physical content of this operation let us consider free electrons. In this case M is the magnetiza- tion corresponding to the Landau diamagnetisnr of free electrons. This phenomenon is due to the incona- plete compensation of bulk and surface contributions and vanishes in the classical limit hco -+ 0 (co = eB/m) is the cyclotron frequency). This indicates that (l) does not yield the usual Hall conductivity which is finite in the classical limit and given by (5). 1-9163/83/0000-0000/$ 03.00 © 1983 North-Holland

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Page 1: Comments on the widom Hall effect current formula

Volume 99A, number 8 PHYSICS LETTERS 19 December 1983

COMMENTS ON THE W1DOM HALL EFFECT CURRENT FORMULA

J. HAJDU and Ute GUMMICH Inst imt J~ir Theoretische Ph)'sik, Universitdt zu Kdln, D-5000 Cologne 41, West Germany

Received 12 September 1983 Revised manuscript received 18 October 1983

Recently Widom has derived a formula lor the Hall current. Ihe aim of the following comments is to claril} the re- lation between Widom's formula and the usual one.

Widom [1] defines the Hall current by

j = curl M , (1)

where M is the total magnetization of the system con- sidered per unit volume (respectively unit area). Assum- ing that M depends on the space coordinates only via the chemical potential

j = E x a , (2)

a = e (OM/O~)T ,B = e ( O n / O B ) T , ~ . (3)

Here E = grad ~'/e is the electric field, B is the magnetic field, - e (e > 0) is the electron charge and n is the elec- tron density. In (3) a Maxwell relation has been utiliz- ed which holds for the total magnetization. The con- ductivity corresponding to (2) and (3) is

Oxy : - o), x = e (On/OB) T. ~ , (4)

when B = (0, 0, B) is chosen. In order to see the relation between (4) and the

"usual" Hall conductivity which is in the most simple

cases

Oxy = - e n / B , (5)

we consider a system of electrons interacting with ran- domly distributed impurities. Assuming, in the thermo- dynamic limit, macroscopic rotational and translational invariance, the impurity averaged equilibrium distribu- t ion function (J(H)) I will be diagonal in the Landau representation Is), c~ = (n, k y , kz), and independent o f the quantum number k v selected by the gauge A = (0, 8x, 0).

396 0.03

n = Vol 1 Tr ( f (H) ) I

_ eB ,~ [~<c~l ( , t (H)>i i~> ' (6) 2 n L z k z

(using the usual Landau counting o f degenerate states). Thus,

(On/OB)T,~ = ,z/B + (OML /O~)T,B , (7)

where M L is the average o f the magnetic moment over the Landau states,

M L = Vo1-1 ~ (oeI(f(H)(-OH/OB)) I I&). O~

(8)

Eq. (7) can also be obtained from the relation

M = - S 2 / B + ML, (9)

(pointed out by Zyryanov [2], cf. also refs. [3,4] ), where gZ is the thermodynamic potential. Thus, in order to obtain (5) from (1) we have to change the sign and to

subtract curl M L. To see the physical content of this operation let us

consider free electrons. In this case M is the magnetiza- tion corresponding to the Landau diamagnetisnr of free electrons. This phenomenon is due to the incona- plete compensation o f bulk and surface contributions and vanishes in the classical limit hco -+ 0 (co = e B / m )

is the cyclotron frequency). This indicates that ( l ) does not yield the usual Hall conductivity which is finite in the classical limit and given by (5).

1-9163/83/0000-0000/$ 03.00 © 1983 North-Holland

Page 2: Comments on the widom Hall effect current formula

Volume 99A, number 8 PHYSICS LETTERS 19 December 1983

In order to show that in (9) the first term is the sur- face contribution and the second the bulk contribution we adopt the reasoning of Teller [5] (cf. also ref. [4]) and consider a semi-finite system by imposing vansish- ing boundary conditions on the wave functions at x

1 = +- ~ L x and periodic boundary conditions in the y and z directions. Expressing now the magnetic moment

M = - l e r × o (10)

by the center and relative coordinates o f the cyclotron orbit according to Kubo et al. [6],

x = X + ~ , y = Y + ~ , ~ = m l 2 v y ,

7? = - ml2v x , (11)

12 = 1/eB, we obtain

M = - (e/Vol) <Xv d ) - (e/Vol) (~@), (12)

where ( ) denotes averaging with respect to the distri- bution function f (H) and v d and n and off-diagonal parts o f th)e vj, are the diagonal velocity Vy in the energy representation. The diagonal elements o f v y are

(a lvy ta) = (a l ag /aPy la ) = a (a I g l a ) / a k y

= a%/aky = - 1 2 a % / a x . (13)

For a system infinite in the x direction e a are the de- generate Landau levels enk z and (aloy l a) vanishes. Since, for a finite system, this quantity is nonvanishing in a thin layer at X = +- ½L x we denote the first term (12) by M s:

