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Monolayer to Bulk Properties of Hexagonal Boron Nitride Darshana Wickramaratne, Leigh Weston, and Chris G. Van de Walle J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b09087 • Publication Date (Web): 11 Oct 2018 Downloaded from http://pubs.acs.org on October 15, 2018

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The Journal of Physical Chemistry

Monolayer to Bulk Properties of Hexagonal Boron Nitride Darshana Wickramaratne,∗ Leigh Weston, and Chris G. Van de Walle Materials Department, University of California, Santa Barbara, California 93106-5050, USA E-mail: [email protected]

Abstract

ultraviolet regions of the spectrum, 7–9 with potential applications in quantum communication and quantum information science. These various studies and applications sometimes rely on bulk (or thick layers) of h-BN, other times on samples as thin as a single monolayer or a few layers. Hexagonal boron nitride is an sp2 -bonded layered material in which individual layers are held together by van der Waals forces. The bulk unit cell is comprised of two layers, with one B and one N atom in each monolayer. The ground-state stacking configuration of h-BN is AA0 [Fig. 1(a)-(b)], in which the B atoms in one h-BN layer are positioned above the N atoms in the second h-BN layer, and vice versa for the N atoms. Transmission electron microscopy (TEM) on h-BN layers grown by chemical vapor deposition 10 or obtained by exfoliation 11 has revealed the presence of few-layer films with an AB stacking in addition to the ground-state AA0 stacking. In the AB stacking configuration; the B and N atoms in one h-BN layer are translated with respect to the second h-BN layer in the unit cell (Fig. 1(c)-(d)) by a fraction of the in-plane lattice vector. One expects interlayer interactions to affect the band structure of the material, as has been observed in other layered semiconductors. 12 There are currently no experimental reports on the band gap of a single monolayer or the evolution of the electronic structure as a function of the number of layers starting from a single monolayer. Such information is essential for characterization and design

Hexagonal boron nitride (h-BN) is widely used in two-dimensional electronics and serves as a host for single-photon emitters. We study the electronic structure of h-BN as a function of the number of layers and take into account different stacking configurations. Using first-principles calculations based on a hybrid functional, we find that the band gap of a single monolayer is direct, while for thicknesses above a monolayer the band gap is indirect. By examining the positions of the band edges with respect to the vacuum level we find this directto-indirect transition to be driven by a shift in the conduction-band minimum at the M point; This shift changes the band gap by 0.5 eV going from a single monolayer to bulk. We analyze these results in terms of the orbital composition of the band edges at different high-symmetry points in the Brillouin zone.

Introduction Hexagonal boron nitride (h-BN) is widely employed in devices comprised of two-dimensional (2D) materials. 1–3 In the bulk structure h-BN has a band gap of 6.08 eV 4 which has led to its use as a dielectric or tunnel barrier in electronic devices. 5 It is also being explored for optoelectronic devices that operate in the deep ultra-violet region of the spectrum. 6 Research has also been motivated by its use as a host for single photon emitters in the infrared and

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(a)

AA stacking

(b)

fects in h-BN. However, GW calculations do not selfconsistently predict structure or energetics. For instance, Paleari et al. 19 studied the change in the quasiparticle band gap of AA0 hBN from one to five monolayers and in the bulk, and found the band gap becomes indirect above a single monolayer. However, their work relied on experimental lattice parameters of bulk h-BN, and no geometry optimization was performed. Our tests indicate this can introduce errors in the band gap. In particular, the direct band gap at K of the bulk h-BN structure where no geometry optimization is performed is 0.35 eV larger than the band gap of the optimized structure. Furthermore, the evolution of the absolute position of the band edges and the role of different stacking configurations was not addressed. Detailed results for the electronic structure as a function of the number of layers are thus lacking. In the present work we perform firstprinciples calculations of the band structure of h-BN using a hybrid functional, which provides reliable results for both atomic and electronic structure. 22 We focus on fundamental band gaps; i.e., excitonic effects are not included. Excitons affect absorption and emission spectra of h-BN, 4,23 but knowledge of the fundamental gaps is a prerequisite for studying these effects. We acknowledge that quasiparticle calculations in the GW approach might reveal additional effects: weaker screening in fewer-layer structures may increase the gap. However, as noted above, the lack of geometry optimization in the GW calculations introduces other uncertainties. We examine the position of band edges with respect to the vacuum level, from one to eight layers and up to the bulk limit. Along with inspections of orbital compositions, we thus provide a comprehensive description of the evolution of the band structure.

