Ground-State Electronic Structure of Quasi-One-Dimensional Wires in Semiconductor Heterostructures

We apply density-functional theory, in the local-density approximation, to a quasi-one-dimensional electron gas in order to quantify the effect of Coulomb and correlation effects in modulating and, therefore, patterning, the charge-density distribution. Our calculations are presented specifically for surface-gate-defined quasi-one-dimensional quantum wires in a GaAs-(AlGa)As heterostructure, but we expect our results to apply more generally for other low-dimensional semiconductor systems. We show that at high densities with strong confinement, screening of electrons in the direction transverse to the wire is efficient and density modulations are not visible. In the low-density, weak-confinement regime, the exchange-correlation potential induces small density modulations as the electrons are depleted from the wire. At the weakest confinements and lowest densities, the electron density splits into two rows, thereby forming a pair of quantum wires that lies beneath the surface gates. An additional double-well external potential forms at very low density which enhances this row-splitting phenomenon. We produce phase diagrams that show a transition between the presence of a single quantum wire in a split-gate structure and two quantum wires. We suggest that this phenomenon can be used to pattern and modulate the electron density in low-dimensional structures with particular application to systems where a proximity effect from a surface gate is valuable.

Figure 1

(a) Hartree and (b) LDA calculations of the lateral potential at 4.2 K on a cross section 6 nm below the $\mathrm{GaAs}/(\mathrm{AlGa})\mathrm{As}$ interface for the device described in Sec. 3.

Figure 2

(a) Hartree and (b) LDA calculations of the electron charge density at 4.2 K along the same cross section as in Fig. 1. The electron screening is efficient, and density modulations in the transverse direction vanish for this strong-confinement regime. The difference between the Hartree and local-density approximations is minimal.

Figure 3

(a) Hartree and (b) LDA calculations of the integrated transverse density for $V_{\mathrm{SG}}=-0.50\mathrm{V}$ at 50 mK. The electrons do not form a double-row density, but the exchange-correlation potential generates small transverse density modulations, which do not occur in the Hartree simulations.

Figure 4

(a) Hartree and (b) LDA calculations of the integrated transverse density for $V_{\mathrm{SG}}=-0.46\mathrm{V}$ at 50 mK. Traces show the density for different top-gate voltages from $V_{\mathrm{TG}}=-1.70\mathrm{V}$ to $V_{\mathrm{TG}}=-1.40\mathrm{V}$ in steps of $\mathrm{\Delta }{V}_{\mathrm{TG}}=0.02\mathrm{V}$. The system is in the weak-confinement regime where electrostatic repulsion is dominant and double-row formation occurs. For high densities, the electrons screen themselves, but as the density decreases, two rows form in both the Hartree and LDA calculations. For $\mathrm{\Delta }{V}_{\mathrm{TG}}<-1.58\mathrm{V}$, the external potential $V_{\mathrm{ext}}(\overrightarrow{r})$ acquires a double-well structure which enhances row separation. When the wires are completely decoupled, the electron density lies almost entirely underneath the surface gates.

Figure 5

Number of peaks in the transverse density as a function of $V_{\mathrm{TG}}$ and $V_{\mathrm{SG}}$ using (a) the Hartree approximation and (b) the LDA. For weak confinement and low density, double-row formation occurs for small $V_{\mathrm{SG}}$ and large $V_{\mathrm{TG}}$ in both the Hartree and LDA calculations. Transverse density modulations in the LDA calculations at low density create a peaked structure, which extends to a less weakly confined regime where $V_{\mathrm{SG}}$ is more negative. Below the dashed green line, the $V_{\mathrm{ext}}(\overrightarrow{r})$ forms a double-well potential below the $\mathrm{GaAs}/(\mathrm{AlGa})\mathrm{As}$ interface.

Figure 6

Number of peaks in the transverse density as a function of $r_0$ and $r_s$ in the local-density approximation. A typical split-gate device, for example in Sec. 3, corresponds to the small-$r_s$ and -$r_0$ limit. The blue dotted line defines the transition above which the electrons separate into two rows in the Hartree approximation which corresponds to zigzag or double-row configurations and is due to electrostatic repulsion. Modulations in the transverse density generated by the exchange-correlation potential occur for larger $r_s$ and $r_0$. Calculations where $V_{\mathrm{ext}}(\overrightarrow{r})$ has two minima correspond to $r_0>10$ and are not shown in this figure.

Figure 7

Number of Kohn-Sham bands below the chemical potential as a function of $V_{\mathrm{TG}}$ and $V_{\mathrm{SG}}$ using (a) the Hartree approximation and (b) the LDA. A double jump in the number of conducting bands is seen for very weak confinement and is driven by the formation of a double-well potential in the external gate potential $V_{\mathrm{ext}}(r)$ below the dashed green line. The conduction bands do not correlate with either the formation of a double row in the density or the density modulations generated by the exchange-correlation potential.