Direct Measurement of the Local Density of States of a Disordered One-Dimensional Conductor

One-dimensional electron systems (1DESs) containing two and one occupied subbands are found below the different types of $[1\overline{1}2]$ steps of the InAs(110) surface. Using low-temperature scanning tunneling spectroscopy, we determined the subband energies, the disorder potential, and the local density of states of these 1DESs. The rather complete knowledge of the 1DES allowed us to compare the measured LDOS with a single-particle calculation. Surprisingly, we did not find significant deviations from the calculation albeit the electron-electron interaction in the 1DESs is stronger than the electron-disorder interaction.

Figure 1

(a) Constant-current image of $n\mathrm{\text{−}}\mathrm{I}\mathrm{n}\mathrm{A}\mathrm{s}(110)$, $V=200\mathrm{m}\mathrm{V}$, $I=90\mathrm{p}\mathrm{A}$; steps are numbered and the red line marks the direction of the line section in (c). (b) Magnification of the area marked by the white rectangle in (a), $V=200\mathrm{m}\mathrm{V}$, $I=500\mathrm{p}\mathrm{A}$; crystallographic directions indicated. (c) The line section along the red line in (a) in comparison with step edge morphology (grey line); step potential (+) is indicated. (d) Deduced potential across step 2 (black line) [11]; subband energies $E_n$ are marked (blue lines). (e) $dI/dV$ spectra measured on step 2 (blue line) and 200 nm away from each step (black line), $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=50\mathrm{m}\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=1.2\mathrm{n}\mathrm{A}$, $V_{\mathrm{m}\mathrm{o}\mathrm{d}}=0.4\mathrm{m}\mathrm{V}$; the blue curve is offset for clarity; light blue line: fit of the 1DES DOS using $E_1=-76\mathrm{m}\mathrm{e}\mathrm{V}$, $E_2=-23\mathrm{m}\mathrm{e}\mathrm{V}$ and a Gaussian broadening of $\sigma =3\mathrm{m}\mathrm{e}\mathrm{V}$; grey line at lower curve: expected 3DES DOS. (f) $dI/dV$ curves taken at different distances from step 2 as given on the right. Measurement positions are marked as dots in the constant-current image of the inset, $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=50\mathrm{m}\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=1.2\mathrm{n}\mathrm{A}$, $V_{\mathrm{m}\mathrm{o}\mathrm{d}}=0.4\mathrm{m}\mathrm{V}$. (g) Schematic representation of expected wave functions ${\Psi }_n$ (dotted lines) and squared wave functions $|{\Psi }_n{|}^2$ (solid lines) inside the confinement potential of step 2. (h) $dI/dV(x)$ at $V=-60\mathrm{m}\mathrm{V}$ [first subband (SB), blue] and $V=-20\mathrm{m}\mathrm{V}$ (first and second SB, magenta); to deduce the shape of the second SB, intensity of the first SB at $V=-20\mathrm{m}\mathrm{V}$ is subtracted (red line).

Figure 2

(a) Constant-current image of step 2, $V=200\mathrm{m}\mathrm{V}$, $I=1.2\mathrm{n}\mathrm{A}$. (b) The same after subtracting the spatially averaged apparent height of the step edge. (c) Potential along the step edge deduced from (b) according to the model discussed in Ref. 11; the arrow marks a position discussed in the text. (d)–(o) $dI/dV$ images recorded at different $V$ as indicated in mV; $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=50\mathrm{m}\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=1.2\mathrm{n}\mathrm{A}$, $V_{\mathrm{m}\mathrm{o}\mathrm{d}}=0.4\mathrm{m}\mathrm{V}$.

Figure 3

(a) Fourier transformation (FT) of the $dI/dV$ images shown in Figs. 2d, 2e, 2f, 2g, 2h, 2i, 2j, 2k, 2l, 2m, 2n, 2o; dispersion curves of unperturbed InAs corresponding to the two subbands are shown as yellow lines [23] and expected dispersion of a LL is shown as a red line [5]. (b) Calculated FT corresponding to (a) (see text). (c) FT determined for the upward step 3 ($g\simeq 0.71$); lines have the same meaning as in (a). Inset: line profile of $dI/dV(y)$ at $V=-13\mathrm{m}\mathrm{V}$; $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=50\mathrm{m}\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=1.2\mathrm{n}\mathrm{A}$, $V_{\mathrm{m}\mathrm{o}\mathrm{d}}=0.4\mathrm{m}\mathrm{V}$. (d) Calculated FT corresponding to (c) with the line section of the LDOS at $E=-13\mathrm{m}\mathrm{e}\mathrm{V}$ as the inset. Vertical lines in the insets of (c) and (d) highlight similarities.