B N

y

y

z

x

AB stacking

(c)

(d)

y

y

z

x b3

A

L

Γ 1

M

H

T1 K

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b2

Figure 1: Schematic view of stacking configurations in h-BN. (a) Top view and (b) side view in the ground-state AA0 stacking configuration. (c) Top view and (d) side view in the AB stacking configuration. has been translated with respect to the second layer. (e) Brillouin zone of bulk h-BN

of applications that utilize h-BN. The electronic structure of bulk, 13–15 monolayer 16 and bilayer h-BN 17 has been previously examined in first-principles calculations. Density functional calculations within the local density approximation (LDA) 18 or the generalized gradient approximation (GGA) 17 were performed for monolayers, and also used to examine the impact of interlayer interactions on the band gap. 17,18 However, the LDA and GGA functionals severely underestimate the band gap, raising uncertainty about these predictions. Quasiparticle calculations within the GW approximation 13,16,18–21 in principle provide reliable results for the band structures and the fundamental band gaps. GW calculations have also been combined with solutions of the BetheSalpeter equation 13,19,20 to address excitonic ef-

Methods Our calculations are based on density functional theory 24 as implemented in the Vienna Ab initio Simulation Package (VASP), 25 using the hybrid functional of Heyd, Scuseria,


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and Ernzerhof (HSE). 26,27 Van der Waals interactions, which are essential to obtain the correct description of the interlayer distance, are included using the Grimme-D3 approach. 28 We use projector augmented wave potentials 29 and a plane-wave basis set with a 500 eV cutoff. Setting the mixing parameter α in the hybrid functional equal to 0.31 leads to a band gap and structural parameters that are in close agreement with experiment for bulk h-BN. We keep this mixing parameter fixed in our studies as a function of number of layers. It has been shown that the mixing parameter should be inversely proportional to the electronic dielectric constant of the material. 30 A recent study has shown that the dielectric constant of h-BN remains almost unchanged as a function of number of layers, 31 supporting our use of a single mixing parameter. For the monolayer and few-layer calculations a slab geometry with 18 ˚ A of vacuum is used. The Brillouin zone is sampled with a 9 × 9 × 5 k-point grid in the case of the bulk structure, and a 9 × 9 × 1 k-point grid for the monolayer and few-layer calculations. In the slab calculations, the valence-band and conduction-band edges are determined with respect to the vacuum level by examining the macroscopic average of the planar-averaged electrostatic potential.

group of P6/mmm. We find that the in-plane lattice parameters in the AB stacking are the same as in the AA0 structure, while the out-ofplane lattice parameter decreases very slightly to a value of c = 6.54 ˚ A (interlayer separation ˚ 3.27 A). The calculated band structures for the ground-state AA0 stacking configuration for one to five layers of h-BN and for the bulk are shown in Fig. 2. The band gap of the bulk structure is indirect, with the valence-band maximum (VBM) at the T1 point (along K – Γ) and the conduction-band minimum (CBM) at M. Our calculated indirect band gap of 5.95 eV is identical to the value obtained in GW calculations, 13 and in good agreement with the experimentally measured low-temperature indirect band gap of 6.08 eV. 4 At a thickness of five layers, the band gap remains indirect. The VBM occurs at K and the CBM at M; henceforth we will use the notation Kv and Mc to indicate the high-symmetry point as well as the conduction- or valence-band character of the state. As the number of layers decreases the fundamental band gap increases. Monolayer hBN exhibits the largest band gap, 6.47 eV. The monolayer band gap is direct and located at the K point. In Fig. 3(a) we plot the direct band gap at K and the indirect Kv – Mc gap for film thicknesses ranging from one to eight layers, as well as for bulk. The band-gap values as a function of the number of layers, n, are also listed in Table 1. Both direct and indirect band gaps decrease monotonically with increasing n, with the indirect band gap decreasing more rapidly than the direct gap. For both stacking configurations, the crossover from direct to indirect gap occurs between n=1 and n=2, but the direct gap at K has a stronger dependence on n in AB than in AA0 stacking [Fig. 3(a) and Table 1]. The decrease in the Kv -Mc indirect gap is more similar between AA0 and AB stacking. To understand the trends, we inspect the band edges on an absolute scale, i.e., with respect to the vacuum level, as shown in Fig. 3(b). On an absolute scale the valence band at K remains almost unchanged (to within 0.1 eV) as n increases from one to eight. Kc exhibits larger

