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Hall conductivity of amorphous semiconductors in the random phase model
JOURNALOF NON-CRYSTALLINESOLIDS6 (1971) 329--341 © North-Holland Publishing Co.
H A L L C O N D U C T I V I T Y OF A M O R P H O U S S E M I C O N D U C T O R S IN THE RANDOM PHASE MODEL LIONEL FRIEDMAN* Cavendish Laboratory, University of Cambridge, Cambridge CB2 3RQ, England Received 26 April 1971 Kubo's formula for the electrical conductivity is used to evaluate the Hall conductivity in the Random Phase Model of Amorphous Solids appropriate to the diffusive, short mean free path regime. The Hall mobility is found to be small in magnitude, temperature insensitive, and to exhibit the same anomalies in sign as found in the localized regime: namely, negative for a non-degenerate distribution of holes on electrons due to diffusive motion about a closed path of three sites.
1. Introduction The nature of the conducting state above the mobility edge of amorphous solids has been a matter of considerable speculation. In this regime, the crystal wave function is extended, though the mean free path is the order of a lattice spacing. There is likely to be cellular disorder t), in which the local energies of the individual atoms are spread over a wide range, and topological disorder characterized by random variations in bond lengths and orientations 1). Since the solution of Schrtidinger's equation for this case is prohibitive, various plausible models have been proposed for this regime. One such model is the R a n d o m Phase Model (RPM) suggested by Cohen 2) and Mott3). Here, the crystal wavefunction is represented as a linear superposition of atomic wavefunctions with coefficients which have no phase correlation from one site to the next. This model has been used by Hindley 4) to calculate electrical conductivity from the K u b o - G r e e n w o o d formula. His result is in agreement with that of Mott3). In the present paper, K u b o ' s formula for the antisymmetrical part of the electrical conductivity is evaluated in the same approximation in order to obtain the Hall mobility in this regime. An expression [c.f. (29)] is obtained of comparable validity to the ordinary conductivity. The Hall mobility is small, less than the drift mobility #o calculated in the same model, and temperature insensitive. In addition, for the case of three-fold local coordi* Science Research Council Visiting Fellow. Permanent address: RCA Laboratories, Princeton, New Jersey, U.S.A. 329
330
LIONEL FRIEDMAN
nation, the sign of the Hall effect for holes is expected to be of the same sign as for electrons, just as in the localized regime. These predictions are in agreement with the bulk of the experimental data on the Hall effect in amorphous solids 5), for which explanations based only on energy band and Fermi surface concepts 6), or two band models 7), have been proposed. 2. Basic H a m i l t o n i a n in the R P M
In the RPM, the basic states are taken as a superposition of local atomic (or Wannier) states Ix) = ~ a~. In>, (1) n
where the absence of phase correlation among the amplitudes aK. will be used later. Since the RPM states are extended, the electronic coupling between sites must be retained in zeroth order; rather, it is the modification of the Hamiltonian due to the external electric and magnetic fields which is the perturbation. Thus, in the absence of applied fields, the matrix elements of the Hamiltonian are (2) = ~ a,,,,, * a,,, , ?1', n
where the matrix elements in the local atomic basis may be written (n'l H ° In) = (Eo + W.) 6.,. + J~,°)(1 - 6.,n).
(3)
Here, Eo is the eigenvalue of the isolated atom. The quantity I4I. is the expectation value of energy at the nth atomic site due to the neighbouring external potentials. These quantities may be expected to be spread over a range of energies. The transfer integrals _.,,.t ~°) are taken to be different from zero only between neighbouring sites, and equal to a single constant, - J . Thus, the model takes into account only cellular disorder and is the same as the model of Anderson 8). Thus we have <~'1 H ° Ix> = Z
a,.,. * a~, { ( E 0 + Wn)(~n'n + J('On)(1 - (~n'.)}
n', n
= 8x ~x'x = 8x ~
n t, n
a,,,., * axn
~n'n"
The diagonalization of this system gives the eigenvalues ~. The existence of a solution in p r i n c i p l e is sufficient for the purposes of the present paper. As has been discussed in some detail 9), the effect of an applied magnetic field is to modify the phases of the transfer integrals according to the formula j~,n) = j~no)exp (i~,,,),
(4)
331
HALL CONDUCTIVITY IN THE RANDOM PHASE MODEL
where ~.,. = -
e
hc
H . ½ In x n ' ] .
(5)
The effect of an applied electric field is simply to shift the energy at the nth site by eE. n. Thus, the matrix elements of the perturbed Hamiltonian AH = H - H (°) are
= 2 a~,,,* a~. t , - . ' . t ( - l ("n ) -.#, ° )}' b (-l - f . ' , )'+ e "E ' n 6 ,
.
(6)
n', n
We shall also need the matrix elements of the velocity operator in the local site representation in the presence of applied fields. Defining the velocity operator as the commutator of the Hamiltonian with the position operator 10), we find i
H A L L C O N D U C T I V I T Y OF A M O R P H O U S S E M I C O N D U C T O R S IN THE RANDOM PHASE MODEL LIONEL FRIEDMAN* Cavendish Laboratory, University of Cambridge, Cambridge CB2 3RQ, England Received 26 April 1971 Kubo's formula for the electrical conductivity is used to evaluate the Hall conductivity in the Random Phase Model of Amorphous Solids appropriate to the diffusive, short mean free path regime. The Hall mobility is found to be small in magnitude, temperature insensitive, and to exhibit the same anomalies in sign as found in the localized regime: namely, negative for a non-degenerate distribution of holes on electrons due to diffusive motion about a closed path of three sites.
