Fig. 12: Effect of disorder and impurities on the valley Hall response.
From: Quantized valley Hall response from local bulk density variations
Effect of lattice imperfections on the valley Hall coefficient of the triaxially strained flake with strain intensity τ = 0.07 > 0 (ℓB = 4.6a) and chemical potential μF = 0.25 t in units of the hopping parameter t. a, b, c Short-range on-site potential \({V}_{{{{{{{{\rm{imp}}}}}}}}}^{\delta }({{{{{{{\bf{r}}}}}}}})\) with Nimp = Ntot, where Nimp is the number of impurities and Ntot is the number of sites in the flake. d, e, f Gaussian impurities potential described by \({V}_{{{{{{{{\rm{imp}}}}}}}}}^{\lambda }({{{{{{{\bf{r}}}}}}}})\) with an impurity concentration of Nimp/Ntot = 0.01 and λ = 3 a, where λ is the width of each Gaussian and a is the lattice spacing. g, h, i Off-diagonal bond disorder. a, d, g We show a corresponding cut of the local valley Hall marker \({{\mathfrak{S}}}_{\alpha }({{{{{{{\bf{r}}}}}}}})\), discriminated by sublattice (α = A, B), along the direction \({{{{{{{\bf{r}}}}}}}}={x}_{c}\hat{{{{{{{{\bf{x}}}}}}}}}+y\hat{{{{{{{{\bf{y}}}}}}}}}\) for each particular disorder realization. a, d W = 0.2 t and in (g), Γ = 0.606. The insets schematically show the deviations of the flake’s on-site energies or the tunneling amplitudes with respect to the pristine case. b, e, h Deviations of the corresponding valley Hall response from the pristine case. The response (solid blue line) is obtained by averaging along a bulk region of size Lbulk = 4.6 a and over 50 different disorder configurations. The standard deviation is indicated in light blue shaded area. c, f, i The corresponding bulk density of states \({\bar{\rho }}_{{{{{{{{\rm{bulk}}}}}}}}}\), also averaged over the different disorder realizations, as a function of energy and the disorder intensity W or Γ. Dashed gray lines indicate the energy of the pseudo Landau levels in the pristine case. The dashed blue-line shows the chemical potential μF = 0, 25 t chosen to compute the valley Hall response.
