Fig. 2: Observation of the half-quantized layer Hall effect in Device A1.
From: Half-quantized layer hall effect as a probe of quantized axion field
a Dependence of the Hall conductance \({\sigma }_{{xy}}\) on the magnetic field \({\mu }_{0}H\), measured at different bottom gate voltage \({V}_{{\rm{bg}}}\). b The dependence of the Hall conductance on Vbg with parallel (p, orange at zero field) and antiparallel (ap, blue at \(\pm 0.4\ {\rm{T}}\), pink at zero field) magnetization configuration. The schematics denotes the 4 main states, i.e. insulating/hole-doped bottom surface with parallel/antiparallel magnetization configurations. The red and blue dashed arrows on the side surface illustrate the Hall currents from top and bottom surfaces, respectively. The black arrow indicates the chiral edge state of the Chern insulator. c Schematic of the HQLHE in a sandwich magnetic TI heterostructure with antiparallel magnetization configuration. Berry curvature concentrated near the band edges. When the Fermi level at bottom surface is tuned deep into the hole-doped regime by \({V}_{{\rm{b}}}\), the Berry curvature vanishes, leaving only the single-surface Hall current (dashed red arrows). d Numerical calculations of the layer-resolved AHC \({\sigma }_{{xy}}^{{\rm{t}}}\) (red) and \({\sigma }_{{xy}}^{{\rm{b}}}\) (purple) as a function of fermi energy (\({E}_{{\rm{F}}}\)). The bulk Hall signal (black) is vanishingly small throughout the whole range of \({E}_{{\rm{F}}}\). e Numerical results of the total \({\sigma }_{{xy}}\) at zero field as a function of \({V}_{{\rm{b}}}\) with parallel/antiparallel magnetization configurations. Error bars correspond to the standard deviation obtained from 30 independent disorder configurations. f Renormalization group flow in \(\left({\sigma }_{{xy}},{\sigma }_{xx}\right)\) plane with varying \({V}_{{\rm{b}}{\rm{g}}}\) (−200 V to 150 V) and \({\mu }_{0}H\) (\(\pm 1.2\ {\rm{T}}\) loop as in a). The purple dashed lines are guide to the eye for the flows of the fixed points from \({e}^{2}/h(\pm 1.0,0)\) and (0, 0) to points along \({\sigma }_{{xy}}=\pm {e}^{2}/2h\).
