Fingerprints of spin-current physics on magnetoelectric response in the spin-$\frac{1}{2}$ magnet ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{Ge}}_2{\mathrm{O}}_7$

As is well known, the single-site anisotropy vanishes in the spin-$\frac{1}{2}$ compounds as a consequence of the fundamental Kramers degeneracy. We argue, rather generally, that a similar property holds for the magnetically induced electric polarization $\mathit{P}$, which should depend only on the relative orientation of spins in the bonds but not on the direction of each individual spin. Thus, for insulating multiferroic compounds, $\mathit{P}$ can be decomposed in terms of pairwise isotropic, antisymmetric, and anisotropic symmetric contributions, which can be rigorously derived in the framework of the superexchange (SE) theory, in analogy with the spin Hamiltonian. The SE theory also allows us to identify the microscopic mechanism that stands behind each contribution. The most controversial and intriguing one—concerning the form, appearances, and implications for the properties of real compounds—is the antisymmetric or spin-current mechanism. In this work, we propose that, within the SE theory, the disputed magnetoelectric (ME) properties of tetragonal ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{Ge}}_2{\mathrm{O}}_7$, representing the lattice of magnetic ${\mathrm{Cu}}^{2+}$ ions in the tetrahedral environment, can be explained solely by the spin-current mechanism, while other contributions are either small or forbidden by symmetry. First, after analysis of the symmetry properties of the SE Hamiltonian and corresponding parameters of electric polarization, we explicitly show how the cycloidal spin order induces the experimentally observed electric polarization in the direction perpendicular to the tetragonal plane, which can be naturally explained by the spin-current mechanism operating in the out-of-plane bonds. Then, we unveil the previously overlooked ME effect, where the application of the magnetic field perpendicular to the plane not only causes the incommensurate-commensurate transition, but also flips the electric polarization into the plane due to the spin-current mechanism operating in the neighboring bonds within this plane. In both cases, the magnitude and direction of $\mathit{P}$ can be controlled by rotating the spin pattern in the tetragonal plane. Our analysis is based on a realistic spin model, which was rigorously derived from first-principles electronic structure calculations and supplemented with a new algorithm for the construction of localized Wannier functions obeying the crystallographic symmetry of ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{Ge}}_2{\mathrm{O}}_7$.

Figure 1

Crystal structure of ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{Ge}}_2{\mathrm{O}}_7$ in the tetragonal plane. The Cu and O atoms are indicated by the medium red and yellow spheres, respectively, the Ba atoms are indicated by the big green spheres, and the Ge atoms are indicated by the small blue spheres. The ${\mathrm{CuO}}_4$ and ${\mathrm{GeO}}_4$ tetrahedra are colored red and green, respectively. The regular unit cell ($xy$, with the lattice parameter $a$) containing two formula units is shown by blue lines. The smaller unit cell ($x^{\prime }{y}^{\prime }$, with the lattice parameter $a_0=\frac{1}{\sqrt{2}}a$), containing one formula unit and describing the periodicity of the in-plane components of the DM interactions, is shown by black lines.

Figure 2

Left panel: Electronic band structure of ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{Ge}}_2{\mathrm{O}}_7$ with the spin-orbit coupling calculated within the QE method as well as in the Wannier basis for the 1- and 5-orbital models. Right panel: Total and partial $\mathrm{Cu}\text{−}3d$ densities of states as obtained in the QE method. The Fermi level ($E_{\mathrm{F}}$) is at zero energy. Notations of the high-symmetry points of the Brillouin zone are taken from Ref. [64].

Figure 3

Schematic view on the superexchange theory for exchange interactions and electric polarization: In the atomic limit, the hole is localized in the Wannier state $|{\alpha }_0\rangle$ of the central site 0. Then, in the 1st order of perturbation theory with respect to the transfer integrals, ${\widehat{t}}_{0j}$, this Wannier function acquires tails $|{\alpha }_{0\to j}\rangle$ spreading to the neighboring sites $j$. By knowing the wave functions to the 1st order in ${\widehat{t}}_{0j}$, one can evaluate the energy to the 2nd order in ${\widehat{t}}_{0j}$, which constitutes the basis of the superexchange theory of the exchange interactions. Equivalently, the electric polarization can be related to the change of the Wannier density to the 2nd order in ${\widehat{t}}_{0j}$. ${\mathit{\tau }}_{ji}$ denotes the vector connecting the atomic site $i$ with the site $j$.

Figure 4

(a) Fragment of the crystal structure of ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{Ge}}_2{\mathrm{O}}_7$ in the tetragonal plane, explaining the environment of the Cu sites $0–4$. (b) The in-plane components of Dzyaloshniskii-Moriya interactions ${\mathit{D}}_{0j}$ operating in the nearest-neighbor Cu-Cu bonds $0\text{−}j$ (the transfer integrals ${\mathit{t}}_{0j}$ in the 1-orbital model obey the same symmetry rules). (c) The vectors $\mathcal{P}$ describing the antisymmetric part of electric polarization in the same bonds. (d) The symmetry properties of vectors $\mathcal{P}$ in the next-nearest-neighbor bonds between the plane.

Figure 5

(a) Side view and (b), (c) top views on cycloids with the propagation vectors $\mathit{q}$ (b) and ${\mathit{q}}^{*}$ (c). By applying the magnetic field perpendicular to the spin rotation plane, one can switch between magnetic domains with $\mathit{q}$ and ${\mathit{q}}^{*}$, and thus reverse the electric polarization $P^z$.

Figure 6

Summary of incommensurate-commensurate transition in ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{Ge}}_2{\mathrm{O}}_7$ induced by the magnetic field $\mathit{H}\parallel z$: The incommensurate cycloidal phase (a) is transformed into the $C$-type antiferromagnetic phase, which is illustrated in the inset of (b). The magnetic field leads to the canting of spins as explained in (b) and (c). The transition is accompanied by the flip of electric polarization from $\mathit{P}\parallel z$ to $\mathit{P}\perp z$, which is driven by the spin-current mechanism. The magnetic sites surrounding the central atom 0, which contribute to the electric polarization in the cycloidal and $C$-type antiferromagnetic phases, are denoted by the cyan and green colors, respectively. $\delta \mathit{q}$ specifies the direction of the propagation vector.

Figure 7

Magnetoelectric effect in ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{Ge}}_2{\mathrm{O}}_7$: Absolute value of electric polarization in the $xy$ plane versus magnetic field.

Figure 8

Magnetoelectric effect in ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{Ge}}_2{\mathrm{O}}_7$: Directions of electric polarization in the $xy$ plane, ${\mathit{P}}^{xy}$, corresponding to different types of antiferromagnetic domains for the same direction of the external magnetic field $\mathit{H}=(0,0,H)$. The directions of spin moments are shown by cyan arrows.