Interference of the Bloch phase in layered materials with stacking shifts

In periodic systems, electronic wave functions of the eigenstates exhibit the periodically modulated Bloch phases and are characterized by their wave numbers $\mathbf{k}$. We theoretically address the effects of the Bloch phase in general layered materials with a stacking shift. When the interlayer shift and the Bloch wave vector $\mathbf{k}$ satisfy certain conditions, interlayer transitions of electrons are prohibited by the interference of the Bloch phase. We specify the manifolds in the $\mathbf{k}$ space where the hybridization of the Bloch states between the layers is suppressed in accord with the stacking shift. These manifolds, named stacking-adapted interference manifolds (SAIM), are obviously applicable to general layered materials regardless of a detailed atomic configuration within the unit cell. We demonstrate the robustness and usefulness of the SAIM with first-principles calculations for layered boron nitride, transition-metal dichalcogenide, graphite, and black phosphorus. We also apply the SAIM to general three-dimensional crystals to derive special k-point paths for the respective Bravais lattices, along which the Bloch-phase interference strongly suppresses the band dispersion. Our theory provides a general view on the anisotropic electronic motion intrinsic to the periodic-lattice structure.

Figure 1

Interference of electronic wave function, demonstrated by simple model wave functions whose values at sites are represented by $\pm 1$. (a) Spatially confined electronic eigenstate on the triangle-chain lattice [1]. $t$ denotes the intersite hopping amplitude. The combination of the hopping geometry and the spatial phase structure of the wave function yields zero transition amplitude to the neighboring sites. (b) Spatial phase configuration of the Bloch state on the rectangular lattice. The corresponding crystal wave numbers are indicated in the BZ. (c) Interlayer hybridization between the Bloch states at the neighboring layers. When the second layer is shifted from the original one, the hybridization is canceled for some Bloch states because of the phase interference.

Figure 2

Hybridization between the Bloch states at one-dimensional chains. (a) The single-band tight-binding model on a one-dimensional chain, which is invariant under mirror reflections with respect to $m_1$ and $m_2$. $t$ denotes the intersite hopping amplitude and $\mathit{a}$ denotes the primitive translation vector. The phase configuration of the Bloch state with $k=\pi$ is displayed. (b) Hybridization between the $k=\pi$ Bloch states at neighboring chain. The two chains are parallelly stacked with a shift $\tau$ so that the inequivalent mirror planes of the neighboring chains coincide with each other. $t^{\prime }$ denotes the intersite hopping amplitude for the neighboring sites belonging to different chains.

Figure 3

Example of the theorem on the interlayer hybridization. (a) The hexagonal net has three inequivalent $C_3$ axes, which are connected by fractional vectors ${\mathbf{\tau }}_1$ and ${\mathbf{\tau }}_2$. (b) ${\mathcal{K}}_{\mathrm{in}}(C_3;\mathbf{\tau })$ with $\mathbf{\tau }={\mathbf{\tau }}_1$ or ${\mathbf{\tau }}_2$, indicated by red or dark points in the hexagonal BZ. (c) When an identical layer is stacked along the original one with in-plane shift $\mathbf{\tau }={\mathbf{\tau }}_1$ or ${\mathbf{\tau }}_2$, the global $C_3$ axis corresponds to inequivalent ones for the respective layers. The interlayer hybridization is consequently suppressed for $\mathbf{k}\in {\mathcal{K}}_{\mathrm{in}}(C_3;\mathbf{\tau })$.

Figure 4

${\mathcal{K}}_{\mathrm{in}}(S;\mathbf{\tau })$ for the rectangular net. (a) The primitive translation vectors $a_1$ and $a_2$, the corresponding reciprocal vectors $b_1$ and $b_2$, and the BZ. (b)–(d) Possible lists of ${\mathcal{K}}_{\mathrm{in}}(S;\mathbf{\tau })$ when (b) $S=C_2$, (c) $m_x$, and (d) $m_y$, respectively. The red or dark points and bold lines in the BZ indicate ${\mathcal{K}}_{\mathrm{in}}(S;\mathbf{\tau })$. For (c) and (d), the same ${\mathcal{K}}_{\mathrm{in}}(S;\mathbf{\tau })$ applies for an arbitrary setting of $\mathbf{\tau }$ with respect to the component parallel to the reflection plane. The case of $S=g_x$ ($g_y$) is trivially the same as that of $S=m_x\left(m_y\right)$.

Figure 5

Electronic structure of bilayer BN. (a) The monolayer crystal structure, the corresponding BZ, and calculated band structure. Primitive lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ and possible shift vectors ${\mathbf{\tau }}_1$ and ${\mathbf{\tau }}_2$ are also shown. In the BZ, $k$-point path and ${\mathcal{K}}_{\mathrm{in}}(C_3;{\mathbf{\tau }}_1)$ and ${\mathcal{K}}_{\mathrm{in}}(C_3;{\mathbf{\tau }}_2)$ are indicated. (b) Calculated Bloch wave functions for the conduction (C) and valence (V) states at the $K$ point. The spatial dependence of the plane-wave part $\exp (i\mathbf{k}·\mathbf{r})$ has been ignored. (c) The bilayer structures and their calculated band structures. (d) Calculated Bloch wave functions for the states indicated in panel (c). Those for the degenerate states are not shown.

