Dynamics of KH2PO4 type ferroelectric phase transitions

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Dynamics of KH2PO4 type ferroelectric phase transitions

C. Tsallis

To cite this version:

C. Tsallis. Dynamics of KH2PO4 type ferroelectric phase transitions. Journal de Physique, 1972, 33

(11-12), pp.1121-1127. �10.1051/jphys:019720033011-120112100�. �jpa-00207339�

DYNAMICS OF KH2PO4 TYPE FERROELECTRIC PHASE TRANSITIONS

C. TSALLIS

Service de

Physique

^{du Solide et} de Résonance

Magnétique

Centre d’Etudes Nucléaires de

Saclay,

^{BP n°}

2, 91, Gif-sur-Yvette,

^France

(Reçu

^{le 14 avril}

1972,

révisé le 28

juillet 1972)

Résumé. 2014

L’énergie

^de

configuration

^est

ajoutée

à la théorie de

Kobayashi

pour les cristaux

ferroélectriques

^du type ^{KDP. La théorie est} ainsi rendue cohérente du

point

^{de vue}

thermodyna- mique, puisque

^le paramètre ^{d’ordre et la}

fréquence

^{du mode mou} associé s’annulent à la même température. ^Il

apparaît

^la

possibilité

d’existence d’un autre type ^de

ferroélectriques

^{où le}

phéno-

mène

displacif

declenche la transition et induit un

phénomène

d’ordre-désordre des ions

légers (protons, deutérons).

^Une

grande

sensibilité à la

pression,

de diverses

grandeurs physiques statiques

et

dynamiques, suggère

^{des travaux}

expérimentaux

intéressants sur des substances

ferroélectriques

mixtes.

Abstract. ²⁰¹⁴ We include the

configuration

energy in

Kobayashi’s theory

^for

KDP-type

^ferroelec-

tric

crystals.

^The

theory

is then shown to be coherent from the

thermodynamical point

^of

view,

^as

the order parameter ^{and the}

frequency

^{of the} associated soft mode vanish at the same temperature.

The results show that it is

possible

^to have another kind of mixed

ferroelectric,

^{where the}

displacive phenomenon triggers

the transition and induces an order-disorder

phenomenon

^{for the}

light

^ions

(protons, deuterons).

^Great

sensitivity

^to pressure of various static and

dynamic

^relevant

physical quantities

suggests

interesting experimental

^{work on} mixed ferroelectrics.

Classification

Physics ^Abstracts

17.29

1. Introduction. ^- A

great

^{amount of}

experimental

and theoretical work has been done on the ferroelec- tric transition of

KH2PO4 (KDP)

^and

isomorphous crystals (refer

^to

^{[1], [2]}

^for

^general characteristics)

and the convergence of the results leaves ^no doubt about the

microscopic

mechanism of the transition

(refer

^{to the excellent paper}

^{Kobayashi published}

in 1968

[3]).

^The

crystal instability

^is

triggered by

an order-disorder

phenomenon

^{in the}

proton assembly (each proton

^{moves in a} double-well

potential

^essen-

tially

^created

by

^the

heavy

^ions

K+, ^p+5

^and

0-2).

Simultaneously, displacements

^{of the}

heavy

^ions

are induced

by

^the

strong coupling

^between

protons

and the

[K-P04] system ([3]

^to

[13]).

^{We may thus}

consider this transition to be of ^a mixed

type,

^as order-disorder and

displacive

characteristics _appear

together.

To take account

theoretically

^of the collective motion of the

protons,

de Gennes introduced in 1963

[14]

^{a formalism}

(widely

used since

then)

^where

the ^«

right-left » positions

^{of a}

proton

in its double well are

represented by

^{the «}

up-down » projections

of ^a

-1-pseudo-spin.

^From this stand

point

^{there are}

many formal

analogies

^{between a} mixed-ferroelectric and ^a

magnetic displacive-ferroelectric crystals (parti- cularly

the collective vibration modes of the

protons

in KDP may be

regarded

^as

pseudo-magnons).

Recently

^{some work has been done}

[15]

^to

[17]

^on

the

dynamics

^{of the}

magnetic ferroelectrics,

^{and the}

conclusions _may

be,

^{to a certain} extent,

applied

^to

the mixed ferroelectrics. In these latter

crystals

^the

transition is

characterized,

^{as is shown in}

Kobayashi’s

paper,

by

^the

softening

^{of a}

particular

^{mode of}

vibration

(which

^{is shown to be a}

mixing

^of

pseudo-

magnon and

phonon)

^of the whole

system,

^{in a} similar _way to what

happens

ⁱⁿ

displacive

^ferroelec-

trics

([18]

^to

[21]).

