Confinement-induced depletion of the enhanced $g$-factor in quantum wires

The spin splitting of the electronic subbands in a parabolically confined quantum wire in a strong magnetic field is calculated using the self-consistent Hartree-Fock approximation. The effective $g$-factor and the critical density at which the subbands become almost spin-degenerate are determined. The results are compared with recent experimental data.

Figure 1

Hartree-Fock energies of a quantum wire in a magnetic field for the four lowest subbands at densities where the second (spin ↓, left) and third (spin ↑, right) subbands start to be occupied (bullets: $s=\uparrow$, circles: $s=\downarrow$, dashed: Fermi level).

Figure 2

Spin polarization $\gamma$ (top) and effective $g$-factor $g^{*}$ (bottom) as functions of the density $\rho$ obtained from the self-consistent Hartree-Fock method taking into account the three lowest Landau bands (parameters: $B=14\mathrm{T}$, $\hslash {\omega }_0=6\mathrm{meV}$, ${\rho }_{\mathrm{c}}=1.1∕l$ in the lowest subband, screening parameter $\kappa =2∕l$). Inset: Energy $ϵ\left(k\right)$ (dashed) in the noninteracting limit; self-energies $\Sigma$, ${\Sigma }^F$ (Fock), and ${\Sigma }^H$ (Hartree) at ${\rho }_{\mathrm{c}}$, respectively, as functions of the wave number $kl$; solid: perturbative, bullets: self-consistent results.

Figure 3

Crossover density as a function of the screening length ${\kappa }^{-1}$. Data points: self-consistent Hartree-Fock approximation with $B=14\mathrm{T}$, $\hslash {\omega }_0=6\mathrm{meV}$ in the lowest subband; wire width $16l$ and wire length $50l$. Inset: self-energy with $\kappa =0.5∕l$ (dashed), $\kappa =2.0∕l$ (full), and $\kappa =6.0∕l$ (dotted).

Figure 4

Crossover density ${\rho }_{\mathrm{c}}$ at magnetic fields as indicated as a function of the confinement energy $\hslash {\omega }_{\mathrm{p}}$. Full lines: Hartree-Fock results with screening length $l∕2$; open symbols: experimental data of Ref. 9 corresponding to side gate voltages $V_{\text{side}}$ from $-1.5\text{to}-4\mathrm{V}$. Inset: results for $B=9\mathrm{T}$ and $B=14\mathrm{T}$ obtained self-consistently (dots), and perturbatively (full lines) compared with the results obtained by neglecting the Hartree term (dashed).