Domain switching dynamics in topological antiferroelectric vortex domains
K. L. Yang ,1H. L. Lin,1L. Lin ,1,2Z. B. Yan,1J.-M. Liu,1and S.-W. Cheong3
1Laboratory of Solid State Microstructures and Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
2Department of Applied Physics, College of Science, Nanjing Forestry University, Nanjing 210037, China
3Rutgers Center for Emergent Materials and Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA
(Received 16 December 2021; revised 9 June 2022; accepted 30 June 2022; published 25 July 2022) Hexagonal manganites can exhibit intriguing ferroelectric (FE) and partially undistorted antiferroelectric (PUA) Z6-vortex domain patterns, which are dual to each other. Due to the same origin, it can be expected that the two real-space topological structures will behave the same way in terms of nonelectric properties such as domain topology and scaling of vortex density. However, their electric properties would be very different from each other, while these differences are rarely understood due to the absence of investigation on PUAZ6-vortex domain patterns. In this work, we study the response of the PUAP-3c1 structure to external electric fields and the switching dynamics of PUAZ6-vortex domain patterns by combining Landau theory and phase-field simulations. We find that an electric field along thecaxis can tune the trimerizing tilting angle of the MnO5
trigonal bipyramids in PUAP-3c1 structure while no such effect is present in the FEP63cmstructure. As a result, a continuous phase transition sequenceP-3c1→P3c1→P63cmcan be induced. For both the PUA Z6-vortex domain pattern and the single PUA domain, the variation of the total polarization with a sinusoidal electric field manifests as a double-hysteresis loop. We find out that this is because the transition fromP63cm back toP3c1 is a first-order transition. The switching dynamics of PUAZ6-vortex domain patterns is found to be very different from that observed on FEZ6-vortex domain patterns. Our results advance the understanding of real-space topological structures and suggest that the PUAP-3c1 structure may exhibit excellent performance in dielectric energy storage.
DOI:10.1103/PhysRevB.106.024110
I. INTRODUCTION
Materials with several coexisting orders such as electric, magnetic, charge, orbit, and chiral orders and so on, tend to exhibit extraordinary functionalities. The well-known ex- amples are the electric-field control of magnetization and magnetic-field control of polarization in multiferroics. Mul- tiferroics represent a broad class of polar insulators with coexisting ferroelectric (FE) order and magnetic order below certain temperatures, and promising magnetoelectric (ME) coupling has been promised [1–4]. Such application potentials have been expected for a decade since 2003 when BiFeO3as the representative type-I multiferroic was revisited and the magnetism-induced ferroelectricity in TbMnO3 as the first type-II multiferroic was discovered [5,6]. BiFeO3 exhibits excellent ferroelectricity but only negligible ME coupling. For TbMnO3 and other type-II systems, the magnetocontrol of polarization reversal was evidenced, but polarization was too small. Given these advantages and deficiencies, a third class of materials, the so-called improper magnetic ferroelectrics such as layered perovskites and hexagonal manganites and ferrites, are receiving attention [7–9].
The FE hexagonal manganites h-RMnO3 (R=Dy−Lu, In, Y, and Sc) can exhibit intriguing topological Z6-vortex domain pattern, which is generated by a trimerization- type structural phase transition [10,11]. All members in the h-RMnO3 family have similar crystal structures, and each
consists of layers of corner-sharing MnO5trigonal bipyramids separated by layers of Rions, as shown in Fig. 1(a). These members usually adopt the high-symmetry paraelectric (PE) aristo-type structure with P63/mmcsymmetry at T >TC∼ 1000 K, the phase transition point. On the other hand, the FE P63cmstructure is the ground state for most members. Due to the size mismatch between theRions in theRlayer and the MnO5 trigonal bipyramids in the Mn-O layer, the structural trimerization is triggered belowTC, resulting in a trimerizing tilting of the MnO5 trigonal bipyramids at one of six equiv- alent anglesφ=30◦∗n (nis an even integer). As defined in Fig.1(b),φrepresents the trimerizing tilting angle and is usually called trimerization phase (TP). The corresponding tilting amplitudeQas defined in Fig.1(c)is called trimeriza- tion amplitude (TA). In this sense, the supercell becomes three times larger in size. The structural trimerization gives rise to threefold degenerate antiphase (AP) domains (denoted asα, β, andγ) described byZ3symmetry.
Microscopically, the structural trimerization is driven pri- marily by the condensation of nonpolarK3mode. On the other hand, the polar–2 mode arises due to the anharmonic cou- pling with theK3 mode [12–14]. In the FEP63cmstructure, the R ionic displacements along the c axis align in the up- up-down configuration or the down-down-up configuration.
A nonzero net polarization along thecaxis (denoted by P) spontaneously arises due to the unequal population of upward and downward R ions. The polarization is a by-product of
FIG. 1. (a) Schematic of the crystal structure ofh-RMnO3in the high-symmetry paraelectric (PE)P63/mmcphase. (b) Projection of the displacements of apical oxygen ions on theabplane associated with the trimerizing tilting of the MnO5trigonal bipyramids in the trimerized state. The displacement direction of an apical oxygen ion is selected to define the trimerization phaseφ. (c) Side view of the MnO5trigonal bipyramid tilting. The tilting amplitudeQfrom thecaxis is called trimerization amplitude. (d) Two sets of tilting angles which are dual to each other. Each set contains six angles. The angles marked with red belong to the FEP63cmphase, while the angles marked with blue the PUAP-3c1 phase. (e) The calculated lattice structure symmetries forh-InMn1−xGaxO3as a function ofx.
the structural trimerization and thus the ferroelectricity is im- proper. One major consequence of this lattice distortion is that the AP domains are coupled with FE domains and they share the same walls. It is noted that each AP domain can supply two opposite polarizations (+, –) along thecaxis described byZ2 symmetry, and now we have six types of AP+FE do- mains (denoted asα+,α–,β+,β–,γ+,γ–respectively). Their relationship to the six TPs (φ=30◦∗n,n=even) of the FE P63cmphase is shown in Fig.1(d). The six distinct domains are arranged in sequence around a central point, constructing the so-called Z6 vortex. The coupled AP+FE domain struc- ture allows only two types of vortex configurations, i.e., the so-called sixfold vortex(α+-β–-γ+-α–-β+-γ–)and antivortex (α+-γ–-β+-α−-γ+-β–) with opposite winding orders. From the viewpoint of self-organization, the vortex-antivortex pairs can form complicated topological domain networks over the whole space.
