Time evolution of scattering states and velocity increase due to nonlinear processes in the quantum Hall regime

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Time evolution of scattering states and velocity increase due to nonlinear processes in the quantum Hall regime

J. Riess, C. Duport

To cite this version:

J. Riess, C. Duport. Time evolution of scattering states and velocity increase due to nonlinear pro- cesses in the quantum Hall regime. Journal de Physique I, EDP Sciences, 1991, 1 (4), pp.515-521.

�10.1051/jp1:1991149�. �jpa-00246347�

J.

Phys.

I1

(1991)

515-521 AVRIL 1991, PAGE 515

Classification

Physics

Abstracts

72.20M 73.20D 73.20J

Time evolution of scattering states and velocity increase due to nonfinear processes in the quantum ^Han regime

J. Riess and C.

Duport (*)

Centre de Recherches sur [es Trds Basses

Tempbratures,

C.N.R.S., B.P. 166X, 38042 Grenoble Cedex, France

(Received23

November 1990,

accepted

3 January J991)

Abstract. We report ^{the first} numerical results

(with

realistic parameter

values)

for the time evolution of a scattered Landau function in _a model system

recently proposed by

one of the authors.

They give

_a

striking

illustration for the Hall

velocity

increase

beyond

the classical value of the conduction electrons in the quantum Hall

regime.

This

phenomenon,

which is crucial for the

integer

_quantum Hall effect, is caused by a

special

kind of nonclassical

particle dynamics

induced

by

disorder and cannot be described

by

linear response

theory.

1. Inwoducfion.

The

integer

_quantum Hall effect

[I] (IQHE)

in

large samples

results from _a localization- delocalization process caused

by

disorder in the presence of _a

higli magnetic

field. The

microscopic

details of this process are stiff not

fully understood,

and

despite

considerable effort in the last decade the theoretical situation remains controversial

(recent

reviews _are

given

in

Refs.[2, 3]).

Most

recently

substantial _progress has been achieved

[4-6]

in

understanding

the

scattering

mechanism which is

responsible

for the

IQHE (associated

with bulk

states).

It _was found that the nature of the time-evolution of the electron states is crucial for the

IQHE

and

further,

that in the quantum ^Hall

regime

_an essential part of the individual

particle

currents is non-linear with

respect

to the electric field E

(while

the

macroscopic

Hall

current is linear in

E).

These results open new

perspectives

for the

theory

of the

IQHE.

In this article the

scattering

_process in the

quantum

Hall

regime

is illustrated in this _new theoretical framework. We

investigate numerically

the time evolution of _a scattered Landau function in _a model system, ^{where the} current

carrying

states can be calculated

explicitly.

This enables _one to understand the detailed

mechanism,

which _causes the increase of the Hall

velocity

of the

conducting

electrons in the presence of disorder. This increase of the

single- particle velocity beyond

the classical value results from contributions which _are nonlinear in the electric field

[5, 6].

^Since most

previous

theories of the

IQHE

_are based _on finear _response

approximations,

it is

important

to

investigate

the order of

magnitude

^of these nonlinear terrns in _a model with realistic

physical

parameters. In this article _we will calculate the time

(*)

Present address: Ecole Normale

Supdrieure

de

Lyon,

46, Allde d'Italie, F-69364 Cedex 07,

France.

516 JOURNAL DE

PHYSIQUE

I bt 4

evolution of _a

conducting

state in the center of the broadened Landau

band,

and _we will _see that the nonlinear contribution to the

velocity by

far exceeds the linear term. Therefore the present calculation confirms

previous

results

[5,

6] ^which suggest ^{that linear} response

theory

is not

adequate

for _a

microscopic description

of bulk states in the quantum ^Hall

regime.

2.

Description

of the model system.

We consider electrons _{on a}

long strip (of

width

L~)

^{in the}

x-direction, subject

to a

magnetic

field B

=

(0, 0, B),

and to _an electric field E

=

(0, E~, 0).

^{In addition} a substrate

potential V(x, y)

=

V(x)

₊

V~(x, y)

is

present,

where

V(x)

is _{a sequence} of barriers and wells which vary

slowly

_{over a}

magnetic length (Fig. I)

and

V~(x, y)

is _a

homogeneous

disorder

potential (the y-dependence

is

important).

