Pairing of holes via vortex/antivortex attraction in doped La 2-x(Sr)xCuO4

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Pairing of holes via vortex/antivortex attraction in

doped La 2-x(Sr)xCuO4

Gerard Corsten, Cliff Liem, Raphaël Blumenfeld, Naeem Jan

To cite this version:

2229

LE

JOURNAL

DE

_PHYSIQUE

Short Communication

Pairing

of holes

via

vortex/antivortex

attraction

in

_{doped La2-x(Sr)xCuO4}

Gerard Corsten

_(1),

Cliff Liem

_(1),

_Raphaël

Blumenfeld

₍₂₎

and Naeem Jan

_(1,2)

(1)

Physics Department,

St. Francis Xavier

_{University, Antigonish,}

Nova _Scotia, B2G 1CO,

Canada

(2)

Cavendish

_Laboratory,

_Madingley

_Road,

_University

of

_{Cambridge, Cambridge,}

CB3 _OHE, G.B.

(Received

27

_July

199Q

accepted

6

_{August 1990)}

Abstract. 2014 The

consequences of delocalised

magnetic

frustration in the _pure

_{antiferromagnetic}

XY(n =

2)

spin

system are studied

_numerically.

This frustration is found to enhance

_{significantly}

the formation of vortex excitations, which tend to bind antivortices below the Kosterlitz-Thouless transi-tion _temperature. We find a net attraction between frustrated

_plaquettes

mediated

_by

the attraction between the vortices.

_Applied

to the

_{understanding}

of

_high

_temperature

_{superconducting}

_materials, like

_{La2-x(Sr)xCuO4,}

this can serve as a real _space

_pairing

mechanism with a

_{high binding}

_energy

between the _holes, induced

_by

_doping.

There are common features between this

_pairing

model and the

_{"spinon-holon" picture,}

which _may

_provide

a link between these different

_approaches.

1

_Phys.

France

_{SI (1990)}

2229-2233 15 OCTOBRE _1990,

Classification

Physics

Abstracts 74.20 - 75. 10J

Theoretical

_arguments

_{(1, 2]}

and

_experiments

_{[3, 4]}

indicate that _pure

La2Cu04

is an

anti-ferromagnet

because of the

_spin

associated with the holes localised on the Cu atoms

_(Cu++).

The _energy

_gained

from Pauli’s exclusion favours

_opposite

states of the

_spins

on

_neighbouring

Cu atoms.

_Emery

_{[1] (1)}

has

_presented

an

_appealing

_simple

ionic

_picture

_illustrating

the

_ground

state of

La2Cu04.

Aharony

et aL

_[2]

_(II)

focused on the

_{magnetic properties}

of this

_system

and the effects of

_doping

on the

_{antiferromagnetic ordering.}

There is

_strong

_experimental

evidence

[5]

that the

_doping

in

_{La2_x(Sr)xCu04}

induces holes which are more or less localised on the

planar

oxygen atoms. These holes are distinct from those localised on the _{copper atoms,} which

are

_primarily

_responsible

for

_{antiferromagnetism.}

The holes associated with the _oxygen atoms

are

_responsible

for the

_{supercurrent.}

In II it has been noted that the net

_magnetic

effect of the holes induced

_by

_doping

is a

ferromagnetic

interaction between

neighbouring

_copper atoms on an otherwise

_{antiferromagnetic}

_system.

Thus the

La2Cu04

superconductors

may be considered

as

_{weakly coupled layers}

of two-dimensional

_{(2d) antiferromagnetic spin}

_systems

with the

_doping

2230

forming ferromagnetic

bonds and hence

_leading

to frustration within the

_planes.

It has been

_suggested

_{[1, 2]}

that the

_spins

at each _copper site are most

_likely

to behave as

Heisenberg

quantum s

=1/2

_spins.

Nevertheless,

following

I,

II and reference

_[6],

we treat them as

classical

_spins

since in this work we are

_primarily

interested in the

_long

_range

_magnetic

_cooperative

effects. Both 1 and II introduced frustration

_{by confining}

the hole created

_{by doping}

to one _oxygen atom,

although they acknowledged

that this hole is shared

_by

at least four _{oxygen atoms,} and

probably

more. This artificial confinement was _necessary because

_simply

_letting

the hole

_spread

on four bonds around a _copper atom and

_making

them

_{ferromagnetic,}

_may remove the frustration

altogether.