1. Introduction The nature of the conducting state above the mobility edge of amorphous solids has been a matter of considerable speculation. In this regime, the crystal wave function is extended, though the mean free path is the order of a lattice spacing. There is likely to be cellular disorder t), in which the local energies of the individual atoms are spread over a wide range, and topological disorder characterized by random variations in bond lengths and orientations 1). Since the solution of Schrtidinger's equation for this case is prohibitive, various plausible models have been proposed for this regime. One such model is the R a n d o m Phase Model (RPM) suggested by Cohen 2) and Mott3). Here, the crystal wavefunction is represented as a linear superposition of atomic wavefunctions with coefficients which have no phase correlation from one site to the next. This model has been used by Hindley 4) to calculate electrical conductivity from the K u b o - G r e e n w o o d formula. His result is in agreement with that of Mott3). In the present paper, K u b o ' s formula for the antisymmetrical part of the electrical conductivity is evaluated in the same approximation in order to obtain the Hall mobility in this regime. An expression [c.f. (29)] is obtained of comparable validity to the ordinary conductivity. The Hall mobility is small, less than the drift mobility #o calculated in the same model, and temperature insensitive. In addition, for the case of three-fold local coordi* Science Research Council Visiting Fellow. Permanent address: RCA Laboratories, Princeton, New Jersey, U.S.A. 329
330
LIONEL FRIEDMAN
nation, the sign of the Hall effect for holes is expected to be of the same sign as for electrons, just as in the localized regime. These predictions are in agreement with the bulk of the experimental data on the Hall effect in amorphous solids 5), for which explanations based only on energy band and Fermi surface concepts 6), or two band models 7), have been proposed. 2. Basic H a m i l t o n i a n in the R P M
In the RPM, the basic states are taken as a superposition of local atomic (or Wannier) states Ix) = ~ a~. In>, (1) n
where the absence of phase correlation among the amplitudes aK. will be used later. Since the RPM states are extended, the electronic coupling between sites must be retained in zeroth order; rather, it is the modification of the Hamiltonian due to the external electric and magnetic fields which is the perturbation. Thus, in the absence of applied fields, the matrix elements of the Hamiltonian are (2)
where the matrix elements in the local atomic basis may be written (n'l H ° In) = (Eo + W.) 6.,. + J~,°)(1 - 6.,n).
(3)
Here, Eo is the eigenvalue of the isolated atom. The quantity I4I. is the expectation value of energy at the nth atomic site due to the neighbouring external potentials. These quantities may be expected to be spread over a range of energies. The transfer integrals _.,,.t ~°) are taken to be different from zero only between neighbouring sites, and equal to a single constant, - J . Thus, the model takes into account only cellular disorder and is the same as the model of Anderson 8). Thus we have <~'1 H ° Ix> = Z
a,.,. * a~, { ( E 0 + Wn)(~n'n + J('On)(1 - (~n'.)}
n', n
= 8x ~x'x = 8x ~
n t, n
a,,,., * axn
~n'n"
The diagonalization of this system gives the eigenvalues ~. The existence of a solution in p r i n c i p l e is sufficient for the purposes of the present paper. As has been discussed in some detail 9), the effect of an applied magnetic field is to modify the phases of the transfer integrals according to the formula j~,n) = j~no)exp (i~,,,),
(4)
331
HALL CONDUCTIVITY IN THE RANDOM PHASE MODEL
where ~.,. = -
e
hc
H . ½ In x n ' ] .
(5)
The effect of an applied electric field is simply to shift the energy at the nth site by eE. n. Thus, the matrix elements of the perturbed Hamiltonian AH = H - H (°) are
.
(6)
n', n
We shall also need the matrix elements of the velocity operator in the local site representation in the presence of applied fields. Defining the velocity operator as the commutator of the Hamiltonian with the position operator 10), we find i
--h J"~"') R,.,, (7) where
Rnn, =
n
-
n'
is the intersite vector. The basic approach of the present paper is an extention to the RPM of a previous calculation 1°) of the Hall mobility in an appropriate local site (small polaron) representation. The starting point is the Kubo-type formula for the frequency dependent electrical conductivity tensor
e2h a,j (co) = iQl/[~-ij (f2) - ~ , j (0)],
(8)
where V is the sample volume, £2---h (to + is), and s is a positive infinitesimal. ~'ij (O) is the analytic continuation to the complex Q-plane of the thermodynamic velocity correlation function
~ , j (hto,) =
f
1
du exp (hto, u) Z Tr { e x p ( - H (fl - u)) v, e x p ( - Hu) vj}, (9)
where
fl = 1/kT, co, = 2nir/flh,
r integral,
and z = Tr {exp ( - fin)}.
The equivalence of (8) to the standard Kubo formula is established in Ap-
332
LIONELFRIEDMAN
pendix B of ref. I0. It is therefore also equivalent to the Kubo=Greenwood formula as shown in ref. 4. 3. Electrical conductivity
We first calculate the electrical conductivity and show that it agrees with the results of Mott 3) and Hindley4). Setting i = j = x in (9), taking the trace in the RPM and replacing H by H 0 in the exponential operator, we get 1
Z Tr { e x p ( - (fl - u) H) vx e x p ( - uH) v~}
'X
=~
exp ( - fle~)
~¢
X
exp(u(e,,-e,,,))l(glv,,[~')l 2 •
(10)
xt
Following Hindley, we find from (1) that (g] vx ]u') = ~ a,,n ax,n, (n[ Vx In'), n, n t
and
Il 2 =
axn* a~,n, axm a*,,,,, (hi Vx In'5 (ml vx Ira'>. n~ ?1' /11) ?11'
Now a~, = [a~. I exp (iq~n) . Averaging over the individual phases under the assumption of no correlation gives zero unless the coefficients match up; in particular, we require that rn =n, m' =n', and (l(xl Vx Ix')lz)~ = ~ la.n[ 2 [a~,n,I2 I(n[ Vx [n')l 2 • n , ?1'
Furthermore, from the orthonormality of the a~/s and the fact that the magnitudes are random as well, the average occupation probability at any site must be the reciprocal of the number of unit cells, ([axnl2)mag = a3/V,
(11)
where a is the lattice constant so that a-3 is the number of atoms per unit volume. Thus, denoting the average over both the phases and magnitudes of the a's by single brackets,
(l(×[ Vx lu')l 2)
= (a3"]z V a3~ \ V / ~ I(nl Vx In')l 2 = V I(nl Vx [n')l 2, H, n '
(12)
n'
where the last equality follows from the assumed topological order in spite of cellular disorder.
