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Force induced phase transition of honeycomb-structured ferroelectric thin film
Physica A 392 (2013) 3570–3577
Contents lists available at SciVerse ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Force induced phase transition of honeycomb-structured ferroelectric thin film C.D. Wang a,b,∗ , R.G. Ma b a
Center of Super-Diamond and Advanced Films (COSDAF), City University of Hong Kong, Hong Kong Special Administrative Region
b
Department of Physics and Materials Science, City University of Hong Kong, Hong Kong Special Administrative Region
highlights • Phase diagrams of the honeycomb ferroelectric thin film depend sensitively on the force. • Three different functions for the exchange interaction under the external force are proposed. • Force effects on pseudo-spin polarization in 3D and 2D phase diagrams are revealed.
article
info
Article history: Received 19 February 2013 Available online 20 April 2013 Keywords: Ferroelectrics Force Phase transition
abstract The external force induced phase transition properties of a honeycomb-structured ferroelectric thin film described by a spin- 21 transverse Ising model were investigated by use of the effective field theory with correlations (EFT). The forces were introduced into the systems by performing three different interaction functions. The results demonstrate that characteristic phase diagrams and phase regions are strongly dependent on the added external force. In particular, the whole process of phase transition from ferroelectric to paraelectric and then back to ferroelectric was revealed in a 3-D (dimensional) phase diagram vividly for the first time suggesting that forces can be used to control pseudo-spin polarizations in the hexagonal lattice sensitively. © 2013 Elsevier B.V. All rights reserved.
1. Introduction In recent years, with the development of the technology of growth and characterization of nanostructured systems, ferroelectrics has been attracted considerable interest both experimentally and theoretically. In experiments, by using scanning force microscopy (SFM), Gruverman et al. [1] and Shvartsman et al. [2] studied the asymmetric nanoscale switching and nonlinear local piezoelectric deformation in ferroelectric thin films, respectively. Furthermore, piezoresponse force microscopy was developed to study the retention loss in the ferroelectric (SrBi2 Ta2 O9 )-insulator (HfO2 )-silicon structure by Zhang et al. [3], and by combining it with electron backscatter diffraction, Lowe et al. [4] revealed the crystallographic mapping of ferroelectric thin films. In addition, atomic force microscopy (AFM) was also adopted to investigate the ferroelectric thin films, for example, Wang et al. [5] and Molotskii et al. [6] revealed that AFM can induce an electric field and generate ferroelectric domains in ferroelectric thin films. In theory, the transverse Ising model (TIM) is generally considered to describe the phase transition of order–disorder type ferroelectrics, as it can successfully describe numerous physical systems, such as magnetic spin systems, binary alloys, lattice gases, and even proteins. In ferroelectric thin films or its superlattices, the surface and size effects, the influences
∗ Correspondence to: Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong Special Administrative Region. Tel.: +852 27844366. E-mail address: [email protected] (C.D. Wang). 0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.04.017
C.D. Wang, R.G. Ma / Physica A 392 (2013) 3570–3577
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Fig. 1. Schematic diagrams of the honeycomb-structured ferroelectric thin film. (a) The surface of the honeycomb-structured ferroelectric thin film. For only considering the nearest-neighbor lattices, to the black atom, we only need to take into account the exchange interactions between the three red atoms in this layer. (b) The cross-section of the five-layer honeycomb-structured ferroelectric thin film, which shows that after adding the external force, it was bent and the exchange interactions between different layers have been labeled as Ji (i = 1, . . . , 5). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
of exchange interaction and transverse field in the phase diagrams, dielectric properties of the ferroelectric thin film and the dynamical properties of the TIM were all systemically studied by mean-field approximation (MAF), Fermi-type Green’s function (using Bogolyubov’s equation of motion method), and Green’s function [7–13]. Recently, the effective field theory with correlations which is superior to the mean-field theory (MFA) and corresponds to the Zernike approximation (ZA) [14] was widely applied to investigate the phase transition properties and magnetic properties of ferroelectric nanowires and nanotubes [15–20]. Except for the critical phenomena in a mixed spin-1 and spin-2 Ising model on honeycomb and square lattices, the mixed spin-1 and spin- 32 Ising system with alternative layers of honeycomb were also investigated under first and second phase transitions. However, as far as we know there is seldom reported the force induced phase transition properties of honeycomb-structured ferroelectric thin films until now through micro-force there may be induced dramatic changes in ferroelectrics in experiments as mentioned above. In the present work, we investigate the influence of the external force on the phase diagrams of honeycomb-structured ferroelectric thin films under the EFT with correlations based on differential operators. In addition, the vivid processes of phase transition from order to chaos to order are demonstrated in 3-D diagrams. 2. Model and formulation The surface of the spin- 12 Ising model of the honeycomb-structured ferroelectric thin film is shown in Fig. 1(a). Fig. 1(b) shows the schematic cross-section diagram of the curved multilayer honeycomb-structured ferroelectric thin film under external added force. (Indeed, the honeycomb-structured ferroelectric thin film can be any number of layers. Here, for simplicity while maintaining its generality, we discuss the five-layer thin film.) The Hamiltonian of this system, described by the Ising model in a transverse field (TIM) can be expressed as follows [7–13]: H =−
1 2 ⟨i , j ⟩
Ji,j µZi µZj −
Ωi µxi ,
(1)
i
where Ωi is the transverse field, and µk (k = i, j) is the Ising spin operator with µk = ± 21 at sites i and j. J is the nearestneighbor spin pair interaction coupling constant of the intralayer or the adjacent spin pair of the interlayers. The indexes ⟨i, j⟩ denote summations over all pairs of nearest-neighbor spins. To the intralayer, Ji,j = Jζ , while to interlayers, Ji,j = Jζ +1 , where ζ denotes the ζ th layer of the film. When the external force was introduced into the ferroelectric thin film, the nearest-neighbor spin pair interaction coupling constant should be changed. Generally, as the thin film bent, the exchange interaction of the intralayer will be weakened. In addition the exchange interaction should also be a little different from the bottom layer to the top layer. Based on such considerations, we propose three different functions for the exchange interaction by introducing the external force as 1
one structure parameter: (i) Jn = J0 (1 + F1 )(1 + 1n ), (ii) Jn = J0 (1 + e− F )(1 + 1n ), and (iii) Jn = J0 (1 − ln(1 + F ))(1 + 1n ). By introducing the differential operator technique within the effective-field theory (EFT) [14–20], we can get the average polarization value of pseudo-spins in the ith (i = 1, 2, 3, . . . , n) layer as follows: mi = µZi
[cosh(Ji,j+δ ∇) + ξj+δ sinh(Ji,j+δ ∇)] fi (x)|x=0 = δ
(2)
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where runs over only the nearest neighbors of site i, and z is the lattice coordination number, ∇ = ∂∂x is the differential operator and one mathematical relation exists as the following:
exp(γ ∇)f (x) = (x + γ ).
