Real-Space Observation of Drift States in a Two-Dimensional Electron System at High Magnetic Fields

The local density of states of the adsorbate-induced two-dimensional electron system is studied in magnetic fields up to $B=6\mathrm{T}$. Landau quantization is observed and drift states with a width of about the magnetic length are found in agreement with theoretical predictions. At the tails of the Landau levels the states form closed paths indicating localization. These states show the expected energy dependence. A multifractal analysis applied to the data results in a nice parabolic shape of the characteristic $f(\alpha )$ spectra, but we find only a slight displacement of the origin from $\alpha =2.0$ for the states in the center of the Landau level.

Figure 1

$dI/dV$ curves spatially averaged over $100\times 100{\mathrm{n}\mathrm{m}}^2$: (a) $2.7\%$ $\mathrm{F}\mathrm{e}/n\mathrm{\text{−}}\mathrm{I}\mathrm{n}\mathrm{A}\mathrm{s}(110)$, $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=100\mathrm{m}\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=300\mathrm{p}\mathrm{A}$, $V_{\mathrm{m}\mathrm{o}\mathrm{d}}=1.8\mathrm{m}\mathrm{V}$; different $B$ fields, subband energies $E_0$, $E_1$, Fermi level $E_F$, and expected Landau level splittings $\hslash {\omega }_c$ are indicated; (b) $4.5\%$ $\mathrm{F}\mathrm{e}/n\mathrm{\text{−}}\mathrm{I}\mathrm{n}\mathrm{A}\mathrm{s}(110)$, $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=100\mathrm{m}\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=500\mathrm{p}\mathrm{A}$, $V_{\mathrm{m}\mathrm{o}\mathrm{d}}=2\mathrm{m}\mathrm{V}$; labels as in (a); QD marks the peak corresponding to the tip induced quantum dot [13].

Figure 2

(a)–(d) $dI/dV$ images of $0.8\%$ $\mathrm{F}\mathrm{e}/n\mathrm{\text{−}}\mathrm{I}\mathrm{n}\mathrm{A}\mathrm{s}(110)$ recorded at different $V$ as indicated, $B=6\mathrm{T}$, $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=100\mathrm{m}\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=300\mathrm{p}\mathrm{A}$, $V_{\mathrm{m}\mathrm{o}\mathrm{d}}=1.8\mathrm{m}\mathrm{V}$; the average number of contributing states $\overline{N}$ is given above each image; (e) spatially averaged $dI/dV$ curve, $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=100\mathrm{m}\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=300\mathrm{p}\mathrm{A}$, $V_{\mathrm{m}\mathrm{o}\mathrm{d}}=1\mathrm{m}\mathrm{V}$; quantum dot peak (QD), Landau level ($\mathrm{L}\mathrm{L}n$), and Landau energy $\hslash {\omega }_c$ are indicated; dots mark the voltages corresponding to the images (a)–(d); (f) disorder potential of measured area determined with the help of the QD [7]; (g) histogram of determined FWHM of drift states. White rings mark the same area in (c) and (f).

Figure 3

$dI/dV$ images at $B=6\mathrm{T}$: (a)–(j) $2.7\%$ Fe, $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=100\mathrm{m}\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=300\mathrm{p}\mathrm{A}$, $V_{\mathrm{m}\mathrm{o}\mathrm{d}}=1.8\mathrm{m}\mathrm{V}$; (k)–(r) $0.8\%$ Fe, $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=100\mathrm{m}\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=300\mathrm{p}\mathrm{A}$, $V_{\mathrm{m}\mathrm{o}\mathrm{d}}=1\mathrm{m}\mathrm{V}$; sets in (a)–(f), (g)–(j), (k)–(n), and (o)–(r) show the same area, respectively; (a) $V=-82.6\mathrm{m}\mathrm{V}$, (b) $V=-85.6\mathrm{m}\mathrm{V}$, (c) $V=-88.6\mathrm{m}\mathrm{V}$, (d) $V=-91.6\mathrm{m}\mathrm{V}$, (e) $V=-94.6\mathrm{m}\mathrm{V}$, (f) $V=-97.6\mathrm{m}\mathrm{V}$; (g) $V=-82.6\mathrm{m}\mathrm{V}$; (h) $V=-85.6\mathrm{m}\mathrm{V}$, (i) $V=-88.6\mathrm{m}\mathrm{V}$, (j) $V=-91.6\mathrm{m}\mathrm{V}$; (k) $V=-26.9\mathrm{m}\mathrm{V}$; (l) $V=-24.5\mathrm{m}\mathrm{V}$, (m) $V=-23.4\mathrm{m}\mathrm{V}$, (n) $V=-21.6\mathrm{m}\mathrm{V}$; (o) $V=-0.5\mathrm{m}\mathrm{V}$, (p) $V=-2.9\mathrm{m}\mathrm{V}$, (q) $V=-5.7\mathrm{m}\mathrm{V}$, (r) $V=-7.6\mathrm{m}\mathrm{V}$; bright spikes in the images are Fe atoms.

Figure 4

(a) $\sum _i{\mu }_i(q,\delta )\mathrm{l}\mathrm{n}[{\mu }_i(1,\delta )]$ versus $\mathrm{l}\mathrm{n}(\delta )$ at different $q$ for the data of Fig. 2b (see text); (b) $f(\alpha )$ spectra for the data set shown in Fig. 2 ($B=6\mathrm{T}$) and for a data set from the same 2DES at $B=0\mathrm{T}$ [Figs. 2(i)–2(k) of [7] ]; inset: $f(\alpha )$ for $B=6\mathrm{T}$, $V=-20\mathrm{m}\mathrm{V}$ (points) and parabolic fit (line).