Quantum transmission in disordered insulators: random matrix theory and transverse localization

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matrix theory and transverse localization

Yshai Avishai, Jean-Louis Pichaxd, Khandker Muttalib

To cite this version:

Yshai Avishai, Jean-Louis Pichaxd, Khandker Muttalib. Quantum transmission in disordered insula-

tors: random matrix theory and transverse localization. Journal de Physique I, EDP Sciences, 1993,

3 (11), pp.2343-2368. �10.1051/jp1:1993249�. �jpa-00246872�

Classification

Physics

Abstracts

05.60 72.10B 72.15R

Quantum transmission in disordered insulators: random matrix

theory and transverse localization

Yshai Avishai (~>~), Jean-Louis Pichard

(~)

^{and Khandker A. Muttalib}

(3)

(~)

C-E-A-,

Service de

Physique

de

I'#tat _Condensd,

_Centre

d'#tudes

_de

_Saclay,

_{91191 Gif}

sur

Yvette

Cedex,

France

(~) Department of

physics,

Ben Gourion university, ^Beer Sheva, Israel (~)

Physics Department, University

of

Florida, Gainesville,

Fl 32611, U-S-A-

(lleceived

20

April

1993, received in final form 12

July1993, accepted

15

July 1993)

R4sum6, Nous considdrons les interfdrences

quantiques

de

trajectoires dlectroniques

clas-

siquement permises

_ou interdites dans des

didlectriques

ddsordonnds. Sans faire _une

approxima-

tion de chemins

dirigds,

_nous

reprdsentons

_un dilfuseur

dlastique

trbs ddsordonnd par sa matrice de transmission t. Nous

rappelons

comment la distribution des valeurs propres de

t.tt

_{peut ktre}

obtenue I

partir

d'un certain ansatz conduisant £ _une

analogie

_{avec un gaz} de Coulomb £

une

temperature

fl~~ qui depend

des

symdtries

du

systbme.

Nous

rappelons

les

consdquences

de

cette th40rie de matrices aldatoires _pour des isolants

quasi-id

et _nous dtendons notre 4tude £

des modbles

microscopiques

tridimensionnels oh la localisation transverse est

prdsente.

Pour des cubes de taille

L,

suivant la valeur de la

longueur

de localisation

(,

_nous trouvons deux

rdgimes

pour les spectres ^de

t.tt. Quand

L

If

m 1-5, la distibution des espacements entre valeurs _propres

reste

proche

de celle de

Wigner (repulsion

des valeurs

propres).

Le _cross-over habituel _entre

les _cas

orthogonal

et unitaire _a lieu pour un

champ magndtique appliqud

dlevd AB _m

4lo/(~

oh 4lo est le quantum de flux. Ce mkme

champ

rdduit les fluctuations de conductance _g ainsi que la _moyenne de

log(g) (augmentation

de

()

et induit _{sur un} dchantillon donna de

grandes

fluctuations de la

magndtoconductance

d'une

ampleur comparable

£ celles d'4chantillon £ dchan- tillon

(comportement ergodique). Quand

( est de l'ordre de la maille du

rdseau, (L/(

»

5-6),

la

rdpulsion

des valeurs _propres est

plus

faible et l'elfet du

champ magndtique

_cesse d'ktre im- portant. Dans _ces deux

rdgimes,

les fluctuations d'dchantillon h dchantillon de

log(g)

sont bien ddcrites _{par une}

gaussienne

^£ un

parambtre

et

(~~

varie lindairement _en fonction du

parambtre

de ddsordre. Dans _ces modbles

microscopiques,

la

ddpendance

_en fonction de

l'energie

de Fermi de la conductance _est trbs

comparable

£ celle rdcemment observde _sur de petits fits de GaAs:Si.

Abstract. We consider quantum interferences of

classically

allowed _or forbidden electronic

trajectories

in disordered dielectrics. Without

assuming

_a directed

path approximation,

_we represent _a

strongly

disordered elastic scatterer

by

its transmission matrix t. We recall how the

eigenvalue

distribution of

t.tt

can be obtained from _a certain ansatz

leading

to _a Coulomb gas

analogy

at _a temperature

fl~~

which

depends

_on the system symmetries. We recall the

consequences of this random matrix

theory

for

quasi-id

^insulators and _we extend _our

study

to microscopic three-dimensional models in the presence of transverse localization. For cubes of size L, _we find two

re#mes

for the spectra of

t.tt

as a function of the localization

length

(. For

L/(

_@ 1-5, the

eigenvalue spacing

distribution remains close to the

Wigner

surmise

(eigenvalue repulsion).

The usual

orthogonal-unitary

_cross-over is observed for

large magnetic

field

change

AB _m

4lo/(~

where 4lo denotes the flux quantum. This field reduces the conductance

fluctuations and the average

log-conductance (increase

of

()

and induces

on a

given sample large magneto-conductance

fluctuations of

typical magnitude

similar to the

sample

to

sample

fluctuations

(ergodic behaviour).

When

(

is of the order of the lattice

spacing (L/(

_»

5-6),

the

eigenvalue repulsion

is weaker and the removal of _a time reversal symmetry has _{a more}

negligible

role. In those two

regimes,

the

sample

to

sample

fluctuations of the

log-conductance

_are close to _a normal one-parameter

distribution,

and

(~~ depends linearly

_on the disorder parameter.

