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Phase diagram of ferroelectric thin film with diluted surface
Superlattices and Microstructures 49 (2011) 307–313
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Phase diagram of ferroelectric thin film with diluted surface H. Arhchoui a,∗ , Y. El Amraoui a , I. Luk’yanchuk b , D. Mezzane c a
Laboratoire de Physique Statistique et Modélisation des systèmes, Département de Physique, Errachidia, Morocco
b
Laboratoire de Physique de la Matière Condensée de l’Université de Picardie, France
c
LMCN, Département de Physique, FSTG Marrakech, Morocco
article
info
Article history: Received 13 April 2010 Accepted 5 June 2010 Available online 30 June 2010 Keywords: Ferroelectric thin film Transverse Ising model
abstract We study the influence of the surface of the ferroelectric films on the properties of the intrinsic ferroelectric polarization state in frame of statistical Transverse Ising Model (TIM). The quality of the surface is modelled by the dilution parameter q that reflects the relative weight of the paraelectric impurities vs. ferroelectric ions. Using the effective field approach the variation of the average value of spontaneous polarization as function of temperature for different film thicknesses has been investigated as function of dilution parameter q. The phase diagram of the system and crossover from the ferroelectric to the paraelectric phase is discussed. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Fabrication of nano-scale ferroelectric memory storage devices initiated a great deal of interest to study ferroelectric materials confined to the restrictive geometry of a microelectronic environment. A special emphasis is given to understanding the properties of thin ferroelectric films and to the role of the surface and interface effects on the average spontaneous polarization [1–3]. Modelling of ferroelectric systems is usually based on the complimentary methods: statistical physics approach, ab initio simulations and minimization of the phenomenological Ginzburg–Landau functional. In the current article, we apply the microscopic statistical approach to investigate the influence of the surface effects on the ferroelectric phase in thin films. The consideration is based on the pseudo-spin Transverse Ising Model (TIM) that is the adequate tool to investigate the bulk properties of KH2 PO4 -type ferroelectrics. This model has also been applied to study the critical properties of ferroelectric thin films [4–12]. Particularly, in [4] the surface properties were taking into account by
∗
Corresponding author. E-mail address: [email protected] (H. Arhchoui).
0749-6036/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2010.06.005
308
H. Arhchoui et al. / Superlattices and Microstructures 49 (2011) 307–313
surface variation of the transverse field coupling constant. It is however more realistic to consider the film surface as the ensemble of diluted ferro/paraelectric ions. Phase diagram of such a system will be obtained in the present work. The outline of this work is as follows. A brief formulation of the effective field theory which is based on the introduction of a differential operator technique [13] is given in Section 2. The numerical results are obtained and discussed in Section 3. In Section 4 we present the principal conclusions. 2. The model We consider a surface-diluted transverse Ising thin film with a thickness L having simple cubic symmetry in which each layer is parallel to the diluted (001) surfaces. The Hamiltonian of the system is given by: H =−
−
Jij Siz Sjz ξi ξj −
−
(ij)
Ωi ξi Six .
(1)
i
Here Siz and Six are the z and x components of spin-1/2 operator at site i, the transverse field Ωi represents a quantum tunnelling between potential minima, and Jij is the interaction between nearestneighbor dipoles. Physically Jij describes both the direct electrostatic dipolar interaction and the indirect dipolar interaction provided by the lattice deformation potential, screened at large distances. Unlike magnetic systems the numerical value of phenomenological constants Jij can be found only from ab initio calculations. Approximately they are estimated as ferroelectric transition temperature i.e. several hundreds of K (∼0.1–1 eV). The tunnelling integrals Ωi are of the order of 200 cm−1 (0.03 eV) for proton ions [14] and much lower for more heavy ions. We assume that parameters Jij , Ωi takes the value J and Ω in the bulk layers and Js and Ωs at the surface layer. Parameter ξi is the site occupancy parameter equal to one or zero, depending on whether the site is occupied or not by ferroelectric ion. Since only the surfaces are considered to be diluted in the present system, ξi = 1 with a probability q when the surface site i is ferroelectric and ξi = 0 with a probability (1 − q) when the surface site i is paraelectric. In bulk ξi = 1. The standard effective field method of treatment of such system is based on the differential operator technique [13] in which the spatial distribution of polarization across the ferroelectric layers is based on the system of coupled equations, that in application to our case can be written as: For first and last layer with (k = 1, i = 2 or k = L, i = L − 1) pk = [q {cosh (Js D) + pk sinh (Js D)} + (1 − q)]4 [cosh (JD) + pi sinh (JD)] Fs (x)|x=0 .
