Wigner crystallization in inhomogeneous one-dimensional wires

We explore the theory of electrons confined by one-dimensional power-law potentials. We calculate the density profile in the high-density electron gas, the low-density Wigner crystal, and the intermediate regime. We extract the momentum-space wave function of the electron at the Fermi surface, which can be measured in experiments on tunneling between parallel wires. The onset of localization leads to a dramatic broadening of the momentum-space wave function together with pronounced sharpening (in energy) of the tunneling spectrum.

Figure 1

(Color online) Model of potential. (a) Two-dimensional gray-scale plot, where darker colors represent lower potential energy. (b) Potential along one wire [dashed (blue) line in (a)] for $\beta =2,3,6,10,50,1000$. (c) Potential transverse to the two wires [dotted (red) line in (b)].

Figure 2

Interaction-induced change of dimensionless density $\overline{n}$ of a spin-$1∕2$ electron gas confined to a one-dimensional potential $V\left(x\right)\propto x^6$. Interaction parameters are $\overline{a}=10$ and $\overline{d}={10}^{-3}$. Solid line, Eq. (15) using regularization $f^{\left(2\right)}$; dashed line, approximation described above Eq. (16); dotted line, noninteracting profile. Inset: Renormalized dimensionless cloud radius $t$ as a function of $1∕\overline{a}$ from solving Eq. (16).

Figure 3

Peak wave vector $k_f=t{k}_0$ of wave function at Fermi surface as a function of chemical potential ${\mu }_0={\hslash }^2{k}_0^2∕2m^{*}$ within the Thomas-Fermi approximation. Dotted line shows noninteracting result, $t=1$. Solid line includes Coulomb interactions with $a=10\mathrm{nm}$, $d_{\perp }=20\mathrm{nm}$, in a power-law trap with $\beta =6$ and $w=3\mu \mathrm{m}$. This approximation will break down when $k_f\lesssim 1∕a$.

Figure 4

Interaction dependence of the momentum-space wave function of last single-particle state within the Hartree-Fock approximation. Eighteen particles are confined to the wire. Lighter colors correspond to larger values of ${\mid \varphi \left(k\right)\mid }^2$. The horizontal axis shows different interaction strengths parametrized by the ratio of the Bohr radius $a$ to the wire length $w$ (note the logarithmic scale). The potential has exponent $\beta =6$ and a cutoff $d=0.07w$.

Figure 5

(Color online) Real-space electron density $n$. Parameters coincide with Fig. 4. Solid, (blue) dotted, and (red) dashed lines correspond to total density and the densities of up- and down-spin electrons [in (a), the latter two curves coincide]. Panels (a), (b), and (c) correspond to $a∕w=0.5$, 0.15, and 0.05.

Figure 6

(Color online) (a) Density of 13 electrons with $a∕w=0.1$; see caption of Fig. 5 for key. (b) Wave function ${\varphi }_i\left(x\right)$ (solid) and ${\varphi }_j\left(x\right)$ (dashed) of the two highest-energy occupied states. (c) Momentum-space wave functions: (red) dashed, ${\mid {\varphi }_i\left(k\right)\mid }^2$; (blue) dotted, ${\mid {\varphi }_j\left(k\right)\mid }^2$; solid, ${\mid {\varphi }_i\left(k\right)\mid }^2+{\mid {\varphi }_j\left(k\right)\mid }^2$.

Figure 7

(Color online) Energy of two electrons in a one-dimensional harmonic well. Thick line, exact singlet; thin line, exact triplet; (red) dotted line, variational singlet; (blue) dashed line, exact Hartree-Fock calculation. Variational Hartree-Fock result is not shown as it is indistinguishable from the (blue) dashed line. Note logarithmic scale.

Figure 8

Relative wave function for two particles in a harmonic well for $a=\left(\mathrm{a}\right)$ 1000; (b) 10; (c) 0.1. All lengths are measured in units of $d=\sqrt{\hslash ∕m\omega }$. Solid line shows exact results, while dashed lines show variational approximation.

Figure 9

(Color online) Tunneling matrix element ${\mid f\left(k\right)\mid }^2$ for two harmonically trapped particles in a singlet state: exact $a=100$ (solid), 0.05 (dashed); Hartree-Fock $a=100$ [(red) dotted], 0.05 [(green) dotted, scaled by 0.32]. All lengths are measured in units of $d=\sqrt{\hslash ∕m\omega }$. Inset shows triplet states.

Figure 10

(Color online) Top: Double-well potential along wire generated by gates. Bottom: Hartree-Fock electron density in this potential (see Fig. 5 for key). As illustrated, the characteristic length scale of the potential is $w$. The interactions are characterized by $a=0.1w$ and $d=0.07w$, yielding a low-density crystal-like central region surrounded by a high-density liquidlike region.