From the

thermodynamical point

^{of view it is}

clear

that,

^{in a} second-order transition due to a

dynamical softening,

the order

parameter

^{and the}

frequency

of the mode

responsible

for the transition must vanish at the ^same

temperature (which

^is

preci- sely

^the definition of the transition

temperature),

as

they

^{are « two faces of a}

single

^{coin ».}

Kobayashi’s theory

^{fails on this}

point,

^{and we} show in this paper that the consideration of the

configuration

energy of the

heavy

ions raises _up this incoherence. The

approximations

made in this kind of

theory

^{and the}

possibility

of another

type

^{of mixed} ferroelectric

(where

^the

heavy

^ions

trigger

the transition and induce

an order-disorder

phenomenon

^{in the}

protons)

appear

clearly

^{in the} boson creation-annihilation formalism used in section 4. Great

sensitivity

^on

pressure of various static and

dynamic

^relevant

physical quantities suggests interesting experimental

work on

KDP-type

ferroelectrics.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019720033011-120112100

1122

2. Model. ^- Let us consider ^a

crystal

which may be

regarded

^{as the}

superposition

of several sublattices :

one of them made

by light

^ions

(like protons

^or

deuterons)

and the others

by heavy

^ions

(like p+5, K+, 0-2).

^{Let us} suppose there is one

light

ion per unit cell and that it is allowed to move in a double well

potential (created by

^{all the other ions and}

almost

temperature independent) along

the z-axis.

Strictly speaking

^the

light

^{ions do not build} up ^a sublattice unless all of them are on the same side of the double well. KDP has

actually eight protons

per unit

cell ;

nevertheless the mean

ordering

^is

along

the z-axis. Our

simplified

model looses

nothing

essential in

considering only

^one

light

^{ion per unit}

cell,

^{and the}

generalization

^{of this}

point

^demands

nothing

^but

patience.

^{Let us call m}

(where - f m f)

the reduced mean

position

^of

protons

^with

respect

to the center of the double

well, and il

^{a reduced}

characteristic

displacement

^{between the}

heavy

^ion

sublattices ;

^{both m}

and il

^{are assumed}

parallel

^to

the z-axis. The

crystal

^{is assumed to}

undergo

^a

phase

transition in such a way ^{as to}

have,

^{in a certain}

temperature domain,

^a

paraelectric (disordered) phase

^{where m} = il ⁼

0, and,

^{out of this}

temperature domain,

^a ferroelectric

(ordered) phase

where m i= 0

and q

^{= 0.}

The Hamiltonian _may then be written as follows

(our

notation is close to that of Blinc

[7]) :

where the

light

ion Hamiltonian _may be written

[12]

as follows

where T

( > 0)

^{is the}

tunnelling frequency

^of

protons through

the barrier of the double

well,

^and

Jjj, depends only on 1 Ri - Rj’ 1 (Jjj

⁼

0). By Ri

^we

denote the

position

^{of the}

jth

unit cell with

respect

to an

arbitrary origin, hence j

^{runs from 1 to}

N,

the number of unit cells of the

crystal.

^The

spin- operators obey

the usual commutation relations.

Only

the essential terms have been considered in this Hamiltonian. However more refined interactions may be

easily

included in the formalism. For

example,

the

influence,

^on the

tunnelling frequency,

^{due to}

the other

light

^{ions may} be introduced

by

^{a term}

an

asymmetrical

double well

potential

may be account- ed for

by

^a

term - A L Sj [22], [23] ;

^{and even inter-}

j

actions between ^more than two

spins

^{could be includ}

ed with no

great

^formal difficulties

[9], [12].

In the lattice Hamiltonian ^we shall

only

^consider

the harmonic contribution of the

optical

^vibration

strongly coupled

^{to the}

light-ion system (1),

^{so we}

have

where

Zq

^are the Fourier transformed coordinates associated to the vibrations

(parallel

^{to the}

z-axis) strongly coupled

^{to the}

light ions,

^and

Pq

^{are their}

canonical momenta

(we

^{consider the mass}

equal

^to

unity). They obey

^{of course} the usual relation

[Zq, PQ-,] - ibqq’, ^{(with h}

⁼

^1).

^{Let us add that}

Wq

⁼ Co_q, where ^q runs over all allowed modes in the first Brillouin zone.

