Abstract
In this colloquium, we review the research on excitons in van der Waals heterostructures from the point of view of variational calculations. We first make a presentation of the current and past literature, followed by a discussion on the connections between experimental and theoretical results. In particular, we focus our review of the literature on the absorption spectrum and polarizability, as well as the Stark shift and the dissociation rate. Afterwards, we begin the discussion of the use of variational methods in the study of excitons. We initially model the electron–hole interaction as a soft-Coulomb potential, which can be used to describe interlayer excitons. Using an ansatz, based on the solution for the two-dimensional quantum harmonic oscillator, we study the Rytova–Keldysh potential, which is appropriate to describe intralayer excitons in two-dimensional (2D) materials. These variational energies are then recalculated with a different ansatz, based on the exact wavefunction of the 2D hydrogen atom, and the obtained energy curves are compared. Afterwards, we discuss the Wannier–Mott exciton model, reviewing it briefly before focusing on an application of this model to obtain both the exciton absorption spectrum and the binding energies for certain values of the physical parameters of the materials. Finally, we briefly discuss an approximation of the electron–hole interaction in interlayer excitons as an harmonic potential and the comparison of the obtained results with the existing values from both first-principles calculations and experimental measurements.
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N. Mounet, M. Gibertini, P. Schwaller, D. Campi, A. Merkys, A. Marrazzo, T. Sohier, I.E. Castelli, A. Cepellotti, G. Pizzi et al., Nat. Nano. 13, 246 (2018)
R.F. Frindt, A.D. Yoffe, Proc. R. Soc. Lond. A 273, 69 (1963)
E. Fortin, W. Sears, J. Phys. Chem. Solids 43, 881 (1982)
T. Mueller, E. Malic, npj 2D Mat. Appl. 2, 29 (2018)
G. Wang, A. Chernikov, M.M. Glazov, T.F. Heinz, X. Marie, T. Amand, B. Urbaszek, Rev. Mod. Phys. 90, 021001 (2018)
H. Yu, X. Cui, X. Xu, W. Yao, Natl. Sci. Rev. 2, 57 (2015)
M.V. Durnev, M.M. Glazov, Physics-Uspekhi 61, 825 (2018)
R.S. Knox, inCollective Excitations in Solids (Springer, USA, 1983), pp. 183–245
N. Rytova, Moscow Univ. Phys. Bull. 3, 30 (1967)
L.V. Keldysh, Sov. J. Exp. Theor. Phys. Lett. 29, 658 (1979)
A.K. Geim, I.V. Grigorieva, Nature 499, 419 (2013)
X.Y. Dong, R.Z. Li, J.P. Deng, Z.W. Wang, J. Phys. Chem. Solids 134, 1 (2019)
K.S. Thygesen, 2D Mater. 4, 022004 (2017)
M.N. Brunetti, O.L. Berman, R.Y. Kezerashvili, J. Phys.: Condens. Matter 30, 225001 (2018)
D. Van Tuan, M. Yang, H. Dery, Phys. Rev. B 98, 125308 (2018)
J.Z. Zhang, J.Z. Ma, J. Phys.: Condens. Matter 31, 105702 (2019)
B. Scharf, D.V. Tuan, I. Žutić, H. Dery, J. Phys.: Cond. Matter 31, 203001 (2019)
H.C. Kamban, T.G. Pedersen, Sci. Rep. 10, 5537 (2020)
L.S.R. Cavalcante, A. Chaves, B.V. Duppen, F.M. Peeters, D.R. Reichman, Phys. Rev. B 97 (2018)
K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Science 306, 666 (2004)
K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Nature 438, 197 (2005)
