Graphene kirigami as a platform for stretchable and tunable quantum dot arrays

The quantum transport properties of a graphene kirigami similar to those studied in recent experiments are calculated in the regime of elastic, reversible deformations. Our results show that, at low electronic densities, the conductance profile of such structures replicates that of a system of coupled quantum dots, characterized by a sequence of minibands and stopgaps. The conductance and I-V curves have different characteristics in the distinct stages of deformation that characterize the elongation of these structures. Notably, the effective coupling between localized states is strongly reduced in the small elongation stage but revived at large elongations that allow the reestablishment of resonant tunneling across the kirigami. This provides an interesting example of interplay between geometry, strain, spatial confinement, and electronic transport. The alternating miniband and stopgap structure in the transmission leads to I-V characteristics with negative differential conductance in well defined energy/doping ranges. These effects should be stable in a realistic scenario that includes edge roughness and Coulomb interactions, as these are expected to further promote localization of states at low energies in narrow segments of graphene nanostructures.

Figure 1

(a) Pattern schematics of the graphene kirigami indicating the most relevant geometric parameters. Snapshots of the actual kirigami used in the calculations for deformations of (b) 0%, (c) 15.5%, and (d) 34.7%. The figures were generated by VMD [27].

Figure 2

Stress-strain curve for the kirigami, showing the first two stages corresponding to the elastic and reversible deformations.

Figure 3

(a) Conductance of the kirigami for different values of deformation. (b) DOS of the undeformed structure. The lower row of panels contains a magnification of the energy range comprising the first group of resonances ($0.03<E/t_0<0.09$) for the undeformed kirigami (c), kirigami deformed by 15.5% (d), and deformed by 34.7% (e).

Figure 4

Normalized LDOS for the kirigami (a) undeformed at $E=0.067t_0$, (b) deformed by 15.5% at $E=0.074t_0$, and (c) deformed by 34.7% at $E=0.073t_0$.

Figure 5

LDOS at one of the transmission peaks ($E=0.044t_0$) that contributes to the lowest transmission miniband of the undeformed kirigami, as discussed in the main text. The corresponding kirigami structure is shown underneath the density plot.

Figure 6

Normalized C-C bond length for the kirigami deformed by 15.5% [panel (a), stage 1] and 34.7% [panel (b), stage 2], where the normalization is by the undeformed C-C bond length.

Figure 7

The distribution of the nearest-neighbor hopping amplitudes $t_{ij}/t_0$ calculated according to Eq. (1) for the same structures represented in Fig. 6.

Figure 8

Current vs gate voltage for the kirigami deformed by 0%, 15.5%, and 34.7%. We used a bias voltage of $V_{SD}=100$ meV, the current is expressed in units of $I_0=2e/h$, and the gate voltage in units of $t_0$.

Figure 9

Effective conductance in the presence of (a) weak ($\eta =0.01t_0,{\tau }_{\varphi }\approx 6.1\times {10}^{-15}\mathrm{s}$) and (b) strong ($\eta =0.5t_0,{\tau }_{\varphi }\approx 1.2\times {10}^{-16}\mathrm{s}$) phase-breaking scatterers for different deformations. The dashed blue line corresponds to the conductance of a pristine zigzag nanoribbon with the same width of our contacts. Be aware of the different scales in the vertical axes.

Figure 10

(a) Conductance of the armchair kirigami deformed by 0% and 16.28%. (b) LDOS for the undeformed armchair kirigami at $E=0.121t_0$.