An intrinsic mechanism breaking universality in the conductance fluctuations of realistic mesoscopic disordered metals

HAL Id: jpa-00211095

https://hal.archives-ouvertes.fr/jpa-00211095

Submitted on 1 Jan 1989

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

An intrinsic mechanism breaking universality in the

conductance fluctuations of realistic mesoscopic

disordered metals

M. T. Béal-Monod, G. Forgacs

To cite this version:

An

intrinsic mechanism

_{breaking universality}

in

the

conductance fluctuations of realistic

_mesoscopic

disordered

metals

M. T. Béal-Monod

₍₁₎

and G.

_Forgacs

₍₂₎

(1)

Physique

des Solides

_(*),

Université de Paris-Sud, 91400

_Orsay,

France

(2)

Physics,

Clarkson

_University,

Potsdam N.Y.

13376,

U.S.A.

(Reçu

le 6 mars _1989,

_accepté

sous

_{forme définitive}

le 2 mai

1989)

Résumé. 2014 Des fluctuations universelles de conductance

(UCF)

de l’ordre de

_e2/h,

ont été récemment observées à basse

température

dans des métaux désordonnés

_{mésoscopiques,}

indépendamment

de la taille de l’échantillon et du

degré

de désordre.

L’explication

théorique

qui

suivit

_supposait

_{que les}

_impuretés,

source du _désordre, étaient

_{indépendantes}

les unes des autres.

Nous réexaminons ce

_problème

avec

_{l’hypothèse plus}

_{réaliste que les}

_impuretés

sont en fait corrélées et d’autant

_plus

_{que leur concentration} est

_plus

élevée. Nous montrons _{ainsi que les}

UCF sont _{modifiées par} ces interactions : le résultat

e2/h

est

_multiplié

_par un facteur

_numérique

plus grand

ou

_plus

_petit

_{que un, suivant que les}

_impuretés

se _repoussent ou s’attirent. Les modifications

_dépendent,

en

_particulier,

de la concentration en

_impuretés

et donc du

degré

de désordre. Par

_conséquent

le caractère universal des fluctuations de conductance n’est

_plus

vrai dans les

_systèmes

réalistes. De nouvelles

_expériences

seraient utiles pour tester cette théorie en

comparant des cas où les interactions sont

_répulsives

ou attractives. Abstract. 2014 Low

temperature universal conductance fluctuations

_(UCF)

of the order of

e2/h

have been

recently

observed in

mesoscopic

disordered metals,

independent

of

sample

size and

_degree

of disorder. The theoretical

_explanation

which followed assumed that the

_impurities,

at the source of the disorder, were

_independent

of each other. We reexamine this

_{problem using}

the more realistic

_assumption

that the

_impurities

are, in fact, correlated and more and more so for

increasing

concentration. We show that the UCF are modified

_by

these interactions : the

e2/h

result is

_{multiplied by}

a numerical factor

_larger

or smaller than one,

depending

on whether

the

_{impurities repel}

or attract each other. The modifications do

depend,

in

_particular,

on the

impurity

concentration and thus on the

degree

of disorder. Therefore the universal character of the conductance fluctuations breaks down in realistic _systems. Further

_experiments

would be useful to test the present

theory by comparing

cases where the interactions are

_repulsive

or

attractive. Classification

Physics

Abstracts 71.55J -

72.15R 72.70

2710

Introduction.

The recent _{progress in research} on disordered electronic

_systems

is marked

_by

several

important

milestones

_{[1] :}

the

_prediction

and

_experimental

observation of deviations from the standard Boltzmann

_theory

in metals and the

_concept

of Anderson localization of the electrons

_yielding

metal-insulator transitions in disordered electronic

_systems...

The

latest

discovery

was the existence of universal conductance fluctuations

(UCF)

in

systems

of

«

mesoscopic

» sizes

[2, 3].

Such sizes are intermediate between

_microscopic

and

_macroscopic

and are relevant to

_{technological}

_{applications.}

It was thus shown that in the

_mesoscopic

systems,

new

_quantum

mechanical

fluctuations,

as a function of the chemical

_potential

or of an

_applied

_magnetic

_field,

_{show up in the form of time}

_{independent, reproducible, sample}

specific, aperiodic

and noiselike structures. The

fluctuations,

of the order of

_e2/h,

are the

consequence of

quantum

interference in the

transport

properties

of disordered metals at low

temperature.