HALL CONDUCTIVITY IN THE RANDOM PHASE MODEL
333
Substituting (12) into (10) and then into (9), carrying out the indicated integration, using (7), and taking the real part of axx in the limit ~0~ 0, s ~ 0, we get 2 2a3
1
~'~
Re (a~ °)) = n ~e - a x ~ z J2 -Z fl L exp ( - fie,,)
Here a x is the x-component of the intersite vector, and z is the nearest neighbour coordination number. The quantity in square brackets is the density of electron states, N(e,`). In essence, the averaged square matrix element is reduced by a factor of (aa/V) and is independent of x', this being compensated for by an unrestricted sum over all 5~, giving the density of states factor. The indicated thermodynamic average is carried out by introducing the electron distribution function. In this connection, it is assumed that the Fermi level is sufficiently far from the energy 5o separating localized from extended states [perhaps pinned near the centre of the pseudogap as suggested by Davis and Mott11)] that the Fermi-Dirac function may be replaced by a Boltzmann distribution. Thus, oo
Z
exp ( - fie,,) N (5,`) = exp (fl~) 2V
de,`[N (e,`)] 2 exp ( - fie,`).
(13) Taking N(e,,) to be constant within kT of ec, one gets 2n e 2 Re (a) = ~ ha {za6j2 IN
(ec)] 2} exp ( - fl (e¢ -/~)),
(14)
where an average over directions has been taken. To put this into the form obtained by Mott and Hindley, the dimensionless parameter 2 is introduced via the relation z½d = 2h2/ma 2 ,
giving Re(a)-
2n e2haa 3 m 2 22[N(e¢)]2exp(-fl(e~-#))"
(15)
The drift mobility is obtained by dividing (14) by ne, where the electron density is given by oo
n =
I de,` N (e,`) exp (fl (# ge
N(.o)
e,`)) ~ ~ f l ~ exp ( - fl (e, -/~)).
334
LIONEL FRIEDMAN
One gets
#o-
23 eah
z ~a3J
l
N(ec) •
(16)
4. Hall conductivity The Hall conductivity is obtained from the antisymmetric components of the conductivity tensor (8). Thus one must consider
Tr
~xy ( hco,)
exo - uH) vy} .
0
(17) Now it is known from previous studies that a magnetic field dependent conductivity enters to third (or higher) order in the magnetic field dependent transfer integrals (4). In view of (7), the exponential operators of (17) must be expanded to first order in the applied fields. Use is made of the standard relations (u] exp ( - uH) ]u') = exp ( - ue.) 6~., exp ( - ue,) - exp ( + uex') (u] AH Ix'b, /~x - - 8 x '
where the last matrix element is given by (6). Inserting this and the similar expansion for e x p ( - (fl-U) H) into the argument of the trace of (17) gives two terms, a typical one being: exp(-- (fl -- u)e~) (u[ Vx Ix') exp(-- ue~,) -- exp(-- ue~,,)
1 ~
8x' -- 8x" X, ~¢', X"
x (u'l AH Ix") (u"] vy Ix). Using the basic expansion 1) of the RPM states in terms of the localized states,
(xl v~ Ix') (x'l At I Ix") (x"l v, Ix) = ~ (a,,* a.,,., a,~.., a,,,..., a...,,., a..) m, in' n', n"
p", p
x (m] Vx [m') (n'[ AH [n") (p"[ v, [p). Just as for the electrical conductivity, the average over the random phases of the coefficients restricts one to the terms for which n' =m', p"=n", 1)=m, leaving lax,m,[2 [aw,,,,,I2 [axm[2 (ml vx [rn') (m'[ AH In") (n"] vy ]m). m, m ' , n"
335
HALL CONDUCTIVITY IN THE RANDOM PHASE MODEL
Averaging over the magnitudes of the coefficients and using (11), one obtains the analog of (12)
(a~)2 ~ (J~',)', - -'t°) ,'
.
(18)
m',/1"
Now the Hall conductivity ate) x y = ½(axy - aye)
(19)
is obtained from the antisymmetrie component of the correlation tensor
°~-~) (~) = ½[,~'~y (Q)
-
~'yx (Q)].
(20)
From (18) and the analogous expression for the second term in the expansion of the trace, one finds, following the method of calculation of ref. 10, that
~
(a)
=
,.~.~
~djk
_ .~o~
"jk ]
j,k
(21) where 1 " ~ x ' , J¢"; ~¢ - 8 x - - 8~,,
[exp( - / / ~ , , ) - e x p ( - ] / e ~ )
exp(-//~,,)-exp(-//e~)]
(22)
That (21) is finite to this order clearly depends on the existence of three sites i, j, and k which are mutual nearest neighbours for which there is finite electronic overlap between all pairs of sites. Next, the velocity matrix elements in (21) are eliminated by use of (7). This gives
.~;~o~(~)
=
(s#,
_
--xy
#o,) j~., J[#' 2 .~:~.oL
j..
h 2 "*kJ~ Z
j, k
,,,.
,
,x"
~, x', x"
(23) Here A~Z) kji :
½(XkiYij-
YkiX~j) = ½ [ R j i × R k , ] :
(24)
is the z-component of the area of the triangle substended by the sites i, j, and k. In the spirit of the RPM, this will at the end be replaced by an appropriate average area. Following ref. 10, it is noted that "akji ,jr,) is antisymmetric with
LIONELFRIEDMAN
336
respect to interchange of the indices j and k. Hence, one may make the replacement (Jj(kn) _ jjo))jk(n) j,~.) + + {(jj.,
_
j?,)
s:o,) .<.,
= ½H ( k t",j"k )
--
j(o,]jk(g, j i ( j . , _ [(j(km _ j(Ok))J(k~) j(n,].} jk /
.<., .,.,_
Since only the linear response is relevant to the Hall effect, the difference (JJkH) --JJk°)) is expanded to first order in the magnetic field: 1] = iJi °)
( JJ~) -- °jkl(O)~ = JJO) [exp ( ictjk)
O~jk .
