Journal of Applied Physics

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7 March 2018 Volume 123 Number 9

jap.aip.org

Journal of

Applied Physics

Effects of temperature and electric field on order parameters in ferroelectric hexagonal manganites

DOI: 10.1063/1.5010063

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2018 03 03

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Effects of temperature and electric field on order parameters in ferroelectric hexagonal manganites

C. X.Zhang,¹K. L.Yang,^1,a)P.Jia,¹H. L.Lin,¹C. F.Li,¹L.Lin,¹Z. B.Yan,¹and J.-M.Liu^1,2

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

2Institute for Advanced Materials, Hubei Normal University, Huangshi 435002, China

(Received 23 October 2017; accepted 23 January 2018; published online 2 March 2018)

In Landau-Devonshire phase transition theory, the order parameter represents a unique property for a disorder-order transition at the critical temperature. Nevertheless, for a phase transition with more than one order parameter, such behaviors can be quite different and system-dependent in many cases. In this work, we investigate the temperature (T) and electric field (E) dependence of the two order parameters in improper ferroelectric hexagonal manganites, addressing the phase transition from the high-symmetryP63/mmcstructure to the polarP63cmstructure. It is revealed that the trimerization as the primary order parameter with two components: the trimerization amplitudeQand phaseU, and the spontaneous polarization P emerging as the secondary order parameter exhibit quite different stability behaviors against variousTandE. The critical exponents for the two param- etersQandPare 1/2 and 3/2, respectively. As temperature increases, the window for the electric fieldE enduring the trimerization state will shrink. An electric field will break the Z₂part of the Z₂Z3 symmetry. The present work may shed light on the complexity of the vortex-antivortex domain structure evolution near the phase transition temperature.Published by AIP Publishing.

https://doi.org/10.1063/1.5010063

I. INTRODUCTION

Very different from typical ferroelectrics where spontaneous polarization is the primary order parameter,^1,2improper ferroelectrics usually have the ferroelectricity that is induced by phase transitions such as structural transitions, charge orders, and so on, rather than the primary displacive mode.^3–8 Recently, these improper ferroelectrics have been receiving substantial attention due to the possible coexistence and inter-coupling of ferroelectricity and magnetism, i.e., multi- ferroicity.⁹ Among these improper ferroelectrics, hexagonal manganites (h-RMnO3: R¼rare earths) have been of particular concern, and the coupled lattice distortion and ferroelectric polarization are of high interest.^10–12 Especially, the finding of topological domain patterns in such systems,^13,14as shown in Fig.1(c)for guiding the eye, makesh-RMnO3the focus of research on multiferroics. For the method used to obtain (c), the reader can refer to previous works.^15,16

It is now well established that such a real space topological domain pattern in ferroelectrich-RMnO3essentially correlated with the trimerization-type structural phase transition.^13,17,18 Such a trimerization occurs below certain temperatureTsand it originates from the periodic tilting of the MnO₅trigonal bipyra- mids from the high-symmetryP63/mmc structure. This collec- tive tilting leads to the lattice distortion with the effective unit cell being three times the basic unit cell in size. Consequently, the lattice symmetry is lowered to the polar P63cm space group which accommodates the three degenerate trimerization antiphases (a,b,c). It is described by the Z3symmetry. Here, each antiphase can accommodate two equivalent polarization

states, markedP"andP#for the upward and downward polar- izations along thec-axis, respectively. Consequently, there are generated six interlocked structural antiphase and ferroelectric domains satisfying the so-called Z2Z3 symmetry. These domains would meet in a cloverleaf arrangement that can be characterized by two winding orders (a^þ, b^–,c^þ,a^–,b^þ, c^–) and (a^þ,c^–,b^þ,a^–, c^þ,b^–). Geometrically, these two orders correspond to a vortex and an antivortex which constitute a vortex-antivortex pair. This pair appears to be the basic unit for constructing the real space topological domain structure, as observed experimentally over the whole space. Such a topological domain pattern has been sufficiently discussed using the graph theory.¹⁹

