- Email: [email protected]
Polarization reversal behavior of an asymmetric ferroelectric thin film
Chinese Journal of Physics 61 (2019) 80–85
Contents lists available at ScienceDirect
Chinese Journal of Physics journal homepage: www.elsevier.com/locate/cjph
Polarization reversal behavior of an asymmetric ferroelectric thin film
T
Cui Lian School of Electronic Information Engineering, Yangtze Normal University, Chongqing, 408100, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Asymmetric ferroelectric thin film Surface transition layer Polarization
Using the Landau–Khalatnikov equation of motion, the polarization reversal behavior in an asymmetric ferroelectric thin film has been studied. Our model first introduces a third power of polarization to describe the asymmetry of a ferroelectric thin film with surface transition layer, which originates from the difference between the surfaces. Interestingly, vertical drift of polarization switching behaviors was found in this system. The properties consisting of hysteresis loop, spontaneous polarization, switching current of an asymmetric ferroelectric thin film with surface transition layer are discussed.
1. Introduction Ferroelectric thin films as well as their artificially composite materials have attracted great research interest for many years because of their uncommon physical properties [1–3]. The size and surface effects are the longest-standing issues for meeting the need of miniaturization of ferroelectric materials [4]. Hence, the researchers are dedicated to studying the size and surface effects of the ferroelectric materials both experimentally and theoretically. In these works, Landau theory and transverse Ising model are often used to investigate the size and surface effects. The two surfaces of ferroelectric materials have always been assumed to be identical, namely, the symmetrical surfaces have been used. Nevertheless, asymmetrical situation are often met in practice due to the different substrates, impurities, top and bottom electrodes, defects etc. in the process of preparation for the film [5,6]. For example, the polarization has a larger value when it points up from substrate to air than that when it points down as reported in Ref. [7]. Experimentally, some interest abnormal phenomena of ferroelectric thin films appear, e.g. asymmetric hysteresis loop has been observed in lead-zirconate-titanate (PZT) film on silicon substrate [8] and PZT/Pt films [9]. But, on theory, the report about physical properties of an asymmetric ferroelectric thin film is very shortage. Thus, the limited understanding of physical mechanism of an asymmetric ferroelectric thin film motivates us to study the properties of asymmetric ferroelectric thin films. In particular, their polarization reverse behaviors are important in applications such as in ferroelectric memories. In this paper, we use the Landau–Khalatnikov equation of motion and introduce an odd power polarization term in the expression of the Landau free energy to study the polarization reversal properties of an asymmetric ferroelectric thin film. The vertical drift of the hysteresis loop was obtained at different parameters of the surface transition layer. As far as we know, the method and the interesting results obtained in our paper were not concerned in the reported references. 2. The model and theory Simulation of the polarization-reversal process has also been performed based on the Landau-type free energy for homogeneous
E-mail address: [email protected]. https://doi.org/10.1016/j.cjph.2019.06.016 Received 5 April 2019; Received in revised form 8 June 2019; Accepted 24 June 2019 Available online 06 August 2019 0577-9073/ © 2019 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
Chinese Journal of Physics 61 (2019) 80–85
L. Cui
Fig. 1. Geometric structure of a ferroelectric thin film under study.
