Abstract

The electronic and optical properties of 2D hexagonal boron nitride are studied using first principle calculations. GW and Bethe–Salpeter equation (BSE) methods are employed in order to predict with better accuracy the excited and excitonic properties of this material. We determine the values of the band gap (7.32 eV, indirect), optical gap (5.58 eV), and excitonic binding energies (2.19 eV) and analyze the excitonic wave functions. We also calculate the exciton energies following an equation of motion formalism and the Elliot formula and find good agreement with the GW+BSE method. The optical properties are studied for the TM and TE modes, showing that 2D hexagonal boron nitride (hBN) is a good candidate for polaritonics in the UV range. In particular, it is shown that a single layer of hBN can act as an almost perfect mirror for ultraviolet electromagnetic radiation.

© 2019 Optical Society of America

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2018 (3)

P. Back, S. Zeytinoglu, A. Ijaz, M. Kroner, and A. Imamoğlu, “Realization of an electrically tunable narrow-bandwidth atomically thin mirror using monolayer MoSe2,” Phys. Rev. Lett. 120, 037401 (2018).
[Crossref]

G. Scuri, Y. Zhou, A. A. High, D. S. Wild, C. Shu, K. De Greve, L. A. Jauregui, T. Taniguchi, K. Watanabe, P. Kim, M. D. Lukin, and H. Park, “Large excitonic reflectivity of monolayer MoSe2 encapsulated in hexagonal boron nitride,” Phys. Rev. Lett. 120, 037402 (2018).
[Crossref]

Y. Gutiérrez, R. Alcaraz de la Osa, D. Ortiz, J. M. Saiz, F. González, and F. Moreno, “Plasmonics in the ultraviolet with aluminum, gallium, magnesium and rhodium,” Appl. Sci. 8, 64 (2018).
[Crossref]

2017 (8)

F. Cadiz, E. Courtade, C. Robert, G. Wang, Y. Shen, H. Cai, T. Taniguchi, K. Watanabe, H. Carrere, D. Lagarde, M. Manca, T. Amand, P. Renucci, S. Tongay, X. Marie, and B. Urbaszek, “Excitonic linewidth approaching the homogeneous limit in MoSe2-based van der Waals heterostructures,” Phys. Rev. X 7, 021026 (2017).
[Crossref]

T. Q. P. Vuong, G. Nbois, P. Valvin, E. Rousseau, A. Summerfield, C. J. Mellor, Y. Cho, T. S. Cheng, J. D. Albar, L. Eaves, C. T. Foxon, P. H. Beton, S. V. Novikov, and B. Gil, “Deep ultraviolet emission in hexagonal boron nitride grown by high-temperature molecular beam epitaxy,” 2D Mater. 4, 021023 (2017).
[Crossref]

T. Low, A. Chaves, J. D. Caldwell, A. Kumar, N. X. Fang, P. Avouris, T. F. Heinz, F. Guinea, L. Martin-Moreno, and F. Koppens, “Polaritons in layered two-dimensional materials,” Nat. Mater. 16, 182–194 (2017).
[Crossref]

K. Ba, W. Jiang, J. Cheng, J. Bao, N. Xuan, Y. Sun, B. Liu, A. Xie, S. Wu, and Z. Sun, “Chemical and bandgap engineering in monolayer hexagonal boron nitride,” Sci. Rep. 7, 45584 (2017).
[Crossref]

F. Ferreira and R. M. Ribeiro, “Improvements in the gw and Bethe-Salpeter-equation calculations on phosphorene,” Phys. Rev. B 96, 115431 (2017).
[Crossref]

A. Chaves, R. Ribeiro, T. Frederico, and N. Peres, “Excitonic effects in the optical properties of 2d materials: An equation of motion approach,” 2D Mater. 4, 025086 (2017).
[Crossref]

K. Zhang, Y. Feng, F. Wang, Z. Yang, and J. Wang, “Two dimensional hexagonal boron nitride (2d-hbn): synthesis, properties and applications,” J. Mater. Chem. C 5, 11992–12022 (2017).
[Crossref]

K. Zimmermann, A. Jordan, F. Gay, K. Watanabe, T. Taniguchi, Z. Han, V. Bouchiat, H. Sellier, and B. Sacépé, “Tunable transmission of quantum Hall edge channels with full degeneracy lifting in split-gated graphene devices,” Nat. Commun. 8, 14983 (2017).
[Crossref]

2016 (9)

J. Duan, X. Wang, X. Lai, G. Li, K. Watanabe, T. Taniguchi, M. Zebarjadi, and E. Y. Andrei, “High thermoelectric power factor in graphene/hbn devices,” Proc. Natl. Acad. Sci. 113, 14272–14276 (2016).
[Crossref]