They

are

_independent

of both the

_degree

of disorder and

_sample

size

_(as

_long

as

the inelastic diffusion

_length

exceeds the

_sample

_dimensions)

and,

in this sense, are universal. It was

_emphasized

that such fluctuations thus reveal a new and fundamental

_aspect

of

quantum transport,

rather than

_just

a

_simple

finite size effect.

So far in the theories of

_mesoscopic

_{[2, 3]}

as well as infinite

[1],

_systems

the

_randomly

spread impurities

have been considered as

_independent.

This may be so

_only

in the case of

extreme dilution. But at finite concentration the

_impurities

are, in

fact,

correlated and more

and more so as their concentration increases. We indeed

_recently

showed

_[4]

that even in the

weakly

localized

_regime

of infinite disordered

_metals,

interactions between

_impurities

are

expected

to

_play

an

_important

role on the Anderson localization of the electrons. Such

interactions

produce,

locally,

an atomic

ordering

which is

accompanied by

oscillating

functions of

_{(kF d ), d}

_being

the nearest

_neighbor

distance in the lattice and

_kF

the Fermi

momentum. This local

_order,

in turn, affects the

_conductivity

of the infinite

system.

It may induce a metal-insulator transition for a value of the

impurity

concentration for which the

independent impurity

theories

_[1]

would lead to a metallic behavior. We showed

[4]

that our

interacting impurity

model can

_{explain satisfactorily}

two different kinds of

_experiments.

The purpose of our

_present

_{paper is} to reexamine the UCF in

_mesoscopic

_sysems

_taking

into account the interactions between the

_impurities.

We use the same

_simple

mean field

model to describe the atomic local order induced

_by

the

_impurity

interactions. Such a model was introduced

_long

_ago

_[5]

to describe the local order between

_spins

and

_adapted

later on

[6]

to the case of metallic

_alloys

with non

_magnetic

_{components ;}

in this last case the interactions were of chemical

_origin.

Reference

_[4]

_extensively

recalled the model so that we will not

discuss it further here. Let us

_{just point}

out

_that,

_already

at the

_early

_stages

of

references

_{[5, 6],}

the Drude formula for

_transport

in disordered metals was shown

[7]

to be modified

_by

the

_impurity

interactions in such a _{way that the}

(Drude)

_{conductivity always}

decreased when the interactions assumed the screened Coulomb form between excess

_charges

on the

_impurity

sites. This was recovered in reference

[4]

where,

in

addition,

the

multiply

crossed

_{conductivity diagram}

_(the

localization

_term)

was also shown to be affected

_by

these interactions.

In the next

_section,

we

_compute

the main contribution to the conductance variance which takes the form of the correlation function of two static conductances related

_by

two diffusion

propagators,

shown in

_figure

1 ;

reference

_[3]

indeed remarked

that,

although

many other

diagrams

are

_involved,

the

_qualitative

behavior of the final result is the same as that obtained

by only evaluating

the above one. That is

_why

we restrict our

_study

here to that

_{diagram only.}

Fig.

1. - A two diffuson

_diagram

_(the

two shaded

_areas)

_contributing

to the conductance fluctuations and identical to

_figure

5a of reference

_[3],

but where interactions between

_impurities

are taken into

account as in reference

[4] :

the individual Green’s functions are renormalized

_{by interacting impurity}

scattering

as well as the two diffusons as indicated in

_figure

2.

did in reference

_[4].

We show that the result of reference

_[3]

is

_{multiplied by}

a function of

(kF d )

involving

also the

_impurity

concentration and the interaction

_strength,

as was the case

in our

_conductivity

calculation in reference

_[4].

The net result is thus that the universal character of the conductance fluctuations breaks down. We

_give

the details of the calculation in three dimensions

_(3D)

but the

_procedure

is the same in 2D for which case we

_only

_present

the result. We then end with a discussion and

_implications

for other

_properties

of

_mesoscopic

systems.

Calculation of the two-diffuson conductance fluctuation

_diagram.

In this

_section,

we evaluate the

_diagram

drawn in

_figure

1,

that we call K and which

represents,

as recalled above from reference

_[3],

a

_typical

contribution to the correlation function of two static conductances and thus to the variance of the

conductance,

var

(g ),

where g

is the conductance in units of

_{e2/h, g}

=

G /

(e2/h).

