Hall effect near the mobility edge

Journal of Non-Crystalline North-Holland

Solids 164-166 (1993) 445-448

Hall effect near the mobility edge H. Okamoto, K. Hattori and Y. Hamakawa Faculty of Engineering Science, Osaka University Toyonaka, Osaka 560, Japan

Perturbative renormalization-group procedure has been employed to discuss the behavior of weakfield Hall conductivity near the mobility edge in amorphous semiconductors. It is found that the anomalous sign can occur for carrier mean free paths less than a certain critical maginitude. Under limited, but presumably practical conditions, the conductivity and Hall mobilities are linked by a simple relation, which may lead to reassessment of the value of the Hall measurements. to the regular electron system. The weak-field Hall conductivity o,, for the system of infinitely large scale is given in the guage-independent on-shell form due to ltoh [2];

1. INTRODUCTION A large part of the behavior of electrons in disordered systems is still left behind a mysterious veil, among which the most striking is the Hall effect involving the sign anomaly [l]. Theories developed in the context of a hopping model can be used to interpret the anomaly; however, their validity in various classes of amorphous semiconductors is questionable, particularly for a-Si:H. In terms of the mobility edge model, no satisfactory explanation of the phenomenon have appeared. The failure is likely due to an unconscious neglect of the random interference of electron waves leading to the occurrence of a mobility edge E,. This work is aimed at giving a quantitative description of the anomalous Hall effect near the mobility edge in accordance with a microscopic model in which interference effects are properly taken into account. In addition we try to decode the message from Hall measurements regarding to carrier mobility.

o

h

x ~m(Tr[v,G- v,G- v,G-v,G+])

2. THEORY OF THE HALL CONDUCTIVITY

+c

q lixil

i

of which the regular transfer term defines the ck dispersion-relation (we assume = k*) of the virtual-crystal system, while the site-diagonal term bears a zero-range random scattering potential 0022-3093/93/$06.00

0 1993 - Elsevier Science Publishers

lim

L-W

d& -z I()

a&

C+“(E) h&kJ

,

(3)

h

with hL(EI.$ being newly introduced dimensionless spectral function which involves quantum corrections arising from multiple scattering processes to the classical Hall conductivity oh(O).The quantity is obviously size (L) dependent, and described as a function of energy E normalized by the mobility edge energy E, (hereafter designated as u), where the energies are referred to the shifted band edge [Sl in the sense of the second order renormalized perturbation. The following discussions are restricted to the case of zerotemperature, infinitely large system size and in-

The electron system of our present concern in the absence of magnetic field is modelled by a simple one-band TB Hamiltonian;

itj

t2)

where R denotes the volume of cubic sample of size L, B the magnetic field, fthe Fermi function, and vX,v,,the velocity operators. G + (G-) represents the retarded (advanced) one-electron Green operator being relevant to the TB Hamiltonian, eq.(l), and -Z*W indicates the ensemble average over a distribution of random site-energy vr,which is here assumed to be a Gaussian. The central issue of our approach is to cast the Hall conductivity into the form; o,, =

H = Vc lixjl

1

B.V.

All rights reserved.

446

H. Okamoto et al. / Hall effect near the mobility edge

f n e s m a ma0neic so ha h e m o b i y edge structure remains unchanged and well-defined,

2.1. Perturbative A p p r o a c h We start by evaluating the Hall conductivity ,eq.(2), in a perturbation series in terms of standard diagram technique. The Tr part is described by Feynman graphs including four velocity (current) verteces, of which the leading (classical) term is illustrated in Fig.1 (a), where solid lines with an arrow indicate renormalized Green function Gk in the momentum representation. The quantum correction terms are comprised of Cooperon (wavy line with an arrow) and diffusion (wavy line) propagators defined in Figs.1 (b) and (c), in the context of the Langer-Neal correction [4]. These propagators are associated with a common factor

/-(q) = (4g'2A O'i2/~'2U) /q2 ,

(4)

A being the electron mean free path in the vicinity of the mobility edge, and ¢~=the imaginary part of the self-energy [3, 5].

~

vy ~

~

v,

(a) ~

Tr

v

= x

y+

-k+q

k'

=

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~

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~

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I

=

k-q rd (q)

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+

-

~

= ~

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=

-k+q (a2)

cP'-',7--"~=

(a4)

(a3) _ ....,...,~,._~ _

_ ....._:,,_..~ +

(b)

(c)

Figure 2. Quantum correction terms of first order (a), and second order (b) and (c) in 1/u. which are different in the configuration concerning two additional velocity verteces and single scatter lines. Direct evaluations of these graphs yield the Hall conductivity with quantum correction function; up to second order in the form of the spectral

I A 2

U

I.~_~

~

k

(al)

,.