In these

microscopic models,

the Fermi energy

dependence

of the conductance is _very similar to the _one

recently

observed in small GaAs:Si wires.

1. Introduction.

When _one has to tackle the

problem

of electronic

quantum transport,

one of the usual micro-

scopic starting points

is the familiar Anderson

tight-binding

Hamiltonian:

H =

~ _e~a)a~

₊

~ _h~~a)a~ (i)

i <i,j>

where the _{e, are} the random site

energies

and the

hq

_represent the nearest

neighbor couplings

or transfer _terms. For

instance,

the _{e~ can} be random with _a

rectangular

distribution of width W

(Anderson model)

_{or can} take at random two values _{ea or eb}

(binary alloy model).

The

hq

can take for

simplicity

a uniform value h if

I, j

_are nearest

neighbors,

0 otherwise. Perturbations could be done when _one of the two terms of H

(random

substrate electrostatic

potential

_or

kinetic

energy)

is

negligible compared

to the other. The

scaling theory

of localization

iii

assumes the existence of _a universal

fl(g)-function

of conductance _g

relating

those two limits

through

the

mobility edge

in three dimensions.

For disordered metals whose

quantum

conductance _g, measured in unit of

e~/h,

is

larger

than _one, the conventional

perturbation approach

starts from

plane-wave

like states which _are

only weakly perturbed by

the

scattering

caused

by

the random substrate

(W/h

«

I).

In this

limit,

electronic

transport essentially

results from

quantum interferences of classically

allowed

diffusive paths

and

quantum theory

is close to its semi-classical limit.

(I)

We know for the three different

possible symmetries (orthogonal, _unitary

and

symplectic cases)

the four

loop

order

expansions

_{[2] in 2 + e} dimensions of the

scaling

functions

fl(g).

One

gets

^after

integration

the weak localization corrections to Boltzmann

conductance,

I.e. how

< g >

depends

_on the

sample

size

L,

the brackets

denoting

the ensemble _average value.

(ii)

The

sample

to

sample

conductance fluctuations _around the ensemble _average

essentially satisfy

Gaussian distributions whose variance

depends only

_on the basic

symmetries

and

slightly

on the

dimensionality

of the

system

_[3]

(Universal

Conductance

Fluctuations).

(iii)

The conductance fluctuations induced _{on a}

given sample by

the variation of _an

applied magnetic

field _or of the Fermi enegy ^are of similar

magnitude

_[4]

(ergodic behaviour).

An

applied magnetic

^field removes time reversal

symmetry, suppressing

the weak

(anti)

localization correction for _{< g >} in the absence

(in

the

presence)

of

spin-orbit scattering, halving

the variance of the universal conductance fluctuations. The _cross-over field is in

general

small in

metals, corresponding

to the

application

of _a flux quantum 4lo

through

the

temperature

dependent

_area where electrons

keep

their quantum ^coherence.

(iv)

The conductance fluctuation

yielded by

_a local

reorganization

of the substrate

potential (single impurity move)

have been understood

mostly

from known results _on random

walks, yielding

_a

possible explanation

for universal

amplitude

of observed

II f-noises _[5].

For Anderson

insulators, W/h

_» I and it is better to

perturb

^the

point-like-localized eigenstates

of the random term in

(I) by

the kinetic transfer _term. The

perturbation

_removes

the

possible degeneracies

of the _e~

(which

is

huge

in the

binary alloy model)

and the

eigenstates

of

(I)

_are still localized _over domains of

typical

size

(

which _can extend _over many

impurities

in the

vicinity

of the

mobility edge.

The Green function

G(E)

between two sites I and

j

_can be

expanded

in powers of the

hopping integral

h

<

ilG(E) Ii

>=

~ fl II (

(2)

Here _we have also introduced _a

magnetic phase

factor

Ar

for

generality.

The

paths

r

going

from I to

j

_are in this _{case a}

priori highly

_non

classical,

and electronic

transport mainly

results from

classically forbiden trajectories

which

interfere

due to the random

sign of

the denominator

and to the vector

potential.

When all the factors

h/(E e,)

<

I,

^{the shortest directed}

paths

rd

need

only

to be considered [6] and electron

trajectories

with self intersections have _no

significant

contribution in this limit:

~,Ar~

<

ilG(E)lJ

_>m

^h~ ~ fl

~ _~

~~ (3)

rd ~ird) ^{~ ~}

Here,

the

products

_are restricted to the ordered sequences of sites I

belonging

to the directed

paths rd.

The combinatorics of the shortest directed

paths rd

of

length

L _can be

simply

obtained if _{some cases}

(e.g

if the sites I and

j

stand _at the

opposite

corners of _a square

lattice).

More

generally,

these directed

paths relating

two

points separated by

L have transverse fluctuations

extending

_over

(L()(. _Assuming

standard

diffusion,

_one

_gets (

=

1/2

but _{a more}

detailed

analysis

shows that dominant

paths

_are

"superdiffusive", yielding

different values _[7]

for

(.

One finds that:

(I)

the localization

length ( characterizing

the

asymptotic decay

of the

propagator:

< logy ^<

ijG(E)jj

_{> j2} >=

-)

=

-2L(p

+

iii) ₍₄₎

depends

_on the disorder parameter ^W

through

_a local contribution

to

=

[log(W/h)]~~

which

comes from the

exponentially

small

hopping

term

[h/W]~

in

(3) (binary alloy

model with

e~ = +W at E ₌

0).

