Chern insulating phases and thermoelectric properties of EuO/MgO(001) superlattices

The topological and thermoelectric properties of ${\left(\mathrm{Eu}\mathrm{O}\right)}_n/{\left(\mathrm{Mg}\mathrm{O}\right)}_m$(001) superlattices (SLs) are explored using density functional theory calculations including a Hubbard $U$ term together with Boltzmann transport theory. In ${\left(\mathrm{Eu}\mathrm{O}\right)}_1/{\left(\mathrm{Mg}\mathrm{O}\right)}_3$(001) SL at the lattice constant of MgO a sizable band gap of 0.51 eV is opened by spin-orbit coupling (SOC) due to a band inversion between occupied localized Eu $4f$ and empty $5d$ conduction states. This inversion between bands of opposite parity is accompanied by a reorientation in the spin texture along the contour of band inversion surrounding the $\mathrm{\Gamma }$ point and leads to a Chern insulator with $C$ = –1, also confirmed by the single edge state. Moreover, this Chern insulating phase shows promising thermoelectric properties, e.g., a Seebeck coefficient between 400 and $800\mu {\mathrm{VK}}^{-1}$. A similar SOC-induced band inversion takes place also in the ferromagnetic semimetallic ${\left(\mathrm{Eu}\mathrm{O}\right)}_2/{\left(\mathrm{Mg}\mathrm{O}\right)}_2$(001) SL. Despite the vanishing band gap, it leads to a substantial anomalous Hall conductivity with values up to –1.04 $e^2/h$ and somewhat lower Seebeck coefficient. Both cases emphasize the relation between nontrivial topological bands and thermoelectricity also in systems with broken inversion symmetry.

Figure 1

(a) Schematic view of the ${\left(\mathrm{Eu}\mathrm{O}\right)}_n/{\left(\mathrm{Mg}\mathrm{O}\right)}_m$(001) superlattice with $n=2$ and (b) the corresponding Brillouin zone.

Figure 2

Element- and orbital-projected band structure of ferromagnetic ${\left(\mathrm{Eu}\mathrm{O}\right)}_n/{\left(\mathrm{Mg}\mathrm{O}\right)}_m$(001) (a) for $n=1$ and $m=3$ with an out-of-plane lattice parameter $c=2a_{\mathrm{MgO}}$ and (b) for $n=2$ and $m=2$ with optimized $c$. The corresponding projected density of states (DOS) are displayed in (c) and (d), respectively. The orbital character is color coded, highlighting that the states below/above the Fermi level have predominantly Eu-$4f$ (blue)/Eu-$5d$ (green) as well as O-$2p$ (red) character, respectively.

Figure 3

$\mathrm{GGA}+U$ band structures of ferromagnetic ${\left(\mathrm{Eu}\mathrm{O}\right)}_n/{\left(\mathrm{Mg}\mathrm{O}\right)}_m$(001) at a fixed lateral lattice constant of MgO and (a) constrained or (b) optimized out-of-plane lattice constant $c$ for $n=1$, $m=3$, as well as (c) optimized $c$ for $n=2$, $m=2$. Majority and minority channels are shown in dark blue/light orange. The corresponding $\mathrm{GGA}+U+\mathrm{SOC}$ band structures are displayed in panels (d)–(f).

Figure 4

(a), (b) $\mathrm{GGA}+U+\mathrm{SOC}$ band structures for FM ${\left(\mathrm{Eu}\mathrm{O}\right)}_1/{\left(\mathrm{Mg}\mathrm{O}\right)}_3$(001) and ${\left(\mathrm{Eu}\mathrm{O}\right)}_2/{\left(\mathrm{Mg}\mathrm{O}\right)}_2$(001) with magnetization along the [001] direction; (c), (d) Berry curvatures along the same $k$-path. The corresponding anomalous Hall conductivities (AHC) ${\sigma }_{xy}$ in units of $e^2/h$ as a function of the chemical potential are displayed in (e), (f).

Figure 5

(a) The band-decomposed spin-texture in $k$ space from the $\mathrm{GGA}+U+\mathrm{SOC}$ calculation with magnetization along [001] for the topmost occupied and lowest unoccupied bands [cf. Fig. 4] of the FM ${\left(\mathrm{Eu}\mathrm{O}\right)}_1/{\left(\mathrm{Mg}\mathrm{O}\right)}_3$(001) superlattice. The color scale denotes the projection on the $\widehat{z}$ axis with red (blue) indicating parallel (antiparallel) orientation. (b) edge states for ${\left(\mathrm{Eu}\mathrm{O}\right)}_1/{\left(\mathrm{Mg}\mathrm{O}\right)}_3$(001) and (c) ${\left(\mathrm{Eu}\mathrm{O}\right)}_2/{\left(\mathrm{Mg}\mathrm{O}\right)}_2$(001). Red/white represent higher local DOS, blue regions denote the bulk energy gap and red lines mark the edge state connecting the valence and conduction band.

Figure 6

The in- and cross-plane components of the electrical conductivity tensor divided by the relaxation time $\tau$, the Seebeck coefficient, $PF/\tau$ and the electronic contribution to the figure of merit ${ZT|}_{el}$ are shown as a function of the chemical potential in (a), (b) at 300 K and 600 K for FM ${\left(\mathrm{Eu}\mathrm{O}\right)}_1/{\left(\mathrm{Mg}\mathrm{O}\right)}_3$(001) superlattice at $c$ = 8.4 Å, whereas (c), (d) show the thermoelectric properties with the optimized out-of-plane parameter of $c_{opt}$ = 9.6 Å, and (e), (f) the thermoelectric performance of ${\left(\mathrm{Eu}\mathrm{O}\right)}_2/{\left(\mathrm{Mg}\mathrm{O}\right)}_2$(001) SL.