Dynamic hysteresis of tetragonal ferroelectrics

(1)

Dynamic hysteresis of tetragonal ferroelectrics: The resonance of 90°- domain switching

D. P. Chen and J.-M. Liu

Citation: Appl. Phys. Lett. 100, 062904 (2012); doi: 10.1063/1.3683549 View online: http://dx.doi.org/10.1063/1.3683549

View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v100/i6 Published by the American Institute of Physics.

Additional information on Appl. Phys. Lett.

Journal Homepage: http://apl.aip.org/

Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors

(2)

Dynamic hysteresis of tetragonal ferroelectrics: The resonance of 90

-domain switching

D. P. Chen¹and J.-M. Liu^1,2,a)

1Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China

2Institute of Advanced Materials, School of Physics, South China Normal University, Guangzhou 510006, China and International Center for Materials Physics, Chinese Academy of Sciences, Shenyang, China (Received 6 December 2011; accepted 22 January 2012; published online 8 February 2012) The dynamic hysteresis of ferroelectric lattice with 90-domain structure in response to time-varying electric field of frequencyxand amplitudeE0is investigated using Monte Carlo simulation based on the Ginzburg-Landau phenomenological theory. A resonance mode of the polarization switching at low frequency range, associated with cluster dipole switching, beside the dipole switching resonance mode, is revealed, characterized by two separate peaks in the hysteresis area spectrumA(x). It is indicated that the power law scaling behaviorsA(x)x^aforx!0 andA(x)x^bforx! 1 remain applicable.V^C 2012 American Institute of Physics. [doi:10.1063/1.3683549]

When a ferroelectric (FE) system below its Curie point (Tc) is submitted to a time-varying electric field E(t)¼E0sin(2pxt), where E0is the amplitude, x the frequency, andtis time, a counter-clock loop-like polarization (P)-Ehysteresis is generated in one cycle of theP-switching in response toE(t), as shown in Fig.1(a). This hysteresis is a hallmark of ferroelectricity, and its area A(x, E0) together with remnant polarization (Pr) and coercivity (Ec) serve as the three core parameters of FE performance.^1,2 The so- called area spectrumA(x) is usually complicated, given an E0at temperatureT, and is coined the dynamic hysteresis.^3,4 A typicalA(x) at a given E0 is shown by the dashed blue line in Fig.1(b), which exhibits a single-peaked pattern orig- inating from dipole switching.

In fact, attention to the dynamic hysteresis is of interest not only from fundamental physics but also for application significance.²SpectrumA(x) reflects the energy dissipating during one cycle ofP-switching and is a measure of the dynamics of phase transitions.^5,6For many FE applications, the speed of P-switching is receiving attentions,^7–13 and spectrum A(x) offers helpful information on the dynamics and kinetics ofP-switching. It is believed that theP-switching is realized via the nucleation-and-growth process based on the individual dipole switching events.^11,14,15 At E0>Ec, there appear a characteristic time s_n for domain nucleation and time s_g for domain growth, so that an effective time seff p

ðsnsgÞcan be defined to scale the switching over the whole lattice.^7,8It implies that a resonance of the switching in response toE(t) atx_max1/s_eff, whereA(x) reaches its maximal occurs. This spectrum is denoted asAdi(x), and the resonance mode may be called as thedipole switching mode.

Consequently, the following power law scaling behaviors are proposed:^5,16

AðxÞ ¼ þ

PdE)

AðxÞ /x^aE^m₀ atx!0

AðxÞ /x^bEⁿ₀ atx! 1; (1)

where exponentsa,b,m, andnare all positive and materials dependent. Yang et al.¹¹ extended the dynamic hysteresis model proposed by Nattermannet al.¹⁷for magnetic materials to FE materials in which quenched disorders and defects are inevitable. Thus, the P-switching is associated with domain relaxation, wall creep, and wall flow.^18–20

The dynamic hysteresis in realistic FE materials seems even more complicated. Experiments^7–13,21reported scattered power law exponents in the low-xand high-xlimits. These inconsistencies raise two questions: (1) the models used so far are over-simplified, excluding those essential contributions involved in FE materials.^22–24 (2) Structural and FE anomalies such as lattice defects and quenched disorders impose impact on the dynamic hysteresis.^11,24 Conventional theory on FE dynamic hysteresis stems from the spin Hamil- tonian for magnetic matters and deals with ferroic lattice without considering complicated domain structures. For tetragonal PbTiO₃ or BaTiO₃, the microstructure is dominant with the 90-domains. This specific structure adds an additional spatial scale and probably an additional excitation mode for P-switching: avalanche-like switching of those dipoles. As a straightforward argument, this mode, coined as the cluster switching mode, is schematically shown in Fig.