_po is _a

global

contribution

embodying

the

W-independent _quantum

interferences of the directed

paths

in this

particular

model

[8]. (

increases _{as a} function of _an

applied

field B _as

B~/~

_with

a universal

factor, giving

rise to _a

positive magnetoconductance

_[9].

This

sign

^{of the}

magnetoresistance

is not reversed

by

the _presence of _a

strong spin-orbit scattering [10],

as it is in disordered metals.

(ii)

The

sample

to

sample

conductance fluctuations _are

lognormally distributed,

but with anomalous size

dependence

of the fluctuations.

var(log(g))

=

C.L~~,

where the

prefactor

C does not

depend

_on the _mean <

log(g)

>c~

L,

_w

=

1/3

in two dimensions and _{w m}

1/5

in three dimensions

[7, 8].

^The distribution of

log(g)

then contains _more than _a

single parameter.

(iii)

The

application

of _a flux

quantum

4lo

through

the

"cigare-shape domain",

which _con- tains the forward scatterered

trajectories, yields

_a fluctuation

6(g)

cz g much smaller than the

typical sample

to

sample

fluctuations

ill] (non ergodicity).

No other characteristic

magnetic

field scale _appears if

self-intersecting trajectories

are

neglected.

(iv)

The effect of the motion of _a

single

scatterer has _not been addressed in directed

path

theories. Broad distributions of the

log-conductance

with _a

quasi-universal

variance have been

numerically

^{observed without} a directed

path approximation [12].

The restriction of

quantum

interferences _to this class of directed

trajectories

is

usually

made in ad hoc models

(binary alloy

distribution at _an energy far from the e~) where one assumes that the

path

contribution is

always

_an

exponentially decreasing

function of its

length.

This does not take into account situations where small denominators could

compensate

^the

exponentially

small value of

h~.

Such quantum resonances

necessarily

_occur in _a device where carrier _concen- tration can be varied

by capacitive

technics. This

approximation

does not describe also _more

general

cases where

transport

involves

complicated

interferences between

classically

forbidden and allowed

trajectories (vicinity

of the

mobility edge).

In _an Anderson model with _a broad

distribution of

hlej,

there is _a finite

probability

to have clusters where the electron _can diffuse

easily,

be backscattered and selfintersects its _own

trajectory

^before

having

to tunnel

through large

_energy barriers. Such metallic clusters _can

yield.

_non trivial

magnetic

field _or

spin-orbit

scattering dependence

in the

vicinity

of the

mobility edge.

An alternative _non

perturbative theory

for the

analysis

of

quantum

transmission in insu- lators is

provided by

_a maximum

entropy

ansatz

[13] using

the transfer matrix M. This

approach quantitatively

describes

quantum

transmission

through quasi-Id

wires _over

length

scale smaller or

larger

than their localization

lengths, leading

_us to

predict

novel universal sym-

metry breaking

^effects

[14].

^We are interested to know its

degree

of

validity

outside

quasi-one

dimension in the presence of _a

strong

transverse

localization,

and to compare its

predictions

from

perturbation theory

restricted to directed

paths.

We do _not restrict ourselves _to the limit of _an

elementary

localization of the conduction electrons around

randomly

located donors _or

acceptors,

I.e. to _a

purely geometrical disorder,

but _we consider also

systems

closer _{to a mo-}

bility edge

where localization results from

complez interferences

between

classically

allowed _or

forbidden trajectories.

In this later _case, it is

likely

that the relevant

quantum

interferences do not

only

involve

simple

^forward scatterered directed

paths,

and that the effect of _a

strong

spin-orbit coupling

_or of _a

large applied magnetic

field have to be revisited

[15, 16]. Being

outside _a

simple perturbative limit,

_we

rely

_on numerical simulations which include all

possible path

contributions without _{any a}

priori simplification.

We _{use as}

microscopic

models either _a

random

quantum

wire network _{or an} Anderson model with

rectangular

distribution of _e~.

This work

belongs

to _a series of three _papers devoted _to

quantum

interferences in disordered insulators. One of them is theoretical

ii?]

and

analyzes

how the

purely quantum

interferences which _we consider _are truncated

by

activated _process at low

temperature.

^{The second}

[18]

^is

experimental

and

fives

the _energy and

magnetic

field

dependence

of the conductance. One could then _compare the results of _our simulations to their results and _{note a}

striking (at

least

qualitative) similarity.

The

difficulty

for _{a more}

quantitative comparison

is due to the activated processes.

We recall in sections 2 and 3 the maximum

entropy description

of

quantum

transmission and its

implications

for

quasi-one-dimensional

insulators

(typical

conductance and

sample

to

sample fluctuations, _symmetry breaking effects).

We

study

in section 4 the modification of the

physical

constraint used in this maximum

entropy description

for three-dimensional insulators.

The

persistence

of the short _range

repulsion

in _presence of transverse localization is shown in section 5

through

_a

study

of the

eigenvalue

nearest

spacing

distribution of

t.tt, though

the

long

_range interactions _are weaker than usual random matrix behaviour

(study

of the

spectral rigidity).

In section

6,

the

sample

to

sample

fluctuations of

In(g)

_are studied in three dimensions for

large

^{disorder. We show that}

(~~

varies _{as a} function of the disorder

parameter linearly

from the

mobility edge

to the

strongly

^localized

regime.