(3)
The function fθ (x) is defined as: fθ (x) =
x
(2Ωθ )2 + x2
tanh
1 4β
(2Ωθ ) + 2
x2
(θ = s, i)
(4)
with β = k 1T (kB is the Boltzmann constant) and T as the absolute temperature. B For the honeycomb-structured ferroelectric thin film depicted in Fig. 1, the longitudinal (z-direction) polarizations of each curved layer under external force can be written as follows (here, to simplify the calculation while maintaining its generality, we discuss a five-layer honeycomb-structured ferroelectric thin film): m1 = [cosh(A) + m1 sinh(A)]3 × [cosh(B) + msinh (B)]f1 (x)|x=0 ,
(5)
m2 = [cosh(B) + m1 sinh(B)] × [cosh(B) + m2 sinh(B)] × [cosh(C ) + m3 sinh(C )]f2 (x)|x=0 ,
(6)
m3 = [cosh(C ) + m2 sinh(C )] × [cosh(C ) + m3 sinh(C )] × [cosh(D) + m4 sinh(D)]f3 (x)|x=0 ,
(7)
m4 = [cosh(D) + m3 sinh(D)] × [cosh(D) + m4 sinh(D)] × [cosh(E ) + m5 sinh(E )]f4 (x)|x=0 ,
(8)
m5 = [cosh(E ) + m4 sinh(E )] × [cosh(E ) + m5 sinh(E )] f5 (x)|x=0
(9)
3
3
3
3
where A, B, C , D and E are defined as A = J1 ∇, B = J2 ∇, C = J3 ∇, D = J4 ∇ and E = J5 ∇ . Based on Eqs. (5)–(9), we can obtain the following matrix by neglecting the nonlinear terms of mi :
3p1 − 1 p2
Q1 3p2 − 1 p3
Q2 3p3 − 1 p4
Q3 3p4 − 1 p5
= 0, Q4 3p5 − 1
(10)
with p1 = sinh(A) cosh2 (A) cosh(B)f1 (x)|x=0 ,
(11)
p2 = sinh(B) cosh (B) cosh(C )f2 (x)|x=0 ,
(12)
p3 = sinh(C ) cosh (C ) cosh(D)f3 (x)|x=0 ,
(13)
p4 = sinh(D) cosh (D) cosh(E )f4 (x)|x=0 ,
(14)
p5 = sinh(E ) cosh (E )f5 (x)|x=0 ,
(15)
Q1 = sinh(B) cosh (A)f1 (x)|x=0 ,
(16)
Q2 = sinh(C ) cosh (B)f2 (x)|x=0 ,
(17)
Q3 = sinh(D) cosh(C )f3 (x)|x=0 ,
(18)
Q4 = sinh(E ) cosh(D)f4 (x)|x=0 .
(19)
3
3
3
3
3
4
The phase transition properties of the honeycomb-structured ferroelectric thin film under the influence of the external force can be determined from Eq. (10) combined with branch Eqs. (11)–(19). In the following discussions, let us define the exchange interaction J as J = . 3. Numerical results and discussion As the external force could influence the exchange interaction between the nearest-neighbor lattices, it was introduced as an important parameter into the exchange interaction functions like the three interaction functions proposed above. In this section, we would like to investigate the influence of the external force in different phase diagrams. In the model, we have considered the difference of the exchange interaction . To be more accurate in the calculation, we also differentiate the transverse field between the surface and the inner layer as ΩS and Ωb . For comparison, the results of the usual MFA are also presented in some phase diagram figures. In addition, the whole dynamic variation process of the average pseudo-spin polarizations (P) will be depicted vividly. Fig. 2 shows the variation of the phase diagrams (Π ∼ TC ) as a function of external force, F in the honeycomb-structured ferroelectric thin film system with fixed values of ΩS /J = 1, Ωb /J = 2 and n = 5. In Fig. 2(a) and (c) (function (i) and function (iii)), it demonstrates clearly that with the increase of the force from 3 to 0.5, the range of ferroelectric phase (FE)
C.D. Wang, R.G. Ma / Physica A 392 (2013) 3570–3577
Fig. 2. Effect of external force on the phase diagram Π vs. TC with ΩS /J = 1, Ωb /J = 2 and n = 5 (Π = J0 ). (a) Jn = J0 (1 +
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1 F
)(1 + 1n ), (b) Jn =
1
J0 (1 + e− F )(1 + 1n ), (c) Jn = J0 (1 − ln(1 + F ))(1 + 1n ). For comparison, except for the results of effective-field theory (EFT) (solid lines), the results of the mean-field approximation (MFA) (dashed lines) are also shown. All curves are for n = 5.
