Semimetallic and semiconducting graphene-$h\mathrm{BN}$ multilayers with parallel or reverse stacking

We theoretically investigate three-dimensional (3D) layered crystals of alternating graphene and $h\mathrm{BN}$ layers with different symmetries. Depending on the hopping parameters between the graphene layers, we find that these synthetic 3D materials can feature semimetallic, gapped, or Weyl semimetal phases. Using first-principles calculations to parametrize the low-energy Hamiltonians we establish the most likely electronic phases. Our results demonstrate that 3D crystals stacked from individual 2D materials represent a synthetic materials class with emergent properties different from their constituents.

Figure 1

Possible electronic phases for 3D crystals of graphene and $h\mathrm{BN}$ monolayers periodically stacked in the $z$ direction (sketch on the top left, where the dashed orange line indicates the unit cell). Depending on the hopping parameters ${\gamma }_5$ and ${\gamma }_2$ between graphene sheets, this artificial material may exhibit different semimetal phases illustrated by the dispersion sketches. The black mark in the parameter map indicates parameter values we obtain from DFT calculations yielding the range ${\gamma }_5\in \{0.48{\gamma }^2/V_B,0.64{\gamma }^2/V_B\}$ and ${\gamma }_2\in \{-0.10{\gamma }^2/V_B,-0.07{\gamma }^2/V_B\}$ with increasing interlayer distance $d$, identifying overlapping bands as the most likely phase (pink regime).

Figure 2

Parameters obtained by fitting the model Hamiltonian [see Appendix pp2, Eq. (B1)] to the DFT-calculated low energy band structure, as a function of the layer spacing $d$. In the rightmost figure, we explicitly compare the DFT and model low energy band structure for the lowest energy distance $d=3.15$ Å.

Figure 3

Fermi surface for $E_F=0$ of the overlapping bands of the parallelly stacked Gr/$h\mathrm{BN}$ 3D crystals that we identify using the parameters obtained from DFT.

Figure 4

Possible dispersions for 3D Gr/$h\mathrm{BN}$ crystals with antiparallel stacking, where adjacent $h\mathrm{BN}\mathrm{s}$ in every second layer are rotated with respect to each other (sketch on the left). The dashed orange box illustrates that the unit cell in the $z$ direction is twice as large for antiparallel stacking compared to parallel arrangements. We chose similar parameters as for the phases (blue, magenta) in Fig. 1. The phenomenological low-energy description in the magenta phase reproduces our DFT results (rightmost dispersion for a cut along the turquoise path in momentum space where the two electron bands and the two hole bands are degenerate, respectively).

Figure 5

Conical crossings between the 3D bands of antiparallelly stacked Gr/$h\mathrm{BN}$ stacks at $k_z=0$ that we determine for the parameters obtained from DFT. The white ring of radius $k_0^{\prime }$ indicates the $E_F=0$ Fermi line.

Figure 6

Geometries of the parallel stacking configurations. The dashed lines represent the unit cell. (a) ${\mathrm{C}}_{\text{A}}$ and B are dimer atoms, while ${\mathrm{C}}_{\text{B}}$ and N are nondimer atoms. The interlayer distance $d=3.35$ Å. (b) ${\mathrm{C}}_{\text{A}}$ and N are dimer atoms, while ${\mathrm{C}}_{\text{B}}$ and B are nondimer atoms. The interlayer distance $d=3.50$ Å. (c) ${\mathrm{C}}_{\text{A}}$ and B, as well as ${\mathrm{C}}_{\text{B}}$ and N, are dimer atoms. The interlayer distance $d=3.55$ Å. Initial interlayer distances from Ref. [9].

Figure 7

Top: DFT and fit results for the geometry in Fig. 6. Bottom: the corresponding DFT-calculated projected band structure.

Figure 8

Top: DFT and fit results for the geometry in Fig. 6. Bottom: the corresponding DFT-calculated projected band structure.

Figure 9

Top: DFT and fit results for the geometry in Fig. 6. Bottom: the corresponding DFT-calculated projected band structure.

Figure 10

Evolution of the fit parameters as a function of the interlayer distance, summarizing the results from Table 1, Table 2, and Table 3.