In the past several years, various aspects of such inter- esting domain patterns have been intensively investigated, including the topological geometry, the microscopic mechanism, and the dynamics.^16,20–25 Nevertheless, it is a bit strange that less comprehensive studies on the phase transitions in the space of temperatureTand the electric fieldEin terms of order parameters have been reported. Landau theory on the structural and ferroelectric phase transitions in such systems must include two order parameters. One is the structural trimerization as described by the trimerization amplitude Q and the phase U and the other is the spontaneous polarization (P). For details, readers may refer to earlier lit- erature.^3,26Certainly, for a system with more than one order parameter, the temperature dependence of these order parameters and the critical behaviors have also been rarely investigated. It is of fundamental importance for understanding the mechanism of phase transitions occurring in such systems. In this paper, particular focus is placed on the TandEdepen- dence of the two order parameters in a mode improper

a)Author to whom correspondence should be addressed: klyang.phy@

gmail.com

0021-8979/2018/123(9)/094102/6/$30.00 123, 094102-1 Published by AIP Publishing.

JOURNAL OF APPLIED PHYSICS123, 094102 (2018)

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ferroelectric h-RMnO3. The coupled behaviors of the two order parameters in the phase transition sequence can be illustrated to some extent.

The remaining part of this article is organized in the following. In Sec.II, the phenomenological Landau theory used in this paper is introduced. The main results on the variation of the order parameters against varyingTandEwill be presented in Sec. III with relevant discussions, including the critical behaviors, although the critical behaviors as pre- dicted by Landau theory on second-order phase transitions are not accurate. A brief conclusion is given in Sec.IV.

II. LANDAU THEORY

We start from the well-established phenomenological Landau theory in which the two order parameters, trimerization and polarization (P), are included. As mentioned above, the trimerization can be described by its amplitude Q and phaseU, withQdescribing the tilting amplitude of a bipyra- mid triangular from thec-axis andUthe azimuthal angle of the tilting, as shown in Fig.1(b). Here, only the amplitudeQ is discussed and the phaseU is found to be independent of stimuli in many cases. In fact, it is understood that the phase U is determined by the trimerization geometry and it may take one of the six constant values (0,6p/3, 62p/3, p). A combination of the two order parameters can characterize the lattice distortion from the high-symmetryP63/mmcstructure to the polarP63cmstructure.^11,12In the framework of lattice dynamics, the trimerization can be viewed as the condensation of the zone-boundary K₃ mode and the ferroelectric polarization as the secondary order parameter can be characterized by the amplitude of the polarC₂^–mode. So,QandP can also represent the amplitude of the zone-boundary K₃ mode and the zone-center C2– mode, respectively. The

nonlinear coupling between the K₃ and C2– modes induces ferroelectricity in h-RMnO3.^11,12,18 This is the reason why the ferroelectricity in h-RMnO3 is categorized as the improper ferroelectricity. For many improper ferroelectrics, the two polarization states may not be degenerate, but here they are.

The expression of the system free energy (density) for an h-RMnO3system has been well established according to the transformation properties of the trimerization and the polarization based on Landau theory.^11,12In consideration of the effect ofTandEon the crystal in a mono-domain state, the free energy can be written in the following form:

F¼ a0

2Ts

ðTTsÞQ²þb0

4Q⁴þc0

6Q⁶þc⁰0

6 Q⁶cos 6U gQ³Pcos 3Uþg⁰

2Q²P²þaP

2P²EzP; (1)

where the term -EzPis the electrostatic energy that is rooted in the fact that only the polarization along thec(z)-axis exists in these ferroelectric hexagonal manganites and it is sufficient to consider the effect of the electric field along the z-axis Ez.^13,16 According to Artyukhin et al., Qand P in Eq. (1), respectively, represent the amplitude of the zone-boundary K₃mode and the zone-centerC₂^–mode. BothQandPwere measured in displacement, while the unit used in this work is pm. The real polarizationPcis related to the amplitude of the polar zone-centerC2–modeP byPc¼V¹Z^*P, where Z^*is the effective charge of the polar modeC₂^–and V is the volume of the unit cell. For the sake of simplification, the amplitude of the polar zone-centerC2–modePis used to scale the polarization in this work. Parameters a0, b0 and c0 are the constants for the free energy polynomial onQextended up to

FIG. 1. The lattice structure of ferroelectric hexagonal YMnO3 as seen from the side view (a) and the top view (b). The topological domain patterns (c) obtained from the phase-simulation method are shown to guide the eye.

For the method used to obtain (c), the reader can refer to previous works.^15,16

094102-2 Zhanget al. J. Appl. Phys.123, 094102 (2018)

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the sixth-order,c⁰₀ is the anisotropic coupling factor between QandU,gis the nonlinear coupling factor between modeK3

and modeC2–,g⁰is the coupling factor betweenQandP, and aPis the self-energy factor ofP.