K.F. Mak, C. Lee, J. Hone, J. Shan, T.F. Heinz, Phys. Rev. Lett. 105, 136805 (2010)
A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.Y. Chim, G. Galli, F. Wang, Nano Lett. 10, 1271 (2010)
U. Wurstbauer, B. Miller, E. Parzinger, A.W. Holleitner, J. Phys. D: Appl. Phys. 50, 173001 (2017)
A. Chernikov, T.C. Berkelbach, H.M. Hill, A. Rigosi, Y. Li, O.B. Aslan, D.R. Reichman, M.S. Hybertsen, T.F. Heinz, Phys. Rev. Lett. 113, 076802 (2014)
K.F. Mak, K. He, C. Lee, G.H. Lee, J. Hone, T.F. Heinz, J. Shan, Nat. Mat. 12, 207 (2012)
K. He, N. Kumar, L. Zhao, Z. Wang, K.F. Mak, H. Zhao, J. Shan, Phys. Rev. Lett. 113 (2014)
D. Xiao, G.B. Liu, W. Feng, X. Xu, W. Yao, Phys. Rev. Lett. 108 (2012)
A.K. Geim, I.V. Grigorieva, Nature 499, 419 (2013)
Z. Yin, H. Li, H. Li, L. Jiang, Y. Shi, Y. Sun, G. Lu, Q. Zhang, X. Chen, H. Zhang, ACS Nano 6, 74 (2012)
Z. Yin, H. Li, H. Li, L. Jiang, Y. Shi, Y. Sun, G. Lu, Q. Zhang, X. Chen, H. Zhang, ACS Nano 6, 74 (2011)
A. Pospischil, M.M. Furchi, T. Mueller, Nat. Nano. 9, 257 (2014)
M.M. Furchi, A. Pospischil, F. Libisch, J. Burgdörfer, T. Mueller, Nano Lett. 14, 4785 (2014)
M. Amani, D.H. Lien, D. Kiriya, J. Xiao, A. Azcatl, J. Noh, S.R. Madhvapathy, R. Addou, S. KC, M. Dubey et al., Science 350, 1065 (2015)
X.X. Zhang, T. Cao, Z. Lu, Y.C. Lin, F. Zhang, Y. Wang, Z. Li, J.C. Hone, J.A. Robinson, D. Smirnov et al., Nat. Nano. 12, 883 (2017)
M.R. Molas, C. Faugeras, A.O. Slobodeniuk, K. Nogajewski, M. Bartos, D.M. Basko, M. Potemski, 2D Mat. 4, 021003 (2017)
B. Scharf, T. Frank, M. Gmitra, J. Fabian, I. Žutić, V. Perebeinos, Phys. Rev. B 94, 245434 (2016)
S. Haastrup, S. Latini, K. Bolotin, K.S. Thygesen, Phys. Rev. B 94, 041401 (2016)
M. Massicotte, F. Vialla, P. Schmidt, M.B. Lundeberg, S. Latini, S. Haastrup, M. Danovich, D. Davydovskaya, K. Watanabe, T. Taniguchi et al., Nat. Commun. 9, 1633 (2018)
A.F. Rigosi, H.M. Hill, Y. Li, A. Chernikov, T.F. Heinz, Nano Lett. 15, 5033 (2015)
T.C. Berkelbach, D.R. Reichman, Annu. Rev. Condens. Matter Phys. 9, 379 (2018)
N. Aquino, G. Campoy, A. Flores-Riveros, Int. J. Quant. Chem. 103, 267 (2005)
F. Grasselli, Am. J. Phys. 85, 834 (2017)
M. Iñarrea, V. Lanchares, J.F. Palacián, A.I. Pascual, J.P. Salas, P. Yanguas, Commun. Nonlinear Sci. Numer. Simul. 68, 94 (2019)
V. Shahnazaryan, I. Iorsh, I.A. Shelykh, O. Kyriienko, Phys. Rev. B 96, 115409 (2017)
S. Aas, C. Bulutay, Opt. Express 26, 28672 (2018)
C. Jin, E.C. Regan, A. Yan, M.I.B. Utama, D. Wang, S. Zhao, Y. Qin, S. Yang, Z. Zheng, S. Shi et al., Nature 567, 76 (2019)
A. Castellanos-Gomez, J. Phys. Chem. Lett. 6, 4280 (2015)
M.Y. Li, Y. Shi, C.C. Cheng, L.S. Lu, Y.C. Lin, H.L. Tang, M.L. Tsai, C.W. Chu, K.H. Wei, J.H. He et al., Science 349, 524 (2015)
A.S. Pawbake, M.S. Pawar, S.R. Jadkar, D.J. Late, Nanoscale 8, 3008 (2016)
W. Choi, N. Choudhary, G.H. Han, J. Park, D. Akinwande, Y.H. Lee, Mater. Today 20, 116 (2017)
P.V. Ratnikov, Phys. Rev. B 102, 085303 (2020)
A.J. Chaves, R.M. Ribeiro, T. Frederico, N.M.R. Peres, 2D Mat. 4, 025086 (2017)
J. Have, N.M.R. Peres, T.G. Pedersen, Phys. Rev. B 100, 045411 (2019)