system. Imperfections like impurities and defects play an important role in the switching process, but since it is difficult to include such imperfections within the continuum form of free energy. So in 1990, Ishibashi proposed a lattice model—free energy in a discrete form [10]. Basically, the study of the surface and size effects on the properties of ferroelectric thin films can be divided into two categories [11]. For the first one, the thin film is modeled by the stacking of layered ferroelectrics where the total energy can be expressed by the summation of Landau-type free energies of individual layers. The influence of the surface on the properties of ferroelectric thin films is embodied by the boundary condition, which is called extrinsic. For the second one, the transverse Ising model whose size effects are attributed to the collective cooperative effect of dipoles inside the film. It must be large enough to withstand against thermal agitation or erasure against external field, which is called intrinsic type. In our paper, we study the asymmetry surface effects of the film using the first category theory, within the framework of the Landau–Khalatnikov theory. The advantage of using Landau–Khalatnikov theory is that it can tackle rapid transient problems [12,13]. For example, a time scale is much shorter than the Maxwell relaxation time of the medium. Thus, the switching properties can be evaluated. Hence, the ferroelectric thin film with a surface transition layer located between two metallic electrodes is considered as the stacking of thin layers, as shown in Fig. 1. The properties of every layer with thickness Δz are homogeneous in planes parallel to the surface. The variation of polarization is only along the z direction. The number of this layer stack is N. Thus, the whole thickness of the ferroelectric thin film is L = N Δz . If the origin is located at the left surface, we set z = 0 then an arbitrary layer located at a position z can be recognized by the index i, i.e. z = iΔz (1 ≤ i ≤ N ). To study this system, an amendatory Landau-type free energy for the system is needed to be established. Based on the Ref. [14], for the sake of describing the asymmetry of the film due to the structure loss of inversion symmetry caused by two different surfaces, we introduce a third power of polarization for each layer in the Landau-type free energy and assume its coefficient to be function of position in order to reflect the asymmetric degree arose from the surfaces. So the free energy of the film can be expressed by Landautype free energy expression [14–16] N
G=
1
∑ ⎡ 2 A (T − Tc ) Pi2 − i=1
⎣
1 1 Φψ (z ) Pi3 + CPi4 + K (Pi − Pi − 1)2 − EPi ⎤, 3 4 ⎦
(1)
with the boundary condition [12]
⎛ dP ⎞ = 0 ⎝ dz ⎠
when z = 0 and z = L,
(2)
where Pi is the polarization of the nth layer; the coefficients A, C, and K are independent of temperature T and position z; Tc is the transition temperature of the bulk ferroelectric; E is the applied field. A new term (the second term) is added in Eq. (1) in order to reflect the asymmetric degree arose from the surface transition layer. Thereinto, Φ > 0 implies the positive polarization state is more stable than the negative polarization state. In this paper, we only discuss the case of Φ > 0. The distribution function ψ(z) reflects the effect of surface transition layer. The K (Pi − Pi − 1)2 term in Eq. (1) represents the coupling effect between neighboring layers. The polarization gradient (dP/dz)2 can be included in the finite difference expansion of the K (Pi − Pi − 1)2 term [16]. The time variation of the polarization during switching can be described by the Landau–Khalatnikov equation as follows:
γ
∂Pi ∂G =− = −A (T − Tc ) Pi + Φψ (z ) Pi 2 − CPi3 + K (Pi + 1 + Pi − 1 − 2Pi ) + E ∂t ∂Pi
(3)
where γ is the coefficient of viscosity, which causes a delay in domain motion. It is convenient to rescale the variables into dimensionless forms. We set fi = Pi/ P0 with P0 = ATc / C , tT = T / Tc , δ = B / B0 with B0 = ATc , ζ = z / ξ0 with ξ0 = K / ATc , l = L/ ξ0 , η = tATc / γ , φ = ΦP0/ ATc , k = K / ATc , σ = (ε0 ATc )−1, e = E / E0 with E0 = P0/ ε0 , where ɛ0 is the dielectric constant of vacuum. Finally, we can obtain the normalized equation below: 81
Chinese Journal of Physics 61 (2019) 80–85
L. Cui
∂fi ∂η
= (1 − tT ) fi + φψ (ζ ) fi2 − fi3 + k (fi + 1 + fi − 1 − 2fi ) + σe
(4)
The corresponding rescaling boundary condition is
⎛⎜ df ⎞⎟ = 0 ⎝ dζ ⎠
when ζ = 0 and ζ = l.