Results and Discussion We examine h-BN in the stacking configurations depicted in Fig. 1. We find the AB stacking configuration to be 1.2 meV/(formula unit) higher in energy than the AA0 stacking, consistent with previous hybrid functional calculations. 32 For the AA0 stacking (space group P63 /mmc) we obtain an in-plane lattice parameter of a = 2.49 ˚ A and an out-of-plane lattice parameter c = 6.55 ˚ A (corresponding to an interlayer separation of 3.28 ˚ A). The calculated in-plane lattice parameter is 0.4% lower than the experimental value (a = 2.50 ˚ A) 33 and the calculated out-of-plane lattice constant is 1.5% lower than the experimental value (c = 6.65 ˚ A). 33 The AB stacking configuration retains hexagonal symmetry and has a space


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Table 1: The Kv -Kc direct gap, Kv -Mc indirect gap, valence-band energies Kv , and conduction-band energies Kc and Mc with respect to the vacuum level (all in eV) as a function of the number of layers n for the AA0 and AB stacking configurations in h-BN. In bulk h-BN the VBM occurs at T1 , resulting in a fundamental indirect gap of 5.95 eV for AA0 and 5.98 eV for AB.

AA0 stacking

n Kv -Kc

Kv -Mc

1

6.47

6.50

2

6.44

3

Kc

AB stacking Mc

Kv

Kv -Kc

Kv -Mc

-0.32 -0.29 -6.79

6.47

6.50

-0.32 -0.29 -6.79

6.19

-0.38 -0.63 -6.82

6.33

6.19

-0.49 -0.63 -6.82

6.41

6.05

-0.43 -0.79 -6.84

6.27

6.10

-0.57 -0.74 -6.84

4

6.39

5.96

-0.46 -0.89 -6.85

6.23

6.02

-0.62 -0.83 -6.85

5

6.39

5.95

-0.47 -0.91 -6.86

6.22

6.00

-0.64 -0.86 -6.86

6

6.39

5.95

-0.48 -0.92 -6.87

6.22

6.00

-0.65 -0.87 -6.87

7

6.38

5.94

-0.49 -0.93 -6.87

6.22

5.99

-0.65 -0.88 -6.87

8

6.38

5.94

-0.49 -0.94 -6.87

6.22

5.98

-0.65 -0.89 -6.87

∞

6.42

6.01

-0.45 -0.86 -6.87

6.23

6.02

-0.64 -0.85 -6.87

shifts: by 0.17 eV in AA0 and by 0.33 eV in AB stacking. But the largest shift is observed at Mc : a change of 0.65 eV in AA0 and by 0.60 eV in AB stacking. It is this larger shift at Mc relative to Kc that leads to the direct-toindirect crossover in the band gap between n=1 and n=2. Our analysis of the band-edge states on an absolute scale also allows us to determine the ionization potential and electron affinity of hBN. For this purpose we assume the valence band at K remains unchanged on an absolute scale going form the n=8 structure to the bulk limit (n=∞). This leads to the valence band at K being 6.87 eV below the vacuum level in bulk h-BN in the AA0 and AB stacking configurations. Using the Kv -Mc band gap of bulk h-BN, we then find the conduction band at M to be 0.86 eV below the vacuum level in AA0 h-BN and 0.85 eV below the vacuum level in AB h-BN. 0.45 eV below the vacuum level in AA0 h-BN and 0.64 eV below the vacuum level in AB h-BN. One might think that the increase in band gaps with decreasing film thickness can be attributed to quantum confinement. However, we can rule out quantum confinement as the origin of the band-gap change, based on our