J.C.G. Henriques, G. Catarina, A.T. Costa, J. Fernández-Rossier, N.M.R. Peres, Phys. Rev. B 101, 045408 (2020)
J.C.G. Henriques, G.B. Ventura, C.D.M. Fernandes, N.M.R. Peres, J. Phys.: Condens. Matter 32, 025304 (2020)
M.R. Molas, A.O. Slobodeniuk, K. Nogajewski, M. Bartos, L. Bala, A. Babiński, K. Watanabe, T. Taniguchi, C. Faugeras, M. Potemski, Phys. Rev. Lett. 123, 136801 (2019)
M. Van der Donck, M. Zarenia, F.M. Peeters, Phys. Rev. B 97, 195408 (2018)
A. Castellanos-Gomez, L. Vicarelli, E. Prada, J.O. Island, K.L. Narasimha-Acharya, S.I. Blanter, D.J. Groenendijk, M. Buscema, G.A. Steele, J.V. Alvarez et al., 2D Mater. 1, 025001 (2014)
J.Z. Zhang, J.Z. Ma, J. Phys.: Condens. Matter 31, 105702 (2019)
J. Stark, Ann. Phys. 348, 965 (1914)
B. Simon, Phys. Lett. A 71, 211 (1979)
C.W. McCurdy, C.K. Stroud, M.K. Wisinski, Phys. Rev. A 43, 5980 (1991)
X. Hong, J. Kim, S.F. Shi, Y. Zhang, C. Jin, Y. Sun, S. Tongay, J. Wu, Y. Zhang, F. Wang, Nat. Nano. 9, 682 (2014)
L. Meckbach, T. Stroucken, S.W. Koch, Phys. Rev. B 97, 035425 (2018)
R. Kumar, I. Verzhbitskiy, F. Giustiniano, T.P.H. Sidiropoulos, R.F. Oulton, G. Eda, 2D Mater. 5, 041003 (2018)
S. Ovesen, S. Brem, C. Linderälv, M. Kuisma, T. Korn, P. Erhart, M. Selig, E. Malic, Commun. Phys. 2, 23 (2019)
H.C. Kamban, T.G. Pedersen, Phys. Rev. B 100, 045307 (2019)
B. Miller, A. Steinhoff, B. Pano, J. Klein, F. Jahnke, A. Holleitner, U. Wurstbauer, Nano Lett. 17, 5229 (2017)
F. Vialla, M. Danovich, D.A. Ruiz-Tijerina, M. Massicotte, P. Schmidt, T. Taniguchi, K. Watanabe, R.J. Hunt, M. Szyniszewski, N.D. Drummond et al., 2D Mater. 6, 035032 (2019)
H. Wang, C. Zhang, W. Chan, C. Manolatou, S. Tiwari, F. Rana, Phys. Rev. B 93, 045407 (2016)
G. Moody, C.K. Dass, K. Hao, C.H. Chen, L.J. Li, A. Singh, K. Tran, G. Clark, X. Xu, G. Berghäuser et al., Nat. Commun. 6, 8315 (2015)
B. Han, C. Robert, E. Courtade, M. Manca, S. Shree, T. Amand, P. Renucci, T. Taniguchi, K. Watanabe, X. Marie et al., Phys. Rev. X 8, 031073 (2018)
E. Courtade, M. Semina, M. Manca, M.M. Glazov, C. Robert, F. Cadiz, G. Wang, T. Taniguchi, K. Watanabe, M. Pierre et al., Phys. Rev. B 96, 085302 (2017)
C. Zhang, H. Wang, W. Chan, C. Manolatou, F. Rana, Phys. Rev. B 89, 205436 (2014)
M. Van der Donck, M. Zarenia, F.M. Peeters, Phys. Rev. B 96, 035131 (2017)
M. Van der Donck, F.M. Peeters, Phys. Rev. B 98, 115104 (2018)
C. Ruppert, O.B. Aslan, T.F. Heinz, 14, 6231 (2014)
M.A. Semina, Phys. Sol. Stat. 61, 2218 (2019)
J. Planelles, Theor. Chem. Accounts 136, 81 (2017)
M. Van der Donck, F.M. Peeters, Ph.D. thesis, Universiteit Antwerpen, 2019
V. Shuvayev, L. Deych, A.A. Lisyansky, I.V. Ponomarev, Self-Consistent Approach for Calculations of Exciton Binding Energy in Quantum Wells, inMarch Meeting of the APS, Los Angeles (2005)
T.G. Pedersen, Phys. Rev. B 94, 125424 (2016)
N. Lundt, E. Cherotchenko, O. Iff, X. Fan, Y. Shen, P. Bigenwald, A.V. Kavokin, S. Höfling, C. Schneider, Appl. Phys. Lett. 112, 031107 (2018)
R. Bhat, J. Sound Vibrat. 114, 65 (1987)
K.M. Liew, K.Y. Lam, J. Vibrat. Acoustics 113, 182 (1991)
M.F.C.M. Quintela, J.M.B.L. dos Santos, J. Phys.: Condens. Matter 33, 035901 (2020)
J.W. Brown, H.N. Spector, Phys. Rev. B 35, 3009 (1987)