The Hamiltonian of _an electron has the form

H

=

(1/2 m) )

^~

+

) _{(q/c) jBx}

₊

_(t)/Lyj ⁾

₊

_{v(x, y)}

,

with

# (t)

₌

cE~ L~

t and

periodic boundary

conditions in

y-direction (this

restricts the

description

to bulk

states).

For further details and motivation of this model _see references

[4- 6].

ll~~ev)

~ Vlx)

j Eb

Q o j ^ED

# -Ea

~ l-Eb

_3

---~---~ -_ ~---~---~ ~

~---~--j

-800 Xo 0 XlX2 800 Xl~)

Fig.

I. Smooth substrate

potential

V

(xi

in the

region

of _a barrier. Also indicated is the nature of the orbitals in the _case, where in addition _to

V(x)

_a weak disorder

potential V'(x, _y),

a sufficiently small electric field E~ ^and a strong

magnetic

field B in z-direction _are present. The orbitals _are characterized

by

the

position

of the localization centres x~ of the

corresponding unperturbed (V~

= 0) ^orbitals

~~(x -x~),

see text. Full line

regions fully

nonadiabatic

(classically conducting)

orbitals; shaded

regions:

intermediate nonadiabatic orbitals

(composed

of

classically

and

nonclassically conducting

parts) _; dashed

region

_: adiabatic

(nonconducting)

orbitals. Figure I

corresponds

to the parameter

values d(~, = 0.5 _x 10~^~

(eV

)~, L~ =

0.I _cm, B

= 6T, E~ = 2.37 _x 10~^~ V _cm ^' V

(xi

also represents the energy E~ ^{of the}

unperturbed

orbitals

~~,

see

(I).

Outside the dashed

region unperturbed

and

perturbed energies

coincide _on the scale of the

figure.

In the absence of

V'

_{the solutions of the}

time-dependent Schroedinger equation

have the

approximate

form

[4] (in

the k-th Landau

band)

_:

~~

_~

^~~'

~' ~~Y~ ^~~~ ~~~ ~

"~~~~Y~

_~P,k(X> t

>

k,

_p

integer

,

where

u~,~ (x, t)

is the

product

of _a Herrnite

polynomial

and of _a Gaussian

g~(x, t)

centered at

xp(t)

₌

chP/(qBLy) 4 (t)/(BL~).

The

energies

_are

(in good approximation) E~

~ ⁼

hw

(k

₊

1/2)

₊

V(x~) (I)

(w

=

iqB/(mc)().

In the

following

_we consider _a

single

band and

drop

^the index k. Each

M 4 SCATTERING IN THE

QUANTUM

HALL REGIME 517

#~ (x,

y, ^t

)

describes _a

particle

localized at

x~(t)

and

moving

in x-direction with the constant, classical

velocity

_v

=

cE~/B.

^{For centers}

x~(t)

on the left

(right)

hand side of _a barrier in

figure

I

E~(t)

increases

(decreases)

with time. Therefore the

spectrum

^{consists of}

intersecting

levels

(Fig. 2).

In the presence of the disorder

potential ^V'

the _energy levels anticross and become

individually periodic

with

period

_r

=

hi iqE~ L~[.

^In our model the

physical parameters

are

chosen such that the

perturbed (V'

_#

₀₎

adiabatic states

(denoted w~(x,

_y,

t))

in the center

[- E~, E~]

of the band _can be described

by

_a weak disorder

approximation [4-6].

This enables

an

explicit

calculation of the states

w~(x,

_y,

t),

which here become linear combinations

c~(t) #~ (t)

+ c~,

(t) _#~, (t)

of

only

two

unperturbed

states at _a

given

value of _t

(but

the

pair

^of

indices

~p, p' ) changes periodically

whenever _t increases

by r/2),

and

they

_are identical with _a

single unperturbed

function

#~(x,

_y,

t)

for times

exactly

half way between two successive

anticrossings.

Such _an adiabatic state

w~(x,

_y,

t)

describes _{a wave}

packet moving

with the classical

velocity

_v, but which is

alternately

localized in _a small

neighbourhood

of _one of the

two fixed sites

x~(t*)

^and

x~,(t*)

situated _on

d@ferent

sides of _a barrier

(well).

(t*

is the time where

E~(t)

^and

_E~, (t) intersect.)

