Quasiparticle mass enhancement and Fermi surface shape modification in oxide two-dimensional electron gases

We propose a model that is intended to qualitatively capture the electron-electron interaction physics of two-dimensional electron gases formed near transition-metal oxide heterojunctions containing $t_{2g}$ electrons with a density much smaller than one electron per metal atom. Two-dimensional electron systems of this type can be described perturbatively using a $GW$ approximation, which predicts that Coulomb interactions enhance quasiparticle effective masses more strongly than in simple two-dimensional electron gases, and that they reshape the Fermi surface, reducing its anisotropy.

Figure 1

A schematic summary of the $t_{2g}$ two-dimensional electron-gas model. In (a) the energy offset $\mathrm{\Delta }$ between the anisotropic $xz$ and $yz$ (red) band edge and the isotropic $xy$ (blue) band edge is emphasized. In (b) the anisotropy of the elliptical $xz$ and $yz$ Fermi surfaces (red) is contrasted with the $xy$ band's circular Fermi surface (blue). The lower panel highlights the difference in $\widehat{\mathit{z}}$-direction confinement between the $xz$ and $yz$ bands and the $xy$ band, which can be crudely characterized by a separation distance $d$.

Figure 2

Feynman diagrams representing the $G_{0}W$-RPA for the self-energy of a quasiparticle in one of the elliptical bands of the $t_{2g}$ 2DEG. In the RPA, the bare (i.e., unscreened) Coulomb interaction is dressed (i.e., screened) by an infinite series of bubble diagrams representing the density fluctuations of each component (e.g., each band) of the system. When applied to the three-band $t_{2g}$ 2DEG, each bubble represents a density fluctuation in either the circular $xy$ band, ${\chi }_{\mathrm{C}}\equiv {\chi }_{xy}^{\left(0\right)}$, or the elliptical $yz$ and $xz$ bands, ${\chi }_{\mathrm{E}}\equiv {\chi }_{yz}^{\left(0\right)}+{\chi }_{xz}^{\left(0\right)}$. The circular $xy$ band and the elliptical $xz$ and $yz$ bands have different $\widehat{\mathit{z}}$-direction confinement, which we have crudely accounted for with the separation distance $d$. As a result, thin wavy lines representing the bare Coulomb interaction are equal to either $v_q$ or $v_q{e}^{-qd}$ depending on whether the density fluctuations attached to each thin wavy line's end points are in the same layer or different layers, respectively. Directed lines represent the noninteraction Green's function of the elliptical-band quasiparticle in question, as in Eq. (16) with $\alpha =yz\mathrm{or}xz$. The thick wavy line represents the fully screened RPA interaction given by Eq. (13).

Figure 3

Fractional change in the $xz$ band's Fermi wave vectors, $k_{\mathrm{F}x}^{*}/k_{\mathrm{F}x}$ (top) and $k_{\mathrm{F}y}^{*}/k_{\mathrm{F}y}$ (bottom), as a function of interaction strength parameter $r_{\mathrm{s}}$. The red curve is for band offset $\mathrm{\Delta }=35\mathrm{meV}$ and layer separation $d=10a$, where $a=3.9$ Å is the ${\mathrm{SrTiO}}_3$ lattice constant. The blue curve is for band offset $\mathrm{\Delta }=200\mathrm{meV}$ and layer separation $d=2a$. The dashed green curve is for a single-band anisotropic 2DEG whose $\widehat{\mathit{x}}$-direction and $\widehat{\mathit{y}}$-direction noninteracting masses are the same as those of the $xz$ band in the $t_{2g}$ 2DEG model. For this curve, $r_{\mathrm{s}}$ is defined from the total density in the single anisotropic band.

Figure 4

Same as Fig. 3, but calculated by the self-consistent solution of Eqs. (42) and (43), rather than using the linearized Eqs. (26) and (27).

Figure 5

The leading order in $r_{\mathrm{s}}$ quasiparticle mass enhancement factor [i.e., the inverse of Eq. (55)] is plotted for $r_{\mathrm{s}}\le 1$. The dashed red line shows the result for an isotropic single-band 2DEG, and the solid blue line shows the result for the $xz$ band of the $t_{2g}$ 2DEG, which also includes the effect of screening by electrons in the $yz$ and $xy$ bands. The Thomas-Fermi screened electron-electron interaction gives the leading-order contribution to quasiparticle mass enhancement at small $r_{\mathrm{s}}$ (high density). The presence of multiple filled bands in the $t_{2g}$ 2DEG yields a larger Thomas-Fermi screening wave vector, and therefore a larger quasiparticle mass enhancement than in a single-band 2DEG. Inset: Same as the main figure but plotted over $r_{\mathrm{s}}\le 0.1$. Orange circles and green triangles label data points from the numerical evaluation of the full $G_{0}W$-RPA equations for the effective mass, demonstrating that expression (55) is correct, but it is only quantitatively useful for small $r_{\mathrm{s}}$.

Figure 6

The quasiparticle mass enhancement factor of an isotropic single-band 2DEG (brown dashed line) compared against the light (purple) and heavy (green) quasiparticle mass enhancement factors of an anisotropic single-band 2DEG.

Figure 7

Panel (a) shows the light quasiparticle mass of the $t_{2g}$ 2DEG's $xz$ band plotted against $r_{\mathrm{s}}$ for several values of the band offset $\mathrm{\Delta }$ and separation $d$. For the red curve, $\mathrm{\Delta }=35\mathrm{meV}$ and $d=10a$, where $a=3.9$ Å is the ${\mathrm{SrTiO}}_3$ lattice constant. For the blue curve, $\mathrm{\Delta }=200\mathrm{meV}$ and $d=2a$. The dashed green curve is for a single-band anisotropic 2DEG whose $\widehat{\mathit{x}}$-direction and $\widehat{\mathit{y}}$-direction noninteracting masses are the same as the $xz$ band in the $t_{2g}$ model. For this curve, $r_{\mathrm{s}}$ is defined from the total density in the single anisotropic band. Panel (b) is the same, but for the heavy effective mass.