Except the FE P63cm structure, the other two types of low-symmetry structures, i.e., non-FE P-3c1 and FE P3c1, are also allowed to emerge from the high-symmetryP63/mmc structure via theK3-mode condensation in terms of the space- group analysis [15]. In theP-3c1 structure, the trimerizing tilting angle of the MnO5 trigonal bipyramids adopts one of six intermediate values, i.e.,φ=30◦∗n(nis an odd integer), as shown by a set of angles with blue font in Fig.1(d): (τ◦, υ∗,μ◦,τ∗,υ◦,μ∗) [16,17]. This means theP-3c1 phase can also supply six types of domains like the FEP63cm phase.
TheRionic displacements show an up-no-down pattern and thus the non-FE P-3c1 is also called a partially undistorted antiferroelectric (PUA) state. From the symmetry argument, the PUAP-3c1 phase is expected to exhibit PUA Z6-vortex domain patterns dual to FEZ6-vortex domain patterns. In the P3c1 structure, the trimerizing tilting angle adopts an arbitrary
value satisfying φ=30◦∗n, wheren is an integer. The net displacement of all theseRions along thecaxis is nonzero, indicating a nonzero polarization [18]. Thus, theP3c1 phase is ferroelectric. The symmetry consideration suggests that P63cmandP-3c1 are direct subgroups ofP63/mmc, andP3c1 as the most asymmetric structure of the three hettotype space groups is a subgroup of both P63cm and P-3c1. A proper substitution at Mn site is an effective way to make the ground state of an h-RMnO3 transition from the FE P63cm phase to the PUA P-3c1 phase. It is very likely that the FEP3c1 phase arises as an intermediate phase during this transition.
For example, the Ga substitution inh-InMnO3, as verified ex- perimentally and theoretically [17–19], drives such a sequence of phase transitions, as shown in Fig.1(e). Since the trimer- izing tilting angle in the P3c1 phase is not fixed, the P3c1 phase can be very similar in the crystallographic structure to the FE P63cm phase or the PUA P-3c1 phase. As a result, the critical regions between the FE P63cm and PUAP-3c1 phases are hard to identify, although distinct boundaries are drawn in Fig.1(e)as a guide for the eye. Using first-principles calculations, Griffin et al. obtained a set of critical points:
x=0.25, 0.4, 0.6 [18]. By the way, PUAZ6-vortex domain patterns have only been observed inh-InMn1−xGaxO3 so far [17].
Due to the same origin, FE and PUA Z6-vortex domain patterns may behave the same way in terms of nonelectric property. Several cases can be highlighted. (1) An external shear strain can impose the so-called Magnus-type force, driving the neighboring vortex-antivortex pairs to separate from each other [20]. (2) The complicated real-space do- main network can be neatly analyzed with graph theory [21].
(3) The vortex/antivortex density as a function of cooling rate is described by the Kibble-Zurek mechanism [22–24].
When it comes to the electric properties, however, we must be careful because they are very likely to behave differently.
In this regard, there is currently a lack of a comprehensive understanding. Comparatively speaking, FEZ6-vortex domain patterns have gone through intensive research and we now have a good knowledge of them. For instance, an electric field may drive the domain walls to move but the vortex/antivortex cores remain immobile [25]; instead of complete poling, an electric field can only result in narrow domains due to the incommensurate sum of the partial unit-cell-shift vectors for each pair of domain walls [25,26]; the charged tail-to-tail domain walls show enhanced conductance, while the charged head-to-head domain walls reduced conductance [27–29]. By the way, charged domain walls are energetically unfavor- able, but they are stabilized by topological protection in FE h-RMnO3. It is a pity that there is very little study on PUA Z6-vortex domain patterns at present. One fundamental issue with the PUAZ6-vortex domain pattern is its switching dy- namics with external electric fields, which is not clear yet. In particular, the role the PUAP-3c1 structure plays therein is of great interest.
In this work, we study the response of the PUA P-3c1 structure to external electric fields and the switching dynam- ics of PUA Z6-vortex domain patterns. Based on Landau phase transition theory, we analyze the ground-state phase of h-RMnO3and the effect of an external electric field along the caxis on the ground-state phase, and calculate the dependence of order parametersφ,Q,andPon this external electric field.
We find that the electric-field dependence of φ in the PUA P-3c1 structure can be very different from that in the FEP-3c1 structure. We then model the switching dynamics of both a PUAZ6-vortex domain pattern and a single PUA domain. We find that PUA Z6-vortex domain patterns can exhibit many extraordinary and promising features which are not present on FEZ6-vortex domain patterns.
II. ANALYTICAL AND NUMERICAL ANALYSES BASED ON LANDAU THEORY
A. Landau phase transition theory forh-RMnO3
For the trimerization-type structural phase transition oc- curring in h-RMnO3, Artyukhin et al. first established the free-energy expansion in powers of order parameters Q,φ, P, and their gradients [30]. Cano later proposed an additional symmetry-allowed higher-order termQ12cos43φ[31]. The ad- dition of this term enables the FEP3c1 phase to arise as a result of a residual symmetry breaking in equilibrium. The final free-energy expression for theh-RMnO3family can be written as
f = fbulk+fgrad
fbulk = a 2Q2+b
4Q4+c 6Q6+c
6Q6cos 6φ +d
6Q12cos43φ−gQ3Pcos 3φ+g
2Q2P2+aP
2 P2 fgrad= 1
2
l=x,y,z
slQ
∂Q
∂l 2
+Q2 ∂φ
∂l 2
+slP ∂P
∂l 2
, (1)
where fbulk is the bulk free-energy term and fgrad is the gra- dient free-energy term.a,b, andcare the coefficients for the free-energy polynomial onQextended up to the sixth order.
c measures the initial Z6 anisotropy of h-RMnO3. d is the coefficient for the higher-order term. To ensure a positive- definite fbulk,d must be positive.gis the nonlinear coupling factor between the nonpolarK3 and polar2–modes,gis the higher-order coupling factor betweenQandP, andaP is the self-energy factor of P. slQ andsPl scale the energy cost for the spatial variations ofQ,φ,andP. As is usually done, we assume that all the coefficients are constants independent of temperature except a. The temperature dependence of a is generally given bya=–a0(T–TC)/TC, wherea0is a negative constant.
The equilibrium values ofQ,φ, andPare determined from the condition that fbulk should be a minimum with respect to all these order parameters. In practice, the first step is to find all the extrema of fbulk by solving the system of equations
∂fbulk/∂Q=0,∂fbulk/∂φ=0, and∂fbulk/∂P=0. We have Q[a+bQ2+(ce−ce)Q4+gP2
+2(dQ6cos23φ+ce)Q4cos23φ]=0 2(dQ6cos23φ+ce)Q6sin 3φcos 3φ=0
P=gQ3cos 3φ/(gQ2+aP)1, (2) where ce=c–3g2/2(gQ2+aP) and ce=c–3g2/2(gQ2+ aP). Parameter ce measures the eventual Z6 anisotropy and plays a key role in determining the symmetry of the low- symmetry structure. Its significance will be shown below. It turns out that Eq. (2) has the following four types of solutions:
1.Q=0,P=0→P63/mmc (3)
2.
