Strong electron-electron interactions in a dilute weakly-localized metal near a metal-to-insulator transition

TL;DR

The study investigates strong electron-electron interactions in ultra-thin, dilute silicon δ-doped layers near the 2D metal-insulator transition. By fabricating As and P δ-layers as thin as 0.4–1.8 nm and with $n_{2D}$ up to ~2×10^{14} cm^{-2}, the authors realize a half-filled disordered 2D Hubbard regime and measure magneto-conductance at milliKelvin temperatures. They separate weak localization and Altshuler-Aronov corrections, finding that near the insulating state the Zeeman-driven AA contribution dominates and yields a negative correction to conductance, with $ ext{d} olinebreak[ ule{0pt}{0pt}] abla \sigma abla ext{propto} abla(h)$ where $h = rac{g\mu_B B}{k_B T}$, and isotropic in field orientation, arguing against a Kondo regime. The results support a picture of a strongly interacting, disordered half-filled 2D Hubbard system in silicon, with $r_s oughly 1.4$–$5.5$ and a thickness $ ( ext{delta}) oughly 0.4$–$1.8$ nm, revealing the essential role of electron-electron interactions near MIT for these dilute 2D layers.

Abstract

Because it is easily switched from insulator to metal either via chemical doping or electrical gating, silicon is at the core of modern information technology and remains a candidate platform for quantum computing. The metal-to-insulator transition in this material has therefore been one of the most studied phenomena in condensed matter physics, and has been revisited with considerable profit each time a new fabrication technology has been introduced. Here we take advantage of recent advances in creating ultra-thin layers of Bohr-atom-like dopants to realize the two-dimensional disordered Hubbard model at half-filling and its metal-to-insulator transition (MIT) as a function of mean distance between atoms. We use gas-phase dosing of dopant precursor molecules on silicon to create arsenic and phosphorus $δ$-layers as thin as 0.4~nm and as dilute as 10$^{13}$~cm$^{-2}$. On approaching the insulating state, the conventional weak localization effects, prevalent at high dopant densities and due to orbital motion of the electrons in the plane, become dominated by electron-electron interaction contributions which obey a paramagnetic Zeeman scaling law. The latter make a negative contribution to the conductance, and thus cannot be interpreted in terms of an emergent Kondo regime near the MIT.

Figures (4)

• Figure 1: Experimental setup. (a) Hall-bar geometry of the contacts allowing for 4-point resistance measurements. (b) Schematic of the sample composition. The donor layer is encapsulated in the silicon lattice at a depth $d \sim$ 20 nm below the surface, covering a width $\delta$ between 0.4 nm and 1.8 nm having a 2D electron density $n_\text{2D} = (1$–$19)\times10^{13}$cm$^{-2}$.
• Figure 2: Typical magneto-conductance data. (a) At temperature $T= 115 {\rm}$ mK, the magnetic field is set parallel to the plane of the $\delta$-layer (blue points - with no dependence on the relative orientation to the current) and perpendicular to it (green points). The black points are the difference between in- and out-of-plane magneto-conductance per square, the solid black line being a fit with weak-localization corrections, Eq. (\ref{['eq_HLN']}) minus Eq. (\ref{['eq_Sullivan']}). The orange data (right y-axis) is a Hall measurement ($R_{xy}$) taken at 1.75 K, fitted linearly by the solid line. (b) Difference between out-of-plane and in-plane magneto-conductance per square for various temperatures. The solid lines are fits with Eq. (\ref{['eq_HLN']}) minus Eq. (\ref{['eq_Sullivan']}). (c) Conductance per square versus temperature in black, displaying a logarithmic dependence as expected for weak localization down to $T_\text{c}$$\approx$ 300 mK. The coherence length $\ell_\phi$ obtained from the fits in (b) is shown in orange. It follows a power law $T^{-0.31}$ down to $T_\text{c}$. All data in (a), (b), and (c) are obtained from a sample of electron density $n = (1.18 \pm 0.01) \times 10^{13}$ cm$^{-2}$.
• Figure 3: Influence of dopant density on the magneto-conductance. (a) Magneto-conductance per square at 600 mK for samples with different free carrier densities $n_\text{2D}$. As $k_\text{F}\ell$ decreases with decreasing density the effect of the perpendicular magnetic field (full dots) becomes smaller. The conductivity for $B_\parallel$ is denoted by crosses. (b) Magnitude of the weak-localization part of the magneto-conductance ($\Delta\sigma_{xx}(B_\perp)-\Delta\sigma_{xx}(B_\parallel)$ shown on the left $y$-axis) at 600 mK in a 2 T magnetic field as a function of $k_\text{F}\ell$. The right $y$-axis indicates the free carrier density. The out-of-plane magneto-conductance is seen to be dominated by weak localization corrections which depend on $k_\text{F}\ell$. In (b) the dots correspond to arsenic $\delta$-layers while the squares are from phosphorus $\delta$-layers, showing that the magneto-conductance does not depend on the dopant species.
• Figure 4: $B/T$ scaling of the AA correction. (a) Change in conductivity due to the Zeeman splitting in units of the conductance quantum $e^2/\hbar$, as a function of the logarithm of the reduced field $h=g\mu_\text{B} B/k_\text{B}T$. The data are shown for multiple temperatures and stem from a single sample with free carrier density $n = (2.14 \pm 0.02) \times 10^{13}$ cm$^{-2}$. The data collapse down to 250 mK. (b) Same as in (a), except that an effective temperature $T_{\rm eff}$ is used to obtain a full collapse of the data. $T_\text{eff}$ is plotted for all samples in (c). (c) Effective temperature $T_{\rm eff}$ determined for all samples so as to collapse the AA correction as in (a) and (b).