Finally

^{we shall}

consider,

in the interaction Hamil-

tonian, only

the first order

(essentially dipolar)

^inter-

action between

light

^and

heavy

^ions

(particularly

^no

phonon-assisted tunnelling

^{of the}

light

^{ions is consi-}

dered) ;

^we may then write

where

Vq

^{is real and even in q, and}

Let us

point

^out that in the total Hamiltonian se

no

damping

^{effects are} considered. Let us now decom- pose

Zq

^{into its thermal average}

(noted >)

^and

fluctuating parts,

^{i. e.}

Zq = Qq

⁺ llq ^where Ilq ⁼

Zq

> and

Q9 - Zq - llq (hence Qq

> ⁼

0).

As we are

considering

^a ferroelectric

instability

^we

have

Let us now introduce the

phonon

creation and annihi- lation

operators

with the usual transformation

where

We _may now rewrite the total Hamiltonian as fol- lows :

1

(1) ^{It is clear} that, ^{in a} particular substance where an acous-

tical mode is strongly coupled ^{to the} light-ion system, additio- nal terms must be included in KL ^{as well as in} J~pL

(see [28]).

Except

^{for the three last terms, this}

expression

^is

essentially

^{the same as}

Kobayashi’s [3], [7].

^To

study

the thermal statistics of this Hamiltonian we need to know the ^exact

partition

^{function Z = Tr}

e - fla (where fl

⁼

^{1/kB T),}

which is very

complicate

^to

obtain. So we have to use an

approximate ^method, namely

^a Hartree-Fock

type

variational method. This is to say ^we propose ^a class of trial Hamiltonians

£0 (which depend

^{on some} variational

parameters)

^and

the

corresponding

^trial

density

^{matrix :}

and we minimize with

respect

^{to the} variational para- meters the

approximate

^free energy :

where

Fo

^{= -}

1 /fi Log

^Tr

e-pJeo

and

>o

^denotes the thermal _average over the states of

Xo -

We shall _use, in the

following section,

this method to

study

the statics of our

system.

3. Statics. ^- Let us choose as the trial Hamilto- nian the

following

^{static one}

(2)

where the molecular field

(h.,, hz) and il

^{are the varia-}

tional

parameters.

First of all we

diagonalize Jeo expressing

^{it in a}

rotated reference

system (ç, Q

^{such as to have}

h,

^{= 0.}

Next we

minimize,

^with

respect to r¡, hx

^and

hz,

the free energy

F previously

defined and we obtain :

or

(ferroelectric phase),

^{where J* =}

Jo

⁺

Vo 0

As we see, the static role

played by

^the

coupling

between

light

^and

heavy

^{ions is}

essentially

^to

replace Jo by

^J*.

The transition

temperature

^is

given by

2

(2) ^{The omission of the term N}

mo

2 ^{112 in Jeo does not change} the results, ^{as it} gives ^no contribution to the entropy.

where

In

figure 1,

^{where we have}

represented

^this

^function,

we may ^see the

great sensitivity

^of

To

^{on i, which}

is itself very sensitive ^on pressure. This fact

suggests interesting experimental

^work

[24].

Existence of the ferroelectric solution at

positive temperatures implies

J* > 0 and r 1.

Stability

^{of the}

crystal

^{in the}

paraelectric phase

^{demands that the ordered}

phase

be on the low

temperature

^{side of}

To.

Discussion of

m(T)

shows that the transition is ^a second order one

for all allowed values of

T, Jo, Vo

^and coo. As usual in Hartree-Fock methods we

verify

that the free energy is

analytic,

^which

implies

^a

Landau-type

behaviour for m, i. ^e.

m2

^oc

(To - T)

^{near and}

below

To.

FIG. 1. ^- Dependence of the transition temperature ^{on the} microscopic parameters ^{rand J* .}

In

figures

2 and 3 the behaviour of some relevant

quantities

^{is shown.}

We should like to call attention to an

important point :

^{the order}

parameter

^{is not m} but the electric

polarization

^P

(linear

combination of m and

’1).

In some

respects (the

determination of

To,

^for

example)

^the differentiation is irrelevant because

FIG. 2. ^- Temperature dependence ^{of the} light-ion (proton) contribution (in ^reduced units) ^{to the} polarization.

1124

FIG. 3. ^- Temperature dependence ^{of total}

( si, >)

^and

transverse

« sj > )

^{mean values of the} fictitious spin polari-

zation (in ^reduced units).

17

(V 0

^{m, hence} il, ^m and P vanish simulta-

neously at To.