X. Li, S. Sundaram, Y. El Gmili, T. Ayari, R. Puybaret, G. Patriarche, P. L. Voss, J. P. Salvestrini, and A. Ougazzaden, “Large-area two-dimensional layered hexagonal boron nitride grown on sapphire by metalorganic vapor phase epitaxy,” Cryst. Growth Des. 16, 3409–3415 (2016).
[Crossref]

J. Bao, M. Edwards, S. Huang, Y. Zhang, Y. Fu, X. Lu, Z. Yuan, K. Jeppson, and J. Liu, “Two-dimensional hexagonal boron nitride as lateral heat spreader in electrically insulating packaging,” J. Phys. D 49, 265501 (2016).
[Crossref]

L. Banszerus, M. Schmitz, S. Engels, M. Goldsche, K. Watanabe, T. Taniguchi, B. Beschoten, and C. Stampfer, “Ballistic transport exceeding 28 in cvd grown graphene,” Nano Lett. 16, 1387–1391 (2016).
[Crossref]

G. Cassabois, P. Valvin, and B. Gil, “Hexagonal boron nitride is an indirect bandgap semiconductor,” Nat. Photonics 10, 262–266 (2016).
[Crossref]

P. Cudazzo, L. Sponza, C. Giorgetti, L. Reining, F. Sottile, and M. Gatti, “Exciton band structure in two-dimensional materials,” Phys. Rev. Lett. 116, 066803 (2016).
[Crossref]

T. Galvani, F. Paleari, H. P. C. Miranda, A. Molina-Sánchez, L. Wirtz, S. Latil, H. Amara, and F. M. C. Ducastelle, “Excitons in boron nitride single layer,” Phys. Rev. B 94, 125303 (2016).
[Crossref]

D. Basov, M. Fogler, and F. G. de Abajo, “Polaritons in van der Waals materials,” Science 354, aag1992 (2016).
[Crossref]

Y. Gutierrez, D. Ortiz, J. M. Sanz, J. M. Saiz, F. Gonzalez, H. O. Everitt, and F. Moreno, “How an oxide shell affects the ultraviolet plasmonic behavior of ga, mg, and al nanostructures,” Opt. Express 24, 20621–20631 (2016).
[Crossref]

2015 (4)

A. M. Watson, X. Zhang, R. Alcaraz de La Osa, J. M. Sanz, F. González, F. Moreno, G. Finkelstein, J. Liu, and H. O. Everitt, “Rhodium nanoparticles for ultraviolet plasmonics,” Nano Lett. 15, 1095–1100 (2015).
[Crossref]

R. Alcaraz de la Osa, J. Sanz, A. Barreda, J. Saiz, F. González, H. Everitt, and F. Moreno, “Rhodium tripod stars for UV plasmonics,” J. Phys. Chem. C 119, 12572–12580 (2015).
[Crossref]

Y. Stehle, H. M. Meyer, R. R. Unocic, M. Kidder, G. Polizos, P. G. Datskos, R. Jackson, S. N. Smirnov, and I. V. Vlassiouk, “Synthesis of hexagonal boron nitride monolayer: control of nucleation and crystal morphology,” Chem. Mater. 27, 8041–8047 (2015).
[Crossref]

J. Jung, A. M. DaSilva, A. H. MacDonald, and S. Adam, “Origin of band gaps in graphene on hexagonal boron nitride,” Nat. Commun. 6, 1 (2015).
[Crossref]

2014 (3)

G. Shi, Y. Hanlumyuang, Z. Liu, Y. Gong, W. Gao, B. Li, J. Kono, J. Lou, R. Vajtai, P. Sharma, and P. M. Ajayan, “Boron nitride-graphene nanocapacitor and the origins of anomalous size-dependent increase of capacitance,” Nano Lett. 14, 1739–1744 (2014).
[Crossref]

M. B. Ross and G. C. Schatz, “Aluminum and indium plasmonic nanoantennas in the ultraviolet,” J. Phys. Chem. C 118, 12506–12514 (2014).
[Crossref]

A. Rodin, A. Carvalho, and A. C. Neto, “Excitons in anisotropic two-dimensional semiconducting crystals,” Phys. Rev. B 90, 075429 (2014).
[Crossref]

2013 (5)

J. M. McMahon, G. C. Schatz, and S. K. Gray, “Plasmonics in the ultraviolet with the poor metals al, ga, in, sn, tl, pb, and bi,” Phys. Chem. Chem. Phys. 15, 5415–5423 (2013).
[Crossref]

