Orderedisorder phase transitions in thin films described by transverse Ising model

The orderedisorder phase transition in thin films at finite temperature and zero temperature (quantum phase transition) is discussed within the transverse Ising model using molecular field [r]

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Original Article

Orderedisorder phase transitions in thin ﬁlms described by transverse

Ising model

Nguyen Tu Niem, Bach Huong Giang, Bach Thanh Cong*

VNU University of Science, 334 Nguyen Trai, ThanhXuan, Hanoi, Viet Nam

a r t i c l e i n f o Article history:

Received 11 August 2016 Accepted 21 August 2016 Available online 26 August 2016 Keywords:

Orderedisorder phase transition Transverse Ising model Thinﬁlm Curie temperature Ferroelectric perovskite Critical transverseﬁeld

a b s t r a c t

The orderedisorder phase transition in thin films at finite temperature and zero temperature (quantum phase transition) is discussed within the transverse Ising model using molecularfield approximation Experimentally, it is shown that the Curie temperature TCof perovskite PbTiO3ultra-thinfilm decreases

with decreasingﬁlm thickness We obtain an equation for TCof thin ﬁlm in external magnetic and

transverseﬁelds Our equation explains well for the case of strong transverse strain ﬁeld this behaviour © 2016 The Authors Publishing services by Elsevier B.V on behalf of Vietnam National University, Hanoi This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/)

1 Introduction

Nanoscale materials like ferroelectric (FE) and ferromagnetic (FM) ultra-thinﬁlms now are important classes of materials which have been used for making of new electronic devices (see reviews

[1,2]) In order to understand properties of thinfilms, some tradi-tional models for bulks like Heisenberg, XY, Ising one are applied and solved by different theoretical methods (see for example, re-view [3] on the case of frustrated thin films) According to the MermineWagner theorem[4], 2D Heisenberg model with isotropic short range exchange interaction has no long range order atfinite nonzero temperature Thin films are quasi two dimension case when the condition of the MermineWagner theorem may be violated by presence of anisotropic exchange between layers, crystallographic anisotropy … Among anisotropic models, the transverse Ising model (TIM) plays essential role because of its simplicity and usefulness for explanation of wide classes of phase transitions including quantum phase transition [5] De Gennes firstly introduced the transverse Ising spin 1/2 model for descrip-tion of FE phase of KDP[6] TIM is solved exactly for one dimen-sional spin 1/2 chain[5], but not for the 2D and thinfilm cases Several authors have used TIM for calculation of such as: thinfilms

and FE particles within MFA[7,8]; FM magnetization in a thinfilm within effectivefield approximation[9]; influence of layer defect on the damping in FE thin films[10] In previous works, nature of the transversal field that plays important role in damping of orderedisorder phase transition temperature was briefly investi-gated Quantum phase transition (QPT) in transverse Ising model for thinfilms is also not well examined according to our awareness, even in MFA Aim of this research is to use TIM for study order-edisorder phase transitions in thin films at finite and zero (QPT) temperatures and to describe thickness dependence of the Curie temperature in ultrathin PbTiO3films within MFA The QPT case is

derived fromﬁnite temperature results in the limit T/0 Film model and meanﬁeld approximation

Following [11], we consider cubic spin lattice of a thin ﬁlm, which consists of n spin layers and there are N spins in every layer The Oz axis of the crystallographic coordinate system is directed perpendicularly to theﬁlm surfaces and the spin layers are parallel to xOy plane (seeFig 1)

A spin position in the lattice is shown by the lattice vector (denoted bynj) wherenis the layer indexðn¼ 1; …; nÞ, Rjis the

two-component vector denoting the position of the jth spin in this layer Vectorbz is unit vector directed along Oz axis, and a is the spin lattice constant (in the rest of this paper, this quantity is taken to be and all the lengths are measured in unit of lattice constant) The transverse Ising Hamiltonian for the spinﬁlm system is written as: * Corresponding author

E-mail address:congbt@vnu.edu.vn(B.T Cong)