(c) k

+

hL(U) = 1 +

I

,.,

Vy

• • •

(b) k

vx i

i + ...

~=...~,.L~~

Figure l. Feynman graph for the leading term of the Hall conductivity (a), structure of Cooperon (b) and diffusion (c) propagators, where broken lines represent single scattering process,

Graphs giving the singular contribution to first order correction consist of one Cooperon propagator, as sketched in Fig.2 (a). Second order correction is due to 8 groups of graphs which contain two or three propagators, among which the structures of two representative groups are shown in Figs.2 (b) and (c). Note here that each group contains more than 20 contributing graphs

2

U2 '

with coefficients ci=-2, c.=-I/4 and c '2=-5/4. " " ~ " " The coeffzczent c~ agrees wzth that achzeved wzth different procedures[6], while the second order correction terms are identified first in this work. 2.2. R e n o r m a l i z a t i o n - G r o u p Approach As is easily recognized, eq.(5) is valid only for large u, that is, the energy region far above the mobility edge, whereas our primary focus is placed on the Hall effect near the mobility edge. It is then required to establish quantum correction terms beyond second order in 1/u, which however involves considerable elaboration if attempted in the perturbation procedure. An alternative way is to rely on the scaling hypothesis for the Hall effect, as does in the localization problems, which has been proven correct from the microscopic point of view [4].

H. Okamoto et al.

/

Hall effect near the mobility edge

On the basis of the perturbative result eq.(5), the scaling hypothesis is formulated into two coupled renormalization-group equations; d Inh'L _ 1 - 1__ + ;LL, N d In L gL hL

(6)

dln;LL l + d =l - 1n~-L L(

(7)

~-L5[ LA-~gL]) '

where

"hL(U) = (U 2 L~ A)hL(U ) ,

(8)

and gL denotes the dimensionless conductance which obeys [4] d IngL _ 1 - 1__ (9) d In L gL" For eq.(7) to hold universality, the right-hand side should not contain the system size L explicitly. It is then assumed that, if higher order correction terms are fairly taken into account, the term [ LU/AgL] is to be replaced by unity. By integrating these coupled renormalization-group equations with initial conditions ;L.A = -u-5/2, h. = 1 and gA = U, the spectral function h(u) for t~e infinitely large system is found h(u) = lira hL(U) L-->oo u - ul (- 1 + 2 u 0.5

....

, ....

mobility z

551l nu- [ u _ - - ~ U l ] ) 2 , ~ ....

, ....

, ....

, ....

(10)

while thewell-known conductivity (o)spectrum is written 1-1/u. Expanding eq.(10) up to terms of order 1/u 2 immediately reproduces the original perturbative result, eq.(5), indicating the selfconsistency of our scaling procedure. More important is that higher correction terms are incorporated here in a natural way, so that the spectral function h(u) can encompass the electron behavior near the mobility edge. How the spectral function h(u), that is the microscopic Hall conductivity oh(e), behaves is illustrated in Fig.3. The negative sign of h(u) implies "anomalous" Hall sign; positive oh(e) for electron conduction. It turns out that o,(e) vanishes as o(e) does when the mobility edge ec (u=l) is approached from above, and there the Hall sign is invariably anomalous. The sign reverts to normal for an energy beyond u = de c = 2.5. An equivalent argument stands for hole conduction near the valence band edge (negative for hole), providing a successful interpretation for the double sign anomaly [1] for the mobility edge transport. 3. MACROSCOPIC HALL MOBILITY Putting eq.(10) into eq.(3) permits us to predict the magnitude of the Hall mobility ~h against the carrier mean free path A via the relation ~ = (c/B)(crh/c), with o being the conductivity. The resuit is illustrated in Fig.4 for various values of Vl kBT, which might cover practical range of the transfer energy V (or the effective mass) and measurement temperature T; to be specific, V~ kBT=50 is compatible with the case of electrons

.

edge

0

o z

447

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>~

0.15

"~

0.1

>-

0.05

• • , • • • , . . . ~ • • . , . . . , . .

5

"

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//

-1

. . . .

t 1.5

. . . .

, . . . .

2 REDUCED

, . . . .

2.5 ENERGY

-I -0.05 0 ~_

, . . . . . ,. . .

3

3.5

~

4

u

Figure 3. Hall conductivity spectral function h(u) against reduced energy u = e/e c.

