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arXiv:0707.2992 · Physical Review B

Hartree-Fock calculations of a finite inhomogeneous quantum wire

Jiang Qian, Bertrand I. Halperin

Published Feb 21, 2008• 2 citations
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Abstract

We use the Hartree-Fock method to study an interacting one-dimensional electron system on a finite wire, partially depleted at the center by a smooth potential barrier. A uniform $1\phantom{\rule{0.2em}{0ex}}\mathrm{T}$ Zeeman field is applied throughout the system. We find that with the increase in the potential barrier, the low density electrons under it go from a nonmagnetic state to an antiferromagnetic state and then to a state with a well-localized spin-aligned region isolated by two antiferromagnetic regions from the high density leads. At this final stage, in response to a continuously increasing barrier potential, the system undergoes a series of abrupt density changes, corresponding to the successive expulsion of a single electron from the spin-aligned region under the barrier. Motivated by the recent momentum-resolved tunneling experiments in a parallel wire geometry, we also compute the momentum-resolved tunneling matrix elements. Our calculations suggest that the eigenstates being expelled are spatially localized, consistent with the experimental observations. However, additional mechanisms are needed to account for the experimentally observed large spectral weight near $k=0$ in the tunneling matrix elements.

DOI
10.1103/physrevb.77.085314

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We use the Hartree-Fock method to study an interacting one-dimensional electron system on a finite wire, partially depleted at the center by a smooth potential barrier. A uniform 1T1\phantom{\rule{0.2em}{0ex}}\mathrm{T}1T Zeeman field is applied throughout the system. We find that with the increase in the potential barrier, the low density electrons under it go from a nonmagnetic state to an antiferromagnetic state and then to a state with a well-localized spin-aligned region isolated by two antiferromagnetic regions from the high density leads. At this final stage, in response to a continuously increasing barrier potential, the system undergoes a series of abrupt density changes, corresponding to the successive expulsion of a single electron from the spin-aligned region under the barrier. Motivated by the recent momentum-resolved tunneling experiments in a parallel wire geometry, we also compute the momentum-resolved tunneling matrix elements. Our calculations suggest that the eigenstates being expelled are spatially localized, consistent with the experimental observations. However, additional mechanisms are needed to account for the experimentally observed large spectral weight near k=0k=0k=0 in the tunneling matrix elements.
Motivated by recent tunneling experiments in the parallel wire geometry, we calculate results for momentum resolved tunneling into a short one-dimensional wire, containing a small number of electrons. We derive some general theorems about the momentum dependence, and we carry out exact calculations for up to N=4N=4N=4 electrons in the final state, for a system with screened Coulomb interactions that models the situation of the experiments. We also investigate the limit of large NNN using a Luttinger-liquid type analysis. We consider the low-density regime, where the system is close to the Wigner crystal limit, and where the energy scale for spin excitations can be much lower than for charge excitations, and we consider temperatures intermediate between the relevant spin energies and charge excitations, as well as temperatures below both energy scales.
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We consider a long quantum wire at low electron densities. In this strong-interaction regime a Wigner crystal may form, in which electrons comprise an antiferromagnetic Heisenberg spin chain. The coupling constant JJJ is exponentially small, as it originates from tunneling of two neighboring electrons through the segregating potential barrier. We study this exponential dependence, properly accounting for the many-body effects and the finite width of the wire.
The low-energy properties of a homogeneous one-dimensional electron system are completely specified by two Tomonaga-Luttinger parameters ${K}_{\ensuremath{\rho}}$ and ${v}_{\ensuremath{\sigma}}.$ In this paper we discuss microscopic estimates of the values of these parameters in semiconductor quantum wires that exploit their relationship to thermodynamic properties. Motivated by the recognized similarity between correlations in the ground state of a one-dimensional electron liquid and correlations in a Wigner crystal, we evaluate these thermodynamic quantities in a self-consistent Hartree-Fock approximation. According to our calculations, the Hartree-Fock approximation ground state is a Wigner crystal at all electron densities and has antiferromagnetic order that gradually evolves from spin-density wave to localized in character as the density is lowered. Our results for ${K}_{\ensuremath{\rho}}$ are in good agreement with weak-coupling perturbative estimates ${K}_{\ensuremath{\rho}}^{\mathrm{pert}}$ at high densities, but deviate strongly at low densities, especially when the electron-electron interaction is screened at long distances. ${K}_{\ensuremath{\rho}}^{\mathrm{pert}}\ensuremath{\sim}{n}^{1/2}$ vanishes at small carrier density n, whereas we conjecture that ${K}_{\ensuremath{\rho}}\ensuremath{\rightarrow}1/2$ when $\stackrel{\ensuremath{\rightarrow}}{n}0,$ implying that ${K}_{\ensuremath{\rho}}$ should pass through a minimum at an intermediate density. Observation of this nonmonotonic dependence could be used to measure the effective interaction range in a realistic semiconductor quantum wire geometry. In the spin sector we find that the spin velocity decreases with increasing interaction strength or decreasing n. Strong correlation effects make it difficult to obtain fully consistent estimates of ${v}_{\ensuremath{\sigma}}$ from Hartree-Fock calculations. We conjecture that ${v}_{\ensuremath{\sigma}}{/v}_{\mathrm{F}}\ensuremath{\propto}{n/V}_{0},$ where V0{V}_{0}V0​ is the interaction strength, in the limit $\stackrel{\ensuremath{\rightarrow}}{n}0.$