The whole motion is

periodic

in time

(with period r),

and therefore

corresponds

to _a

vanishing

dc-current.

For very low

E~

^all wave functions _are adiabatic. At

higher

values of

E~

nonadiabatic

/ _f/

~,

~qP' ^~

Wm

~~qS

,

P 7

r/

~y

~j 0

Ep

Epj~

_1 0 2 3 k

t/(t/2)

Fig.

2.-

Intersecting

energy levels

E~(t)

^of

unperturbed (V'

= 0) Landau functions localized _on different sides of _a barrier of

figure

I.

Ascending (descending)

levels represent localisation centers

J~(t)

_on the left

(right)

hand side of the barrier. In the absence of V'

an electron follows its level

continuously

_across intersections. In the presence of V'

a

splitting

of the _wave function _occurs at each intersection _one part ^{follows the}

unperturbed

function with

probability

_P~,~,

(nonadiabatic,

classical

part),

the other follows the

anticrossing

curve with

probability

I P~,~,

(adiabatic,

nonclassical

part).

Each thick dot represents ^the square of _an

expansion

coefficient

c~.(t

=

nT/2)

_(~ of _a scattered _wave function

#i(x,

_y,

t)

=

z c~,(t) ~~,(x,

y,

t)

with initial condition

c~,(t

=

0)

= 3~~,,

E~(0)

=

E~.

These values are calculated according to the scheme indicated

by

arrows

(e.g.

~ (x, _y, 3

~/2

) is distributed _on three _{centers on} the

right

and two centers on the left hand side of the barrier). The

separation

between

adjacent

centers x~ ^{is 0.7} x 10~ ^" _cm, which is not visible _on the scale of the figure.

518 JOURNAL DE PHYSIQUE I M 4

transitions

(Zener tunnellings)

become

possible (see Fig. 2).

The

probabilities

for these _non- adiabatic transitions _are

(see

Ref.

[5],

where _a factor 4 is

missing)

P~~,

= exp

(-

4 _w ^~

d(~,

exp

[-

2

(x~ x~,)~/(2 )~] El V'(x~) _chE~]

Here d~~, ^{is a Fourier} coefficient of

V'

_{as defined in reference}

[4],

A

=

(hc/qB['/~,

and

J~,

_x~, are situated _on

opposite

sides of _a barrier

(well)

such that

E~(t)

₌

E~,(t).

Since

P~~,

=

P~~j,~,~j

^the

probabilities only depend

_on the energy of intersection E of the

unperturbed

levels

E~(t), E~,(t).

3. Time evolution of _a scanered Landau function.

For Ee

[E~, E~]

or e

[- E~, E~]

the

probabilities P~

~, decrease from

practicafly

_one at E=

±E~

to

practically

_{zero at} E=

±E~.

In

these'intermediate

energy ^zones

[-E~, -E~]

and

[E~,E~]

each _wave function

undergoes

_a series of

splittings

at each time

nr/2,

n

integer,

_see

figure

2. Due to _a succession of such

splitting

events

(caused by

the

disorder

V')

_any initial state

#~(x,y,t)

^with _energy

E~(t)

ⁱⁿ the nonadiabatic

region

[- E~, E~] eventually

becomes _a

time-dependent

linear combination

£c~,(t) #~,(x,

_y,

t)

of

unperturbed

functions

#~,(x,

_y,

t)

with

energies _E~, (t)

_e

[- E~, E~].

States outside

[E~, E~]

remain

adiabatic,

since their nonadiabatic transition

probabilities

_are

negligible. They

do not mix with the states of the non-adiabatic

region [- E~, E~]

in the _course of time.

For the

present

calculations

E~

^and

E~

are defined such that

P~~,(± ^E~)

₌ ^{0.9999 and}

P~~,(±E~)

= 0.01. Further _we set

P~~>(E)

₌ ^{I for Ee}

^[-E~,E~]

^and

P~~,(E)

=

0 for

(E( ~E~.

The

parameter

^{values used} can be taken from

figure

I. The

magnetic

field B

=

6T

corresponds

to a

cyclotron

_energy hw

= l0meV. This _means that the smooth

potential V(x)

of

figure

^I fluctuates in space with _an

amplitude

of about 0.3 hw and the

rapidly varying potential V'(x, y)

with _an

amplitude

of the order of 0.I hw

(determined by

id~~,

[).

