Can excitons produce superdiamagnetic currents ?

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Can excitons produce superdiamagnetic currents ?

P. Nozières, D. Saint-James

To cite this version:

P. Nozières, D. Saint-James. Can excitons produce superdiamagnetic currents ?. Journal de Physique

Lettres, Edp sciences, 1980, 41 (8), pp.197-199. �10.1051/jphyslet:01980004108019700�. �jpa-00231758�

L-197

Can

excitons

_produce

_{superdiamagnetic}

currents ?

P. Nozières and D. Saint-James

_(*)

Institut _{Laue-Langevin,} 156X, 38042 Grenoble _Cedex, France

(Re~u le 8 janvier 1980, accepte le 26 fevrier 1980)

Résumé. 2014 Nous montrons _que la condensation de Bose des excitons ne _peut pas produire de courants

diamagné-tiques. Ceux-ci sont nuls dans une _{approximation} de champ moyen cohérente, qui inclut à la fois les termes nor-maux et anormaux.

Abstract. 2014 It is shown that Bose condensation of excitons

cannot _produce

_{superdiamagnetic}

currents. The latter vanish within a consistent mean field _{approximation} when both normal and anomalous _pairings are retained.

J. _Physique - LETTRES 41

(1980) L-197 - L-199 15 AVRIL _1980,

Classification

Physics Abstracts

71.35

It has been claimed

_recently

that Bose condensation of

_optically

active excitons could

_give

rise to electric

currents -

yielding

a

_{superdiamagnetism}

reminis-cent of

_{superconductivity [1],}

_[3].

Such a conclusion is

_puzzling,

since it amounts to

_producing

a

charged

current via the condensation of neutral entities

_(the

excitons).

In the

_present

note, we _{argue that such}

diamagnetic

currents do not exist :

_they

_appear

_only

as a _consequence of inconsistent

approximations.

Let ~

be the net electric

_dipole

moment of the electron _system

An

_unambiguous

definition of the current

_operator

is

J

_{= ~}

_{Usually, ~}

will contain a free carriers unbound-ed _part

_together

with an atomic

_polarization

term.

More

_{specifically,}

we use a basis

of orthogonal Wannier

functions

~(r 2013 Ri) (n

is the band

_{index, Ri}

the lat-tice

_site).

Then

Because the _qJn are

_orthogonal,

The first term J.1f in

₍₁₎

is the band

_diagonal

usual

(*) On leave of absence from _Groupe de _Physique des Solides de 1’Ecole Normale Superieure, Universite Paris VII.

w

position

operator,

while ~

is the

bounded

atomic

polarization

term.

For

instance,

let us consider a

_highly

simplified

model : a two band _{system of}

_spinless

electrons,

in

which each site is a _symmetry centre. Let one band

have _s-symmetry

_(creation

_operator

_a~1),

the other one

being

pz-type

(creation

operator

b*)

(1).

To the extent that the

_overlap

between

_neighbouring

site is

_small,

the dominant contribution _{to p may} be written

(d

is the usual interband

_dipole

matrix

_element,

_parallel

to

_Oz).

The free carrier and atomic

contributions,

~f

and

_~,

are

clearly separated.

We now return to the

_general

case. The

Hamilto-nian is written as

The first term of

₍₃₎

embodies kinetic energy and the

one

_body

lattice

potential;

U is the

_particle

interac-tion. Since U

_{depends only}

on

_positions,

it must commute with _tt. Thus

_only

the first term in

₍₃₎

contri-butes to the

_{current A}

_(actually,

_only

the kinetic

energy). Making

use of

_(1),

we see that the current

may be

split

into two _parts

e) We consider a _{single p-subband,} which _implies either uniaxial

symmetry, or excitons all polarized in a single direction, here the

z-axis.

L-198 JOURNAL DE _{PHYSIQUE -} LETTRES

Jf

is the usual intraband conduction current :

It is the

_quantity

one deals with in

_transport

pheno-mena.

J,

instead,

arises from the atomic

polarization

~ ;

it involves interband _processes,

_supposedly

respon-sible for

_{superdiamagnetism.}

We now _argue that _any

_quantity

which is the time

derivative of a bounded

_operator

has a zero

expecta-tion value in _any

_{~~~~ ~ ~ )}

of the Hamiltonian.

Consequently

There cannot exist a

_{steady superdiamagnetic}

current

(note

that such a conclusion would not

_apply

to the

ordinary

current

_{J f,}

_{since J1r}

is not a bounded

opera-tor : one must _worry about

_{boundary conditions).}

If an

_approximate

calculation

_yields

a finite value of

J,

it means that either J is

inaccurately

defined,

or that

the

_{approximations}

violate the condition

_{[t7, ~}

=

0,

or both. In _any case, a

finite ( J >

should be an

artefact.

In a recent

letter,

Batyev

[4]

considered the same

problem.

He noted that the current _operator used

_by

Volkov et al. was

_inaccurate,

as it violates the

conti-nuity

equation

(equivalent

to our J =

/t).

Within a

BCS

_type

effective

Hamiltonian,

he restored

_charge

conservation

_by

_modifying

the current

_operator,

introducing

as an extra term the commutator of

position

with the BCS

_potential.

He then showed that if the

_{continuity equation}

is

_obeyed,

_{then J ~}

vanishes,

a conclusion which is

essentially equivalent

to our

_general

_argument

_(4).

