Imaging localized states in graphene nanostructures

We present scanning-gate images of a single-layer graphene quantum dot which is coupled to source and drain via two constrictions. We image and locate conductance resonances of the quantum dot in the Coulomb-blockade regime as well as resonances of localized states in the constrictions in real space, which are interpreted in light of previous transport experiments. In addition our technique allows us to estimate the extent of these localized states, namely, radii of about 10–13 nm.

Figure 1

(Color online) (a) Atomic force micrograph of the graphene sample after reactive ion etching obtained under ambient conditions. The QD is connected to source and drain via two constrictions. The nearby nanoribbon can be used as a charge detector but it was not connected in the measurements presented here. (b) In situ atomic force micrograph of the sample after cooldown at $T\approx 2.6\text{K}$. This image was taken after positioning the tip above the sample with our home-built AFM (Ref. 28). The scale bar in (a) denotes 500 nm, in (b) $1\mu \text{m}$. (c) Backgate trace taken at $T\approx 2.6\text{K}$ and $V_{\text{bias}}=500\mu \text{V}$. The charge neutrality point is shifted to $V_{\text{BG}}\approx 30\text{V}$. (d) Charge-stability diagram of the quantum dot. The charging energy is found to be $\Delta E_C=3.5\text{meV}$ at $V_{\text{BG}}=15\text{V}$ and $T=90\text{mK}$. The speckling inside the diamonds is due to calculating the derivative $G_{\text{dot}}=\partial I_{\text{dot}}/\partial V_{\text{bias}}$ numerically.

Figure 2

(Color online) Scanning-gate image in the hole regime, $V_{\text{BG}}=12\text{V}$. The tip voltage was $V_{\text{tip}}=2\text{V}$, left gate voltage $V_{\text{LG}}=0.15\text{V}$, and the scan frame has a size of $1.4\times 1.4\mu {\text{m}}^2$. A symmetric bias of $V_{\text{bias}}=300\mu \text{V}$ was applied across source and drain and the tip was scanned at a constant height of $\Delta z\approx 120\text{nm}$ above the sample. Coulomb resonances of the quantum dot show up as concentric rings denoted by arrow (QD). The center of the Coulomb resonances is offset from the topographic center of the dot by approximately 240 nm. Such a behavior, known from previous scanning-gate experiments, is of minor importance here (Ref. 31). The outline of the quantum dot and its connection to source and drain via the two constrictions, depicted with dashed, black lines, is corrected for the offset, assuming that the Coulomb resonances are centered in the quantum dot. Most striking, however, is the appearance of two more sets of concentric rings which are highlighted by arrows (A) and (B) and which are centered around points in the constrictions. The black, dashed-dotted line between points P and Q denotes the line along which the linescan of Fig. 3 was taken. The scale bar denotes 500 nm.

Figure 3

(Color online) (a) Linescans along the dashed-dotted line between points P and Q depicted in Fig. 2 while the left gate was stepped from 5 to 10 V. The other parameters are the same as in Fig. 2. We can clearly distinguish between features of resonances A and B and the quantum dot. Two resonances of A are highlighted with white, dotted lines labeled (1) and (2), a quantum-dot resonance is highlighted with a white, dashed line, and a resonance of B is denoted with a white, dashed-dotted line. The upper, broad resonance (1) of A does not show any charging effect, whereas the lower, sharp one (2) is accompanied with avoided crossings of the Coulomb resonances. (b) Shifting resonance (1) of A along the $x$ axis and scaling it by 1.67 lead to the determination of relative lever arms between quantum dot and resonance A as explained in the text.

Figure 4

(Color online) (a) Model for explaining the essential features induced by localized states in the constrictions. For simplicity, we consider just one localized state (upper red puddle next to the source lead) which is coupled to the source lead via the tunnel coupling ${\Gamma }_{\text{loc}}$. $C_{\text{dot-loc}}$ denotes the capacitive coupling between the localized state and the quantum dot. (b) Scanning-gate measurements at $T\approx 90\text{mK}$ measured at the mixing chamber. The image was taken at $V_{\text{BG}}=12\text{V}$, $V_{\text{bias}}=35\mu \text{V}$, $V_{\text{tip}}=-100\text{mV}$, and a tip height of $\Delta z=40–45\text{nm}$; the scan frame covers an area of $0.3\times 0.3\mu {\text{m}}^2$. Compared to 2, Coulomb resonances are much sharper due to the lower temperature. Although this measurement and 2 and 3 were taken at the same backgate voltages, a direct comparison of them is difficult because of several charge rearrangements in between. The crossings of Coulomb resonances and resonances of B lead to a modulation of $G_{\text{dot}}$ and no avoided crossing of Coulomb resonance for arrow (1) and to avoided crossings for arrow (2). The scale bar denotes 100 nm.