One of the _purposes of this Letter is to

_explore

the

_magnetic

_consequences of

_sharing

a hole between four _oxygens, while

_retaining

the essential frustration. The main result we

_report

is that the

_doped

holes enhance the formation of vortex and antivortex excitations which attract,

when at close _range,

_leading

to a real _space

_pairing

mechanism.

We have taken a few liberties with the standard heat-bath

_algorithm

_[7]

for the

_planar

model.

The energy of interaction between a

_target

_spin

and its

_{nearest-neighbours}

is

_computed

as the dot

product

of this

_spin

and the resultant

_spin

vector from its

_{nearest-neighbours,}

but this resultant vector, whose

_length

can _vary between 0 and

4,

is

_{approximated by}

40 discretised values. Our

sec-ond

_{approximation}

consists of

_allowing

increments of 1.8

_degrees

in the

_angle

of a

_spin

_{vector, A 0.}

We

_expect

both these

_{approximations}

to become

_{insignificant}

as the

_{temperature increases,}

and

to check this

_assumption

we further divided the interval 0 - 4 into 80 sections.

_Doing

this we

_fing

that at T =

_0.2,

the results are

_unchanged,

thus

_indicating

that for T >

_0.2,

these

_{approximations}

are indeed

_{insignificant.}

We believe that even with these

_{approximations}

the heat-bath

_algorithm

approaches equilibrium

faster and more

_efficiently

than the Glauber method. The

_following

prob-abilities were

_precomputed

and strored for each one of the discretised resultant vectors:

where

Equilibrating

the unfrustrated

_system

at T = 0.2 as a reference

_point

for a 20 x 20 _system, we

observe 4.6 x

10^ 6

vortex/antivortex

_(V/A)

_pairs

_per site _per Monte Carlo time

_step,

_agreeing

with

Thbochnik and Chester

_[8],

who

_reported

a value of ~B0 at this

_temperature.

With

the introduction of one frustrated

_plaquette

we find that this number

_jumps

to 0.007

V/A

per site per Monte Carlo time

_step.

At T = 0.3 the

corresponding

values are

10-4

and 0.02 and at T = 0.4 we find 5.4

x

10-4

and

0.03,

respectively.

In 88% of all the cases

_observed,

the

_frustrating

bond is a

_part

of the

_plaquette

on which the vortex is excited. This leads us to conclude that the _presence of the

frustrated

_plaquettes

enhances,

by

orders of

_magnitude,

the

_probability

to form V/A

_pairs.

The V/A excitation can be understood in the context of the Kosterlitz-Thouless

_{[9] (K-T)}

_theory,

where

a frustrated

_plaquette

creates a

_{region of high}

_energy relative to its

_{neighbourhood,}

which acts as a nucleus for the création of an antivortex or a vortex. The unscreened _energy of this excitation grows

logarithmically

with the

system’s

size,

favouring

the formation of a _V/A

_pair,

as we observe in our

_simulations,

in the dilute limit.

_{Assuming Boltzmann-type}

_{probabilities,}

we infer from the

above data:

_i)

the _energy of a

_spontaneous

_V/A

excitation,

E1/JAF

= 1.9:1:

_0.1;

_ii)

the

energy of a _V/A excitation with one frustrated

_{plaquette, EZ/JAF=}

0.55

_0.1;

and

_iii)

the

_binding

energy between a frustrated

_plaquette

and a vortex,

Ehv/JAF

= 1 .35 + 0.1. As the K-T critical

temperature

is

_approached

there is an increase in

_the

number of _V/A

_pairs,

as

_expected

near the

transition in the XY model. Our simulations also reveal that the _presence of two

_neighbouring

such a

_pair

is lower than both those of

_separate

vortices or

_{antivortices,}

and than that of two V/A

pairs.