Q=0,P=0
sin 3φ=0, φ=30◦∗n(n=even) →P63cm, (4) 3.
Q=0,P=0
cos 3φ=0, φ=30◦∗n(n=odd) →P−3c1, (5) 4.
⎧⎪
⎨
⎪⎩
Q=0,P=0,cos23φ= −ce/dQ6 cos 3φ=cos 3(φn±φ)= ±
−dQce6
φ=φn±φ, φn=30◦∗n(n=odd)
→P3c1. (6)
The first solution obviously corresponds to the high- symmetry PE P63/mmcphase, since there is no trimerizing tilting and no polarization. According to Eq. 1, sin3φ=0 and cos3φ=0 give zero and nonzero polarization, respec- tively. In addition, both sin3φ=0 and cos3φ=0 have six solutions. The six solutions to sin3φ=0 are exactly the six TPs of the FE P63cm phase, while the six solutions to cos3φ=0 are the six TPs of the PUAP-3c1 phase. In the light of the previous introduction to h-RMnO3’s three low- symmetry structures, we can tell that the second and third solutions correspond to the FEP63cmand PUAP-3c1 phases, respectively. Unlike the second and third solutions, the fourth solution supports 12 TPs. Furthermore, none of these TPs satisfy φ=30◦∗n, wheren is an integer. Therefore, it can be confirmed that the fourth solution corresponds to theP3c1 phase, which can be accompanied by 12 symmetry-equivalent domains. Then, we need to find the stability conditions of
TABLE I. Values of physical parameters used in the present calculations.
a0(eV Å–2) b(eV Å–4) c(eV Å–6) c(eV Å–6) aP(eV Å–2) g(eV Å–4) g(eV Å–4)
−0.82 1.13 0.81 0.56 3.48 1.02 4.85
sxQ(eV) syQ(eV) szQ(eV) sxP(eV) syP(eV) szP(eV) L
5.14 5.14 12 8.88 8.88 29 1
each phase. Using the method introduced in the Appendix, we derived the stability conditions for the four phases:
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩
1.P63/mmc ifa>0 2.P63cm if
a<0 ce<−dQ6 3.P-3c1 if
a<0 ce>0
4.P3c1 if
a<0
−dQ6<ce<0
. (7)
It can be seen that the phase transition from the high- symmetryP63/mmcphase to any of the three low-symmetry phases,P63cm,P-3c1, andP3c1, always occurs ata=0, i.e., T =TC. It should be noted thatTCis not a constant but varies with the type of R. Below TC, the value of ce determines which low-symmetry phase is stabilized in equilibrium. The FE P63cmphase is stabilized ifce<–dQ6, while the PUA P-3c1 phase is stabilized ifce>0. In the in between range (i.e., –dQ6<ce <0), theP3c1 phase is stabilized. There is an important feature regarding the in between range. AtTC, Qis zero and thus the in between range does not exist. This means the high-symmetryP63/mmcphase cannot transition into theP3c1 phase directly, which is consistent with the fact thatP3c1 is not a direct subgroup ofP63/mmc.
B. Effect of electric field
In the framework of Landau theory, the way to study the effect of an external electric field is adding an electrostatic energy term to the free-energy expression. In this work, the electrostatic energy term is given by
felec= −E P, (8) whereE is the external electric field along thecaxis. Since the polarization in h-RMnO3 can only be along the c axis, it is sufficient to consider such an external electric field. In Eq. (1), the unit of energy is given in eV and bothQandP are measured in Å. Thus, the unit ofE is eV/Å, which is not a common unit of electric field. According to Ref. [30], the real electric fieldEris related toE byEr =2E/Z, whereZ is the effective charge of the polar2– mode. With the addition of felec, Eq. (2) and its solutions need to be modified. The modified Eq. (2) can be written as
Q{a+bQ2+(ce−ce)Q4+gP2+[2(dQ6cos23φ +ce)Q3cos 3φ−λE]Qcos 3φ} =0
[2(dQ6cos23φ+ce)Q3cos 3φ−λE]Q3sin 3φ=0 P= gQ3cos 3φ+E
gQ2+aP , (9)
where λ=3g/(gQ2+aP). We already know the solutions to Eq. (2) correspond to the four phases (P63/mmc,P63cm, P-3c1, andP3c1) thath-RMnO3usually adopts, so the modi- fied solutions [i.e., the solutions to Eq. (9)] actually represent the changes in these phases under the effect of E and are written as
1.P63/mmc→Q=0,P=E/aP, (10) 2.P63cm→
Q=0,P=0
sin 3φ=0, φ=30◦∗n(n=even), (11) 3.P−3c1→
⎧⎨
⎩
Q=0,P=0,cos 3φ=h0(E)2 h0(E)
=0 ifE =0
=0 ifE =0
, (12)
4.P3c1→
Q=0,P=0,cos 3φ=h1(E) orh2(E) h1(E)= −h2(E) if E=0 .
(13) Here, h0(E), h1(E), and h2(E) are the functions of E.
Several main features associated with the effect of E should be highlighted here. First,Qis still zero andPshows a typical linear dependence onE for the high-symmetry PEP63/mmc phase. Second, there is a significant difference between the FE P63cm and PUA P-3c1 phases in response to E. For the FE P63cm phase, sin3φ=0 still holds and its six TPs (φ=30◦∗n,n=even) remain the same. However, this is not the case for the PUAP-3c1 phase. With the application ofE, cos3φchanges from zero to a nonzero value. Accordingly, the solutions to Eq. 2 are no longer the six TPs (φ=30◦∗n, n=odd) of the PUA P-3c1 phase. This means that E can change the trimerizing tilting angle of the MnO5 trigonal bipyramids in the PUAP-3c1 structure while no such effect is present in the FE P63cm structure. Third, the 12 TPs of the P3c1 phase are divided into two groups by E, each of which contains six TPs. As for the difference between the two groups, it is not clear yet in the present case. Further research is needed on this.
C. Numerical analysis
HowQ,φ,andPvary withEcannot be given in an analytic form due to the complexity of Eq. (9), but can be numeri- cally calculated if the values of the coefficients in Eq. (1) are known. In this work, all these coefficients exceptd take the values listed in Table I, which were obtained by Småbråten et al.through first-principles calculations onh-InMnO3[32].