The effects of _a variation of the

applied

magnetic

field and of the Fermi _{energy are} considered in sections 7 and 8. We _{compare our}

results for the

eigenvalues

of

t.tt

_{with those which}

are understood for the

system

Hamiltonian

(Sect. 9),

and with the

experimental

results of reference

[18] (Sect. 10).

2. Maximum

entropy description

of

quantum

scatterers.

We consider _a N-channel

quantum

scatterer that _we

represent by

its transfer matrix M. This matrix

gives

the electronic flux

amplitudes

at the

right

hand side of the _scatterer in terms of the flux

amplitudes standing

at its left hand side. The statistical ensemble for _a matrix

X,

related to M

by

the relation

X =

(Mt.M

₊

(M.Mt)~~ l)/4, (5)

is chosen

[13]

as the most random matrix ensemble

given

the

density a(I)

of the

eigenval-

ues of X. This statistical ensemble of maximum information

entropy

for X

implies

that the

X-eigenvectors

_are

statistically independent

of the

eigenvalues

and _are distributed

totally

at random. We denote

by

t the transmission matrix at the Fermi energy.

Using

_a

two-probe

Lan- dauer formula _{we express} the conductance g =

2.tr(t.tt)

_{as a} linear statistics of the

eigenvalues Ta

of

t.tt

_or the

eigenvalues (la)

of X via the relations:

9 = 2.

f

Ta

= 2.

f

~

,

a=i a=1

~ +

~a (6)

the factor two

coming

from _a

possible spin

_or Kramers

degeneracy. Sometimes,

_we refer _to the

eigenvalues (Aa)

of X _as the radial

parameters

of

M,

since

they

_can ^be also introduced

through

a natural

"polar" decomposition

of the transfer matrix.

The

joint probability

distribution

P((la))

_can be

easily

calculated from this maximum

entropy

ensemble. One

gets

an

expression formally

identical to the Boltzmann

weight

^of a set of classical

charges

free to move on a half line

(the positive part

^{of the real}

axis)

at a

well-defined

"temperature"equal

to

fl~~.

P(jlj)

c~ exp

-fl.H(play), (7)

and

governed by

_an Hamiltonian:

N N

H((la))

=

~j

In

(la 16

+

~j _{V(la ). (8)}

a<b a=i

The

confining potential

is

given

in terms of the

input density a(I) by v(>)

=

/ ~ _a(>')

_in

_ii >'(d>'. (9)

In usual random matrix

theories,

this is the classical Coulomb gas

analogy

which introduces

a

temperature,

_a

one-body potential

and _a

two-body

interaction.

(I)

The

temperature

is

given by

the fundamental

symmetries

which leave the

system

invariant

(time

reversal

symmetry, spin

rotation

symmetry), fl

=

1, 2,

4

respectively

for the

orthogonal, unitary

and

symplectic

_case. ^{In the absence of}

spin-crbit scattering, fl

₌ I without

applied magnetic

field and becomes

equal

to 2 if the

applied magnetic

field is

large enough (removal

of time reversal

symmetry).

(ii)

The

confining potential V(I)

_can be

adjusted

in order to attract most of the

la

_near the

origin, yielding

_a

large

value for g

(metal),

_{or on} the

contrary V(I)

_can be _very

weakly

attracting, leaving

the

"charges" spread

out

along

the

positive

real

axis, yielding

_g < I

(insu- lator).

In other

words,

the

spectrum

of X

depends

_on the different

physical

parameters

(Fermi

momentum

kf,

disorder measured

by I, system

size

L...) through

_a

simple one-body potential V(I)

in this maximum

entropy

ansatz.

(iii)

As the value of

fl,

the crucial

logarithmic

interaction between the

la

is obtained from the invariant _measure of the _space of matrices X which _are invariant under certain

symmetries

that each member of the statistical ensemble must

satisfy.

The consequences of the _ansatz

(8)

on the

spacing

distribution of successive radial _param-

eters and _on their two

point

correlation functions have been

carefully investigated _{[19, 20]}

for disordered conductors where the

la

_are in

part

accumulated close to the

origin.

Detailed studies confirm both the _presence of the infinite _range

logaritmic

interactions and the

change

of

temperature

^associated with

symmetry breaking

effects when the

sample shape

is

nearly quasi-one

dimensional. For

higher

dimensions

(d

= 2 and

3),

the transverse diffusion

slightly

reduces the

rigidity

^{of the}

)-spectra

for

high

transmission _or reflection

(edges

of the

spectra),

albeit the random matrix ansatz remains _a

good approximation.

In the

quasi-Id limit,

the ansatz

(3)

is consistent with _a diffusion

equation _[13]

which

implies

the

quantitatively

correct behaviors for the average and the variance of _{g in} the metallic

limit,

if the

input density a(I)

satisfies _a certain

integro-differential equation.

More

important

for the

questions investigated here,

this ansatz remains valid when the

sample length

is

larger

than the

quasi-Id

localization

length, establishing

the

validity

of this "Coulomb gas model"

in the presence of

longitudinal

localization. In other

words,

the Coulomb _gas model remains

essentially

stable with respect to matrix

multiplication

for _a fixed number N of channels.

3.

Longitudinal

localization without transverse localization

(quasi-ld limit).

