Origin of up-up-down-down magnetic order in ${\mathrm{Cu}}_2{\mathrm{GeO}}_4$

We use density-functional band-structure calculations to explore the origin of the up-up-down-down (UUDD) magnetic order in ${\mathrm{Cu}}_2{\mathrm{GeO}}_4$ with the frustrated $J_1\text{−}{J}_2$ spin chains coupled into layers within the spinel-like crystal structure. In contrast to earlier studies, we find that the nearest-neighbor coupling $J_1$ should be negligibly small, due to a nearly perfect compensation of the ferromagnetic direct exchange and antiferromagnetic superexchange. Under this condition, weak symmetric anisotropy of the exchange couplings gives rise to the UUDD order observed experimentally and also elucidates the nontrivial ordering pattern between the layers, whereas a small Dzyaloshinskii-Moriya interaction causes a spin canting that may generate local electric polarization. We argue that the buckling of the copper chains plays a crucial role in the suppression of $J_1$ in ${\mathrm{Cu}}_2{\mathrm{GeO}}_4$ and sets this compound apart from other $J_1\text{−}{J}_2$ chain magnets.

Figure 1

(a) Crystal structure of ${\mathrm{Cu}}_2{\mathrm{GeO}}_4$. (b) Structural chains of copper atoms with the exchange integrals following the notation of Ref. [25]. The green arrows show Dzyaloshinskii-Moriya vectors, while the red arrows represent electron spins that form the UUDD pattern according to Ref. [24]. Crystal structures were visualized using the vesta software [36].

Figure 2

Electronic density of states for ${\mathrm{Cu}}_2{\mathrm{GeO}}_4$ obtained on the DFT (GGA) level.

Figure 3

Band structure of ${\mathrm{Cu}}_2{\mathrm{GeO}}_4$ near the Fermi level calculated within DFT and $\mathrm{DFT}+\mathrm{SO}$. The inset shows the magnified view to highlight the band splitting.

Figure 4

LT wave vector ${\mathbf{Q}}^{\text{LT}}$ depending on the ratio $J_1/J_2$ within isotropic $J_1\text{−}{J}_2$ model.

Figure 5

Luttinger-Tisza eigenvalues along $\mathbf{q}=(\frac{\pi }{2},q)$ obtained using the $J_{ij}$ values from the fplo and vasp codes, respectively (Table 2).

Figure 6

Magnetic order in the $J_2\text{−}J$ model stabilized under (a) the action of anisotropic terms $\mathrm{\Gamma }$; (b) the combination of anisotropic terms $\mathrm{\Gamma }$ and antiferromagnetic interlayer coupling $J_c$. The ordering pattern in (b) is identical to the experimental magnetic structure of ${\mathrm{Cu}}_2{\mathrm{GeO}}_4$, depicted in (c) [24].

Figure 7

Small deviation from the collinear UUDD order caused by the nearest-neighbor DM interactions in ${\mathrm{Cu}}_2{\mathrm{GeO}}_4$. The green arrows represent the DM vectors, while the red arrows depict the spins of the magnetic ${\mathrm{Cu}}^{2+}$ ions forming the UUDD pattern. Black circles schematically show the direction of spin canting along $c$.

Figure 8

The Wannier orbitals of $x^2\text{−}{y}^2$ symmetry obtained (a) within the one-band model that includes the states at the Fermi level only, and (b) by taking all Cu $d$ and oxygen $p$ states into account (Fig. 2). Different colors denote different phases of the Wannier orbital. The lower graphs are three-dimensional magnetic form factors represented on the same isosurface level. (c) A comparison between the ionic ${\mathrm{Cu}}^{2+}$ form factor obtained within the three-Gaussian approximation [64] and the covalent form factors calculated by powder-averaging of (a) and (b).

Figure 9

Experimental magnetic susceptibility of ${\mathrm{Cu}}_2{\mathrm{GeO}}_4$ [23] and its fits with the spin-chain ($J$-only) and rectangular-lattice ($J\text{−}{J}_2$) models.