5. Magnetoelectric topology: monopole and vortex
5.3. Ferroelectric domain vortex in hexagonal RMnO 3
Besides skyrmion as a real-space topological particle (defect), other topological defects are pos- sible in ferroelectrics. A recent hot topic is the ferroelectric domain vortex as first observed and then well investigated in hexagonalRMnO3(R=Y, Ho, Lu,. . .). In 2010, Choi et al. reported an interlocked ferroelectric and structural antiphase domain structure in multiferroic YMnO3[123].
As discussed in Section3.3.1, the structural trimerization (so-calledK3 mode distortion) of Mn triangles gives rise to three types of ferroelectric/antiphase domains:α,β,γ, which break the 60◦rotation symmetry of hexagonal lattice. The ferroelectric polarization, which is also a result of trimerization, comes from the↑ − ↑ − ↓shifts of the Rspin trimers and aligns along the c-axis (+ or −). This configuration breaks an additional space-inversion symmetry. Thus, the ferroelectric state belongs to theZ3×Z2group, resulting in the cloverleaf pattern of six domains emerging from one point, as shown in Figure42.
For a sample grown below the ferroelectric Curie temperature, the trimerization-induced fer- roelectric domains can be in a stripe-like pattern [473]. In contrast, if the sample is grown above the ferroelectric Curie temperature, the vortex patterns due to the structural trimerization appear above the ferroelectric Curie temperature. When temperature passes across the ferroelectric Curie point, these vortex patterns interlock the ferroelectric domains, leading to a frozen domain struc- ture during the quenching process. Around the center point of six domains, the domain circling sequence restricted by very high-energy barriers is alwaysα+−β−−γ+−α−−β+−γ−or α+−γ−−β+−α−−γ+−β−, corresponding to a vortex or an anti-vortex, respectively. The cloverleaf domains, or namely the vortex structure, is robust against thermal fluctuations over high temperature.
Theoretically, the complex topological ferroelectric domain structures can be understood in the framework of the Landau theory with the help of density functional calculations [475]. The clamping between the polarization domains and the structural antiphase domains can be under- stood by a nonlinear coupling between structural trimerization (characterized by the amplitude QK3and phase) and polarization along thez-axis (Pz):
ftrimer∼ −Q3K3Pzcos(3), (30)
whereftrimerdenotes the free energy term. Given a distortion mode, a spontaneous ferroelectric polarization is generated. To lower the energy, the preferred values are: 0, ±π/3,±2π/3, and π. In other words, six structural antiphase domains will be formed if a trimerization appears (QK3>0), giving sixvalues, as sketched in Figure43(a). For=0 and±π/3, the polarization should align upward, and it would align downward for= ±2π/3 andπ.
Considering the other energy terms and proper coefficients extracted from the density func- tional calculations, the energy landscape in the (QK3,) plane is indeed a Mexican hat shape like with six minima (six preferredvalues), as sketched in Figure43(b). The direct paths con- necting any two nearest-neighbor minima give the lowest energy barriers comparing with other paths connecting the non-nearest-neighbor minima. Therefore, parameteracross any domain
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Figure 42. (Color online) Vortex domain structure in hexagonal RMnO3. (a) A dark-field transmission electron microscopic image (R=Y). The six structural antiphase domains emerge from one central point (so-called vortex core), ordered in the α−β−γ−α−β−γ sequence. (b) A transmission electron microscopic image of an almost fully electric-poled region. The three domains of negative ferroelectric polarizations become seriously shrunk into narrow stripes but cannot be completely melted away due to the robustness of the domain walls, demonstrating that the vortex core is topologically protected. (c) A schematic of the evolution of a cloverleaf ferroelectric domain driven by a downward electric fieldE. Arrow Edenotes the electric field, arrowQrefers to a quenching and arrowSCrefers to a slow cooling, both from high temperature (above the ferroelectric Curie temperature). (a)–(c) Reprinted by permission from Macmil- lan Publishers Ltd: Choi et al. [123]. Copyright c(2010). (d) A piezoelectric force microscope image of ErMnO3. (e)–(h) The representation of domain structure using the famous four-color theorem. Theα,β, and γ regions are colored using three colors: red, blue, and green. The two directions of polarization up (face) and polarization down (edge) are distinguished by normal colors and light colors. (d) Reprinted figure with permission from Chae et al. [473]. Copyright c(2012) by the American Physical Society. (e) All adjacent edges or faces have different colors. Inset: an enlarged view of the pink rectangle area. (f)–(h) The schematic two-, four-, and six-gons with one, two, and three vortex–antivortex pairs, respectively. Insets: the corre- sponding real examples imaged using an optical microscope. (e)–(h) Reprinted figure with permission from Chae et al. [474] Copyright c(2010) by the National Academy of Sciences.
wall must be stepped by ±π/3, while all domain walls with other steps (±2π/3 and π) are much higher in energy and will decay to multiple domain walls with a±π/3 step. In this sense, once a cloverleaf vortex emerges, its six domain walls cannot merge to change the sequence of
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(a) (b)
Figure 43. (Color online) Structural trimerization and energy landscape of topological domain structure.