-0.1

0

2 NORMALIZED

4

6 MEAN

8 FREE

PATH

10

12

..Va

Figure 4. Hall mobility t~ plotted against the normalized mean free path Ala.

448

H. Okamoto et al. / Hall effect near the mobility edge

in a-Si:H at room temperature [5]. For 1< A/a <5 ("a" being the averaged atomic spacing), ~ is found positive (anomalous) and increases with A/a, reaching about 0.1 - 0.15 cm2/Vs. Also found in this range of A is a weak but positive temperature dependence for V/kaT>50. These behaviors meet with experimental observations on a wide range of a-Si:H alloys. It follows then that typical magnitude of P-h= 0.1 cm2Ns for electron conduction in a-Si:H [7] implies A/a = 4 - 5, or A = 10 - 12A for a=2.4/~, Another feature of significance read from Fig.4 is a reversal to normal Hall sign, which occurs at 7< A/a <14, depending on V/kBT. T h e critical mean free path corresponds to 17A< A <34A, if a is set 2.4A (a-Si:H). This kind of sign switch from anomalous to normal has been actually observed in p-c-Si:H with crystalline size greater than 20A [7]. Provided that the crystalline size determines A in such material system, then an agreement between the experimental and theoretical results would provide evidence in favour of our theoretical approach, The conductivity mobility Pc is a fundamental quantity of our greatest interest. However it is hardly accessed by standard transport measurements owing to carrier interactions between the band edge and shallow-lying states. If a definite relation is established between I.~ and Pc, then p-c is to be evaluated through I-~ measured directly from the Hall measurements like in crystalline semiconductors. To examine this issue, the mobility ratio p.h/pc is calculated and displayed against N a in Fig.5, again with a parameter V~ kaT. Both the mobilities appear to be linked in a 1°"1

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'

'

'

"

' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

ANOMALOUSHALLSIGN

O

" -=.,

~

~

o

o

complicated manner, particularly for large A region, where I~ tends to change its sign. However, if only the region of 2< N a <5 is considered, an approximate relation p-all1c = 2.8 (kBT/V) (a/A) is likely to hold. Employing a typical value !.~ = 0.1 cm2/Vs for electron conduction in a-Si:H with V/kBT=50 and A/a = 5, suggests Pc = 9 cm2/ Vs, whicT= does not contradict that inferred from the temperature dependence of the drift mobility. Finally, we comment on the relation between the magnitude of the Hall mobility and the characteristic energy of the Urbach tail Eu [8] which is widely accepted as a measure of structural disorder involved in the material. Noting a proportionality Eu =: V (a/A) [3,5], it is straightforward to see from Fig.4 that the Hall mobility p-h roughly scales as 1/E u when1< AJa <5. 4. CONCLUSIONS Microscopic theory of the weak-field Hall effect near the mobility edge has been developed on the basis of perturbative renormalizationgroup procedure in a one-electron scheme. The results satisfactorily account for the sign and magnitude of the Hall conductivity for mobility edge transport in both the qualitative and quantitative aspects. A simple relation has been found between Hall and conductivity mobilities, which would offer an extended opportunity for the determination of conductivity mobility from Hall measurements. REFERENCES 1.

reviewed b y N . F . Mott, Philos. M a g . , B 6 3

2. 3.

M. Itoh, Phys. Rev.B, 45 (1992) 4241. E.N. Economou, C.M. Soukoulis, M.H. Cohen and A.D. Zdetsis, Phys. Rev.B, 31 (1985) 6712. reviewed by A. Kawabata, Prog. Theor. Phys., suppl., 84 (1985) 16. H. Okamoto, K. Hattori and Y. Hamakawa, J. Non-Cryst. Solids, 137&138 (1991)627. B. Shapiro and E. Abrahams, Phys. Rev.B, 24

4.

1°'= 5. _. J

O

V/k.T=25 /

6.

50 ~5 " ~

7. lo~

' ' ' J ~ 0 2

............. 4 6 8

10

NORMALIZEDMEANFREEPATH AJa

12

Figure 5. Ratio of the Hall mobility It, to the conductivity mobility Ix=as a function of the normalized mean free path A/a.

(1991) 3.

(1981)4025.

W.E. Spear and P.G. LeComber, in "The Physics of Hydrogenated Amorphous Silicon I", ed. by J.D. Joannoupoulos and G. Lucovsky, Sprin-

ger-Verlag (1984) chapt.3. 8.

C.E. Nebel and R.A. Street, MRS Symposium on Amorphous Silicon Technology, San Francisco (1993) in press.