(2)
For the second or the (L − 1) layer p2 = [cosh (Js D) + p2 sinh (Js D)]4 [q {cosh (JD) + p1 sinh (JD)} + (1 − q)] × [cosh (JD) + p3 sinh (JD)] F (x)|x=0 p(l−1) =
(3)
4
cosh (Js D) + p(l−1) sinh (Js D) [q {cosh (Js D) + p1 sinh (Js D)} + (1 − q)] × [cosh (JD) + pl−2 sinh (JD)] F (x)|x=0 .
(4)
For each layer with polarization pL 3 ≤ k ≤ L − 2 pk = [cosh (JD) + pk sinh (JD)]4 [cosh (JD) + pk−1 sinh (JD)]
× [cosh (JD) + pk+1 sinh (JD)] F (x)|x=0 .
(5)
Here the differential operator D = d/dx acts on the generating functions Fs (x) and F (x) that are defined by: F S ( x) =
x
ys
tanh (β y)
and F (x) =
x
y
with ys = x2 + Ωs2
1/2
,
y = x2 + Ω 2
1/2
.
tanh (β y)
H. Arhchoui et al. / Superlattices and Microstructures 49 (2011) 307–313
309
After some algebra the above equations can be developed as follows pk =
4 −
C4l
l =0
p2 =
1 −
1 −
C1k
4 −
C4l
l =0
k
P(l−1) =
l Cm A1,L (l, m) pls1 pm 2
(6)
m=0
1 −
C1k
1 −
(7)
m=0
4 −
C4l
l =0
k
1 Cm A2 (k, l, m) pks1 pl2 pm 3
1 −
1 Cm Al−1 (k, l, m) pksl pl(l−1) pm l −2
(8)
m=0
where the coefficients Ai are given by: A1l = [q cosh (Js D) + (1 − q)]4−l [sinh (Js D)]l [cosh (JD)]1−m [sinh (JD)]m tanh(β x)|x=0 [sinh (JD)] [cosh (JD)] An = [cosh (JD)] × [sinh (JD)]m tanh(β x)|x=0 . 1−k
4 −l
k
(9)
[sinh (JD)] [cosh (JD)] l
1 −m
(10)
Near the critical point we can linearize the coupled equations (6)–(8) as: pk = A1,L (1, 0) p1 + A1,L (0, 1) p2
(11)
p2 = A2 (1, 0, 0) ps1 + A2 (0, 1, 0) p2 + A2 (0, 0, 1) p3
(12)
pi = Ai (1, 0, 0) pi−1 + Ai (0, 1, 0) pi + Ai (0, 0, 1) pi+1 .
(13)
This system can be symbolically written in the matrix form:
⃗z = 0 Mp
(14)
⃗n = (p1 , p2 , . . . , pL ) and M is a matrix of elements where p Mij = Aij − 1 δij + Aij δij−1 + δij+1 .
(15)
All the information about the critical temperature of the system is contained in Eq. (14). In general case, for arbitrary exchange interaction Rs , surface field Ωs and transverse field Ω , the evaluation of the critical temperature relies on numerical solution of the system of linear equations (11)–(13). These ⃗z when: equations have the nontrivial solution p det M = 0.