decreases sensitively. Note that as it will be outside the visible range when F = 3 for interaction function (iii), 1.05 was chosen instead of 3 while the other two force values are the same. Based on that, it reveals that the interaction function (iii) is more sensitive in the phase diagram compared with function (i) as increasing the force value from 1 to 1.05, the FE also shrinks dramatically. Fig. 2(b) presents the case of function (ii). It shows that with the increase of the force, the FE range increases which is in contrast to the other two cases as demonstrated in Fig. 2(a) and (c). Moreover, in comparison with Fig. 2(a)–(c), it is found that the variation of phase range is most significant in Fig. 2(c), suggesting that the force plays the most important role in the structure (iii) honeycomb-structured ferroelectric thin film. As a reference, the results based on MFA are also demonstrated as indicated by the dashed line, and the same tendency as that originated from the EFT is observed. It is obvious that MFA results overestimate the FE in the phase diagrams to some extent, which agrees with the previous literature [19,20].
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Fig. 3. Phase diagram of the honeycomb-structured ferroelectric thin film with Ωb /J = 4, Π /J = 3, and n = 5 plotted as a function of F for different ΩS . (a) Jn = J0 (1 +
1 F
1
)(1 + 1n ), (b) Jn = J0 (1 + e− F )(1 + 1n ), (c) Jn = J0 (1 − ln(1 + F ))(1 + 1n ).
Fig. 3 presents the effect of the force on the phase diagram of ΩS /J vs. TC under three different exchange interaction functions. In Fig. 3(a), the FE range decreases with increasing the added force from 3 to 8. Note that, here the FE ranges are all in bump-structure. For interaction function (ii) as shown in Fig. 3(b), it is in contrast to the above observations as the FE range enlarges with the added force increasing from 0.1 to 1 accompanied with a changing regular shape of the FE range to bump-structured shape slowly. In Fig. 3(c), i.e. interaction function (iii), it clearly demonstrates that with the increase of the force from 0.5 to 0.8, the FE range decreases sensitively with regular shape. From the above observations, it verifies the accuracy of the conclusions, as the tendency of the results of the three functions in Fig. 3 is consistent with those of Fig. 2. Furthermore, comparing the three phase diagrams, function (iii) is still the most sensitive one among the three functions, because the FE range changes dramatically as the force only changes slightly. Based on the observations, we will only talk about interaction function (iii) in the following text. Fig. 4(a) shows the phase diagram of ΩS /J vs. Ωb /J for different fixed values of Π0 /J. It is interesting to find that with the increase of Ωb , ΩS decreases, suggesting that it is easier to transfer from FE to PE at the moment. In addition, a critical
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Fig. 4. (a) Phase diagram of ΩS /J and Ωb /J with the fixed values of F /J = 0.8, T = 0.2 and n = 5 plotted as a function of Π /J. The exchange interaction function was chosen as the most sensitive one, Jn = J0 (1 − ln(1 + F ))(1 + 1n ). (b) The linear relationship between the crossover value Ωbc /J and Π /J.
