Kondo screening of a finite, non-uniform one-dimensional Wigner crystal in a quantum point contact: a microscopic many-body theory problem

Scanning gate microscopy (Brun et al., Nature Communications 5, 4290 (2014)) showed that continuously deforming a GaAs quantum point contact (QPC) with a charged tip produces repetitive splittings of the zero-bias conductance anomaly correlated with appearances of the 0.7 anomaly — evidence for a small, non-uniform one-dimensional Wigner crystal of localized electrons in the channel, whose odd/even parity alternates equilibrium Kondo screening (spin-1/2 ground state, zero-bias peak) with non-equilibrium singlet-triplet Kondo physics (spin-singlet ground state, finite-bias peaks at $eV=J$). The authors' classical Thomas–Fermi simulations could estimate only the size of the low-density crystallization region and explicitly left open the theoretical problem: how many charges localize, with what spatial arrangement and spin couplings, and how Kondo temperatures, singlet-triplet splittings, and non-equilibrium screening evolve with chain length and non-uniform spacing in the curved QPC potential. No dedicated quantitative many-body theory of the finite non-uniform Wigner chain and its parity-dependent Kondo phenomenology has yet been reported; the problem remains open.

Background

A quantum point contact (QPC) is a short, gate-defined constriction between two reservoirs of a two-dimensional electron gas (2DEG); its linear conductance is quantized in units of $G_{0}=2e^{2}/h$ when subbands are occupied one by one. Experiments since the 1990s show two robust anomalies on the lowest plateau: the 0.7 anomaly, a shoulder near $0.7\,G_{0}$ whose height grows with magnetic field and often saturates on cooling, and the zero-bias anomaly (ZBA), a narrow peak at zero source–drain bias in the nonlinear differential conductance (Cronenwett et al., PRL 88, 226805 (2002)). The Kondo effect — the many-body screening of a localized spin by conduction electrons below a characteristic Kondo temperature $T_{K}$, producing a zero-bias resonance — was early proposed as the mechanism, with the localized spin itself created by interactions in the open channel (Meir, Hirose and Wingreen, PRL 89, 196802 (2002); Hirose, Meir and Wingreen, PRL 90, 026804 (2003); Rejec and Meir, Nature 442, 900 (2006)). A complementary view attributes the anomalies to a van-Hove ridge — an enhanced density of states at the top of the QPC barrier — amplifying interaction effects without local moments (Bauer et al., Nature 501, 73 (2013)).

In a low-density one-dimensional (1D) electron gas, Coulomb repulsion can overcome kinetic energy, and electrons order into a Wigner crystal — a periodic array of localized charges first described by Wigner (1934) and, for 1D, by Schulz (Phys. Rev. Lett. 71, 1864 (1993)). In a Wigner crystal, neighboring spins couple antiferromagnetically with an exchange $J$ that is exponentially small in the inter-particle spacing (Klironomos, Ramazashvili and Matveev, Phys. Rev. B 72, 195343 (2005)), and a finite Wigner-crystal wire coupled to reservoirs shows Kondo-like behavior depending on the parity of the electron number (Matveev, Phys. Rev. Lett. 92, 106801 (2004)). A parallel experiment on length-tunable QPCs (Iqbal et al., Nature 501, 79 (2013)) observed odd/even alternation of Kondo peaks attributed to emergent localization of one or two charges.

Brun et al. (Nature Communications 5, 4290 (2014)) used scanning gate microscopy (SGM), in which a negatively charged atomic-force tip scanned above the surface depletes the 2DEG and locally deforms the QPC potential. Sweeping the tip toward the QPC at 20 mK, they observed up to four successive splittings of the ZBA, each correlated with a reappearance of the 0.7 anomaly. They interpreted this as successive states of a small, non-uniform 1D Wigner crystal (up to about eight localized electrons): an odd number $N$ gives a spin $S=1/2$ ground state screened by equilibrium Kondo effect (ZBA), while an even $N$ gives an $S=0$ singlet ground state with the $S=1$ triplet entered only at finite bias $eV=J$, i.e. non-equilibrium Kondo screening (split ZBA). Three-dimensional classical electrostatics in the Thomas–Fermi approximation — a self-consistent mean-field treatment of the electron gas — reproduced the size of the low-density region where the 1D Wigner-crystallization criterion ($n_{1D}\,a_{B}<1$, with $a_{B}$ the effective Bohr radius) is met, and its growth under tip approach matched the observed oscillation count. But, as the authors stress, the classical simulation 'cannot be used to calculate the number of localized charges, as quantum mechanics dominates at such a low density'; charges in the strongly curved QPC potential are not uniformly spaced, 'an extension of the concept of Wigner crystal' is needed, possible disorder pinning of the crystal is unexplored, and 'the particular case of a 1D chain of localized charges in the Kondo regime still requires theoretical investigations'. Understanding this problem requires settling whether the interaction-localized chain picture, with its parity-dependent Kondo phenomenology, is microscopically correct — which also adjudicates the long-standing competition with the van-Hove-ridge interpretation of the 0.7 and zero-bias anomalies.

Problem Statement

For a realistic quantum point contact potential — optionally deformed by a scanning-gate tip — whose low-density channel region satisfies the one-dimensional Wigner-crystallization criterion, construct a quantitative microscopic (quantum many-body) description of the finite chain of interaction-localized electrons and of its coupling to the reservoirs, such that the theory:

1. predicts the number of localized charges and their spatial arrangement, including the odd/even parity alternation, from the confinement itself — going beyond the classical Thomas–Fermi estimate of the crystallization-region size;

2. computes the antiferromagnetic exchange couplings between neighboring localized spins, the singlet-triplet splitting $J$ for an even number of charges, and the Kondo temperature $T_{K}$ for an odd number, including their dependence on chain length and on the non-uniform spacing imposed by the strongly curved QPC potential;

3. describes the non-equilibrium Kondo screening at finite source–drain bias (finite-bias triplet Kondo peaks at $eV=J$ for even occupancy, zero-bias Kondo peak for odd occupancy) and the associated linear-conductance phenomenology (the 0.7 and zero-bias anomalies), so as to confront scanning-gate and transport experiments quantitatively;

4. delimits how disorder and possible pinning of the crystal, the strong potential curvature, and the occupation of higher transverse modes at the edges of the low-density region modify the simple uniform one-dimensional Wigner-crystal picture.

A resolution of this problem is a single self-consistent microscopic calculation that satisfies all four requirements for one explicit quantum point contact confinement (with or without the scanning-gate tip deformation); requirements (1)–(3) supply the quantitative core and requirement (4) delimits the regime of validity of the resulting description, and a partial result that satisfies only some of the requirements does not resolve the problem.

Brun et al. (Nature Communications 5, 4290 (2014)) also note that their study 'should guide future theoretical works, and will open the way to further experimental investigations involving fine tuning of the QPC potential using various methods'; such fine-tuning experiments — over temperature, magnetic field, gate voltages and tip geometry — could test the predictions of these refined quantum models.

References

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