Kc

Mc

Kv

calculations of effective mass. The calculated out-of-plane electron effective mass along M-L [Fig. 1(e) and Fig. 2(f)] in the AA0 stacking is 1.39 m0 , where m0 is the free electron mass. The corresponding shift due to quantum confinement is much smaller than the increase in the energy of Mc by 0.65 eV in going from n=8 to n=1. The sensitivity of the band edges at Mc , Kc and Kv to changes in n can instead be explained by the orbital composition of these states. As evident from Fig. 2, these states are comprised of pz states that extend out of the plane of a h-BN layer, and thus they are sensitive to the presence of additional h-BN layers. The magnitude of the interlayer coupling for the states at the K and M points shows significant differences, as illustrated in Fig. 4(a). Focusing first on conduction-band states, in a single monolayer the B atom contributes one pz state to the CBM at K. For n=2, the CBM shifts to M. The lowest two states at Mc are pz states that are nondegenerate and derived from equal contributions of the two B atoms in the unit cell with minor contributions from the N pz states. The splitting between these two pz states at Mc is large, as illustrated in Fig. 4(a). The lowest two states at Kc are degenerate and comprised of individual B pz states with no con-


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The Journal of Physical Chemistry (a) 1 layer

6.6

(b) 2 layers

8

(a)

Band gap (eV)

6 Energy (eV)

4 2 0 -2

M

K

(c) 3 layers

M

K

K -K (AA ) v c

6.4

K -K (AB) v

c

6.2 Kv-M c (AB)

6

pz

(d) 4 layers

Kv-M c (AA ) 5.8 -0.2 (b)

M K (e) 5 layers

px,py

M K (f) Bulk

Energy (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

M

K

M

-0.4

Kc (AA )

-0.6

Kc (AB)

-0.8

M c (AB) M c (AA )

-1

Kv (AB and AA )

-6.8

K T1

Figure 2: Evolution of the band structure of h-BN in the AA0 stacking configuration as a function of layer thickness, for (a) one, (b) two, (c) three, (d) four, and (e) five layers, as well as for (f) bulk h-BN. The color of each band indicates the pz and the pxy character of the states, according to the color bar on the right.

-7.0

1

2

3

4

5

6

n

7

8

Figure 3: (a) Evolution of the Kv -Kc direct gap and the Kv -Mc indirect band gaps as a function of the number of layers n. (b) Position of the band edges at Kv , Kc , and Mc with respect to the vacuum level as a function of n for the AA0 and the AB stacking configuration.

tribution from the N atoms. To elucidate the different magnitude of the splitting at the K and M points of the conduction band we examine the spatial distribution of the pz states for increasing number of h-BN layers. For n > 2 we find the same trends in the contribution of states to Mc and Kc and the relative magnitude of splittings between the states. At Mc each of the B atoms in the unit cell contributes a pz state with a minor contribution of pz states by the N atoms. At Kc each state is derived from a pair of B pz states that are localized on h-BN layers that are not adjacent to each other. The wavefunctions at the band extrema [Mc , Kc , and Kv ] for the n=4 structure are illustrated in Figs. 4(b)-(d). The states at Kc and Kv are doubly degenerate; the wave function for the other state in the dou-

bly degenerate pair is simply related by inversion symmetry. The difference in the distribution of pz states at Mc and Kc explains the different magnitude of the splittings observed in Fig. 4(a). The pz states at Mc occur on B atoms in every layer [Fig. 4(b)], and are therefore more sensitive to the presence of additional layers. In contrast, at Kc the pz states interact more weakly due to the h-BN layer that is in-between the pair of B pz states [Fig. 4(c)]. This difference in splitting between Mc and Kc was also observed in tight-binding studies for bulk h-BN 34 and attributed to interlayer coupling and the symmetry of B pz wavefunctions.