To do so we will use

_extensively

the main

_{ingredients describing}

the local order between the

impurities

as derived in reference

_[4]

_(hereafter

referred to as

_BMF),

where the

_system

is

considered as an

_alloy

of two

_types

of atoms A and

B,

the A atoms

_being

the

_impurities

and the B

_being

the host atoms. On the other

_hand,

we also follow the

_{diagrammatic procedure}

of

reference

_{[3] (hereafter}

referred to as

LSF).

We confine our discussion to the metallic

_regime,

where

_{kF e >}

1,

where

f

is the elastic mean free

_path

for

_independent

impurities.

As shown in

2712

VF is the Fermi

velocity ;

T’ is

given

in formula

(22)

of BMF and is recalled here for convenience :

Jo

is the Bessel function of the first kind of index 0

_[8].

_JI

_{of index 1 will appear also later in} formula

_{(12) ;}

z is the number of nearest

_neighbors

in the

lattice,

W =

VAA

+

_{VBB - 2 VAB,}

and the V’s are interaction

_energies

between

_pairs

of nearest

_neighbors

AA,

BB and AB

respectively,

c is the

impurity

concentration.

To

is the

_temperature

where the

sample

has been

quenched

from,

and it thus characterizes the

_degree

of local order. ro is the elastic

scattering

time in the absence of interaction

_{(W = 0 )}

and

e

=

VF TO.

To evaluate the

_diagram

of

_figure

_1,

we first need to calculate the diffuson T exhibited in

figure

2 and

_representing

an infinite ladder of

_scattering

of the electron on

_interacting

Fig.

2. -

The

_{diagrammatic Bethe-Salpeter equation}

for the diffuson r in the

_{particle-hole}

channel

(formula (5)) :

here too, as in

_figure

1, the electron lines should be understood as renormalized with the lifetime T’of formula

_{(2) ;}

the cross is the

scattering potential given by

formula

(3)

involving

the

impurities.

Each cross denotes a

_{scattering potential}

modified

_by

the interactions which has

been calculated in formula

₍₁₅₎

of BMF as

_{being :}

VA

and

_VB

are the

potentials

created

_bys

the atoms A and B

_{respectively, 0}

is the

_scattering

angle

between momenta _{pl and p2.}

As in

BMF,

we rewrite

₍₃₎

as :

Note that our

_present

diffuson vertex T is the

_{particle-hole analogue}

of the

_Cooperon

appearing

in the

_{particle-particle}

channel in the

_multiply

crossed

_{conductivity diagram}

in BMF. The

_{Bethe-Salpeter equation}

for our r reads :

Then,

as in

BMF,

we linearize

(5)

to first order in

_WIT,

and we

_get

a structure for the diffuson

_analogous

to the one of the

_Cooperon

formula

(39)

of BMF :

with the

_quantities

A’ and

B’,

different from the

_{corresponding}

A and B ones in BMF and

given by :

Recall that : Z =

[2 7rN (EF)

TO]-l.

After a

_lengthy

but

_{straightforward algebra quite}

similar to the one

_displayed

in

BMF,

we

2714

Do

is the bare diffusion coefficient

both

_frequency

cases are additive. We may rewrite r as :

One can see that the modifications introduced in r for finite W in the

_present

_{particle-hole}

channel

_(the

_{la 3D}

_factor),

is

_exactly

the same as that

_occurring

in formula

(49)

of BMF for the

Cooperon

in the

_{particle-particle}

channel. This is not

_surprising

since the

_frequency

and

momentum

_dependences

are

_unchanged.

In both cases, the numerical

prefactor

which is involved

_depends

on the

_sample

characteristics

_{(c, z, W, kF d, TO).}

This last

_point

will be

important

for our conclusion.

_If,

as was done in the 2nd of reference

_[2],

we add to

_figure

1 a

similar

_diagram

but with 2

_Cooperons

instead of 2

_diffusons,

the result will thus be the same as

in the

_present

case : each

_Cooperon

is

_{multiplied by}

_{(1 + /33D)-}

_Similarly

in

_2D,

,a 2D may be

extracted from BMF and reads :

The

_preceding

result will enter into the calculation of K in

_figure

1

_(identical

to

_Fig.