(5)
Another index j is introduced to represent the discretized time η = j Δη. The time dependent polarization at each layer is denoted as fi, j. Through the forward difference in time for the left-hand side of Eq. (4), the difference equation for fi, j can be written as:
fi, j + 1 = fi, j + Δη [(1 − tT ) fi, j + φψ (ζ ) fi, j 2 − fi, j 3 + k (fi + 1, j + fi − 1, j − 2fi, j ) + σe]
(6)
According to the finite difference for the derivative (df/dζ) in the second-order smallness, Eq. (5) becomes
ζ=0 ⎧ f0,1 = f1, j = f2, j ⎨ fN , j = fN − 1, j = fN − 2, j ζ = l ⎩
(7a,b)
The overall polarization of the whole film is defined as
fj =
1 N
N
∑ fi,j
(8)
i=1
and the switching current is determined by
Ij =
df j dη
(9)
We choose a simple form for the distribution function ψi(ζ), whose peculiar choice does not affect the generality of the results and conclusions: [15]
(
⎧ iΔζ − u ⎪ λ ψi (ζ ) = ⎨ ⎪0 ⎩
)
2
0≤i< u Δζ
u Δζ
≤i≤N
(10)
where u is the thickness of the surface transition layer. Parameter λ reflects the dissymmetric degree of the surfaces. As stated, the polarization in the initial state is the negative remnant polarization, that is to say, the initial polarization is at zero field; then at the time η = 0 the field is switched to a positive value. 3. Numerical results and discussions In the calculations, we focus on discussing the actions of surface transition layer on the properties of an asymmetric ferroelectric thin film. Fig. 2 shows the hysteresis loops of the film with different u and λ. The bold lines in Fig. 2(a) and (b) represent the case without surface transition layer, which corresponds to the absence of the third power of polarization in Eq. (1). We can see the loop shape is symmetric. When a surface transition layer exists, the distortion of the hysteresis loop appears and the whole hysteresis loop drifts along the perpendicular axis. Moreover, the vertical drift of the hysteresis loop is upward as the surface transition layer thickness increases or λ reduces. The essential reason is that in the positive polarization case it possesses a lower energy than that in the negative polarization case, as can be easily seen from free energy Eq. (1). The stronger the surface effect is, the lower energy the positive polarization state has. It is found that the shapes of the hysteresis loops are strongly dependent on the parameters of surface effect (u and λ). The time dependence of the average polarization is expounded in Fig. 3(a) and (b) with various u and λ. The curves for the average polarization versus the time vertically drift compared to that without surface transition layer. It can be clearly seen that with increasing u or decreasing λ, the curves for the time dependence of the average polarization vertically drift upwards. The reason is similar to that in Fig. 2. The switching current as a function of the time is shown in Fig. 4 with different u. One can distinctly see that only one peak of switching current versus the time emerges without surface transition layer. When the surface transition layer exists, two peaks of the switching current appear, corresponding to the polarization reversal of surface transition layer and interior bulk, respectively. This result is in accordance with that obtained by Chew et al. [17]. They pointed out that these two maxima can be clearly distinguished in a measurement of the total polarization current, for what one might expect from an experimental measurement of switching. Furthermore, the peaks of the switching current shift to the short time with augmenting u. The physical reason is that the surface transition layer with lower polarization is needed to take less time to reverse the negative polarization state. This result is in good agreement with that of Ref. [18]. 82
Chinese Journal of Physics 61 (2019) 80–85
L. Cui
Fig. 2. Hysteresis loops of the film with different values of (a) u; (b) λ.
4. Conclusions We have first introduced a third power of polarization in a ferroelectric thin film owing to asymmetric surfaces. The switching characteristics of polarization of an asymmetric ferroelectric thin film within the framework of the Landau–Khalatnikov theory have been discussed. The results show that the hysteresis loops, evolution with time of average polarization and switching current of an asymmetric ferroelectric thin film exhibit some interesting phenomena. The hysteresis loops present vertical drift when a surface transition layer exists. With enhancing the surface effect, the hysteresis loops drift vertically upwards. The curves for the evolution with time of average polarization also drift perpendicularly. The switching current of the asymmetric ferroelectric thin film presents two peaks, and the peaks shift to the short time for larger surface effect. Our results reveal the polarization reversal mechanism of an asymmetric ferroelectric thin film and provide a valuable guide in applications of ferroelectric thin films.
Declaration of Competing Interest None.