We summarize in this section

previous

results obtained from the

global

ansatz

[13-21]

^and based _{on an}

analytic _expression

for

a(I) yielded

from the diffusion

equation _[22]

valid in the

quasi-Id

limit. The

analysis

is easier in terms of the variables

(ua)

related to the

(la) through

(cosl~(u~) i)

~a

=

2

(1°)

In the

large N-limit,

_one

gets

for the

u-density a(u)

=

(

0

§

_u

§ 2L/1 (11)

and _zero elsewhere which

gives

^{for the}

confining potential:

U(u)

₌

j~j

~~~~

ln

cosh(u) cosh(u')(du'

_m

~'~u~. ₍₁₂₎

2

o 4L

We write

again

the distribution

P((ua))

_{as a} Boltzmann

weight

at _a

temperature fl~~

of _a classical

system, getting

for the Hamiltonian

N N

H((ua))

₌

~jV2(ua,ub)

+

~jVi(uc). (13)

a<b c=1

The two

body

interaction is _now:

V2(ua, ub)

" In

cosh(ua) cosh(ub)(, (14)

and the _one

body potential

contains _a

fl-dependent

term

yielded by

the

change

of variable:

T(u~)

=

-jIn(sinh(u~))

⁺

^{U(u~). (15)}

However,

the

quasi-Id

localized limit is characterized

by

_a

particular

behavior of the

spectral density: increasing

the

system

size L for _a fixed number N of

channels,

_we end up ^to have:

I<ui<U2«.

«UN

(16)

which allowes _us to

simplify V2(ua,ub),

since In

(cosh(ua) cosh(ub)(

_*

max(ua,ub),

to

get approximately independent

Gaussian fluctuations of _ua centered at

and of variance

2L/(flNl).

Since _vi

" 2L

If,

_{we see} that the localization

length

is

proportional

to

fl:

(

₌

flNl. (18)

The

N-dependence

of

(

is characteristic of the

quasi-Id

limit where tranverse localization is absent.

Change

of

(

and

change

^of the

magnitude

of the conductance fluctuations

by

_a

symmetry breaking

effect _are the two sides of the _same

phenomenon,

since

(

is defined from the ensemble _average <

log(g)

>m

2L/(

and not from < g ^>. The characteristic _cross-cver fields

reducing

the

magnitude

of the conductance fluctuations and

increasing (

in the absence of

spin-orbit scattering

_are

typically larger

in insulators than in conductors. The

change

^from

orthogonal

to

unitary symmetry

occurs when _a flux

quantum

4lo ^is

applied through

the

phase

coherent domain in _a conductor

(temperature dependent)

and

through

the localization domain for _an insulator

(temperature independent).

The

corresponding

fields B* _are of order of _m 10-

l00 Gauss in a conductor at very low temperature

(a

few

mK)

and _m 0.5 2 T for _a

large

localization

length

insulator. For insulators with

strong spin-crbit scattering

,

if _{one assumes}

that the removal of the Kramers

degeneracy

^for vi

by

_an

applied magnetic

field B introduces _a

splitting

which is

negligible compared

to the

large

value of 2L

If,

the transition

fl

₌ 4 _-

fl

= 2

essentially

halves the value of

(.

The existence of _a

positive magnetoresistance

for insulators with

strong spin-orbit scattering

is in this

theory

the

counterpart

^of the

suppression

of weak-

anti-localization in

conductors,

in

agreement

^with some

experiments [16]

^{and in}

disagreement

with directed

path

theories

[10].

The second

important

result that _we have derived _concerns the

sample

to

sample

fluctuations of the

logarithm

of the conductance since

log(g)

_CS

2L/f

= vi

(19)

in this

quasi-Id insulating

limit. We obtain _a normal distribution of

log(g)

with

var(log(g))

₌ <

log(g)

>

(20)

We remark here that this is in contrast to Beenakker's result

[23]

^that

var(g)

_m I

(Universal

Conductance

Fluctuations)

for _any

confining potential V(I).

This is due to the

marginally confining potential

characteristic of

quasi-ld

insulators such that g

mainly depends

_{on vi,} and does not

correspond

to the

large

N-limit considered

by

Beenakker. His result

applies

to the continuum limit which does not

probe eigenvalue

fluctuations _at the scale of nearest

neighbor spacings.

^We note also that contrary to

approximations

based _on directed

paths, log(g)

has _a

one-parameter distribution.

In summary, the above results indicate that the

validity

of _a "Coulomb _gas model" for the

(la)

in the

quasi-Id

limit is not at all

equivalent

to the

validity

of

Wigner-Dyson

statistics since the

spacings

between successive

la

become _so

exponentially large

that the

two-body

interaction is reduced _{to an} effective _one

body potential.

In

particular,

this

density

effect is

responsible

for the

non-universality

of

var(g)

in insulators. But _{one cannot} check

easily

in this limit the

underlying

_presence of the

logarithmic

interaction.

If _we consider _now

strongly

^disordered L _* L _* L

cubes,

the increase of their size L will increase the number N

=

(kf.L)~~~

of channels in addition _to the

longitudinal length.

^{Real 3d-}

insulators _are then _more

likely

than

quasi-Id systems

to be characterized

by

_an

exponentially large

value of

li

in the

large

L-limit without

having exponentially large

^intervals between successive

(la).

Does the Coulomb _gas

analogy

which is at least _an excellent

approximation

in the absence of tranverse

localization,

remains of _some interest in _a

strong

^{disorder limit?}

Do two successive

la

still

repel

each other when their

spacing

is not

exponentially large,

_as

implied by

the ansatz? Does the coulomb _gas

"temperature"

take the classical values

implied by symmetry

considerations? This is the first issue that _we have to check in _our numerical

study.

4.

Density

and

confining potential

in the presence of transverse localization

(3d- limit).

We have calculated series of

(la)

for different realizations of _a

microscopic

^{model. We have} used _a 3d-network of disordered

quantum

wires _as

explained

in

appendix

A. Each link

(I, j)