point for ΩS = 0 was observed, while over it, ΩS increased with increasing Ωb . Such asymmetrical ΩS vs. Ωb dependence for different interactions is distinct from the general ferroelectric thin films [21]. Meanwhile, with the increase of Π0 , the critical value Ωb increases, which implies that large exchange interactions would result in transferring from FE to PE more difficultly. The observation corresponds to the result of MFA in Ref. [10]. Fig. 4(b) presents the relationship between the crossover values Ωbc and Π obtained from Fig. 4(a). The fitting curve shows a standard linear relationship between Ωbc and Π , suggesting the phase transition is the competition between the exchange interaction and the transverse field. Fig. 5 presents the influence of force in the phase diagram of ΩS /J vs. Π /J for the type (iii) interaction function (Jn = J0 (1 − ln(1 + F ))(1 + 1n )). As depicted in Fig. 5, the range under the curve is FE while the outside part is PE. It reveals that the FE range decreases with the force increasing from 0.4 to 1 in this type of phase diagram (ΩS /J vs. Π /J), the tendency of which is consistent with other types of phase diagrams as talked about in Figs. 2 and 3. Moreover, more vivid phase transitions, i.e. the dynamic variations process of pseudo-spin polarization were revealed in Fig. 6 for the interaction function Jn = J0 (1 − ln(1 + F ))(1 + 1n ). In Fig. 6(a), the 3-D curve is shown in a regular shape with four up-feet touching the four vertexes nearly, and ordered distinct layers (indicated by different colors) presented under lower temperature when F = 1.71827. It can be considered that all the pseudo-spins in the lattices of the ferroelectric thin film are parallel along one direction, which means that the polarization of the ferroelectric thin film is the largest one, P = 21 . When the force is increased to 1.71828, the regular four up-feet curve start to deform, suggesting that some chaos vibrations are introduced into the ferroelectric system, the corresponding polarization is a little decreased (0 < P ≤ 21 ). As the force is further increased to 1.718281, the four up-feet curve vibrates dramatically (Fig. 6(c)), however, the shape can still be identified which verifies that more chaos vibration (PE) exists in the current FE, while the system is still under the domination of FE, suggesting that the polarization has been dramatically decreased from 12 (0 < P ≪ 21 ). It is interesting to find that F = 1.71828 seems to be the critical value of the phase transition between FE and PE. As the 3-D curve (Fig. 6(d)) is symmetrically distributed up and down, it is difficult to differentiate the tendency of the geometry to be up or down, suggesting it is in a completely chaotic state (PE, P = 0). When the force is added to 1.7182822 as shown in Fig. 6(e), some peripheral parts move up while some move down. The prototype of polarization (P < 0) can be well distinguished accompanied with the enlarged shape of the graph in Fig. 6(e) compared with Fig. 6(d). As the force is further increased to 1.7182823 in Fig. 6(f), the 3-D curve grows to strip-type along
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Fig. 5. Effect of external force on the transition range in the phase diagram of ΩS /J vs. Π /J with Ωb /J = 2, T = 2, n = 5 and Jn = J0 (1 − ln(1 + F ))(1 + 1n ).
Fig. 6. The whole variation process of the average pseudo-spin polarizations shown in 3-D (dimension) under Π /J vs. ΩS /J , Ωb /J for Jn = J0 (1 − ln(1 + F ))(1 + 1n ) with n = 5, T = 10: (a) F = 1.71827, (b) F = 1.71828, (c) F = 1.718281, (d) F = 1.718282, (e) F = 1.7182822, (f) F = 1.7182823, (g) F = 1.7182829, (h) F = 1.71829, (i) F = 1.7184.
C.D. Wang, R.G. Ma / Physica A 392 (2013) 3570–3577
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Ωb , the downward curve can be already well identified. The observation suggests that the polarization was further improved (− 21 < P < 0). In Fig. 6(g) (F = 1.7182823), except for further growth along Ωb , the curve also grows along ΩS . Up to this step, the chaotic pseudo-spin is suggested to turn to be order (− 21 ≤ P). As the force is further increased to 1.71829, the 3-D four down-feet can be clearly observed with only one-strip vibration parallel to the Ωb axis. Up to now, it can be concluded that the system has been in FE (P ≈ − 21 ) only with very faint vibration. Finally, when the force is increased to 1.7184 (shown
in Fig. 6(i)), regular four down-feet touching the four nadirs of the axis with ordered distinct layers was demonstrated. In this step, the polarization of pseudo-spin, P = − 21 has been obtained, implying that the whole process of the phase transition from FE to PE, and then to FE again has been finished. 4. Conclusion In this work, the phase diagrams of the honeycomb-structured ferroelectric thin film described by a spin- 12 transverse Ising model have been examined by changing the external force introduced into the system by performing three different interaction functions under FET. The results reveal that the FE decreases sensitively with the external force enlarging both in Π vs. TC and ΩS vs. TC phase diagrams for type (i), and type (iii) interaction functions, while the tendency is in contrast for type (ii) interaction functions. For comparison, results under the MFA are also presented in some phase diagrams, suggesting that the differential operator technique with correlations reduces some extent of the ferroelectric features of the ferroelectrics. For a fixed force and temperature, the crossover value of the transfer field in type (iii) interaction functions was found and the linear relationship between the crossover value of the transfer field and the constant exchange field was revealed. In addition, the force reducing the phase region for the type (iii) interaction function was particularly depicted in the ΩS vs. Π diagram. In particular, the whole process of the phase transition from FE (P = 21 ) to PE (P = 0), and then
back to FE (P = − 21 ) was revealed both in 3-D phase diagrams vividly for the first time, which suggests that force can be an essential parameter in control of pseudo-spin polarizations in the honeycomb-structured ferroelectric thin film. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
A. Gruverman, A. Kholkin, A. Kingon, H. Tokumoto, Appl. Phys. Lett. 78 (2001) 2751. V.V. Shvartsman, N.A. Pertsev, J.M. Herrero, C. Zaldo, A.L. Kholkin, J. Appl. Phys. 97 (2005) 104105. Z.H. Zhang, X.L. Zhong, Y. Zhang, J.B. Wang, C.J. Lu, W.N. Ye, Y.C. Zhou, Europhys. Lett. 98 (2012) 27011. M. Lowe, T. Hegarty, K. Mingard, J. Li, M. Cain, J. Phys.: Conf. Ser. 126 (2008) 012011. B. Wang, C.H. Woo, J. Appl. Phys. 94 (2003) 4053. M. Molotskii, J. Appl. Phys. 97 (2005) 014109. H.K. Sy, J. Phys.: Condens. Matter 5 (1993) 1231. C.L. Wang, W.L. Zhong, P.L. Zhang, J. Phys.: Condens. Matter 3 (1992) 4743. B.H. Teng, H.K. Sy, Phys. Rev. B 70 (2004) 104115. C.D. Wang, B.H. Teng, X.J. Zhang, Z.X. Lu, L.W. Liu, D.M. Cheng, X.H. Lu, Physica A 388 (2009) 1472. J.M. Wesselinowa, Solid State Commun. 121 (2002) 489. J.M. Wesselinowa, St. Kovachev, Phys. Rev. B 75 (2007) 045411. J.M. Wesselinowa, Phys. Status Solidi B 231 (2002) 187. F. Zernike, Physica 7 (1940) 565. T. Kaneyoshi, J. Magn. Magn. Mater. 323 (2011) 2483. T. Kaneyoshi, Solid State Commun. 151 (2011) 1528. T. Kaneyoshi, Phys. Status Solidi B 248 (2011) 250. T. Kaneyoshi, J. Magn. Magn. Mater. 322 (2010) 3014. C.D. Wang, Z.Z. Lu, W.X. Yuan, S.Y. Kwok, B.H. Teng, Phys. Lett. A 375 (2011) 3405. M. Keshin, N. Sarh, B. Devorem, Solid State Commun. 151 (2011) 1025. T. Kaneyoshi, Physica A 319 (2003) 355.