1(c) left column. The spectrum Ado(x) arising from this mode may be single-peaked too and plotted by the dashed dot line. The spectrum A(x) should be a sum ofAdi(x) and Ado(x).

In this letter we address this issue. We start from the Ginzburg-Landau phenomenological theory for a tetragonal FE lattice. In our calculation, we take lattice constant a0

(1.0 nm), Landau coefficient a₀ in the free energy, and polarizationP0of the object FE material atT¼T1far below Tcas reference, wherea0¼a10(T1Tc) anda10>0. All free energy terms are normalized by term ja0jP02 which has an energy unit. Thus, one has normalized T^*¼T/(TcT1) with T0¼Tc/(TcT1). We perform the Monte Carlo simulation on a two-dimensional (2D)LLsquare lattice with periodic boundary conditions (L and all spatial coordinates are normalized bya0), using the Metropolis algorithm. On each site, an electric dipole P(r)¼(Px,Py), normalized byP0, and an elastic displacement vector u(r)¼(ux,uy), normalized bya0,

a)Author to whom correspondence should be addressed. Electronic mail:

[email protected].

0003-6951/2012/100(6)/062904/5/$30.00 100, 062904-1 VC2012 American Institute of Physics

(3)

are imposed. The model and procedure of the simulation were described in earlier works,^22,24,25 and only a brief description is presented here. The free energyF in dimensionless form includes the Landau potentialfld, dipole gradient energy fgr, dipole-dipole interaction fdip, elastic energy fel, electrostrictive interactionfes, and electrostatic energyfep

F¼ ð

½fldþfgrþfdipþfelþfesþfepdv; (2) wherevis the volume unit. The Landau potentialfldextend- ing to the sixth-order is

fld ¼a₁ðP²_xþP²_yÞ þa₁₁ðP⁴_xþP⁴_yÞ þa₁₂P²_xP²_y þa₁₁₁ðP⁶_xþP⁶_yÞ þa₁₁₂ðP²_xP⁴_yþP⁴_xP²_yÞ; (3) wherea^*iis the normalized values of the potential coefficients a_i, anda^*₁¼a^*₁₀(T^*T0).

The lowest order gradient energy fgr with square sym- metry can be written as

fgr¼1

2g₁₁ðP²_x;xþP²_y;yÞ þg₁₂Px;xPy;yþ1

2g₄₄ðPx;yþPy;xÞ² þ1

2g⁰₄₄ðPx;yPy;xÞ²

Pi;j¼@Pi=@rj; ð4Þ

wherer_j¼(x,y) ifj¼x,y, andg^*11,g^*12,g44^*, and g⁰^*₄₄are the normalized values of the positive gradient coefficients g11, g12,g44andg⁰44.

The third term is the dipole-dipole interaction. In the dimensionless form, this term can be expressed as

fdip¼tdip

ð PðrÞPðr⁰Þ

jrr⁰j³ 3½PðrÞðrr⁰Þ½Pðr⁰Þðrr⁰Þ jrr⁰j⁵

" #

dr⁰; (5)

where dimensionless factor tdip¼(8peja0j)¹ with e the dielectric permittivity and we take tdip¼1;r andr⁰ are the spatial coordinates. This term favors an anti-parallel align- ment of dipoles, and it is of long-range. A realistic calculation is done by either Fourier transform or finite truncation treatment,²² while for 2D lattice this treatment is precise as long as the truncating distance R is big (R¼8 in our simulation).

