Spatial Imaging of Two-Dimensional Electronic States in Semiconductor Quantum Wells

We measure the spatial distribution of the local density of states (LDOS) at cleaved surfaces of $\mathrm{InAs}/\mathrm{GaSb}$ isolated quantum wells and double quantum wells (DQWs) by low-temperature scanning tunneling spectroscopy. Distinct standing wave patterns of LDOS corresponding to subbands are observed. These LDOS patterns and subband energies agree remarkably well with simple calculations with tip-induced band bending. Furthermore, for the DQWs, coupling of electronic states between the quantum wells is also clearly observed.

Figure 1

(a) Schematic band diagram of an $\mathrm{InAs}/\mathrm{GaSb}$ heterostructure. Colored regions show the band gap. (b) Typical small area topographic image (empty state) on a cleaved cross section of $\mathrm{InAs}(6\mathrm{nm})/\mathrm{GaSb}(7\mathrm{nm})$ quantum wells (QWs). Sample bias voltage is $V=+1\mathrm{V}$. Feedback current is $I=0.1\mathrm{nA}$. (c)–(e) Typical large area topographic images (filled state) for $6\mathrm{nm}/7\mathrm{nm}$, $12\mathrm{nm}/16\mathrm{nm}$, $17\mathrm{nm}/23\mathrm{nm}$ QWs. $V=-1\mathrm{V}$, $I=0.1\mathrm{nA}$. The scale is the same for all three images.

Figure 2

Normalized differential tunneling conductance [$(dI/dV)/(I/V)$] as a function of the sample bias voltage ($V$) at the center of an InAs layer (a) and a GaSb layer (b) for $17\mathrm{nm}/23\mathrm{nm}$ QWs. The data were averaged along lines parallel to the interface. Blue regions show the band gap. (c) Color contour plot of the spatial variation of the STS spectra.

Figure 3

(a) High spatial and energy resolution STS spectra for $17\mathrm{nm}/23\mathrm{nm}$ QW, focusing on the white dotted circle in Fig. 2c. $(dI/dV)/(I/V)$ curves are plotted for each $V$ with a step of 0.01 V. (b) Calculated LDOS spectra taken to be the sum of the squared wave functions for subbands. The conduction band potential profile modulated by the electron accumulation in the well is overlaid. (c) Enhanced STS spectra. In the InAs layer, oscillation minima in $(dI/dV)/(I/V)$ curves as a function of the position along the growth direction at each $V$ are set to zero. The subband indices are indicated. (d) Experimentally deduced electron density of states: each $(dI/dV)/(I/V)$ curve in (a) is integrated over the quantum well. (e) Enhanced STS spectra for $12\mathrm{nm}/15\mathrm{nm}$ QW. (f) STS spectra for $6\mathrm{nm}/7\mathrm{nm}$ QW.

Figure 4

(a)–(c) Comparison between observed and calculated subband energies. (d) The differences between observed and calculated subband energies as a function of the observed subband energy. Tip-induced band bending as a function of $V$ calculated for InAs and GaSb independently (dotted lines) and averaged (dashed line).

Figure 5

(a) STM topographic image of a DQW ($V=+1\mathrm{V}$). (b) STS spectra. (c) $(dI/dV)/(I/V)$ curves at voltages arrowed in (b) (offset). Red dashed lines are calculated LDOS. (d) Calculated squared wave functions (solid lines) and subband energies (offset) for the DQW (red). Those for 4.8 nm (green dashed line) and 5.4 nm (blue dashed line) single QWs are also shown for comparison.