Y. ping Feng, H.N. Spector, Phys. Rev. B 48, 1963 (1993)
O. Akimoto, E. Hanamura, J. Phys. Soc. Jpn. 33, 1537 (1972)
T.F. Rønnow, T.G. Pedersen, B. Partoens, Phys. Rev. B 85, 045412 (2012)
R. Loudon, Am. J. Phys. 27, 649 (1959)
J.C.G. Henriques, G.B. Ventura, C.D.M. Fernandes, N.M.R. Peres, J. Phys.: Condens. Matter 32, 025304 (2019)
P. Cudazzo, I.V. Tokatly, A. Rubio, Phys. Rev. B 84, 085406 (2011)
S. Schmitt-Rink, C. Ell, J. Luminesc. 30, 585 (1985)
H.C. Kamban, T.G. Pedersen, N.M.R. Peres, Phys. Rev. B 102, 115305 (2020)
L. Xu, M. Yang, S.J. Wang, Y.P. Feng, Phys. Rev. B 95, 235434 (2017)
J. Zhou, W.Y. Shan, W. Yao, D. Xiao, Phys. Rev. Lett. 115, 166803 (2015)
C.K. Yong, M.I.B. Utama, C.S. Ong, T. Cao, E.C. Regan, J. Horng, Y. Shen, H. Cai, K. Watanabe, T. Taniguchi et al., Nat. Mater. 18, 1065 (2019)
G.H. Wannier, Phys. Rev. 52, 191 (1937)
N.F. Mott, Proc. Phys. Soc. Sec. A 62, 416 (1949)
J. Have, G. Catarina, T.G. Pedersen, N.M.R. Peres, Phys. Rev. B 99, 035416 (2019)
Y.W. Chang, D.R. Reichman, Phys. Rev. B 99, 125421 (2019)
D.V. Tuan, B. Scharf, Z. Wang, J. Shan, K.F. Mak, I. Žutić, H. Dery, Phys. Rev. B 99, 085301 (2019)
G.D. Mahan, Many-particle physics, Physics of solids and liquids, 2nd edn. (Plenum Press, 1990)
S.W.K. Hartmut Haug,Quantum Theory of Optical and Electronic Properties of Semiconductors, 4th edn. (World Scientific Publishing Company, 2004)
H. Ouerdane, J. Appl. Phys. 110, 074905 (2011)
N.M.R. Peres, R.M. Ribeiro, A.H.C. Neto, Phys. Rev. Lett. 105, 055501 (2010)
S. Glutsch,Excitons in Low-Dimensional Semiconductors (Springer, Berlin, Heidelberg, 2004)
T.C. Berkelbach, M.S. Hybertsen, D.R. Reichman, Phys. Rev. B 88, 045318 (2013)
P. Cudazzo, I.V. Tokatly, A. Rubio, Phys. Rev. B 84, 085406 (2011)
D.Y. Qiu, F.H. da Jornada, S.G. Louie, Phys. Rev. Lett. 111, 216805 (2013)
O. Ávalos Ovando, D. Mastrogiuseppe, S.E. Ulloa, J. Phys.: Condens. Matter 31, 213001 (2019)
D.W. Snoke,Solid state physics: essential concepts (Addison-Wesley, San Francisco, 2009)
J. Jackson,Classical electrodynamics (Wiley, New York, 1999)
O.L. Berman, G. Gumbs, R.Y. Kezerashvili, Phys. Rev. B 96, 014505 (2017)
A. Edery, P. Laporte, J. Phys. Comm. 2, 025024 (2018)
B. Cohen-Tannoudji, F. Laloe, D. Claude, inQuantum Mechanics (Wiley, 2019), Vol. 3
J. Horng, T. Stroucken, L. Zhang, E.Y. Paik, H. Deng, S.W. Koch, Phys. Rev. B 97, 241404(R) (2018)
M. Rahaman, C. Wagner, A. Mukherjee, A. Lopez-Rivera, S. Gemming, D.R.T. Zahn, J. Phys.: Condens. Matter 31, 114001 (2019)
I. Kylänpää, H.P. Komsa, Phys. Rev. B 92, 205418 (2015)
K.W. Lau, C. Cocchi, C. Draxl, Phys. Rev. Materials 3, 074001 (2019)
G. Bassani, G. Parravicini, R. Ballinger, Electronic States and Optical Transitions in Solids, International Series of Monographs on solid state physics (Pergamon Press, New York, 1975)
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Martins Quintela, M.F.C., Peres, N.M.R. A colloquium on the variational method applied to excitons in 2D materials. Eur. Phys. J. B 93, 222 (2020). https://doi.org/10.1140/epjb/e2020-10490-9
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DOI: https://doi.org/10.1140/epjb/e2020-10490-9