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The Journal of Physical Chemistry 6.6 Band gaps (eV)

(a)

0.5

0 K c Energy (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-0.5 Mc

K -K v

6.4

c

6.2 K -M v

c

6

-1 3

4

5

Kv

6 d (Å)

7

8

9

-6.8

Figure 5: Evolution of the direct Kv -Kc gap and the indirect Kv -Mc gap in AA0 -stacked bulk h-BN as a function of the interlayer separation distance. The equilibrium separation distance is 3.28 ˚ A.

-7 1

2

(b) - Mc

3

4 n

5

(c) - Kc

6

7

8

∞

(d) - Kv

of the states at Mc than at Kc . We examine the impact of this effect by varying the interlayer separation in bulk h-BN. In Fig. 5 we show the change in the direct band gap at K and the indirect Kv -Mc gap as a function of increasing interlayer separation. We find that when the interlayer separation is increased to 5.90 ˚ A (i.e., by 80% of the equilibrium interlayer distance) the character of the band gap changes from indirect to direct, with a band-gap value of 6.46 eV. Figure 5 clearly shows that the indirect Kv -Mc is much more sensitive to interactions between the layers than the direct Kv -Kc gap, consistent with our observation that the indirect gap is also much more sensitive to to the presence of additional layers.

Figure 4: (a) Energies of valence-band states at Kv (◦) and conduction-band states at Mc (H) and Kc () as a function of the number of layers n for AA0 stacking. The energies are plotted with respect to the vacuum level. (b)-(d) Wavefunctions of the band extrema at (b) Mc , (c) Kc and (d) Kv for the n=4 structure. The color scheme for the atoms is the same as in Fig. 1.

Turning to the valence band, the states at Kv are derived from N pz states, with each state being localized on a single layer. For n=2, the VB at Kv is degenerate with each state localized on a particular layer. For n > 2 the splitting of the pz derived states at Kv is smaller than the splitting of the states at Mc and Kc . This can be attributed to the wavefunction of each state being localized on a single layer [Fig. 4(d)]. In the bulk, the conduction-(resp. valence-) band states are derived from equal contributions of pz states from each of the B (resp. N) atoms in the unit cell. The presence of additional layers has a larger impact on the interlayer coupling and splitting

Conclusions In conclusion, we have examined the evolution of the electronic structure of h-BN as a function of the number of layers. The band structures and band-edge states were analyzed in terms of the orbital composition at the various highsymmetry points in the Brillouin zone. We find the conduction band at M to be the most sensitive to changes in the number of layers, which leads to the direct-to-indirect crossover in the band gap at a thickness above a single monolayer. The ionization potential is 6.87 eV, determined by extrapolating from calculations for the band energies at Kv with respect to vacuum


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for layers with different thicknesses. Combining this information with values for the indirect Kv Mc band gap, we determined an electron affinity of 0.86 eV for AA0 stacking and 0.85 eV for AB stacking. An analysis of orbital composition shows that, among the band edges, the conduction band at Mc is most sensitive to the stacking, and also to the interlayer separation.

nal Boron Nitride Is an Indirect Bandgap Semiconductor. Nat. Photon. 2016, (5) Britnell, L.; Gorbachev, R. V.; Jalil, R.; Belle, B. D.; Schedin, F.; Katsnelson, M. I.; Eaves, L.; Morozov, S. V.; Mayorov, A. S.; et al., Electron Tunneling Through Ultrathin Boron Nitride Crystalline Barriers. Nano Lett. 2012, 12, 1707–1710.

Acknowledgements

(6) Jiang, H.; Lin, J. Hexagonal Boron Nitride for Deep Ultraviolet Photonic Devices. Semicond. Sci. Technol. 2014, 29, 084003.

The work was supported by by the National Science Foundation (NSF) under Grant No. DMR-1434854, and by the NSF MRSEC Program (DMR-1720256). Computational resources were provided by the Center for Scientific Computing at the CNSI and MRL (an NSF MRSEC, DMR-1720256) (NSF CNS-0960316), and by the Extreme Science and Engineering Discovery Environment (XSEDE) supported by the NSF (ACI-1548562).