(16)
Eq. (16) has L independent eigen-modes with characteristic temperatures Ti from which maximal one should be selected as the critical temperature Tc [15]. The corresponding vector solution consists of the positive layer polarizations (p1 , p2 , . . . , pL ) and is compatible with a longitudinal ferroelectric ordering. The other formal solutions correspond in principle to other types of ordering that usually do not occur [15]. 3. Numerical results 3.1. The total longitudinal polarization p Expanding the coupled equations (6)–(8) and using the symmetry between layer polarization (p1 = pL and p2 = pL−2 . . .), one can easily determine the variation of the polarization as function of temperature for different values of the thickness L. We find that there exist two phases: ferroelectric phase (FP) with different from zero longitudinal polarization p and paraelectric phase (PP) for which the total polarization vanishes. The transition from one phase to the other is found to be a continuous one. In Fig. 1, typical variations of polarization in thin films with dilution q = 1 (pure surfaces) and Ω = 1, Rs = 1 are given for L = 4, L = 6 and L = 15. The dashed line represent the bulk longitudinal J
310
H. Arhchoui et al. / Superlattices and Microstructures 49 (2011) 307–313
Fig. 1. The temperature dependence of the total polarization p for a dilution parameter q = 1 (pure surfaces), Rs = 1 and for number of layers L.
Ω J
= 1, and
polarization per site at Ω = 1. The critical temperature Tc increases when the value of L increases. This is closely related to the universal features found in Refs. [8,16–19]. Fig. 2 shows the variation of polarization p in the systems with diluted surfaces, when the value of q is changed. These figures show the behaviour that is qualitatively similar to that observed in Fig. 1. Note also that when q decreases, the polarization decrease rapidly and the critical temperature decrease. In order to prove this result, we have plotted the polarization p as function of temperature T for L = 4 and L = 15 in Fig. 3(a), (b). 3.2. The phase diagram From Eq. (16) we can obtain numerically the phase diagrams of the film. Fig. 4(a) shows the variation of TC with the value of surface dilution q when the values of Rs and Ωs are fixed at Rs = 1, Ωs = 1 and for L = 4, L = 6 and L = 15. The horizontal dash-dotted line represents the bulk transition temperature. The results show that the system exhibits two phases: a ferroelectric phase (FP) and a paraelectric phase (PP). We notice that there exists a critical value qc of q for which the critical temperature of the thin film is independent of its thickness and is equal to For the small values of L (L = 4 and L = 6) and for q ≺ qc , the critical temperature smaller than the bulk critical temperature KB TC J
KB TCB
KB TCB J
K B TC J
KB TCB J
.
of the film is
. On the other hand, for q ≻ qc the critical temperature KB T S
K T
of the film is greater both than J and than J C . The critical temperature BJ C decreases with increase of L. For large value of L (L = 15), the behaviour is qualitatively similar to that for L = 4 and L = 6, but K T
KB T B
for q lower than 0.1, the critical temperature of the film BJ C is higher than J C . Fig. 4(b), (c) shows the variation of TC when variation of the surface dilution q and Rs , when Ωs = 1 and L = 4, L = 6 and L = 15. The horizontal dash-dotted line represents the bulk transition temperature. The observed behaviour is similar to that of Fig. 4(a). We find that Ps is dependent on Rs , qc decrease and Tc increases when Rs increases. 4. Conclusion In this work, we have studied the critical behaviour of the transverse Ising thin films with diluted surface. The variation of the polarization as function of temperature for some values of thickness L
H. Arhchoui et al. / Superlattices and Microstructures 49 (2011) 307–313
a
311
b
c
Fig. 2. The temperature dependence of the total polarization p for a dilution q = 0 (a) q = 0.2 (b) and q = 0.5 (c) with and Rs = 1 for different numbers of layers L.
a
Ω J
=1
b
Fig. 3. The temperature dependence of the total polarization p for L = 4 (a) and L = 15 (b) with different dilution parameters q.
