Ferrroeletri Phase Transitions and the Ising Model
F. C. Sa Barreto
Departamento deFsia,Universidade Federalde MinasGerais,
30.123-970 BeloHorizonte,MG,Brazil
Reeivedon1September,2000
TheIsingmodelhasbeenanimportanttheoretialtoolintheunderstandingofphasetransitions
inferroeletrimaterials. Werstreviewhowitrelatestotheunderlyingphysisoforder-disorder
phasetransitionsinthesesystems,aswellasmean-eldresultsforthespontaneouspolarizationand
thedieletrionstantneartheritialtemperature. Forhydrogen-bondedferroeletris,atermof
interationwithanexternaltransverseeld isneessaryto aountfor protontunnelingbetween
thetwominimaof adouble-wellpotential. Finally,wedisussbothexperimental andtheoretial
resultsforthe problemsofproton-lattieinterations,entral peakdynamis,dynamialbehavior
ofpseudo-one-dimensionalferroeletris andpressureeetsonhydrogenbondedferroeletris.
I Introdution
We begin with the Hamiltonian of a rystal used in
the study of ferroeletri phasetransitions [1℄. A
fer-roeletriphase transition islassied asa
displaive-typeororder-disordertypetransition. Foradisplaive
phasetransitionsomerequirementsareimposedonthe
Hamiltonianparameterssothatatransitiontakesplae
ontherystal. Foranorder-disordertransitiona
disor-deredlattiebeomesorderedwithloweringofthe
tem-perature. Quantumeetsmayalsobeomeimportant
in thisase.
InordertoobtaintheHamiltonianwetakea
single-ion model. Consider a diatomi rystal onsisting of
atoms AandB andassumethat inaphasetransition
AatomsmovewhileB atomsremainxed. Thefores
atingonatom Aaredue totheneighboring B atoms
andalsofromnext-neighborAatoms. Letu( ~
R )bethe
displaementalonganaxis,saythez-axis,ofanatom
A in the unit ellwhose enter is loated at ~
R. The
Hamiltonianis
H = X
~
R (
1
2 au(
~
R) 2
+ 1
4 bu(
~
R) 4
)+ 1
2
X
~
R; ~
R 0
(u( ~
R ) u( ~
R 0
)) 2
(1)
where a<0,b>0and is thestinessoeÆientfor
thefores(strings)linkingthepartiles. Therstterm
in Eq. (1) desribestheinteration between Aand B
atoms whilethe seond termdesribestheinteration
betweenatomsA. Letusexplainhowthetwotypesof
phasetransitionsaredesribedbythisHamiltonian.
(a) Displaive phase transitions (jaj ). For
T >T
,themeanpositionsofatoms Aarein the
en-tersoftheellsand,therefore,u( ~
R )=0,exeptfor ~
R
0 ,
atingonanAatomis
U = 1
2
(a+6)u( ~
R
0 )
2
+ 1
4 bu(
~
R
0 )
4
(2)
The oeÆientoftherst termispositiveand the
in-dividualAatommovesinaneetivepotentialwitha
singleminimum(Fig. 1).
(b) Order-disorder phase transitions (jaj ). In
Eq. (2), ifjaj thersttermis negative,while the
seondispositive. Anindividualatommovesina
sym-metriphasein adouble wellpotential(Fig. 2). Eah
atomoupiesoneofthetwoequilibriumpositions
or-respondingto theenergyminima,
u
1; 1 =
r
jaj
b
=u( ~
R
0
): (3)
ThedisplaementofanAatomfrom theenter ofthe
unit ellisaquantitywhihanhaveonlytwovalues,
u( ~
R)=uS( ~
R
0
); (4)
where S( ~
R )=1. InsertingEq. (4) intoEq. (1)and
onsideringthat S( ~
R ) 2
=1,wendthat
H = 1
2 X
~
R ; ~
R 0
JS( ~
R )S( ~
R 0
) (5)
where J =2u 2
0
and terms not ontaining S( ~
R ) have
Figure1.Eetivepotentialwithasingleminimum,
our-ringindisplaivephasetransitions.
II Order-disorder ferroeletris
Intheseferroeletris,thepartilesperformosillations
relative to their equilibrium positions, and an also
movefromoneequilibriumpositiontotheother,under
the ation of randon thermal fores. Therefore, they
moveadistane of2u( ~
R
0
)and ahargedpartile
per-forming suh amotion,from position 1to 1,orfrom
1to1,anberegardedasareversibleeletridipole.