(7) Chejanovsky, N.; Rezai, M.; Paolucci, F.; Kim, Y.; Rendler, T.; Rouabeh, W.; Favaro de Oliveira, F.; Herlinger, P.; Denisenko, A.; Yang, S. et al. Structural Attributes and Photodynamics of Visible Spectrum Quantum Emitters in Hexagonal Boron Nitride. Nano Lett. 2016, 16, 7037–7045.

References

(8) Jungwirth, N. R.; Fuchs, G. D. Optical Absorption and Emission Mechanisms of Single Defects in Hexagonal Boron Nitride. Phys. Rev. Lett. 2017, 119, 057401.

(1) Dean, C. R.; Young, A. F.; Meric, I.; Lee, C.; Wang, L.; Sorgenfrei, S.; Watanabe, K.; Taniguchi, T.; Kim, P.; Shepard, K. et al. Boron Nitride Substrates for High-quality Graphene Electronics. Nat. Nanotechnol. 2010, 5, 722–726.

(9) Bourrellier, R.; Meuret, S.; Tararan, A.; Stephan, O.; Kociak, M.; Tizei, L. H.; Zobelli, A. Bright UV single photon emission at point defects in h-BN. Nano Lett. 2016, 16, 4317–4321.

(2) Xue, J.; Sanchez-Yamagishi, J.; Bulmash, D.; Jacquod, P.; Deshpande, A.; Watanabe, K.; Taniguchi, T.; JarilloHerrero, P.; LeRoy, B. J. Scanning Tunnelling Microscopy and Spectroscopy of Ultra-flat Graphene on Hexagonal Boron Nitride. Nat. Mater. 2011, 10, 282–285.

(10) Ji, Y.; Calderon, B.; Han, Y.; Cueva, P.; Jungwirth, N. R.; Alsalman, H. A.; Hwang, J.; Fuchs, G. D.; Muller, D. A.; Spencer, M. G. Chemical Vapor Deposition Growth of Large Single-Crystal Mono-, Bi-, Tri-Layer Hexagonal Boron Nitride and Their Interlayer Stacking. ACS Nano 2017, 11, 12057–12066.

(3) Yankowitz, M.; Xue, J.; Cormode, D.; Sanchez-Yamagishi, J. D.; Watanabe, K.; Taniguchi, T.; Jarillo-Herrero, P.; Jacquod, P.; LeRoy, B. J. Emergence of Superlattice Dirac Points in Graphene on Hexagonal Boron Nitride. Nat. Phys. 2012, 8, 382–386.

(11) Warner, J. H.; Rummeli, M. H.; Bachmatiuk, A.; Buchner, B. Atomic Resolution Imaging and Topography of Boron Nitride Sheets Produced by Chemical Exfoliation. ACS Nano 2010, 4, 1299–1304.

(4) Cassabois, G.; Valvin, P.; Gil, B. Hexago-


7


Page 8 of 10

(12) Padilha, J.; Peelaers, H.; Janotti, A.; Van de Walle, C. G. Nature and Evolution of the Band-edge States in MoS2: From Monolayer to Bulk. Phys. Rev. B 2014, 90, 205420.

(21) Wu, F.; Galatas, A.; Sundararaman, R.; Rocca, D.; Ping, Y. First-principles engineering of charged defects for twodimensional quantum technologies. Physical Review Materials 2017, 1, 071001.

(13) Arnaud, B.; Lebegue, S.; Rabiller, P.; Alouani, M. Huge Excitonic Effects in Layered Hexagonal Boron Nitride. Phys. Rev. Lett. 2006, 96, 026402.

(22) Ellis, J. K.; Lucero, M. J.; Scuseria, G. E. The Indirect to Direct Band Gap Transition in Multilayered MoS2 as Predicted by Screened Hybrid Density Functional Theory. Appl. Phys. Lett. 2011, 99, 261908.

(14) Liu, L.; Feng, Y.; Shen, Z. Structural and Electronic Properties of h-BN. Phys. Rev. B 2003, 68, 104102.

(23) Schuster, R.; Habenicht, C.; Ahmad, M.; Knupfer, M.; B¨ uchner, B. Direct observation of the lowest indirect exciton state in the bulk of hexagonal boron nitride. Physical Review B 2018, 97, 041201.

