Nested Fermi surfaces and correlated electronic phases in hole-doped semiconductor quantum wells

We demonstrate the existence of novel interaction effects in hole-doped semiconductor quantum wells which are connected to dramatic changes in the Fermi surface geometry occurring upon variation of the doping. We present band structure calculations showing that quantum wells formed in $p$-type cubic semiconductors develop perfectly nested Fermi surfaces at a critical hole density $p\sim 1/d^2$ set by the width $d$ of the quantum well. Nesting gives rise to competing superconducting and charge or spin density wave order, which we analyze using the perturbative renormalization group method. The correlated phases may be created or destroyed by tuning the hole density towards or away from the critical density. Our results establish $p$-type semiconductor quantum wells as a platform for novel correlated phases, which may be precisely controlled using electrostatic gating and external magnetic fields.

Figure 1

Dispersion of 2D subbands, ${\varepsilon }_n\left(\right|\mathit{k}\left|\right)$ in Ge (left) and GaAs (right) semiconductor quantum wells of width 20 nm along the [100] and [110] directions (solid and dashed lines, respectively). The horizontal blue dashed lines show the values of the Fermi energy for a typical 2D hole density $p={10}^{11} {\mathrm{cm}}^{-2}$, while the horizontal solid red lines show the values of the Fermi energy at the critical density at which the curvature of the Fermi surface changes sign, which in Ge (left) is $p\approx 2.56\times {10}^{11} {\mathrm{cm}}^{-2}$ and in GaAs (right) are $p\approx 1.99\times {10}^{11}/2.92\times {10}^{11} {\mathrm{cm}}^{-2}$ for the larger/smaller spin-split Fermi surfaces. Energy and momentum are plotted in units of meV and $\pi /d$, respectively. The Luttinger parameters ${\gamma }_1=13.38,{\gamma }_2=4.24,{\gamma }_3=5.69$ for Ge and ${\gamma }_1=6.85,{\gamma }_2=2.10,{\gamma }_3=2.90$ for GaAs and bulk inversion asymmetry parameters $b_{41}^{8v8v}=81.93$ eV Å${}^3, C_k=-3.4$ meV Å for GaAs were taken from [32].

Figure 2

Evolution of the Fermi surface of a Ge quantum well with increasing hole doping, with hole densities (in units of ${10}^{11} {\mathrm{cm}}^{-2}$) (left) $p=5.13,1.03,1.54,2.05$; (right) $p=2.18,2.31,2.43,2.56$. These densities correspond to $p=0.2p^{*},0.4p^{*},0.6p^{*},0.8p^{*}$ for the left and $p^{*}=0.85p^{*},0.9p^{*},0.95p^{*},p^{*}$ where $p^{*}$ is the doping at which the curvature of the sides of the Fermi surface changes sign (as defined in the text). The band parameters used were the same as for Fig. 1.

Figure 3

The spin-split Fermi surfaces in a GaAs quantum well in zero magnetic field, with hole densities (a) $p=1.99\times {10}^{11} {\mathrm{cm}}^{-2}$, at which the larger Fermi surface is nested and (b) $p\approx 2.91\times {10}^{11} {\mathrm{cm}}^{-2}$, at which the smaller Fermi surface is nested. The band parameters used were the same as for Fig. 1.

Figure 4

Nested Fermi surfaces for a 20 nm Ge quantum well in a 1 T magnetic field applied along the [010] direction, at hole densities (a) $1.97\times {10}^{11} {\mathrm{cm}}^{-2}$ and (b) $3.00\times {10}^{11} {\mathrm{cm}}^{-2}$. Portions of the Fermi surface on which the normal momentum coordinate varies by less than 0.25% are highlighted in red. The band parameters used were the same as for Fig. 1.

Figure 5

Nested Fermi surfaces for a 20 nm GaAs quantum well in a 1 T magnetic field applied along the [010] direction, at densities (a) $p=1.94\times {10}^{11} {\mathrm{cm}}^{-2}$ and (b) $2.83\times {10}^{11} {\mathrm{cm}}^{-2}$. Portions of the Fermi surface on which the normal momentum coordinate varies by less than 0.25% are highlighted in red. The band parameters used were the same as for Fig. 1.

Figure 6

One-loop corrections to the interaction vertex.

Figure 7

(a) Coupling constants used in the patch RG model $g_1=\nu \mathrm{\Gamma }\left(ABAB\right), g_2=\nu \mathrm{\Gamma }\left(ACAC\right), g_3=\nu \mathrm{\Gamma }\left(ACDB\right)$. (b) RG flow of the couplings $g_1,g_2,g_3$ plotted in the blue, green, and red lines, respectively, for initial values $g_1=g_2=0.3,g_3=0.1$, and the dimensionless susceptibilities ${\mathrm{\Delta }}^{SC}/{\mathrm{\Delta }}_0^{SC}$ (violet, dotted) and ${\mathrm{\Delta }}^{DW}/{\mathrm{\Delta }}_0^{DW}$ (cyan, dotted and dashed).

Figure 8

The spatial modulation of the charge density in the CDW phase for the cases where (a) four sides of the Fermi surface exhibit nesting equally, (b) the left and right sides are nested over a greater region than the top and bottom sides, and (c) only the left and right sides are nested. Darker/brighter regions correspond to higher/lower density.