Localization in an inhomogeneous quantum wire

We study interaction-induced localization of electrons in an inhomogeneous quasi-one-dimensional system—a wire with two regions, one at low density and the other high. Quantum Monte Carlo techniques are used to treat the strong Coulomb interactions in the low-density region, where localization of electrons occurs. The nature of the transition from high to low density depends on the density gradient—if it is steep, a barrier develops between the two regions, causing Coulomb blockade effects. Ferromagnetic spin polarization does not appear for any parameters studied. The picture emerging is in good agreement with measurements of tunneling between wires.

Figure 1

(Color online) Two-dimensional ground-state density $\rho$ for gate potentials of different shape [all with $V_g=0.8$ (in units of $H^{\ast }$) and $N=31$]. Potentials A, B, and C have $s=15$, 4, and 2, respectively. D is obtained using a Gaussian with an angular width of 1.2 radians. Potential B is used in most of the Rapid Communication.

Figure 2

(Color online) For a short constriction, the two-dimensional ground-state density, showing a single localized electron. ($V_g=0.5H^{\ast }$, ${\theta }_0=0.7$, and $N=30$, yielding a length of $0.34\mu \text{m}$.)

Figure 3

Electron density as the low-density region is depleted. $V_g$ (in units of $H^{\ast }$) is varied for potential shape B $\left(s=4,N=31\right)$; each curve is shifted vertically by $20\mu {\text{m}}^{-1}$ with respect to the one below. For $V_g=0.4$, 8 electrons are localized. As $V_g$ increases, localization becomes stronger and a gap forms at the crystal-liquid boundary—the system is in the Coulomb blockade regime.

Figure 4

(Color online) The total effective potential (electrostatic plus external) of bulk electrons in a square well [potential is 1.8 for $x>0$ and $x<-150$, 0 otherwise (other parameters given in Ref. 37)]. Solid line is from a quantum-mechanical single-electron calculation (Inset: density, with Friedel oscillations comparable to those seen in Fig. 1 for potential A). Dashed line is obtained assuming a constant density distribution. Clearly, an electron at $x>0$ would feel a potential barrier separating it from the bulk electrons.

Figure 5

(a) Difference of total energy when electrons in the constriction are fully polarized or antiferromagnetically ordered, as a function of $V_g\left(N=31\right)$. The antiferromagnet is lower in energy for $V_g<0.6$. (b) For a homogeneous ring with 30 electrons, energy difference per electron between fully polarized and antiferromagnetic states. The antiferromagnetic state is always favored for the parameters we have studied. ($V_g=0$; density varied by changing $r_0$.)

Figure 6

(Color online) Comparison of ground-state densities obtained using LSDA, QMC with LSDA orbitals, and QMC with Gaussian orbitals at (a) $V_g=0.7$ and (b) $V_g=0.87$.