This shows explicitly that the perturbation is the magnetic field and not J. Expanding the other two phase factors, one gets ?tH) l(O)~ 1 (H) T(H) " j k - - ~'jk ] "ki " i j = ½i[O~jk
(1 d-
iO~ki)
(1 +
O~ij) -- O~kj (1
+
i~ji ) (1
+ i0~ik)] I(0)1(0)1(0) ° j k "ki "lij
_~ ;*, 1(0) 1 ( 0 ) 1 ( 0 ) x ~'jk "jk "ki ~'ij •
Now, one may write eH e O~kJ = -- 2hC Rj × R k = - 2h~c [gi × Pkj -~- Pig × Pij] ,
where the Pij are the intersite vectors, i.e., R j = R i d- P i j ,
Rg = R i .at- Pig,
and P k j ---- P i j -- Pig.
Noting the double sum over sites j and k in (23), we assume inversional symmetry on the average: for every (j, k) there is a corresponding pair ( j ' , k') for which Pij" =
- -
Pij,
Pik' "~-
-
-
Pik,
so that R i × pg,j, = -
Ri
× Pkj,
but Pig" X Pij' = Pig X P i j "
Thus the first term of ~kj proportional to the vector Ri from the arbitrary choice of coordinates to site i, vanishes in the double sum over j and k, and we are left with i rio) £(o) j fo) e H . A R j i , jk
g~
~J h c
where Agji :
½Pig ×
PU"
(25)
HALL CONDUCTIVITY IN THE RANDOM PHASE MODEL
337
This insures that our results are properly gauge invariant 9) and independent of choice of origin. Furthermore, as shown in some detail in ref. 10 in a local site representation 2n 2 lim o¢.,, .,,;. = - - - fl exp ( - fie.) 6 (s., - e.) 6 (s.,, - e.). (26) a-.0 3 From (26), (25), (23) and (8), one obtains
fle2FeHra(z)~2]fa3~2~ J(°)J~°'jkki Ji~°, 3 V [_ I.'~kjU .J ~V } h
_(a) __ 4n2
o:,y
j,k
Xz1
~
fie.) 6 (e. - e.,) 6 (e. - e.,,)
exp ( -
/A(Z)~2 ( ,j(z)'~2 The quantity V~kjU is replaced by ~'~kj~) , the average square of the projected three-site area on a plane perpendicular to the magnetic field. One takes ( ,4 (z)'~2
Z'~kji]
= t]a 4 ,
where r/is a parameter ~ 1. Making the replacements j~.o)= _ j etc., taking the sums over •' and ~" to give two density of states factors, and finally taking the thermal average as in (13) for the electrical conductivity (this introduces an additional factor of the density of states), one gets 8n 2 e 2 (ell ) xy- 3 h ~,,,~a2 aaj3[N (ec)]3 ~.qexp(- fl(sc - la)),
a(a)
(27)
where ~> 1 is the average number of sites j and k which form a closed path with respect to an arbitrarily chosen site i. The Hall angle On is obtained by dividing (27) by (14):
-(")
OH - ° ~ ' = 4 n
(ell) (:) a 2 [a3.1N(sc)] q
(Txx
and the Hall mobility
(28)
hc
~a=Onc/H is
I~H=4~(e~2)[aajN(8c)](~l:).
(29)
Finally, from (16) one has
~H _ 6kT(__ ~) /~D
J
r/~
.
(30)
338
LIONEL FRIEDMAN
Eqs. (29) and (30) are the principal results. The following features are noted: (1) From (29) and (16), it is seen that both Prl and PD are proportional to N(ec) and are therefore small to the extent that this factor is less than the free electron value. (2) PH is smaller than PD by the ratio (30). Taking 2z J,-, 1 eV, kT,-~ 1/40 eV, q = ] , and 2 = z = 6 , this ratio is 1/10. (3) The explicit temperature dependence of PD, eq. (16), does not appear in Pn. The Hall mobility PH is temperature independent to the extent that N(e,,) may be approximated by N(8¢) within an energy range kT of e¢. (4) From (23), the sign of the Hall effect for holes is the same as in the local representation. As has been shown for that case 12), the sign of the Hall effect for holes is the same as for electrons (i.e., both n-type) for the three-site case. Such a situation corresponds to the complementary case of a nondegenerate distribution of holes just below the energy ev in the valence band separating extended from localized states. This result is consistent with the common observation of a p-type thermopower and n-type Hall effect in amorphous solids and liquids 5, n). (5) To estimate the magnitude of PH, J is eliminated in favour of the bandwidth W=2zJ. Also, N(ec)~-1/aaVo, where Vo is the spread of energies over which N(e,,) is finite. Then,
(ea2~fW__~(tl~) I~n~2r~\ h ]\VoJ z 2 "
(31)
Propagation with short mean free path at e¢ corresponds to 3, 8) W
1
Vo 5.5' Since
(ea2/h) = 1.6 cm2/V sec, for a =3 .~, r/2 cm 2 #r~ ~ 1.8 z2 V sec"
The parameter r/is reasonably taken to be r / = cos 2 0 = ~. Taking ~ = z = 6, we get ktH -~ 10-1 cm2/V sec. 5. "Fermi glass" case
The previous analysis applies to excited carriers, in which energies just above ec (or just below ev in the case of hole conduction) are many kT from
HALL CONDUCTIVITY IN THE RANDOM PHASE MODEL
339
the Fermi energy #. This permits a one-electron approach and the use of non-degenerate statistics in our formulae. Such a situation is applicab'e to the chalcogenide glass and amorphous germanium ~), among other materials. However, a different situation applies to Hall effect measurements in the liquid metals and their alloys of short mean free path (L~a) where conduction occurs at the Fermi level. For this case as well, the thermopower is generally p-type and the Hall coefficient n-type and too small in comparison with that expected from the number of free valence electrons. To treat the degenerate many-electron case, an occupation number representation is adopted I... n~...>, (32) where n, =0, I gives the occupation of the xth state in the RPM. The unperturbed Hamiltonian operator corresponding to the matrix preceding eq. (4) is H° = E e,,b~b,,, (33) where b~, b~ are the usual creation and annihilation operators. The velocity operator is given by v, = ~ b~,b,,,,. (34) ~¢t, xo
The (diagonal) electrical conductivity is considered first. Taking the trace (10) in the representation (32) and using (33) and (34), it is straightforwardly shown that, instead of (10), one gets exp(u (e~ - e~,))I12
b~,b,,,)>,
x, x '
where
-
b~,b,,,)> =
fk (1 -- fk'),
with
A = Eexp (P (e~ - ~t)) + 11-' the Fermi distribution function. In view of the Dirac delta function 6 (~ - ~,) which appears in Retr after integrating over u, one has :k(1 - A , ) - . -
laA ~"
It follows that, instead of (14), one gets Retr = ~- ha
za6jZ
{N(#))2.
