The topologically-disordered square lattice

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The topologically-disordered square lattice

Jean-Marc Greneche, J.M.D. Coey

To cite this version:

The

_{topologically-disordered}

_square

lattice

J. M. Greneche

_{(1, *)}

and J. M. D.

_Coey

₍₂₎

(1) Kernforschungszentrum

Karlsruhe, I.N.F.P., Postfach 3640, D-7501 Karlsruhe, F.R.G.

(2)

Department

of Pure and

_{Applied Physics, Trinity College,}

Dublin 2, Ireland

(Reçu

le 19 mai 1989,

accepté

sous

_{forme définitive}

le 27

_{septembre 1989)}

Résumé. 2014

Nous

_développons

la méthode des défauts

_topologiques

à un réseau carré bidimensionnel: la

_{réorganisation topologique}

_{obtenue par} cassure de liaisons

_puis

création de nouvelles liaisons entre cations, conduit à un

_système

désordonné type

AB2 après

insertion des anions. Deux modèles ont été obtenus à

_partir

de défauts

_topologiques

élémentaires de nature

différente en tenant _compte des contraintes

_{géométriques imposées}

au cours de leur

empilement.

Les valeurs de la densité et de

_{l’énergie élastique}

calculées

_après

relaxation sont

_plus

_{faibles pour}

le modèle

_présentant

un nombre

plus important

de

_{cycles impairs :}

par ailleurs, des

_pics

de traces

cristallines sont observables sur la fonction _{d’interférence, alors que le second modèle,}

_possédant

une

_majorité

de

_{cycles pairs,}

est

_typiquement

_amorphe.

Nous

_présentons

la

_{configuration}

magnétique

de l’état fondamental calculée en _{supposant que} toutes les interactions sont

antiferromagnétiques

et ne

_{dépendent uniquement}

de

_l’angle

de

_{superéchange}

ABA. Abstract. 2014 The

topological

defect method,

consisting

of

_modifying

the

_topology

of a

_crystalline

supercell by breaking

and

_reforming

bonds between cations, is

_applied

to a two-dimensional square lattice to simulate four-coordinated

_AB2

_systems after introduction of anions. Two distinct

elementary

defects are used to _generate two

_{highly-disordered}

networks with

_geometric

constraints

_imposed

on the

topological

defects. After

relaxation,

the lower

density

and lower elastic energy are found for the network with a

_majority

of odd-membered

_rings

which retains

vestiges

of

crystalline peaks

in the

_scattering

function

_{S(q) ;}

the other network, with a

_majority

of even-membered

_rings

is

_{entirely non-crystalline. Finally,}

the

_{A-spin configuration}

in the

_magnetic

ground

state, is calculated

assuming antiferromagnetic

interactions that

depend only

on the ABA

superexchange angle.

Classification

Physics

Abstracts

61.40D - 75.25 - 75.50K

Introduction.

The

_physics

of disordered

_systems

has been

_{receiving growing}

attention

_during

the

_past

two

decades. Considerable effort has been devoted to

_developing

structural

models,

characterizing

their

_topological

disorder and

_evaluating

their

_{physical properties.}

Some

concepts

have been

adapted

from

_{crystallography}

to describe the structure of

amorphous

materials

(amorphog-raphy

[1]).

The idea of frustration

_[2],

used to

_explain

the

_magnetic

behaviour of disordered

(*)

Alexander von Humboldt fellow from Laboratoire de

_Physique

des Matériaux, Université du

Maine, 72017 Le Mans Cedex, France

_{(Permanent address).}

systems

with

antiferromagnetic

interactions,

was

_already

_present

in certain

_crystalline

lattices. Such

concepts

were

_usually

introduced at first for two-dimensional

_(2-D)

_systems

and then extended to three-dimensions

_(3-D).

Relatively

little attention has been

_paid

to 2-D disordered

_systems.

Besides

_offering

an excellent

_approach

to an

understanding

of the

properties

of 3-D disordered

_systems,

_they

provide

a means of

_interpreting

the mechanisms of

_{grain growth}

in

_{polycrystalline}

or sintered

materials

_(see

Weaire et al.

_[3],

_{physisorption}

and

_{chemisorption}

_(for

a

_review,

see

_[4]).

To our

_knowledge,

the

_simplified

2-D

_{Bernal liquid}

was the first 2-D disordered network

discussed in the literature

_{[5] :}

it was advanced to model the order-disorder transition and consisted of a

_{close-packed assembly}

of

_regular

or

_{quasi-regular polygons}

_(triangles

and

squares).

An extension of this

_model,

called « the

_triangular

_{square lattice »} was

_recently

discussed in

_{thermodynamic}

terms

_{[6] ;}

then a frustrated

_topology

was

_applied

to the

_problem

of

_{amorphous antiferromagnetism assuming Ising}

_[7]

and

_Heisenberg

_[8]

_magnetic

moments.

