Microscopic analysis of structure stabilities in the ionic compounds

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Microscopic analysis of structure stabilities in the ionic

compounds

B. Piveteau, Claudine Noguera

To cite this version:

2005

Short Communication

Microscopic

analysis

of

structure

stabilities

in the ionic

_compounds

B. Piveteau and C.

_Noguera

Laboratoire de

_Physique

des

_Solides,

Université de

Paris-Sud,

91405

Orsay,

France

(Received

on

_{July 9, 1990, accepted}

on

_July

_13,1990)

Résumé. 2014 En

associant,

pour la

_première

_fois,

un calcul

_d’énergie

de cohésion et une

_approche

à la Landau des transitions de

_phase,

nous

_présentons

une

_analyse

de la stabilité relative de deux

phases

structurales d’un

_composé

_{ionique à 0}

K. Cette association de deux

méthodes,

en elles mêmes

classiques, permet

de découvrir

_{progressivement}

les interactions

_{microscopiques}

associées à

_chaque

distortion,

et fournit les valeurs

_numériques

des coefficients de Landau dans le

_{développement}

d’éner-gie correspondant.

Cette méthode illustrée ici sur le

_composé

La2CuO4

ouvre un

_{large champ}

d’in-vestigation

dans le domaine des stabilités structurales des

_composés

_ioniques.

Abstract. 2014

Associating

for the first time a total _energy calculation and a

_phase

transition

_approach,

we

_present

an

_analysis

of the relative

_stability

of two structural

_phases

in an ionic

_compound

at zero

temperature.

This new combination of two classical methods allows a

_{progressive discovery}

of the

microscopic

relevant interactions associated with each

distortion,

and

_yields

the value of the Landau coefficients in the

_development

of the _energy. Illustrated on the

_compound

La2CuO4,

this method has a wide

_{applicability}

in the field of the structural

_stability

of ionic

_systems.

LE

JOURNAL

DE

PHYSIQUE

1

_Phys.

France 51

_{( 1990)}

2005-2010 15 SEPTEMBRE

_1990,

1

Classification

Physics

Abstracts

61.SOL - 61.60 - 64.70K - 74.70V

Over the

_past,

there has been a constant interest in

_{understanding why}

a

_given

ionic

_compound

cristallizes in one structure rather than in another one

_{[1] .}

But

_{only recently}

did one

_apply

elabo-rate numerical methods to this

_problem.

livo families of

_approaches

now exist : the first one relies

on electronic structure calculations such as the ab-initio

_{pseudo-potential}

method

_[2]

or the self-consistent

_{tight binding}

one

_{[3] ;}

the second one is based

_upon

the use of ionic

_{pair-potentials [4],}

at various levels of

_{sophistication,}

and was

_proved

to

_reproduce

well static and

_{dynamic properties}

of

_perfect

and defected structures

_[4,5].

Yet,

most of the

time,

the

_heavy

numerical methods hinder the

_{physical insight.}

The calculations

2006

currently

tell which structure is more stable at zero

_{temperature, but,}

not

_why

the other

_possible

structures lie

_higher

in

_energy.

As a

_consequence,

no theoretical

_prediction

can be made on how

a structure evolves

_upon

_replacement

of one kind of atom

_by

another one.

We

_propose

in this letter to associate a total

_energy

calculation with a

_phase

transition

_approach,

in order to

_analyze

the relative

_stability

of two structural

_phases

1 and II in an ionic

_compound

at zero

_temperature.

These two methods

_by

themselves are not _{new, but} we demonstrate that their

combination allows to establish a

_hierarchy

between the various distortions of the

_structure,

in close

_relationship

with the order

_parameters

of the

_{(possibly fictituous)}

_phase

transition between 1 and II

_(assumed

here to be a second order

_one).

It is then

_{possible, by performing partial}

_energy

minimizations,

to make detailed _energy counts associated with each

_primary

order

_distortion,

and thus find the

_microscopic

interactions

_responsible

for the

_symmetry

_breaking,

at the

_origin

of the

higher stability

of one of the

_phases.