(35)
Following the same procedure for the (off-diagonal) Hall conductivity, it
340
LIONEL FRIEDMAN
is found that the degenerate case is obtained by the replacement 1
exp ( - flex) (b~b~ (1 - b*~,b.,) (1 - b~,,b~,,)) 6 (e~, - e.) 6 (e.,, - e~)
= A ( 1 - A.) (1 - A.,) a ( ~ . - ~) a ( ~ , , - ~), which, in view of the product of the two delta functions, becomes fk (1 - fk') (1 - fk-) "-)
exp (2fl (e~ - p))
1 02f~
[exp (fl(~. - ~)) + 1] 3 - 2fl 2 a ~
1 Of~ 2fl a~.
(36) Now
af. 0e x
fl
[exp (½fl (e,, - p)) + exp ( - ½fl (e,, - p))]2
is an even function of (e~-p), while a2f~ _ _ fiE exp (½fl (e~ - #)) - exp ( - ½fl (e~ - #)) 0e2 [exp (½fl (e~ - #)) + exp ( - ½fl (e~ - / 0 ) ] 3 is clearly odd in (e~-p). Thus, in the low temperature limit in which N(e~) is taken to be a constant at the Fermi surface, the first term of (36) gives zero. It follows that the formula for the Hall mobility for this case is PH =
\z/\
h ] [aaJN(P)]"
(37)
The Hall coefficient appropriate to this case is g n -- It ( ~ ) 2
=
ec W N (p)'
(38)
where W = 2zJ is the bandwidth Now, if the electrons were free, their density would be given by n = ~N~.o. (~) ~,
where sf.o. ( 0 = c ~
is the free electron density of states. In addition, ~_-__ Wf , where f is the fractional occupancy of the band (e.g. f = ½ for a monovalent metal). It follows that (38) can be written Rn =
necg '
(39)
HALLCONDUCTIVITYIN THERANDOMPHASEMODEL
341
where g -- (Nf.e. //~
"
Taking 5 = z , t/---½ and f = ½ , the numerical factor multiplying (39) is ]. A similar formula, with the coefficient equal to unity, was given by Straub et a l J 3) for the case o f impurity conduction. The details o f their calculation are not given. The magnetic field dependence o f their Hall coefficient (and magnetoresistance), however, is consistent with the above reciprocal dependence on the density o f states at the Fermi level*. Considerations leading to the same result were also given by Ziman 14).
Acknowledgements The author would like to express his sincere thanks to Professor N. F. M o t t for his cordiality during his stay at Cambridge and for his interest in this work.
References 1) 2) 3) 4) 5)
J. M. Ziman, J. Phys. C [2l 1 (1968) 1532. M. H. Cohen, J. Non-Crystalline Solids 2 (1970) 432. N. F. Mott, Advan. Phys. 16 (1967) 49, section 4.6; Phil. Mag. 22 (1970) 7. N. K. Hindley, J. Non-Crystalline Solids 5 (1970) 17. J. C. Male, Brit. J. Appl. Phys. 18 (1967) 1543; A. H. Clark, Phys. Rev. 154 (1967) 750. 6) R. S. Allgaier, Phys. Rev. 185 (1969) 227. 7) K. W. B6er, J. Non-Crystalline Solids 4 (1970) 583; K. W. B~er and R. Hailslip, Phys. Rev. Letters 24 (1970) 230. 8) P. W. Anderson, Phys. Rev. 109 (1958) 1492. 9) L. Friedman and T. Holstein, Ann Phys. (N.Y.) 21 (1963) 494. 10) T. Holstein and L. Friedman, Phys. Rev. 165 (1968) 1019. 1I) E. A. Davis and N. F. Mott, Phil. Mag. 22 (1970) 903. 12) T. Holstein, private communication; L. Friedman, in: Proc. Intern. Conf. on Low Mobility Materials, Eilat, 1971 (Taylor and Francis, to be published); I. G. Austin and N. F. Mott, Advan. Phys. 18 (1969) 71 ; R. W. Munn and W. Siebrand, Phys. Rev. 2 (1970) 3435. 13) W. D. Straub, H. Roth, W. Bernard, S. Goldstein and J. E. Mulhern, Phys. Rev. Letters 21 (1968) 752. 14) J. M. Ziman, in: Proc. Intern. Conf. on The Properties of Liquid Metals; Advan. Phys. 16 (1967) 551.
* This will be considered in more detail in a future publication in this Journal by N. F. Mott and the present author.
Rnn, =
n
-
n'
is the intersite vector. The basic approach of the present paper is an extention to the RPM of a previous calculation 1°) of the Hall mobility in an appropriate local site (small polaron) representation. The starting point is the Kubo-type formula for the frequency dependent electrical conductivity tensor
e2h a,j (co) = iQl/[~-ij (f2) - ~ , j (0)],
(8)
where V is the sample volume, £2---h (to + is), and s is a positive infinitesimal. ~'ij (O) is the analytic continuation to the complex Q-plane of the thermodynamic velocity correlation function
~ , j (hto,) =
f
1
du exp (hto, u) Z Tr { e x p ( - H (fl - u)) v, e x p ( - Hu) vj}, (9)
where
fl = 1/kT, co, = 2nir/flh,
r integral,
and z = Tr {exp ( - fin)}.