Acknowledgment This work has been supported by the Science Research Project for Advanced Talents of Yangtze Normal University (Grant no. 2018KYQD065). 83
Chinese Journal of Physics 61 (2019) 80–85
L. Cui
Fig. 3. Polarization as a function of the time for different values of (a) u; (b) λ.
Fig. 4. Time dependence of the switching current for different thickness of surface transition layer.
84
Chinese Journal of Physics 61 (2019) 80–85
L. Cui
References [1] Y. Zheng, W.J. Chen, X. Luo, B. Wang, C.H. Woo, Critical properties of nanoscale asymmetric ferroelectric tunnel junctions or capacitors, Acta Mater. 60 (2012) 1857. [2] J. Leist, H. Gibhardt, K. Hradil, G. Eckold, Switching behaviour of modulated ferroelectrics: the kinetics of the field induced lock-in transition in K2SeO4, J. Phys.: Condens Matter 23 (2011) 305901. [3] A. Muliana, Time dependent behavior of ferroelectric materials undergoing changes in their material properties with electric field and temperature, Inter. J. Sol. Stru. 48 (2011) 2718. [4] N. Ng, R. Ahluwalia, D.J. Srolovitz, Depletion-layer-induced size effects in ferroelectric thin films: A Ginzburg-Landau model study, Phys. Rev. B 86 (2012) 094104. [5] L.R. Zheng, C.L. Lin, W.-P. Xu, M. Okuyama, Vertical drift of P—E hysteresis loop in asymmetric ferroelectric capacitors, J. Appl. Phys. 79 (1996) 8634. [6] H.N. Lee, H.M. Christen, M.F. Chisholm, C.M. Rouleau, D.H. Lowndes, Strong polarization enhancement in asymmetric three-component ferroelectric superlattices, Nature 433 (2005) 395–399. [7] C.L. Wang, L. Zhang, W.L. Zhong, P.L. Zhang, Switching characters of asymmetric ferroelectric films, Phys. Lett. A 254 (1999) 297. [8] S. Sun, L. Liu, P.A. Fuierer, B.A. Tuttle, Depolarization and hysteresis loop asymmetry in PZT thin films with self polarization, Ferroelectrics 152 (1994) 187. [9] L.R. Cheng, C.L. Lin, T.P. Ma, Current-voltage characteristics of asymmetric ferroelectric capacitor, J. Phys. D 29 (1996) 457. [10] Y. Ishibashi, Structure and physical properties of domain walls, Ferroelectrics 104 (1990) 299. [11] V.C. Lo, K.T. Li, Thickness dependence of susceptibility and phase transition temperature in ferroelectric thin films, Phys. Stat. Sol. B 244 (2007) 783–793. [12] H.X. Cao, V.C. Lo, Z.Y. Li, Simulation of flexoelectricity effect on imprint behavior of ferroelectric thin films, Solid State Commun. 138 (2006) 404. [13] V.C. Lo, Z.J. Chen, Simulation of the effects of space charge and schottky barriers on ferroelectric thin film capacitor using Landau-Khalatnikov theory, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 49 (2002) 980. [14] C.L. Wang, L. Zhang, W.L. Zhong, P.L. Zhang, Switching characters of asymmetric ferroelectric films, Phys. Letts. A 254 (1999) 297. [15] T. Lü, W. Cao, Influence of imperfect surface on properties of ferroelectric thin film, Microelec. Engin. 66 (2003) 818. [16] V.C. Lo, Simulation of thickness effect in thin ferroelectric films using Landau—Khalatnikov theory, J. Appl. Phys. 94 (2003) 3353. [17] K.H. Chew, J. Osman, R.L. Stamps, D.R. Tilley, F.G. Shin, H.L.W. Chan, Surface aided polarization reversal in small ferroelectric particles, J. Appl. Phys. 93 (2003) 4215. [18] L. Cui, Q. Xu, Z.Y. Han, X. Xu, J.X. Che, H.J. Xue, T. Lü, Influences of surface transition layer on switching time and coercive field of a ferroelectric thin film, Solid State Sci. 16 (2013) 65.