The dipole momentis p=eu( ~
R
0
). So,in the
disor-deredphasetherystalisasystemofrandomlyaligned
dipoles. The partiles oupy either equilibrium
posi-tion with the sameprobability. In a phasetransition
the probabilities of the two equilibrium positions are
unequal. In the ordered phase eah partile oupies
one ofthepositionsforalongerperiodoftimethanin
the other. AswementionedearliertheIsing
Hamilto-nian desribesthismehanism.
In order to get some basi information on
order-disorder ferroeletris we use the results of the mean
eldapproximation,whiharemoreauratethelarger
theradiusofinterationofthepartiles. Themain
re-sults arethefollowing:
(i)Thetemperaturedependeneofthespontaneous
po-larizationisgivenby
P 2
= ( 3p
2
v 2
) (T T
)
T
; (6)
where pis the dipole moment, v is the volume of the
unit ell,T
=J
0 =k
B
is theCurietransition
tempera-ture,andJ
0 =
P
~
R 0
J( ~
R ; ~
R 0
)isaonstantrepresenting
the interation of the partile at ~
R with all the other
partilesin therangeofinterationof radius ~
R
0 .
(ii) The dieletri onstant along the diretion of the
polarizationis
"=1+ 4p
2
J
0 v
T
T T
: (7)
The Curie-Weisslaw is thus obtained, with the Curie
onstant C =4p 2
T =J v. The typeof interation in
thesystemhasto begivensothat thevalueofJ
0 an
be obtained before quantitative alulations are
per-formed.
(iii)Theeletrield(meanvalue)atingoneah
par-tileis
E=E
0 +
J
0 v
p 2
P; (8)
where E
0
is the marosopi eld. For rystals
be-longingtotheorder-disordertype,therelationC=T
=
4= =3isapproximatelyfullled,where =J
0 v=p
2
isknown astheLorentz fator. The following are
ex-amplesoforder-disorderrystals: (a)TGS,T
=322K
and C = 3200K; (b) NaNO
2 , T
= 433K and C =
5000K;()KH
2 PO
4 ,T
=123KandC=3600K.On
theotherhand,in theaseofrystalsofthedisplaive
typethatrelationisnotfullledevenapproximately,as
inBaTiO
3
,whereT
=400KandC=170000K.
(iv)Anothertesttoharaterizeanorder-disorder
tran-sition is the hange of the entropy from T = 0K to
T = T
. In that ase one obtains S = Rln2 per
mole. In orders of magnitudethe value S=R 1 is
anindiationofanorder-disordertyperystal. Infat,
for TGS, S=R = 1:1, while NaNO
2
and KH
2 PO
4
havethesamenumerialvalue,namely0:7.
III Tunneling eets in
ferro-eletris
Anorder-disorderphasetransitionourswhenthe
dis-order eets (k
B
T) are omparable with the ordered
eets (J
0
). However, if the masses of the partiles
responsibleforthedynamis aresmall, suh as
hydro-gen,then quantum-mehanialtunneling of the
parti-lesthroughthebarrier(seeFig. 2)willaddtothe
dis-ordermehanism. Wewillnowskeththededutionof
theHamiltonian whih desribes quantum-mehanial
order-disorderphasetransitions. TheHamiltonianwill
turnouttobethetransverseIsingmodel[2℄.
Figure2. Eetivedouble-wellpotential,whereanatomlies
Thestateofeahpartileinadouble-wellpotential
anberepresentedbythewavefuntion,
'= 1 ' 1 + 2 ' 2 ; (9) where ' 1 and' 2
are thewavefuntionsdesribingthe
statesof thepartile ontheleft and rightsides ofthe
potential,withthenormalizationondition 2 1 + 2 2 =1.