Y. Yang, J. M. Callahan, T.-H. Kim, A. S. Brown, and H. O. Everitt, “Ultraviolet nanoplasmonics: a demonstration of surface-enhanced Raman spectroscopy, fluorescence, and photodegradation using gallium nanoparticles,” Nano Lett. 13, 2837–2841 (2013).
[Crossref]

G. Maidecchi, G. Gonella, R. Proietti Zaccaria, R. Moroni, L. Anghinolfi, A. Giglia, S. Nannarone, L. Mattera, H.-L. Dai, M. Canepa, and F. Bisio, “Deep ultraviolet plasmon resonance in aluminum nanoparticle arrays,” ACS Nano 7, 5834–5841 (2013).
[Crossref]

Y. V. Bludov, A. Ferreira, N. Peres, and M. Vasilevskiy, “A primer on surface plasmon-polaritons in graphene,” Int. J. Mod. Phys. B 27, 1341001 (2013).
[Crossref]

N. Berseneva, A. Gulans, A. V. Krasheninnikov, and R. M. Nieminen, “Electronic structure of boron nitride sheets doped with carbon from first-principles calculations,” Phys. Rev. B 87, 035404 (2013).
[Crossref]

2012 (3)

M. W. Knight, L. Liu, Y. Wang, L. Brown, S. Mukherjee, N. S. King, H. O. Everitt, P. Nordlander, and N. J. Halas, “Aluminum plasmonic nanoantennas,” Nano Lett. 12, 6000–6004 (2012).
[Crossref]

J. Deslippe, G. Samsonidze, D. A. Strubbe, M. Jain, M. L. Cohen, and S. G. Louie, “Berkeleygw: a massively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures,” Comput. Phys. Commun. 183, 1269–1289 (2012).
[Crossref]

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2011 (5)

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Comput. Phys. Commun. (1)

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Figures (12)

Fig. 1.
Fig. 1. Electronic band structure (left) and electronic density of states of hBN (right) for both DFT and G W calculations.
Fig. 2.
Fig. 2. Imaginary part of dielectric function of 2D hBN. The blue (red) lines represent the BSE (RPA) imaginary part of dielectric function.
Fig. 3.
Fig. 3. Real part of the dielectric function of 2D hBN. The blue (red) lines represent the BSE (RPA) real part of dielectric function.
Fig. 4.
Fig. 4. Excitonic energies for the lowest energy exciton states. The system has a C 3 v symmetry with three representations: A 1 , E , and A 2 . The states 1 to 4 have E symmetry and are valley degenerate; states 5 and 6 have A 2 and A 1 symmetries, respectively, and are nondegenerate (see [29]).
Fig. 5.
Fig. 5. Probability density | ϕ ( r e , r h ) | 2 for the exciton states 1 to 8. The hole is localized slightly above the nitrogen atom (light color) at the center of the lattice.
Fig. 6.
Fig. 6. Fit of the Elliot formula to the G 0 W 0 + BSE result. There is a very good agreement for the real part and a small shift in the imaginary part. The exciton linewidth used was γ = 0.1 eV . The parameters of the fitting are shown in Table 6.
Fig. 7.
Fig. 7. Exciton and K K transition energy as function of the environment dielectric constant. We can see that the dependence of the first exciton energy is almost linear, while the K K transition energy has a greater dependence on the dielectric constant.
Fig. 8.
Fig. 8. Exciton–polariton dispersion relation for complex frequency. The results are given as a function of the wavenumber ν ˜ = λ q 1 . The gray dashed-dot line represents the light cone in air. In this approach, the wavenumber can reach large values for both TE and TM modes for either A or B exciton energies. Detail around excitons A and B is shown in the right panels.
Fig. 9.
Fig. 9. Exciton–polariton dispersion relation in the complex wavenumber approach. Panel A (B) shows the TM (TE) mode. The TM mode has a dispersion almost insensitive to the relaxation rate while the TE mode changes significantly: the wavenumber is close to the free-light one and only for γ = 4 meV there is a different behavior.
Fig. 10.
Fig. 10. Exciton–polariton propagation ratio. Panel A (B) shows the TM (TE) mode. The propagation rate of the TM mode is very low except for γ = 4 meV . The peak at ω = 5.48 corresponds to the propagation of radiation. As can be seen in Fig. 9, the wavenumber tends to the free-light wavenumber. The same result appears in the propagation rate for the TE modes: except for γ = 4 meV , all other modes correspond to poorly confined modes (see Fig. 11 also). For γ = 4 meV and the TE mode, the propagation rate decreases with the increasing frequency.
Fig. 11.
Fig. 11. Exciton–polariton confinement ratio. Panel A (B) shows the TM (TE) mode. The confinement of the TM mode increases with the frequency and has a small dependence with the relaxation rate γ . The TE modes for the higher values of γ are poorly confined. For the value, we have a peak in the confinement below the exciton energy.
Fig. 12.
Fig. 12. Reflection coefficient for monolayer hBN and different values of γ with the parameters from Table 6. Panels (a) and (c) show the A and B excitons, respectively, with the G 0 W 0 parameters, while panels (b) and (d) show the result from the equation of motion formalism. As the equation of motion formalism predicts higher excitonic weights, in this case we have broader reflectance peaks around the excitons energies in comparison with the G 0 W 0 result.

