Imaging coherent transport in a mesoscopic graphene ring

Mesoscopic graphene devices often exhibit complex transport properties, stemming both from the peculiar electronic band structure of graphene and from the high sensitivity of transport to local disorder in this two-dimensional crystal. To disentangle contributions of disorder in the different transport phenomena at play in such devices, it is necessary to devise new local-probe methods and to establish links between transport and the microscopic structure of the devices. Here, we present a spatially resolved investigation of coherent transport inside a graphene quantum ring (QR), where Aharonov-Bohm conductance oscillations are observed. Thanks to scanning gate microscopy (SGM), we first identify spatial signatures of the Coulomb blockade, associated with disorder-induced localized states, and of charge-carrier interferences. We then image resonant states which decorate the QR local density of states (LDOS). Simulations of the LDOS in a model disorder graphene QR and temperature dependence of SGM maps confirm the presence of such scarred states.

Figure 1

Structural characterization of the QR and electrical transport measurements. (a) AFM picture of the QR. Dashed lines delineate fluorinated graphene regions. The scale bar is 400 nm. (b) $G$ vs $V_{\mathrm{g}}$ at 30 mK. The inset presents $G$ vs $V_{\mathrm{g}}$ at 7 K. The position of the Dirac point is $\sim 25$ V, as determined by linear fits performed on both sides of the minimum (red dashed lines). (c) Magnetoconductance obtained at 30 mK and $V_{\mathrm{g}}=-15$ V. The inset represents the FFT of the magnetoconductance curve. The shaded region indicates the AB frequency range defined by the outer and inner radii of the ring. The red dashed line is a Lorentzian fit to the data. (d) AB oscillations amplitude as a function of $V_{\mathrm{g}}$.

Figure 2

SGM imaging of the QR. (a) SGM experimental setup: the graphene QR is defined by fluorination (green regions) and lies on an oxidized doped Si substrate, used as a back gate. The yellow areas correspond to ohmic contacts on top of the graphene flake. A metallic tip is approached close to the QR, with an applied bias $V_{\mathrm{tip}}$, and is scanned to produce a SGM map of the device conductance. (b) SGM conductance map obtained at large charge-carrier concentration ($n=7.2\times {10}^{12}{\mathrm{cm}}^{-2}$; $V_{\mathrm{g}}=-5$ V), $V_{\mathrm{tip}}=-2$ V, $B$ = 30 mT, and $T$ = 30 mK. The continuous lines delineate the position of the QR. The red dashes highlight two examples of conductance fringes inside the QR arms. The scale bar represents 400 nm.

Figure 3

SGM images of the QR at low charge-carrier density, close to the neutrality point. (a),(b) SGM map at $V_{\mathrm{g}}$ = +18 V, $B$ = 0.7 T, $V_{\mathrm{tip}}=-2$ V, and $T=30$ and 500 mK, respectively. Continuous lines indicate the position of the QR. The scale bars correspond to 400 nm. (c) Plot of the conductance vs $V_{\mathrm{tip}}$ and the tip position along the ${\mathrm{AA}}^{{}^{\prime }}$ green dashed line in Fig. 3. The positions of both QR apertures are given by the black dashed lines. Red dashed lines correspond to a Gaussian fit to the data, and highlight Coulomb resonances.

Figure 4

SGM line profile, measured along the ${\mathrm{BB}}^{{}^{\prime }}$ green dashed line in Fig. 3, as a function of the magnetic field ($V_{\mathrm{tip}}=-2$ V, $V_{\mathrm{g}}$ = +18 V and $B$ = 0.7 T). Vertical lines correspond to the inner and outer limits of the QR.

Figure 5

SGM mapping and LDOS simulations at large charge-carrier density. (a) SGM map at $V_{\mathrm{g}}=-5$ V, $B$ = 30 mT, $V_{\mathrm{tip}}$ = +0.5 V, and $T$ = 30 mK. Continuous lines indicate the position of the QR. Arrows point toward radial fringes inside the QR. The scale bar corresponds to 400 nm. (b) QR model geometry used for tight-binding simulations including a random disorder potential. (c),(d) LDOS simulations for $E_{\mathrm{F}}/t$ = 0.144 and 0.119, respectively. For each figure, the black scale bar represents $100a$.

Figure 6

$\Delta G_{\mathrm{SGM}}$ extracted from SGM maps calculated over the fluorinated areas enclosed by the red dashed boxes (red circles, left axis), in the regions where CB effects were evidenced at 500 mK enclosed by the blue dashed boxes (blue circles, left axis) and over the graphene QR area enclosed by the green dashed box (green circles, left axis) in Fig. 3 ($V_{\mathrm{tip}}=-2$ V, $V_{\mathrm{g}}$ = +18 V, and $B$ = 0.7 T). $\Delta G_{\mathrm{AB}}$, extracted from the magnetoconductance curves (black circles, right axis), vs $T$, for $V_{\mathrm{g}}$ = +25 V. Red and blue dashed lines are the result of a fit to $\alpha T^{-1/2}+\beta$ of $\Delta G_{\mathrm{SGM}}$ calculated in the red ($\alpha$ = 0.94 and $\beta$ = 0) and blue ($\alpha$ = 0.94 and $\beta$ = 0.0065) boxes, respectively, in Fig. 3. The green dashed line is a fit to $\gamma T^{-1/4}$ ($\gamma =0.4$) of $\Delta G_{\mathrm{SGM}}$ calculated in the green box of Fig. 3.