However the

frequency

^{which becomes}

singular

^at

To is,

^as it is shown in section

4,

^{the one}

associated to

P,

^{and not to m. So it is}

important

^to

avoid confusion between P and m. Similar conside- rations must be made in situations

((NH4)HZP04-type

antiferroelectrics ^{or more}

complicated structures)

where the order

parameter

^{is not P.}

Let us

finally point

^{out another}

peculiarity

^{of the}

mixed-ferroelectric

we

^are

studying

here. For this let us first consider F : we see that

Using

^the

va_rious

^{relations we have} established we

may

rewrite

^{F as}

^F(q,

^m,

^T),

^{where it is}

easy

^to

verify

that

oF/or

⁼

ayam

^{= 0. If we}

develop

^{F in}

(q, m)

and retain

only

the second-order terms we obtain

The

eigenvectors

of this matrix are linear combinations of m

and 17

^with

temperature-dependent

coefficients.

Because of this

peculiarity,

^the

polarization

^P

(which is,

^{in our}

model,

^{a linear} combination of m

and il

with constant

coefficients)

^{will not be one of the}

eigenvectors,

^{as it is}

quite

^{often the case.}

4.

Dynamics.

^{- Let us} introduce the

pseudo-

magnon

operators by

Holstein-Primakoff’s transfor- mation

where

sf = S§ ± iS;

^and

(a;, aj) ^satisfy

^{usual boson}

relations, particularly [a j, a; ^]

⁼

b jj’.

^An

^expansion

of these

expressions

^{leads to}

Let us now

perform

^a Hartree-Fock

type

linearization

(discussed

in section

5),

^{i. e.}

where the factor 2 comes from the fact that there are

two ways of

constructing

^{the mean value}

aj aj

>.

We may ^now rewrite

(4. l’)

^{as follows}

where ^we have used

Let us now introduce the Fourier transform boson

operators by

So if we

perform consecutively

transformations

(3.1), (4.2)

^and

(4.3)

^{in the} Hamiltonian

(2.1),

we

neglect high-order

^{terms and we omit zero- and}

first-order terms

(because they

^{have no influence on}

the boson

frequencies)

^{we obtain}

where

(J,,

^{is real and even in}

q).

Let us now

diagonalize

^the

part

^{of Je}

concerning only

^the

pseudo-magnons.

^{For this we}

perform

^the

Bogolyubov

transformation

where

80 Je becomes

where

A final

diagonalization (see appendix)

^{leads to}

where

and

In this _way the

system

may ^now be

regarded

^{as the}

coexistence of two

types

^{of non}

interacting

^bosons

(quasi-phonons

^and

quasi-pseudo-magnons

for q -

0).

In

figure

^{4 we have}

presented

^an illustration

[7]

^of

realistic

dispersion

^curves for the modes

appearing

here.

The

KDP-type phase

transitions are due ^{to an}

instability

^{related to the}

quasi-pseudo-magnons.

^It

si easy ^to

verify

^{that at}

To (given ^by (3.1))

^and

q ⁼ 0 we obtain

We see then

that,

^{in the case}

Yq

^{= 0}

^{(no light}

^ion-

lattice

interaction)

^{J* =}

Jo,

^and

cv((To)

⁼

wg(To) =

^o.

FIG. 4. ^- Dispersion ^curves of the relevant vibration modes.

These results are very

important

^from the thermo-

dynamical standpoint,

^{for the}

crystal instability

^and

the

phase

transition are one and the same

pheno-

menon, hence the order

parameter

^{and the}

frequency

of the mode

responsible

^{for the}

instability

^must

vanish at the ^same

temperature.

^{In this} way ^{we see} that in the case

Vq

^{= 0 the} transition is

exclusively

due to a

pseudo-magnon

^mode

softening

^{and the}

order

parameter

^{is m. But when}

Vq :0

⁰ the transition

temperature

^is

higher (the light

ion-lattice interaction stabilizes the ordered

phase),

^the

pseudo-magnon frequencies

^{are still different from zero, the transition}

is due to a

quasi-pseudo-magnon instability

^{and the}

order

parameter

^{is a} combination of m and _il.

Let us

finally

^add

^that,

^as is usual in Hartree- Fock

type theories,

^{we obtain near}

To

^{and on both}

sides of it

rog2

⁼

KIT - Tao 1.

^{If we call}

Kp(Kf)

the

proportionality

coefficient in the

paraelectric (ferroelectric) phase

^we find that

KflKp >,

^{2 for all}

allowed values of the

microscopic parameters

^of the

theory (the equality corresponds

^{to the limit}

Vo -

^{0 and 2}

r/J* - 1) ;

furthermore the ratio

Kf/Kp

^is

^extremely

^{sensitive to} the values of _{the para-} meters 2

r/Y*

^and

iFo/a)o.