The equivalence of (8) to the standard Kubo formula is established in Ap-
332
LIONELFRIEDMAN
pendix B of ref. I0. It is therefore also equivalent to the Kubo=Greenwood formula as shown in ref. 4. 3. Electrical conductivity
We first calculate the electrical conductivity and show that it agrees with the results of Mott 3) and Hindley4). Setting i = j = x in (9), taking the trace in the RPM and replacing H by H 0 in the exponential operator, we get 1
Z Tr { e x p ( - (fl - u) H) vx e x p ( - uH) v~}
'X
=~
exp ( - fle~)
~¢
X
exp(u(e,,-e,,,))l(glv,,[~')l 2 •
(10)
xt
Following Hindley, we find from (1) that (g] vx ]u') = ~ a,,n ax,n, (n[ Vx In'), n, n t
and
I
axn* a~,n, axm a*,,,,, (hi Vx In'5 (ml vx Ira'>. n~ ?1' /11) ?11'
Now a~, = [a~. I exp (iq~n) . Averaging over the individual phases under the assumption of no correlation gives zero unless the coefficients match up; in particular, we require that rn =n, m' =n', and (l(xl Vx Ix')lz)~ = ~ la.n[ 2 [a~,n,I2 I(n[ Vx [n')l 2 • n , ?1'
Furthermore, from the orthonormality of the a~/s and the fact that the magnitudes are random as well, the average occupation probability at any site must be the reciprocal of the number of unit cells, ([axnl2)mag = a3/V,
(11)
where a is the lattice constant so that a-3 is the number of atoms per unit volume. Thus, denoting the average over both the phases and magnitudes of the a's by single brackets,
(l(×[ Vx lu')l 2)
= (a3"]z V a3~ \ V / ~ I(nl Vx In')l 2 = V I(nl Vx [n')l 2, H, n '
(12)
n'
where the last equality follows from the assumed topological order in spite of cellular disorder.
HALL CONDUCTIVITY IN THE RANDOM PHASE MODEL
333
Substituting (12) into (10) and then into (9), carrying out the indicated integration, using (7), and taking the real part of axx in the limit ~0~ 0, s ~ 0, we get 2 2a3
1
~'~
Re (a~ °)) = n ~e - a x ~ z J2 -Z fl L exp ( - fie,,)
Here a x is the x-component of the intersite vector, and z is the nearest neighbour coordination number. The quantity in square brackets is the density of electron states, N(e,`). In essence, the averaged square matrix element is reduced by a factor of (aa/V) and is independent of x', this being compensated for by an unrestricted sum over all 5~, giving the density of states factor. The indicated thermodynamic average is carried out by introducing the electron distribution function. In this connection, it is assumed that the Fermi level is sufficiently far from the energy 5o separating localized from extended states [perhaps pinned near the centre of the pseudogap as suggested by Davis and Mott11)] that the Fermi-Dirac function may be replaced by a Boltzmann distribution. Thus, oo
Z
exp ( - fie,,) N (5,`) = exp (fl~) 2V
de,`[N (e,`)] 2 exp ( - fie,`).
(13) Taking N(e,,) to be constant within kT of ec, one gets 2n e 2 Re (a) = ~ ha {za6j2 IN
(ec)] 2} exp ( - fl (e¢ -/~)),
(14)
where an average over directions has been taken. To put this into the form obtained by Mott and Hindley, the dimensionless parameter 2 is introduced via the relation z½d = 2h2/ma 2 ,
giving Re(a)-
2n e2haa 3 m 2 22[N(e¢)]2exp(-fl(e~-#))"
(15)
The drift mobility is obtained by dividing (14) by ne, where the electron density is given by oo
n =
I de,` N (e,`) exp (fl (# ge
N(.o)
e,`)) ~ ~ f l ~ exp ( - fl (e, -/~)).
334
LIONEL FRIEDMAN
One gets
#o-
23 eah
z ~a3J
l
N(ec) •
(16)
4. Hall conductivity The Hall conductivity is obtained from the antisymmetric components of the conductivity tensor (8). Thus one must consider
Tr
~xy ( hco,)
exo - uH) vy} .
0
(17) Now it is known from previous studies that a magnetic field dependent conductivity enters to third (or higher) order in the magnetic field dependent transfer integrals (4). In view of (7), the exponential operators of (17) must be expanded to first order in the applied fields. Use is made of the standard relations (u] exp ( - uH) ]u') = exp ( - ue.) 6~., exp ( - ue,) - exp ( + uex') (u] AH Ix'b, /~x - - 8 x '
where the last matrix element is given by (6). Inserting this and the similar expansion for e x p ( - (fl-U) H) into the argument of the trace of (17) gives two terms, a typical one being: exp(-- (fl -- u)e~) (u[ Vx Ix') exp(-- ue~,) -- exp(-- ue~,,)
1 ~
8x' -- 8x" X, ~¢', X"
x (u'l AH Ix") (u"] vy Ix). Using the basic expansion 1) of the RPM states in terms of the localized states,
(xl v~ Ix') (x'l At I Ix") (x"l v, Ix) = ~ (a,,* a.,,., a,~.., a,,,..., a...,,., a..) m, in' n', n"
p", p
x (m] Vx [m') (n'[ AH [n") (p"[ v, [p). Just as for the electrical conductivity, the average over the random phases of the coefficients restricts one to the terms for which n' =m', p"=n", 1)=m, leaving lax,m,[2 [aw,,,,,I2 [axm[2 (ml vx [rn') (m'[ AH In") (n"] vy ]m). m, m ' , n"
335
HALL CONDUCTIVITY IN THE RANDOM PHASE MODEL
Averaging over the magnitudes of the coefficients and using (11), one obtains the analog of (12)
(a~)2 ~
.
(18)
m',/1"
Now the Hall conductivity ate) x y = ½(axy - aye)
(19)
is obtained from the antisymmetrie component of the correlation tensor
°~-~) (~) = ½[,~'~y (Q)
-
~'yx (Q)].
(20)
From (18) and the analogous expression for the second term in the expansion of the trace, one finds, following the method of calculation of ref. 10, that
~
(a)
=
,.~.~
~djk
_ .~o~
"jk ]
j,k
(21) where 1 " ~ x ' , J¢"; ~¢ - 8 x - - 8~,,
[exp( - / / ~ , , ) - e x p ( - ] / e ~ )
exp(-//~,,)-exp(-//e~)]
(22)
That (21) is finite to this order clearly depends on the existence of three sites i, j, and k which are mutual nearest neighbours for which there is finite electronic overlap between all pairs of sites. Next, the velocity matrix elements in (21) are eliminated by use of (7). This gives
.~;~o~(~)
=
(s#,
_
--xy
#o,) j~., J[#' 2 .~:~.oL
j..
h 2 "*kJ~ Z
j, k
,,,.