These are realistic values for quantum ^Hall devices

(see

_e.g. Sect. 3.2 of

[2]).

For the

following only

the absolute _squares

[c~>(t)

_[~ ^{of the}

expansion

coefficients after _a

large

number _n of the

(extremely short) splitting

intervals

r/2

_are needed. Due to

phase

randomness [4]

they

_can be calculated

using only

the

splitting probabilities P~~,

^and

I

P~,~,

^{at the level}

anti-crossings.

We now consider the time evolution of _a function

# (x,

_y,

t)

=

£ c~,(t) #~, (x,

y,

t),

which

at t ₌ 0 is chosen to be _a

single unperturbed

Landau function

#~(x,

_y,

t) (I.e.,

_c~,

=

8~~,),

^{whose center}

x~(0)

^{is located} at J~ on the left hand side of the

potential

barrier in

figure I,

with _an energy

E~(0)

situated at the _upper

edge E~

^of the

fully

non-adiabatic interval

[- E~, E~ ]. Figure

3 shows the numerical values for the

probability

coefficients

c~,

(nr /2)

^~ of this _wave function for times _t

=

nr/2,

_n

~

0, integer.

The

corresponding

centers

x~,(nr/2),

which _are

occupied

with the

probability ic~,(nr/2)i~,

represent a finite _set of the

fixed,

discrete

points x~(0)

_{w x~} on the x-axis.

Figure

3 shows the

occupation

of these fixed centers

J~

^{with their}

corresponding probabilities

at different times

nr/2.

It illustrates the

spreading

in x-direction of the

single-particle density

# # ^*

(x,

_y,

nr/2 dy

=

£

_c~,

(nr/2

^~ _u~,

(x)~

_{at a}

time

nr/2, I.e.,

after _n

splitting

intervals.

At t

=

0,

when the center

x~(t)

of the

original, unperturbed

function

#~(x,

_y,

t)

enters the interrnediate _zone

[E~, E~]

_on the left of the barrier

(at x~(0)

_{= xo,} ^E=

E~,

where the

probabilities _P~,~,

start to become smaller than

one),

it starts to be

partly

scattered to the

right

hand side of the barrier

(to

_xi at t ₌

0,

and to x _~ xi for _t

~

0) according

to the scheme of

M 4 SCATTERING IN THE

QUANTUM

HALL REGIME 519

1.

n=0

it=01

o

lo.69

n ₌ 2000

o

0.25

n=~000

63%

0

n= 6000

lo

77 o

n=8000 10"

81%

-~~~ -~~~

x ii ^{~~~ ~~~ ~~~}

Fig. 3. Time evolution of the scattered orbital ~ (x, _y, t)

=

jj

_c~.(t) ~~. (x, y, ii with initial condition

as in figure 2 (I.e.

x~(0)

₌ xo = 423

h

see Fig. I). Shown _are the

probabilities c~,(t)

_(~ as a function of the

position

of the centers

x~,(t)

of the

occupied

Landau functions

~~,(x,

y,

t)

at different times t ₌ nr/2, _n

integer

_{; T}

=

h/(qE~L~)

=

1.72 _x 10~~

s.

figure

2. This

implies

that

during

_every

subsequent splitting

interval of

length r/2

_a fraction of each

probability [c~,(nr/2)

_i~ on the

right

hand

side,

which

corresponds

to an

unperturbed

state with _energy

E~,(nr/2)

e

[E~, E~],

is backscattered to the

original (left hand)

side. This

causes a tail in the

probabilities

behind the unscattered front

peak

_on the left hand side of the

barrier

(Fig. 3).

The

position

of this front

peak corresponds

to the _center

x~(nr/2)

^{of the}

unperturbed

Landau function

#~ (x,

_y, t

)

at t

= nr

/2.

The

probability

c~

(nr/2)

^~ of this front

peak

diminishes with

increasing time,

since at each time

nr/2

it is

multiplied

with _{a new} Zener

probability _P~~, (which

itself decreases with

increasing n).

In _our numerical

example

this front

peak probability

is

practically

reduced to zero after _n

=

8 000

splitting

intervals

(correspon- ding

to 8 000

r/2

= 6.9 _x

10~~

_{s, see}

Fig. 3).