We have chosen to illustrate these

_general

_concepts

on two structural

_phases

met in the

com-pound

LaZCu04. La2Cu04,

which became famous because of the occurrence

_{of high}

_Tc

supercon-ductivity

upon

doping,

presents

a

_phase

transition between a

_high

_temperature

_{tetragonal phase}

(HTT ;

space group I4/mmm

called

phase

1 in the

_following)

to an orthorhombic one

_(ORTHO

;

space group

Bmab,

subgroup

of

I4/mmm ;

_phase

_{Il)[7] .}

The transition occurs at 500 K at

am-bient

_pressure

and

_disappears

above a critical

_pressure

of about 60 kbars

_{[8] .}

In the

_pressure

range

in which it was

_studied,

the transition was found to be second order. Earlier studies of the

mechanism of the HTT to ORTHO transition discussed the role of the electronic structure close

to the Fermi level

_{[9, 10].}

Quantitative

total

_energy

calculations were also

_performed

[4, 5,11] .

But none of these

_provided

a detailed

_explanation

of the mechanism of the

_phase

transition. A

thorough

report

of this

_study

will be

_presented

_elsewhere,

_together

with a

_development

of this method for first order transitions

_[12].

’

We have chosen the

_simplest

version of the ionic

_pair

_potential

method

_[5]

as a

_{computationnal}

support

to our

_analysis.

It is worth

_stressing

that the choice of another numerical code would

only modify

some of the details of the

_{specific analysis}

of

La2Cu04.

But it would not alter the

general

conclusion of this

letter, i.e.,

the interest of

_associating

a total

_energy

calculation and

a

_phase

transition

_approach.

The ionic

_{pair potential}

method

_expresses

the total

_energy

E

_per

formula unit as a sum of

_pair

interactions between a central unit and all the other units. Three

contributions are considered : the direct Coulomb

_interaction,

a hard core

_repulsion

of the Born

Mayer

type

and a small Van der waals term. For two ions labelled i

_{and j,}

at a distance

_Rij,

with

charges

equal

respectively

to

_Qi

and

_{Q3 ,}

the interaction

_energy

reads :

The constants

_Bij,

_pij and

_Cij

are derived

_[5]

from atomic values

_BH,

_pis,

Cü

which were shown

to

_{reproduce correctly}

the structures of

_La203

and CuO. Th

_perform

the

summations,

we

_group

the atoms in neutral entities

La2CU06,

as

_displayed

in

_figure

1.

We discuss elsewhere

_[12]

the details of the

_computation.

Let us

_just

_say

that the achieved

convergence

of the summations

_gives

an

_accuracy

of 10 meV on the total

_{energy per}

formula

unit

_{(i.e. 0.006% )}

and an

_accuracy

better than 1 meV on

_energy

différences. We

_improved

the

agreement

with

_experimental

data

_{[13] ,}

_{by adding}

to the ionic contributions

_(Madelung,

Born

Mayer,

Van der

_Waals)

a term

_Ecov=

_-Aexp(-qd2)

_mimicking

the delocalization of electrons in

the

_CU02

_{planes (withA=25}

eV and

_q=1.5

_A-’).

We also studied the

_stability

of the ORTHO

phase

under

_pressure,

_by

_minimizing

the

_enthalpy

E + PV of the

_system.

We found that ORTHO is

_destroyed

above a critical

_pressure

of about 57 kbar in

_agreement

with the

_{extrapolation}

of X

Ray

data

_{[8] ;}

the orders of

_magnitude

for the

_{compressibility}

coefficients

_[14]

are correct and

Fig.

1. -

Representation

of the

La2Cu04

unit in the orthorhombic

_phase

and definition of the

_eight

struc-tural

_parameters

_{dl, d’2J Z, 0’, 60’,}

u and

{3.

The units are titlted with _respect to the

_[110]

direction of the

tetragonal

axis.

_They

include a

_Copper

atom surrounded

_by

four

_{in-plane-oxygen}

atoms

_(called

02 in the

text),

two

_out-of-plane

_oxygen atoms

_(called

₀₁₎

and two lanthanum _atoms, with

_chemically

_strong

La=O1 bonds

_(the

associated interatomic distance is less than the sum of Van der Waals

_radii,

while the

_eight

other La-0 distances are much

_larger

_[13]

_).

structural behaviour of

La2

CU04

is

_correctly

_reproduced

_by

our numerical

_approach,

and that we

may

safely

use it to understand the

_microscopic

_origin

of the ORTHO

_stability.