85
Contents lists available at ScienceDirect
Chinese Journal of Physics journal homepage: www.elsevier.com/locate/cjph
Polarization reversal behavior of an asymmetric ferroelectric thin film
T
Cui Lian School of Electronic Information Engineering, Yangtze Normal University, Chongqing, 408100, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Asymmetric ferroelectric thin film Surface transition layer Polarization
Using the Landau–Khalatnikov equation of motion, the polarization reversal behavior in an asymmetric ferroelectric thin film has been studied. Our model first introduces a third power of polarization to describe the asymmetry of a ferroelectric thin film with surface transition layer, which originates from the difference between the surfaces. Interestingly, vertical drift of polarization switching behaviors was found in this system. The properties consisting of hysteresis loop, spontaneous polarization, switching current of an asymmetric ferroelectric thin film with surface transition layer are discussed.
1. Introduction Ferroelectric thin films as well as their artificially composite materials have attracted great research interest for many years because of their uncommon physical properties [1–3]. The size and surface effects are the longest-standing issues for meeting the need of miniaturization of ferroelectric materials [4]. Hence, the researchers are dedicated to studying the size and surface effects of the ferroelectric materials both experimentally and theoretically. In these works, Landau theory and transverse Ising model are often used to investigate the size and surface effects. The two surfaces of ferroelectric materials have always been assumed to be identical, namely, the symmetrical surfaces have been used. Nevertheless, asymmetrical situation are often met in practice due to the different substrates, impurities, top and bottom electrodes, defects etc. in the process of preparation for the film [5,6]. For example, the polarization has a larger value when it points up from substrate to air than that when it points down as reported in Ref. [7]. Experimentally, some interest abnormal phenomena of ferroelectric thin films appear, e.g. asymmetric hysteresis loop has been observed in lead-zirconate-titanate (PZT) film on silicon substrate [8] and PZT/Pt films [9]. But, on theory, the report about physical properties of an asymmetric ferroelectric thin film is very shortage. Thus, the limited understanding of physical mechanism of an asymmetric ferroelectric thin film motivates us to study the properties of asymmetric ferroelectric thin films. In particular, their polarization reverse behaviors are important in applications such as in ferroelectric memories. In this paper, we use the Landau–Khalatnikov equation of motion and introduce an odd power polarization term in the expression of the Landau free energy to study the polarization reversal properties of an asymmetric ferroelectric thin film. The vertical drift of the hysteresis loop was obtained at different parameters of the surface transition layer. As far as we know, the method and the interesting results obtained in our paper were not concerned in the reported references. 2. The model and theory Simulation of the polarization-reversal process has also been performed based on the Landau-type free energy for homogeneous
E-mail address: [email protected]. https://doi.org/10.1016/j.cjph.2019.06.016 Received 5 April 2019; Received in revised form 8 June 2019; Accepted 24 June 2019 Available online 06 August 2019 0577-9073/ © 2019 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
Chinese Journal of Physics 61 (2019) 80–85
L. Cui
Fig. 1. Geometric structure of a ferroelectric thin film under study.