This _means that

by

this time the

totality

of the

original unperturbed

Landau function

#~(x,

_y,

t)

has been scattered into _{a sum} of other

unperturbed

Landau functions

#~,(x,y,t). Figure

3 illustrates the

fact,

that the total

probability

of the electron _on the

original

side of the

barrier, I.e.,

the _sum of all the

probabilities _ic~,(~ corresponding

to

occupied

centers on the left hand side of the

barrier,

diminishes with

time,

whereas the total

probability

on the

right

hand side of the barrier increases.

The average

position (x(nr/2))

of the electron at t

=

nr/2

is

given by £

_x~,

ic~,(nr/2)

_i~,

I.e.,

it is determined

by

the

occupation probabilities

of the centers x~, ^{at t} =

nr/2.

In the absence of disorder

(V~

=

0) (x(nr/2))

is

equal

to

x~(nr/2)

= J~ +

(cE~/B) nr/2

=

520 JOURNAL DE

PHYSIQUE

I M 4

J~j~~~,

describing

_a motion with the constant, classical

velocity

_v

=

cE~/B.

^{In the} presence of V~ the front

peak

_on the left hand side of the barrier _moves with this

velocity,

but the average

position (x(nr/2))

is ahead of the classical

position

_x~j~~~,

I-e-,

the effective

velocity

u~~ =

(d/dt) (x(t) )

of the electron is increased with

respect

to the classical value u. Thds _can be understood from the

general

nature of the

splitting

_process shown in

figure

2.

Figures

3

and 4 illustrate this

numerically

_: in

figure

4 u~~ increases from _u

=

cE~/B

at t ₌ 0 to

roughly

40 times this value between t _m 0.5 _x

10~~

_s and _t

m 3 _x

10~~

s. At iater times u~~

decreases,

and it .would

asymptotically

reach the classical value u, when

practically

all the

particle density

is _on the

fight

hand side of the barrier.

~oo

<x> (li

~ll"~~~°

v,

,

o I'

II/

,'

/ ,

/ x class.

0 3.G~ 6.88

til0'~sl

Fig.

4.-

Average position (x(t))

of the

particle corresponding

to the calculated _wave function

#(x, _y,

t)

shown in

figure

3. The dashed line _{serves as a}

guideline

for the eye. As _a

comparison

the

position

x~j~~(t) associated with the

unperturbed,

initial Landau function is

represented.

This

asymptotic

situation will _never be

reached,

since the front of the

occupied

centers

J~,

^{on the}

right

hand side of the barrier

eventually

will reach x~,

I-e-,

the

beginning

of the next intermediate _zone

(caused by

_a

potential well,

_see

Fig. I),

and the

corresponding probability

(c~,(~ will then itself be

split

and

gradually

scattered _across the

well,

in the _same way as the

probability (c~(t

=

0)

_(~

= l of the

incoming

Landau function has been scattered _across the barrier for _t

> 0. The _same scenario will go on with all the other

probabilities

_c~,

(t)

_~, which reach this next intermediate _zone at

subsequent

times. As _a consequence, after successive scatterin~s _across many barriers and

wells,

the _wave function

#(x,

_y,

t)

will be

considerably spread

in

x-direction,

and it will be located

simultaneously

_on different sides of di~erent

barriers and wells.

As _a result of the disorder-induced

scattering

_process the «>a,>e function

#(x,

_y,

t)

is _a

superposition

of parts, ^which propa~ate ^{with the classical}

velocity

u ₌

cE~/B (corresponding

to nonadiabatic

transitions)

and of parts, which propagate

non-classically (corresponding

to adiabatic

behaviour).

Since _a fraction of the latter

gives

rise to _an effective

velocity

much faster than _u, the

dc-velocity (I Iv) (x(nr/2

+ r

)) (x(nr/2))

of the electron tends to a

constant value

(for sufficiently large times)

which is

consid@rably higher

than the classical value _u. This increase with

respect

to u of the effective velocities of the

dc.conducting

states

represents

the so-called

compensating

current

[7],

which is

responsible

for the

quantization

of the Hall conAuctance-

4. Discwsion.

The discussed

particle propagation

is

classically impossible (and

_contrary to _common

sense).

In the absence of the static

potential V(x,

_y

),

the classical and

quantum

^{mechanical behaviour}

M 4 SCATTERING IN THE QUANTUM ^HALL REGIME 521

coincide the

particle

_moves with the constant

velocity

u ₌

cE~/B.