In an other

_approach,

a 2-D random

_network,

_adapted

to vitreous

_silica,

was handbuilt

_by

generating

arrays of

triangles equivalent

to extended Zachariasen schematics

_[9].

Topologi-cally

disordered

_hexagonal

networks were obtained

_{by systematically introducing topological}

defects into a 2-D lattice while

_{respecting boundary}

conditions

_{[10] :}

the method consists of

breaking

and

_reconnecting

bonds,

the coordination

_{being kept}

constant. The author concludes in this paper that the

ring

statistics converge to fixed values

_{independently}

of the network size.

In their

approach,

Weaire and Kermode focused their attention on the evolution of 2-D soap froths

[3].

Two processes were introduced to simulate the

_topological

rearrangement

of the cells contained in the network so as to

_reproduce

the time evolution of a _{soap film :} one consists of

_creating

a bond and the other of

_destroying

a small

_cell,

the coordination still

_being

kept equal

to 3. Their model

_reproduces

well the features observed in soap froth

itself,

as well as

_{grain growth}

or

_plastic

deformation in metals

[11].

These studies show the interest in

_{understanding}

the

_{amorphography}

of 2-D disordered

systems.

They

can also be considered as a first

_approach

to

_studying

real 3-D

_amorphous

materials.

Recently,

Wooten and Weaire

_[12]

have

_developed

the

_topological

defect method for

generating amorphous

tetrahedral structures for a-Si or a-Ge. The

procedure

involves successive introduction of defects into a

_{large crystalline supercell,}

each defect

_resulting

in a local

_{rearrangement}

of the

_topology,

the coordination

_remaining

unaltered.

Our aim is to

_develop

the method for octahedral structures. We have

_{previously analysed}

two handbuilt structural models

_[13]

and a

_computed

one

_{[14] representing}

the

_AB3

random network

_composed

of

_{comer-sharing AB6}

octahedra. The

_algorithm

for

_generating

the

computer

model is based on three

_physical

criteria :

_(i)

6-fold A and 2-fold B

coordination,

(ii)

steric encumbrance of the atoms A and B and

_(iii)

_{geometrically}

frustrated

_topology

with the

_presence

of both odd- and even-membered A

_rings.

The latter is indicated

_by

the

_magnetic

behaviour of

_amorphous

_FeF3,

where the

_{antiferromagnetic}

Fe-F-Fe

_{superexchange}

interac-tions are frustrated

_[15].

In this _paper, we

_apply

the

_topological

defect method to a 2-D square

lattice,

_thereby

generating

a

_planar

_{non-crystalline}

network. The structure,

topology

and

_properties

are then

discussed. Extension to 3-D will be

_reported

elsewhere

_[16].

The

_topological

defect method.

We

_begin

with a

_crystalline

_square

_{superlattice containing}

the A

_positions

which was taken to

neighbour

A cations.

_Only

four-membered

_rings

are

_present

in such a network. The

topological

defect method consists in

_modifying

the

_topology

of the

_starting

_{configuration by}

breaking

some cation-cation bonds and then

_reforming

new cation-cation bonds which

respect

the coordination. The local

_{rearrangement}

involved after introduction of two

_types

of defect are illustrated in

_figure

1. The first

_topological

defect

_{(Fig. la)}

consists in

_rotating

the intemal square

ring

defined

_by

cations

A, B,

C and D as follows : the

_BE,

_CF,

DG and AH bonds are broken and then the

_{AE, BF,}

CG and DH bonds are connected. The second one

(Fig. lb)

consists in

_breaking

BC and AD bonds and then

_reconnecting

AC and BD bonds. Defect

₍₁₎

may be considered as a dislocation

_quadrupole

and defect

₍₂₎

as a dislocation

_dipole.

The final

arrangement

in both cases exhibits three and five-membered

_{rings :}

a six-membered

_ring

_can result from the

_{judicious juxtaposition}

of two

_topological

defects.

Furthermore,.

the

topological

environment created

_by

one

_type

of defect can be obtained

_by

_juxtaposing

several of the other ones, and vice versa.

Fig.

1. - Local _{rearrangement} induced

_by

the two _types of

_topological

defects discussed in the text.

The locations of the

_topological

defects are chosen

_{randomly by}

the

_computer

_program. Some of the

_topological

_{defects may be moved} to

_optimize

the

_homogeneity

of the disorder. No

_dangling

bonds or double bonds between two cation nearest

_neighbours

are allowed.

Some rules

_relying

on

_geometrical

considerations to exclude unrealistic situations were established as

_explained

in next section.

When a

_topological

defect is

_generated

near the outer

_edge

_{of the square}

_{superlattice,}

the

neighbourhood

is obtained

_by

consideration of

_{periodic boundary}

conditions.