of this network is characterized

by

_a one-dimensional uniform electrostatic

potential

_ei,j, which is taken at random with _a

rectangular

distribution of width W. Real

(propagating mode)

or

imaginary (decaying mode)

wave-vector k~,j =

(E ei,j)~/~

characterizes the electronic wavefunction

along

the link

(I, j).

The

matching

^conditions at each intersection

satisfy

the basic

symmetry

and conservation laws of

quantum

^mechanics. We have

alternatively

used the Anderson model

(I)

with the _same distribution for the random

potential

_e~,

obtaining

similar

conclusions.

(See

also another recent numerical

study

of 3d Anderson models

by

Markos and Kramer

[24]).

In

figure I,

the

density a(u)

is

given

for 3d-cubes close to the

mobility edge (W

=

4)

and

3.5

3

2.5

~s>

it .5

o.5

~0

_{5 10 15}

~

20 25 30 35

Fig_I. Average density a(u)

calculated for cubes of size L

= 6 for disorder parmneters ^W = 4

(circles)

and W

= 7

(squares).

35

<~ ~> I ~~~~oooooo°°°°°°°°°°°°°11

20

o°°°~ ..:.°°°

~ ~...

~ ..

15 o

:.°°°

~:

jo

~

.

0 0

30 V(~)

25

w=5.0, a=0.294; b=() 045 20

15

lo

5

w=6.5, a=0. I13, b=0 004

~

0 100 200 300 400 500

~ l 0~

Fig.3.

Potential

V(~)

calculated for cubes of size L

= 8 for different values of W

(squares)

and the

analytical

^fit

given

ⁱⁿ equation

(21)

for indicated values of

a and b

=

~p~

o-g Pls)

o 6

1).4

U,2

0 0,5 1.5 2 2.5 3 3.5

s

Fig.4. P(S)

for S ₌ (v2

m)/

< u2 vi >

corresponding

to the

same cubes _as in

figures

1-2 for W = 4

(circles),

W

= 7

(squares)

and

Brody

distribution with q = 0.7

(thick line).

random matrix

theory (Wigner

surmise for

fl

=

I).

This distribution also describes the

spacings

in the bulk of the

spectrum (e.

_{g. u7}

u6)

where the associated

decay lengths 2L/ua

_are

quite

small. For

larger

disorder W

=

7,

< vi >m

10),

level

repulsion

is weaker. The obtained

histogram

can then be fitted

by

_an

empirical

formula

proposed by Brody [25] (with

_q

=

0.7)

Pq(s)

= As~

exp[-os~+~] (23)

in order to

interpolate

from the correlated random matrix

spectra (Wigner

surmise q =

I)

to uncorrelated levels

(Poisson

distribution _q

=

o).

We _{can see} in

figure

2 that the _mean

spacing

u~ vi m

4,

which _can

partly explain

the weakness of the

repulsion (In cosh(u2) cosh(ui

₎₍ is

more sensitive to the

large

value of _u2 than to the value of _u2

vi) However, looking deeper

in the

spectrum,

we can still _{note some} deviation of the actual distribution of

(u7 u6)

^from

the

Wigner

^surmise

(Fig. 5),

while the _mean

spacing

is much smaller. But the

persistence

of

a

strong eigenvalue repulsion

between transmission modes with

extremely

short characteristic

p(si 0,8

0,6

0.

0.5 1.5 2 2.5 3

s

Fig.5. P(S)

for S

= (u7

v6)/

_{< v7 v6 >} for L

= 6, W

= 7

compared

to the

Wigner

surmise.

decay lengths ((7

"

2L/u7

_"

0.5)

is

by

itself

quite

noticeable: the

spacing

distribution

being

much closer from _a

Wigner

surmise than from _a Poisson distribution

characterizing

uncorrelated

eigenvalues.

Since _we have

approximately

obtained

spacing

distributions in the _presence of time reversal

symmetry (orthogonal case)

close to the

Wigner surmise,

it is

interesting

to know what is the value of the _cross-over field B* needed for

changing

this distribution _to the

Wigner

surmise

cllaracteristic of the

unitary

ensemble. It is also

particularly interesting

to _see if B* charac- terizes also the

typical

field scale where _a

change

of

(

_occurs, in

agreement

with _our

theory relating

the value

(

to the

importance

of the fluctuation of the

(la)

which _are controlled

by

the

"temperature" fl~~

We have

applied increasing magnetic

^{field B}

(measured

in units of 4lo per lattice

cell).

We first _note that weak fields

(about

_one flux

quantum through

^the

sample)

have _no clear effects _on the different _{< ua >} and _on the

spacing

fluctuations. Near the

mobility edge (W

₌

4),

we observe _as

expected

both the

Wigner-surmise

of the

unitary

ensemble

(fl

₌

2, Fig. 6)

and

a

slight change off (increase by

_a factor _m

1.3)

for B

= 0.25. One _{can see} also in

figure

6 that

P(u2

_vi does not

depend

_very much _on B for

larger

disorder. We think that this is

partly

due _to the _gauge invariance of _a lattice model which makes

impossible

to remove the coherent

backscattering

^effects within too small localization domains, Two conclusions _emerge from the

magnetic

^field

dependence

of the

spacing

fluctuations.

First,

the _cross-over field B* is much

higher

in insulators than in metals.

Contrary

to _a metal where the

typical

electronic

trajectories

are random walks

exploring

the whole coherent

sample,

the

typical trajectories

in _an insulator

are

enclosing

much smaller _areas characterized

by

much

larger dephazing

fields. Our results agree with the criterion B* m

lbo/(~. Second,

the necessary field in order to

significantly change (

is

precisely

the _same which

changes

the

spacing distribution,

in

agreement

^with the

existence of _a relation between the

change off

and the

change

of

fl.

The

study

of the

spacing

distribution is not the _more accurate test of the

validity

of the random matrix ansatz for

3d-insulators, probing

more the short _range

eigenvalue repulsion

than the

long

_range

spectral rigidity.

The usual statistical _measure of the

long

_range

rigidity

is the

A3-statistics

introduced

by Dyson

dud Mehta

[26].

^{Once the}

spectra

^have been

numerically

unfolded

[27]

^{to series of} constant

averaged density,

this statistics _measure how the rescaled series deviate from _a

regular

_{sequence as a} function of the _average number of considered levels.

pis)

,u

o' 'q

~

D ..