(a) A schematic of a structural trimer domain. The six domains are distinguished by a phaseof theK3 mode distortion. An anticlockwise rotation around the vortex core gives rise to aπ/3 phase step across each domain wall.P: local polarization along thec-axis (perpendicular to the plane). Arrows indicate the direction of partial apical oxygen motion of the trimer distortions. Reprinted by permission from Macmillan Publish- ers Ltd: Das et al. [132]. Copyright c(2014). (b) A contour plot of the free energy for uniformly trimerized states as a function of trimerization (amplitudeQand phase). The six lowest energy positions (=0,
±π/3,±2π/3, π) correspond to the six domains. The lowest-energy structural domain wall is marked using the white dashed line, connecting two neighboring energy minima with a phase shift±π/3. Due to the Mexican hat shape-like potential well, any domain wall beyond±π/3 is much higher in energy and thus unstable against the spontaneous decomposition into multi-domain walls. Reprinted by permission from Macmillan Publishers Ltd: Artyukhin et al. [475]. Copyright c(2014).
domains. Of course, these domain walls can terminate at the surface or an anti-vortex, leading to the vortex–antivortex pairs connected by domain wall strings [123].
Experimentally, an electric field polarizes a ferroelectric polarization via domain wall shift- ing (expanding the aligned domains and shrinking the opposite domains) [123]. Nevertheless, a vortex/antivortex core cannot be annihilated individually until a dielectric breakdown, as shown in Figure42(b) and (c) [123,477]. Even though, a shear strain can impose a Magnus-type force which pulls vortices and antivortices in opposite directions and unfolds them into a topological stripe domain state [478], analogous to the current-driven dynamics of vortices in superconduc- tors and superfluids. In addition, the annihilation between a vortex and an anti-vortex is possible [474], analogous to the annihilation between an elementary particle/antiparticle.
In hexagonal ErMnO3 where Er3+ is magnetic, magnetic signals from these domain walls were observed at low temperature [479]. Although these signals directly originate from the Er3+
spins, they depend on Mn antiferromagnetic structure, implying that the Mn antiferromagnetic domains are also clamped to the structural-antiphase/ferroelectric domains, a manifestation of the domain wall magnetoelectricity.
Besides this domain wall magnetoelectricity, the bulk magnetoelectric effect in hexagonal RMnO3/RFeO3 mediated by the DM interaction may be possible, if the Mn/Fe magnetic order take the so-calledA2pattern (see Figure11) [132]. However, for many hexagonalRMnO3mate- rials, for example, ErMnO3, the magnetic ground state is theB2 pattern (see Figure11), which forbids any bulk magnetoelectricity. Even though, a magnetic field can drive the transition of magnetic state to the A2 pattern. In this case, the magnetoelectric domains (not the magnetic
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Figure 44. (Color online) Magnetoelectric domain structure of a hexagonal ErMnO3single crystal. (a) A room temperature piezo-response force microscopy image. The white and dark colors denote the up and down ferroelectric polarizations, respectively. (b) and (c) The low-temperature (4 K) magnetoelectric force microscopy images: (b) at zero magnetic field and (c) at a magnetic field of 8 T. The images in (a)–(c) were taken at the same location on the (001) surface. (d) A cartoon listing the magnetoelectric coefficient (α) of theA2phase in different ferroelectric domains. It is clear that the sign ofαchanges with polarizationPz
although the canted magnetic momentM is positive in all domains. (e) A cartoon showing the effective magnetoelectric coupling through structural trimerization. (f) Cartoons showing a variation of the buckling of MnO5polyhedra (trimer modeQK3) induced by electric fields via polarization componentPz, resulting in a variation of the canted momentMzof Mn spins. Reprinted by permission from Macmillan Publishers Ltd: Geng et al. [476]. Copyright c(2014).
domains but defined by the magnetoelectric coefficient) can be visualized, as demonstrated in ErMnO3, where the magnetoelectric domains are clamped to the structural-antiphase domains mentioned above (see Figure44for more details) [476].
Finally, it is noted that ferroelectric vortices have also been observed and simulated in BiFeO3
films and heterostructures besidesRMnO3[431,480].
5.4. Magnetoelectricity of topological surface state
In condensed matter physics, a phase of matter can be defined by its response to some stimuli.
The electromagnetic Lagrangian of a topological insulator, which has a form analogous to the theory of axion electrodynamics, contains a scalar product of electric field and magnetic field, E·B[481,482]. This linear magnetoelectric term can generate a number of novel physical phe- nomena, for example, monopole-like behavior. Regarding the magnetoelectric effect, an electric field can induce a magnetic dipole and vice versa. This magnetoelectric effect purely originates from the orbital motion of electrons, which can be highly sensitive and reproducible without fatigue.
Such an intrinsic magnetoelectric effect can be intuitively visualized using a theoretical sce- nario. Although theZ2topological insulators are in principle nonmagnetic, the surface state hides
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the intrinsic noncollinear spin texture in moment space, for example, the spin rotates around the surface Dirac cone. This coupling is glued by the spin–orbit coupling and no net magnetic moment is available by integrating the whole surface state. Nevertheless, an electric field will shift the Dirac surface state to one momentum direction a little bit, resulting in a change of the Fermi surface. Since this surface state is spin-momentum bound, a net magnetization will be generated, implying an intrinsic linear magnetoelectric response [483,484].
In addition, Qi et al. predicted a mirror magnetic monopole which can be generated in a topological insulator by a charge near the surface [485]. Experimentally, the magnetic order and magnetoresistance of a magnetic ion-doped topological insulator(Cr0.15(Bi0.1Sb0.9)1.85Te3) can be tuned using an electrical method [486].