We discuss the measure of the external electric field as done in earlier work.¹⁶ Here, Ez¼EE0is set, where E0

¼1.0 eV/A˚ and E is a dimensionless number. The realistic electric fieldEr is obtained byEr¼2EE0/(9.031e)¼22146 Ewith the unit of kV/cm. In this work, hexagonal YMnO₃is taken as the representative of hexagonal manganites RMnO, and the parameter values for YMnO₃ given by Artyukhin et al.¹²based on the first-principles calculations are taken in practical calculations. These parameters are shown in Table I. For hexagonal YMnO₃, the structural phase transition temperature is about 1270 K, which is used as the value ofTsin the following calculations.

We first concentrate on the possible ground states as given by Landau theory. According to the usually used ana- lyzing method in Landau theory,^3,15,16the equilibrium state of order parameters can be determined from the standard procedure by evaluating the conditions for minimization of free energy densityFwith respect to parameters Q,U, and P. Mathematically, this procedure is equivalent to solving the following set of equations:

@F

@Q¼ a0

Ts

ðTTsÞ þb0Q²þ ðc0þc⁰₀cos 6UÞQ⁴

3gQPcos 3Uþg⁰P²

Q¼0;

@F

@U¼ 2c ⁰₀Q⁶cos 3Uþ3gQ³P

sin 3U¼0;

@F

@P¼ gQ³cos 3Uþ g⁰Q²þaP

PEz¼0: (2) The solution of Eq. (2)can be written in the following form:

a0

Ts

ðTTsÞþb0Q²þðc0þc⁰₀ÞQ⁴3gQcos3U

gQ³cos3UþEz

g⁰Q²þaP

þg⁰ gQ³cos3UþEz

g⁰Q²þaP

!2

¼0;

U¼0;6p=3;62p=3;p;

P¼gQ³cos3UþEz

g⁰Q²þaP

; 8>

>>

<

>>

>: (3)

which can be divided into two types: one with upward polarization (P") and the other with downward polarization (P#), as shown below

a0

Ts

ðTTsÞ þb0Q²þ ðc0þc⁰₀ÞQ⁴3gQgQ³þEz

g⁰Q²þaP

þg⁰ gQ³þEz

g⁰Q²þaP

!2

¼0;

U¼0;62p=3 P¼ gQ³þEz

g⁰Q²þaP

ðP";P>0Þ;

8>

>>

<

>>

>:

a0

Ts

ðTTsÞ þb0Q²þ ðc0þc⁰₀ÞQ⁴3gQgQ³Ez

g⁰Q²þaP

þg⁰ gQ³Ez

g⁰Q²þaP

!2

¼0;

U¼6p=3;p;

P¼ gQ³Ez

g⁰Q²þaP

ðP#;P<0Þ: 8>

>>

<

>>

>:

(4) It can be seen from Eq.(4)that the variation ofQin the

P_" state exposed to a positive electric field is exactly the

same as that in the P# state exposed to a negative electric field, corresponding to a reversal of polarizationP. It accords with the fact that theP_"andP_#states are the two degenerate states described by the Z₂symmetry.

III. RESULTS AND DISCUSSION

A. Temperature dependence of order parameters As the electric field is absent, the following relationship betweenQandTcan be obtained from Eq.(4)

a0

Ts

ðTTsÞþb0Q²þðc0þc⁰₀ÞQ⁴3gQ gQ³ g⁰Q²þaP

þg⁰ gQ³ g⁰Q²þaP

!2

¼0; (5)

which is applicable to theP"andP#states. TheT-dependence of parameter Qcan be obtained by numerically solving Eq.

(5), and the correspondingT-dependence of parameterPcan be evaluated from Eq.(4).

Taking the P" state as an example, we check the T- dependences of parameters Q and P in the absence of an electric field, and the results are plotted in Figs.2(a)and2(b) where the solid dots are numerically calculated data and the solid lines represent the fitting using the scaling relationship for a second-order phase transition.

