Structure of vacancy induced spin polarons in solid 3He

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Structure of vacancy induced spin polarons in solid 3He

G. Montambaux, P. Lederer, M. Héritier

To cite this version:

G. Montambaux, P. Lederer, M. Héritier.

Structure of vacancy induced spin polarons

in solid 3He.

Journal de Physique Lettres, Edp sciences, 1979, 40 (18), pp.499-503.

�10.1051/jphyslet:019790040018049900�. �jpa-00231674�

Structure of vacancy

induced

_{spin polarons}

in

solid

3He

G.

_Montambaux,

P. Lederer and M. Héritier

Laboratoire de _Physique des Solides _(*), Université Paris-Sud, Centre d’Orsay, 91405 _Orsay, France

(Re~u le 27 _fevrier 1979, revise le 18 juillet 1979, accepte le 24 juillet 1979)

Résumé. 2014 On étudie l’existence de

polarons ferromagnétiques

induits par des lacunes, dans les _{phases cubiques} centrées et _hexagonales bidimensionnelles de 3He. On discute _{l’hypothèse} de

_parfaite

localisation et la limite des

_polarons

de _petite taille. Ceux-ci ne

_paraissent

pas

pouvoir expliquer

le défaut

_d’entropie

observé autour

de 1 K dans 3He

_cubique

centré. On discute ensuite la

_possibilité

de la condensation des

_polarons

en une _goutte

ferromagnétique macroscopique.

Abstract. 2014 We

study the existence of _{ferromagnetic polarons} induced _by vacancies, in b.c.c. and 2 D

_hexagonal

3He. We discuss the

_assumption

of

_perfect

vacancy localization within the

polaron

and the small size

_polaron

limit. These models do not seem to account for some

_negative

contribution to the _entropy observed around 1 K in b.c.c. 3He. We then discuss the

_possibility

of _polaron condensation in a

_{macroscopic ferromagnetic drop.}

Classification

Physics Abstracts 67.80 - 75.30

1. Introduction. - Giant

spin

polarons

are

_thought

to form around vacancies in fermion

_quantum

crystals

such as b.c.c.

3He

or 2 D alternate

phases [1, 2].

The _vacancy lowers its kinetic _energy

_by

_propagating

in a

region

of

lined-up spins,

the volume of which

is limited

_by

the _entropy and

_exchange

_energy loss.

It may be that

_{susceptibility}

_[2]

and

_specific

heat

anomalies

_[3]

which have been observed in the mK

region

[4]

in b.c.c.

3He

are

_experimental

manifesta-tions of these

_polarons,

_although

other theoretical

explanations

are available 9in the literature

_[4].

Irres-pective

of the actual

_validity

of theoretical treatments

of the observed

anomalies,

the existence of vacancy

magnetic polarons

under convenient

_physical

condi-tions can

_hardly

be

_questioned,

as it relies on a

few essential features of fermion quantum

crystals.

Theorical treatments until now

_{[1, 2]}

have been

based on a _very

_simple

model which treats the lattice

as a

continuum,

which is

strictly

correct

_only

for

infinitely large polarons.

Up

to now, conditions for the existence of

_polarons

as a function of

tempera-ture and

_strength

of

_exchange

forces have been discussed

_{only briefly}

_[2].

The

_implication

of the alternate or non-alternate character of the lattice have been stressed

_{recently [5].}

It _appears to be a convenient

_time,

in view of the

progress of low temperature

techniques

and of

experiments

on

3He

in the relevant _range of

tempe-(*) Laboratoire associe au C.N.R.S.

1

rature

_{[4, 6]}

to refine

_existing

models,

test their

_validity,

and make a more accurate assessment for

_polaron

thermodynamics

[7].

This _paper deals with :

i)

the effects linked with the

_polaron

_boundary,

ii)

the small size

_polaron

limit which would seem

appropriate

to

_higher

_temperature,

iii)

the

_possibility

of

_polaron

condensation in a

macroscopic ferromagnetic drop

[7].

The first

_topic

has either been

_disregarded

in

previous

work

_[1]

or discussed _very

_briefly

_[2].

The

second

_topic

has not been discussed

_previously.