,
,x"
~, x', x"
(23) Here A~Z) kji :
½(XkiYij-
YkiX~j) = ½ [ R j i × R k , ] :
(24)
is the z-component of the area of the triangle substended by the sites i, j, and k. In the spirit of the RPM, this will at the end be replaced by an appropriate average area. Following ref. 10, it is noted that "akji ,jr,) is antisymmetric with
LIONELFRIEDMAN
336
respect to interchange of the indices j and k. Hence, one may make the replacement (Jj(kn) _ jjo))jk(n) j,~.) + + {(jj.,
_
j?,)
s:o,) .<.,
= ½H ( k t",j"k )
--
j(o,]jk(g, j i ( j . , _ [(j(km _ j(Ok))J(k~) j(n,].} jk /
.<., .,.,_
Since only the linear response is relevant to the Hall effect, the difference (JJkH) --JJk°)) is expanded to first order in the magnetic field: 1] = iJi °)
( JJ~) -- °jkl(O)~ = JJO) [exp ( ictjk)
O~jk .
This shows explicitly that the perturbation is the magnetic field and not J. Expanding the other two phase factors, one gets ?tH) l(O)~ 1 (H) T(H) " j k - - ~'jk ] "ki " i j = ½i[O~jk
(1 d-
iO~ki)
(1 +
O~ij) -- O~kj (1
+
i~ji ) (1
+ i0~ik)] I(0)1(0)1(0) ° j k "ki "lij
_~ ;*, 1(0) 1 ( 0 ) 1 ( 0 ) x ~'jk "jk "ki ~'ij •
Now, one may write eH e O~kJ = -- 2hC Rj × R k = - 2h~c [gi × Pkj -~- Pig × Pij] ,
where the Pij are the intersite vectors, i.e., R j = R i d- P i j ,
Rg = R i .at- Pig,
and P k j ---- P i j -- Pig.
Noting the double sum over sites j and k in (23), we assume inversional symmetry on the average: for every (j, k) there is a corresponding pair ( j ' , k') for which Pij" =
- -
Pij,
Pik' "~-
-
-
Pik,
so that R i × pg,j, = -
Ri
× Pkj,
but Pig" X Pij' = Pig X P i j "
Thus the first term of ~kj proportional to the vector Ri from the arbitrary choice of coordinates to site i, vanishes in the double sum over j and k, and we are left with i rio) £(o) j fo) e H . A R j i , jk
g~
~J h c
where Agji :
½Pig ×
PU"
(25)
HALL CONDUCTIVITY IN THE RANDOM PHASE MODEL
337
This insures that our results are properly gauge invariant 9) and independent of choice of origin. Furthermore, as shown in some detail in ref. 10 in a local site representation 2n 2 lim o¢.,, .,,;. = - - - fl exp ( - fie.) 6 (s., - e.) 6 (s.,, - e.). (26) a-.0 3 From (26), (25), (23) and (8), one obtains
fle2FeHra(z)~2]fa3~2~ J(°)J~°'jkki Ji~°, 3 V [_ I.'~kjU .J ~V } h
_(a) __ 4n2
o:,y
j,k
Xz1
~
fie.) 6 (e. - e.,) 6 (e. - e.,,)
exp ( -
/A(Z)~2 ( ,j(z)'~2 The quantity V~kjU is replaced by ~'~kj~) , the average square of the projected three-site area on a plane perpendicular to the magnetic field. One takes ( ,4 (z)'~2
Z'~kji]
= t]a 4 ,
where r/is a parameter ~ 1. Making the replacements j~.o)= _ j etc., taking the sums over •' and ~" to give two density of states factors, and finally taking the thermal average as in (13) for the electrical conductivity (this introduces an additional factor of the density of states), one gets 8n 2 e 2 (ell ) xy- 3 h ~,,,~a2 aaj3[N (ec)]3 ~.qexp(- fl(sc - la)),
a(a)
(27)
where ~> 1 is the average number of sites j and k which form a closed path with respect to an arbitrarily chosen site i. The Hall angle On is obtained by dividing (27) by (14):
-(")
OH - ° ~ ' = 4 n
(ell) (:) a 2 [a3.1N(sc)] q
(Txx
and the Hall mobility
(28)
hc
~a=Onc/H is
I~H=4~(e~2)[aajN(8c)](~l:).
(29)
Finally, from (16) one has
~H _ 6kT(__ ~) /~D
J
r/~
.