Peer review under responsibility of Vietnam National University, Hanoi

Contents lists available atScienceDirect

Journal of Science: Advanced Materials and Devices

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / j s a m d

http://dx.doi.org/10.1016/j.jsamd.2016.08.007

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H¼ mhX

vj

SznjUX

vj

Sxnj1

2 X

nj;n0j0

Jnn0Rj Rj0SznjSzn0j0; (1)

where theﬁrst (second) term of (1) corresponds to the energy of the spin system in the longitudinal (transversal) ﬁeld h (U) The third term is Ising type exchange interaction between spins

In the meanﬁeld approximation (MFA), where spin ﬂuctuation dSz

nj¼ Sznj hSzni is neglected, Hamiltonian (1) is rewritten as

HMF¼

N

X

n;n0

Jnn00ð0ÞSznSzn0

mX

vj

hnSznjUX

vj

Sxnj; (2a)

Jnn0kị ẳ

X

j

Jnn0RjeikRj: (2b)

Brackets〈…〉 mean the thermodynamic average andb1¼ kBT

The effectiveﬁeld hnacting on the spin at the layernis given by

hnẳ h ỵm1X

n0

Jnn0ð0ÞSzn0

: (3)

Jnn0ðkÞis a Fourier image of the nearest neighbour (NN) spin

ex-change Jnn0ðRjÞ Denoting J (Jp) the exchange strength between

in-plane (out-of-in-plane) NN spins, one has

Jnn0kị ẳ zsJxkịdnn0ỵ Jpdn0;nỵ1ỵdn0;n1; (4a)

xkị ẳ

zs

X

j

eikRj (4b)

zs(2p) stands for the in-plane (out-of-plane) NN spin number and

zsỵ 2p ¼ Z denotes the total NN number for a given spin in the bulk

spin lattice For simple cubic spin lattice zs¼ and p ¼ HMFcan

be diagonalized easily by the well-known unitary transformation of spin operators (see[5])

Sxnjẳhgn

n

Sxn0jỵmgU

n

Szn0j; Sz

njẳ mgU

n

Sxn0jỵhgn

n

Szn0j; gnẳ

h2nỵ

U m 2 s ; (5)

HMF¼

N

X

n;n0

Jnn0ð0ÞSznSzn0

mX

vj

gnSzn0j: (6)

2.1 Equations of state atﬁnite temperature

It is easy to see from the Equation(6)thatgnplays a role of an

effectiveﬁeld acting on the spin Sz0

njsimilar to hnin the Equation

(2a) One gets the free energy in MFA as

F¼ 1

bln

SpebHMF

;

FẳN

2 X

n;n0

Jnn00ịmznmzn0N

b

X

n ln

shS ỵ 1=2ịYn

shYn

2

; (7)

Ynẳbmgn: (8)

Here and in the following parts we denote average of the spin components per site at layer n as mzn¼ hSzni; mxn¼ hSxni MFA

equations for components of order parameter of the spin system at ﬁnite temperature can be found from minimum condition of the free energy (7)

mznẳhn

gn

bsYnị; (9a)

mxnẳ U

mgn

bsðYnÞ: (9b)

Here bS(x) is the Brillouin function

bsxị ẳ

Sỵ12

coth

Sỵ12

x12cothx

2: (10)

MFA equations of state (9a, b) for components of layer magnetic moments of thinﬁlms can be derived in another way by realizing that in new prime0representation (5)

D

Snxj0Eẳ 0; DSnz0jEẳ bSYnị: (11)

Close to the orderedisorder phase transition temperature (the Curie temperature TC), the spin system is unstable and the

mag-netic moment at layern(proportional to internal molecularfield) is small and may be neglected comparing with the longitudinalfield h, and transversalfieldU Then Equation(9a)reduces to

Xn

n0¼1

dnn0bSbcfị

f Jnn00ị

mzn0ẳ 0; (12a)

mz;0ẳ mz;nỵ1ẳ 0; (12b)

fẳ

mhị2ỵU2

q

; bcẳ kBTcị1: (12c)

To have non-trivial solution of the system of linear algebraic Equation (12a), the determinant of the Toeplitz-type tridiagonal matrix Dnmust be zero,

detDn¼ det

2 6 6 6

a c 0 … 0

c a c … 0

0 c a c … 0

: : : : … : :