A small adjustment to the value ofchas been made to stabi- lize the PUAP-3c1 phase. Due to the lack of availabledvalue, here we maked take a set of different values: 0, 0.2, 0.5, and 1. In a free-energy expression, lower-order terms tend to play a bigger role than higher-order terms. This is why the values of higher-order term coefficients are usually smaller than that of lower-order term coefficients. Given that the values of the
FIG. 2. CalculatedE dependence of order parametersφ,Q,and Pwithdtaking a set of different values. (a)E dependence of the solutions to Eq.2. AtE=0, the solutions are the six TPs of the PUAP-3c1 phase. AtE<EcorE >−Ec, the six solutions belong to the FEP3c1 phase. AtE >EcorE<−Ec, the solutions are three TPs of the FEP63cmphase.Ecis the critical electric field for the transitionP3c1→P63cm. (b), (c)Edependence ofQandP. TheE dependence ofQandPfollows a different rule after reaching the FE P63cmstate.
lower-order term coefficients listed in Table I are about 1, it is sufficient for the maximum value ofd to be 1. At this point, the variation of order parametersQ,φ,andP withE can be numerically calculated. In Fig.2, we show the calcu- lation results. It can be seen that the variation of each order parameter withEremains qualitatively the same for different d values. Let us first take a look at the variation ofφwithE shown in Fig.2(a), which actually shows how the solutions to Eq.2 vary withE. It can be seen that every solution to Eq.2 changes continuously from an otherwiseP-3c1 TP to aP63cm TP whetherEincreases in the positive or negative direction. It should be noted that in the end only threeP63cmTPs remain instead of six, as two neighboringP-3c1 TPs become the same FEP63cmTP. Before becoming a FEP63cmTP, no solution is a multiple of 30°, which means theP3c1 phase is favored by E at this moment. After becoming a FE P63cm TP, no solution changes anymore. The above-mentioned variation of φwithEindicates that a continuous phase transition sequence P-3c1→P3c1→P63cm can be induced by applyingE to the PUAP-3c1 phase. Here, we defineEcas the critical elec- tric field for the transitionP3c1→P63cm. It can be seen that Ecincreases with enhancedd. It can be derived from Eq. (9) that theE dependence of bothQandPis independent of φ for the PUAP-3c1 phase. In other words, theE dependence of either of them remains the same for the six solutions to Eq.2. The calculatedE dependence ofQshown in Fig.2(b) reveals thatQ changes little withE, while the calculated E dependence ofP shown in Fig. 2(c)reveals thatP shows a nearly linear dependence onE. By the way, theEdependence
of bothQandPwill follow a different rule after reaching the FEP63cmstate.
D.E-caused change in the contour plot of fbulk+ felect
For the study of uniformed trimerized states, fbulk+felect
is sufficient. A common approach is to draw the contour plot of fbulk+felect as a functionQandφ. Here, Phas been op- timized. The contour plot can show the distribution of energy minima, from which people can gain a great deal of useful information on the domain pattern that may appear such as do- main morphology, domain-wall width, etc. Before modeling the switching dynamics of PUA Z6-vortex domain patterns, we preferred to examine how the contour plot of fbulk+felect varies withEfirst. Here, we madedtake two different values:
0 and 1. We show the results in Fig. 3. It should be noted that fbulk+ felect is expressed as a function ofQx andQy in Fig.3.QxandQyare related toQandφbyQx=Qcosφand Qy=Qsinφ[33]. It can be seen that theE-caused change in the contour plot of fbulk+felectremains qualitatively the same ford =0 andd =1, which indicates that the switching dy- namics of PUAZ6-vortex domain patterns will be qualitatively independent ofd. Two main points regarding the distribution of energy minima should be highlighted. On one hand, the six energy minima that appear atE =0 correspond to the six TPs (φ=30◦∗n„n=odd) of the PUAP-3c1 phase. On the other hand, the change in the distribution of these energy minima is consistent with what was obtained from Fig.2(a), that a con- tinuous phase transition sequenceP-3c1→P3c1→P63cm can be induced byE.
III. MODELING OF DOMAIN SWITCHING DYNAMICS A. Method: Phase-field simulation
For the phase-field simulation, order parameters Q and φ in the polar coordinates need to be transformed into the Cartesian coordinates Qx and Qy. The relationship between the two types of coordinates has been given earlier. The total free energy is expressed in the following form:
F =
V
(fbulk+fgrad+ felec)dV. (14) Following the standard procedure introduced in the previ- ous literature [34,35], we start from the temporal evolution of order parameter fields described by the time-dependent Ginzburg-Landau equations:
∂η(r,t)
∂t = −L δF
δη(r,t), η=Qx,Qy,P, (15) where Lis the kinetic coefficient related to the domain-wall mobility andtis the time measured in iteration step (IS). Com- bined with periodic boundary conditions, the semi-implicit spectral method is employed in practical calculations [36].
The lattice size will be specifically stated below. The initial lattice is set by assigning a zero value for each order param- eter plus a small random noise of uniform distribution. Since the switching dynamics of PUAZ6-vortex domain patterns is qualitatively independent of d, it is sufficient for d to take only one value in the following simulations. Given thatd is preferably smaller thancandc, we maked take 0.2.
FIG. 3. Electric field-caused change in the contour plot of fbulk+felect. Here, fbulk+felectis expressed as a function ofQxandQy, which are related toQandφbyQx=QcosφandQy=Qsinφ. For the contour plots in the first row,d=0; for the contour plots in the second row, d=1.
B. FE and PUAZ6-vortex domain patterns
For a better understanding of the difference between FE and PUA Z6-vortex domain patterns, we preferred to show them together. First, we obtained a FE Z6-vortex domain pattern on a lattice of 1024x×1024y×1zwithx= y=z=0.2 nm using the method introduced in our pre- vious work [37]. A part of the lattice was cut out and is shown in Fig.4(a). Then, we performed the phase-field simulation on a lattice with the same size to obtain a PUA Z6-vortex domain pattern. Similarly, a part of the lattice was cut out and is shown in Fig.4(b). The two intercepted lattices have the same size, which is 320x×320y×1z. The spatial contours of order parametersPandφwere displayed for each of them. It can be seen in Fig.4 that both domain patterns consist of six types of domains and the arrangement of the six types of domains is the same for them. In either of the two domain patterns, six distinct domains always meet at a central point and form a cloverleaf domain configuration. The cloverleaf domain configuration can have only two possible domain sequences, which give rise to vortex and antivor- tex, respectively. For the PUAZ6-vortex domain pattern, the two possible domain sequences areτ◦-υ∗-μ◦-τ∗-υ◦-μ∗ and τ◦-μ∗-υ◦-τ∗-μ◦-υ∗. It is easy to see that the two domain patterns differ in domain and domain wall. In the PUAZ6- vortex domain pattern, the domains are nonpolarized but the domain walls are polarized. Of the six domain walls around a vortex/antivortex core, three are positively polarized (PP) and three are negatively polarized (NP). The PP domain walls alternate with the NP domain walls. Such a phenomenon with nonpolarized domains and polarized domain walls is quite unusual, and certainly polarized domain walls are of interest,
e.g., from the viewpoint of domain-wall conduction. In the FEZ6-vortex domain pattern, the situation is reversed, i.e., the domains are polarized but the domain walls are nonpolarized.