Ω J
= 1, and Rs = 1 for
312
H. Arhchoui et al. / Superlattices and Microstructures 49 (2011) 307–313
a
b
c
Fig. 4. Transition temperature TC as a function of surface dilution parameter q for Rs = 1.5 (a), Rs = 2 (b), Rs = 2.5 (c) and for different numbers of layers L.
has been investigated for different values of dilution parameter q. The temperature dependence of polarization p decreases from the saturation value at T = 0 and reduces to zero at T = TC . On the other hand, the critical temperature Tc increases when the value of L increases. The phase diagrams as function of dilution parameter q has been investigated for different values J of the surface exchange interaction Rs = JS and for a fixed value of the transverse field Ω . We found that the special point Ps, at which the critical temperature is independent of the film thickness L, dependents on the exchange interaction Rs . Note that in this article we neglected the long range interaction between ferroelectric dipoles, which is known to lead to formation of non-uniform ferroelectric domains [20–22] and vortex-like structures [23,24]. Incorporation of such effects into transverse Ising model and clarification of the surface effects would be the next step of the proposed microscopic consideration. Acknowledgements This work was supported by France–Morocco collaborative program ‘‘VOLUBILIS’’ and by FP7 IRSES mobility project ‘‘ROBOCON’’. References [1] J.F. Scott, Ferroelectrics Memories, Springer, New York, 2000. [2] M. Dawber, K.M. Rabe, J.F. Scott, Rev. Modern Phys. 77 (2005) 1083.
H. Arhchoui et al. / Superlattices and Microstructures 49 (2011) 307–313 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
K. Iijima, T. Terashima, Y. Bando, K. Kamigaka, H. Terachi, J. Appl. Phys. 72 (1992) 2840. H. Arhchoui, Y. El Amraoui, D. Mezzane, I. Luk’yanchuk, Eur. Phys. J. Appl. Phys. 48 (2009) 10503. T. Kaneyoshi, J. Magn. Magn. Mater. 321 (2009) 3630. A. Tabyaouia, M. Madania, A. Ainanea, M. Saber, Physica A 358 (2005) 150. A. Tabyaouia, A. Ainanea, B. Movagharb, M. Saber, Physica A 329 (2003) 277. T. Kaneyoshi, Physica A 319 (2003) 355. X.G. Wang, S.H. Pan, G.Z. Yang, Solid State Commun. 113 (2000) 807. X.G. Wang, S.H. Pan, G.Z. Yang, Chinese Phys. Lett. 17 (2000) 132. D.L. Yao, Y.Z. Wu, Z.Y. Li, Phys. Status Solidi B 231 (2002) 3. K. Htoutou, A. Ainane, M. Saber, J.J. De Miguel, J. Phys. Chem. Solids 67 (2006) 182. R. Honmura, T. Kaneyoshi, J. Phys. C 12 (1979) 3979. P.G. de Gennes, Solid State Commun. 1 (1963) 132. A.R. Ferchmin, W. Maciejewski, J. Phys.: Condens. Matter. 12 (1979) 4511. T. Kaneyoshi, J. Magn. Magn. Mater. 264 (2003) 30. T. Kaneyoshi, Phys. Status Solidi B 237 (2003) 592. T. Kaneyoshi, Physica A 328 (2003) 174. T. Kaneyoshi, Phys. Status Solidi B 241 (2004) 213. A.M. Bratkovsky, A.P. Levanuk, Phys. Rev. Lett. 84 (2000) 3177. F. De Guerville, L. Lahoche, I. Lukyanchuk, M. ElMarssi, Mater. Sci. Eng. B 120 (2005) 16. I. Luk’yanchuk, L. Lahoche, A. Sene, Phys. Rev. Lett. 102 (2009) 147601. I.I. Naumov, L. Bellaiche, Fu Huaxiang, Nature 432 (2004) 737. G. Pascoli, L. Lahoche, I. Luk’yanchuk, Integr. Ferroelectr. 99 (2008) 60.