We

discuss the third _aspect either at fixed number of

particles

or at fixed chemical

_potential,

and the

possibility

of an

_experimental

observation. 2. Review. - We

study

a

_crystal

with

_nearly

one

fermion _per

site,

with a

_repulsive

interaction U

which we _suppose is infinite. We consider the case

of a finite _temperature T. We

_neglect

orbital

degene-racy. The relevant Hamiltonian is the Hubbard

Hamiltonian which allows an

approximate

descrip-tion of

_exchange

forces. Elastic

_properties

of the

lattice are not well described. But essential

_properties

due to the

_{antisymmetrization}

of the whole wave

function of the lattice of fermions with a hard core

repulsive

interaction are well accounted

for,

when

hole

_{propagation through}

the

_crystal

is involved. In the infinite

_repulsive

interaction

limit,

configu-rations with two fermions on the same site are

L-500 JOURNAL DE _{PHYSIQUE -} LETTRES

bidden. A hole _propagates

_through

the lattice

_by

exchanging

its

_place

with a nearest

_neighbouring

fermion. Its motion

_depends

on the

_magnetic

confi-guration.

A theorem due to

_{Nagaoka [8]}

shows

_that,

in this case, the

ground

state is the

_{fully ferromagnetic}

state. The hole moves in an _energy band

_stretching

from - zt to + zt where z is the number of nearest

neighbours

and t is the

_{overlap integral}

between

nearest

_neighbours.

An extension of this

_theorem,

due to Brinkman and Rice

_[9],

shows that for a random

configuration

the _vacancy band is narrowed. It extends from

-

Wo zt to _c~o zt where

in the

_{self-retracing-paths approximation}

_[9].

It is a

consequence of the

partial

localization of the vacancy

because of the

_suppression

of

_paths

for its

propa-gation.

Several authors

_{[1, 2]}

_suppose that at finite

tempe-rature, the vacancy decreases its kinetic energy

by

localizing

itself in a

ferromagnetic polaron,

the size

of which is limited

_by

_entropy

and

_exchange

(we

suppose here that

exchange

is

_negligible

if ~7 ~>

t).

In order to reduce _{the many}

_body

_problem

to a

single particle problem,

they

proposed

the model of

a

_sphere

of radius

_R,

within which the

_spin

pola-rization

₍

_{sZ ~}

is

_supposed

to be

_uniform.

_{SZ ~}

varies

abruptly

from 2

to 0

_through

the

_{polaron boundary.}

In fact _{the vacancy} is not

_entirely

localized in the

ferromagnetic

volume since the

_potential

_{energy is}

not infinite outside the

_polaron.

There still exist

paths

in the

_spin

disordered

_region

for which the

vacancy motion does not

_perturb

the

_spin

configu-ration.

_Formally

the

_problem

is somewhat different from the localization of an electron in a bubble in

liquid

4He

_{[10, 11],}

a

physical

situation which offers an

example

of a

nearly perfectly

localized

particle.

In this case the

_depth

of the

_potential

well in which

the electron moves is

_quasi

infinite

_compared

with

the

_ground

state _energy of the

_particle.

Therefore in our _case, we have to take into account

the

_exponential

_decay

of the _vacancy wave function

outside the

_{polaron [2]}

and we shall show the

impor-tance of the

_boundary

effects on the

previously

obtained results. Furthermore we shall extend these

results in the limit of small size

_polarons

(R ~

a)

where the discrete nature of the lattice cannot be

ignored.

In that case, the kinetic energy can be

evaluated

_by

the moment method

_[9].

3. Localization energy. - In

one

dimension,

all

paths

are self

_retracing

and the motion of _{the vacancy} does not

_perturb

the

_{surrounding spins.}

Thus there does not exist

_spin

_polaron

in one dimension.

In two or three

dimensions,

the existence of a

ferromagnetic polaron

is

_possible

in an alternate

lattice

_{[5, 7].}

In this case, the lowest kinetic energy

of the hole is assumed to be the lowest bound state

in the

_potential

This effective

_potential

expresses, in an

_approximate

way, the fact that the kinetic energy of the vacancy

depends

on the

spin configuration.

The

ground

state

corresponds

to the l = 0 orbital number. We have

to solve the

_{Schrodinger equation}

for a

_particle

of mass m

_moving

in

_V(r).