Apparently, theT-dependence ofQclearly demonstrates the typical second-order phase transition, where the

TABLE I. The values of physical parameters used in the present calculations (e.g., YMnO3).¹²

a0(eV A˚^–2) b0(eV A˚^–4) c0(eV A˚^–6) c⁰₀(eV A˚^–6) aP(eV A˚^–2) g(eV A˚^–4) g⁰(eV A˚^–4)

2.626 3.375 0.117 0.108 0.866 1.945 9.931

s^xQ(eV) s^yQ(eV) s^zQ(eV) s^xP(eV) s^yP(eV) s^zP(eV)

5.14 5.14 15.4 8.88 8.88 52.7

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continuous decaying near the critical temperature (Tc) can be well described by the well-established scaling law. Here,Q as the primary order parameter does fit well the power law with the scaling exponentbQ¼0.49666 0.0003, as shown in Fig.2(a), consistent with the theoretical prediction of 1/2 for a typical second-order phase transition. This consequence is reasonable considering the nature of Q as the primary order parameter, while this prediction seems not well applicable to the secondary order parameter P. In fact, if one looks at theP(T) data, as shown in Fig.2(b), the best fitting can be obtained only in the T-range far below the critical pointTc, noting that the critical point forPmust be the same as that for Qin the absence of an electric field (E¼0). It seems that the data atT ! Tcexhibit a behavior different from the 1/2-exponential behavior. We consider two specific cases in the absence of an electric field (E¼0)

P6g

g⁰Q6ðTcTÞ^b^Q

ðE¼0; Q6¼0 and g⁰Q² aPÞ P6 g

aP

Q³6ðTcTÞ^3b^Q ¼6ðTcTÞ^b^P; b_P¼3b_Q ðE¼0;Q!0Þ

8>

>>

><

>>

: (6)

where bPis the scaling exponent for order parameter P. It is thus clearly shown that the order parameterPfollows the same scaling behavior asQifg⁰Q²aPandQ6¼0. However, asT

!Tc-, one hasQ!0, leading to the second scaling law given in Eq. (6). In this case, one has the scaling exponent for P, bP¼3bQ¼3/2 in the absence of an electric field.

In fact, the calculated P(T) data over a broad T-range belowTc, as shown in Fig.2(b), can be approximately fitted using the power law, with the scaling exponentbP¼0.5984 60.0011. It is noted that this fitting produces big uncertain- ties, and such an exponent is not common for a typical second order phase transition. Subsequently, we come to the data very close toTcand the data are re-plotted in Fig.2(c), and the fitting using the power scaling relationship gives an exponent bP¼1.5 3/2, very well consistent with the prediction of Eq.(6).

Here, it noted that Landau theory itself may be inaccurate for correctly describing the critical behavior of a second-order phase transition. The scaling exponent of 1/2 is either not correct. The significance of the present results is to unveil a scaling behavior of the secondary order parameter different from that of the primary order parameter. In the qualitative sense, this is not strange since the two order parameters follow different symmetries on one hand, and on the other hand, polarization P as the secondary order parameter is coupled with the primary order parameter Qvia the specific term given in Eq.

(1), resulting in a different critical behavior. Another point that is noticeable is that the critical exponents forQandPare connected by the scaling law. In fact, the scaling behavior of the secondary order parameter in the system that has more one order parameter has rarely been investigated before. The result of this section provides a clue for answering such questions.

Here, it should also be mentioned that noT-dependence of parameterUis available, and this consequence is straight- forward since no such dependence is considered in Eq. (3).

Therefore, the trimerization phase U is independent of T.

Such a prediction may not be correct in the quantitative sense, but correct qualitatively.

B. Electric field dependence of order parameters Now, we investigate the effect of E on the two order parameters. The calculated results at three different temperatures in the case of two polarization states (P#andP") are plotted in Figs. 3(a)and 3(b). Two features can be highlighted.

First, it is clearly seen that for eachT, parameterQas a function ofEshows a distorted semi-cycle pattern and it is easy to understand that such a semi-cycle pattern shifts toward the E>0 side for theP"state and toward theE<0 side for theP#

state. Second, theE-window for stableQ-states becomes narrow with increasing T. This is also understandable since Q

FIG. 2. The calculated T-dependence and the corresponding fits of order parameters Q (a) and P (b) given by the exponential law. The fittings are performed over a proper temperature range below the critical temperature.

In order to obtain a more accurate critical behavior for the secondary order parameterP, a more accurate fitting is presented in (c), giving a critical exponent 3/2, three times as large as that of the primary order parameterQ.

