A Unified Understanding of the Thickness ... - ACS Publications

Letter pubs.acs.org/JPCL

A Unified Understanding of the Thickness-Dependent Bandgap Transition in Hexagonal Two-Dimensional Semiconductors Joongoo Kang,*,† Lijun Zhang,‡ and Su-Huai Wei§ †

Department of Emerging Materials Science, DGIST, Daegu 711-873, Korea College of Materials Science and Engineering and Key Laboratory of Automobile Materials of MOE, Jilin University, Changchun 130012, China § Beijing Computational Science Research Center, Beijing 100094, China ‡

S Supporting Information *

ABSTRACT: Many important layered semiconductors, such as hexagonal boron nitride (hBN) and transition-metal dichalcogenides (TMDs), are derived from a hexagonal lattice. A single layer of such hexagonal semiconductors generally has a direct bandgap at the highsymmetry point K, whereas it becomes an indirect, optically inactive semiconductor as the number of layers increases to two or more. Here, taking hBN and MoS2 as examples, we reveal the microscopic origin of the direct-to-indirect bandgap transition of hexagonal layered materials. Our symmetry analysis and first-principles calculations show that the bandgap transition arises from the lack of the interlayer orbital couplings for the band-edge states at K, which are inherently weak because of the crystal symmetries of hexagonal layered materials. Therefore, it is necessary to judiciously break the underlying crystal symmetries to design more optically active, multilayered semiconductors from hBN or TMDs.

B

bandgap multilayers have also been proposed, for example, by effectively enlarging the interlayer separation of 2D materials through intercalating atoms22 or inserting different TMD layers to form heterolayered TMDs.23 In this Letter, we provide a unified understanding of the DIBT of hexagonal 2D semiconductors using symmetry analysis and direct first-principles calculations. Taking hBN and MoS2 as examples, we first present a simple perturbation theory of multilayer crystals in the AB-stacking, in which two interlayer interactions, U1 and U2, capture the essence of the multilayer electronic structures (Figure 1). The nearestneighbor interlayer interaction U1 is found to be significantly suppressed for the band-edge states at the K point, while U1 at different crystal momentum (e.g., Γ) is considerably larger (Table 1). The large inhomogeneity of U1 in the momentum space leads to the DIBT of hexagonal 2D semiconductors. Finally, using the symmetry analysis of inter- and intralayer orbital couplings, we provide a microscopic understanding of why the effective interaction U1 is small for the band-edge states at K. If the local stacking order of the neighboring layers is AB (or AC), our DIBT theory is general regardless of the stacking sequences in the hexagonal layered phase. We performed first-principles electronic structure calculations of few-layer MoS2 and hBN using density functional theory (DFT). The calculations were done using the generalized gradient approximation (GGA-PBE24) and the

esides graphene, hexagonal boron nitride (hBN) and transition-metal dichalcogenides (TMDs) are first among the family of two-dimensional (2D) atomic crystals that were isolated from bulk phases by mechanical exfoliation.1 Since the success of their isolation, there has been intense interest in these hexagonal 2D materials because of their unusual physical properties and potential technological applications.2−12 hBN is a wide-bandgap semiconductor and has been demonstrated to be an ideal substrate for 2D materials.3 TMDs exhibit a variety of electronic properties depending on the structure, number of layers, and composition. Recently, hBN and TMDs are widely used as key building blocks of “van der Waals” heterostructures,13,14 in which various 2D atomic crystals are reassembled layer-by-layer to obtain new emergent properties. From prior work,15−22 it has been well-established that a single layer of hBN or group-VI TMD (e.g., MoS2, MoSe2, WS2, WSe2, etc.) has a direct bandgap at the high-symmetry point K, while a multilayer phase becomes an indirect semiconductor. Furthermore, from thickness-dependent evolution of the bandedge energies of TMDs, previous theoretical work22 revealed that there exist different localization prototypes of the bandedge states. Understanding and control of the direct-to-indirect bandgap transition (DIBT) of hexagonal 2D semiconductors is of great scientific and technological importance because the DIBT converts multilayer hBN or TMD into optically less active materials. The origin of the DIBT in TMDs has been controversially attributed to the quantum confinement effect at reduced number of layers,15,18,20 to the weak interlayer interaction,17 or to both.22 Some strategies of generating direct © XXXX American Chemical Society