The final A-site

_topology

is achieved when no more defects that

_satisfy

the rules can be introduced. At this

_stage,

the anions are inserted

_{by placing}

the B-atoms

mid-way

between

two cation nearest

_neighbours.

_Then,

a

_procedure

of structural relaxation

_using

an harmonic

potential

(analogous

to that used in

_[14])

is extended to the whole

_model,

in

_conjonction

with

periodic boundary

conditions. The

_topology

is

_{unchanged by}

such a relaxation

_procedure.

Finally,

different

_physical

_parameters

are deduced from the relaxed structural network :

density,

radial distribution

functions,

interference

functions,

distortion

_parameters,

_ring

statistics,

angular

distribution

functions,

magnetization

and

_spin

correlation function

(as-suming

an interaction scheme and

_allocating

a

_magnetic

moments to A ions

_only).

The

_{geometric problem.}

structure is not

_completely

_{random ;}

a certain

_regularity

is

_{imposed by}

the local atomic

arrangement :

consequently,

some

_geometric

constraints due to this

_short-range

order have to

be taken into account when

_{superimposing}

the

_topological

defects. The

_problem,

in other

words,

is how to fill a

_plane

with somewhat distorted

_polygons,

the 4 : 2 coordination

_being

everywhere

maintained.

In the

_present

_study,

the

_{following quantitative}

limits were

_established,

_{guided by}

the structural data observed on the three

_{polymorphic crystalline phases}

of

_FeF3

_[17]

and those on the

_amorphous

forms

_[18].

First,

the

_type

of

_polygon

is limited to

_triangle,

_square,

_pentagon

or

_hexagon.

In

addition,

the

_polygons

have to be convex and

_weakly

distorted : a concave or a

highly-distorted polygon

would be

_equivalent

to unrealistic

_large-angle

distortions in the local

structure. Then we

_analysed

the different

_{possibilities}

of

_filling

a

_plane

with

_polygons

so as to

establish the rules to be

_applied

in order to avoid unrealistic stituations when

_{superimposing}

the

_topological

defects : 35 cases are found and

_only

the more favorable ones are listed in table I. The lower and upper limits on the sum of the four ideal

_angles

AAA around one A

atom were estimated as follows : first

_{by taking}

the ratio of A and B ionic

_radii,

one obtain a lower limit for the

_angle

ABA

_(135°)

is

_appropriate

for

Fe 31

and

_{F- ), and}

then

_by

_considering

the lower _{and upper values of the AB distance}

_{experimentally}

observed

_{(1.95 (3)}

_Â)

found for

a-FeF3

[18].

From these

_data,

two extreme values were defined

_equal

to 330° and 385°

between which the distortions can be consistent with the structural observed data on the

amorphous

ferric fluorides. The sum of the four real AAA

_angles

is,

of course, 360°.

Table I. Geometric situations encountered in a two-dimensional case.

The two 2-D models.

The

_{starting supercell}

was a _{square lattice}

_containing

50 x 50 A

sites,

which should be

large

enough

to be

_{representative}

of the

_amorphous

state. The distances between two first-nearest

neighbours

was taken as a. Then the method was

_applied

to

_generate

two

_models,

1 and

_2,

using only

the defects a and b

_respectively

_presented

in

_figure

1. The final

_topology

of the two

models is illustrated in

_figures

2 and 3 : 214

_a-type

and 269

_b-type

defects were

_successfully

introduced. One can first remark that no « unrealistic » situation is observable in either

model,

in

_agreement

with the limits

_proposed

in table I.

In

_figures

4 and

5,

the A-site

_topology

of the two models is

_presented

after

_carrying

out the relaxation

_procedure

on whole

_system

₍₂

500 A cations and 5 000 B

_anions)

_taking

into

account the

_{periodic boundary}

conditions. The distance AB was

_initially

taken

_equal

to

a/2 (dAB

=

1.95 Â).

Structural

_analysis

of the 2 models.

The

_ring

statistics

_presented

in table II show that the two models are

_{topologically}

rather different. No six-membered

_rings

were allowed in the second network. The first model contains 64 % of odd-membered

_rings

while the second one contains 44 %. In

addition,

the elastic energy of the second model is found to be four times

_higher

than that of the first one,

after

_applying

the structural relaxation

_procedure

in both cases. At this

_stage,

we conclude that the model with six-membered

_rings

is more realistic.

Table II.

_{- Ring}

statistics

_for

the two-dimensional models

_compared

to a

_no-defect

_squared

lattice.

The structural characteristics of the two relaxed models are

_reported

in table III. Here it would appear that the two models are _very

_similar,

_although

the

_AA,

AB and BB

_peaks

are broader for the second model. Note also that the

_density

calculated for the model 1 is lower than that of the

_{crystalline phase,}

while the

_density

of the model 2 is

_{higher ;}

such

_changes

are

directly

related to the nature of the

_topological

defects.