C. Switching dynamics of a PUAZ6-vortex domain pattern A double-hysteresis loop can usually be observed when a sinusoidal electric field is applied to an antiferroelectric.
But such a loop has never been reported in h-RMnO3 be- fore, primarily because the PUA P-3c1 phase is not easy to stabilize. Here, we can easily do this using the phase-field simulation. We first obtained a well-evolved PUA Z6-vortex domain pattern on a lattice of 1024x×1024y×1zwith x=y=z=0.25 nm. Then, a sinusoidal electric field was applied to this domain pattern. This sinusoidal electric field is given by E =Emsin(2πωt), where Em is the max- imum applied electric field and ω is the frequency. In this work,Em=0.64 eV/Å andω=2.5×10–5(1/IS). As shown in Fig. 5, the variation of Pt withE manifests as a double- hysteresis loop. Here,Ptis the total polarization of this domain pattern. Several main features regarding the double-hysteresis loop should be highlighted. First, the double-hysteresis loop is extremely slim and strikingly reminiscent of the loops observed in nanoscale antiferroelectric domains which have extremely high energy-storage efficiency [38,39]. Second, at the stageE =0→Em(−Em),Ptshows a nearly linear depen- dence on E. Third, at the beginning part of the stage from E =Em(−Em) back toE =0 which is denoted by+(–), the Pt-E curve deviates slightly from its counterpart at the stageE =0→Em(−Em). TheEwindow of this part is very short. Fourth, after passing the+ (–) part, thePt-E curve
FIG. 4. The simulated FE (a) and PUA (b) Z6-vortex domain patterns. The spatial contours of order parametersP(upper) andφ (lower) are displayed for each of them.
becomes nearly linear again and overlaps its counterpart at the stageE =0→Em(−Em).
To understand why the double-hysteresis loop is so slim for PUA h-RMnO3, we tracked the evolution of the PUA Z6-vortex domain pattern with the sinusoidal electric field. For a better illustration, we cut a part with size 300x×300y× 1zfrom the lattice and a set of successive snapshots of the evolution [at the states (A)–(H) of the hysteresis shown in Fig.5] on the intercepted part were extracted and shown in
FIG. 5. The double-hysteresis loop achieved by applying a si- nusoidal electric field E=Emsin(2πωt) to a well-evolved PUA Z6-vortex domain pattern. Here, Em=0.64 eV/Å and ω=2.5× 10–5(1/IS). Various states in the hysteresis are labeled with (A)–(H), respectively.
Figs. 6(a)–6(h). The spatial contours of order parametersP andφ are displayed for each snapshot. Let us first examine the snapshots shown in Figs. 6(a)–6(c), which display the evolution of the PUA Z6-vortex domain pattern at the stage E =0→Em. As obtained from Fig.2(a),φmoves away from an otherwise PUA P-3c1 TP, goes through a series of con- secutive FEP3c1 TPs, and finally becomes a FEP63cmTP in each domain. This means the continuous phase transition sequenceP-3c1→P3c1→P63cmdid happen at this stage.
In the meantime,P’s in all domains increase in the same pace.
According to Fig.2(c), theEdependence ofPis nearly linear in each domain. A very unusual phenomenon occurring at this stage is that the domain walls never move, which has rarely been reported in previous dielectric materials. As a result, the size of each domain remains unchanged and the nearly linearE dependence ofPin each domain leads to the nearly linearPt-E curve that occurs at the stageE =0→Emof the double-hysteresis loop. After reaching the FEP63cmstate, the PP domain walls vanish while the NP domain walls remain, which leads to aZ3-vortex domain pattern. This means PUA type-I (i.e.,Z6-vortex) domain patterns can also be switched to type-II (i.e., Z3-vortex) domain patterns by electric fields like the FE type-I domain patterns [21,40]. By the way, the vanishing of the PP domain walls is in line with the re- sult drawn from Fig. 2(a)that two neighboring PUA P-3c1 TPs will eventually become the same FEP63cmTP under a large enough electric field. Next, let us examine the evolu- tion of the PUAZ6-vortex domain pattern at the stage from E =Emback toE=0. The relevant snapshots are shown in Figs. 6(d)–6(h). Earlier, we have confirmed that the con- tinuous phase transition sequenceP-3c1→P3c1→P63cm happened at the stageE=0→Em. However, the direct tran- sition fromP63cmback to P3c1 in each domain would not happen. Let us take a look at the snapshot shown in Fig.6(d).
At this moment, E is smaller thanEc, which means the FE P3c1 state has replaced the FEP63cmstate to be favored by E. But, the FEP63cmstate is still maintained in each domain.
That is to say, the direct transition fromP63cmback toP3c1 in each domain really did not happen. The most notable feature at this moment is that the remaining NP domain walls become dispersive. A subsequent snapshot shown in Fig.6(e)reveals that these dispersive NP domain walls are where the domains holding the FEP3c1 state come from. Due to being favored energetically, the domains holding the FEP3c1 state expand, as shown in Figs.6(e)and6(f). Conversely, the domains hold- ing the FEP63cmstate are compressed and have to shrink.
In the end, the domains holding the FE P63cmstate shrink to new PP domain walls and the FEP3c1 state dominates all domains. It is noted that the above domain structure evolution process happens at the+part of the double-hysteresis loop.
This process is completed so quickly that theEwindow of the + part is very short. The slight deviation between thePt-E curve at the+part and its counterpart at the stageE =0→ Emis due to the very small polarization gap between the FE P3c1 andP63cmstates at thisE window. This explains why the double-hysteresis loop is so slim. What happens next is the opposite of the evolution of the PUAZ6-vortex domain pattern at the stage E =0→Em, which leads to the superposition of two nearly linear Pt-E curves. Once back to E=0, the state in each domain returns toP-3c1, as shown in Fig.6(h).