Received: December 3, 2015 Accepted: January 22, 2016

597

DOI: 10.1021/acs.jpclett.5b02687 J. Phys. Chem. Lett. 2016, 7, 597−602

Letter

The Journal of Physical Chemistry Letters

and U2(k, n), to describe the first and second nearest-neighbor interlayer interactions, respectively (Figure 1). For a noninteracting N-layer phase, the corresponding band energies are degenerate at ε(0)(k, n). When only U1 is considered, the perturbation potential acting on the N degenerate levels is described by a N × N tridiagonal Toeplitz matrix,29 in which the diagonal and off-diagonal components are ε(0)(k, n) and U1(k, n), respectively. Then, the band energies split into εi(k, n)

( Nπ+i 1 ), with the eigenstate ψ having πiα 2 sin( N + 1 ), for the component on the αth layer ψ (α) = N+1

= ε(0)(k, n) + 2U1(k , n)cos

i

i

1 ≤ i, α ≤ N. For example, for 6-layer MoS2, U1 leads to a symmetric level splitting of the top of the valence band at the Γ point into six levels. When both U1 and U2 turn on, U2 contributes to an asymmetric level splitting. The U1 largely affects the electron distribution across the layers; for 6-layer MoS2, the charge density is higher in the middle layers (i.e., α = 3 and 4) for the highest level ε1, while the electron is more distributed in the outer surface layers for ε3. The electron distributions are in good agreement with previous results.30 The U1 and U2 in the perturbation theory were fitted to the DFT energy levels (εiDFT) of N-layer MoS2 and hBN phases (Table 1). The energy levels were aligned for different N by using the averaged electrostatic potential in the vacuum region as a common energy reference.22 To better fit the DFT results, we used different diagonal components, ε(0)(surface) and ε(0)(bulk), in the N × N perturbation matrix for the two surface layers and the (N − 2) inner layers, respectively. Because the perturbation matrix is diagonalizable, its trace should be equal to the sum of the N eigenvalues of the matrix. Therefore, the “sum rule” requires that (N − 2)ε(0)(bulk) + 2ε(0)(surface) = ∑εiDFT(k, n) for N ≥ 3. The surface effect is found to be negligible for most single-layer states, except for the valenceband edge state of MoS2 at Γ with ε(0)(surface) − ε(0)(bulk) = −0.16 eV. The substantial surface correction arises from the different strength of the intra-layer orbital coupling in the surface layers. It is interesting to note that if the surface effect is not taken into account, the monolayer MoS2 phase becomes an indirect bandgap material (Figure S1 in the Supporting Information). Figure 3 compares the DFT energy levels of multilayer MoS2 with the perturbation theory result. For the band-edge states indicated by dots in Figure 2a, the calculated U1 and U2 are listed in Table 1. The excellent agreement between them indicates that U1 and U2 capture the essence of the multilayer electronic structures. This quantitative analysis of effective interlayer interactions shows that for the band-edge states at K, the U1 and U2 are substantially smaller than for other singlelayer states in Table 1. Consequently, the thickness-dependent level splitting becomes relatively small for the band-edge states at K. For bilayer MoS2, the order of the two valence-band edge energies at K and Γ is reversed because of the small perturbation potential at K. Likewise, the large difference of U1 for the conduction band-edge states at K and T causes the shift of the conduction-band edge position from K to a point near T at around N = 5, in agreement with previous theoretical work.22 For multilayer hBN in the AB-stacking, we found that the DFT result (not shown here) is almost identical to the perturbation theory result (Figure 4a). The U1 and U2 are listed in Table 1 for the band-edge states denoted by dots in Figure 2c. For the Γ point, the third valence-band maximum state is

Figure 1. Interlayer perturbation potentials, U1 and U2, for (a) multilayer TMD and (b) hBN. The nearest-neighbor interlayer interaction, U1, depends on two different types of orbital couplings, h1 and h2, across the interlayer gap.