The spatial strain field is facilitated with elastic energy felyielding

fel¼1

2C₁₁ðu²_x;xþu²_y;yÞ þC₁₂ux;xuy;yþ1 2C₄₄u²_x;y ui;j¼@ui=@rj ifi¼j

ui;j¼ ð@ui=@rjÞ þ ð@uj=@riÞif i6¼j;

(6)

withC11^*,C12^*, andC44^* are the normalized values of the elastic coefficients C11, C12, and C44. Given the linear quadratic coupling of local elastic strain with the local dipole, one has

fes¼ tes½ux;xðq₁₁P²_xþq₁₂P²_yÞ þuy;yðq₁₁P²_yþq₁₂P²_xÞ þq₄₄ux;yPxPy; (7)

wheretes is a dimensionless pre-factor,q^*11,q^*12, andq^*44are the normalized values of the electrostrictive coefficientsq11, q12, and q44. This term is the core ingredient favoring the 90-domain structure. Finally, electrostatic energy fep is given as

fep¼ ðExPxþEyPyÞ; (8) where electric field E¼(Ex, Ey) is normalized by term ja0jP0, takingEx¼E(t)¼E0sin(2pxt) andEy¼0.

We consider two cases: one case withtes¼1, and the lattice is dominated with the 90-domain structure; the other by settingtes¼0, so the single domain or 180-domain structure is preferred. The parameters for the simulation are listed in Table I and were used extensively in literature.^22,23 The Monte Carlo simulation procedure was described previ- ously,^22,24,25and no details are given here. The finite lattice size effect is negligible as long as lattice sizeLis much big- ger than the 90-domain width. The unit for frequency xis inversely measured by the Monte Carlo step (mcs) which counts L² dipole flip attempts. One mcs¹ corresponds to the characteristic ratex0for the dipole flip attempts, which is supposed to follow the thermal-activation law x0¼texp(F0/kBT), withF0the potential barrier andtthe frequency for the basic dipole excitation mode.

We first look at the case offes¼0. Extensive simulation on theP-Eloops at variousxandE0are performed, and the hysteresis evolves from thin and well saturated loop into square and fat loop until unsaturated round one, with increasing x, as shown in Fig.2(a)and discussed in literature.^5,7,9 In Fig. 3(a) are plotted the threeA(x) curves for this case.

All these curves show the single-peaked pattern, with the peak height and location shifting upward and rightward with increasingE0. From thelog-logplot (inset), one observes the power law scaling in both thex!0 andx! 1limits. In correspondence, the domain structures at several states

FIG. 1. (Color online) A schematic ofP-Ehysteresis (a), a typical area spectrum A(x) with its two components Adi(x) and Ado(x) (A(x)¼Adi(x)þAdo(x)) (b). Various states in the hysteresis are labeled with O, B, C, D, F, and G, respectively. The resonance peaks are marked with timessdiandsdo. (c) The proposed two modes forP-switching in a lattice with 90-domain structure: thecluster switching modeanddipole switching mode. Each arrow represents a dipole.

062904-2 D. P. Chen and J.-M. Liu Appl. Phys. Lett.100, 062904 (2012)

(4)

associated with theP-Eloop are presented in the left column of Fig. 3. While the initial state is occupied with 180- domains (state O), it is seen that state C hash0 and state F hashpover the whole lattice, in spite of the local fluctuations. These results show that theP-switching is dominated by thedipole switching mode and the resonance occurs at time s_di.

For fes>0, the lattice is dominated with the 90- domains. At tes¼1, the simulated loops at several x are shown in Fig.2(b), and the area spectra at three differentE0

are presented in Fig.3(b). No substantial difference in the loop pattern between the two cases tes¼0 and tes¼1 is shown. The minor differences are (1) the 90-domain structure shows a rounder loop pattern, while the single-domain lattice generates a more square-like loop; (2) with decreasing x, the nominal coercive field shows slower shrinking for the 90-domain structure than that for the single-domain structure.

