Spectroscopy of the local density of states in nanowires using integrated quantum dots

In quantum dot (QD) electron transport experiments, additional features can appear in the differential conductance $dI/dV$ that do not originate from discrete states in the QD, but rather from a modulation of the density of states (DOS) in the leads. These features are particularly pronounced when the leads are strongly confined low-dimensional systems, such as in a nanowire (NW) where transport is one dimensional and quasi-zero-dimensional lead states can emerge. We study such lead states in InAs NWs. We use a QD integrated directly into the NW during the epitaxial growth as an energetically and spatially well-defined tunnel probe to perform $dI/dV$ spectroscopy of discrete bound states in the “left” and “right” NW lead segments. By tuning a sidegate in close proximity of one lead segment, we can distinguish transport features related to the modulation in the lead DOS and to excited states in the QD. We implement a noninteracting capacitance model and derive expressions for the slopes of QD and lead resonances that appear in two-dimensional plots of $dI/dV$ as a function of source-drain bias and gate voltage in terms of the different lever arms determined by the capacitive couplings. We discuss how the interplay between the lever arms affects the slopes. We verify our model by numerically calculating the $dI/dV$ using a resonant tunneling model with three noninteracting quantum dots in series. Finally, we use the model to describe the measured $dI/dV$ spectra and quantitatively extract the tunnel couplings of the lead segments. Our results constitute an important step towards a quantitative understanding of normal and superconducting subgap states in hybrid NW devices.

Figure 1

(a) False color scanning electron microscopy (SEM) image of device I that consists of an InAs NW with two short InP segments (white) that form an epitaxially defined quantum dot (QD). Between the epitaxially defined QD and the source S (blue) and drain D (yellow) contacts are bare InAs segments, which we refer to as the NW lead segments L1 and L2. Adjacent to the NW lead on L2 at the drain side, a local sidegate is placed. In addition, the substrate acts as a global backgate. (b) Illustration of the sharp density of states of the QD and modulated density of states in the two NW lead segments L1 and L2. Note that we indicated the barriers between the source contact and L1, and between the drain contact and L2, by a fuzzy box to indicate that in practice, these barriers are ill defined. This is in contrast to the barriers forming the QD. They are precisely known, both in position and size.

Figure 2

(a) Differential conductance $dI/dV$ as a function of the voltage bias $V_{sd}$ (vertical axis) and backgate voltage $V_{bg}$ (horizontal axis) at $V_{sg}=0$ V and (b) $dI/dV$ for sidegate voltage $V_{sg}$ at $V_{bg}=22$ V. White arrows indicate excited-state (ES) and green arrows lead-state resonances originating from the L2 lead segment. Black arrows show how the additional energy for the first ($E_{\text{add}}$) and second (charging energy $E_C$) electron in an orbital is estimated. From these values, the orbital energy spacing $\mathrm{\Delta }E$ can be estimated.

Figure 3

(a) Equivalent circuit used to model the electrostatics of the device in Fig. 1. Here, “L1” and “L2” are the left and right lead QDs and “QD” is the integrated QD. The gray boxes represent a tunnel coupling (resistor) and a capacitive coupling. The capacitors $C_{m,N}$ are labeled by two subscripts, where the first $m\in \left\{bg\right|sg\left|s\right|d\}$ refers to the metallic parts consisting of backgate and sidegate, source, and drain, and the second $N\in \left\{\mathrm{QD}\right|\mathrm{L}1\left|\mathrm{L}2\right\}$ refers to the integrated QD and the lead segments L1 and L2. (b) A stability diagram illustrating the resonances that arise when the electrochemical potential ${\mu }_{QD}$ in the integrated QD (sometimes loosely denoted as the “state” or “level” in the QD) aligns with either the electrochemical potential of the source ${\mu }_s$ or drain ${\mu }_d$ contacts (solid black). The lead-state resonances appear instead when ${\mu }_{QD}$ aligns with either ${\mu }_{L1}$ (red) or ${\mu }_{L2}$ (green). In the latter case, the solid lines represent the case when the lever arms ${\alpha }_{sL1}$ (red) or ${\alpha }_{dL2}$ (green) are equal to one, whereas for the dashed lines, ${\alpha }_{sL1},{\alpha }_{dL2}<1$. The alignment of the electrochemical potentials is depicted for five points indicated by the roman numerals. Here, the density of state for one lead shows a pronounced peak, while for the opposite lead a constant DOS is assumed.

Figure 4

Differential conductance $dI/dV$ as a function of voltage bias $V_{sd}$ and (a) backgate voltage $V_{bg}$ and (c) sidegate voltage $V_{sg}$ for device I. (b),(d) The modeled $dI/dV$ where the lever arms were extracted from the slopes of the measured data seen in (a) and (c). (e) $dI/dV$ as a function of $V_{sd}$ and $V_{bg}$ measured for device II. The lead resonances are indicated with green (L1) and blue (L2) lines and arrows. (f) Results from the model with input values extracted from (e). (g) and (f) show line cuts along the green and blue curves in (e) and (f), respectively. Here, the solid colored lines are the model values and the black markers are the experimental data.