Table 1. Interlayer Interactions, U1 and U2, for MoS2 and hBN at the Selected k-Points in the Brillouin Zone (Figure 2)a system

k

n

U1 (eV)

U2 (eV)

MoS2

Γ Γ K K K T

vbm 4 vbm 1 vbm 1 cbm 1 cbm 2 cbm 1

0.53 0.26 0.039 0.001 0.25 0.17

0.044 −0.070 0.001 0.000 0.008 0.006

hBN

Γ K K M M

vbm 3 vbm 1 cbm 1 vbm 1 cbm 1

0.50 0 0 0.24 0.32

0.027 0.024 0.048 −0.009 0.049

The band index “vbm n” and “cbm n” denote the nth valence-band maximum (VBM) state and the nth conduction-band minimum (CBM) state of a single-layer phase, respectively. a

projector-augmented wave method,25 as implemented in the VASP code.26,27 We used an energy cutoff of 500 eV for the plane wave part of the wave function and a (30 × 30 × 1) kpoint sampling. The van der Waals interlayer interaction was included using the DFT-D2 method of Grimme.28 Figure 2 compares the calculated band structures of the hexagonal 2D semiconductors for single-layer and bilayer phases. In our DFT calculations, the bandgaps of MoS2 and hBN are underestimated. However, this does not affect our main findings on the underlying mechanism of DIBT. A single layer of MoS2 or hBN has a direct bandgap at the K point. In contrast, it becomes an indirect semiconductor as the number of layers increases to two or more. For the bilayer MoS2, the top of the valence band appears at the Γ point. For the bilayer hBN, the conduction band edge lies at the M point. Because of relatively weak interlayer interactions, the electronic structure of a multilayered hexagonal semiconductor can be obtained from the band energies of a single-layer phase by using a first-order degenerate perturbation theory. For a single-layer state with a crystal momentum k and a band index n, we introduce two effective perturbation potentials, U1(k, n) 598


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Figure 2. Calculated electronic band structures for hexagonal layered semiconductors: (a) monolayer and (b) bilayer phases of MoS2 and (c) monolayer and (d) bilayer phases of hBN. For comparison, the top of the valence band at the K point is set to zero for each case. The inset of panel a shows the first Brillouin zone of the hexagonal atomic crystals.

involves the orbitals at the same sublattice sites. The energy levels of hBN in the AA-stacking in Figure 4b show that the hypothetical multilayer hBN remains as a direct bandgap semiconductor because of the large h1-induced level splitting at K. Because h1 is symmetry-forbidden at the K point in the AB stacking, the U1 is then mediated solely by the h2-type interlayer orbital coupling (Figure 1). In our first-order degenerate perturbation theory, the orbital-coupling matrix element is (0) (0) given by ⟨ψ(0) layer1(K, n)|h2|ψlayer2(K, n)⟩, where ψlayer1(K, n) and (0) ψlayer2(K, n) are the single-layer states of the neighboring layers 1 and 2, respectively. Because h2 is the orbital coupling between different types of atoms, a nonzero matrix element requires that the single-layer state of each layer should be a hybridized state of cation and anion orbitals. For the h2-type interlayer orbital coupling in TMD, the single-layer state should be a TM d−VI p hybridized state. Similarly, for hBN, the single-layer state should be a B p−N p hybridized state to enable the h2-type orbital coupling. For the rest of this paper, we will show that h2 is intrinsically small (or exactly zero) for the band-edge states at K by investigating the intralayer orbital coupling in a single-layer hexagonal lattice. For a single-layer hBN, the pz-orbitals of the band-edge states at K are exclusively located on either B or N sites because of the 3-fold rotational symmetry, leading to the completely suppressed h2-type interlayer orbital couplings. Because both h1