Contents lists available at SciVerse ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Force induced phase transition of honeycomb-structured ferroelectric thin film C.D. Wang a,b,∗ , R.G. Ma b a
Center of Super-Diamond and Advanced Films (COSDAF), City University of Hong Kong, Hong Kong Special Administrative Region
b
Department of Physics and Materials Science, City University of Hong Kong, Hong Kong Special Administrative Region
highlights • Phase diagrams of the honeycomb ferroelectric thin film depend sensitively on the force. • Three different functions for the exchange interaction under the external force are proposed. • Force effects on pseudo-spin polarization in 3D and 2D phase diagrams are revealed.
article
info
Article history: Received 19 February 2013 Available online 20 April 2013 Keywords: Ferroelectrics Force Phase transition
abstract The external force induced phase transition properties of a honeycomb-structured ferroelectric thin film described by a spin- 21 transverse Ising model were investigated by use of the effective field theory with correlations (EFT). The forces were introduced into the systems by performing three different interaction functions. The results demonstrate that characteristic phase diagrams and phase regions are strongly dependent on the added external force. In particular, the whole process of phase transition from ferroelectric to paraelectric and then back to ferroelectric was revealed in a 3-D (dimensional) phase diagram vividly for the first time suggesting that forces can be used to control pseudo-spin polarizations in the hexagonal lattice sensitively. © 2013 Elsevier B.V. All rights reserved.
1. Introduction In recent years, with the development of the technology of growth and characterization of nanostructured systems, ferroelectrics has been attracted considerable interest both experimentally and theoretically. In experiments, by using scanning force microscopy (SFM), Gruverman et al. [1] and Shvartsman et al. [2] studied the asymmetric nanoscale switching and nonlinear local piezoelectric deformation in ferroelectric thin films, respectively. Furthermore, piezoresponse force microscopy was developed to study the retention loss in the ferroelectric (SrBi2 Ta2 O9 )-insulator (HfO2 )-silicon structure by Zhang et al. [3], and by combining it with electron backscatter diffraction, Lowe et al. [4] revealed the crystallographic mapping of ferroelectric thin films. In addition, atomic force microscopy (AFM) was also adopted to investigate the ferroelectric thin films, for example, Wang et al. [5] and Molotskii et al. [6] revealed that AFM can induce an electric field and generate ferroelectric domains in ferroelectric thin films. In theory, the transverse Ising model (TIM) is generally considered to describe the phase transition of order–disorder type ferroelectrics, as it can successfully describe numerous physical systems, such as magnetic spin systems, binary alloys, lattice gases, and even proteins. In ferroelectric thin films or its superlattices, the surface and size effects, the influences
∗ Correspondence to: Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong Special Administrative Region. Tel.: +852 27844366. E-mail address: [email protected] (C.D. Wang). 0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.04.017
C.D. Wang, R.G. Ma / Physica A 392 (2013) 3570–3577
3571
Fig. 1. Schematic diagrams of the honeycomb-structured ferroelectric thin film. (a) The surface of the honeycomb-structured ferroelectric thin film. For only considering the nearest-neighbor lattices, to the black atom, we only need to take into account the exchange interactions between the three red atoms in this layer. (b) The cross-section of the five-layer honeycomb-structured ferroelectric thin film, which shows that after adding the external force, it was bent and the exchange interactions between different layers have been labeled as Ji (i = 1, . . . , 5). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