5a in

_LSF,

which was said in LSF to be the essential

_result).

Now if we _{go back} to the various other

diagrams

considered in

_LSF,

with more than 2

_diffusons,

it is clear that the

_recipe

to calculate them reduces to the

_{following :}

(i)

multiply

each diffuson

_{(or Cooperon)}

_by

_{(1 + 8 ) ;}

(ii)

change

the lifetime of the electron from TO to T’ with T’

_given

above

_by

_{(2) ;}

(iii)

change

the limits in the

_{final q integral}

from with

Actually,

as

_long

as L >

1 ’ ,

the

_{upper’limit (1/f’)}

in

_{the q integral}

does not matter in the result as well as

(1/f)

did not matter in LSF when L »

1. Therefore,

provided

that :

only

(i)

and

_(ii)

above will affect the LSF result.

In

_figure

1

_(ii)

will

_modify

the

_integration

of the various Green’s functions

_yielding

a factor

Similarly,

in

_2D,

we

_{get :}

Discussion.

We first

_remark,

in connection with

₍₁₄₎

and

_(15),

that K will be enhanced or decreased

_by

the

impurity

interactions

_depending

on whether W is

_positive

or

_negative,

i.e. whether the

impurities

repel

each other while

_they

are attracted

_by

the host atoms, or if

_they

attract each other while

_{being repelled by}

the host atoms.

_If,

as in reference

_[7],

we _express

W in terms of the screened Coulomb interactions between excess

_charges

on the

_impurities

sites,

proportional

to cos

(2 kF

d)l (2

kF

d )3

in 3D and to - sin

_{(2 kF}

_{d)l (2}

_kF

_{d )2}

in

_2D,

the

change

in K will be an

_oscillating

function of

_{(2 kF d).}

Now in order to

_complete

the calculation to

_get

the variance of the conductance

var

_{(g ),}

it would remain to

_perform

the same kind of calculation with

_diagrams

with more

than 2 diffusons

_{(or Cooperons)}

with the above

_recipe.

We will not do it here but it is

_already

clear that var

(g )

will assume the form :

where y will be a function of c, z,

W,

_To,

(kF d),

both in 3D and 2D similar to the corrections in formulas

₍₁₄₎

and

_(15).

Therefore,

and this is the crucial

_point

of our

_present

_{paper, since}

y does

depend

on the various above

_parameters,

the conductance fluctuations are not

universal

_[9]

in realistic

_mesoscopic

disordered

systems,

when the interactions between the

impurities

are taken into account. We believe this could be

_usefully

tested

_{by varying sensibly}

the above

_parameters,

for instance

_{by comparing}

the results obtained when

_dealing

with

impurities

repelling

or, on the

_contrary,

_attracting

each other.

Aside from the

_complete

calculation of var

_{(g )}

sketched

above,

another

_interesting

_aspect

of this

_study

would be to examine how the

_impurity

interaction would

_modify

the result of reference

_[10].

It was shown in reference

[10]

_that,

for a

_{given degree}

of weak

_disorder,

the conductance was

_extremely

sensitive even to the motion of one

_single

_impurity.

If one

impurity

moved a distance

_{6r:::. kF 1,}

the conductance

_changed,

in

_3D,

_by

5G-(f2 lh) (k2ÎL )- "2

x osc.

_{(kF 6 r)}

where osc.

_{(kF â r )}

denotes an

_oscillating

function of

(kF,6r).

In

2D,

the

_{corresponding expression}

is 6G -

_{(f2/h )(kp P)-nz}

x osc.

Thus,

for

_2D,

the motion of a

_{single impurity}

leads to

_changes

as

_big

_(if

kF e -

₁₎

as if the entire

_sample

were

changed.

This has been

_understood,

on

_{physical grounds,}

since conductance is known

[11]

to

be

_proportional

to the

_quantum

mechanical transmission

_probability

_through

the

_system,

which itself can be understood in terms of interference between classical

_{Feynmann paths}

through

the

_sample

_[3].

The

_question

which arises in the

_present

context is how much ÔG may be

expected

to

_change

if several

_{interacting impurities}

are moved ?

Likewise,

how will the

_{oscillating dependence}

in

_{(kF d )}

due to

_impurity

interactions combine with the

oscillating dependence

in

_(kF,6r)

found in reference

_{[10] ?}

Does the combination of these two

2716

Acknowledgments.