0 0 … a c

0 0 … c a

3 7 7 7

ẳ 0; (13a)

where aẳ bsbcfịzsJ

f ; and c ẳ bsbcfị

Jp

f : (13b)

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Determinant Equation(13a)reduces to the eigenvalue problem of tridiagonal matrix Dn(see for example[12]) and one has

1ẳ bSbcfị

J f h

zs 2phcos

pn

nỵ

i

: (14)

heren¼ 1,2,…n In order to have corresponding expression for 3D limiting case, when n/∞, it is necessary to chosen¼ n in (14) Finally, one obtain the equation for Curie temperature

1ẳ bSbcfị

J f h

zsỵ 2phcos

p

nỵ

i

: (15)

Equation(15)is the explicit MFA equation for the Curie tempera-ture of TIM with arbitrary spin comparing with the S¼ 1/2 case[8] It is seen from(15)that the Curie temperature is a function of the lon-gitudinal and transverseﬁeld f (see(12c)) and anisotropic exchanges For the case of small transversaleld UkBTcị and zero

longitu-dinal eld h ẳ 0; f ẳUị, an expansion for the Brillouin function bSxị ẳ SS ỵ 1ịx=3 may be used, and the formula(15)reduces to

MFA result for Tcof Heisenberg ferromagnetic thinﬁlms given by[13]

kBTc

J ẳ

SS ỵ 1ị

zsỵ 2phcos

p

nỵ

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Formula(16)is also correct for TIM when bothﬁeld energies are small in comparison with Curie temperature energyðmh;U≪kBTcÞ

At some critical value of the transversalﬁeld, the Curie tem-perature of the n-layerﬁlm reduces to zero, TcUcị ẳ One gets for

hẳ case Uc

JS ẳ zsỵ 2phcos

p

nỵ

(17)

DenotingDTcẳ Tcb TcDUcẳUbcUcị, where the Curie

tem-perature Tb

c (the critical transversalﬁeldUbc) of bulk is obtained

from Equation(16)(Equation(17)) in the n/∞ limit, we can get for the weak transversalﬁeld case

kBDTc

J z

2SS ỵ 1ị

3 h

1 cos

p

nỵ

(18a)

kBDTczSỵ 13 DUc; (18b)

U≪kBTc: (18c)

According to(18aec),(19)for small transversalﬁeld, changes of Curie temperature and critical transversal ﬁeld from their bulk values are mutual linear dependent

2.2 Ground state at zero temperature

In order to examine QFT in thinﬁlms, one needs to obtain the ground state free energy and equations of states at zero tempera-ture Taking limit T/0ðb/∞; bSðbmgnÞ/SÞ in the formulae

(7)e(9a, b), we have

F0¼

N

X

n;n0

Jnn00ịmznmzn0 NSm

X

n gn; (19)

mznẳShn

gn ;

(20)

mxn¼ SU

mgn:

(21)

We note that expression mzn,mxnﬁgured in the formulae(19) to

(21)are zero temperature components of the spin moment From Equation(20)we can obtain the same formula(17)for the critical

Fig Temperature dependence of the relative components of the spin moment per siteðmz=S; mx=SÞ for double layer thin ﬁlm with two identical surfaces withh¼ 0:8; S ¼

Fig Dependence of the components of the average spin moment per site of monolayer (n¼ 1) or symmetric double layer (n ¼ 2) ﬁlms on the relative transversal ﬁeld strength.UðnÞ

c is critical transversalﬁeld given by the formula(17)when h¼ (see

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transversalﬁeld using condition mznUcị ẳ The formula(17)is

obtainedrstly for the critical values of transversal field of TIM, it is valid for description offinite temperature orderedisorder transi-tion or QPT in both bulk or thinfilms at MFA level

3 Numerical calculation and comparison with experiment for Tc

In this part we perform the numerical calculation for cubic spin lattice ðzs¼ 4; p ẳ 1ị to show inuence of the elds and other

factors like thickness, anisotropic behaviour of exchanges on the phase transition in simple cubic spin lattice ultra-thin ﬁlms All energy quantities inﬁgures are expressed in unit of the in-plane exchange energy J