(30)
338
LIONEL FRIEDMAN
Eqs. (29) and (30) are the principal results. The following features are noted: (1) From (29) and (16), it is seen that both Prl and PD are proportional to N(ec) and are therefore small to the extent that this factor is less than the free electron value. (2) PH is smaller than PD by the ratio (30). Taking 2z J,-, 1 eV, kT,-~ 1/40 eV, q = ] , and 2 = z = 6 , this ratio is 1/10. (3) The explicit temperature dependence of PD, eq. (16), does not appear in Pn. The Hall mobility PH is temperature independent to the extent that N(e,,) may be approximated by N(8¢) within an energy range kT of e¢. (4) From (23), the sign of the Hall effect for holes is the same as in the local representation. As has been shown for that case 12), the sign of the Hall effect for holes is the same as for electrons (i.e., both n-type) for the three-site case. Such a situation corresponds to the complementary case of a nondegenerate distribution of holes just below the energy ev in the valence band separating extended from localized states. This result is consistent with the common observation of a p-type thermopower and n-type Hall effect in amorphous solids and liquids 5, n). (5) To estimate the magnitude of PH, J is eliminated in favour of the bandwidth W=2zJ. Also, N(ec)~-1/aaVo, where Vo is the spread of energies over which N(e,,) is finite. Then,
(ea2~fW__~(tl~) I~n~2r~\ h ]\VoJ z 2 "
(31)
Propagation with short mean free path at e¢ corresponds to 3, 8) W
1
Vo 5.5' Since
(ea2/h) = 1.6 cm2/V sec, for a =3 .~, r/2 cm 2 #r~ ~ 1.8 z2 V sec"
The parameter r/is reasonably taken to be r / = cos 2 0 = ~. Taking ~ = z = 6, we get ktH -~ 10-1 cm2/V sec. 5. "Fermi glass" case
The previous analysis applies to excited carriers, in which energies just above ec (or just below ev in the case of hole conduction) are many kT from
HALL CONDUCTIVITY IN THE RANDOM PHASE MODEL
339
the Fermi energy #. This permits a one-electron approach and the use of non-degenerate statistics in our formulae. Such a situation is applicab'e to the chalcogenide glass and amorphous germanium ~), among other materials. However, a different situation applies to Hall effect measurements in the liquid metals and their alloys of short mean free path (L~a) where conduction occurs at the Fermi level. For this case as well, the thermopower is generally p-type and the Hall coefficient n-type and too small in comparison with that expected from the number of free valence electrons. To treat the degenerate many-electron case, an occupation number representation is adopted I... n~...>, (32) where n, =0, I gives the occupation of the xth state in the RPM. The unperturbed Hamiltonian operator corresponding to the matrix preceding eq. (4) is H° = E e,,b~b,,, (33) where b~, b~ are the usual creation and annihilation operators. The velocity operator is given by v, = ~
The (diagonal) electrical conductivity is considered first. Taking the trace (10) in the representation (32) and using (33) and (34), it is straightforwardly shown that, instead of (10), one gets exp(u (e~ - e~,))I
b~,b,,,)>,
x, x '
where
-
b~,b,,,)> =
fk (1 -- fk'),
with
A = Eexp (P (e~ - ~t)) + 11-' the Fermi distribution function. In view of the Dirac delta function 6 (~ - ~,) which appears in Retr after integrating over u, one has :k(1 - A , ) - . -
laA ~"
It follows that, instead of (14), one gets Retr = ~- ha
za6jZ
{N(#))2.
(35)
Following the same procedure for the (off-diagonal) Hall conductivity, it
340
LIONEL FRIEDMAN
is found that the degenerate case is obtained by the replacement 1
exp ( - flex) (b~b~ (1 - b*~,b.,) (1 - b~,,b~,,)) 6 (e~, - e.) 6 (e.,, - e~)
= A ( 1 - A.) (1 - A.,) a ( ~ . - ~) a ( ~ , , - ~), which, in view of the product of the two delta functions, becomes fk (1 - fk') (1 - fk-) "-)
exp (2fl (e~ - p))
1 02f~
[exp (fl(~. - ~)) + 1] 3 - 2fl 2 a ~
1 Of~ 2fl a~.
(36) Now
af. 0e x
fl
[exp (½fl (e,, - p)) + exp ( - ½fl (e,, - p))]2
is an even function of (e~-p), while a2f~ _ _ fiE exp (½fl (e~ - #)) - exp ( - ½fl (e~ - #)) 0e2 [exp (½fl (e~ - #)) + exp ( - ½fl (e~ - / 0 ) ] 3 is clearly odd in (e~-p). Thus, in the low temperature limit in which N(e~) is taken to be a constant at the Fermi surface, the first term of (36) gives zero. It follows that the formula for the Hall mobility for this case is PH =
\z/\
h ] [aaJN(P)]"
(37)
The Hall coefficient appropriate to this case is g n -- It ( ~ ) 2
=
ec W N (p)'
(38)
where W = 2zJ is the bandwidth Now, if the electrons were free, their density would be given by n = ~N~.o. (~) ~,
where sf.o. ( 0 = c ~
is the free electron density of states. In addition, ~_-__ Wf , where f is the fractional occupancy of the band (e.g. f = ½ for a monovalent metal). It follows that (38) can be written Rn =
necg '
(39)
HALLCONDUCTIVITYIN THERANDOMPHASEMODEL
341
where g -- (Nf.e. //~
"
Taking 5 = z , t/---½ and f = ½ , the numerical factor multiplying (39) is ]. A similar formula, with the coefficient equal to unity, was given by Straub et a l J 3) for the case o f impurity conduction. The details o f their calculation are not given. The magnetic field dependence o f their Hall coefficient (and magnetoresistance), however, is consistent with the above reciprocal dependence on the density o f states at the Fermi level*. Considerations leading to the same result were also given by Ziman 14).
Acknowledgements The author would like to express his sincere thanks to Professor N. F. M o t t for his cordiality during his stay at Cambridge and for his interest in this work.
References 1) 2) 3) 4) 5)
J. M. Ziman, J. Phys. C [2l 1 (1968) 1532. M. H. Cohen, J. Non-Crystalline Solids 2 (1970) 432. N. F. Mott, Advan. Phys. 16 (1967) 49, section 4.6; Phil. Mag. 22 (1970) 7. N. K. Hindley, J. Non-Crystalline Solids 5 (1970) 17. J. C. Male, Brit. J. Appl. Phys. 18 (1967) 1543; A. H. Clark, Phys. Rev. 154 (1967) 750. 6) R. S. Allgaier, Phys. Rev. 185 (1969) 227. 7) K. W. B6er, J. Non-Crystalline Solids 4 (1970) 583; K. W. B~er and R. Hailslip, Phys. Rev. Letters 24 (1970) 230. 8) P. W. Anderson, Phys. Rev. 109 (1958) 1492. 9) L. Friedman and T. Holstein, Ann Phys. (N.Y.) 21 (1963) 494. 10) T. Holstein and L. Friedman, Phys. Rev. 165 (1968) 1019. 1I) E. A. Davis and N. F. Mott, Phil. Mag. 22 (1970) 903. 12) T. Holstein, private communication; L. Friedman, in: Proc. Intern. Conf. on Low Mobility Materials, Eilat, 1971 (Taylor and Francis, to be published); I. G. Austin and N. F. Mott, Advan. Phys. 18 (1969) 71 ; R. W. Munn and W. Siebrand, Phys. Rev. 2 (1970) 3435. 13) W. D. Straub, H. Roth, W. Bernard, S. Goldstein and J. E. Mulhern, Phys. Rev. Letters 21 (1968) 752. 14) J. M. Ziman, in: Proc. Intern. Conf. on The Properties of Liquid Metals; Advan. Phys. 16 (1967) 551.
* This will be considered in more detail in a future publication in this Journal by N. F. Mott and the present author.