FIG. 6. Switching dynamics of a PUAZ6-vortex domain pattern. (a)–(h) Snapshots of the evolution of the PUAZ6-vortex domain pattern with a sinusoidal electric field at the states (A)–(H) of the double-hysteresis loop shown in Fig.4in sequence. The spatial contours of order parametersP(upper) andφ(lower) are displayed for each snapshot.
Since the evolution of the PUAZ6-vortex domain pattern at the stage E=0→ −Em→0 is similar to what described above, we will not discuss it here. To sum up, (1) the double- hysteresis loop happens because the direct transition from P63cm back toP3c1 does not happen in each domain like its opposite, i.e., the transitionP3c1→P63cm; (2) the evo- lutions of the PUA Z6-vortex domain pattern at the stages E =0→Em(−Em) andE=Em(−Em)→0 are two nearly opposite processes without the parts involving the mutual transition betweenP63cmandP3c1.
D. Switching dynamics of a single PUA domain There is a statement made by Levanyuk and Sannikov (LS) that the double-hysteresis loop observed in antiferroelectrics is a result of the first-order transition induced in the electric field [41]. In Sec.II, we draw a conclusion that a continuous phase transition sequence P-3c1→P3c1→P63cm can be induced by applyingE to the PUA P-3c1 phase. According to the statement made by LS, we should not observe double- hysteresis loops in PUAh-RMnO3. But, a double-hysteresis
FIG. 7. Switching dynamics of a single PUA domain. (a) The double-hysteresis loop achieved by applying the sinusoidal electric fieldE =Emsin(2πωt) to a single domain.Ptis the total polarization in the single domain. The variation ofφwithEin the single domain can take two different forms. One is a closed loop (b) while the other is not (c).
loop was still observed when a sinusoidal electric field was applied to a PUAZ6-vortex domain pattern. The reasons for this remain to be explored. In fact, we only know that this phase transition sequence is continuous at the stageE =0→ Emor−Em. Whether the opposite phase transition sequence P63cm→P3c1→P-3c1 at the stageE =Em or−Em→0 is continuous is not clear yet. In particular, we did not observe the direct transition from P63cm back to P3c1 during the E-caused evolution of a PUA Z6-vortex domain pattern, so it is necessary to strictly confirm whether the transition from P63cmback toP3c1 is continuous. To do this, we applied the same sinusoidal electric fieldE =Emsin(2πωt) to a single PUA domain to detect the variation of Pt. Here, Pt is the total polarization in the single PUA domain. In the meantime, the variation of φ with E in the single PUA domain was also recorded. We show them in Fig.7. It can be seen from Fig. 7(a) that the variation of Pt with E still manifests as a double-hysteresis loop. This is mainly because there is a sudden change inPt at the stage fromE =Emor−Emback to E=0. The variation ofφwithEcan take two different forms.
As shown in Figs.7(b)and7(c), one is a closed loop while
the other is not. There is one thing both forms share, which is that φ always shows a sudden change at the stage from E =Emor−Emback toE =0. By the way,Pt andφshow a sudden change at the same positions. The above clearly shows that the transition fromP63cmbackP3c1 is discontinuous or of the first order. Therefore, our results on both a PUAZ6- vortex domain pattern and a single PUA domain are actually consistent with the statement made by LS.
IV. DISCUSSION
It is found that the experimentally observed configura- tion of PUA domains in h-InMn1−xGaxO3 appears to be a Z3-vortex domain pattern instead of aZ6-vortex domain pat- tern due to the existence of three hidden walls [17]. Such a phenomenon has been a mystery. In terms of the results obtained in this work, this phenomenon can be well under- stood. A so-called self-poling effect has been reported in many FEh-RMnO3. It is found that the effect is induced by an oxygen vacancy gradient, which can produce an effective electric field. It is the effective electric field that leads to the appearance of type-II domain patterns in FE h-RMnO3
[21,40]. We believe that there is also a self-poling effect in PUAh-InMn1−xGaxO3. According to Fig.2(a), the effective electric field will drive the TPs of two neighboring domains in a PUA vortex to approach the same FEP63cmTP. As a result, the crystallographic structures of the two neighboring do- mains become very similar and the domain boundary between the two neighboring domains becomes blurred and difficult to identify. This is the reason for the hidden domain-wall formation. So far, we have explained why the configuration of PUA domains inh-InMn1−xGaxO3appears to be aZ3-vortex domain pattern instead of aZ6-vortex domain pattern.
Intensive research on antiferroelectrics is primarily due to its candidacy for dielectric energy storage. According to the standard judging the performance of a dielectric energy storage material, an extremely slim double-hysteresis loop means an extremely high energy-storage efficiency. In con- ventional antiferroelectrics, such a double-hysteresis loop can be achieved only by tailoring domain size into micrometer or nanometer scales, which is not easy to implement [38,39].
In this way, antiferroelectric domains can quickly respond to electric fields, while PUA h-RMnO3 responds to electric fields mainly by changing the trimerizing tilting angle of the MnO5 trigonal bipyramids. Since the PUA P-3c1 and FE P63cmstates are linked by a series of consecutive TPs of the FE P3c1 state, the transition P-3c1→P63cmis a continu- ous process without energy dissipation due to the absence of domain-wall motion. When reaching a polarization saturation state, conventional antiferroelectrics usually manifest as a ho- mogeneous single-domain state where no domain walls exist.
However, the presence of domain walls in PUAh-RMnO3 is enforced due to topological protection. Owing to the existence of domain walls, the FEP3c1 state can replace the FEP63cm state to dominate all domains so quickly that the evolutions of PUAZ6-vortex domain pattern at the charging and discharging processes are two nearly opposite processes. This means the stored energy can be almost completely released. That is why PUA h-RMnO3 can have an extremely high energy-storage efficiency. It is a pity that PUA h-RMnO3 seems unable to
have a high energy-storage density in the current situation due to small saturation polarization and low breakdown electric field. Nevertheless, the PUA P-3c1 structure still represents an unusual antiferroelectric mechanism. Combining some ad- vanced fabricating techniques such as doping that can improve saturation polarization and breakdown electric field, people have a chance to utilize this mechanism to explore energy- storage materials with excellent performance in the future.
V. CONCLUSIONS
In summary, we have studied the response of the PUA P-3c1 structure to external electric fields and the switching dynamics of PUA Z6-vortex domain patterns. We find that an electric field along thecaxis can tune the trimerizing tilting angle of the MnO5trigonal bipyramids in PUAP-3c1 struc- ture while no such effect is present in the FEP63cmstructure.