Notethatexitedstatesarenotonsideredhere. Upon
onsidering the eets of tunneling arossthe
double-well internal barrier, the ground state energy "
0 will
split into two levels, with energies "
1 = " 0 and " 2 =" 0
+,orrespondingtosymmetri(
1
)and
anti-symmetri(
2
)ombinations. Henethewavefuntion
desribingthesystemis
=a 1 1 +a 2 2 ; (10) where a 1 = 1 p 2 ( 1 2
) and a
2 = 1 p 2 ( 1 + 2 ): (11)
The Hamiltonian of the many-partile system is
givenby H = X ~ R H 0 ( ~
R )+ 1 2 X ~ R; ~ R 0
V(~r ~ r 0 ); (12) where ~ r 0 = ~
R+u( ~ R)and H 0 = 0 0 =S z ( ~
R ): (13)
Notie that the energies have been shifted by "
0 . HereS z ( ~
R)arethespin-1=2Paulimatries,andV(~r
~
r 0
) is the interation operator. The latter an be
ex-panded in terms of the displaements of the partiles
u( ~
R)=u,foru ~
R ,asfollows,
V(~r ~
r 0
)=V( ~
R ~
R 0
)+(~u( ~
R ) u( ~ R 0 )V 0 ( ~ R ~ R 0 )+ 1 2 (u( ~ R) u( ~ R 0 )) 2 V 00 ( ~ R ~ R 0
)+::: (14)
Thematrixrepresentation ofthe partile
displae-mentis,
u( ~
R )= u 11 u 12 u 21 u 22 ; (15) where u ij = Z i u j dv: (16)
Bytakingintoaountthesymmetriesofthe
displae-ments(uisanoddfuntionoftheoordinates),ofthe
potential(evenfuntion oftheoordinates)andofthe
funtions
1 and
2
(even and odd, respetively), we
obtainafterastraightforwardalulationthe
Hamilto-nian in terms of spin-1=2 Pauli operators,S x
( ~
R ) and
S z
( ~
R ),
H =V
0 + 0 X ~ R S z ( ~ R ) 1 2 X J( ~ R ~ R 0 )S x ( ~ R )S x ( ~ R 0 ); (17) whereV 0 = P ~ R ; ~ R 0 V( ~ R ~ R 0 ), 0 =+ 1 4 P ~ R 0 V 00 ( ~ R ~ R 0 )(u 2 11 u 2 22
)andJ( ~
R ~
R 0
)=2V 00 ( ~ R ~ R 0 )u 2 12 . Inthe
representationofthefuntions'
1 and'
2
,weobtain,
H = X ~ R S x ( ~ R ) 1 2 X ~ R; ~ R 0 J( ~ R ~ R 0 )S z ( ~ R )S z ( ~ R 0 ); (18)
whihisthetransverseIsingmodelHamiltonian.
The polarization is proportional to < S z
> and,
in themeaneld approximation,itanbeobained by
solvingthetransendentalequation,
p ( 2 +J 2 0 <S z > 2 ) J 0 =tanh( p ( 2 +J 2 0 <S z > 2 ) k B T ): (19)
The transition temperature obtainedwithin the mean
eld approximationis
T = 2J 0 k B = =J 0 ln 1+=J 0 1 =J0 : (20)
Withtheinreaseoftunneling(aswearesetting~=1,
thetunneling frequenyis 2)thetransition
tempera-turedropsandT
!0as!J
0
. Therefore,tunneling
worksasadisordermehanism.
IV Disussion of experimental
and theoretial results
We shall present now some experimental results that
havebeenexplainedbytheuseofthemodelsdisussed
previously. They are the resultof investigations done
inthePhysisDepartmentoftheFederalUniversityof
Minas Gerais, Brazil. More reent experimental and
theoretialresultsanbefoundelsewhere[3℄.
(a)Proton-LattieInterations.
Theaddition of aB
ij S x i S x j
{typeoupling between
thetunnelingmotionofoneprotonandthatofanother
to the Ising model in a transverse-eld Hamiltonian,
or the addition of the probably larger S x i F x i Q i {type
pseudospin phonon oupling (desribing the
modula-tion of thedistane between thetwoequilibrium sites
in a O-H-O bond by nonpolar phonons), results in a
temperature-dependent renormalization of the proton
tunneling integral [4℄. This is important lose to T
,
where the soft-mode frequeny vanishes, !
rit ! 0.
This mayleadtolargeisotopeshiftsinT
on
deutera-tionevenforsmallvaluesofthetunnelingintegraland
may explain some phenomena observed in PbHPO
4
[5,6℄and squari aid[7℄ aswell asthedependene of
theeetiveproton-lattieinteration onstanton
eet may be also due to the presene of a S x
i D
x
i Q
2
i
termin additiontotheKobayashitermS z
i F
z
i Q
i when
ouplingwithpolaroptiphononsistakenintoaount
[4℄. When the lattie motion is so anharmoni that
thelattieionsmoveinadouble-wellpotentialandthe
proton-lattie oupling are so strongthat theprotons
an tunnel in only one outof the two possible lattie
ongurations,twoCurietemperaturesmayappear[4℄.
(b)DynamiCentralPeak.
