Semiclassical deconstruction of quantum states in graphene

We present a method for bridging the gap between the Dirac effective field theory and atomistic simulations in graphene based on the Husimi projection, allowing us to depict phenomena in graphene at arbitrary scales. This technique takes the atomistic wave function as an input, and produces semiclassical pictures of quasiparticles in the two Dirac valleys. We use the Husimi technique to produce maps of the scattering behavior of boundaries, giving insight into the properties of wave functions at energies both close to and far from the Dirac point. Boundary conditions play a significant role to the rise of Fano resonances, which we examine using the processed Husimi map to deepen our understanding of bond currents near resonance.

Figure 1

A magnified view of a boundary on a graphene flake. The orientation of the cut relative to the orientation of the lattice can produce two edge types, zigzag (highlighted in blue) and armchair (highlighted in red). The two sublattices of the unit cell are indicated in black (A sublattice) and gray (B sublattice).

Figure 2

The two-dimensional dispersion relation for graphene demonstrates the two inequivalent valleys as cones where the edges of the Brillouin zones (black lines) meet. Dashed white lines indicate the one-dimensional dispersion surface at $E=0.5t$, while solid white lines indicate $E=0.98t$, demonstrating extreme trigonal warping.

Figure 3

Processed Husimi maps (left and central plots) and eigenstates (right plots) of the closed graphene stadium billiard with 20 270 orbital sites at energies $E=0.974t\text{(a)},0.964t\text{(b)},\text{and}0.951t\text{(c)}$. All three calculations use coherent states with relative uncertainty $\Delta k/k=30\%$, whose breadth is indicated by the double arrows on the right. Only the upper-right quarter of each stadium is shown. The left plots present the multimodal analysis for the $K^{\prime }$ valley. The magnitude of the divergence of the processed Husimi map [Eq. (11)] is indicated in green (red) for positive (negative) values. The central panels present the magnitude of the angular deflection indicated in blue [Eq. (13)]. Red boxes indicate scattering regions magnified in Fig. 4.

Figure 4

Magnified views of the divergence and angular deflection in Fig. 3 (red boxes). The sources and drains in the $K^{\prime }$-valley processed Husimi map are actually intervalley scattering points, which occur along nonzigzag boundaries. In contrast, points of angular deflection that are not sources or drains correspond to intravalley scatterers and occur along pure or nearly pure zigzag boundaries.

Figure 5

A closed-system eigenstate at $E=0.72t$ for the smaller graphene stadium. At top, the filtered multimodal analysis with relative momentum uncertainty $\Delta k/k=30\%$ along with the wave function (right). The spread of the coherent state is indicated by the double arrows. Bottom, higher-resolution calculations of the divergence (green for positive, red for negative) and the angular deflection (blue) are shown against the graphene structure. The black circle indicates where the system boundary is perturbed in the original paper[27] as discussed in Sec. 3c.

Figure 6

Schematic indicating the locations of armchair (blue) and zigzag (red) edges in the circular system (left) and the Wimmer system (right).

Figure 7

Low-energy graphene states require additional tools to fully grasp the classical dynamics. The processed Husimi map for the $K^{\prime }$ valley is plotted for four eigenstates of a closed circular system with 71 934 orbital sites at energies around $E=0.2t$. All three calculations use coherent states with relative uncertainty $\Delta k/k=20\%$, with breadth indicated by the double arrows on the right. From left to right: the Husimi flux, multimodal analysis, and the wave function. The divergence of the Husimi flux is indicated in green (red) for positive (negative) values. In blue, the angular deflection.

Figure 8

In parts (a) and (b), the same information is plotted as in Fig. 7, but for the Wimmer system (see Fig. 6), with 96 425 orbital sites. These states also have energies near $E=0.2t$ and are represented by coherent states of uncertainty $\Delta k/k=20\%$ with breadth indicated by the double arrows.

Figure 9

An extremely small rooftop graphene flake at energy $E=0.0015735t$ showing two edge states at the top and bottom boundaries, which tunnel into each other. At top, the full wave function, at middle, divergence is indicated in green and red, and a schematic of the Husimi flux for the $K^{\prime }$ valley is shown. At bottom, the Fourier transform of the state is shown with the contour line used to generate the processed Husimi map in white, set at an arbitrary energy in order to maximize the intersection of the contour with the Fourier-transform amplitude. The double arrows indicate the spread of the wave packet used to generate this map.

Figure 10

System properties of the scattering density matrix $\rho$ around the Fano resonance centered at $E=1.9582\text{eV}$ for the open system in the inset. Top: The transmission profile across the two leads, with the closed-system eigenstate energy at $E=1.9579\text{eV}$, corresponding to the eigenstate at index 1483 (below), indicated by the vertical gray line. Middle: Diagonalizing the density matrix produces a handful of nontrivial scattering wave functions in its eigenvectors. The eigenvalues of these vectors, which correspond to their measurement probability, are graphed. The wave function associated with the closed-system eigenstate hybridizing with the direct channel peaks strongly around the Fano resonance. Bottom: The density matrix is projected onto the closed-system eigenstates, showing that eigenstate 1483 strongly peaks at the Fano resonance.

Figure 11

Above and below the Fano resonance in Figs. 10 (inset), the time-reversal symmetry between the $K$ and $K^{\prime }$ valleys is lifted, making it possible to add the Husimi flux for both valleys to measure valley-polarized current. Above, the Husimi flux maps of both valleys are added for the scattering wave function at energies $E=1.9582t$ and $1.9586t$, with $\Delta k/k=30\%$. Below, the probability flux, convolved with a Gaussian kernel of the same size as the coherent state. At energies this close to resonance, the wave function does not visually change from the closed-system eigenstate in the inset, but the residual current that occurs near these resonances switches direction across resonance.