Anderson transition symmetries at the band-edge of a correlated Sn/Si monolayer

Abstract

Anderson localization is predicted to enhance the critical temperature of disordered superconductors. Despite a huge body of theoretical work based on non-linear sigma models, experiments are lacking to understand correlated electrons in disordered potentials. In this study, we investigate a tin monolayer on silicon, a material known for its likely antiferromagnetic Mott-correlated groundstate. We analyze the statistical properties of tunneling conductance maps of increasingly localized states as we approach the edge of the valence band. Using multifractal analysis, we show that the system follows an exact symmetry relation based on the algebraic structure of nonlinear sigma-models (NLsMs). We anticipate that this symmetry may be broken in specific - e.g. chiral electronic phases. Finally, we point out that multifractal analysis can equally be applied to universal conductance fluctuations in magneto-transport experiments, thus providing a powerful tool to probe the underlying symmetries of disordered electronic phases.

Figures (4)

• Figure 1: Local-density of state maps through the valence-band edge of the Sn/Si monolayera Topographic map of the tin monolayer deposited on silicon. For both topography and spectroscopic data, the bias voltage is -1V and the current setpoint 100 pA. Except is specified otherwise, the temperature is fixed at 300 mK throughout the paper.b dI/dV spectra taken at the lower band-edge of the tin monolayer along a vertical line (cf. the black arrow indicates the X position on a). The color bar codes for the dI/dV values. c Crystalline structure of the tin $\sqrt{3}\times \sqrt{3}$ tin reconstruction on a Si(111) surface. d dI/dV spectroscopy at 4K reveals a large gap of roughly 0.6 eV. e dI/dV map at -0.48 eV, very close to the band-edge. The colour bar codes for the differential conductance at a given energy and position in pS.f dI/dV maps at fixed energy at E = $\{$ -0.65, -0.6, -0.55, -0.5 $\}$ eV along with their Fourier transform (quasi-particle interference patterns). The Relative Standard Deviation (RSD) of the dI/dV map is given on each panel and the colour bar codes for the differential conductance dI/dV.
• Figure 2: Local density of states spatial correlation functionsa Iso-energy radial correlation functions C(E,R) of dI/dV maps at fixed energy. The x-axis denotes the energy of the maps and the y-axis the log of distance. The inset represents the same data but with a linear R axis. The black dashed line is a power law $\vert E-E_{\rm c}\vert^{-\nu_{\rm app}}$ with $E_{\rm c}=-0.65$ eV and $\nu_{\rm app} = 0.75$. bRadial correlations as function of energy, with E ranging from -0.8 eV to -0.45 eV. The black and red lines follow power laws of exponents -0.03 and -0.05. The colour bar accounts for the energy of the LDOS map.cPower law exponent of $C(E,R)$ fits as a function of energy plotted along with the fractal scaling exponent $\Delta_2$ measured by multifractal analysis. The inset shows multifractal spectra taken at all energies (zooming on the region where $f(\alpha) \sim 2$). The color codes for energy as on the main plot. This panel shows that the exponent of correlation functions fitted from panel b perfectly correspond to fractal exponent $\Delta_2$, as predicted by theory burmistrov2013multifractality.
• Figure 3: Energy-dispersion of the local density of states variancea Mean normalized dI/dV spectrum $\langle\eta_{\Lambda}(E) \rangle_r$ as a function of energy. The grey area denotes the standard deviation on the map. For $E>-0.45$ eV, a large part of the grid's pixels have negative dI/dV, corresponding to a dominance of noise in this highly insulating region. We shall therefore focus on the $E<-0.45$ eV region. b Lin-lin and lin-log plots of the normalized variance computed on an iso-energy LDOS map $\sigma^2 = \langle\delta \rho^2\rangle/\langle \rho \rangle^2$. On c, we shift the origin of energies to the band edge and show that adding a magnetic field of 6.5 T slightly changes the scaling law. d Variance $\sigma^2(E)$versus mean $\langle \eta_{\Lambda}(E)\rangle_r$ of normalized tunneling conductance. e Dimensionless conductance $g(E)$ in the parabolic diffusive band model ($g(E)=\hbar \rho(E) D(E)$) as a function of energy, for different elastic scattering rates $\gamma_{\rm el}$. The Thouless energy is set at $E_{\rm Th} = 30 \mu$eV. f Comparison of the experimental $\sigma^2(E)$ with the weak-disorder model $1/4\pi g(E)$ for $\gamma_{\rm el} =\{100,150,250\}$ THz.
• Figure 4: Distribution functions and multifractal analysisa Solid lines denote the dI/dV distribution functions (or histogram) $P(\tilde{\eta})$ of differential conductance values$\tilde{\eta}(E)$ at energy E = $\{-0.9, -0.8, -0.7, -0.6 ,-0.57, -0.54, -0.5, -0.48\}$ eV. Black dashed lines correspond to $\tilde{\eta}^{-3}P(1/\tilde{\eta})$. We recall that the mobility edge is at -0.65 eV. b Solid lines denote the multifractal spectra $f(\alpha)$ of dI/dV maps at energy E = $\{-0.9, -0.8, -0.7,-0.6,-0.57, -0.54, -0.5, -0.48\}$ eV. Red dashed lines account for $f(4-\alpha)-2+\alpha$. Multifractal spectra $f(\alpha)$ are fitted with the weak-multifractality solution Eq.\ref{['Weak_MF_fit']} (black dash-dotted lines). $\alpha_0$ is given in Supplementary Note 1 and Supplementary Figure 2 along with a polynomial interpolation in Table S3.