chosen because it has the same pz orbital character as for the band-edge states at K and M. As we will discuss later, the U1 for the band-edge states at K is actually zero for hBN. A singlelayer hBN has a direct bandgap at K. For N ≥ 2, however, multilayer hBN has the conduction-band edge at M, because the level splitting at M is significantly larger than at K. The results of multilayer TMD and hBN point to the same conclusion that the weak U1 for the band-edge states at the K point causes the DIBT. A microscopic understanding of the DIBT thus boils down to understanding why U1 is small in terms of interatomic orbital couplings in the hexagonal 2D materials. The interlayer interaction U1 involves two types of interlayer orbital couplings (Figure 1): h1, which is the orbital coupling between the atoms of the same type (dashed line), and h2, which is the coupling between different types (solid line). For the AB-like local stacking, two same-type atomic orbitals interacting via h1 are positioned at dif ferent sublattice sites of the hexagonal lattice. Therefore, the h1-type orbital coupling becomes zero at the K point, just as the nearestneighbor orbital coupling in a single-layer graphene becomes zero at K because of the 3-fold rotational symmetry.31 Thus, the h1-type interlayer orbital coupling at the K point is symmetryforbidden for both TMD and hBN. Importantly, the symmetry argument above applies only for the K point, and the h1 orbital coupling is symmetry-allowed at other k-points (e.g., Γ). For the AA-stacking, however, large h1-type interlayer orbital coupling is still possible for the K point, because h1 now 599


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Table 2. Orbital Coupling Selection Rules for Monolayer HTMD at High-Symmetry k-Points, Γ and ±Ka

Figure 3. Evolution of the energy levels of MoS2 as a function of the number of layers (N): (a) DFT-PBE and (b) perturbation theory results. The VBM of bulk MoS2 (i.e., infinite N) is set to zero. The groups of the levels, each originating from a single-layer state at N = 1, are highlighted by shaded areas. For the band-edge states at K, the levels are too close in energy to be plotted by dots. Therefore, only the shaded areas are shown. The dashed line denotes the average values of the N energy levels for vbm 1 at Γ.

For a pair of TM d-orbital and group-VI p-orbital, “Yes” in the table denotes that the corresponding p−d orbital coupling is symmetryallowed, while “No” denotes that it is symmetry-forbidden.

a

(m, m′) pairs among 15 possible pairs from m = −2, −1, 0, 1, 2 and m′ = −1, 0, 1 (Table 2). The p−d orbital coupling at the K point is symmetry-allowed if −m + m′ = 3n + 1 for an integer n. At the Γ point, the orbital-coupling selection rule is −m + m′ = 3n. The band-edge state of TMD at K or Γ has a definite m for the constituent TM d-orbital, which is essentially determined by the strengths of the crystal field splitting at the TM site and the k-dependent d−d couplings.32 For instance, the valenceband edge state at the K point is characterized by m = −2, while the valence-band edge state at the Γ point has the d-orbital character of m = 0. The conduction-band edge state at K also has the d-orbital character with m = 0. According to the selection rule at the K point in Table 2, the orbital coupling is allowed for (m, m′) = (−2, −1) for the VBM and (m, m′) = (0, 1) for the CBM. The selection rule at the Γ point dictates that the p−d coupling for the VBM at the Γ point is allowed for (m, m′) = (0, 0). Although they are symmetry-allowed, the p−d orbital couplings in TMD turn out to be weak for the band-edge states at K, leading to small h2. Figure 5a shows that the valence-band edge state at K, which is supposed to be the p−d antibonding state with (m, m′) = (−2, −1), consists nearly exclusively of φd−2. Likewise, the conduction-band edge state at K is mainly composed of φd0 (Figure 5b), indicating that the p− d orbital coupling is largely suppressed for the band-edge states. In sharp contrast, the second CBM state at K has both TM dand group-VI p-orbital characters (Figure 5c), and the associated h2-type orbital coupling is thus responsible for the large U1 in Table 1. The suppressed p−d couplings in TMD for the band-edge states at K can be ascribed to the mismatch of the bonding characters of the TM−VI bonds. For the VBM state at K, the