M s - Vol ~ X ( - 1 2 aea/OX)fa

_ el 2 ~ ~" - ~ a j . a e J a x = - a / B . (14)

In deriving (14) we have used ky = - l 2 X , e a = e nkz ( X )

= 0 for X -+ + ~, and

g2 = - ~-1 ~ ln[1 + exp/3(f - %)1 • (15) a

In the calculation of the second term in (12) - which we denote b y M b - we can replace ~ by its off-diagonal part ~n = m l 2 @ and use the Landau wave functions

(cf. ref. [5] ) for which the only non-vanishing matrix elements are

(c~[V.vl~ + 1) = 2 - 1 / 2 / c o ~ + 1,

(a i ry Is - 1) = 2 -1/2 l t v ~ , (16)

[c~ -+ 1 = (n +- 1, k y , k z ) ] . Consequently

Mb - V-ol f~ enk z /B = M L . (17)

Eqs. (12), (14) and (17) establish (9). Thus, the quan- tity we have to subtract from - curl M, in order to ob- tian the usual Hall conductivity (5) is the contribution o f the bulk term (17). In the classical limit (OML/

Of)T,B approaches - n / B and [by (7)] (On/aB)r ,~ vanishes.

In the case o f a 2d electron impurity system show- ing quantum Hall effect the energy spectrum consists o f Landau bands (broadened Landau levels). If, at zero temperature, the Fermi energy lies in a band gap, (OML/ Of)T,B vanishes [cf. (8)] and consequently

(On/OB)r,¢ = (n/B) (e F in a band gap). (18)

Eq. (1) can also be seen from a different point of view. It can be shown that in local thermal equilibrium the average current density is given by (1). We shall, therefore, label the corresponding conductivities by a subscript l. The local thermal equilibrium state is char- acterized by a density operator p l which is nonstation- ary, [H, pl] 4= O, and which gives rise to an inhomoge- neous density n(r) = Tr { J n ( r ) } . In practice, however, one is usually interested in a stationary state with ho- mogeneous distribution. It is possible to find, for an isolated system, an asymptotic stationary density oper- ator which develops from a local equilibrium state im- posed as initial condition [7]. It has been demonstrated by several authors [3,8,9] that in this state the average current density is given by

] = curl M + E × a i , (19)

where E* = grad ~'*/e, f* = f - e~ being the electro- chemical potential and the components o f a i are the isolated Hall conductivities given by the Kubo formula, a i (Oiyz, i i = Ozx, exy ). If eq. (19) is inserted into the for- mula for the macroscopic current density which ap- pears in Ohm's law in Maxwell's equations,

397

Page 3: Comments on the widom Hall effect current formula

Volume 99A, number 8 PHYSICS LETTERS 19 December 1983

J = ] - curl M, (20)

the local equilibrium contribution to (19) drops out; the local equilbrium current density is exactly com- pensated by the current associated with the discontin- uity of the magnetization at the surface of the system [2,3 ,8-12] . [By this reason (1) is often refered to as diamagnetic surface current.] Thus, from this point of view (1) is an auxiliary quantity with no direct ex- perimental significance.

The quantity (On/OB)r , ~ appears also in a version of the Kubo formula derived by St~eda [13],

a iy = atxy + °Ilxy , (21)

where

alxl), = - e ( On/3B )T , f . (22)

For free electrons

aIy = e (OML/Of )T , B . (23)

Eqs. ( 2 1 ) - ( 2 3 ) and (7) establish (5) as the Hall con- ductivity of an isolated system of free electrons.

We are indebted to Dr. P. Fazekas for stimulating discussions.

Re fe rences

[1] A. Widom, Phys. Lett. 90A (1982) 474. [2] P.A. Zyryanov, Soy. Phys. Solid State 6 (1965) 2853. [3] P.A. Zyryanov and V.I. Okulov, Soy. Phys. Solid State

7 (1966) 1411. [4] M. Heuser and J. Hajdu, Z. Phys. 270 (1974) 289. [5] E. Teller, Z. Phys. 67 (1931) 311. [6] R. Kubo, H. Hasegawa and N. Hashitsume, J. Phys. Soc.

Japan 14 (1959) 56. [7] S. Nakajima, Prog. Theoret. Phys. 20 (1958) 948. [8] V.G. Bar'yakhtar and S.V. Peletminskii, Soy. Phys. J ETP

21 (1965) 126. [9] S.'V. Peletminskii, Sov. Phys. Solid State 7 (1966) 2157.

[10] L. SmrEka and P. Sffeda, J. Phys. C10 (1977) 2153. [ 11 ] P.S. Zylyanov and G.l. Guseva, Sov. Phys. Usp. 11

(1969) 538. [12] P.S. Zyryanov and M.I. Klinger, Quantum theory of

electronic transport phenomena in cristalline semicon- ductors (Nauka, Moscow, 1976) ch. IX (in russian).

[13] P. St~eda, J. Phys. C15 (1982) L717.

398