In a

_{totally ferromagnetic}

medium,

the

problem

would be a

single particle problem

with k a

_good

quantum

number. The effective mass

_depends

on the

transfer

_{integral t through}

the

_dispersion

relation obtained

_by

the

_{tight-binding}

method. An

_expression

of these relations is

_possible

near the zone

_{boundary :}

1ï2/2 m

= 4

a2 t for the 2

D

hexagonal

lattice

(2)

11212 m

_ 4 3

~

for the b.c.c. lattice

(2)

where a is the distance between nearest

_neighbours.

In fact the nuclear

_spin

_system is

_perturbed

when the

vacancy tunnels from site to site. For that reason,

the

_study

of _{the vacancy} cannot be reduced to a

single particle problem

and k is not a

_good

_quantum

number. Therefore

_considering

the evolution of a

particle

in an effective

potential

is a serious

approxi-mation,

in

_particular

when the

_polaron

size is compa-rable with the lattice

_spacing.

The kinetic _energy of the

_particle

is written

In the

_perfect

localization

_{limit, y equals n}

in three

dimensions and 2.405

(first

zero of

_Jo,

the first Bessel function of the first

_kind)

in two dimensions.

_Figure

1

shows the variation of E versus R

_by

solving

Schrö-dinger equation

for the

_{potential (1)}

in 3 D. In two

dimensions,

there

_always

exists a bound state

what-ever the

_polaron

radius R _may be. The _vacancy

energy goes to - ₍₀₀ zt when R _goes to zero. In

three dimensions the bound state

_disappears

if

for the b.c.c. lattice. For R

_Ro,

the kinetic _energy is

_equal

to _{the energy of the upper}

_edge

of the

well,

i.e.,

the _energy in the random

medium, -

(00 zt. This difference is

_probably

_unphysical

and stems

from the continuum

_{approximation.}

One can see that the exact solution of the continuum

problem

differs

_appreciably

from the

_simplest

one in

which a

_perfect

localization is assumed

_{[1]. Edge}

effects lower the kinetic energy,

particularly

for small

_polaron

radii. For R N

2,

a case of

physical

interest

_{[2, 3],}

the exact _energy is smaller

_by

a factor

of 2.

However,

for these small size

_polarons,

the

discre-teness of the lattice cannot be

_ignored.

Therefore,

we have used a moment method. We have

_computed

the first few moments of _{the vacancy}

_density

of

states in two dimensions

_(up

to the 10th

_moment)

and in three dimensions

_(up

to the 8th

_moment)

for

polarons

of different sizes. The

difficulty

of the method increases

_tremendously

with the order of the moment, because we need to

_know,

for each

_path,

the

_probability

that the initial

_{spin configuration}

is restored at the end of the

_{path [9, 12]. By}

extrapo-lation of the first moments, we have obtained the

lower band

_edge

_{energy of the vacancy.} Given the

small number of moments we have

_computed,

the

determination

_{implies large}

error bars

(see

_Fig.

1).

We find that the continuum

_{approximation}

_is,

_by

chance,

surprisingly good

at _{very small} radii.

How-ever, for R ~

2,

the kinetic _{energy in the} discrete lattice is smaller than in the continuum.

Fig. 1. - Localization

energy of a _vacancy in a _polaron in the

b.c.c. lattice versus the number N of spins in the polaron. One particle continuum approximation : a) in a _perfect localization

model, E = ~2 x _{1i2/2 mR2, b)} E _{= n2/4} x 1i2 2 mR2, c) exact

calculation. The bound state disappears if N N 12. The bars _give

an estimation of _{polaron energy} calculated _{by extrapolation} of the first moments of the density of states.

4. Existence of the

_polaron

state. - Mott

[13]

esti-mates that the _vacancy band shows _strong

_peaks

due

to the

_{heavy spin polaron.}

An exact calculation of the free energy

requires

the

_knowledge

of the form of the

_density

of states for one

_particle.

In a

_simple

calculation,

various authors

_[1,

2,

5, 7,

13]

assumed that vacancies are all in the

ground

state. We also make this

_assumption

which we shall discuss below.

Taking

these

_{approximations}

and in a model of

independent

vacancies,

the free _energy is

For each structure, the

_depth

of the

_potential

well V

is

_{given, y}

and E

_depend

_only

on the radius R. The

first term describes the lower _energy state for the vacancy in the

ferromagnetic

medium. In the

conti-nuum

_{approximation,}

the second one is the energy

to localize

_partially

the _vacancy in the

_sphere

(finite

potential

well).