The substantial difference between the two cases is illus- trated byA(x). Fortes¼1, it is surprising thatA(x) at each E0demonstrates two well defined peaks instead of one peak in the case oftes¼0, similar to the solid line in Fig.1(b). For details, first, the peak at the high-xside is located ats_di, similar to the case of tes¼0, while the low-x peak is located

roughly at s_do10⁵mcs. Time s_do would be longer if tes is larger. Second, the peaks ats_doands_diincrease in height and rightward with increasing E0, while this response is much more significant for the peak ats_dothan that ats_di, indicating that the dynamic response associated with the peak at s_dois non-robust. Third,A(x) at both the low-xand high-xlimits still follows the power-law scaling, as shown in the inset of Fig.3(b). The scaling exponents are roughlyE0-independent but stronglyT-dependent.

To understand the origin of peak ats_do, one snapshots the domain structures at several states. Referring to state C and state F, it is observed that even passing forward to state B and

TABLE I. Physical parameters chosen for the simulation.^22,23All these parameters appear in the text in the dimensionless form.

Parameter (unit) Value Parameter (unit) Value Parameter (unit) Value

L(a0) 64–256 T0(Tc/(TcT1)) 4.00 tes 0, 1

a^*₁₀(a₁₀) 0.10 a^*₁₁(a₁₁P²0/ja0j) 0.24 a₁₂^* (a₁₂P²0/ja0j) 4.50 a^*111(a111P⁴0/ja0j) 0.49 a^*112(a112P⁴0/ja0j) 1.20 g11^* (g11/a²0ja0j) 1.60 g^*12(g12/a²0ja0j) 0.00 g^*44(g44/a²0ja0j) 0.80 g⁰44^* (g⁰44/a²0ja0j) 0.80 C^*11(C11/ja0jP²0) 2.75 C^*12(C12/ja0jP²0) 1.79 C44^* (C44/ja0jP²0) 0.54

q^*11(q11/ja0j) 0.142 q^*12(q12/ja0j) 0.0074 q44^* (q44/ja0j) 0.0157

FIG. 2. (Color online) SimulatedP-Eloops for the FE lattice withtes¼0 (a) andtes¼1 (b). The value ofxfor each loop is labeled numerically with unit mcs¹.T^*¼0.05.

FIG. 3. (Color online) EvaluatedA(x) for (a)tes¼0 and (b)tes¼1 respectively, at three differentE0.T^*¼0.05. The snapshots of the FE domains for tes¼0 (left column) andtes¼1 (right column) at states O, C, and F (see Fig.

1(a). The domain orientations are marked by colors, and anglehis defined with respect to thex-axis. The blue and red arrows indicate the domain orientations.

(5)

state C from state O, the 90-domain pattern can be somehow maintained, blueprinted by the green stripes in state C. Inside these stripes are accommodated with a number of dipole clusters. At state F, the reversed domain structure still shows the yellow stripe-like pattern which is opposite in polarization to state C. Also, inside these stripes are accommodated with a number of dipole clusters which align in the direction opposite to those clusters in state C. This demonstrates that the dipoles in these clusters reverse in an avalanche/collective manner, i.e., the cluster switching mode rather than the individual dipole switching. This is an important result, revealing the cluster switching of the 90-domains as an additional sequence makes substantial contribution to the dynamic hysteresis. In fact, quite earlier experiment already revealed the so-called trans-domain cooperative switching in PbZr_xTi_1xO₃ceramics with the 90-domain structure.²⁶

To separate the contributions from the two modes, one consults to Fig.4(a)where the evaluatedA(x) fortes¼1 and tes¼0 atE0¼4.0 are plotted. Although the two spectra are different in the overall sense, their behaviors at the high-x limit remain identical. Keeping in mind that A(x)¼Adi(x)þAdo(x), one hasA(x)¼Adi(x) atx! 1. It is reasonable to assume thatAdi(x) at fes>0 has the same distribution asA(x) atfes¼0, although their magnitudes are different but proportional to each other. Therefore, Adi(x) andAdo(x) at tes¼1 can be simply obtained, as shown in Fig.4(b).

As expected, both Adi(x) and Ado(x) show the single- peaked pattern with the peaks located, respectively, at s_di ands_do. The ratios_do/s_diis630, indicating that the cluster switching occurs at much lowerxthan the dipole switching.

