Electric field induced phase transition in first order ferroelectrics with large zero point energy

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Physica A 387 (2008) 115–122 www.elsevier.com/locate/physa

Electric field induced phase transition in first order ferroelectrics with large zero point energy C.L. Wanga,, J.C. Lia, M.L. Zhaoa, J.L. Zhanga, W.L. Zhonga, C. Arago´b, M.I. Marque´sb, J.A. Gonzalob a

School of Physics and Microelectronics, State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, PR China b Departamento de Fı´sica de Materiales, Universidad Auto´noma de Madrid, 28049 Madrid, Spain Received 14 January 2007; received in revised form 8 August 2007 Available online 29 August 2007

Abstract An electric field induced phase transition in first order ferroelectrics with very large zero point energy is studied on the framework of the effective field approach. It is well known that when the zero point energy of a system is relatively large, the ferroelectric behaviour is depressed and no phase transition can be observed. The critical value Ocf of zero point energy for whom the phase transition disappears turns out to be dependant on the order of transition. For zero point energies larger than this critical value, a phase transition may be induced applying an external electric field. This temperature dependence of the induced polarization shows a discontinuous step when the applied electric field is weak, but becoming a continuous one at a strong applied electric field. Another critical value of zero point energy Ocp4Ocf is deduced for which no phase transition at all can be attained. r 2007 Elsevier B.V. All rights reserved. PACS: 77.80.Bh; 77.22.Gm; 05.70.Fh Keywords: Ferroelectrics; Phase transition; Zero point energy

1. Introduction The effective field approach has proved to be a simple but valuable way to describe phase transitions [1]. The main supposition of this model is that each individual dipole is influenced, not only by the applied electric field, but also by every dipole of the system. In its simplest version, which takes into account only dipole interactions, describes fairly well the main features of continuous ferroelectric phase transitions, named second order. The inclusion of quadrupolar and higher order terms into the effective field expression is necessary for describing the properties of discontinuous or first order phase transitions [2–4]. The effective field approach has turned out to be successful explaining the composition dependence of the Curie temperature in mixed ferroelectrics systems [5,6]. A quantum effective field approach has also been developed for phase transitions Corresponding author. Tel.: +86 531 88377035; fax: +86 531 88377031.

E-mail address: [email protected] (C.L. Wang). 0378-4371/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2007.08.033

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at very low temperature [7–12] in ferro-quantum paraelectric mixed systems. The depression of ferroelectricity by the zero point energy has been simulated using Monte Carlo methods in an Ising lattice for second order phase transitions [13]. In this paper, we will use the quantum effective field approach to investigate for the first time the influence of the zero point energy on first order phase transitions. When the zero point energy of the system is large enough and the ferroelectric order is suppressed, a phase transition-like temperature dependence of the polarization can be observed applying an electric field. The discussion will be focused on this kind of field induced phase transitions. 2. Effective field approach The effective field, as described in detail in Ref. [1], can be expressed as E eff ¼ E þ bP þ gP3 þ dP5 þ    ,

(1)

where E is the external electric field, and the following terms correspond to the dipolar, quadrupolar, octupolar, etc., interaction. In this work we keep just the first two terms, that is, dipolar and quadrupolar interaction which give account of the first order transition. From statistical considerations we obtain the implicit equation of state as     E eff m ðE þ bP þ gP3 Þm P ¼ Nm tanh ¼ Nm tanh , (2) kB T kB T where N is the number of elementary dipoles per unit volume, m is the electric dipole moment, kB is the Boltzmann’s constant, and T is the absolute temperature. The Curie temperature is given by kB T C ¼ bNm2 ! T C ¼

bNm2 . kB

The explicit form of the equation of state can be obtained from Eq. (2),   kB T P 1 tanh E¼  bP  gP3 . m Nm

(3)

In order to handle easier this expression we can introduce the following normalization: e

E ; bNm

p

P ; Nm

t

T kB T ¼ ; T C bNm2

g

gN 2 m2 , b

we can rewrite Eq. (3) as e ¼ t  tanh1 p  p  gp3 ,

(4)

and as in absence of external field e ¼ 0, p ¼ pS: t¼

pS þ gp3S . tanh1 pS

(5)

