Formation mechanism of bound states in graphene point contacts

Electronic localization in narrow graphene constrictions is theoretically studied, and it is found that long-lived ($\sim$1 ns) quasibound states (QBSs) can exist in a class of ultrashort graphene quantum point contacts (QPCs). These QBSs are shown to originate from the dispersionless edge states that are characteristic of the electronic structure of generically terminated graphene, in which pseudo-time-reversal symmetry is broken. The QBSs can be regarded as interface states confined between two graphene samples, and their properties can be modified by changing the sizes of the QPC and the interface geometry. In the presence of bearded sites, these QBSs can be converted into bound states. Experimental consequences and potential applications are discussed.

Figure 1

(a) Schematic of the QPC connecting samples near the middle. (b) Schematic of the QPC connecting samples near the edge, where edge states can play an important role in electron transmission. (c) Schematic of the generic graphene QPCs connecting samples near the edge. Both samples infinitely extend along the edges $e_2$ and $e_2^{\prime }$. The angles $\theta$ and ${\theta }^{\prime }$ can be taken to be the same without loss of generality.

Figure 2

Classification of QPCs connecting samples near the edges. (a) Example of class I, which has all AC edges and no edge state. QPCs in this class are similar to QPCs connecting near the middle of samples. (b) Example of class II, for which the interface edges are ZZ but the extending edges are AC. In this class the edge states exponentially decay from the interface and are localized near the interfaces, i.e., bound states. Here the white (black) circles indicate the charge density of edge states on the A (B) sublattice sites. The schematic charge densities of the edge state are for isolated samples [41]. (c) Example of class III, for which the extending edges are ZZ, whereas the interfacing edges can be either ZZ (as in the example shown here) or AC (as in the example shown in Fig. 3). In this class, edge states also exist. However, they are not bound states as indicated by the schematic wave functions of semi-infinite isolated samples (bottom panel). Nevertheless, bound states emerge when QPCs are present. (d) Enlarged view of QPC [indicated by the dashed rectangle in panel (c)]. The QPC contains $N_c$ connecting bonds.

Figure 3

(a) Class-III QPC with rectangular corners with $N_c=5$ connecting bonds. In this QPC, the edge states have amplitude on sublattice A (indicated by the white circle) for the left-hand graphene sample, whereas they have amplitude on sublattice B (indicated by the black circle) for the right-hand graphene sample. The SCBs and WCBs alternate with each other. (b) Energy spectrum for a semi-infinite ribbon consisting of $N_z=6$ zigzag chains. Shaded areas indicate the spectrum of bulk graphene. Edge states appear in $k_c<k\le \pi /a$ (red segments). The black lines indicate the extended states. (c) Magnitude of the QBS wave function $\langle j|\text{QBS}\rangle$ in the vicinity of a SCB assuming a symmetric boundary condition, which can be observed in STM. The calculation was done for $N_z=50$ and ${\varepsilon }_{\text{QBS}}\approx 0.04$.

Figure 4

Theory vs numerical calculations: energy dependence of conductance $g$ for the QPC shown in Fig. 3. Circles represent numerical calculations while solid lines indicate fitting according to Eq. (1). The resonances are labeled $(N_c,i_L)$, in the same manner as in Table 1. In panel (a), only one SCB exists, whereas in panel (c) two SCBs exist. Each SCB leads to a peak in $g$.

Figure 5

Theory vs numerical calculations: $N_z$ dependences of (a) peak conductance, (b) QBS energy, and (c) QBS broadening parameter (both in units of ${\gamma }_0$). The numerical calculations were done for the resonance $(3,2)$ seen in Fig. 4. Our theory predicts roughly constant $g_m$, convergence of ${\varepsilon }_{\text{QBS}}$ to ${\varepsilon }_{\infty }$, and ${\Gamma }_{\text{QBS}}\sim N_z^{-3}$, all of which are consistent with numerical calculations. In panel (c), the solid curve has a slope of $\sim$0.3 while the dashed curve has a slope of $\sim$0.2. The dashed line is to guide the eye. The solid line shows results of theory, and circles show results of numerical calculations.

Figure 6

Bearded sites, represented by the circles attached to the dashed bonds, which turn a WCB into a SCB. A pair of real bound states form on the bond (see text). Geometrically, the SCB (blue bond) shown here is similar to the SCB $(N_c=3,m=2)$ without bearded sites.