We first

_study

the

_origin

of the cohesion in the HTT

_phase.

The total _{energy is}

_conveniently

analyzed

as a sum of a

_Madelung

contribution

_EM

and a Born

_{Mayer plus}

Van der Waals one

EBM.

Each of them can be further

_decomposed

into terms

_EnM

_{(respectivcly EnBM) equal}

to

the Coulomb

_(resp.

Born

_Mayer

_plus

Van der

_Waals)

_{interaction energy} of the central unit with units located in

_planes

_{f(n -1)z}

_{(n >}

_1).

The

_complete

calculation

_(lhb.

I,

column

₁₎

reveals that the cohesion in the HTT

_phase

is

_mainly

due to attractive

_interplane

Coulomb interactions

(E2M

0),

whereas the

_intraplane

Coulomb forces are

_{highly repulsive (EIM}

>

_0).

The

_sign

of these interactions can be understood

_by

_making

a

_development

in

_multipolar

mo-ments : the

La2Cu04

units are neutral and have a zero

_dipolar

_{momentum ;} their

_quadrupolar

mo-. mentum is

_highly

_anisotropic

and

_{elongated along}

the

_[001]

direction of the

_{tetragonal phase.}

In

2008

lhble 1. - Calculated contributions to the total

energy :

EnM

and

_EnBM

_{refer respectively}

to the

Madelung

(M)

and to the Born

_{Mayer plus}

Van der Waals

_(BM)

interaction

_{energies of}

a central

La2CU04

unit with all units located in the

_planes

at a distance

_{:1:( n -1 )z.}

Column 1 : HTT cohesion energy

(in

eV)

at

_{equilibrium ,.}

the set

_of

_{structural parameters}

_{which minimizes the energy}

_{is found}

equal

to

_{{C} = {d2}

= 1.882

Â,

_dr

= 2.4i2

Â,

d = 2.379

Â,

z = fi.fi0i

A} ;

Column 2 : energy

difference

En ( 0:) -

El

in

_meV,

between tire

_{HTT phase}

and an

_{ORTHO phase}

characterized

_by

a

rigid

tilt a=3.6° and the

_{structural parameters}

_{{C} ;}

_{column 3 : energy}

_difference

_{Eu( 0’,}

u,

ba) -

El

in meV between the

_{HTT phase}

and an

_{ORTHO phase}

characterized

_by

the

_{three primary}

order

parameters

a4 =

3.6°,

u = 0.284

Â

and ba

= -0.5°,

and the set

_{of parameters}

_{C}.

These values minimize

_EJI(A,

_u,

_ba)

when

_{C}

is

_kept

constant. Column 4 : _energy

_difference

in meV between

the actual ORTHO

_phase

and the HTT one when

_{all primary}

and

_{secondary order parameters}

are considered. The

_{structural parameters}

which minimize

_En

_{are found equal}

to

_d2

_=1.906,

_dl

= 2.415

A, d = 2.375 A, z = 6.554 A, 0: = 6.23°,60: = -0.18°, fi = 89.0°, u = 0.557 A.

répulsion

is however

_{compensated by}

a

_large

attraction of the central unit with the

_quadrupoles

of the

_{neighbouring planes}

_placed

in a

_body

centered

_{arrangement (E2M large}

and

_{negative ;}

EIM

+

E2M

0).

This

_{representation}

of

La2CU04

units as

_quadrupoles

thus

_gives

a

_physical

support

to the results of the

_energy

calculation.

The orthorhombic distortion

_[13]

involves a

_rigid

:f:a tilt of the entities relative to the

_[110]

axis of the HTT

_structure,

_{accompanied by}

a small additionnal ±ba tilt of the Cu-01

bond,

and

_by

a

rotation of the 01-La bond

_{(displacement}

u of the La

_{atoms ;}

the

Cu,

01 and La atoms are no

longer

aligned) (Fig. 1).

Minor modifications of the HTT

_{parameters 6d, 6d1, bd2,}

6z and

_{a square}

to

_rectangle

shear distortion of the

_CU02

basal

_planes

of the units

_{(,0 # 7r/2)}

are also observed.

