Semimetallic features in quantum transport through a gate-defined point contact in bilayer graphene

We demonstrate that, at the onset of conduction, an electrostatically defined quantum wire in bilayer graphene with an interlayer asymmetry gap may act as a one-dimensional semimetal, due to the multiple-minivalley dispersion of its lowest subband. Formation of a nonmonotonic subband coincides with a near-degeneracy between the bottom edges of the lowest two subbands in the wire spectrum, suggesting an $8e^2/h$ step at the conduction threshold, and the semimetallic behavior of the lowest subband in the wire would be manifest as resonance transmission peaks on an $8e^2/h$ conductance plateau.

Figure 1

Parametric ranges for two quantum transport regimes in a BLG channel with a zigzag orientation along the transport axis. For a small channel width, $W$, or small interlayer gap, ${\mathrm{\Delta }}_0$, the subbands in the BLG wire are approximately quadratic and well separated, leading to the regular ladder of $4e^2/h$ conduction steps (white insets). For a larger ${\mathrm{\Delta }}_0$ or wider $W$, the lowest conduction subbands feature band inversions with multiple band minima/additional degeneracies (black insets). This enables coherent forward scattering resonances, resulting in additional conductance spikes (solid black line) above an initial step of $8e^2/h$ specific to the adiabatic limit, as indicated by the dotted black line.

Figure 2

Top: Potential profile, $V_{+}(x,y)$, confining electrons to the upper layer in gapped BLG along a quasi-1D channel between two leads. Crosses in the profile illustrate the edges of the semimetallic segment of the wire which supports standing waves in the ‘holelike’ part of the nonmonotonic lowest-energy subbands, resulting in transmission resonances. Middle: One-dimensional energy spectra (plotted vs traveling momentum $p_y$) for cuts along a zigzag (ZZ)- and an armchair (AC)-oriented channel. Bottom left: Minivalleys A–C are identified in the contour plot illustrating the trigonally warped low-energy band structure [Eq. (2)] of gapped BLG around the $K^{-}$ point. Bottom right: Overlap between subband states at the Fermi energy in the leads and the top of the inverted branch of the lowest-energy subband at the edge of the semimetallic wire. Overlaps computed using the continuum model of gapped BLG in the presence of a confining potential and spatially modified gap (similar to the methods used in Ref. [20]).

Figure 3

(a) Conductance, $G$, through a channel with a zigzag lattice orientation for a range of smoothing distances, $\mathrm{\Lambda }$. The corresponding channel spectrum is illustrated below. Parameters used are $U_0=-20\mathrm{meV}, {\mathrm{\Delta }}_0=150\mathrm{meV}, \beta =0.3, \delta =10\mathrm{meV}, V_0=-200\mathrm{meV}, L=300$ nm, and $W=50$ nm. Inset at top: The frequency of the resonance peaks on top of the first conductance plateau changes with the length of the semimetallic region, $\ell$. The average lateral energy spacing of the resonance peaks is proportional to the inverse of the effective channel length ($\mathrm{\Delta }E\propto {\ell }^{-1}$), while the wave number, $k$, of standing waves within the channel varies linearly with the channel length (crosses show data extracted from the LDOS images, which are fit with a calculated curve assuming hard wall boundary conditions at the wire ends). (b) Transport characteristics for a weakly gapped system $[{\mathrm{\Delta }}_0=60\mathrm{meV}$; other parameters the same as in (a)] which removes the nonmonotonic features in the dispersion of the lead, resulting in regular, adiabatic conductance steps of $4e^2/h$.

Figure 4

(a) Momentum-averaged local density of states (LDOS) at energy cuts along the first conductance plateau for a zigzag-oriented wire. Device parameters are the same as in Fig. 3, with $\mathrm{\Lambda }=20$ nm. Numerals I–VI identify the energy of each LDOS image in the channel's low-energy subband spectra shown in (b) and in the corresponding conductance profile in the inset in Fig. 3. LDOS image V is for an energy on the first plateau (between peaks), while the other cuts correspond to the energies of resonance peaks. (c) The sum of ‘electronlike’ states in (d) (black curve) compared to the renormalized profile extracted from LDOS image V along the dashed white path. (d) Continuum wave functions calculated for the energy used in LDOS image V. Solid (dashed) wave functions are right-moving (left-moving) states corresponding to filled (open) circles in the $K^{-}$ valley subband spectra in (b).

Figure 5

(a) Conductance, $G$, through a channel with an armchair lattice orientation for a range of smoothing distances, $\mathrm{\Lambda }$. The spectrum within the channel is demonstrated below. The parameters used are the same as in Fig. 3. (b) Transport characteristics for a weakly gapped channel with an armchair orientation [parameters the same as in Fig. 3]. This again removes the band inversions, resulting in an approximately quadratic energy spectrum and a regular series of $4e^2/h$ conductance steps.

Figure 6

Top: Schematic of how the device (represented by the potential profile) is split up into slices in order to implement the recursive algorithm. Bottom: Sketch of the atomic arrangement in each slice for both zigzag and armchair orientations along the transport axis. Slices are constructed by tiling a minimal rectangular unit cell (e.g., gray zone) containing eight atoms along the nontransport (vertical) axis. Intralayer (${\gamma }_0$) hops are shown as solid (dashed) lines for the lower (upper) layer. Intralayer hops within a slice are shown in black, while hops between slices are shown in red. Skew interlayer (${\gamma }_3$) hops within the slice are shown in blue and those between slices are shown in green. The thick black vertical lines define the edges of the $n\mathrm{th}$ slice.

Figure 7

Sketch of how the Rubio-Sancho method operates. Each iteration, $j$, removes every other cell from the lead, updating the values of the local Hamiltonians, ${\mathbf{h}}_{sj}$ and ${\mathbf{h}}_j$, coupling matrices, ${\mathbf{V}}_j^{(†)}$, and surface self-energy, ${\mathrm{\Sigma }}_{sj}$. The algorithm converges when remaining cells are sufficiently distant such that ${\mathbf{V}}_j^{(†)}\to 0$.

Figure 8

Further conductance data (middle) and differential conductance data (bottom) with corresponding channel spectra (top) for zigzag and armchair lattice orientations along the transport axis (left and right columns, respectively) for different values of smoothing length, $\mathrm{\Lambda }$. Channel parameters are $U_0=-20\mathrm{meV}, {\mathrm{\Delta }}_0=150\mathrm{meV}, \beta =0.3, \delta =10\mathrm{meV}, V_0=-200\mathrm{meV}, W=50$ nm, and $L=240$ nm.