system. Imperfections like impurities and defects play an important role in the switching process, but since it is difficult to include such imperfections within the continuum form of free energy. So in 1990, Ishibashi proposed a lattice model—free energy in a discrete form [10]. Basically, the study of the surface and size effects on the properties of ferroelectric thin films can be divided into two categories [11]. For the first one, the thin film is modeled by the stacking of layered ferroelectrics where the total energy can be expressed by the summation of Landau-type free energies of individual layers. The influence of the surface on the properties of ferroelectric thin films is embodied by the boundary condition, which is called extrinsic. For the second one, the transverse Ising model whose size effects are attributed to the collective cooperative effect of dipoles inside the film. It must be large enough to withstand against thermal agitation or erasure against external field, which is called intrinsic type. In our paper, we study the asymmetry surface effects of the film using the first category theory, within the framework of the Landau–Khalatnikov theory. The advantage of using Landau–Khalatnikov theory is that it can tackle rapid transient problems [12,13]. For example, a time scale is much shorter than the Maxwell relaxation time of the medium. Thus, the switching properties can be evaluated. Hence, the ferroelectric thin film with a surface transition layer located between two metallic electrodes is considered as the stacking of thin layers, as shown in Fig. 1. The properties of every layer with thickness Δz are homogeneous in planes parallel to the surface. The variation of polarization is only along the z direction. The number of this layer stack is N. Thus, the whole thickness of the ferroelectric thin film is L = N Δz . If the origin is located at the left surface, we set z = 0 then an arbitrary layer located at a position z can be recognized by the index i, i.e. z = iΔz (1 ≤ i ≤ N ). To study this system, an amendatory Landau-type free energy for the system is needed to be established. Based on the Ref. [14], for the sake of describing the asymmetry of the film due to the structure loss of inversion symmetry caused by two different surfaces, we introduce a third power of polarization for each layer in the Landau-type free energy and assume its coefficient to be function of position in order to reflect the asymmetric degree arose from the surfaces. So the free energy of the film can be expressed by Landautype free energy expression [14–16] N
G=
1
∑ ⎡ 2 A (T − Tc ) Pi2 − i=1
⎣
1 1 Φψ (z ) Pi3 + CPi4 + K (Pi − Pi − 1)2 − EPi ⎤, 3 4 ⎦
(1)
with the boundary condition [12]
⎛ dP ⎞ = 0 ⎝ dz ⎠
when z = 0 and z = L,
(2)
where Pi is the polarization of the nth layer; the coefficients A, C, and K are independent of temperature T and position z; Tc is the transition temperature of the bulk ferroelectric; E is the applied field. A new term (the second term) is added in Eq. (1) in order to reflect the asymmetric degree arose from the surface transition layer. Thereinto, Φ > 0 implies the positive polarization state is more stable than the negative polarization state. In this paper, we only discuss the case of Φ > 0. The distribution function ψ(z) reflects the effect of surface transition layer. The K (Pi − Pi − 1)2 term in Eq. (1) represents the coupling effect between neighboring layers. The polarization gradient (dP/dz)2 can be included in the finite difference expansion of the K (Pi − Pi − 1)2 term [16]. The time variation of the polarization during switching can be described by the Landau–Khalatnikov equation as follows:
γ
∂Pi ∂G =− = −A (T − Tc ) Pi + Φψ (z ) Pi 2 − CPi3 + K (Pi + 1 + Pi − 1 − 2Pi ) + E ∂t ∂Pi
(3)
where γ is the coefficient of viscosity, which causes a delay in domain motion. It is convenient to rescale the variables into dimensionless forms. We set fi = Pi/ P0 with P0 = ATc / C , tT = T / Tc , δ = B / B0 with B0 = ATc , ζ = z / ξ0 with ξ0 = K / ATc , l = L/ ξ0 , η = tATc / γ , φ = ΦP0/ ATc , k = K / ATc , σ = (ε0 ATc )−1, e = E / E0 with E0 = P0/ ε0 , where ɛ0 is the dielectric constant of vacuum. Finally, we can obtain the normalized equation below: 81
Chinese Journal of Physics 61 (2019) 80–85
L. Cui
∂fi ∂η
= (1 − tT ) fi + φψ (ζ ) fi2 − fi3 + k (fi + 1 + fi − 1 − 2fi ) + σe
(4)
The corresponding rescaling boundary condition is
⎛⎜ df ⎞⎟ = 0 ⎝ dζ ⎠
when ζ = 0 and ζ = l.