Figure 6

(a) The hexagonal BZ. (b), (e), and (h) crystal structures of the bulk BN with the rhombohedral, hexagonal, and AA stacking. (c), (f), and (i) The selection rules for interlayer hybridizations, where the allowed hybridizations are indicated by arrows. (d), (g), and (j) The first-principles band structures along the $K\text{−}H$ line. The shade on the band indicates the twofold degeneracy.

Figure 7

First-principles band structures in the bulk BN for the different stackings.

Figure 8

(a) Unit cell of ${\mathrm{MoS}}_2$ monolayer, (b) side view, (c) top view, where ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ denote the primitive lattice vectors. Inequivalent $C_3$ axes are indicated by vectors ${\mathbf{\tau }}_1$ and ${\mathbf{\tau }}_2$. (d) [(e)] the 3R (2H) stacking and corresponding bulk band structure along the $K\text{−}H$ line. The shade on the band indicates the twofold degeneracy. The band structures are taken from Ref. [32].

Figure 9

(a) Site-selective nets for inequivalent C atoms in graphite monolayer. (b) Bilayer with shift $\mathbf{\tau }$ (left) and corresponding relations of the nets (right). The bottom layer is depicted in lighter colors. [(c) and (d)] The interlayer hybridization rules for the $K$-($K^{\prime }$-)point states derived from the site-selective nets. The allowed (forbidden) hybridizations are depicted in solid (dashed) arrows.

Figure 10

[(a), (d), and (g)] Crystal structures of the bulk graphite with the rhombohedral, hexagonal, and AA stacking. [(b), (e), and (h)] The selection rules for interlayer hybridizations, where we show with gray arrows the allowed minimal hybridizations necessary for connecting all the states through the layers. The states forming channels dominated by the nearest-neighbor hybridization are depicted with thick horizontal lines. [(c), (f), and (i)] The first-principles band structures along the $K\text{−}H$ line, where thick lines indicate twofold degeneracy.

Figure 11

(a) Structure of the black phosphorene. In the (i) top and (ii) side views, symmetry elements are shown: mirror $m$, glide-mirror $g$, and twofold screw axis $S_2$. Vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ denote the two-dimensional primitive translation vectors. (b) The corresponding BZ, where ${\mathcal{K}}_{\mathrm{in}}(m;{\mathbf{a}}_2/2)$ and ${\mathcal{K}}_{\mathrm{in}}(g;{\mathbf{a}}_1/2)$ are indicated by wavy and zigzag lines, respectively. (c) Band structure of the black phosphorene. [(d) and (e)] Top (upper) and side views (lower) of bulk black phosphorus with AB (interlayer shift $\mathbf{\tau }={\mathbf{a}}_2/2$) and AA ($\mathbf{\tau }=0$) stacking, respectively. (f) BZ for the orthorhombic structure, where ${\mathcal{K}}_{\mathrm{in}}(m;{\mathbf{a}}_2/2)$ in the monolayer form is indicated by the wavy lines. (g) Band structures of the bulk black phosphorus for the AB and AA stackings. Note that our calculated band structure shows gap closing at the Z point, at odds with the experiments [50, 51]. This is due to a drawback of the generalized gradient approximation to the exchange-correlation functional [52] (see Appendix pp4 for detailed conditions of the calculation) and supposed to be remedied by including the exchange effect more accurately [53].

Figure 12

(a) The tight-binding model on the FCC lattice. Up to third-nearest-neighbor intersite hoppings are represented with arrows. (b) FCC BZ. [(c), (f), and (i)] Decomposition of the FCC lattice into nets with a shift perpendicular to various axes. [(d), (g), and (j)] The two-dimensional BZ corresponding to the auxiliary lattice vectors depicted in (c), (f), and (i), respectively. ${\mathcal{K}}_{\mathrm{in}}(S;\mathbf{\tau })$ and the paths perpendicular to the net through ${\mathcal{K}}_{\mathrm{in}}(S;\mathbf{\tau })$ are depicted by arrows. [(e), (h), and (k)] Top views of the paths.

Figure 13

(a) The NaCl structure for $AB$ binary compounds, which belongs to the FCC lattice. (b) (c) Auxiliary hexagonal and tetragonal BZs reflecting the layer decompositions in Figs. 12 and 12. Note that the $Q_1$ and $Q_2$ points are introduced for convenience, not being special points in the tetragonal BZ. (d) [(e)] Band structures in LiH (NaCl) calculated from first principles. The $k$-point paths are indicated in (b) and (c), respectively.

Figure 14

Compatibility relations for SAIM.

Figure 15

SAIM for the oblique and diamond nets. (a) The primitive translation vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$, the corresponding reciprocal vectors ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$, and the BZ. (b) Possible lists of SAIM ${\mathcal{K}}_{\mathrm{in}}(S;\mathbf{\tau })$ when the crystal structure is $C_2$-symmetric.

