Quantum Hall Transition in Real Space: From Localized to Extended States

Using scanning tunneling spectroscopy in an ultrahigh vacuum at low temperature ($T=0.3\mathrm{K}$) and high magnetic fields ($B\le 12\mathrm{T}$), we directly probe electronic wave functions across an integer quantum Hall transition. In accordance with theoretical predictions, we observe the evolution from localized drift states in the insulating phases to branched extended drift states at the quantum critical point. The observed microscopic behavior close to the extended state indicates points of localized quantum tunneling, which are considered to be decisive for a quantitative description of the transition.

Figure 1

(a) Color scale plot of $dI/dV(V_s)$ measured along a straight line at $B=0\mathrm{T}$ (left); corresponding spatially averaged $dI/dV(V_s)$ (right); dashed lines mark the Fermi level $E_F$ and the subband energies $E_i$ ($i=1,2$); $V_{\mathrm{stab}}=150\mathrm{mV}$, $I_{\mathrm{stab}}=0.13\mathrm{nA}$, $V_{\mathrm{mod}}=2\mathrm{mV}$. (b) Self-consistently calculated band bending at the InSb surface (black line) and electron-density distribution of $i$th subband (red areas); $E_i$ and electron areal densities $n_i$ are marked. (c) $dI/dV(V_s)$ at different $B$ recorded at the same lateral position; $E_i$, $E_F$, ($dI/dV=0$) lines on the right, LLs and spin directions are marked; $V_{\mathrm{stab}}=150\mathrm{mV}$, $I_{\mathrm{stab}}=0.13\mathrm{nA}$, $V_{\mathrm{mod}}=2.0\mathrm{mV}$ (0 T), 1.3 mV (6 T), 1.0 mV (9 T), 0.9 mV (12 T). (d) Experimentally determined Landau fan diagram; green (red) dashed lines: spin-down LLs of the 1st (2nd) subband; arrows: spin levels of the lowest LL; $V_{\mathrm{stab}}=150\mathrm{mV}$, $I_{\mathrm{stab}}=0.10\mathrm{nA}$, $V_{\mathrm{mod}}=1.5\mathrm{mV}$.

Figure 2

LDOS of lowest LL. (a)–(g) Measured $dI/dV$ $(x,y)$; $B=12\mathrm{T}$, $V_s=-116.3\mathrm{mV}$ (a), $-111.2\mathrm{mV}$ (b), $-104.4\mathrm{mV}$ (c), $-100.9\mathrm{mV}$ (d), $-99.2\mathrm{mV}$ (e), $-92.4\mathrm{mV}$ (f), $-89.0\mathrm{mV}$ (g), $I_{\mathrm{stab}}=0.1\mathrm{nA}$, $V_{\mathrm{stab}}=150\mathrm{mV}$, $V_{\mathrm{mod}}=1.0\mathrm{mV}$; same $dI/dV$ color scale in each image; white (green) arrows in (a), (b), (f), and (g) mark drift states encircling potential minima (maxima); red, yellow arrows in (c) and (e) mark tunneling connections existing at identical positions; crosses in (d) mark extended LDOS areas at saddle points. (h) Spatially averaged $dI/dV$ curve with circles at the $V_s$ used in (a)–(g). (i) Calculated LDOS at the center of $\mathrm{LL}0\downarrow$ at $B=12\mathrm{T}$; red arrows mark tunneling connections at the saddle points; white crosses mark extended areas. (j) $dI/dV$ image close to the center of $\mathrm{LL}0\uparrow$ at $B=6\mathrm{T}$, $V_s=-99\mathrm{mV}$; image includes the area of (a)–(g) within the marked rectangle.

Figure 3

(a) Calculated LDOS $(E,x)$ at $B=6\mathrm{T}$; $V_{\mathrm{tip}}=0$ and $\pm 20\mathrm{meV}$ as marked; $m^{*}/m_e≔0.02$, $|g|≔28$ [23] (b), (c) Measured $dI/dV(V_s,x)$ at $B=6$ and 12 T; $x$: lateral position along straight line; $I_{\mathrm{stab}}=0.13\mathrm{nA}$, $V_{\mathrm{stab}}=150\mathrm{mV}$, $V_{\mathrm{mod}}=1.3\mathrm{mV}$ (b), 0.9 mV (c). (d),(e) Calculated LDOS $(E,x)$ at $B=6$ and 12 T; $V_{\mathrm{tip}}=0\mathrm{meV}$.