of exchange interaction and transverse field in the phase diagrams, dielectric properties of the ferroelectric thin film and the dynamical properties of the TIM were all systemically studied by mean-field approximation (MAF), Fermi-type Green’s function (using Bogolyubov’s equation of motion method), and Green’s function [7–13]. Recently, the effective field theory with correlations which is superior to the mean-field theory (MFA) and corresponds to the Zernike approximation (ZA) [14] was widely applied to investigate the phase transition properties and magnetic properties of ferroelectric nanowires and nanotubes [15–20]. Except for the critical phenomena in a mixed spin-1 and spin-2 Ising model on honeycomb and square lattices, the mixed spin-1 and spin- 32 Ising system with alternative layers of honeycomb were also investigated under first and second phase transitions. However, as far as we know there is seldom reported the force induced phase transition properties of honeycomb-structured ferroelectric thin films until now through micro-force there may be induced dramatic changes in ferroelectrics in experiments as mentioned above. In the present work, we investigate the influence of the external force on the phase diagrams of honeycomb-structured ferroelectric thin films under the EFT with correlations based on differential operators. In addition, the vivid processes of phase transition from order to chaos to order are demonstrated in 3-D diagrams. 2. Model and formulation The surface of the spin- 12 Ising model of the honeycomb-structured ferroelectric thin film is shown in Fig. 1(a). Fig. 1(b) shows the schematic cross-section diagram of the curved multilayer honeycomb-structured ferroelectric thin film under external added force. (Indeed, the honeycomb-structured ferroelectric thin film can be any number of layers. Here, for simplicity while maintaining its generality, we discuss the five-layer thin film.) The Hamiltonian of this system, described by the Ising model in a transverse field (TIM) can be expressed as follows [7–13]: H =−
1 2 ⟨i , j ⟩
Ji,j µZi µZj −
Ωi µxi ,
(1)
i
where Ωi is the transverse field, and µk (k = i, j) is the Ising spin operator with µk = ± 21 at sites i and j. J is the nearestneighbor spin pair interaction coupling constant of the intralayer or the adjacent spin pair of the interlayers. The indexes ⟨i, j⟩ denote summations over all pairs of nearest-neighbor spins. To the intralayer, Ji,j = Jζ , while to interlayers, Ji,j = Jζ +1 , where ζ denotes the ζ th layer of the film. When the external force was introduced into the ferroelectric thin film, the nearest-neighbor spin pair interaction coupling constant should be changed. Generally, as the thin film bent, the exchange interaction of the intralayer will be weakened. In addition the exchange interaction should also be a little different from the bottom layer to the top layer. Based on such considerations, we propose three different functions for the exchange interaction by introducing the external force as 1
one structure parameter: (i) Jn = J0 (1 + F1 )(1 + 1n ), (ii) Jn = J0 (1 + e− F )(1 + 1n ), and (iii) Jn = J0 (1 − ln(1 + F ))(1 + 1n ). By introducing the differential operator technique within the effective-field theory (EFT) [14–20], we can get the average polarization value of pseudo-spins in the ith (i = 1, 2, 3, . . . , n) layer as follows: mi = µZi
[cosh(Ji,j+δ ∇) + ξj+δ sinh(Ji,j+δ ∇)] fi (x)|x=0 = δ
(2)
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where runs over only the nearest neighbors of site i, and z is the lattice coordination number, ∇ = ∂∂x is the differential operator and one mathematical relation exists as the following:
exp(γ ∇)f (x) = (x + γ ).
(3)
The function fθ (x) is defined as: fθ (x) =
x
(2Ωθ )2 + x2
tanh
1 4β
(2Ωθ ) + 2
x2
(θ = s, i)
(4)
with β = k 1T (kB is the Boltzmann constant) and T as the absolute temperature. B For the honeycomb-structured ferroelectric thin film depicted in Fig. 1, the longitudinal (z-direction) polarizations of each curved layer under external force can be written as follows (here, to simplify the calculation while maintaining its generality, we discuss a five-layer honeycomb-structured ferroelectric thin film): m1 = [cosh(A) + m1 sinh(A)]3 × [cosh(B) + msinh (B)]f1 (x)|x=0 ,
(5)
m2 = [cosh(B) + m1 sinh(B)] × [cosh(B) + m2 sinh(B)] × [cosh(C ) + m3 sinh(C )]f2 (x)|x=0 ,
(6)
m3 = [cosh(C ) + m2 sinh(C )] × [cosh(C ) + m3 sinh(C )] × [cosh(D) + m4 sinh(D)]f3 (x)|x=0 ,
(7)
m4 = [cosh(D) + m3 sinh(D)] × [cosh(D) + m4 sinh(D)] × [cosh(E ) + m5 sinh(E )]f4 (x)|x=0 ,
(8)
m5 = [cosh(E ) + m4 sinh(E )] × [cosh(E ) + m5 sinh(E )] f5 (x)|x=0
(9)
3
3
3
3
where A, B, C , D and E are defined as A = J1 ∇, B = J2 ∇, C = J3 ∇, D = J4 ∇ and E = J5 ∇ . Based on Eqs. (5)–(9), we can obtain the following matrix by neglecting the nonlinear terms of mi :
3p1 − 1 p2
Q1 3p2 − 1 p3
Q2 3p3 − 1 p4
Q3 3p4 − 1 p5
= 0, Q4 3p5 − 1
(10)
with p1 = sinh(A) cosh2 (A) cosh(B)f1 (x)|x=0 ,
(11)
p2 = sinh(B) cosh (B) cosh(C )f2 (x)|x=0 ,
(12)
p3 = sinh(C ) cosh (C ) cosh(D)f3 (x)|x=0 ,
(13)
p4 = sinh(D) cosh (D) cosh(E )f4 (x)|x=0 ,
(14)
p5 = sinh(E ) cosh (E )f5 (x)|x=0 ,
(15)
Q1 = sinh(B) cosh (A)f1 (x)|x=0 ,
(16)
Q2 = sinh(C ) cosh (B)f2 (x)|x=0 ,
(17)
Q3 = sinh(D) cosh(C )f3 (x)|x=0 ,
(18)
Q4 = sinh(E ) cosh(D)f4 (x)|x=0 .