Measurementsofthetemperatureandfrequeny
de-pendene of the proton and deuteron spin-lattie
re-laxation, T
1
, show the presene of a dynami
en-tral peak in paraeletri KH
2 PO
4
, whih exhibits a
dramati narrowing on deuteration. The results of
an NMR study of the frequeny and temperature
de-pendene of the proton and deuteron T
1
in
paraele-tri KH
2 PO
4
and deuterated KD
2 PO
4
, represented
the rst diret observation of the narrowing of a
dy-nami entral peak on deuteration [8℄. The entral
peak in deuterated KD
2 PO
4
is morethan two orders
of magnitudenarrowerthanthedynami entral peak
in paraeletri KH
2 PO
4
. A ontinued-fration
alu-lation of the order-parameter utuation spetrum of
KH
2 PO
4
as a funtion of deuterium ontent
qualita-tivelyreproduestheobserved entral-peaknarrowing
on deuteration. The two studies together represent
strong evidene that small dynami intrinsi lusters
ourin KDP-typerystals in additionto large stati
defet-indued lusters observedin light-sattering
ex-periments. The study showed as well that naturally
abundant deuterium annot be the ause of the
ob-servedstati(light-sattering)anddynami(T
1 NMR)
entralpeaksinKH
2 PO
4 .
()DynamisofPseudo-One-DimensionalCrystals.
Cesium dihydrogen phosphate CsH
2 PO
4
is a
hy-drogen bonded ferroeletri rystal whih undergoes
an order-disorderphase transition at T
= 152K.The
shift of the phase transition temperature T
= 267K
in theisomorphdeuteratedompoundCsD
2 PO
4
indi-ates that the motion of the proton in the hydrogen
bonds hasan importantrole forthe ferroeletriityin
these materials. Results of X-ray diration [9℄ and
neutron sattering experiments [10℄ showed that the
hydrogenbondsin CsH
2 PO
4
runalonghains. Ithas
been also shown [11℄ that the intrahain interations
dueto thehydrogenbondsaremuhstrongerthanthe
interhain ones, a fat whih haraterizes the
quasi-one-dimensional behaviorof these rystals. Therefore,
in ordertostudytheirpropertiestheoretiallyitis
de-sirable to take into aount the strong intrahain
in-terations as exatly as possible while the interhain
interationsanbetreatedwithinameaneld
approx-imation. Thedynamisofthestrongly-anisotropiIsing
modelinatransverseeldwasusedwiththepurposeof
explainingthedieletriritialslowingdownobserved
experimentallyinthequasi-one-dimensional
hydrogen-bonded ferroeletri rystal CsH PO . Good
agree-ment with the experimental data on the temperature
dependene of the dieletri onstant and relaxation
timehasbeenobtained[12℄.
(d)PressureEets.
Experimentalstudies[13,14℄indiatethatthe
tran-sition temperature of both C
s H
2 PO
4 and C
s D
2 PO
4
dereases as the pressure is inreased. For C
s D
2 PO
4
theritialtemperatureoftheparaeletri-ferroeletri
transitiondereasesataratedT d
=dp= 8:5 o
C/Kbar,
and for pressures larger than about 5:0{6:0kbar the
system orders antiferroeletrially. The Neel
tem-perature ofthe paraeletri-antiferroeletritransition
also dereases with pressure at a rate dT d
n =dp =
2:5 o
C/Kbar. Theferroeletri-antiferroeletri
tran-sitionlineintheT-pphasediagramwasobservedtobe
assoiatedwithaslopeofabout35 o
C/kbar. Thetriple
pointwasloatedatP d
=5:2kbarandT d
= 55:2 o
C.
A ompressible pseudo-spin Ising model Hamiltonian
isusedtoalulatethepressure-temperaturephase
di-agram of quasi-one-dimensional hydrogen-bonded
fer-roeletri rystals suh as CsD
2 PO
4
[15℄. Strong
ef-fetive interations, whih are treated exatly along
hains,andweakvolumedependentinterations,whih
are treatedin the mean eld approximation, between
hains, were assumed. In agreement with the
exper-imental ndings, the phase diagram exhibits a triple
pointandtransitionlinesbetweenferroeletri,
antifer-roeletriandparaeletriphases. However,theresults
suggestedthat the Ising model may be too simple to
aount forthe detailed form of thephase diagram of
CsD
2 PO
4 .
I thankmyolleaguesando-authorsR. Blinand
B.Zeks,fromtheInstitutJ.Stefan(Ljubljana)andJ.F.
Sampaio,A.S.T.PiresandJ.Plasak,fromthe
Depar-tamentodeFsia,UFMG(BeloHorizonte). Thiswork
waspartiallysupportedbyCNPq,FAPEMIG,FINEP,
andMCT(Brazilianagenies).
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