Fig. 1 shows the plot of the normalized spontaneous polarization pS versus normalized temperature obtained from Eq. (5) for several values of the parameter g. As it is shown in the discussion of the role of the quadrupolar interaction in the order of the phase transition (see Refs. [1–3]), values of g under 13 correspond to a second order, continuous phase transition and values higher than 13 indicate that the transition is discontinuous, that is, first order. In this case, a spontaneous polarization py exists at temperature Ty4TC, and then ty41, being Dt ¼ ty 1 the corresponding reduced thermal hysteresis, which is the signature of the first order transitions. In order to determine py(g), we derive in Eq. (5),  @t  ¼0 @pS py

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Fig. 1. Temperature dependence of the spontaneous polarization for different quadrupolar interaction coefficient g. The inset shows the polarization discontinuity py(g) and the thermal hysteresis temperature Dt(g) in a first order transition.

and we obtain py(g) from ð1  p2y Þð1 þ 3gp2y Þ tanh1 py  ðpy þ gp3y Þ ¼ 0.

(6)

Substituting py(g) in Eq. (5) again we obtain ty(g). The inset of Fig. 1 shows py(g) and Dt(g), respectively. We see that Dt grows almost linearly with g and that py approaches to a saturation value that, resolving the equation (6), turns out to be py ¼ 0.8894 when g-N. 3. Quantum effective field approach When a phase transition takes place at very low temperature it is necessary to consider quantum effects. The energy of the system is no longer the classical thermal energy kBT, but the corresponding energy of the quantum oscillator is E ¼ _o0 ð12 þ hniÞ, being E 0 ¼ _o0 =2 the zero point energy, and /nS the average number of states for a given temperature T. From this quantum energy expression we can obtain a new temperature scale TQ (see Ref. [11]) defined as   1 1 _o0 þ _o =k T , _o0  kB T Q ) T Q  2 e 0 B 1 2kB tanhð_o0 =2kB TÞ and then, the corresponding quantum normalized temperature, tQTQ/TC. If we introduce a new normalization for the zero point energy O

E0 _o0 =2 ¼ , bNm2 kB T C

so we can rewrite tQ ¼ O= tanhðO=tÞ,

(7)

and Eqs. (4) and (5) become, respectively, e ¼ tQ tanh1 p  p  gp3 ,

(8)

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Fig. 2. Temperature dependence of the spontaneous polarization at different zero point energy. All the curves correspond to a quadrupolar interaction coefficient g ¼ 0.8.

tQ 

O p þ gp3   ¼ S 1 S ; tanh O=t tanh pS

e ¼ 0.

(9)

From Eq. (9) we can find the temperature dependence of the spontaneous polarization for a given value of g (g ¼ 0.8, to ensure it is a first order transition) and different values of the parameter O. Fig. 2 plots pS(t). The influence of the zero-point energy is quite obvious: when it is small, the phase transition is still of normal first order one. As the zero point energy increases, both the transition temperatures and the spontaneous polarization decrease. The Curie temperature goes to zero for O ¼ 1, but not yet ty, neither the saturation spontaneous polarization does. From the definition of the normalized zero point energy O, we can see then that the Curie temperature goes to zero when the zero point energy is the same as the classical thermal energy kBTC. Imposing again the condition of the zero slope, ðqt=qpS Þpy ¼ 0, we can obtain py(O) (for g ¼ 0.8), and then ty(O), which must be zero when the ferroelectric behaviour will be completely depressed. In this way we work out the zero point energy critical value (Ocf ¼ 1.1236) that would not allow any ordered state. Furthermore, from the condition ty (Ocf,g) ¼ 0, we will find the relationship between the critical zero point energy Ocf and the strength of the quadrupolar interaction given by the coefficient g. Fig. (3) plots Ocf (g) that indicates that Ocf grows almost linearly with g, specially for larger values of g. This means that ferroelectrics with strong first order phase transition feature needs a relative large critical value of zero point energy to depress the ferroelectricity. 4. Field induced phase transition We have just shown that a ferroelectric material, with strong quadrupolar interaction, undergoes a first order transition ðg413Þ unless its zero point energy reaches a critical value, Ocf, because in such case the phase transition is inhibited. However, an induced phase transition must be reached by applying an external electric field. Let be a system with g ¼ 0.8, and O ¼ 1.6, that is a first order ferroelectric with a zero point energy above the critical value and, hence, no phase transition observed. And let us apply a normalized electric field e that will produce a polarization after Eq. (8). Fig. 4 plots p(t) for different values of the electric field. It can be seen that when it is weak (see curves corresponding to 0.01 and 0.02), the polarization attains quickly a saturation value, similar to what is found in quantum paraelectrics. The curve of e ¼ 0.03 (see the dashed line) is split into two parts. The lower part would represent a quantum paraelectric state, but the upper part stands for a kind of ferroelectric state. Therefore there exists a critical value between e ¼ 0.02 and 0.03, which is the minimum electric field for inducting a phase transition. For 0.04o eo0.06 the induced polarization curve