One of us

_(M.

T.

B-M)

_{enjoyed interesting}

comments from B. L. Al’tshuler and S.

_Feng.

References

[1]

See the review

_by

LEE P. A. and RAMAKRISHNAN T. V., Rev. Mod.

_Phys.

57

₍₁₉₈₅₎

287.

[2]

Theories :

AL’TSHULER B. L., J. E. T. P. Lett. 41

₍₁₉₈₅₎

_{648 ;}

AL’TSHULER B. L. and KHMEL’NITSKII D. E., J.E.T.P. Lett. 42

₍₁₉₈₅₎

359 ;

STONE A. D.,

Phys.

Rev. Lett. 54

₍₁₉₈₅₎

2692 ;

LEE P. A. and STONE A. D.,

Phys.

Rev. Lett. 55

₍₁₉₈₅₎

1622 ;

IMRY Y.,

Europhys.

Lett. 1

₍₁₉₈₆₎

249.

Experiments

in Au and Pd

rings :

UMBACH C. P. et al.,

Phys.

Rev. B 30

₍₁₉₈₄₎

4048 ;

WEBB R. A. et

_{al. , Phys.}

Rev. Lett. 54

₍₁₉₈₅₎

_{2696 ;}

WASHBURN S. et al.,

Phys.

Rev. B 32

₍₁₉₈₅₎

4789.

Experiments

in narrow Si inversion

_{layers :}

LICINI J. C. et al.,

Phys.

Rev. Lett. 55

₍₁₉₈₅₎

2987 ;

SKOCPOL W. J. et al.,

Phys.

Rev. Lett. 56

₍₁₉₈₆₎

2865 ; 58

₍₁₉₈₇₎

2347 ;

KAPLAN S. B. and HARSTEIN A.,

Phys.

Rev. Lett. 56

₍₁₉₈₆₎

2403.

[3]

LEE P. A., STONE A. D. and FUKUYAMA H.,

Phys

Rev B 35

₍₁₉₈₇₎

1039.

[4]

BÉAL-MONOD M. T. and FORGACS

_{G., Phys.}

Rev. B 37

₍₁₉₈₈₎

6646.

[5]

DE GENNES P. G. and FRIEDEL J., J.

_Phys.

Chem. Sol. 4

₍₁₉₅₈₎

71.

[6]

BÉAL M. T., J.

_Phys.

Chem. Sol. 15

₍₁₉₆₀₎

72 ; Ph. D. Thesis, Paris

₍₁₉₆₃₎

_unpublished.

[7]

BÉAL M. T. and FRIEDEL J.,

Phys.

Rev. 135

₍₁₉₆₄₎

A466.

[8]

See for instance in Table of

_Integrals,

Series and Products,

by

I. S.

_Gradshteyn

and I. W.

_Ryzhik

(Academic Press)

1965.

[9]

Actually

the « universal » label should not be taken stricto sensu even in the non

_interacting

impurity

case

_(we

_acknowledge

a remark from one of our referees

reminding

that

point

to

_us).

Indeed, even in that case,

only

the central _part of the conductance distribution may be described

by

a Gaussian of universal variance but not the tails, as shown in AL’TSHULER B. L.

et _{al. ,} J.E. T.P. Lett. 43

₍₁₉₈₆₎

441 ; J.E. T.P. 64

₍₁₉₈₆₎

1352. Now, if one took into account the interaction between the

impurities,

as we _{suggest here,} it would

modify

the entire conductance

distribution. In

_particular,

the universal character would

disappear

even in the central _part.

[10] AL’TSHULER

B. L. and SPIVAK B. _{Z., J.E.T.P.} Lett. 42

₍₁₉₈₅₎

_{447 ;}

FENG S., LEE P. A. and STONE A. D.,

Phys.

Rev. Lett. 56

_(1986),1960

and erratum 56

₍₁₉₈₆₎

2772.

[11] LANDAUER

R., I.B.M.J. Res. Dev. 1

₍₁₉₅₇₎

_{223 ; Localization,} Interaction and

_Transport

Phenomena in

_Impure

Metals, Eds. B. Kramer, G.

_Bergman

and Y.

_Bruynseraede

_(Springer