Fig 2presents the thermomagnetic-plots of the relative spin components of the symmetric two layer ﬁlms (the plots for monolayer have similar shapes, but with different TC) One sees that

the increasing transversalﬁeld leads to a reduction of mzbut an

increase in mx.Fig shows these relative spin components as

functions of the relative transversalﬁeld at T ¼ or QPT case The critical transversalﬁeld for monolayer (double layer with aniso-tropic exchanges) is U1ịc ẳ 4JS U2ịc ẳ ỵhịJSị according to

Equation(17) It is clear thatFig 3has general feature for mono-and double layerﬁlms (all plots not depend on the spin S, J,h, and Z)

Fig 4shows theﬁlm thickness dependence of Curie tempera-ture calculated by (15) for given ratio of out-of-plane and in-plane exchanges h Increase of the transversal ﬁeld causes strong damping of Curie temperature

Fig shows the dependence of the Curie temperature on transversefield strength calculated by (15) forh ¼ 1.2 On sees increase of the transversalfield leads to suppression of order in thin films, and there is no order for given thin film whenUUC

Fig 6illustrates dependence of the critical transversalﬁeldUCon

the ﬁlm layer number n for different spin values S ¼ and 3/2 calculated according to Equation(17) The tendency ofUCto increase

withﬁlm thickness is similar to that of the Curie temperature Orderedisorder phase transition described by TIM can be used for description of ferroelectric-paraelectric (FE-PE) phase transition in FE perovskites where the pseudo-spin has meaning the electrical dipole moment Equation (15) and its numerical consequence expressed inFigs and 5may be used to interpret the measured thickness dependence of the Curie temperature of lead titanate (PbTiO3) ferroelectric thin ﬁlms (see [14] and cited references

therein) It is well-known that stoichiometric unstrained PbTiO3

bulk has order (FE)edisorder (paraelectric-PE) phase transition around 763 K But in the thinﬁlms where thickness consists from few to 100 unit cells, there is strong deviation of Tcfrom its

un-strained bulk value [14] (Tc of ﬁlms varies in the interval

900e500 K) Surface reconstruction of atomic layers observed in experiment has origin of increasing intrinsic strain with reduction of thefilm thickness and it is probably sufficient large in few-layer Fig Thickness dependence of the Curie temperature of cubic spin lattice thinfilms

Parameters are S¼ 1,h¼ 1.8, h ¼ (dashed lines connecting points are drawn for better view)

Fig The Curie temperature of several thinﬁlms as a function of the transversal ﬁeld (S¼ 1,h¼ 1.2, h ¼ 0)

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ultra-thinfilm Because of that strain, the in-plane exchange be-tween spins is smaller than the perpendicular one During frame-work of TIM, one can suggest the film in-plane strain to be equivalent to large, constant transversalfieldUalong x direction infilm plane.Fig 7presents good coincidence between the theo-retical MFA curve and experimental data for the PbTiO3perovskite

measured in[14]when parameters are chosen asU/J¼ 6.1,h¼ 1.75, S¼

Investigation on influence of fluctuation on the local moment inside thinfilms and Tcbeyond MFA using method of[11]is aim of

our future work Conclusion

In this contribution we have applied the transverse Ising model for description of the orderedisorder phase transition, QPT in thinﬁlms within MFA The expressions for Curie tem-perature, and critical transversalﬁeld are given more explicitly

comparing with previous results Its usefulness is shown by application to describe well thickness dependence of the Curie temperature observed experimentally in PbTiO3perovskite thin

ﬁlms

Acknowledgement

The authors thank NAFOSTED grant 103.02-2012.73 forﬁnancial support

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Fig Theoreticalﬁtting curve for experimental data of thin perovskite PbTiO3ﬁlms

In this contribution we have applied the transverse Ising model for description of the orderedisorder phase transition, QPT in thin? ??lms within MFA The expressions for... described by the transverse Ising model, Phys Rev B 62 (2000) 11423e11427

[9] T Kaneyoshi, Ferrimagnetic magnetizations in a thin? ??lm described by the transverse Ising model, Phys Stat Solidi... 1133e1136

[5] S Suzuki, Jun-ichi Inoue, B.K Chakrabarti, Quantum Ising phases and Transi-tions in Transverse Ising Models, second ed., Spinger-Verlag, Berlin-Heidel-berg, 2013

[6] P.G

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