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Fig. 3. Quadrupolar interaction dependence of the critical zero point energy Ocf that depress ferroelectricity.

Fig. 4. Temperature dependence of the field induced polarization for O ¼ 1.15 and g ¼ 0.8. The parameter is the strength of the electric field.

shows a discontinuous step, but as the electric field increases, 0.07, 0.08 and so on, the polarization changes continuously from a large value at low temperature to a relative small value at high temperature, showing a continuous step. So there is another critical electric field somewhere in between 0.05o eo0.07, separates the discontinuous step and the continuous step of the induced polarization. In fact, this tri-critical point would be around e ¼ 0.06. It is also important to remark that the above-mentioned features of the field induced phase transitions have been observed in lead magnesium niobate [14], which is a well-known ferroelectric relaxor. For further understanding the field induced phase transitions, we plot in Fig. 5, the hysteresis loops obtained numerically from Eq. (8) and corresponding to the curves displayed in Fig. 4. Double hysteresis loops can be observed when the temperature is lower than a critical point, around t ¼ 0.6, suggesting that ferroelectricity can always be induced at very low temperature. When the temperature is higher than this critical value, there is no hysteresis loop and we can just observe a non-linear p–e behaviour (see for instance the case for t ¼ 0.7). However, when the electric field is lower than eE0.025 (as indicated by the

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Fig. 5. Hysteresis loops at different temperatures for O ¼ 1.15 and g ¼ 0.8. The dashed arrow indicates the minimum electric field needed to induce a ferroelectric state at t ¼ 0.

Fig. 6. Hysteresis loops at zero temperature for different zero point energy values. The critical value of Ocf is the minimum zero point energy for the system to have ferroelectricity, while the critical value of Ocp is the maximum zero point energy for the system to get field induced ferroelectricity.

dashed arrow in Fig. 5), no hysteresis loop can be observed. That is the case corresponding to the curves e ¼ 0.01 and 0.02 in Fig. 4. The critical electric field able to induce a phase transition logically increases with the increasing of temperature, so at lower temperature region in Fig. 4, we can always have field induced ferroelectricity when the applied electric field is strong enough. Let us investigate the hysteresis loops for g ¼ 0.8 and different zero point energy values at t ¼ 0. The corresponding quantum temperature, after Eq. (7), is tQ ðt ¼ 0Þ ¼ O and then, Eq. (8) becomes e ¼ O tanh1 p  p  gp3 .

(10)

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Fig. 6 plots the hysteresis loops calculated after Eq. (10). When the zero point energy is smaller than the critical value Ocf (Ocf ¼ 1.1236, in our case) a normal hysteresis loop is obtained, where the coercive field decreases as zero point energy increases. For the critical value we find a double hysteresis loop with zero coercive fields. But if we continue increasing the zero point energy, we arrive to a point where no hysteresis loop is found at all. That suggests that above this other critical value, be Ocp, there is no way to induce a phase transition, even applying a strong field, and the system remains always in a paraelectric state. From the analysis of the hysteresis loops in Fig. 6 we can determine the critical electric field needed to induce phase transition imposing the conditions,    2  qe q e ¼ 0; 40, qp pc qp2 and deriving in Eq. (10) we get 8 O > >  1  3gp2c ¼ 0 > > < 1  p2c ! , 2O > >  6g p 40 > c > : ð1  p2c Þ2

(11)

hence we obtain p2c ¼

ð3g  1Þ þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3g  1Þ2  6gðO  2Þ 6g

.

