Direct Comparison between Potential Landscape and Local Density of States in a Disordered Two-Dimensional Electron System

The local density of states (LDOS) of the adsorbate-induced two-dimensional electron system (2DES) on $n\mathrm{\text{−}}\mathrm{I}\mathrm{n}\mathrm{A}\mathrm{s}(110)$ is studied by scanning tunneling spectroscopy. In contrast to a similar 3DES, the 2DES LDOS exhibits 20 times stronger corrugations and rather irregular structures. Both results are interpreted as consequences of weak localization. Fourier transforms of the LDOS reveal that the $k$ values of the unperturbed 2DES still dominate the 2DES, but additional lower $k$ values contribute. To clarify the origin of the LDOS patterns, we measure the potential landscape of the 2DES area. We use it to calculate the expected LDOS and find reasonable agreement between calculation and experiment.

Figure 1

(a) ARUPS spectrum of 4.5% $\mathrm{F}\mathrm{e}/n\mathrm{\text{−}}\mathrm{I}\mathrm{n}\mathrm{A}\mathrm{s}(110)$, $h\nu =10\mathrm{e}\mathrm{V}$ (points) compared with fits for different subband energies $E_1$, $E_2$ as indicated (lines). Only the central curve fits the data. (b) Spatially averaged $dI/dV(V)$ curves of $n\mathrm{\text{−}}\mathrm{I}\mathrm{n}\mathrm{A}\mathrm{s}(110)$ (lower curve) and 4.5% $\mathrm{F}\mathrm{e}/n\mathrm{\text{−}}\mathrm{I}\mathrm{n}\mathrm{A}\mathrm{s}(110)$ (upper curve); both curves taken with the same tip, $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=100\mathrm{m}\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=500\mathrm{p}\mathrm{A}$; peaks of the tip-induced quantum dot (QD) and $E_1$, $E_2$ of the 2DES determined by ARUPS as well as the 3DES are indicated. (c) Grey scale plot of $dI/dV(V)$ intensity as a function of position along a scan line, $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=100\mathrm{m}\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=500\mathrm{p}\mathrm{A}$; sample and tip as in upper curve of (b); $E_1$, $E_2$ and QD peak are indicated. (d) Spatially averaged $dI/dV(V)$ curve of 2.7% $\mathrm{F}\mathrm{e}/n\mathrm{\text{−}}\mathrm{I}\mathrm{n}\mathrm{A}\mathrm{s}(110)$, $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=100\mathrm{m}\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=300\mathrm{p}\mathrm{A}$; note the absence of QD peaks.

Figure 2

(a) Potential landscape as determined from laterally fluctuating peak voltage of the lowest-energy QD state, 4.5% $\mathrm{F}\mathrm{e}/n\mathrm{\text{−}}\mathrm{I}\mathrm{n}\mathrm{A}\mathrm{s}(110)$. (b) Constant-current image of the area marked in (a) $V=100\mathrm{m}\mathrm{V}$, $I=50\mathrm{p}\mathrm{A}$; dark spots are Fe atoms. (c) Potential landscape at 0.8% $\mathrm{F}\mathrm{e}/n\mathrm{\text{−}}\mathrm{I}\mathrm{n}\mathrm{A}\mathrm{s}(110)$. Both potential images cover a potential range of 20 meV.

Figure 3

(a)–(g) $dI/dV$ images (LDOS images) of 2.7% $\mathrm{F}\mathrm{e}/n\mathrm{\text{−}}\mathrm{I}\mathrm{n}\mathrm{A}\mathrm{s}(110)$ at different $V$ as indicated; $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=100\mathrm{m}\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=300\mathrm{p}\mathrm{A}$; bright spikes in the images are Fe atoms. Insets: Fourier transformations (FT) of $dI/dV$ images. (h) Dominating $k$ values corresponding to rings in FT’s in comparison with dispersion curve of unperturbed InAs (lines) [16]. (i)–(l) Same as (a)–(h) but for 0.8% $\mathrm{F}\mathrm{e}/n\mathrm{\text{−}}\mathrm{I}\mathrm{n}\mathrm{A}\mathrm{s}(110)$; surface area belongs to the potential in Fig. 2c.

Figure 4

(a) LDOS calculated from the potential landscape in Fig. 2c [18]; $E=-50\mathrm{m}\mathrm{e}\mathrm{V}$. (b) Normalized $dI/dV$ image of the same area; $V=-50\mathrm{m}\mathrm{V}$, $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=100\mathrm{m}\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=300\mathrm{p}\mathrm{A}$. Insets are FT’s. Dots mark identical sample positions as deduced from constant current images. (c) Intensity distribution of the LDOS in (a) and (b); for the sake of comparison the experimental curve is stretched by 5%. (d) Cross correlation function between experimental and calculated image.