313
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Phase diagram of ferroelectric thin film with diluted surface H. Arhchoui a,∗ , Y. El Amraoui a , I. Luk’yanchuk b , D. Mezzane c a
Laboratoire de Physique Statistique et Modélisation des systèmes, Département de Physique, Errachidia, Morocco
b
Laboratoire de Physique de la Matière Condensée de l’Université de Picardie, France
c
LMCN, Département de Physique, FSTG Marrakech, Morocco
article
info
Article history: Received 13 April 2010 Accepted 5 June 2010 Available online 30 June 2010 Keywords: Ferroelectric thin film Transverse Ising model
abstract We study the influence of the surface of the ferroelectric films on the properties of the intrinsic ferroelectric polarization state in frame of statistical Transverse Ising Model (TIM). The quality of the surface is modelled by the dilution parameter q that reflects the relative weight of the paraelectric impurities vs. ferroelectric ions. Using the effective field approach the variation of the average value of spontaneous polarization as function of temperature for different film thicknesses has been investigated as function of dilution parameter q. The phase diagram of the system and crossover from the ferroelectric to the paraelectric phase is discussed. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Fabrication of nano-scale ferroelectric memory storage devices initiated a great deal of interest to study ferroelectric materials confined to the restrictive geometry of a microelectronic environment. A special emphasis is given to understanding the properties of thin ferroelectric films and to the role of the surface and interface effects on the average spontaneous polarization [1–3]. Modelling of ferroelectric systems is usually based on the complimentary methods: statistical physics approach, ab initio simulations and minimization of the phenomenological Ginzburg–Landau functional. In the current article, we apply the microscopic statistical approach to investigate the influence of the surface effects on the ferroelectric phase in thin films. The consideration is based on the pseudo-spin Transverse Ising Model (TIM) that is the adequate tool to investigate the bulk properties of KH2 PO4 -type ferroelectrics. This model has also been applied to study the critical properties of ferroelectric thin films [4–12]. Particularly, in [4] the surface properties were taking into account by
∗
Corresponding author. E-mail address: [email protected] (H. Arhchoui).
0749-6036/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2010.06.005
308
H. Arhchoui et al. / Superlattices and Microstructures 49 (2011) 307–313
surface variation of the transverse field coupling constant. It is however more realistic to consider the film surface as the ensemble of diluted ferro/paraelectric ions. Phase diagram of such a system will be obtained in the present work. The outline of this work is as follows. A brief formulation of the effective field theory which is based on the introduction of a differential operator technique [13] is given in Section 2. The numerical results are obtained and discussed in Section 3. In Section 4 we present the principal conclusions. 2. The model We consider a surface-diluted transverse Ising thin film with a thickness L having simple cubic symmetry in which each layer is parallel to the diluted (001) surfaces. The Hamiltonian of the system is given by: H =−
−
Jij Siz Sjz ξi ξj −
−
(ij)
Ωi ξi Six .
(1)
i
Here Siz and Six are the z and x components of spin-1/2 operator at site i, the transverse field Ωi represents a quantum tunnelling between potential minima, and Jij is the interaction between nearestneighbor dipoles. Physically Jij describes both the direct electrostatic dipolar interaction and the indirect dipolar interaction provided by the lattice deformation potential, screened at large distances. Unlike magnetic systems the numerical value of phenomenological constants Jij can be found only from ab initio calculations. Approximately they are estimated as ferroelectric transition temperature i.e. several hundreds of K (∼0.1–1 eV). The tunnelling integrals Ωi are of the order of 200 cm−1 (0.03 eV) for proton ions [14] and much lower for more heavy ions. We assume that parameters Jij , Ωi takes the value J and Ω in the bulk layers and Js and Ωs at the surface layer. Parameter ξi is the site occupancy parameter equal to one or zero, depending on whether the site is occupied or not by ferroelectric ion. Since only the surfaces are considered to be diluted in the present system, ξi = 1 with a probability q when the surface site i is ferroelectric and ξi = 0 with a probability (1 − q) when the surface site i is paraelectric. In bulk ξi = 1. The standard effective field method of treatment of such system is based on the differential operator technique [13] in which the spatial distribution of polarization across the ferroelectric layers is based on the system of coupled equations, that in application to our case can be written as: For first and last layer with (k = 1, i = 2 or k = L, i = L − 1) pk = [q {cosh (Js D) + pk sinh (Js D)} + (1 − q)]4 [cosh (JD) + pi sinh (JD)] Fs (x)|x=0 .
(2)
For the second or the (L − 1) layer p2 = [cosh (Js D) + p2 sinh (Js D)]4 [q {cosh (JD) + p1 sinh (JD)} + (1 − q)] × [cosh (JD) + p3 sinh (JD)] F (x)|x=0 p(l−1) =
(3)
4
cosh (Js D) + p(l−1) sinh (Js D) [q {cosh (Js D) + p1 sinh (Js D)} + (1 − q)] × [cosh (JD) + pl−2 sinh (JD)] F (x)|x=0 .
