Microscopic origin of the electric Dzyaloshinskii-Moriya interaction

The microscopic origin of the electric Dzyaloshinskii-Moriya interaction (eDMI) is unveiled and discussed by analytical analysis and first-principles based calculations. As similar to the magnetic Dzyaloshinskii-Moriya interaction (mDMI), eDMI also originates from the electron-mediated effect and more specifically from certain electron hoppings that are being activated due to certain local inversion-symmetry breaking. However, the eDMI energy is found to be at least a third-order interaction in atomic displacements instead of bilinear in magnetic dipole moments for mDMI. Furthermore, the eDMI energy form is presented, and we find that novel electrical topological defects (namely, chiral electric bobbers) can arise from this eDMI. Thus unraveling the microscopic origin of eDMI has the potential to lead to, and explain, the discovery of novel polar topological phases.

Figure 1

(a) Tetragonal phase of ${\mathrm{PbTiO}}_3$, with titanium ions displaced along the $z$ direction. (b) Illustrative plot of some atomic displacements under certain force constants $F_{\alpha \beta }(i,j)$ between titanium pairs with $i={\mathrm{Ti}}_0$ and $j={\mathrm{Ti}}_{+1}$ along the $x$ direction: $F_{xz}({\text{Ti}}_0,{\text{Ti}}_{+1})$ and $F_{zx}({\text{Ti}}_0,{\text{Ti}}_{+1})$. Note that $F_{zx}({\text{Ti}}_{-1},{\text{Ti}}_0)$ is equal to $F_{zx}({\text{Ti}}_0,{\text{Ti}}_{+1})$ due to the translational symmetry along the $x$ direction. The red vectors illustrate one possible set of the atomic displacement pattern that could result from a negative $F_{xz}({\text{Ti}}_0,{\text{Ti}}_{+1})$ and positive $F_{zx}({\text{Ti}}_0,{\text{Ti}}_{+1})$.

Figure 2

(a) Top view of the surface bubble defect achieved with negative eDMI. (b) Cross-section view of the (top) surface bubble defect from negative eDMI. (c) Cross-section view of the (bottom) surface bubble defect from positive eDMI. (d) Schematic illustration to show the charge and chirality of the surface bubble defects with respect to the sign of eDMI; the black arrow is to indicate the depolarization field ($E_d$) direction. In addition, the red domains have polarization along the negative $z$ direction, and the blue domains (defects) have polarization along the positive $z$ direction.

Figure 3

(a) Diagram plot of orbital-resolved force constant density ${\xi }_{\alpha ,\beta }^{m,n}$ defined as ${\sum }_{m^{\prime },n^{\prime }}\langle m,i|U_{i,\alpha }|m^{\prime },i\rangle \langle m^{\prime },i|{\widehat{G}}^0\left(\varepsilon \right)|n^{\prime },j\rangle \langle n^{\prime },j|U_{j,\beta }|n,j\rangle \langle n,j|{\widehat{G}}^0|m,i\rangle$ (see Eq. (6) and Eq. (32) in Sec. IV of the Supplemental Material [43]). Panels (b), (c), (d), and (e) are plots of orbital-resolved force constant density (${\xi }_{x,z}^{m,n}$ in blue line and ${\xi }_{z,x}^{n,m}$ in red line) and total force constant density [$f_{x,z}(\varepsilon ,i,j)$ in blue area and $f_{z,x}(\varepsilon ,i,j)$ in red area], as defined in Eqs. (4) and (5). More specifically, panel (c) contains ${\xi }_{x,z}^{m,n}$ and ${\xi }_{z,x}^{n,m}$ where $m=d_{xz}$ and $n=d_{xz}$; panel (d) contains ${\xi }_{x,z}^{m,n}$ and ${\xi }_{z,x}^{n,m}$ where $m=d_{x^2-y^2}$ and $n=d_{xz}$; panel (e) contains ${\xi }_{x,z}^{m,n}$ and ${\xi }_{z,x}^{n,m}$ where $m=d_{xz}$ and $n=d_{z^2}$; panel (f) contains ${\xi }_{x,z}^{m,n}$ and ${\xi }_{z,x}^{n,m}$ where $m=d_{x^2-y^2}$ and $n=d_{z^2}$; and panel (b) contains the sum of all the ${\xi }_{x,z}^{m,n}$ and ${\xi }_{z,x}^{n,m}$ plotted in panels (c), (d), (e), and (f). Panels (g) and (h) are density of states (DOS) projected on the titanium atom (blue ball in the subset) and oxygen atom (red ball in the inset). Note that the Fermi level is set at the zero of energy in all the plots.