~~~

,' 'u

0.2

0,5 1,5 2

s

Fig-G- P(S)

for S

= (v2 vi

)/

< _v2 vi > for cubes of size L

= 8 W

= 4

(circles

_{< vi >@}

2.02)

and W

= 7

(Squares

< vi >m

10.97) compared

with the

Wigner

surmise for p ₌ 1,2.

Applied magnetic

field B

= 0.25.

1,o

o~a

o~

0. 8

(

o. 6

~o.i

o

0.

°~

o

0 5 10 15 20 25 30 35

number of gigenvalues _n

Fig.?.

A3-statistics _{as a} function of the _mean number _n of

eigenvalues

~a. Unfolded spectra obtained in

a 3d-Anderson-model for W ₌ 30 and L

= 7 at the band centre

((

m

I).

The continuous

curves are

given by

_n

II

5

(uncorrelated spectra)

and

Jn(2Yr

*

n)

+ _~ 5

/4

Yr~

/8]

_/Yr~

(random

matrix

theory).

We show in

figure

7 how the

A3-statistics

calculated from the

(la)

of _a 3d Anderson model deviates from the random matrix behaviour when

(

_m I. Similar behaviours characterize the network of random

quantum

wires

previously

considered.

The deviation

of P(S)

and of the

A~-statistics

to the usual random matrix

prediction

can be

partly

due to the

highly

non-uniform

density

of the relevant variables

(la )

which could

strongly

weaken the effect of the

logarithmic ^repulsion

^{for the rescaled variables. In the}

^quasi-Id

^limit

0.8

0.6

0,4

a

0.2

0

0 2 3 4 5

Fig.8.

The inverse localization

length

_{as a} function of W Wc for Wc ₌ 4.5. Values of L beween 6 and 12 _are used to evaluate

(.

for

instance,

we have shown that the ansatz

(8) yields quasi-independent

level

fluctuations, giving

_a

P(S)

_{or a}

A~-statistics

closer to Poisson than to

Wigner.

In references

[28],

a crossover

in

P(S)

and in

A3

from

Wigner

to Poisson is obtained within the random matrix

framework, assuming

^infinite _range

logarithmic repulsion

and _a

one-body potential qualitatively

similar to

equation (21). However,

since _we have shown for disordered conductors

[20]

^{that the 3d-value} of the

amplitude

of the conductance fluctuations is

contradictory

with infinite _range

logarithmic

repulsion

for the

(la),

_we

speculate

that the

long

_range part of this

repulsion

must be _even

more

severely damped by

transverse localization than

by

transverse diffusion.

But,

^from the

study

of those two

microscopic models,

_{we see} that _a short _range

repulsion persists

in the presence of transverse localization. The shorter is

(,

the shorter is this range.

We note that _a

dimensionality dependent damping

of the

logarithmic repulsion

has been shown

[29]

^{to be consistent with the known}

density-density

correlations of Hamiltonian

spectra

in disordered conductors.

Similarly,

the

long

_range

part

of the two

body

interaction in

(8)

_may

need to be modified in order to still write the distribution

P((la))

_as the Gibbs distribution of _a fictitious Hamiltoniau.

6. Disorder

dependence

of the

typical

conductance and of its fluctuations.

In this

section,

we compare the actual behavior of <

In(g)

> and of its fluctuations to _an exten- sion of the results of section 3 and to the

predictions implied by

_a direct

path approximation _[6- lo].

We first note that

contrary

to the

quasi-Id

limit where

In(g)

_{m vi,}

In(g)

is

slightly larger

than _vi in three dimensions

(u2,

_u3,

weakly

^contribute to

£$~~ (cosh(ua 2)~~ ).