(4)
For each layer with polarization pL 3 ≤ k ≤ L − 2 pk = [cosh (JD) + pk sinh (JD)]4 [cosh (JD) + pk−1 sinh (JD)]
× [cosh (JD) + pk+1 sinh (JD)] F (x)|x=0 .
(5)
Here the differential operator D = d/dx acts on the generating functions Fs (x) and F (x) that are defined by: F S ( x) =
x
ys
tanh (β y)
and F (x) =
x
y
with ys = x2 + Ωs2
1/2
,
y = x2 + Ω 2
1/2
.
tanh (β y)
H. Arhchoui et al. / Superlattices and Microstructures 49 (2011) 307–313
309
After some algebra the above equations can be developed as follows pk =
4 −
C4l
l =0
p2 =
1 −
1 −
C1k
4 −
C4l
l =0
k
P(l−1) =
l Cm A1,L (l, m) pls1 pm 2
(6)
m=0
1 −
C1k
1 −
(7)
m=0
4 −
C4l
l =0
k
1 Cm A2 (k, l, m) pks1 pl2 pm 3
1 −
1 Cm Al−1 (k, l, m) pksl pl(l−1) pm l −2
(8)
m=0
where the coefficients Ai are given by: A1l = [q cosh (Js D) + (1 − q)]4−l [sinh (Js D)]l [cosh (JD)]1−m [sinh (JD)]m tanh(β x)|x=0 [sinh (JD)] [cosh (JD)] An = [cosh (JD)] × [sinh (JD)]m tanh(β x)|x=0 . 1−k
4 −l
k
(9)
[sinh (JD)] [cosh (JD)] l
1 −m
(10)
Near the critical point we can linearize the coupled equations (6)–(8) as: pk = A1,L (1, 0) p1 + A1,L (0, 1) p2
(11)
p2 = A2 (1, 0, 0) ps1 + A2 (0, 1, 0) p2 + A2 (0, 0, 1) p3
(12)
pi = Ai (1, 0, 0) pi−1 + Ai (0, 1, 0) pi + Ai (0, 0, 1) pi+1 .
(13)
This system can be symbolically written in the matrix form:
⃗z = 0 Mp
(14)
⃗n = (p1 , p2 , . . . , pL ) and M is a matrix of elements where p Mij = Aij − 1 δij + Aij δij−1 + δij+1 .
(15)
All the information about the critical temperature of the system is contained in Eq. (14). In general case, for arbitrary exchange interaction Rs , surface field Ωs and transverse field Ω , the evaluation of the critical temperature relies on numerical solution of the system of linear equations (11)–(13). These ⃗z when: equations have the nontrivial solution p det M = 0.
(16)
Eq. (16) has L independent eigen-modes with characteristic temperatures Ti from which maximal one should be selected as the critical temperature Tc [15]. The corresponding vector solution consists of the positive layer polarizations (p1 , p2 , . . . , pL ) and is compatible with a longitudinal ferroelectric ordering. The other formal solutions correspond in principle to other types of ordering that usually do not occur [15]. 3. Numerical results 3.1. The total longitudinal polarization p Expanding the coupled equations (6)–(8) and using the symmetry between layer polarization (p1 = pL and p2 = pL−2 . . .), one can easily determine the variation of the polarization as function of temperature for different values of the thickness L. We find that there exist two phases: ferroelectric phase (FP) with different from zero longitudinal polarization p and paraelectric phase (PP) for which the total polarization vanishes. The transition from one phase to the other is found to be a continuous one. In Fig. 1, typical variations of polarization in thin films with dilution q = 1 (pure surfaces) and Ω = 1, Rs = 1 are given for L = 4, L = 6 and L = 15. The dashed line represent the bulk longitudinal J
310
H. Arhchoui et al. / Superlattices and Microstructures 49 (2011) 307–313
Fig. 1. The temperature dependence of the total polarization p for a dilution parameter q = 1 (pure surfaces), Rs = 1 and for number of layers L.