As a result, a continuous phase transition sequenceP-3c1→ P3c1→P63cmcan be induced. For both a PUA Z6-vortex domain pattern and a single PUA domain, the variation of the total polarization with a sinusoidal electric field manifests as a double-hysteresis loop. We find out that this is because the transition fromP63cmback toP3c1 is a first-order transition.
The switching dynamics of PUA Z6-vortex domain patterns shows several extraordinary features. First, the domain walls never move during the transition from PUA P-3c1 to FE P63cm. Second, like FE type-I domain patterns, PUA type-I domain patterns can also be switched to type-II domain pat- terns by electric fields. Third, the evolutions of PUAZ6-vortex domain pattern at the charging and discharging processes are two nearly opposite processes. Our results advance the under- standing of real-space topological structures and suggest that the PUAP-3c1 structure may exhibit excellent performance in dielectric energy storage.
ACKNOWLEDGMENT
This work was financially supported by the National Nat- ural Science Foundation of China (Grants No. 92163210, No.
11874031, No. 11834002, No. 51721001, and No. 11974167).
APPENDIX: CONSTRUCTION OF PHASE DIAGRAM In the Landau-Devonshire theory, the total free energy for a system consists of several terms including the bulk energy, gradient energy, and so on. It is usually expressed in the following form:
Ftot =
V
(fbulk+fgrad+ · · ·)dV, (A1) where fbulkand fgradare called the bulk free-energy term and gradient free-energy term, respectively.
To construct the phase diagram, it is sufficient to consider only term fbulk, which is usually expressed as a polynomial expansion in terms of one or several order parametersημ:
fbulk= f0+ f2(ημ)+f3(ημ)+f4(ημ)+ · · ·, (A2) where fn(hμ) is a polynomial of degreenfor order parameter ημ. Mathematically, constructing a phase diagram is to bring out all the minima of fbulk as a function of order parameters ημ. In practice, the first step is to find all the extrema of fbulk by solving a set of equations:
∂fbulk
∂ημ =0, (μ=1,2, . . . ,l), (A3) wherelis the number of order parameters.
The next step is to find the solutions to Eq. (A3) that make fbulka minimum, which calls for constructing the determinant of second-order derivatives of fbulk with respect to all order parametersημ. Subsequently, one substitutes all solutions to Eq. (A3) into the principal minors. The solutions that make fbulk a minimum should satisfy the condition that all the prin- cipal minors should be positively definite, as expressed in the following form:
Pi=
a11 a12 · · · a1i
a21 a22 · · · a2i
... ... aμν ...
ai1 ai2 · · · aii
>0,
aμν= ∂2fbulk
∂ημ∂ην(i=1,2,· · · ,l). (A4)
[1] S. Dong, J.-M. Liu, S.-W. Cheong, and Z. Ren, Multiferroic ma- terials and magnetoelectric physics: Symmetry, entanglement, excitation, and topology,Adv. Phys.64, 519 (2015).
[2] R. Ramesh and N. A. Spaldin, Multiferroics: Progress and prospects in thin films,Nat. Mater.6, 21 (2007).
[3] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y.
Tokura, Magnetic control of ferroelectric polarization,Nature (London)426, 55 (2003).
[4] S. Dong and J.-M. Liu, Recent progress of multiferroic per- ovskite manganites,Mod. Phys. Lett. B26, 1230004 (2012).
[5] T. Choi, S. Lee, Y. J. Choi, V. Kiryukhin, and S.-W. Cheong, Switchable ferroelectric diode and photovoltaic effect in BiFeO3,Science324, 63 (2009).
[6] J. Wang, J. B. Neaton, H. Zheng, V. Nagarajan, S. B. Ogale, B. Liu, D. Viehland, V. Vaithyanathan, D. G. Schlom, U. V.
Waghmare, N. A. Spaldin, K. M. Rabe, M. Wuttig, and R.
Ramesh, Epitaxial BiFeO3 multiferroic thin film heterostruc- tures,Science299, 1719 (2003).
[7] T. Lottermoser, T. Lonkai, U. Amann, D. Hohlwein, J. Ihringer, and M. Fiebig, Magnetic phase control by an electric field, Nature (London)430, 541 (2004).
[8] H. Das, A. L. Wysocki, Y. Geng, W. Wu, and C. J. Fennie, Bulk magnetoelectricity in the hexagonal manganites and ferrites, Nat. Commun.5, 2998 (2014).
[9] W. Wang, J. Zhao, W. Wang, Z. Gai, N. Balke, M. Chi, H. N. Lee, W. Tian, L. Zhu, X. Cheng, D. J. Keavney, J. Yi, T. Z. Ward, P. C. Snijders, H. M. Christen, W.
Wu, J. Shen, and X. Xu, Room-Temperature Multiferroic Hexagonal LuFeO3 Films, Phys. Rev. Lett. 110, 237601 (2013).
[10] T. Choi, Y. Horibe, H. T. Yi, Y. J. Choi, W. Wu, and S. W.
Cheong, Insulating interlocked ferroelectric and structural an- tiphase domain walls in multiferroic YMnO3,Nat. Mater.9, 253 (2010).
[11] B. B. Van Aken, T. T. M. Palstra, A. Filippetti, and N. A.
Spaldin, The origin of ferroelectricity in magnetoelectric YMnO3,Nat. Mater.3, 164 (2004).
[12] C. J. Fennie and K. M. Rabe, Ferroelectric transition in YMnO3 from first principles, Phys. Rev. B 72, 100103(R) (2005).
[13] T. Lonkai, D. G. Tomuta, U. Amann, J. Ihringer, R. W.
A. Hendrikx, D. M. Többens, and J. A. Mydosh, Develop- ment of the high-temperature phase of hexagonal manganites, Phys. Rev. B69, 134108 (2004).
[14] M. Lilienblum, T. Lottermoser, S. Manz, S. M. Selbach, A.
Cano, and M. Fiebig, Ferroelectricity in the multiferroic hexag- onal manganites,Nat. Phys.11, 1070 (2015).
[15] J. Li, F.-K. Chiang, Z. Chen, C. Ma, M.-W. Chu, C.-H. Chen, H.
Tian, H. Yang, and J. Li, Homotopy-theoretic study and atomic- scale observation of vortex domains in hexagonal manganites, Sci. Rep.6, 28047 (2016).
[16] F. T. Huang, X. Wang, Y. S. Oh, K. Kurushima, S. Mori, Y.
Horibe, and S. W. Cheong, Delicate balance between ferroelec- tricity and antiferroelectricity in hexagonal InMnO3,Phys. Rev.
B87, 184109 (2013).