Figure 16

SAIM for the rectangular net. (a) The primitive translation vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$, the corresponding reciprocal vectors ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$, and the BZ. (b) [(c)] ${\mathcal{K}}_{\mathrm{in}}(S;\mathbf{\tau })$ when the structure is $m_x$- or $g_x$-symmetric ($m_y$- or $g_y$-symmetric).

Figure 17

SAIM for the hexagonal net. (a) The primitive translation vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$, the corresponding reciprocal vectors ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$, and the BZ. (b) ${\mathcal{K}}_{\mathrm{in}}(S;\mathbf{\tau })$ when the structure is $C_3$-symmetric.

Figure 18

SAIM for the square net. (a) The primitive translation vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$, the corresponding reciprocal vectors ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$, and the BZ. (b) ${\mathcal{K}}_{\mathrm{in}}(S;\mathbf{\tau })$ when the structure is $C_4$-symmetric.

Figure 19

(a) Face-centered cubic lattice and (b) its corresponding BZ. (c) [(f)] Decomposition of the lattice into layers perpendicular to the (1 1 1) [(1 0 0)] direction. The primitive lattice vectors for each layer are represented by arrows. The interlayer shift and possible in-plane symmetry elements having two or more symmetry references are shown in the right. (d) and (e) [(g) and (h)] BIFP yielded by the decomposition of (c) [(f)]. Note that we show only the BIFP when the crystal structure has maximal symmetry compatible to the lattice and we do not show the BIFP whose $d_{\mathrm{th}}$ is not larger than the distance of the nearest-neighbor atoms. These points also apply to the later figures.

Figure 20

(a) Body-centered cubic lattice and (b) its corresponding BZ. (c) [(f)] Decomposition of the lattice into layers perpendicular to the (1 1 0) [(1 0 0)] direction. (d) and (e) [(g) and (h)] BIFP yielded by the decomposition of (c) [(f)].

Figure 21

(a) Simple cubic lattice and (b) its corresponding BZ. (c) [(f)] Decomposition of the lattice into layers perpendicular to the (1 1 1) [(1 1 0)] direction. (d) and (e) [(g) and (h)] BIFP yielded by the decomposition of (c) [(f)].

Figure 22

(a) Primitive hexagonal lattice and (b) its corresponding BZ. (c) Decomposition of the lattice into layers perpendicular to the (1 0 0) direction. (d) and (e) BIFP yielded by the decomposition of (c).

Figure 23

(a) Rhombohedral lattice and (b) its corresponding BZ. (c) [(f)] Decomposition of the lattice into layers perpendicular to the (0 0 1) direction. (d) and (e) BIFP yielded by the decomposition of (c).

Figure 24

(a) Body-centered tetragonal lattice and (b) its corresponding BZ. [(c), (f), and (i)] Decomposition of the lattice into layers perpendicular to the (1 0 0), (1 1 0), and (0 0 1) directions, respectively. [(d) and (e)], [(g) and (h)], and [(j) and (k)] BIFP yielded by the decomposition of (c), (f), and (i), respectively.

Figure 25

(a) Primitive tetragonal lattice and (b) its corresponding BZ. (c) Decomposition of the lattice into layers perpendicular to the (1 1 0) direction. (d) [(e)] BIFP yielded by the decomposition of (c) [(f)].

Figure 26

(a) Base-centered orthorhombic lattice and (b) its corresponding BZ. (c) [(f)] Decomposition of the lattice into layers perpendicular to the (1 0 0) [(0 1 0)] direction. The primitive lattice vectors for each layer are represented by arrows. The interlayer shift and possible in-plane symmetry elements having two or more symmetry references are shown in the right. (d) and (e) [(g) and (h)] BIFP yielded by the decomposition of (c) [(f)].

Figure 27

(a) Face-centered orthorhombic lattice and (b) its corresponding BZ. [(c), (f), and (i)] Decomposition of the lattice into layers perpendicular to the (1 0 0), (0 1 0), and (0 0 1) directions, respectively. [(d) and (e)], ](g) and (h)], and ](j) and (k)] BIFP yielded by the decomposition of (c), (f), and (i), respectively.

Figure 28

(a) Body-centered orthorhombic lattice and (b) its corresponding BZ. [(c), (f), and (i)] Decomposition of the lattice into layers perpendicular to the (1 0 0), (0 1 0), and (0 0 1) direction. [(d) and (e)], [(g) and (h)], and [(j) and (k)] BIFP yielded by the decomposition of (c), (f), and (i), respectively.

Figure 29

(a) Base-centered monoclinic lattice and (b) its corresponding BZ. (c) [(f)] Decomposition of the lattice into layers perpendicular to the (1 0 0) [(0 0 1)] direction. (d) and (e) [(g) and (h)] BIFP yielded by the decomposition of (c) [(f)].

Figure 30

Schematic demonstration of the infinite number of decompositions yielded by the mirror symmetry. Red and blue graphs represent two-dimensional layers having identical structure. When the lattice can be regarded from a certain direction (1) as the layers with uniform shift so that the different types of mirror planes ($m_1$ and $m_2$) of the neighboring layers are shared, there is always another direction [(2), for example] where the similar decomposition applies.