Figure 4

(a) Two ${\mathrm{PbTiO}}_3$ ($P4mm$ phase) unit cells along the $x$ direction. The two blue balls on sites $i$ and $j$ are titanium atoms, and their intermediate oxygen atom is indicated by a red ball on site $k$. The vertical and horizontal blue planes represent mirrors $m_{yz}$ and $m_{xy}$, respectively. A green triangle is used to indicate the relative displacements between the two titanium atoms and their intermediate oxygen atom. Panel (b) shows all the orbitals $m$ and $n$ in the orbital-resolved force constant expression. Expression of ${\xi }_{x,z}^{m,n}$ is sketched by a multiplication between the expressions in (c1) and (c2). The blue vertical lines in (c1) and (c2) are mirror $m_{yz}$ operations and transform the expressions in (c1) and (c2) to the expressions in (d1) and (d2), respectively, and whose multiplication gives rise to $-{\xi }_{z,x}^{n,m}$. In all the expression sketches, the yellow circles and squares represent orbital $m$ and $n$, respectively, the arrows are used to indicate effective perturbation potentials, and the green triangles [as also indicated in (a) among sites $i, j$, and $k$] are for Green's function ${\widehat{G}}^0$.

Figure 5

If sites $i, j$, and $k$ in Fig. 4 are collinear, expression of ${\xi }_{x,z}^{m,n}$ is sketched by a multiplication between expressions in (a1) and (a2). The blue horizontal lines in (a1) and (a2) are mirror $m_{xy}$ operations and transform the expressions in (a1) and (a2) to the expressions in (b1) and (b2), respectively, and whose multiplication gives rise to $-{\xi }_{x,z}^{m,n}$. When site $k$ is on top of sites $i$ and $j$ in the triangle, the expression of ${\xi }_{x,z}^{up,m,n}$ for such case is sketched by a multiplication between (c1) and (c2) which can be transformed by mirror $m_{xy}$ (blue horizontal lines) to the expressions in (d1) and (d2) whose multiplication gives rise to $-{\xi }_{x,z}^{dn,m,n}$ which is for the case that site $k$ is below sites $i$ and $j$. In all the expression sketches, the yellow circles and squares represent orbital $m$ and $n$, respectively, the arrows are used to indicate effective perturbation potentials, and the green dots and triangles are for Green's function ${\widehat{G}}^0$.

Figure 6

Illustration plot of the crystal structure with oxygen octahedral tiltings. Three 5-atom cells along [100] direction are plotted. The cell indexes are marked on the titanium atoms (blue ball) as ${\mathit{R}}_1=(-1,0,0), {\mathit{R}}_2=(0,0,0)$, and ${\mathit{R}}_3=(1,0,0)$, respectively. Note that the atomistic basis is defined in a unit cell in which the titanium atom locates at the origin (0,0,0) (see Fig. 5 of the Supplemental Material [43]). Thus the oxygen atoms O1($X$) and O2($X$) site at positions ${\mathit{r}}_1={\mathit{R}}_1+(0.5,0,0)$ and ${\mathit{r}}_2={\mathit{R}}_2+(0.5,0,0)$ before the octahedra tilting displacements, respectively. The oxygen octahedra tiltings around titanium atoms in cells ${\mathit{R}}_1, {\mathit{R}}_2$, and ${\mathit{R}}_3$ are anticlockwise, clockwise, and anticlockwise, respectively. Note that the atomic displacement variables for titanium atom ($\mathit{\mu }\mathit{B}$) and oxygen atom along [100] directions ($\mathit{\mu }\mathit{X}$) are labeled and their definition in unit cell can be found in Fig. 5 of the Supplemental Material [43]. Note also that we use $\mathit{R}$ for cell index and $\mathit{r}$ for atom position in this paper.