Ω J
= 1, and
polarization per site at Ω = 1. The critical temperature Tc increases when the value of L increases. This is closely related to the universal features found in Refs. [8,16–19]. Fig. 2 shows the variation of polarization p in the systems with diluted surfaces, when the value of q is changed. These figures show the behaviour that is qualitatively similar to that observed in Fig. 1. Note also that when q decreases, the polarization decrease rapidly and the critical temperature decrease. In order to prove this result, we have plotted the polarization p as function of temperature T for L = 4 and L = 15 in Fig. 3(a), (b). 3.2. The phase diagram From Eq. (16) we can obtain numerically the phase diagrams of the film. Fig. 4(a) shows the variation of TC with the value of surface dilution q when the values of Rs and Ωs are fixed at Rs = 1, Ωs = 1 and for L = 4, L = 6 and L = 15. The horizontal dash-dotted line represents the bulk transition temperature. The results show that the system exhibits two phases: a ferroelectric phase (FP) and a paraelectric phase (PP). We notice that there exists a critical value qc of q for which the critical temperature of the thin film is independent of its thickness and is equal to For the small values of L (L = 4 and L = 6) and for q ≺ qc , the critical temperature smaller than the bulk critical temperature KB TC J
KB TCB
KB TCB J
K B TC J
KB TCB J
.
of the film is
. On the other hand, for q ≻ qc the critical temperature KB T S
K T
of the film is greater both than J and than J C . The critical temperature BJ C decreases with increase of L. For large value of L (L = 15), the behaviour is qualitatively similar to that for L = 4 and L = 6, but K T
KB T B
for q lower than 0.1, the critical temperature of the film BJ C is higher than J C . Fig. 4(b), (c) shows the variation of TC when variation of the surface dilution q and Rs , when Ωs = 1 and L = 4, L = 6 and L = 15. The horizontal dash-dotted line represents the bulk transition temperature. The observed behaviour is similar to that of Fig. 4(a). We find that Ps is dependent on Rs , qc decrease and Tc increases when Rs increases. 4. Conclusion In this work, we have studied the critical behaviour of the transverse Ising thin films with diluted surface. The variation of the polarization as function of temperature for some values of thickness L
H. Arhchoui et al. / Superlattices and Microstructures 49 (2011) 307–313
a
311
b
c
Fig. 2. The temperature dependence of the total polarization p for a dilution q = 0 (a) q = 0.2 (b) and q = 0.5 (c) with and Rs = 1 for different numbers of layers L.
a
Ω J
=1
b
Fig. 3. The temperature dependence of the total polarization p for L = 4 (a) and L = 15 (b) with different dilution parameters q.
Ω J
= 1, and Rs = 1 for
312
H. Arhchoui et al. / Superlattices and Microstructures 49 (2011) 307–313
a
b
c
Fig. 4. Transition temperature TC as a function of surface dilution parameter q for Rs = 1.5 (a), Rs = 2 (b), Rs = 2.5 (c) and for different numbers of layers L.
has been investigated for different values of dilution parameter q. The temperature dependence of polarization p decreases from the saturation value at T = 0 and reduces to zero at T = TC . On the other hand, the critical temperature Tc increases when the value of L increases. The phase diagrams as function of dilution parameter q has been investigated for different values J of the surface exchange interaction Rs = JS and for a fixed value of the transverse field Ω . We found that the special point Ps, at which the critical temperature is independent of the film thickness L, dependents on the exchange interaction Rs . Note that in this article we neglected the long range interaction between ferroelectric dipoles, which is known to lead to formation of non-uniform ferroelectric domains [20–22] and vortex-like structures [23,24]. Incorporation of such effects into transverse Ising model and clarification of the surface effects would be the next step of the proposed microscopic consideration. Acknowledgements This work was supported by France–Morocco collaborative program ‘‘VOLUBILIS’’ and by FP7 IRSES mobility project ‘‘ROBOCON’’. References [1] J.F. Scott, Ferroelectrics Memories, Springer, New York, 2000. [2] M. Dawber, K.M. Rabe, J.F. Scott, Rev. Modern Phys. 77 (2005) 1083.
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