In other

_respects,

_{by looking}

at the radial distribution functions

_{(RDF) (see}

for

_example

the

AA

_partial

RDF and the total RDF

presented

in

_Figs.

6 and 7

_{respectively),}

one can conclude that the two models are

_structurally

_{very different : the first} structure model looks like that of a distorted

_crystalline

lattice while the second one exhibits the

_{typical signs}

of an

_amorphous

structure.

Fig.

2. -

Representative

view of the square model 1 before

_applying

the structural relaxation

procedure.

Fig.

3. -

Fig.

4. -

Representative

view of the square model 1 after

applying

the structural relaxation

_procedure.

Fig.

5. -

Table III. - Structural characteristics

_of

the two-dimensional models

_compared

to the

no-defect

square lattice

(a

=

3.90 Â ).

(*)

« and {3 are the distortion _parameters of the

_AB2

units

are the

_equilibrium

distance between A and B and two first-nearest B atoms

_{respectively.}

Fig. 6.

Fig. 7.

Fig. 8.

Fig.

6. - AA radial distribution functions of a _{no-defect square lattice}

_(a),

of model 1

_(b)

and of model

2 (c).

Fig.

7. - Total radial distribution functions of a no-defect square lattice

(a),

of model 1

(b)

and of model

2 (c).

Fig.

8. - Interference functions of _a

no-defect square lattice

(a),

of model 1

_(b)

and of model 2

_(c).

Magnetic

behaviour.

The calculation of the

_magnetic

structure consists of

_determining

the

_magnetic

moment

directions after

_minimizing

_{the energy for} classical

_spins.

The initial state with directions of the A site

_magnetic

moments oriented in the

_plane

was

_randomly

_{generated by}

the

computer

for the two models. An XY

_spin

interaction was considered between the 4 nearest

_neighbours

Fig.

9. -

Fig.

10. -

where

_Jij

_represents the

_{integral exchange coupling}

_(negative

in the case of

_{antiferromagnetic}

interactions)

which is related to the

_{superexchange angle}

ABA but assumed

_independent

of the distance between first-nearest

_neighbours

AA. The

_magnetic

structure is then relaxed

_by

minimizing

the

_magnetic

_energy

_according

a

_gradient

method which was

_{previously developed}

for

crystalline

fluorides

_{[19] :}

each

_spin

is examined

_individually

and rotated

_according

to the

torque

due to its

_magnetic

_neighbours

at each

iteration ;

in addition the

_{periodic boundary}

conditions were also taken into account.

The final

_{magnetic configurations}

are

_presented

in

_figures

9 and 10

_respectively

and the

magnetic

characteristics are

_reported

in table IV. The frustration

_degree

can be estimated from the

_magnetic

_{energy per}

_spin

_{which appears} to _{be very similar in both} cases

_although

their

_topologies

are rather different

_(see

the

_percentage

of odd-membered

_rings

in Tab.

_IV).

In

_addition,

the

_spin

correlation functions were estimated for both models and

_compared

to

that of a no-defect

_{antiferromagnetic}

_square

_lattice,

as shown in

_figure

11.

Table IV.

_{- Magnetic}

characteristics

_{o f}

the two-dimensional models

_compared

to the

_no-defect

squared

lattice.

Fig.

11. -

The

_shape

of the

spin

correlation function calculated for the first model confirms that it looks like a distorted

_crystalline

lattice. In both cases, the difference between the A-A radial distribution function

_(presented

in

_Fig.

₆₎

and the

_spin

correlation function shows that the

magnetic

moments are

isotropically

distributed in the

_plane

and indicates an

_{antiferromagnetic}

coupling

between first-nearest

_magnetic

moments.

Conclusions.

The

_application

of the

_topological

defect method to _{the 2-D square lattice shows that the}

superimposition

of

_topological

_{defects may lead}

_packing

difficulties with

_physically

reasonable atomic sizes. The two 2-D models obtained

_{by generating}

two

_types

of

_topological

defect

subject

to

_{plausible geometric}

constraints are

_structurally

and

_{topologically}

rather

different ;

one retains traces of

_crystalline

order,

while the other does not but

_they

exhibit similar

magnetic

behaviour : the

_magnetic

moments are

_{isotropically}

distributed in the

_plane.

Acknowledgments.

The authors thank Prof. D. Weaire for

_stimulating

discussions

_during

the

_present

_study.

One of us

(J.

M.

_G.)

wishes to thank Dr G.

_Czjzek

and Prof. H. Rietschel for their

_hospitality

in the

_{Kernforschungszentrum}

of Karlsruhe and for the

_facility

of

_computing.

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