(12)

Substituting pc (g, O) obtained after Eq. (12) into Eq. (10) we can get the coercive field ec (pc) at zero temperature. Besides, as it is showed in Fig. 6, when the zero point energy attains its critical value Ocf , the coercive field turns out to be zero, so this is another way to check the quadrupolar interaction dependence of the critical zero point energy Ocf (g) as displayed in Fig. 3. It must be noted the full accordance between the two calculations. At the second critical value of the zero point energy, Ocp, the hysteresis loop becomes an inflexion e–p curve as it can also be observed in Fig. 6; so in this case both Eq. (11) must be equal to zero, and by solving the

Fig. 7. Phase diagram of zero point energy critical value Oc versus quadrupolar interaction coefficient g at zero temperature. The second order phase transition region is denoted as ‘‘FE:2nd order’’, ‘‘FE:1st order’’ indicates a normal first order transition, ‘‘iFE’’ is the region of induced ferroelectric phase and on the top, ‘‘PE’’ corresponds to the paraelectric phase.

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system of equations it results ð3g þ 1Þ2 , 12g sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3g  1 pcp ¼ : 6g

Ocp ¼

ð13Þ

The phase diagram at zero temperature is shown in Fig. 7. It displays the role of both, the quadrupolar interaction strength and the zero point energy, on first order phase transitions. On the top of this diagram we find just paraelectric state (PE), while on the bottom left it appears only second order phase transitions (FE:2nd order) that correspond to go13 and Op1. On the bottom right (g413) there are the first order transitions (FE:1st order) as the critical values of the zero point energy (Ocf 41) that depress the ferroelectricity grow monotonously with g. Above the curve Ocf (g) there are induced electric field phase transitions (first order also). They are limited by another curve Ocp (g) given by Eq. (13) indicating that no phase transition can be observed when the zero point energy of the system is greater than this value. 5. Conclusions To conclude, it is possible to induce a phase transition by applying an electric field in a first order quantum paraelectric material. It is possible as well to determine two critical values of the zero point energy, Ocf, that depress ferroelectricity, and Ocp above which it is impossible to induce any kind of phase transition independently of the value of the electric field applied. A phase diagram is presented to display the role of both the quadrupolar interaction strength and the zero point energy on the feature of phase transitions. Acknowledgements We gratefully acknowledge the financial support of National Basic Research Program of China, 2007CB607504, and Natural Science Fund of China, Grant 10474057. Also we acknowledge support from DGICyT through Grant FIS2004-00437. References [1] J.A. Gonzalo, Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics, second ed., World Scientific, Singapore, 2006. [2] B. Noheda, G. Lifante, J.A. Gonzalo, Ferroelectrics 15 (1993) 109. [3] J.A. Gonzalo, R. Ramı´ rez, G. Lifante, M. Koralewski, Ferroelectrics Lett. 15 (1993) 9. [4] B. Noheda, M. Koralewski, G. Lifante, J.A. Gonzalo, Ferroelectrics Lett. 17 (1994) 25. [5] R.A. Ali, C.L. Wang, M. Yuan, Y.X. Wang, W.L. Zhong, Solid State Commun. 129 (2004) 365. [6] C. Arago´, C.L. Wang, J.A. Gonzalo, Ferroelectrics 337 (2006) 233. [7] J.A. Gonzalo, Phys. Rev. B 39 (1989) 12297. [8] J.A. Gonzalo, Ferroelectrics 168 (1995) 1. [9] M. Yuan, C.L. Wang, Y.X. Wang, R.A. Ali, J.L. Zhang, Solid State Commun. 127 (2003) 419. [10] S.A. Prosandeev, W. Kleemann, B. Westwanski, J. Dec, Phys. Rev. B 60 (1999) 14489. [11] C. Arago, J. Garcia, J.A. Gonzalo, C.L. Wang, W.L. Zhong, X.Y. Xue, Ferroelectrics 301 (2004) 113. [12] C. Arago, M.I. Marques, J.A. Gonzalo, Jpn. J. Appl. Phys. 45 (2006) 5892. [13] C.L. Wang, J. Garcia, C. Arago, J.A. Gonzalo, M.I. Marques, Physica A 312 (2002) 181. [14] Z. Kutnjak, J. Petzelt, R. Blinc, Nature 441 (2006) 956.