First-principles modeling of the thermoelectric properties of SrTiO${}_3$/SrRuO${}_3$ superlattices

Using a combination of first-principles simulations, based on density functional theory and Boltzmann's semiclassical theory, we have calculated the transport and thermoelectric properties of the half-metallic two-dimensional electron gas confined in single SrRuO${}_3$ layers of SrTiO${}_3$/SrRuO${}_3$ periodic superlattices. Close to the Fermi energy, we find that the semiconducting majority-spin channel displays a very large in-plane component of the Seebeck tensor at room temperature, $S\sim$ 1500 $\mu$V/K, and the minority-spin channel shows good in-plane conductivity, $\sigma =2.5$ (m$\Omega$ cm)${}^{-1}$. However, we find that the total power factor and thermoelectric figure of merit for reduced doping is too small for practical applications. Our results support that the confinement of the electronic motion is not the only thing that matters to describe the main features of the transport and thermoelectric properties with respect the chemical doping, but the shape of the electronic density of states, which in our case departs from the free-electron behavior, is also important. The evolution of the electronic structure, electrical conductivity, Seebeck coefficient, and power factor as a function of the chemical potential is explained by a simplified tight-binding model. We find that the electron gas in our system is composed by a pair of one-dimensional electron gases orthogonal to each other. This reflects the fact the physical dimensionality of the electronic system (1D) can be even smaller than that of the spacial confinement of the carriers (2D).

Figure 1

(a) Schematic representation of the unit cell, periodically repeated in space, of (SrTiO${}_3$)${}_5$/(SrRuO${}_3$)${}_1$ superlattices. Ti atoms (in blue) and Ru ones (in gray) are situated at the center of the octahedra, as denoted by the layer labeling, while O (in red) is placed at the vertex and Sr (in green) at the interstices. (b) Layer-by-layer PDOS on the atoms at the Sr$B$O${}_3$ ($B$ $=$ Ti or Ru) for the corresponding layer at the same height as in (a). Majority (minority) spin is represented in the upper (lower) half of each panel.

Figure 2

Total DOS (solid black lines) and PDOS on some Ru(4$d$) orbitals, showing the main character of the bands around the Fermi energy in (SrTiO${}_3$)${}_5$/(SrRuO${}_3$)${}_1$ superlattices. Green dot-dashed lines are the projections on the 4$d_{xy}$ orbitals, blue dashed lines those on the 4$d_{xz,yz}$ orbitals (these curves are exactly degenerate), and red dotted lines those on the 4$d_{x^2-y^2}$ orbitals. The zero of energies is aligned at the Fermi level. The wiggles around zero are caused by the finite $k$ resolution.

Figure 3

Results of the calculation of the electronic DOS and transport properties at 300 K [conductivity ($\sigma$), Seebeck coefficient ($S$), and power factor (PF)] of (SrTiO${}_3$)${}_5$/(SrRuO${}_3$)${}_1$ superlattices as obtained by Boltzmann's semiclassical transport theory. On the left column we show results for the majority-spin bands and on the right for the minority-spin ones.

Figure 4

Total PF for the (SrTiO${}_3$)${}_5$/(SrRuO${}_3$)${}_1$ superlattices at 300 K as a function of the chemical potential.

Figure 5

Graphical representation of (a) one-dimensional tight-binding and (b) free-electron bands that illustrate, respectively, Eqs. (9) and (8). In (a) we show with red dashed parabolae the free-electron band approximation to the tight-binding model at the band edge while the blue dash-dotted line depicts the group velocity at the midband ($\varepsilon =0$) and the band bottom ($\varepsilon =-2\gamma$).

Figure 6

Density of states (DOS), electrical conductivity ($\sigma$), Seebeck coefficient (changed sign, $-S$), and power factor (PF) for a free-electron model as described in Eq. (8) (left panels), and the tight-binding model of Eq. (9) (right panels). Different colors represent the dimensionality of the system: solid black lines denote three dimensional, blue dashed lines denote two dimensional, and red dash-dotted lines denote one dimensional.

Figure 7

Illustration of the 4$d$ orbitals of Ru in the RuO${}_2$ layer and hopping parameters, $\gamma$, for (a) the $d_{xz}$ (an equivalent schema, rotated 90${}^{\circ }$, could be made for the $d_{yz}$), (b) the $d_{xy}$, and (c) the $d_{x^2-y^2}$ orbitals. In (a), the hopping along $z$ is strongly hindered as it involves moving into Ti levels which are much higher in energy. From the overlap of the orbitals it is clear than $\gamma \gg {\gamma }^{\prime }$. Meaning of the colors for the atoms as in Fig. 1a.