(15) Dreyer, C. E.; Lyons, J. L.; Janotti, A.; Van de Walle, C. G. Band Alignments and Polarization Properties of BN Polymorphs. Appl. Phys. Express 2014, 7, 031001.

(24) Kohn, W.; Sham, L. J. Self-consistent Equations Including Exchange and Correlation Effects. Phys. Rev. 1965, 140, A1133.

(16) Berseneva, N.; Gulans, A.; Krasheninnikov, A. V.; Nieminen, R. M. Electronic Structure of Boron Nitride Sheets Doped with Carbon from First-principles Calculations. Phys. Rev. B 2013, 87, 035404.

(25) Kresse, G.; Furthm¨ uller, J. Efficient Iterative Schemes for Ab Initio Total-energy Calculations Using a Plane-wave Basis Set. Phys. Rev. B 1996, 54, 11169–11186.

(17) Kang, J.; Zhang, L.; Wei, S.-H. A Unified Understanding of the Thicknessdependent Bandgap Transition in Hexagonal Two-dimensional Semiconductors. J. Phys. Chem. Lett. 2016, 7, 597–602.

(26) Heyd, J.; Scuseria, G.; Ernzerhof, M. Hybrid Functionals Based on a Screened Coulomb Potential. J. Chem. Phys. 2003, 118, 8207–8215.

(18) Blase, X.; Rubio, A.; Louie, S. G.; Cohen, M. L. Quasiparticle Band Structure of Bulk Hexagonal Boron Nitride and Related Systems. Phys. Rev. B 1995, 51, 6868.

(27) Heyd, J.; Scuseria, G. E.; Ernzerhof, M. Erratum: Hybrid Functionals Based on a Screened Coulomb Potential [J. Chem. Phys.118, 8207 (2003)]. J. Chem. Phys. 2006, 124 .

(19) Paleari, F.; Galvani, T.; Amara, H.; Ducastelle, F.; Molina-Sánchez, A.; Wirtz, L. Excitons in Few-layer Hexagonal Boron Nitride: Davydov Splitting and Surface Localization. 2D Materials 2018,

(28) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A Consistent and Accurate Ab Initio Parametrization of Density Functional Dispersion Correction (DFT-D) for the 94 elements H-Pu. J. Chem. Phys. 2010, 132, 154104.

(20) Galvani, T.; Paleari, F.; Miranda, H. P.; Molina-Sánchez, A.; Wirtz, L.; Latil, S.; Amara, H.; Ducastelle, F. Excitons in boron nitride single layer. Phys. Rev. B 2016, 94, 125303.

(29) Blöchl, P. E. Projector Augmented-wave Method. Phys. Rev. B 1994, 50, 17953– 17979.


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(30) Marques, M. A.; Vidal, J.; Oliveira, M. J.; Reining, L.; Botti, S. Density-based Mixing Parameter for Hybrid Functionals. Physical Review B 2011, 83, 035119. (31) Li, L. H.; Santos, E. J.; Xing, T.; Cappelluti, E.; Roldán, R.; Chen, Y.; Watanabe, K.; Taniguchi, T. Dielectric screening in atomically thin boron nitride nanosheets. Nano letters 2014, 15, 218– 223. (32) Constantinescu, G.; Kuc, A.; Heine, T. Stacking in Bulk and Bilayer Hexagonal Boron Nitride. Phys. Rev. Lett. 2013, 111, 036104. (33) Gu, Y.; Zheng, M.; Liu, Y.; Xu, Z. Low-Temperature Synthesis and Growth of Hexagonal Boron-Nitride in a Lithium Bromide Melt. J. Am. Ceram. Soc. 2007, 90, 1589–1591. (34) Doni, E.; Parravicini, G. P. Energy bands and optical properties of hexagonal boron nitride and graphite. Il Nuovo Cimento B (1965-1970) 1969, 64, 117–144.


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Graphical TOC Entry 6.8 Band gap (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

6.6 6.4

Direct gap

6.2

Indirect gap

6.0 1

2

3

4

5

6 n

7

∞

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Page 10 of 10

Monolayer to Bulk Properties of Hexagonal Boron Nitride - The

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