Wave-Function Mapping of InAs Quantum Dots by Scanning Tunneling Spectroscopy

Scanning tunneling spectroscopy is used to investigate the single-electron states and the corresponding squared wave functions of single and freestanding strain-induced InAs quantum dots grown on GaAs(001). Several peaks are found in $dI/dV$ curves, which belong to different single-electron states. Spatially resolved $dI/dV$ images reveal (000), (100), (010), (200), and (300) states, where the numbers describe the number of nodes in $[1\overline{1}0]$, [110], and [001] directions, respectively. The total number and energetic sequence of states is different for different dots. Interestingly, the (010) state is often missing, even when (200) and (300) states are present. We interpret this anisotropy in electronic structure as a consequence of the shape asymmetry of the dots.

Figure 1

(a) Sketch of STS measurement of the sample with freestanding InAs QDs; tunneling path $I$ along $z$ is indicated; (b) band profile along the $z$ direction marked in (a) as calculated with a 1D-Poisson solver [14], $V_{\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}=1.05\mathrm{V}$, CB: conduction band, VB: valence band; a confined QD state is marked as a full black line; (c) constant-current image of the InAs-QD sample, $V_{\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}=3\mathrm{V}$, $I=70\mathrm{p}\mathrm{A}$; left inset: 3D representation of a typical QD with the $[1\overline{1}0]$ direction marked; central inset: wetting layer with atomic resolution; ($2\times 4$) unit cell (white rectangle) is marked.

Figure 2

Single QD, W tip: (a) constant-current image, $V_{\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}=1.7\mathrm{V}$, $I=50\mathrm{p}\mathrm{A}$, height $H$ and crystallographic directions marked; (b) $I(V)$ curves recorded on the QD (black line) and on the wetting layer (grey line), $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=1.6\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=50\mathrm{p}\mathrm{A}$; (c) $dI/dV$ curves recorded simultaneously with (b), $V_{\mathrm{m}\mathrm{o}\mathrm{d}}=28\mathrm{m}\mathrm{V}$; (d) $dI/dV$ curves recorded at different positions above the QD as marked in (a); (e),(f) spatially resolved $dI/dV$ data at $V_{\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}=0.89\mathrm{V}$ and $V_{\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}=1.14\mathrm{V}$, respectively; all data in Fig. 2 are raw data.

Figure 3

STS data of three different QDs: (a1)–(c1): 3D representation of constant-current images with heights $H$ indicated; (a2),(b2),(c2): $dI/dV(V)$ curves spatially averaged over the QD area, peak positions are marked by vertical lines; (a3)–(a5), (b3)–(b6), (c3)–(c6): $dI/dV$ images recorded at $V_{\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}$ as indicated; data are partly smoothed in order to enhance picture quality; the state in (a5) exhibits an energy close to the onset of the wetting layer, which results in a bright surrounding of the wave function; crystallographic directions are marked; (a1)–(a5): W tip, $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=1.85\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=50\mathrm{p}\mathrm{A}$, $V_{\mathrm{m}\mathrm{o}\mathrm{d}}=15\mathrm{m}\mathrm{V}$; (b1)–(b6): W tip, $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=1.6\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=50\mathrm{p}\mathrm{A}$, $V_{\mathrm{m}\mathrm{o}\mathrm{d}}=27\mathrm{m}\mathrm{V}$; (c1)–(c6): PtIr tip, $V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=2.4\mathrm{V}$, $I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=70\mathrm{p}\mathrm{A}$, $V_{\mathrm{m}\mathrm{o}\mathrm{d}}=15\mathrm{m}\mathrm{V}$; constant-current images at $V_{\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}=V_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$ and $I=I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$; left column: calculated squared wave functions taken from 3 and labeled by the number of nodes in different directions.