^This

suggests (see

^also

[3])

interesting experiments

^with variable pressure.

5. Conclusion. ^- In the

previous

^{section we}

have,

without

justification,

linearized the

equations by using

and

.. , ...

However we believe that a more

rigorous procedure

would not

essentially change

^this

approximation.

1126

If we

perform,

^on Hamiltonian

(2.1), consecutively

transformations

(3.1), (4.1) (or (4. l’», (4.3), (4.4)

and

finally (A. 2),

^{we obtain}

(if

^{we omit lower than}

harmonic

terms,

^as

they

^{do not}

modify

^the

dynamics)

Je ⁼

co"(0) Aq Aq

⁺

cv((0) B; Bq

⁺

q

+

(higher-order

^{terms in}

Aq, AQ , Bq, Bq ) ,

where

cv£(0)

^and

COB 0)

^denote

respectively

^{the 0 cK-}

values of

quasi-pseudo-magnon

^and

quasi-phonon frequencies.

^{A treatment of} this Hamiltonian with the variational formalism

proposed

in Section

2, using

the trial Hamiltonian

ico { OA (T) Aq Aq

⁺

Q((T’) B$ Bq} ,

q

(where QA

^and

Q:

^{are the} variational

parameters)

seems to prove that the linearization we have made in the

previous

section consists of a renormalization of

cvÉ(0)

^and

00;(0) ^(to

make them

temperature dependent) by considering only

the low-order inter- actions between

quasi-pseudo-magnons

^and

quasi- phonons.

A more

important objection

^{to the} way ^we have introduced and renormalized

pseudo-magnons

ⁱⁿ

the

previous

section is the

injustified

overextension

(from

^{T -- 0 to T}

To)

^{of the}

temperature

^domain of

validity

^{of the}

approximation.

^{So our}

procedure

does not avoid the usual criticism of most methods

introducing finite-temperature

renormalized _magnons.

If however ^a

rigorous procedure

^was found for per-

forming

^this

renormalization,

^the

thermodynamic

coherence we want to

point

^out

(this

^{is to} say the

simultaneity

^{of a} second-order

phase

transition and the

dynamical instability inducing it)

should remain.

The case we have examined in this paper is ^a transition

essentially triggered by

^an order-disorder

instability (protons

ⁱⁿ

KDP)

^and

accompanied by

^{an induced}

displacive phenomenon (heavy

^{ions in}

KDP).

^The

opposite

^situation

(where

^a

displace instability triggers

the transition and induces the order-disorder

phenomenon ;

^{or in an}

equivalent

way, when the transition is due to a

quasi-phonon instability)

^is

not

theoretically

^forbidden

(anharmonic

^{lattice terms}

should of course be included in the

model)

^{and the}

experimental

observation of such mixed cases should be _very

interesting.

Let us

finally

^{add that neutron}

scattering experiments performed

^on

KD2PO, [25], [26]

^and

(NH4)D2P04 [27]

^{show a}

quasi-elastic scattering intensity strongly dependent

^on

temperature. However,

if the soft-mode

picture

^is

right,

^the

important damping

effects should be included in _any best

theory

^febore

valuable

comparison

^with

experience

could be done.

The author would like to express his sincere thanks to Dr. N. Boccara for various

suggestions

and critical

reading

^{of the}

manuscript,

^to Dr. G. Sarma for _very fruitful advice and to Dr. R. Bidaux for useful dis- cussions.

Appendix

Let us consider

(from

^the Hamiltonian

(4. 5)

^when

we do not include constant

terms)

where

and

where

(+)

^denotes the hermitian

adjoint

^matrix.

Let us now consider the

following diagonalization problem

The existence of non trivial solutions

implies

hence

(we

^note

cvt

^for

( - )

^and

wq

^for

( + )).

The four

eigenvalues

^{are noted}

and the

eigenvectors

^{are noted}

Let us define the scalar

product

^{in the}

following

way

and let ^us

denote 1 e’ ¹

^_

+ ale% eq

^{the norm of the}

vector

eq.

Replacement

of the four

eigenvalues

ⁱⁿ

(A .1)

with the

condition 1 e’ q ¹

^{= 1}

(for

all values of

i )

determine

completely

^{the four real}

eigenvectors eq,

which constitute an orthonormal basis of the vecto- rial space.

If we note

we may

verify that 1 Tq ¹

^{= 1 and}

(Tq +)-1 = (Tq 1)+.

^.

Hence we may rewrite

where

and

(Aq Bq A _ q B_ q)

^{its hermitian}

^adjoint.

So if ^we omit constant terms we have

where all the boson relations are

satisfied, particu- larly

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