The third one is the _entropy increase with _respect to a

_totally

random

region.

The

spherical

model considered here will be

_{quantitatively}

valid if R is

_large

_enough.

In this case the number N of

spins

in the

_polaron

is in the 2 D

_hexagonal

lattice :

and in the b.c.c. lattice :

where a is the distance between nearest

_neighbours.

If N is

_large

_enough

so that the continuum

approach

is

_valid,

we

_neglect

the term In

_(N

+

_{1 ).}

The free _energy is a function

F(R,

T )

of the radius R

and of the

_temperature

T. It

_always

has a local

mini-mum at R = 0 which does not

appear in a

_perfect

localization model. It

_generally

exhibits another minimum for R =

Rm(T)

which describes the

polaron

state. This second minimum is below the first one

( -

Wo

zt)

above a critical

_temperature

_7~.

Then if

T

_7~

the stable state is a

_polaron

of radius

Rm(T).

The

_parameters

_Tc

and

_Rm(T)

are

_{given by}

the varia-tional treatment. The

_polaron

exists for

tempera-ture T as

- 2 D

hexagonal

lattice : the self

_retracing

_paths

approximation

is rather valid since the relative error

in the sixth moment is 4 x

10- 3.

Then with a

_good

approximation,

we _get

The

_validity

of this

_{approximation}

is due to the small number of nearest

_neighbours

which

_implies

large loops.

The exact variational calculation has to

take into account the variation

_{of y}

versus the radius

_{R ;}

y is

always

smaller than 2.405 found

by

Feigel’man

[2].

The minimization of the free _energy with _respect

to R

_gives

the variation

_{of Fm(T)}

versus T. The critical

temperature below which the

polaron

becomes

ther-modynamically

stable is

At this

_temperature

the radius of the

_sphere

is about

L-502 JOURNAL DE _{PHYSIQUE -} LETTRES

approximation

is

_good

enough

for this

_temperature

region.

- b.c.c. lattice : the

self-retracing-paths

approxi-mation is less

_good.

It leads to

A moment calculation

_gives

more

_exactly

the random

bandwidth and leads to

In a better treatment we have to use this result. As

in two

dimensions,

the critical conditions are obtained with the condition

₍₄₎

The true value of

_7~

is

_expected

to be

_{slightly larger.}

We find

_by

the moment method that a

_polaron

of

58

_spins

is stable for T 3 x

10-2

t.

Another

_difficulty

is due to the fact that the

_ground

state is not the

_{only occupied}

state. So we have to

know the form of the

density

of states in the two

configurations.

In the random one, Brinkman and Rice

_[9]

found a band of the form

If we consider that the band

structure

is not

signifi-cantly

altered

_by

_passing

from the random

configu-ration to the

_{polaron configuration,}

the

_temperature

_Tc

is

_{nearly unchanged.}

But if all states condense in a

polaron peak,

Tc

increases

_by

one order of

magnitude.

The exact determination of

_Tc

_requires

a detailed

knowledge

of the _vacancy band.

5. Remarks on the condensation in a

_{ferromagnetic}

macroscopic

drop.

-

Feigelman [7]

studies

submono-layers

films of

3He

adsorbed on a

_graphite

surface.

He assumes that in a lattice commensurable with

the

_substrate,

the number of vacancies is fixed.

He

_predicts

that the condensation of the

_polarons

in

a

_{ferromagnetic macroscopic drop}

decreases the free _energy of the vacancies

_by

a

_{multiplicative}

term

of order 1 :

_Fm

_Fm.

In his treatment he considers

that y

= 2.405 is _constant. This has _to be corrected

by boundary effects. y

is in fact a

decreasing

func-tion

_y(T)

of the

_temperature

_always

lower than 2.405.

A

_rigorous

calculation

_gives

If

_y(To)

=

1,

the

polaron

_state becomes stable. In

fact this

_happens

at a

_temperature

_To

where neither

polarons

nor the

_drop

exist.

Such a condensation is

_impossible

with thermal vacancies. At fixed chemical

_potential,

the vacancies

induce either uniform

_{ferromagnetism}

or

ferro-magnetic

polarons.

Even in a commensurate

_lattice,

in

_{thermodynamic}

_equilibrium,

the fixed

_quantity

is

the vacancy chemical

_potential,

i.e. the external pressure, under usual

experimental

conditions.