In addition, both Adi(x) and Ado(x) exhibit the power-law frequency scaling atx!0 andx! 1, respectively, with similar exponents aand b, thus allowing the same scaling behavior and exponents forA(x). More simulations on cases whereE(t) is applied along directions other than the x-axis

reveal similar behaviors although the scaling exponents may be different. These results suggest that thecluster switching mode can not be ignored for FE materials with 90-domain structure. However, as E0is extremely big orT is high, this mode will be much less pronounced, as identified in some experiments.^8–11 In evidencing the above argument, one looks at the effect of T. We present the A(x) data with E0¼4.0ja0jP0at severalT^*in Fig.5(a). It is seen that with increasingT^*, the two-peaked pattern transits gradually into single-peaked one. The peak or anomaly atsdonearly disap- pears atT^*¼0.30.

We evaluate the power-law scaling exponents (a,b) at different T^* and present them in Fig. 5(b). While b¼1.0 is roughly T-independent, a increases with increasing T^*, from a0.23 atT^*¼0.01 toa0.52 atT^*¼0.50, and then tends to be saturated. In fact, for the low-xbehavior, the high quality PbZr_xTi_1xO₃epitaxial thin films showed Ec(x)x^c with c increasing with increasingT, while the measuredPr(x) remains well saturated,¹¹consistent with the present simulation. Similar results were obtained in PbZr_xTi_1xO₃ceramics.^13,27Theoreti- cally, it was argued thataisT-dependent in the similar way as revealed experimentally.¹⁴On the other hand, exponentbis insensitive to T^* and b¼1.0 seems to be universal. In the present case, no substantial contribution from the cluster switching modein the high-xlimit is observed, thusb¼1.0 applies. It is also understood that thermal fluctuations would have more remarkable impact on thecluster switching mode than on the dipole switching mode, resulting in disappear- ance of the former mode in the highT^*range. Regarding the amplitude exponents m and n, it is obtained that exponent n2.0 applies to all the cases, which is nearlyT-independent either. However, given a fixed T, the evaluated m at fes>0 is smaller than that atfes¼0. It increases slightly with increasingT.

FIG. 4. (Color online) (a) SimulatedA(x) fortes¼0 andtes¼1. (b) Simu- latedA(x) fortes¼1, and its two componentsAdi(x) andAdo(x).T^*¼0.05 andE0¼4.0ja0jP0.

FIG. 5. (Color online) (a) SimulatedA(x) fortes¼1 at differentT^*as labeled. (b) As-evaluated scaling exponentsaandbas a function ofT^*(the dashed lines indicate the theoretically predicted exponents).E0¼4.0ja0jP0.

062904-4 D. P. Chen and J.-M. Liu Appl. Phys. Lett.100, 062904 (2012)

(6)

So far, no clear evidence supporting the existence of the cluster switching modeand its significance is available. How- ever, it is still possible to unveil some indirect evidences.

Consulting to available data on the dynamic hysteresis in several FE materials,^8–11,13 one finds that exponentais smaller whenE0is relatively lower. Keeping in mind that it is techni- cally difficult to access very low frequency for measuring A(x), those data used for evaluating exponentawere obtained atx>0.1 Hz. Thisx-range may overlap with the resonance frequency for thecluster switching mode. Therefore, the eval- uatedawould be smaller at lowerE0sinceAdo(x) becomes more remarkable. Surely, in realistic FE materials, the domain structure is more complicated than pure 90-domains, making an experimentally identifying ofAdo(x) tougher.

This work was supported by the Natural Science Foun- dation of China (11074113, 50832002), the National 973 Projects of China (2011CB922101, 2009CB623303), and the PAPD Program, China.

1M. Dawber, K. M. Rabe, and J. F. Scott,Rev. Mod. Phys.77, 1083 (2005).

2J. F. Scott,Ferroelectric Memories(Springer, Berlin, 2000).

3G. Bertotti,Hysteresis in Magnetism(Academic, New York, 1998).

4B. K. Chakrabarti and M. Acharyya,Rev. Mod. Phys.71, 847 (1999).

5M. Rao, H. R. Krishnamurthy, and R. Pandit,Phys. Rev. B42, 856 (1990).