(5)
Another index j is introduced to represent the discretized time η = j Δη. The time dependent polarization at each layer is denoted as fi, j. Through the forward difference in time for the left-hand side of Eq. (4), the difference equation for fi, j can be written as:
fi, j + 1 = fi, j + Δη [(1 − tT ) fi, j + φψ (ζ ) fi, j 2 − fi, j 3 + k (fi + 1, j + fi − 1, j − 2fi, j ) + σe]
(6)
According to the finite difference for the derivative (df/dζ) in the second-order smallness, Eq. (5) becomes
ζ=0 ⎧ f0,1 = f1, j = f2, j ⎨ fN , j = fN − 1, j = fN − 2, j ζ = l ⎩
(7a,b)
The overall polarization of the whole film is defined as
fj =
1 N
N
∑ fi,j
(8)
i=1
and the switching current is determined by
Ij =
df j dη
(9)
We choose a simple form for the distribution function ψi(ζ), whose peculiar choice does not affect the generality of the results and conclusions: [15]
(
⎧ iΔζ − u ⎪ λ ψi (ζ ) = ⎨ ⎪0 ⎩
)
2
0≤i< u Δζ
u Δζ
≤i≤N
(10)
where u is the thickness of the surface transition layer. Parameter λ reflects the dissymmetric degree of the surfaces. As stated, the polarization in the initial state is the negative remnant polarization, that is to say, the initial polarization is at zero field; then at the time η = 0 the field is switched to a positive value. 3. Numerical results and discussions In the calculations, we focus on discussing the actions of surface transition layer on the properties of an asymmetric ferroelectric thin film. Fig. 2 shows the hysteresis loops of the film with different u and λ. The bold lines in Fig. 2(a) and (b) represent the case without surface transition layer, which corresponds to the absence of the third power of polarization in Eq. (1). We can see the loop shape is symmetric. When a surface transition layer exists, the distortion of the hysteresis loop appears and the whole hysteresis loop drifts along the perpendicular axis. Moreover, the vertical drift of the hysteresis loop is upward as the surface transition layer thickness increases or λ reduces. The essential reason is that in the positive polarization case it possesses a lower energy than that in the negative polarization case, as can be easily seen from free energy Eq. (1). The stronger the surface effect is, the lower energy the positive polarization state has. It is found that the shapes of the hysteresis loops are strongly dependent on the parameters of surface effect (u and λ). The time dependence of the average polarization is expounded in Fig. 3(a) and (b) with various u and λ. The curves for the average polarization versus the time vertically drift compared to that without surface transition layer. It can be clearly seen that with increasing u or decreasing λ, the curves for the time dependence of the average polarization vertically drift upwards. The reason is similar to that in Fig. 2. The switching current as a function of the time is shown in Fig. 4 with different u. One can distinctly see that only one peak of switching current versus the time emerges without surface transition layer. When the surface transition layer exists, two peaks of the switching current appear, corresponding to the polarization reversal of surface transition layer and interior bulk, respectively. This result is in accordance with that obtained by Chew et al. [17]. They pointed out that these two maxima can be clearly distinguished in a measurement of the total polarization current, for what one might expect from an experimental measurement of switching. Furthermore, the peaks of the switching current shift to the short time with augmenting u. The physical reason is that the surface transition layer with lower polarization is needed to take less time to reverse the negative polarization state. This result is in good agreement with that of Ref. [18]. 82
Chinese Journal of Physics 61 (2019) 80–85
L. Cui
Fig. 2. Hysteresis loops of the film with different values of (a) u; (b) λ.
4. Conclusions We have first introduced a third power of polarization in a ferroelectric thin film owing to asymmetric surfaces. The switching characteristics of polarization of an asymmetric ferroelectric thin film within the framework of the Landau–Khalatnikov theory have been discussed. The results show that the hysteresis loops, evolution with time of average polarization and switching current of an asymmetric ferroelectric thin film exhibit some interesting phenomena. The hysteresis loops present vertical drift when a surface transition layer exists. With enhancing the surface effect, the hysteresis loops drift vertically upwards. The curves for the evolution with time of average polarization also drift perpendicularly. The switching current of the asymmetric ferroelectric thin film presents two peaks, and the peaks shift to the short time for larger surface effect. Our results reveal the polarization reversal mechanism of an asymmetric ferroelectric thin film and provide a valuable guide in applications of ferroelectric thin films.
Declaration of Competing Interest None.