We _propose that this mechanism of

_pairing,

mediated

_by

the

_magnetic

attraction of the

V/A,

may lead to the formation of a

_tightly

bound real _{space copper}

_pair.

A recent work

_by

dos Santos et aL

_[10]

showed that in two dimensions and for n =1

(but

for

annealed

_{ferromagnetic}

bonds

_[10]

_),

there is no

_{antiferromagnetic long}

_range

_order,

at _any finite

temperature,

if x > 0.3. This concentration is

_extremely

close to the

_{experimentally}

observed value

for the

_{disappearance}

of

_{superconductivity}

for

_{La2_x(Sr)xCu04}

_{(zc =}

0.32

_[ll]

_),

which _may also

support

the

_present

_conjecture.

The value 0.3 fits with

_{existing knowledge}

about these

_systems

via the

_following

hand

_waving

_argument.

_{Superconductivity}

should

_disappear

when the

_pairing

concept

loses its

_meaning,

i.e.,

when the concentration x increases such that _any two ’nearest

neighbour’

holes are forced to be less than a distance of L lattice

_spacings

_apart.

In 2d this

implies

that there

is,

at _most, one hole for _every

2L2 bonds

_(L

x L

_plaquettes).

If we

_naively

calculate for a uniform distribution in the

_plane,

this

_corresponds

to x =

1/2L2

_giving

L~ N 2.5 _{at xc,}

which should be

_compared

with

_existing

estimates of the coherence

_{length, e}

=21 . This

_general

argument

gives

the

_{typical length}

scale for _any

_pairing

mediated

_by

an excitation that is located

on one

_plaquette.

In this

_picture,

above this

_length,

a vortex and an antivortex are

_{only weakly}

correlated,

while to draw nearer would mean to lose their vortical nature. Hence if holes are

compressed

to a smaller distance

_by

_increasing

_{concentration,}

one

_expects

the

_system

to behave

as a _gas of such excitations rather than as an ensemble of distinct

_pairs.

In the usual K-T

_theory

V/A

_pairs

are formed

_{spontaneously}

and

_{their.unbinding}

initiates the

dissipative

effects which

_destroy

the

_{superfluid ground}

state. The number of these

_pairs

is limited

only by

the size of the

_system.

Our case is somewhat

_{different ;}

at low

_temperatures

the

_density

of V/A

_pairs

is

_similarily

_small,

but their

_density

is limited

_by

the

_doping

concentration. It is

_exactly

the V/A

_pairs

that are created due to nucleation around these

_doped

_"impurities"

that

_participate

in the formation of

_{Cooper pairs.}

In this

_respect

the behaviour in the

_plane

_may resemble the

superfluid

transition in thin films of

4He

and

_{3He mixtures,}

where the

4He

molecules

_represent

our V/A

_{pairs. Experiments}

on such

_systems

exist in the literature

_[12]

and show that for low

concentrations of

4He

the critical

_temperature

increases with

_increasing

concentration of

’hue.

Thus the

_qualitative

features

_(increase

in

Tc

with x for small x and

decrease

in

_Te

with x for

_large

x)

of the

_{superconducting phase}

of

_{La2_x(Sr)xCu04}

are

_compatible

with our

_analysis.

The

_present

_description

of

_high

_temperature

_{superconductivity}

resembles the

_{spin-bipolaron}

theory

discussed

_by

Mott

_{[13] ,}

_though

our

_polarons

are somewhat more

_complex.

The vortices

(or

antivortices)

induced

_by

the frustration are

_analogous

to

_polarons

which

_pair

into

_bipolarons

below the transition

_temperature.

A theoretical calculation of the

_binding

_energy of two

_polarpns,

including

the Coulomb

_repulsion

was

_attempted

for n = 1

[14],

_showing

no

_positive

_binding

en-ergy. If the

pictures

are

_equivalent,

one should not

_expect

_pairing

because annealed

_{ferromagnetic}

frustrating

bonds in an otherwise

_{antiferromagnetic Ising spin}

_system

tend to

_phase

_separate

into domains rather than

_pair.

For n = 2 the

binding

_energy is

larger

than for n =

_1,

due to the

availability

of more states for the

_spins

to settle

_into,

as also

_{predicted by}

the K-T

_theory.

