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]
(1)Original Article
Order
edisorder 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 by
n
j) wheren
is the layer indexðn
¼ 1; …; nÞ, Rjis thetwo-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
(2)H¼
m
hXvj
Sznj
U
Xvj
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 spinsIn the meanfield approximation (MFA), where spin fluctuation
d
Sznj¼ Sznj hSzni is neglected, Hamiltonian (1) is rewritten as
HMF¼
N
X
n;n0
Jnn00ð0ÞSznSzn0
m
Xvj
hnSznj
U
Xvj
Sxnj; (2a)
Jnn0kị ẳ
X
j
Jnn0RjeikRj: (2b)
Brackets〈…〉 mean the thermodynamic average and
b
1¼ kBTThe effectivefield hnacting on the spin at the layer
n
is given byhnẳ h ỵ
m
1Xn0
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 zsJ
x
kịd
nn0ỵ Jpd
n0;nỵ1ỵd
n0;n1; (4a)x
kị ẳzs
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])
Sxnjẳh
g
nn
Sxn0jỵ
mg
U
n
Szn0j; Sz
njẳ
mg
U
n
Sxn0jỵh
g
nn
Szn0j;
g
nẳ
h2nỵ
U
m
2 s ; (5)HMF¼
N
X
n;n0
Jnn0ð0ÞSznSzn0
m
Xvj
g
nSzn0j: (6)2.1 Equations of state atfinite temperature
It is easy to see from the Equation(6)that
g
nplays a role of aneffectivefield acting on the spin Sz0
njsimilar to hnin the Equation
(2a) One gets the free energy in MFA as
F¼ 1
b
ln
SpebHMF
;
FẳN
2 X
n;n0
Jnn00ịmznmzn0N
b
X
n ln
shS ỵ 1=2ịYn
shYn
2
; (7)
Ynẳ
bmg
n: (8)Here and in the following parts we denote average of the spin components per site at layer
n
as mzn¼ hSzni; mxn¼ hSxni MFAequations for components of order parameter of the spin system at finite temperature can be found from minimum condition of the free energy (7)
mznẳhn
g
nbsYnị; (9a)
mxnẳ
U
mg
nbsðYnÞ: (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 layer
n
(proportional to internal molecularfield) is small and may be neglected comparing with the longitudinalfield h, and transversalfieldU
Then Equation(9a)reduces toXn
n0¼1
d
nn0bSb
cfịf Jnn00ị
mzn0ẳ 0; (12a)
mz;0ẳ mz;nỵ1ẳ 0; (12b)
fẳ
m
hị2ỵU
2q
;
b
cẳ 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 bs
b
cfịzsJf ; and c ẳ bs
b
cfịJp
f : (13b)
(3)Determinant Equation(13a)reduces to the eigenvalue problem of tridiagonal matrix Dn(see for example[12]) and one has
1ẳ bS
b
cfịJ f h
zs 2p
h
cos
pn
nỵ
i
: (14)
here
n
¼ 1,2,…n In order to have corresponding expression for 3D limiting case, when n/∞, it is necessary to chosen
¼ n in (14) Finally, one obtain the equation for Curie temperature1ẳ bS
b
cfịJ f h
zsỵ 2p
h
cos
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
U
kBTcị and zerolongitu-dinal eld h ẳ 0; f ẳ
U
ị, an expansion for the Brillouin function bSxị ẳ SS ỵ 1ịx=3 may be used, and the formula(15)reduces toMFA result for Tcof Heisenberg ferromagnetic thinfilms given by[13]
kBTc
J ẳ
SS ỵ 1ị
zsỵ 2p
h
cos
p
nỵ
(16)
Formula(16)is also correct for TIM when bothfield energies are small in comparison with Curie temperature energyð
m
h;U
≪kBTcÞAt some critical value of the transversalfield, the Curie tem-perature of the n-layerfilm reduces to zero, Tc
U
cị ẳ One gets forhẳ case
U
cJS ẳ zsỵ 2p
h
cos
p
nỵ
(17)
Denoting
D
Tcẳ Tcb TcDU
cẳU
bcU
cị, where the Curietem-perature Tb
c (the critical transversalfield
U
bc) of bulk is obtainedfrom Equation(16)(Equation(17)) in the n/∞ limit, we can get for the weak transversalfield case
kB
D
TcJ z
2SS ỵ 1ị
3
h
1 cos
p
nỵ
(18a)
kB
D
TczSỵ 13DU
c; (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/0ð
b
/∞; bSðbmg
nÞ/SÞ in the formulae(7)e(9a, b), we have
F0¼
N
X
n;n0
Jnn00ịmznmzn0 NS
m
X
n
g
n; (19)mznẳShn
g
n ;(20)
mxn¼ S
U
mg
n:(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
(4)transversalfield using condition mzn
U
cị ẳ The formula(17)isobtainedrstly 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
U
1ịc ẳ 4JSU
2ịc ẳ ỵh
ịJSị according toEquation(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 temperatureFig shows the dependence of the Curie temperature on transversefield strength calculated by (15) for
h
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 whenU
U
CFig 6illustrates dependence of the critical transversalfield
U
Conthe film layer number n for different spin values S ¼ and 3/2 calculated according to Equation(17) The tendency of
U
Cto increasewithfilm 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)
(5)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 transversalfield
U
along x direction infilm plane.Fig 7presents good coincidence between the theo-retical MFA curve and experimental data for the PbTiO3perovskitemeasured in[14]when parameters are chosen as
U
/J¼ 6.1,h
¼ 1.75, S¼Investigation 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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http://creativecommons.org/licenses/by/4.0/ ScienceDirect 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 J.F Scott, Application of modern ferroelectrics, Science 315 (2007) 954e959 B Heinrich, J.A.C Bland (Eds.), Ultrathin Magnetic Structures IV, Springer,2005 H.T Diep, Theoretical methods for understanding advanced magnetic mate-rials: the case of frustrated thin N.D Mermin, H Wagner, Absence of ferromagnetism or antiferromagnetismin one- or two-dimensional isotropic Heisenberg models, Phys Rev Lett 17 S Suzuki, Jun-ichi Inoue, B.K Chakrabarti, Quantum Ising phases and Transi-tions in Transverse Ising Models, second ed., Spinger-Verlag, P.G de Gennes, Collective motions of hydrogen bonds, Solid State Commun 1(1963) 132e137 C.L Wang, S.R.P Smith, D.R Tilley, Theory of Isingfilms in a transverse field, 11423e11427 T Kaneyoshi, Ferrimagnetic magnetizations in a thinfilm described by the J.M Wessenlinowa, T Michael, S Trimper, K Zabrocki, Influence of layerdefects on the damping in ferroelectric thin Bach Thanh Cong, Pham Huong Thao, Thickness dependent properties ofmagnetic ultrathin J Borowska, L Lacinska, Eigenvalues of 2-tridiagonal toeplitz matrix, Jour ofAppl Math Comput Mech 14 (4) (2015) 11e17 R Rausch, W Nolting, The Curie temperature of thin ferromagneticfilms, J. Dillon D Fong, G Brian Stephenson, Stephen K Streiffer, Jeffrey A Eastman,Orlando Auciello, Paul H Fuoss, Carol Thompson, Ferroelectricity in ultrathin ... future work ConclusionIn 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