6M. Acharyya and B. K. Chakrabarti,Phys. Rev. B52, 6550 (1995).

7J.-M. Liu, H. L. W. Chan, C. L. Choy, Y. Y. Zhu, S. N. Zhu, Z. G. Liu, and N. B. Ming,Appl. Phys. Lett.79, 236 (2001).

8B. Pan, H. Yu, D. Wu, X. H. Zhou, and J.-M. Liu,Appl. Phys. Lett.83, 1406 (2003).

9R. Yimnirun, R. Wongmaneerung, S. Wongsaenmai, A. Ngamjarurojana, S. Ananta, and Y. Laosiritaworn, Appl. Phys. Lett.90, 112906 (2007);

Appl. Phys. Lett.90, 112908 (2007).

10Y.-H. Kim and J.-J. Kim,Phys. Rev. B55, R11933 (1997).

11S. M. Yang, J. Y. Jo, T. H. Kim, J.-G. Yoon, T. K. Song, H. N. Lee, Z.

Marton, S. Park, Y. Jo, and T. W. Noh,Phys. Rev. B82, 174125 (2010).

12I. S. Poperechny, Y. L. Raikher, and V. I. Stepanov, Phys. Rev. B82, 174423 (2010).

13Y. Y. Guo, T. Wei, Q. Y. He, and J. M. Liu,J. Phys.: Condens. Matter21, 485901 (2009).

14Y. Ishibashi and H. Orihara,Integr. Ferroelectr.9, 57 (1995).

15X. Du and I. W. Chen, in Ferroelectric Thin Film VI, MRS Symposia Pro- ceedings (Materials Research Society, Pittsburgh, 1998), vol. 493, p. 311.

16M. Rao and R. Pandit,Phys. Rev. B43, 3373 (1991).

17T. Nattermann, V. Pokrovsky, and V. M. Vinokur,Phys. Rev. Lett.87, 197005 (2001).

18G. Gerra, A. K. Tagantsev, and N. Setter,Phys. Rev. Lett. 94, 107602 (2005).

19S. Jesse, B. J. Rodriguez, S. Choudhury, A. P. Baddorf, I. Vrejoiu, D.

Hesse, M. Alexe, E. A. Eliseev, A. N. Morozovska, J. Zhang, L. Q. Chen, and S. V. Kalinin,Nat. Mater.7, 209 (2008).

20J. Y. Jo, S. M. Yang, T. H. Kim, H. N. Lee, J. G. Yoon, S. Park, Y. Jo, M.

H. Jung, and T. W. Noh,Phys. Rev. Lett.102, 045701 (2009).

21N. Wongdamnern, A. Ngamjarurojana, Y. Laosiritaworn, S. Ananta, and R. Yimnirun,J. Appl. Phys.105, 044109 (2009).

22H. L. Hu and L. Q. Chen,Mater. Sci. Eng. A238, 182 (1997); B. L. Li and J. M. Liu,Phys. Rev. B73, 014107 (2006).

23J. Wang, Y. L. Li, L. Q. Chen, and T. Y. Zhang,Acta Mater.53, 2495 (2005).

24L. F. Wang and J. M. Liu,Appl. Phys. Lett. 89, 092909 (2006); Appl.

Phys. Lett.90, 062905 (2007);Appl. Phys. Lett.91, 092908 (2007).

25F. Xue, X. S. Gao, and J. M. Liu,J. Appl. Phys.106, 114103 (2009).

26W. W. Cao and C. A. Randall,J. Phys. Chem. Solids57, 1499 (1996).

27X. F. Chen, X. L. Dong, H. L. Zhang, N. B. Feng, H. C. Nie, F. Cao, G. S.

Wang, Y. Gu, and H. L. He,J. Appl. Phys.105, 096104 (2009).

Dynamic hysteresis of tetragonal ferroelectrics

Dynamic hysteresis of tetragonal ferroelectrics: The resonance of 90°- domain switching

Related Articles

Additional information on Appl. Phys. Lett.

Dynamic hysteresis of tetragonal ferroelectrics: The resonance of 90

-domain switching