While we _agree with the

result,

we claim that such a

_procedure

is incorrect.

Since the

_position

commutes with

U,

the mean field

BCS

_potential

should not, on _{the average,} affect the current

_operator

_(which

is defined from first

prin-ciples).

The violation of

_charge

conservation must be cured not

_by

an ad hoc redefinition of

_J,

but

_by

a

consistent mean field

_{approximation.}

In order to see

explicitly

how the cancellation

occurs, we return to our

_spinless

two band model.

We assume a local interaction :

l

The bands are assumed to be

_symmetric

with _respect to some

_{ener~r ~,}

so that

The

_{constant ~}

is irrelevant : it may be absorbed in a

redefinition _{of the energy}

_origin.

It is

_readily

verified

that

_{[tI, at b~]}

= 0 : U does _commute

with ~,

_as it

should. The two contributions to the current are

They

are best

_expressed

in terms of the Fourier trans-formed Bloch waves

where

is the Bloch state _energy

Note that J involves the

_full

_{energy Sk-}

_Replacing

the latter

_by

the

_local

_atomic

_{quantity 7~}

would violate the

_relationship

J

=

~, thereby introducing

_a

spurious

current.

In order to

_proceed,

we treat U within a molecular field

_{approximation.}

We _may assume an

_equal

number of electrons and holes. In order to _preserve electron

hole symmetry, we write U as

The last term of

₍₇₎

_may be

_incorporated

into the one

body

Hamiltonian : we assume that has been done

and we discard it. We moreover

_{set ~}

= 0 via _an

appropriate

choice of the

_origin.

The

_resulting

mean field hamiltonian

_displays

full electron hole symme-try :

In

_{(8), V}

and A are the self consistent

potentials

Bose condensation of excitons is

_signalled

_by

the anomalous

_pairing

A. The hamiltonian

Heff

is

L-199

CAN EXCITONS PRODUCE SUPERDIAMAGNETIC CURRENTS ?

The

_resulting

self

_{consistency equations}

read

Ek

=

(Ek -

V)2

+

1 L1 12

is here the

_{quasiparticle}

energy. We note that

₍₁₀₎

leaves the

_phase

of L1

arbi-trary : this is a result of the _{gauge invariance} of

_{(7) :}

the interaction is

_unchanged

if one shifts the relative

phase

of ai and bi

by

an

_arbitrary

amount. Such an

invariance would not

_persist

if one were to include

non local interactions

(terms

such as a* a* bb are

then

_{quite possible).}

From

_(6),

we see that

If it

_{existed, ( J >}

would

_depend

on the choice of the

gauge, a

_surprising

result. In ref.

[1],

the authors

consider the atomic

quantity

in

_{which Ek}

is

_{replaced by}

Tii

in

_(11).

We saw that such a term

_only

_represents

part of the story.

Restoring Ek

in

_(11),

we

_might

follow

the usual

_practice

of

ignoring

V,

on the

grounds

that it

just renormalizes Ek :

then there is no clear reason

why

(11)

should vanish. The

_{paradox disappears}

if

we consider the self

_{consistency equations}

_together.

Then,

by

subtraction,

we see that V

_adjusts

in such

a way that

Thus,

the

_diamagnetic

current J vanishes

_identically,

irrespective

of the _gauge, as it should.

We see that a _proper solution

_requires

a consistent calculation of both the anomalous and normal

poten-tials,

L1 and V. The

_origin

of that fact is clear. When we

replace

U

_by

its linearized form

_(8),

the commutator

[U, a1

hi]

is no

_longer

zero :

Still,

if we retain both

pairings,

and use

(9),

we see

that

₍₁₂₎

vanishes on the _{average :} mean field

lineari-zation preserves the conservation law

[U,

~]

= 0 _as

far as

_{physical expectations}

are concerned. If however we discard the last term of

_(12),

our conservation law

is

_definitely

ruined. The current

J given

_{by (6)}

is no

longer

the commutator

_{of ~}

with

_H,

and a

_spurious

finite result ensues.

The above model could be

_generalized

to more

complicated

situations,

for instance non local

interac-tions U. In such a case,

however,

the commutation rule

_[U,

_p]

= _{0 is no}

_longer

automatic. It results on

constraints on the various

coupling

matrix elements : these constraints will be vital in

_{guaranteeing J}

= 0. Over

_again,

we

_expect

that

steady diamagnetic

cur-rents will cancel as a result of subtle

_{compensations}

between

_{selfconsistency equations}

for the numerous

pairings

a~

ajm

).

Any approximation

that

distroys

these

_{compensations}

would introduce a

_spurious

commutator

_[U,

_p],

_responsible

for the

_expectation

value

J

).

Thus,

considerable care must be

_exerted

in order to preserve the correct definition J

_{= ~.}

An innocent

_looking

mean field linearization may prove very

misleading.

We did not consider the extension to

_{spin 1/2}

par-ticles. We do not _expect

_major

corrections to the above

_simple

_arguments.

References

[1] VOLKOV, B. _{A., KOPAEV,} Yu. V., J.E.T.P. Lett. 27 (1978) 7.

[2] VOLKOV, B. A., GINZBURG, V. L., KOPAEV, Yu. _V., J.E.T.P.

Lett. 27 (1978) 206.

[3] VOLKOV, B. A., KOPAEV, Yu. V., TUGUSHEV, V. V., J.E.T.P. Lett.

27 (1978) 615.