^If we add _a

potential V(x, y)

which describes obstacles and disorder _one

classically

_{expects a} reduced

averaged velocity.

But the

quantum

^mechanical

analysis

shows that for

sufficiently

small

E~

^the

opposite

can be true : the average Hall

velocity

is increased. This is shown in _a

striking

manner

by

_our

calculation,

which is based _on realistic

physical parameters.

^{Here the}

velocity

increases

by

_a factor up ^to

40, I-e-,

it is not a small effect. The described mechanism

leading

to this

phenomenon

is

time-dependent

and reflects the

fact,

that in

quantum

^mechanics a

particle

localized at one site may

disappear

and

simultaneously

_reappear at a different site.

Classically

such _a behaviour is

impossible.

The

importance

for the

IQHE

of this kind of nonclassical

propagation

has first been

pointed

out in reference

[4].

^{For its} correct

quantum

^mechanical

description

^the

full

time-

dependence

of the _wave function is _necessary

[5, 6],

^linear _response

theory

^is not sufficient.

These results _seem to be in contradiction with most

previous

theories of the

quantum

^Hall

effect,

which _are based _on linear response

theory.

We should like to make the

following

comment _on this

point (suggested by

_our present ^and

previous [4-6] results)

: In the Hall

plateau region

the total

macroscopic

Hall current is linear in the electric field. Therefore the

quantization

of the Hall conductance _can be described

by

_a linear response

theory,

But it appears that this is

only possible

if additional

assumptions

are introduced

(this

is consistent with _a recent

analysis [8, _9]

made in _a different

context).

On the other hand _a linear response

approach

based _on the

Schroedinger equation

alone should not be able to

correctly

describe the

quantization

of the Hall conductance. A detailed reexamination of all such theories

published previously

is not the purpose of _our article.

(This

would be _an immense

piece

of work and may lead to delicate mathematical

discussions.)

We _are however _aware of _{one case} of such _a

reexamination,

which is

significant

in _our

opinion.

It _concerns the

topological approach [10]

to the quantum ^Hall effect. This

theory

is based _on the Kubo

formula,

and it has been considered to be _an

important

contribution to the

understanding

of the

IQHE.

However _a recent

analysis

has

shown,

that the

topological

_argument is

inacceptable

for

purely

mathematical _reasons

[11].

Therefore in this _{case no} contradiction with _our

present

and

previous [4-6]

results remains.

In conclusion _we have calculated the time evolution of _a scattered Landau function in _a model with realistic

parameters

^and

found,

that nonlinear _processes cannot be

neglected

in _a

microscopic description

of the

particle

velocities of bulk states in the quantum ^Hall

regime.

Such processes should therefore be

investigated

_more

thoroughly

in the future.

References

[ii

VON KLITzING K., DORDA G. and PEPPER M.,

Phys.

Rev. Lett. 45

(1980)

494.

[2] ^The Quantum Hall Effect, R. E.

Prange

and S. M. Girvin Eds.

(Springer-Verlag,

New

York)

1987.

[3] ^HAJDU J., Ten Years of Quantum Hall Effect _:

Development

and present state of

Theory

in The

Application

of

High Magnetic

Fields in Semiconductor

Physics

III,

Springer

Ser. in Solid State Sci., G. Landwehr Ed.

(Springer Verlag, Berlin),

to be

published.

[4] RIESS J., Z.

Phys.

B 77

(1989)

69.

[5] RIESS J., Phys. Rev, B

41(1990)

5251.

[6] RIESS J., J. Phys. France 51

(1990)

815.

[7] Current

compensation

_was first discussed

by

PRANGE R. E.,

Phys.

Rev. B 23

(1981)

4802 _; for _a

recent discussion _see references

[8,

5].

[8] ^{JANSSEN M.} and HAJDU J., Z.

Phys.

B 70

(1988)

461.

[9] HAJDU J., JANSSEN M. and VIEHWEGER O., Z.

Phys.

B 66

(1987)

433.

[10] For _a review _see THouLEss D. J.,

chapter

4 of reference [2].

[I

Ii

RlEss J.,

Europhys.

Len. 12

(1990)

253 _; Solid Stale Commun. 74 (1990) 1257.