Ex-

tracting

the localization

length

from the size

dependence

^of <

In(g)

> and

locating

the critical value

Wc

from the criterion d

In(g) Id In(L)

= 0 _we obtain

(Fig. 8)

_a

simple

linear

dependence:

,

(~~(W)

_c~

(W Wc) (24)

of the inverse localization

length

for the 3d-network of quantum ^{wires. In the}

vicinity

of the critical

point,

this linear law is consistent with the value

reported

in the

experimental

litterature for the critical

exponent,

^but lower thou most of the other numerical results

given

by

^finite size

scaling

studies of various

microscopic

models

(for

_a

review,

_see Ref.

[30] ).

We do

16

14 ." .

12

vat(Ing) ,,'

10 ^,"

.,,

-<lng>

8

(..

6

. _."

~ ^.

."

~4

_{5 6 7 8 9}

Fig.9. Average (circles)

and variance

(diamonds) oflog(g)

_{as a} function of W for 3d cubes

(L

₌

8).

not focus _on the close

vicinity

of the

mobility edge,

and do not rule _{out a power} law behavior different from linear in _a small critical

region.

But _we

mainly point

out in this

study

that this linear

dependence persists

when the localization

length

reduces to the lattice

spacing,

without

showing

_a

logarithmic

behavior that _one could

naively

_guess from the

length dependence ^h~

of the transfer term in a directed

path approximation (Eq. (4)).

Considering

the

sample

to

sample

fluctuations of

In(g),

_we have obtained

histograms

in

rough agreement

^with a normal distribution of the

log-conductance.

The variance of

In(g)

varies _{as a} function of the disorder and of

system

size such that relation

(20)

continue to be observed

(Fig.

9)

in three dimensions. A similar result have been noticed in the 2d-localized

regime _[21]

and is consistent with

one-parameter scaling,

and inconsistent with directed

path

theories for the

investigated

_range ^of parameters. This

one-parameter

law is observed in different

microscopic

models

(Anderson

model and network of random

quantum wires).

The

comparison

between the "random matrix

prediction"

and the directed

path

result can be

sligthly

biased since _we

consider the side _to side transmission

through

_a disorder

cube,

and not how electrons _are transmitted from _{a corner} to the

opposite

_corner ^of _a cube. We have noted also that the

equality

between the _mean and the variance of _vi is _{even more} accurate than for

In(g).

7.

Magnetoconductance

fluctuations and

sample

to

sample

fluctuations,

In disordered

conductors,

a

magnetic

field

B.L~

cz 4lo _removes time-reversal

symmetry (cross-

over from the

orthogonal

ensemble _to the

unitary ensemble), suppressing

weak-localization corrections and

halving

the variance of the universal conductance fluctuations. A

(typically weak) magnetic

field

change

of this order induces also

magnetoconductance

fluctuations of

magnitude equal (to

first order in

I/g)

to the

sample

to

sample

fluctuations under

applied magnetic

field

[4].

^{We have}

already

noticed [15] ^{in two dimensions that this}

ergodicity

remains

essentially

correct in the localized

regime

for

(large)

field

change ~AB

m B*

=

4lo/(~,

at least when

-log(g)

_m 2 7. This

"ergodic"

behavior _was

proposed

for

explaining

the

large reproducible magnetoconductance

fluctuations observed in the Mott

hopping regime

_even for

macroscopic

^{size insulators}

(L

_m I

mm).

It is then

interesting

to compare in three dimensions the

typical magnitude

^{of the}

magnetoconductance

fluctuations _{on a}

given sample

to the

sample

t~

sample

fluctuations at

given

field. We do not mean here _a

rigorous

statement about

possible

20

v 5

15

~i

V 3 lo

v

5

v

°0

_0.3

0.4 B

Fig.10.

The five lowest

eigenvalues

_va of _a

single sample (L

=

10)

at moderate disorder

(W

=

5.5)

as a function of the

applied magnetic

field B.

lo

v

8

6

4

2

0.1 0.2 0.3

B

Fig.ll. Magnetic

field

dependence

of _vi for five different

samples (L

= 10, W

=

5.5).

exact

ergodicity

in insulators but _{a mere}

comparison

of the

typical

orders of

magnitude.

Due

to the gauge invariance of _our system ^under

change

of

applied

flux

through

lattice cell

by

4lo and due to the

large

value of B* in

insulators,

a

large

statistics of those

magnetofluctuations

cannot be obtained.

In

figure 10,

we show how the first

(ua) (a

=

1, 2,..., 5)

fluctuate for _a

given sample

with W = 5.5 _{as a} function of B. The fluctuations of _vi

essentially

_agree with _an

ergodic behavior,

as illustrated in

figure

II. In addition to the

large ergodic

fluctuations induced

by

AB _m

B*,

one can notice smaller _non

ergodic

fluctuations which characterizes also the other _ua

deeper

in the spectrum

(Fig. 10)

or vi itself for

larger

disorder

(Fig. 12).

We note that

larger

disorder

increases the scale B* of the

"ergodic"

fluctuations and

prevents

us to observe them when vi »

I, leaving only

smaller fluctuations of characteristic field scale which does _{not seem to}

depend

_on W.

We

point

out that B* in _a conductor is

temperature dependent (the

relevant _area

being