Imaging the percolation of localized states in a multisubband two-dimensional electronic system subject to a disorder potential

The electronic local density of states (LDOS) in an ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}$ surface quantum well (QW) grown by molecular beam epitaxy was measured by low-temperature scanning tunneling spectroscopy under ultrahigh vacuum. The LDOS in the conduction band has a clear steplike energy dependence, revealing that several subbands are formed in the QW. At a given energy, the LDOS shows strong spatial fluctuations in the QW plane due to the presence of a disorder potential. Percolation of localized states with increasing energy is observed in each subband tail. The percolation threshold is determined for each subband by using a semiclassical model. The origin of the disorder potential is discussed, and is ascribed to a random distribution of native point defects located at the QW surface.

Figure 1

(a) ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}∕{\mathrm{In}}_{0.52}{\mathrm{Al}}_{0.48}\mathrm{As}$ QW structure grown by MBE on lattice-matched $\mathrm{In}\mathrm{P}\left(111\right)A$ substrate. Electronic states in the $10\text{−}\mathrm{nm}$-thick ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}$ surface QW are confined on one side by vacuum and on the other side by the $5\text{−}\mathrm{nm}$-thick ${\mathrm{In}}_{0.52}{\mathrm{Al}}_{0.48}\mathrm{As}$ barrier. STS measurements were performed at $5\mathrm{K}$ under UHV on the clean epitaxial surface of the ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}$ QW. (b) Typical $dI∕dU$ spectrum acquired at a given point of the ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}$ QW surface. The hatched region represents the band gap. The subband minima ($E_1$, $E_2$, and $E_3$) found from the spectrum are indicated. (c) Potential profile used to calculate the energy of subband minima in the ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}$ QW. $l$ is the QW thickness. $\chi$ is the electron affinity of ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}$. $\Delta E_C$ is the CB offset between ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}$ and ${\mathrm{In}}_{0.52}{\mathrm{Al}}_{0.48}\mathrm{As}$.

Figure 2

(Color) (a) STM topography of a $214\times 214{\mathrm{nm}}^2$ area of the ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}$ QW surface at $U=+1.1\mathrm{V}$. A $dI∕dU$ spectrum was acquired on each pixel of a $128\times 128$ grid of this area, for $U$ ranging from $0\mathrm{V}\text{to}+1\mathrm{V}$. (b) $dI∕dU$ spatial map of the same area as in (a), at $U=+0.376\mathrm{V}$. (c) $dI∕dU$ spectra averaged over the square areas A (solid curve) and B (dashed curve) indicated in (b). (d) $dI∕dU$ spectrum averaged over the whole area of (a) (black curve), and fit of each subband by Eq. (1) (yellow curve). The percolation thresholds ($E_{C1}$, $E_{C2}$, and $E_{C3}$) determined by the fit are indicated.

Figure 3

(Color) $dI∕dU$ spatial maps of the same area as in Fig. 2a at several values of $U$ covering the transition (a) from the band gap to the first subband, (b) from the first to the second subband, and (c) from the second to the third subband. The percolation thresholds ($E_{C1}$, $E_{C2}$, and $E_{C3}$), determined by fitting the $dI∕dU$ spectrum of Fig. 2d by Eq. (1), are indicated.

Figure 4

(Color) (a) STM topography of a $10.7\times 10.7{\mathrm{nm}}^2$ area of the ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}$ QW surface at $U=+1.2\mathrm{V}$. An acceptorlike point defect (dark spot) is visible. (b) $dI∕dU$ spectrum acquired on the acceptorlike point defect of (a) (solid curve), and $dI∕dU$ spectrum acquired at a point located $4\mathrm{nm}$ away from the defect (dashed curve). In the $dI∕dU$ spectrum acquired on the defect, a peak appears near the valence-band maximum, as indicated by an arrow. (c) STM topography of a $10.7\times 10.7{\mathrm{nm}}^2$ area of the ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}$ QW surface at $U=+1.1\mathrm{V}$. A donorlike point defect (bright spot) is visible. (d) $dI∕dU$ spectrum averaged over a disk area of $2\mathrm{nm}$ radius centered on the donorlike point defect of (c) (solid curve), and $dI∕dU$ spectrum averaged over points located $11.5\mathrm{nm}$ away from the defect (dashed curve). In the $dI∕dU$ spectrum close to the defect, a peak appears near each subband minimum, as indicated by arrows.

Figure 5

(Color) (a) STM topography of a $96\times 96{\mathrm{nm}}^2$ area of the ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}$ QW surface at $U=+2.5\mathrm{V}$. A $dI∕dU$ spectrum was acquired on each pixel of a $64\times 64$ grid of this area, for $U$ ranging from $-1.2\text{to}+1\mathrm{V}$. The positions of the acceptorlike and donorlike point defects found in this area are indicated by blue and red squares, respectively. (b) $dI∕dU$ spectra averaged over the square areas AD (acceptorlike point defect, blue curve) and DD (donorlike point defect, red curve) indicated in (a). (c) Calculated spatial map of the potential $V\left(\mathbf{r}\right)$ in the same area as in (a). See the text for details about the calculation of $V\left(\mathbf{r}\right)$. (d) $dI∕dU$ spatial maps of the same area as in (a) at two different values of $U$. The values of $U$ are given with respect to the experimentally determined percolation threshold for the second subband, $E_{C2}$. (e) Calculated spatial maps of the LDOS of the $n\text{th}$ subband, in the same area as in (a), at two different energies $E$. The energies $E$ are given with respect to the theoretical percolation threshold for the $n\text{th}$ subband, $E_{Cn}=E_n+\overline{V}$. White and black regions correspond to nonzero and zero LDOS, respectively. See the text for details about the calculation of the LDOS.