[17] F.-T. Huang, X. Wang, S. M. Griffin, Y. Kumagai, O. Gindele, M.-W. Chu, Y. Horibe, N. A. Spaldin, and S.-W. Cheong, Dual- ity of Topological Defects in Hexagonal Manganites,Phys. Rev.
Lett.113, 267602 (2014).
[18] S. M. Griffin, M. Reidulff, S. M. Selbach, and N. A. Spaldin, Defect chemistry as a crystal structure design parameter: Intrin- sic point defects and Ga substitution in InMnO3,Chem. Mater.
29, 2425 (2017).
[19] D. A. Rusakov, A. A. Belik, S. Kamba, M. Savinov, D. Nuzhnyy, T. Kolodiazhnyi, K. Yamaura, E. Takayama- Muromachi, F. Borodavka, and J. Kroupa, Structural evolution and properties of solid solutions of hexagonal InMnO3 and InGaO3,Inorg. Chem.50, 3559 (2011).
[20] X. Wang, M. Mostovoy, M. G. Han, Y. Horibe, T. Aoki, Y.
Zhu, and S. W. Cheong, Unfolding of Vortices into Topological Stripes in a Multiferroic Material,Phys. Rev. Lett.112, 247601 (2014).
[21] S. C. Chae, Y. Horibe, D. Y. Jeong, S. Rodan, N. Lee, and S. W. Cheong, Self-organization, condensation, and annihila- tion of topological vortices and antivortices in a multiferroic, Proc. Natl. Acad. Sci. USA107, 21366 (2010).
[22] S. C. Chae, N. Lee, Y. Horibe, M. Tanimura, S. Mori, B. Gao, S.
Carr, and S. W. Cheong, Direct Observation of the Proliferation of Ferroelectric Loop Domains and Vortex-Antivortex Pairs, Phys. Rev. Lett.108, 167603 (2012).
[23] S.-Z. Lin, X. Wang, Y. Kamiya, G.-W. Chern, F. Fan, D. Fan, B. Casas, Y. Liu, V. Kiryukhin, W. H. Zurek, C. D. Batista, and S.-W. Cheong, Topological defects as relics of emergent continuous symmetry and Higgs condensation of disorder in ferroelectrics,Nat. Phys.10, 970 (2014).
[24] Q. N. Meier, M. Lilienblum, S. M. Griffin, K. Conder, E.
Pomjakushina, Z. Yan, E. Bourret, D. Meier, F. Lichtenberg, E. K. H. Salje, N. A. Spaldin, M. Fiebig, and A.
Cano, Global Formation of Topological Defects in the
Multiferroic Hexagonal Manganites,Phys. Rev. X 7, 041014 (2017).
[25] M.-G. Han, Y. Zhu, L. Wu, T. Aoki, V. Volkov, X. Wang, S.
C. Chae, Y. S. Oh, and S.-W. Cheong, Ferroelectric switching dynamics of topological vortex domains in a hexagonal man- ganite,Adv. Mater.25, 2415 (2013).
[26] T. Jungk, Á. Hoffmann, M. Fiebig, and E. Soergel, Electrostatic topology of ferroelectric domains in YMnO3,Appl. Phys. Lett.
97, 012904 (2010).
[27] D. Meier, J. Seidel, A. Cano, K. Delaney, Y. Kumagai, M. Mostovoy, N. A. Spaldin, R. Ramesh, and M. Fiebig, Anisotropic conductance at improper ferroelectric domain walls,Nat. Mater.11, 284 (2012).
[28] J. Schaab, A. Cano, M. Lilienblum, Z. Yan, E. Bourret, R.
Ramesh, M. Fiebig, and D. Meier, Optimization of elec- tronic domain-wall properties by aliovalent cation substitution, Adv. Electron. Mater.2, 1500195 (2016).
[29] G. F. Nataf, M. Guennou, J. M. Gregg, D. Meier, J. Hlinka, E. K. H. Salje, and J. Kreisel, Domain-wall engineering and topological defects in ferroelectric and ferroelastic materials, Nat. Rev. Phys.2, 634 (2020).
[30] S. Artyukhin, K. T. Delaney, N. A. Spaldin, and M. Mostovoy, Landau theory of topological defects in multiferroic hexagonal manganites,Nat. Mater.13, 42 (2014).
[31] A. Cano, Hidden order in hexagonal RMnO3 multiferroics (R=Dy - Lu, In, Y, and Sc),Phys. Rev. B89, 214107 (2014).
[32] D. R. Småbråten, Q. N. Meier, S. H. Skjærvø, K. Inzani, D.
Meier, and S. M. Selbach, Charged domain walls in improper ferroelectric hexagonal manganites and gallates, Phys. Rev.
Mater.2, 114405 (2018).
[33] P. Tolédano and J.-C. Tolédano, Order-parameter symmetries for improper ferroelectric nonferroelastic transitions,Phys. Rev.
B14, 3097 (1976).
[34] L.-Q. Chen, Phase-field models for microstructure evolution, Annu. Rev. Mater. Res.32, 113 (2002).
[35] K. L. Yang, Y. Zhang, S. H. Zheng, L. Lin, Z. B. Yan, J. M. Liu, and S. W. Cheong, Spatial anisotropy of topological domain structure in hexagonal manganites, Phys. Rev. B95, 024114 (2017).
[36] L. Q. Chen and J. Shen, Applications of semi-implicit Fourier- spectral method to phase field equations, Comput. Phys.
Commun.108, 147 (1998).
[37] K. L. Yang, Y. Zhang, S. H. Zheng, L. Lin, Z. B. Yan, J. M. Liu, and S. W. Cheong, Electric field driven evolution of topological domain structure in hexagonal manganites,Phys. Rev. B 96, 144103 (2017).
[38] H. Qi, R. Zuo, A. Xie, A. Tian, J. Fu, Y. Zhang, and S.
Zhang, Ultrahigh energy-storage density in NaNbO-based lead- free relaxor antiferroelectric ceramics with nanoscale domains, Adv. Funct. Mater.29, 1903877 (2019).
[39] J. Li, F. Li, Z. Xu, and S. Zhang, Multilayer lead-free ce- ramic capacitors with ultrahigh energy density and efficiency, Adv. Mater.30, 1802155 (2018).
[40] X. Wang, F.-T. Huang, R. Hu, F. Fan, and S.-W. Cheong, Self- poling with oxygen off-stoichiometry in ferroelectric hexagonal manganites,APL Mater.3, 041505 (2015).
[41] A. P. Levanyuk and D. G. Sannikov, Anomalies in dielec- tric properties in phase transitions, Sov. Phys. JETP28, 134 (1969).