(19)
3
3
3
3
3
4
The phase transition properties of the honeycomb-structured ferroelectric thin film under the influence of the external force can be determined from Eq. (10) combined with branch Eqs. (11)–(19). In the following discussions, let us define the exchange interaction J as J = . 3. Numerical results and discussion As the external force could influence the exchange interaction between the nearest-neighbor lattices, it was introduced as an important parameter into the exchange interaction functions like the three interaction functions proposed above. In this section, we would like to investigate the influence of the external force in different phase diagrams. In the model, we have considered the difference of the exchange interaction . To be more accurate in the calculation, we also differentiate the transverse field between the surface and the inner layer as ΩS and Ωb . For comparison, the results of the usual MFA are also presented in some phase diagram figures. In addition, the whole dynamic variation process of the average pseudo-spin polarizations (P) will be depicted vividly. Fig. 2 shows the variation of the phase diagrams (Π ∼ TC ) as a function of external force, F in the honeycomb-structured ferroelectric thin film system with fixed values of ΩS /J = 1, Ωb /J = 2 and n = 5. In Fig. 2(a) and (c) (function (i) and function (iii)), it demonstrates clearly that with the increase of the force from 3 to 0.5, the range of ferroelectric phase (FE)
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Fig. 2. Effect of external force on the phase diagram Π vs. TC with ΩS /J = 1, Ωb /J = 2 and n = 5 (Π = J0 ). (a) Jn = J0 (1 +
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1 F
)(1 + 1n ), (b) Jn =
1
J0 (1 + e− F )(1 + 1n ), (c) Jn = J0 (1 − ln(1 + F ))(1 + 1n ). For comparison, except for the results of effective-field theory (EFT) (solid lines), the results of the mean-field approximation (MFA) (dashed lines) are also shown. All curves are for n = 5.
decreases sensitively. Note that as it will be outside the visible range when F = 3 for interaction function (iii), 1.05 was chosen instead of 3 while the other two force values are the same. Based on that, it reveals that the interaction function (iii) is more sensitive in the phase diagram compared with function (i) as increasing the force value from 1 to 1.05, the FE also shrinks dramatically. Fig. 2(b) presents the case of function (ii). It shows that with the increase of the force, the FE range increases which is in contrast to the other two cases as demonstrated in Fig. 2(a) and (c). Moreover, in comparison with Fig. 2(a)–(c), it is found that the variation of phase range is most significant in Fig. 2(c), suggesting that the force plays the most important role in the structure (iii) honeycomb-structured ferroelectric thin film. As a reference, the results based on MFA are also demonstrated as indicated by the dashed line, and the same tendency as that originated from the EFT is observed. It is obvious that MFA results overestimate the FE in the phase diagrams to some extent, which agrees with the previous literature [19,20].
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Fig. 3. Phase diagram of the honeycomb-structured ferroelectric thin film with Ωb /J = 4, Π /J = 3, and n = 5 plotted as a function of F for different ΩS . (a) Jn = J0 (1 +
1 F
1
)(1 + 1n ), (b) Jn = J0 (1 + e− F )(1 + 1n ), (c) Jn = J0 (1 − ln(1 + F ))(1 + 1n ).
Fig. 3 presents the effect of the force on the phase diagram of ΩS /J vs. TC under three different exchange interaction functions. In Fig. 3(a), the FE range decreases with increasing the added force from 3 to 8. Note that, here the FE ranges are all in bump-structure. For interaction function (ii) as shown in Fig. 3(b), it is in contrast to the above observations as the FE range enlarges with the added force increasing from 0.1 to 1 accompanied with a changing regular shape of the FE range to bump-structured shape slowly. In Fig. 3(c), i.e. interaction function (iii), it clearly demonstrates that with the increase of the force from 0.5 to 0.8, the FE range decreases sensitively with regular shape. From the above observations, it verifies the accuracy of the conclusions, as the tendency of the results of the three functions in Fig. 3 is consistent with those of Fig. 2. Furthermore, comparing the three phase diagrams, function (iii) is still the most sensitive one among the three functions, because the FE range changes dramatically as the force only changes slightly. Based on the observations, we will only talk about interaction function (iii) in the following text. Fig. 4(a) shows the phase diagram of ΩS /J vs. Ωb /J for different fixed values of Π0 /J. It is interesting to find that with the increase of Ωb , ΩS decreases, suggesting that it is easier to transfer from FE to PE at the moment. In addition, a critical
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Fig. 4. (a) Phase diagram of ΩS /J and Ωb /J with the fixed values of F /J = 0.8, T = 0.2 and n = 5 plotted as a function of Π /J. The exchange interaction function was chosen as the most sensitive one, Jn = J0 (1 − ln(1 + F ))(1 + 1n ). (b) The linear relationship between the crossover value Ωbc /J and Π /J.