In this

respect

our

_{analysis improves}

somewhat on the

_bipolaron

model. The

_present

_description

_may

also be

_compatible

with the fractional

_quantisation

model

_{[15] .}

This model results in the oc-currence of three vortice-like

_{geometrical configurations}

created

_by

combinations of

_spinon

and

holon

_excitations,

_analogous

to those we observe:

_holon-holon,

_equivalent

to

_pairing

between V/A

that are created

_by

two frustrated

_plaquettes;

_{spinon-spinon,}

_equivalent

to

_pairing

between V/A

due to

_spontaneous

_magnetic

excitations near the K-T transition

_{(without involving}

frustrated

pla-quettes) ; spinon-holon,

equivalent

to

_pairing

between V/A in the _presence of a

_single

frustrated

plaquette. Although

this model is believed to have no classical

_analog,

the identical

_shape

of

exci-tions in real _space

_tempts

one to

_speculate

whether there is a

_deeper

connection

_(i.e.,

one-to-one

2232

Let us conclude and discuss some of the

_implications

of our results. We have studied the

magnetic

behaviour of an classical XY frustrated

_{antiferromagnetic spin}

_system

that models the

sturcture of

_{La2-x(Sr)xCu04.}

We find that the formation of

_{V/A pairs}

is

_{greatly enhanced by}

frus-tration and

_consequently

_suggest

that the

vortex/antivortex

attraction between

_plaquettes

serves as a

_real-space

non-retarded

_pairing

mechanism between holes.

An

_intriguing

_implication

of this

_picture

is the

_{following: prior}

to

_pairing

and at

_{low enough}

concentrations,

it has been

_argued

that a

_single

hole sees an effective

_symmetric

double well po-tential formed

_by

the vortex/antivortex excitation

_[16].

As discussed in reference

_{[16] ,}

this means

that there _appears an oscillation in the

_probability

to find the hole in either the vortex or the

antivortex. The

_period

of oscillation is accessible to

_experiment

and

_provides

data on the

bar-rier

_height

between the wells.

_Futhermore,

this

_picture

also allows for a novel mechanism that enables a

_single

hole to

_perform

a random-walk-like movement in the

_{antiferromagnetic}

back-ground. Through

this movement a hole encounters another hole to combine into a

_{Cooper pair}

and the

_typical

_pairing

time also relates to the barrier

_{height, providing}

an

_{independent probe}

of this

_{quantity [16] .}

Numerical evidence from simulations

_{being currently}

carried out, confirm

this

_{implication [17].}

In

_addition,

these simulations also indicate a

_decoupling

between frustrated

plaquettes

and vortices in the

_vicinity

of the K-T

_temperature.

This

_decoupling

has _consequences

on the normal

_{conductivity just}

above the

transition,

which are

_currently

under

_study

_[17].

The

_{analysis presented}

here should be

_interpreted

as a

_suggestion

for

_focusing

the attention on

vortices as

_pairing

_agents,

rather than as a

_{complete theory}

of

_{superconctivity.}

It is also

_subject

to

limitations

_{imposed by}

our basic

_assumptions

that

_i)

the

_planar

model describes the

_spin

behaviour

faithfully,

and that

_ü)

these observations hold for

_quantum

_spins

as well.

Acknowledgements.

We thank Prof. Sir N.

_Mott,

Prof. LJ. de

_Jongh,

Prof. A.

_Aharony,

Prof. H.E.

_Stanley,

Dr.

T. Duke and B.D. Simons for fruitful discussions. This research is

_supported

in

_parts

_by

a

_grant

from the Natural Sciences and

_Engineering

Research Council of Canada

_(G.C.,

C.L. and

_N.J.)

and

_grant

No. RG9358 of the Science and

_Engineering

Research

_Council,

_UK(R.B.).

Note added.

After the

_completion

of this _paper, it was

_brought

to our attention that Schmeltzer and

_Bishop

[18]

carried out

_recently

an

_analytic

_work,

_suggesting

the same

_pairing

mechanism.

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_V.J.,

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_{practically high}

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_plaquettes.

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_(for

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_drastically

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