Many-flavor electron gas approach to electron-hole drops

A many-flavor electron gas (MFEG) is analyzed, such as could be found in a multivalley semiconductor or semimetal. Using the rederived polarizability for the MFEG, an exact expression for the total energy of a uniform MFEG in the many-flavor approximation is found; the interacting energy per particle is shown to be $-0.574447\left(E_h{a}_0^{3/4}{m}^{\ast 3/4}\right)n^{1/4}$, with $E_h$ being the Hartree energy, $a_0$ being the Bohr radius, and $m^{\ast }$ being the particle effective mass. The short characteristic length scale of the MFEG motivates a local-density approximation, allowing a gradient expansion in the energy density and the expansion scheme is applied to electron-hole drops, finding a new form for the density profile and its surface scaling properties.

Figure 1

The Si band structure in the [100] direction generated with a LDA-DFT approximation by a plane-wave pseudopotential method (Ref. 4). The Fermi energy is $E=0\text{eV}$. Below are valence bands with holes at H; above are conduction bands; and the bold parabolic curve signifies the first conduction-band valley with electrons at E.

Figure 2

The dark gray ellipsoids show Fermi surfaces of electrons in the six degenerate conduction-band valleys in Si. $E\left(\mathbf{q}\right)$ is the energy with momentum $\mathbf{q}$, measured with respect to the $\Gamma$ point. ${ϵ}_i\left(\mathbf{p}\right)$ is energy with momentum $\mathbf{p}$, measured with respect to the center of the $i\text{th}$ valley.

Figure 3

The density profile of a 12-flavor electron-hole drop with density parameter $r_s=1$. The numerical solution is shown by the dotted line, analytical approximations to the inside (outside) of the drop are shown by the dashed (dot dashed) lines, and a best fit model fitted to the numerical solution is shown by the solid line.

Figure 4

The surface tension $\gamma$ of the 12-flavor drop of radius $r_m$ is shown using the solid line and pluses based on the left-hand $y$ axis. The variation of the surface thickness $D$ is shown using the dashed line and crosses based on the right-hand axis, each point represents a separate simulation.

Figure 5

The variation of surface tension (dashed line and crosses) and surface thickness (solid line and pluses) with number of flavors present. The straight-line fits are used to give the exponents of the scaling parameters.