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arXiv:cond-mat/0306654 · Journal of the Physical Society of Japan

Tunneling Exponents in Realistic Quantum Wires Using the Mean Field Approximation

Wolfgang Häusler, Allan H. MacDonald

Published Jan 3, 2003• 0 citations
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Abstract

It is demonstrated that the charge Tomonaga-Luttinger parameter $K_ρ$ of quantum wires can be estimated accurately using the Hartree-Fock approximation if carried out self consistently. The dependence of $K_ρ$ on the carrier density distinguishes different regimes of importance of correlations.

DOI
10.1143/jpsjs.72sa.195

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It is demonstrated that the charge Tomonaga-Luttinger parameter KρK_ρKρ​ of quantum wires can be estimated accurately using the Hartree-Fock approximation if carried out self consistently. The dependence of KρK_ρKρ​ on the carrier density distinguishes different regimes of importance of correlations.
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 $\stackrel{\ensuremath{\rightarrow}}{n}0.$
The spin sector of charge-spin separated single mode quantum wires is studied, accounting for realistic microscopic electron-electron interactions. We utilize the ladder approximation (LA) to the interaction vertex and exploit thermodynamic relations to obtain spin velocities. Down to not too small carrier densities our results compare well with existing quantum Monte Carlo (QMC) data. Analyzing second-order diagrams we identify logarithmically divergent contributions as crucial which the LA includes but which are missed, for example, by the self-consistent Hartree-Fock approximation. Contrary to other approximations the LA yields a nontrivial spin conductance. Its considerably smaller computational effort compared to numerically exact methods, such as the QMC method, enables us to study overall dependences on interaction parameters. We identify the short distance part of the interaction to govern spin sector properties.
Using the self-consistent Hartree-Fock approximation for spinless electrons at zero temperature, we study tunneling of the interacting electron gas through a single δ barrier in a finite one-dimensional wire connected to contacts.Our results exhibit features known from correlated many-body models.In particular, the conductance decays with the wire length as ∝ L -2α , where the power α is universal.We also show that a similar result for a wire conductance can be extracted from the persistent current (I) through the δ barrier in a one-dimensional ring, where it is known that I ∝ L -1-α .
Luttinger liquids occupy a notable place in physics as one of the most understood classes of quantum many-body systems. The experimental mission of measuring its main prediction, power laws in observable quantities, has already produced a body of exponents in different semiconductor and metallic structures. Here, we combine tunneling spectroscopy with density-dependent transport measurements in the same quantum wires over more than two orders of magnitude in temperature to very low electron temperatures down to ∼40 mK. This reveals that, when the second 1D subband becomes populated, the temperature dependence splits into two ranges with different exponents in the power-law dependence of the conductance, both dominated by the finite-size effect of the end-tunneling process. This result demonstrates the importance of measuring the Luttinger parameters as well as the number of modes independently through spectroscopy in addition to the transport exponent in the characterization of Luttinger liquids. This opens a pathway to unambiguous interpretation of the exponents observed in quantum wires.
Low-energy properties of the homogeneous electron gas in one dimension are completely described by the group velocities of its charge (plasmon) and spin collective excitations. Because of the long range of the electron-electron interaction, the plasmon velocity is dominated by an electrostatic contribution and can be estimated accurately. We report on Quantum Monte Carlo simulations which demonstrate that the spin velocity is substantially decreased by interactions in semiconductor quantum wire realizations of the one-dimensional electron liquid.
We study the ground-state properties of ferromagnetic quasi-one-dimensional quantum wires using the quantum Monte Carlo (QMC) method for various wire widths bbb and density parameters rs{r}_{\text{s}}rs​. The correlation energy, pair-correlation function, static structure factor, and momentum density are calculated at high density. It is observed that the peak in the static structure factor at k=2kFk=2{k}_{\text{F}}k=2kF​ grows as the wire width decreases. We obtain the Tomonaga-Luttinger liquid parameter ${K}_{\ensuremath{\rho}}$ from the momentum density. It is found that ${K}_{\ensuremath{\rho}}$ increases by about 10% between wire widths b=0.01b=0.01b=0.01 and b=0.5b=0.5b=0.5. We also obtain ground-state properties of finite-thickness wires theoretically using the first-order random phase approximation (RPA) with exchange and self-energy contributions, which is exact in the high-density limit. Analytical expressions for the static structure factor and correlation energy are derived for $b\ensuremath{\ll}{r}_{\text{s}}<1$. It is found that the correlation energy varies as b2{b}^{2}b2 for $b\ensuremath{\ll}{r}_{\text{s}}$ from its value for an infinitely thin wire. It is observed that the correlation energy depends significantly on the wire model used (harmonic versus cylindrical confinement). The first-order RPA expressions for the structure factor, pair-correlation function, and correlation energy are numerically evaluated for several values of bbb and ${r}_{\text{s}}\ensuremath{\le}1$. These are compared with the QMC results in the range of applicability of the theory.