The ORTHO to HTT

_phase

transition is thus a

_complex

_one, with

_primary

as well as

_secondary

order

_{parameters :}

we find that

_(a,

6a,

_u),

which alternate in

_space,

are the three

_primary

order

parameters,

while the other uniform distortions are the

_secondary

order ones, in

_agreement

with

simple

symmetry arguments.

The distortion a, associated with a

_rigid

tilt

_{(OE #}

_0,

6a =

0,

u =

0,

_unchanged

HTT

_parameters);

induces an

_{energyvariation DE(a)}

which has the form of a Landau

_{development : DE(a)}

=

Aa2+

Ba4.

We find that both A and B are

_positive,

which shows that the tilt alone is not stable. The

fraction of the

_intraplane

Coulomb

_{repulsions (EIM)}

is

relaxed ;

_b)

since

_coplanar

units share

oxygen

atoms

_by

their _{corner, the}

_alternating

tilt induces a contraction of the Cu-Cu distances in the direction of the

_CU02

_plane

normal to the tilt

axis ;

this increases

_{significantly}

the

_interplane

Coulomb attraction

_(E2M).

For medium or

_long

_range

interactions both

_processes

_a)

and

_b)

lower the

_energy

of the

_system.

_Yet,

at

_very

short

_distances,

two other effects act

_against

this

_gain

of energy :

c)

two La atoms of

_{coplanar neighbouring}

units come

_{significantly}

closer to each

_other,

and induce

_{large intraplane}

Coulomb

_{repulsions (bE,m}

> 0 in Thb. 1 column

_{(2)) ; d)}

there is an

important

decrease of

_{interplane La-oxygen (01}

and

₀₂₎

distances which induces

_{predominantly}

an increase of Born

_{Mayer repulsion (6E2BM}

>

0,

in ’Iab. 1 column

_(2)).

Thé coefficient A in the Landau

_development

is thus

_equal

to the sum of two _terms, A 1 small and

negative

due to

a)

and

_b),

andA2

large

and

_positive

due to

_c)

and

_d).

We conclude that the

_stability

of the ORTHO

phase

is not driven

_by

the

_rigid

tilt alone

_(A

>

_0),

because this tilt induces too

_large

short _range Coulomb and Born

_{Mayer repulsions.}

We now

_analyse

the role of the

_primary

order

_parameter

u associated with the rotation of the

La-01 bond

_[15].

_{The energy variation}

due to u and a takes the form :

_{AE(a, u)}

=

Aa2+Cau+Du2

to lowest order. HereA and D are

_{positive, showing}

that each distortion alone is

_unstable,

but the

structure with

_{both Q’ =F}

0 and

_{u =F}

0 is nevertheless more stable than HTT because

C2

> 4AD. In

order to understand the

_{microscopic origin}

of the

_large

linear

_coupling

between u and a, we use

the numerical

_program

to calculate

_{the quantity}

_{AE(a, u) -AE(a,}

tt =

0) -AE(OE

=

0,

u) N

Cau

and

_decompose

it into

_elementary

contributions. We find that it is

_{unambiguously}

due to Coulomb and Born

_Mayer

interactions between

_coplanar

La-La

_pairs

and non

_coplanar

La-0

_neighbouring

atoms. These same short _{range interactions} were

_responsible

for the unfavourable contribution

A2a 2 noted

above. It is thus not

_surprising

to find

_that,

since the rotation of the La-0 bound

_puts

the La atoms close to their HTT

_positions,

the _energy increase due to effects

_c)

and

_d)

is erased and the

_energy

_lowering

due to effects

_a)

and

_b)

becomes efficient. This

_may

be seen in table

_I,

in

comparing

columns

₍₂₎

and

_(3).

This

_analysis

demonstrates that the ORTHO

_phase

is stabilized with

_respect

to the HTT

_phase,

thanks to a

_disalignment

of the

_quadrupole

axis

_{accompanied by}

a

contraction of the lattice. The

_alternating

tilt of the

_La2Cu04

entities is

_responsible

for this effect but a rotation of the La-01 bonds is also _necessary to

_impeede

too

_strong

short _range Coulomb and Born

_{Mayer repulsions.}

This

_study

of the

_primary

order

_parameters

answers the

_{question :}

why

is ORTHO more stable than HTT?