point for ΩS = 0 was observed, while over it, ΩS increased with increasing Ωb . Such asymmetrical ΩS vs. Ωb dependence for different interactions is distinct from the general ferroelectric thin films [21]. Meanwhile, with the increase of Π0 , the critical value Ωb increases, which implies that large exchange interactions would result in transferring from FE to PE more difficultly. The observation corresponds to the result of MFA in Ref. [10]. Fig. 4(b) presents the relationship between the crossover values Ωbc and Π obtained from Fig. 4(a). The fitting curve shows a standard linear relationship between Ωbc and Π , suggesting the phase transition is the competition between the exchange interaction and the transverse field. Fig. 5 presents the influence of force in the phase diagram of ΩS /J vs. Π /J for the type (iii) interaction function (Jn = J0 (1 − ln(1 + F ))(1 + 1n )). As depicted in Fig. 5, the range under the curve is FE while the outside part is PE. It reveals that the FE range decreases with the force increasing from 0.4 to 1 in this type of phase diagram (ΩS /J vs. Π /J), the tendency of which is consistent with other types of phase diagrams as talked about in Figs. 2 and 3. Moreover, more vivid phase transitions, i.e. the dynamic variations process of pseudo-spin polarization were revealed in Fig. 6 for the interaction function Jn = J0 (1 − ln(1 + F ))(1 + 1n ). In Fig. 6(a), the 3-D curve is shown in a regular shape with four up-feet touching the four vertexes nearly, and ordered distinct layers (indicated by different colors) presented under lower temperature when F = 1.71827. It can be considered that all the pseudo-spins in the lattices of the ferroelectric thin film are parallel along one direction, which means that the polarization of the ferroelectric thin film is the largest one, P = 21 . When the force is increased to 1.71828, the regular four up-feet curve start to deform, suggesting that some chaos vibrations are introduced into the ferroelectric system, the corresponding polarization is a little decreased (0 < P ≤ 21 ). As the force is further increased to 1.718281, the four up-feet curve vibrates dramatically (Fig. 6(c)), however, the shape can still be identified which verifies that more chaos vibration (PE) exists in the current FE, while the system is still under the domination of FE, suggesting that the polarization has been dramatically decreased from 12 (0 < P ≪ 21 ). It is interesting to find that F = 1.71828 seems to be the critical value of the phase transition between FE and PE. As the 3-D curve (Fig. 6(d)) is symmetrically distributed up and down, it is difficult to differentiate the tendency of the geometry to be up or down, suggesting it is in a completely chaotic state (PE, P = 0). When the force is added to 1.7182822 as shown in Fig. 6(e), some peripheral parts move up while some move down. The prototype of polarization (P < 0) can be well distinguished accompanied with the enlarged shape of the graph in Fig. 6(e) compared with Fig. 6(d). As the force is further increased to 1.7182823 in Fig. 6(f), the 3-D curve grows to strip-type along
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Fig. 5. Effect of external force on the transition range in the phase diagram of ΩS /J vs. Π /J with Ωb /J = 2, T = 2, n = 5 and Jn = J0 (1 − ln(1 + F ))(1 + 1n ).
Fig. 6. The whole variation process of the average pseudo-spin polarizations shown in 3-D (dimension) under Π /J vs. ΩS /J , Ωb /J for Jn = J0 (1 − ln(1 + F ))(1 + 1n ) with n = 5, T = 10: (a) F = 1.71827, (b) F = 1.71828, (c) F = 1.718281, (d) F = 1.718282, (e) F = 1.7182822, (f) F = 1.7182823, (g) F = 1.7182829, (h) F = 1.71829, (i) F = 1.7184.
C.D. Wang, R.G. Ma / Physica A 392 (2013) 3570–3577
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Ωb , the downward curve can be already well identified. The observation suggests that the polarization was further improved (− 21 < P < 0). In Fig. 6(g) (F = 1.7182823), except for further growth along Ωb , the curve also grows along ΩS . Up to this step, the chaotic pseudo-spin is suggested to turn to be order (− 21 ≤ P). As the force is further increased to 1.71829, the 3-D four down-feet can be clearly observed with only one-strip vibration parallel to the Ωb axis. Up to now, it can be concluded that the system has been in FE (P ≈ − 21 ) only with very faint vibration. Finally, when the force is increased to 1.7184 (shown
in Fig. 6(i)), regular four down-feet touching the four nadirs of the axis with ordered distinct layers was demonstrated. In this step, the polarization of pseudo-spin, P = − 21 has been obtained, implying that the whole process of the phase transition from FE to PE, and then to FE again has been finished. 4. Conclusion In this work, the phase diagrams of the honeycomb-structured ferroelectric thin film described by a spin- 12 transverse Ising model have been examined by changing the external force introduced into the system by performing three different interaction functions under FET. The results reveal that the FE decreases sensitively with the external force enlarging both in Π vs. TC and ΩS vs. TC phase diagrams for type (i), and type (iii) interaction functions, while the tendency is in contrast for type (ii) interaction functions. For comparison, results under the MFA are also presented in some phase diagrams, suggesting that the differential operator technique with correlations reduces some extent of the ferroelectric features of the ferroelectrics. For a fixed force and temperature, the crossover value of the transfer field in type (iii) interaction functions was found and the linear relationship between the crossover value of the transfer field and the constant exchange field was revealed. In addition, the force reducing the phase region for the type (iii) interaction function was particularly depicted in the ΩS vs. Π diagram. In particular, the whole process of the phase transition from FE (P = 21 ) to PE (P = 0), and then
back to FE (P = − 21 ) was revealed both in 3-D phase diagrams vividly for the first time, which suggests that force can be an essential parameter in control of pseudo-spin polarizations in the honeycomb-structured ferroelectric thin film. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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