Acknowledgment This work has been supported by the Science Research Project for Advanced Talents of Yangtze Normal University (Grant no. 2018KYQD065). 83
Chinese Journal of Physics 61 (2019) 80–85
L. Cui
Fig. 3. Polarization as a function of the time for different values of (a) u; (b) λ.
Fig. 4. Time dependence of the switching current for different thickness of surface transition layer.
84
Chinese Journal of Physics 61 (2019) 80–85
L. Cui
References [1] Y. Zheng, W.J. Chen, X. Luo, B. Wang, C.H. Woo, Critical properties of nanoscale asymmetric ferroelectric tunnel junctions or capacitors, Acta Mater. 60 (2012) 1857. [2] J. Leist, H. Gibhardt, K. Hradil, G. Eckold, Switching behaviour of modulated ferroelectrics: the kinetics of the field induced lock-in transition in K2SeO4, J. Phys.: Condens Matter 23 (2011) 305901. [3] A. Muliana, Time dependent behavior of ferroelectric materials undergoing changes in their material properties with electric field and temperature, Inter. J. Sol. Stru. 48 (2011) 2718. [4] N. Ng, R. Ahluwalia, D.J. Srolovitz, Depletion-layer-induced size effects in ferroelectric thin films: A Ginzburg-Landau model study, Phys. Rev. B 86 (2012) 094104. [5] L.R. Zheng, C.L. Lin, W.-P. Xu, M. Okuyama, Vertical drift of P—E hysteresis loop in asymmetric ferroelectric capacitors, J. Appl. Phys. 79 (1996) 8634. [6] H.N. Lee, H.M. Christen, M.F. Chisholm, C.M. Rouleau, D.H. Lowndes, Strong polarization enhancement in asymmetric three-component ferroelectric superlattices, Nature 433 (2005) 395–399. [7] C.L. Wang, L. Zhang, W.L. Zhong, P.L. Zhang, Switching characters of asymmetric ferroelectric films, Phys. Lett. A 254 (1999) 297. [8] S. Sun, L. Liu, P.A. Fuierer, B.A. Tuttle, Depolarization and hysteresis loop asymmetry in PZT thin films with self polarization, Ferroelectrics 152 (1994) 187. [9] L.R. Cheng, C.L. Lin, T.P. Ma, Current-voltage characteristics of asymmetric ferroelectric capacitor, J. Phys. D 29 (1996) 457. [10] Y. Ishibashi, Structure and physical properties of domain walls, Ferroelectrics 104 (1990) 299. [11] V.C. Lo, K.T. Li, Thickness dependence of susceptibility and phase transition temperature in ferroelectric thin films, Phys. Stat. Sol. B 244 (2007) 783–793. [12] H.X. Cao, V.C. Lo, Z.Y. Li, Simulation of flexoelectricity effect on imprint behavior of ferroelectric thin films, Solid State Commun. 138 (2006) 404. [13] V.C. Lo, Z.J. Chen, Simulation of the effects of space charge and schottky barriers on ferroelectric thin film capacitor using Landau-Khalatnikov theory, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 49 (2002) 980. [14] C.L. Wang, L. Zhang, W.L. Zhong, P.L. Zhang, Switching characters of asymmetric ferroelectric films, Phys. Letts. A 254 (1999) 297. [15] T. Lü, W. Cao, Influence of imperfect surface on properties of ferroelectric thin film, Microelec. Engin. 66 (2003) 818. [16] V.C. Lo, Simulation of thickness effect in thin ferroelectric films using Landau—Khalatnikov theory, J. Appl. Phys. 94 (2003) 3353. [17] K.H. Chew, J. Osman, R.L. Stamps, D.R. Tilley, F.G. Shin, H.L.W. Chan, Surface aided polarization reversal in small ferroelectric particles, J. Appl. Phys. 93 (2003) 4215. [18] L. Cui, Q. Xu, Z.Y. Han, X. Xu, J.X. Che, H.J. Xue, T. Lü, Influences of surface transition layer on switching time and coercive field of a ferroelectric thin film, Solid State Sci. 16 (2013) 65.
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