Th achieve a

_{complete description}

of the ORTHO

_phase,

we have considered the

_secondary

order

_parameters.

These latter are of no

_importance

as concems the mechanism of stabilization but

_they

_yield

increased values of the stabilization

_energy

and of the

_primary

order

_parameters.

The variations

_ôd2

and az of the Cu-02 and

_interplane

distances are of

_{special importance.}

A

Landau

_development

of the

_{type :}

_{AF(A, c)}

=

Aà2

₊

BA4

₊

Cê2

₊

FE02

_(here

_e

represents

lad2, az}

and à

_{{a, u})}

accounts for the _{energy difference} between ORTHO and HTT. We found

that the

_coupling

term

FA2,-

between first order and

_secondary

order distortions comes

_entirely

from Born

_Mayer

interactions between closest non

_coplanar

La and 01 atoms. These interactions

indeed limited the value of the tilt

_angle

at constant values of the HTT

_parameters.

An increase

in

d 2

relaxes a

_part

of the

_repulsions

and

_consequently

authorizes an additionnal increase of a

and an additionnal stabilization _energy, as seen in table 1 column 4.

We wish to conclude this

letter,

by

stressing

the wide _range of

_{applicability}

of our

_approach

in the

_study

of the relative

_stability

of structural

_phases,

in

_compounds

in which the cohesion is

mainly

due to ionic forces. It

_appears

that the more

_complex

the difference in the structures of 1 and

_II,

the more

_precious

the method of

_analysis

we

_propose.

This

_happens

because it allows

a classification of the effects and a discrimination between the

_{distortion(s)}

which are

_primarily

responsible

for the

_stability

and the ones which are a mere

_consequence

of the others. The method

2010

the

_physical

_parameters

which influence the _{energy balance associated with} the

_primary

order

parameters.

Finally,

our method reveals the

_microscopic

_origin

of

_phase

_transitions,

since it is

able to

_assign

numerical values to the Landau coefhcients and to reveal the

_underlying

_{physics ;}

but at the

_present

time it is restricted to zero

_temperature.

Acknowledgements.

We are

_greatly

indebted to J.R

_Pouget

for _many discussions all

_along

this work. We also

acknowl-edge

fruitful comments from J.

_Friedel,

S.

_Barisic,

M.K.

_Whangbo

and H. Moudden. We thank P. Garoche and R Batail for a critical

_reading

of the

_manuscript.

References

[1]

PHILLIPS

_J.C.,

Bonds and Bands in semiconductors

_(Academic,

New

_York)

1973.

[2]

CHELIKOWSKY J.R. and BURDETT

_J.K.,

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Rev. Lett. 56

₍₁₉₈₆₎

961.

[3]

MAJEWSKY J.A. and VOGL

_{P., Phys.}

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[4]

ALLAN N.L. and MACKRODT

W.C.,

Philos.

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M.H.,

BENO

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GEISER U. and WILLIAMS

_J.M.,

J. Am. Chem. Soc. 109

₍₁₉₈₇₎

7917.

[6]

The

_secondary

order

_parameters

_correspond

to the distortions of the

_{low-symmetry phase,}

which do

not break as _many

_symmetry

elements of the

_{high-symmetry phase}

as the

_primary

order

_parameters.

At a second order

_phase

_{transition, they}

vanish as

₍

_{Tc -}

T)03B4with

an

_{exponent 03B4 larger}

than the

exponent

characterizing

the

_primary

order

_parameter.

_{e.g. :} COWLEY

_{R.A., Adv. Phys.}

29

₍₁₉₈₀₎

1.

[7]

MORET

_R.,

POUGET J.P. and COLLIN

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₍₁₉₈₇₎

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[8]

KIM H.J. and MORET

R.,

Physica

C156

₍₁₉₈₈₎

363.

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POUGET

_J.P.,

NOGUERA C. and MORET

_{R., J.}

_Phys.

France 49

₍₁₉₈₈₎

375 and references herein.

[10]

BURDETI J.K. and KULKARNI

_G.V,

_Phys.

Rev. B40

₍₁₉₈₉₎

8908.

[11]

COHEN

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PICKETT W.E. and KRAKAUER

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