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decreases gradually and theE-range in which a real solution ofQin Eq.(5)exists shrinks with increasingT.

For parameter P, it is a monotonously increasing function of E, as shown in Fig. 3(b). This function could be roughly linear in theE-range whereQ-variation is insignifi- cant. In the two ends, where Q varies sharply, the E-

dependence ofPbecomes severely nonlinear, and eventually the switching ofPoccurs when no more real solution ofQis available at the two ends.

C. Energy contours in the parameter space

Furthermore, it would be beneficial to discuss the variations of order parameters in the free energy space. This energy contour allows us to have an overall understanding of these variations. By substituting the solutions ofQandPgiven in Eq.(3)into Eq.(1), the free energy has the following form:

F¼ a0

2Ts

ðTTsÞQ²þb0

4Q⁴þ1

6ðc0þc⁰₀cos 6UÞQ⁶ 1

2

ðgQ³cos 3UþEzÞ² g⁰Q²þaP

: (7)

By transforming the polar coordinates Q and U into Cartesian coordinates (Qx, Qy) where Qx¼QcosU and Qy

¼QsinU,²⁶Eq.(7)can be rewritten as F¼ a0

2Ts

ðTTsÞðQ²_xþQ²_yÞ þb0

4ðQ²_xþQ²_yÞ²þc0

6ðQ²_xþQ²_yÞ³ þc⁰₀

6ðQ⁶_x15Q⁴_xQ²_yþ15Q²_xQ⁴_yQ⁶_yÞ 1

2

gðQ³_x3QxQ²_yÞ þEz

h i2

g⁰ðQ²_xþQ²_yÞ þaP

: (8)

The calculated energy contours in the (Qx,Qy) planes at differentTandEare summarized in Fig.4. First, the six-fold

FIG. 3. The order parametersQ(a) andP(b) as a function of electric field Eat different temperatures.

FIG. 4. The energy contours in the order parameter space for different temperatures and electric fields.

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symmetry of the contours atE¼0 can be clearly shown, and the six minima appear, corresponding to the six trimerization phases (U¼0,6p/3,62p/3,p). Second, the contour shrinks gradually with increasingTin terms of the energy depth and (Qx,Qy) values of the six valleys, which accords with the previous statement that the trimerization-type structural phase transition starts from the undistorted P63/mmc structure.

Third, for a positive field (E>0), the alternative three of the six valleys, corresponding to the three domains with negative polarization (P#), are equivalently lifted up, damaging the stability of these domains. This character is reversely shown if a negative field is applied. Therefore, the electric field would break the Z₂part of the Z₂Z3symmetry ofh-RMnO3.

Finally, it should be remarked that all the discussions on the phase transition inh-RMnO3manganites against temperature T and electric field E are based on Landau theory described by Eq.(1). This theory represents a simplified ver- sion of the system energy in the framework of spatial symmetry and antiphase-ferroelectric coupling. In recent works,^27,28it has been proposed that further symmetry breaking subsequent to the phase transition from the high-symmetryP63/mmcstructure to the polar P63cm structure is possible. Alternatively, other symmetry breaking paths from theP63/mmcstructure to the polarP63cmstructure may be favored. The corresponding Landau theory remains yet to be developed, due to insufficient experimental data on the physical properties. These possibili- ties deserve additional investigations in near future.

IV. CONCLUSION

The temperature and electric field dependence of the two order parameters for improper ferroelectric phase transitions in hexagonal YMnO₃ have been investigated by numerical calculations based on the phenomenological Landau theory. It has been revealed that the temperature dependence of the trimerization amplitudeQas the primary order parameter fol- lows the scaling behaviors near the critical point of a typical second order phase transition and the scaling exponent is 1/2, while the secondary order parameterPhas a differentT-temperature dependence, giving rise to a critical exponent 3/2.

This difference is suggested to be related to the coupling of the secondary order parameterPto the trimerization-like lattice distortion. The electric field window for the stabilized trimerization state becomes narrow as the temperature rises.

The Z₂part of the Z₂Z3symmetry inh-RMnO3may be bro- ken by applying an external electric field.

ACKNOWLEDGMENTS

This work was supported by the National Key Research Program of China (Grant No. 2016YFA0300101), the 973 projects of China (Grant No. 2015CB654602) and the National Natural Science Foundation of China (Grant Nos.

51431006, 51721001, 51332006, 11504048, and 11774106).

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