Tables (6)

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Table 1. Several Band Gaps Calculated in This Work and by Other Authors Using G W 0 , G 0 W 0 , and BSE, in eV a

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Table 2. Details of DFT Calculations

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Table 3. G 0 W 0 Gap Values for the Transitions K Γ , K K , and Γ Γ , and Optical Gap and Exciton Binding Energy (EBE) Obtained from BSE in This Work a

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Table 4. Width of the Valence Bands a

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Table 5. Effective Masses [in Electron Mass ( m e ) Units] for hBN Calculated Using G 0 W 0 a

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Table 6. Comparison of the Elliot Formula Parameters Used in the G 0 W 0 + BSE Calculation and the Equation of Motion Approach a

Equations (27)

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ϵ 2 ( ω ) = 8 π 2 e 2 ω 2 S | e · 0 | v | S | 2 δ ( ω Ω S ) ,
ϵ 2 ( ω ) = 8 π 2 e 2 ω 2 v c k | e · v k | v | c k | 2 δ ( ω ( E c k E v k ) ) ,
( ω ω ˜ λ k ) p λ ( k , ω ) = ( E 0 d λ ( k ) + B k λ ( ω ) ) Δ f k ,
σ ( ω ) σ 0 = 4 i ω n p n ω E n + i γ ,
ε 1 κ 1 + ε 2 κ 2 + i σ ( ω ) ε 0 ω = 0 ,
κ 1 + κ 2 i ω μ 0 σ ( ω ) = 0 ,
κ i = q 2 ε i ω 2 c 2 ,
τ 1 = γ + 1 p n I [ b n ] | ( ε 1 + ε 2 ) b n 4 π α c q + 1 | 2 ,
c 2 q 2 = ω 2 + c 2 κ α 2 ( ω ) ,
κ TE ( ω ) = i ε 0 ω 2 σ ( ω ) ,
κ TM ( ω ) = i ω μ 0 σ ( ω ) 2 .
R = | π α f ( ω ) 2 + π α f ( ω ) | 2 ,
H 0 ( k ) = v F ( σ · k + σ 3 m v F 2 ) ,
H ^ I ( t ) = e E ( t ) x ^ ,
H ^ ee = e 2 d r 1 d r 2 ψ ^ ( r 1 ) ψ ^ ( r 2 ) V ( r 1 r 2 ) ψ ^ ( r 2 ) ψ ^ ( r 1 ) ,
ψ ^ ( r , t ) = 1 L k , λ ϕ λ ( k ) a ^ k λ ( t ) e i k · r ,
ϕ λ ( k ) = E k + λ m 2 E k ( 1 k x i k y λ E k + m ) ,
E k = k 2 + m 2 ,
V ( q ) = e 2 ε 0 1 q ( r 0 q + ε m ) .
P ( ω ) = i g s e 2 k λ d λ ( k ) p λ ( k , ω ) ,
d λ ( k ) = 1 2 E k ( sin θ + i m E k cos θ ) .
p λ ( k , ω ) = d ω 2 π e i ω t a ^ k , λ ( t ) a ^ k , λ ( t ) ,
( ω ω ˜ λ k ) p λ ( k , ω ) = ( E 0 d λ ( k ) + B k λ ( ω ) ) Δ f k ,
ω ˜ λ k = 2 λ E k + λ Σ k , λ xc ,
Σ k , λ xc = d q ( 2 π ) 2 V ( q ) Δ f k q [ F λ λ λ λ ( k , k q ) F λ λ λ λ ( k , k q ) ] ,
B k λ ( ω ) = d q ( 2 π ) 2 V ( | k q | ) [ p λ ( q , ω ) F λ λ λ λ ( k , q ) + p λ ( q , ω ) F λ λ λ λ ( k , q ) ] .
F λ 1 , λ 2 , λ 3 , λ 4 ( k 1 , k 2 ) = ϕ λ 1 ( k 1 ) ϕ λ 2 ( k 2 ) ϕ λ 3 ( k 2 ) ϕ λ 4 ( k 1 ) .

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