Then,

a

_{macroscopic region}

without thermal vacancies

cannot be

stable,

either in a 2 D commensurate

layer

or in b.c.c.

3He.

The case of metastable vacancies

_might

be different.

Let us assume that

vacancies,

at a

given

_temperature,

become

_trapped

_by

absence of

_{diffusion, by}

_forming

large

magnetic

polarons

[2, 3].

Then,

at lower

tempe-rature, there exists a fixed number of metastable

vacancies in excess of the

_equilibrium

concentration. With an

increasing applied magnetic

field,

the _vacancy

wave function

_overlap

_increases,

and so does the

bandwidth of the sort of

_impurity

band

_they

form. At a critical

magnetic

field,

a Mott-Anderson transi-tion should occur in the

impurity

band,

leaving

the

vacancies free to diffuse.

Then,

under suitable

condi-tions,

the vacancies could condense and form the

Feigelman

drop,

with an

_abrupt

decrease of the

magnetization.

Therefore,

keeping

in mind the number of

_unproved

_assumptions,

we _suggest as a

conjecture

with the desirable

wariness,

that the first order transition

_recently

observed at

_high

_magnetic

fields in b.c.c.

3He

_[14]

is in fact a

magnetic polaron

conden-sation.

At fixed chemical

_potential,

what remains of the

ferromagnetic

drop

is a

tendency

to form

_multiple

vacancy bound states

_[15]

and a finite concentration

of

di-vacancies, tri-vacancies,

etc.

6. Discussion. - The

qualitative

arguments

given

in the literature

_{[1, 2]}

about the existence of _vacancy induced

_magnetic

_polarons

can

_hardly

be refuted.

However,

when one comes to a

_quantitative

descrip-tion of the

_polaron

in order to _compare the

experi-mental data with effects

_{predicted by}

the

_theory,

a

much more refined treatment looks to _{be necessary.}

In

_particular,

we have shown that

_considering

the

vacancy as

completely

localized in the

ferromagnetic

region

is a _poor

approximation.

Obviously,

all the

simple

power laws

predicted

in the crudest model

[1]

for the various derivatives of the free _energy are not

valid.

Therefore,

our results will

_change

the _vacancy contribution to the

_{susceptibility,}

the

_specific

heat,

etc.

In the crudest

_picture

_[1],

a _vacancy

_polaron

with

less than 200

_spins

would not be stable.

_Taking

into

account the

_tailing

of the vacancy wave function in the disordered medium increases in an

_important

_way

the

_stability

of the

_polaron.

In the continuum

approxi-mation,

one finds a minimum size of about 50

spins.

When

_considering

the discrete nature of the

lattice,

this

_figure

is still lowered. We can

_give

an _upper

limit to the formation

temperature

T,

Given _{the range} of values

_proposed

for t

(50

mK-500

_{mK) [16, 17],}

the

_possibility

of existence of small size

_polarons

around 1

_K,

not

_correctly

described in

the continuum

_{approximation,}

is ruled out

_by

our

moment calculation. In

_particular,

_they

could not

explain

a

_negative

_vacancy formation

_entropy

in this

_temperature

_range, as

_recently

_{proposed [18].}

Instead,

we _expect a

_{rapid growth}

of the

_polaron

around

_7~

to a size of 30 to 50

_spins.

_However,

the

complete

discussion of the

_polaron

formation would

require

a variational

_study

in which the

_spin

pola-rization sZ ~

is allowed to _vary

_continuously

with

temperature.

For

_temperature

near 1 mK in b.c.c.

3He,

we

cannot

_neglect

the

_exchange

J = 0.7 mK _on the

melting

curve

_[19].

To take it into account we trans-form terms like T Ln 2 as T Ln 2 +

zJ14.

In the 2 D

_hexagonal

lattice,

the transfer

_{integral t}

is

_probably

_very

small,

nearly

5 mK

_[6].

The

_polarons

will start

_forming

at _very low

_temperature

_{(10-100 ~K).}

This _temperature

_region

is still

_{experimentally}

out

of reach.

References

[1] ANDREEV, A. F., J.E.T.P. Lett. 24 ₍₁₉₇₇₎ 564.

[2] HÉRITIER, M. and _LEDERER, P., J. _Physique Lett. 38 (1977) L-209.

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