Figure 4. Evolution of the energy levels of hBN as a function of the number of layers: (a) AB-stacking and (b) hypothetical AA-stacking. The energy levels were obtained from the perturbation theory of multilayer phases. The VBM of bulk hBN is set to zero.

and h2 are zero for the band-edge states at K, the corresponding U1 should also be zero for hBN in the AB-stacking (Table 1). To explain why the h2-type orbital coupling in TMD is small for the band-edge states at K, we should first understand the selective p−d orbital coupling between the TM d-orbital (φdm) and the group-VI p-orbital (φpm′) in a monolayer TMD. The subscript m or m′ denotes the z-component of the orbital angular momentum. At high-symmetry k such as K and Γ, the orbital coupling between φdm and φpm′ is allowed only for specific 600


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bond center and the hexagonal center. Consequently, the total p−d intralayer couplings can be large, leading to the large h2type interlayer orbital coupling and associated large U1 (Table 1). For the CBM state at K with (m, m′) = (0, 1), the orbital phases associated with the d−d (or p−p) bonding at the hexagonal center add up to zero (Figure 5b). Thus, there exists only a single type of the TM−VI bond at the bond center. However, because the signs of the orbital lobes of the TM dz2orbital (i.e., m = 0) are opposite for the in-plane one and the out-of-plane one, the p−d orbital coupling still involves the mixed bonding characters at the bond center, leading to the suppressed orbital coupling and small h2. In summary, we provide a unified understanding of the DIBT of hexagonal 2D semiconductors based on our symmetry analysis and direct first-principles calculations. Starting from a simple but quantitative explanation of the DIBT within the first-order degenerate perturbation theory of multilayer phases, we show how the phenomenological DIBT theory arises from the selective interorbital couplings in hexagonal 2D semiconductors. When combined with the AB-like local stacking, the 3-fold rotational symmetry is ultimately responsible for the DIBT of hexagonal 2D semiconductors, suggesting that it might be possible to design “hexagonal” but more optically active multilayer semiconductors by judiciously breaking the underlying lattice symmetries (e.g., via interlayer twist or making heterolayered stacks or alloys).

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.5b02687. Figure S1 and the derivation of the selective p−d orbital couplings in H-TMD (PDF)

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Figure 5. Charge density plots of the single-layer states of MoS2 at K for (a) vbm 1, (b) cbm 1, and (c) cbm 2. The contour value is 0.1|e|/ Å3. The right panels of parts a and c show the phases of TM d- and group-VI p-orbitals for different angles around each atom. In the right panel of part b, the TM dz2-orbitals are presented in the side view.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS J.K. thanks R.E. Larsen for helpful discussion on tridiagonal Toeplitz matrix. This work was supported by the DGIST MIREBraiN Program of the Ministry of Science, ICT and Future Planning of Korea (2015010016). This research used capabilities of the DGIST Supercomputing and Big Data Center. Work at Jilin University was supported by National Natural Science Foundation of China under Grant 11404131.

right panel of Figure 5a shows the atomic orbitals’ phases for different angles around each atom, which are determined by adding a position-dependent Bloch phase at K and an angledependent phase associated with the z-component of the orbital angular momentum, i.e., m = −2 for TM d-orbitals and m′ = −1 for group-VI p-orbitals. In the diagram, we assume the phase difference of π between the neighboring TM and groupIV ions, so that the bonding at the bond center has the antibonding character. In this case, at the hexagonal center, the phases of the three TM d-orbitals and the three group-VI porbitals are in phase with −2π/3, thus forming the p−d orbital couplings with the bonding character. Therefore, the p−d orbital couplings at the bond center and the hexagonal center have the opposite bond characters, and this mismatch makes the total p−d orbital coupling largely suppressed. Unlike the case of the VBM state at K, we note that the aforementioned second CBM state at the K point with (m, m′) = (1, −1) has the same antibonding characters both at the bond center and the hexagonal center (Figure 5c). Similarly, the VBM state at the Γ point with (m, m′) = (0, 0) is also characterized by the antibonding TM−VI bonds at both the

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