Electronic structure and magnetic properties of the spin-$1∕2$ Heisenberg magnet ${\mathrm{Bi}}_2\mathrm{Cu}{\mathrm{O}}_4$

The electronic and magnetic structure of the spin-$1∕2$ magnet ${\mathrm{Bi}}_2\mathrm{Cu}{\mathrm{O}}_4$ is investigated by band structure calculations within the local (spin) density approximation [L(S)DA]. The insulating compound shows a complex of narrow, half-filled antibonding bands within the LDA, indicating the importance of strong on-site correlation effects. To describe the lowest-lying excitations, the strong Coulomb correlations have been taken into account by a subsequent mapping of the LDA band structure onto a tight-binding model and via an extended Hubbard model onto an extended Heisenberg model. Alternatively, the leading exchange interactions have been estimated applying the $\mathrm{LSDA}+U$ method for different spin configurations in magnetic supercells. Using our estimated exchange constants, we calculated the spin wave dispersion and the Néel temperature within the random phase approximation. The calculated properties are in good agreement with experimental data, suggesting that the magnetic ground state of ${\mathrm{Bi}}_2\mathrm{Cu}{\mathrm{O}}_4$ is governed by the antiferromagnetic interaction between the structural one-dimensional cuprate chains.

Figure 1

(Color online) The crystal structure of ${\mathrm{Bi}}_2\mathrm{Cu}{\mathrm{O}}_4$: perspective view (top), front view (down left), and lateral view (down right). The oxygen-oxygen bonds are shown only for the visualization of $\mathrm{Cu}{\mathrm{O}}_4$ plaquettes. The chains of isolated plaquettes are stacked along $z$ (top) in a staggered manner.

Figure 2

(Color online) The LDA electronic density of states: total and site-resolved partial density of states (top) and orbital-resolved density of states for $\mathrm{Cu}3d$ states (middle) and $\mathrm{O}2p$ states (bottom). The Fermi level is at zero energy. Note the different scaling in the panels showing site-resolved and orbital-resolved densities of states.

Figure 3

(Color online) The band structure (black lines) and the band weights presented by gray (green) ribbons for $\mathrm{Cu}3d_{x^2-y^2}$ and $\mathrm{O}2p_{\sigma }$ orbitals. The Fermi level is at zero energy. Notation of the high-symmetry points: $\Gamma =\left(000\right)$, $X=\left(\frac{\pi }{a}00\right)$, $S=\left(\frac{\pi }{a}\frac{\pi }{a}0\right)$, $Z=\left(00\frac{\pi }{c}\right)$, $R=\left(\frac{\pi }{a}0\frac{\pi }{c}\right)$, and $A=\left(\frac{\pi }{a}\frac{\pi }{a}\frac{\pi }{c}\right)$.

Figure 4

The transfer integrals, involved in the tight-binding model: first $\left(t_1^A\right)$ and second $\left(t_2^A\right)$ in-chain neighbors, four integrals for the interactions between the neighboring chains of different sublattices ($t_{1u}^{AB}$, $t_{1d}^{AB}$, $t_{2u}^{AB}$, and $t_{2d}^{AB}$), three integrals for the interaction between the neighboring chains of the same sublattice ($t_1^{AA}$, $t_2^{AA}$, and $t_3^{AA}$), and three for the second-neighbor chain of the same sublattice ($t_1^{AAA}$, $t_2^{AAA}$, and $t_3^{AAA}$). The superscripts $A$ and $B$ mean sublattice, the number in the subscript means the order of neighbor, and $u$ and $d$ mean along the positive (up) or negative (down) $c$ direction, respectively.

Figure 5

(Color online) The tight-binding fit presented by gray (red) circles of the LDA band structure (black). The notation of the $k$ points is the same as that in Fig. 3.

Figure 6

(Color online) The exchange integrals, obtained by mapping the tight-binding fit of the LDA band structure to the Heisenberg model (upper value) and derived from total energy $\mathrm{LSDA}+U$ calculations and spin wave dispersion (values shown in brackets). Only five physically important values are shown. The superexchange between the neighboring chains of different sublattices plays a dominant role. All values are given in meV.

Figure 7

(Color online) Different spin arrangements in the ${\mathrm{Bi}}_2\mathrm{Cu}{\mathrm{O}}_4$ structure computed by the $\mathrm{LSDA}+U$ approach. The magnetic ground state observed by neutron diffraction (Ref. 24) is shown in the upper right panel (b).

Figure 8

(Color online) The spin wave dispersion for the model (3) given by two different sets (see Table ) compared with INS + Raman data from Ref. 51.