Abstract

Within a two-band k · p method we analyze different types of strain for the K valley optical characteristics of a freestanding monolayer MoS2, MoSe2, WS2 and WSe2. We predict that circular polarization selectivity for energies above the direct transition onset deteriorates/improves by tensile/compressive strain. Wide range of available strained-sample photoluminescence data can be reasonably reproduced by this simple bandstructure combined with accounting for excitons at a variational level. According to this model strain impacts optoelectronic properties through its hydrostatic component, whereas the shear strain only causes a rigid wavevector shift of the valley. Furthermore, under the stress loading of flexible substrates the presence of Poisson’s effect or the lack of it are examined individually for the reported measurements.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (3)

S. Fang, S. Carr, M. A. Cazalilla, and E. Kaxiras, “Electronic structure theory of strained two-dimensional materials with hexagonal symmetry,” Phys. Rev. B 98, 075106 (2018).
[Crossref]

G. Wang, A. Chernikov, M. M. Glazov, T. F. Heinz, X. Marie, T. Amand, and B. Urbaszek, “Colloquium: Excitons in atomically thin transition metal dichalcogenides,” Rev. Mod. Phys. 90, 021001 (2018).
[Crossref]

O. B. Aslan, I. M. Datye, M. J. Mleczko, K. Sze Cheung, S. Krylyuk, A. Bruma, I. Kalish, A. V. Davydov, E. Pop, and T. F. Heinz, “Probing the optical properties and strain-tuning of ultrathin Mo1−xWxTe2,” Nano Lett. 18, 2485–2491 (2018).
[Crossref] [PubMed]

2017 (2)

M. Feierabend, A. Morlet, G. Berghäuser, and E. Malic, “Impact of strain on the optical fingerprint of monolayer transition-metal dichalcogenides,” Phys. Rev. B 96, 045425 (2017).
[Crossref]

C. Palacios-Berraquero, D. M. Kara, A. R.-P. Montblanch, M. Barbone, P. Latawiec, D. Yoon, A. K. Ott, M. Loncar, A. C. Ferrari, and M. Atatüre, “Large-scale quantum-emitter arrays in atomically thin semiconductors,” Nat. Commun. 8, 15093 (2017).
[Crossref] [PubMed]

2016 (6)

D. Lloyd, X. Liu, J. W. Christopher, L. Cantley, A. Wadehra, B. L. Kim, B. B. Goldberg, A. K. Swan, and J. S. Bunch, “Band gap engineering with ultralarge biaxial strains in suspended monolayer MoS2,” Nano Lett. 16, 5836–5841 (2016).
[Crossref] [PubMed]

A. Branny, G. Wang, S. Kumar, C. Robert, B. Lassagne, X. Marie, B. D. Gerardot, and B. Urbaszek, “Discrete quantum dot like emitters in monolayer MoSe2: Spatial mapping, magneto-optics, and charge tuning,” Appl. Phys. Lett. 108, 142101 (2016).
[Crossref]

Q. Zhang, Z. Chang, G. Xu, Z. Wang, Y. Zhang, Z.-Q. Xu, S. Chen, Q. Bao, J. Z. Liu, Y.-W. Mai, and et al., “Strain relaxation of monolayer WS2 on plastic substrate,” Adv. Funct. Mater. 26, 8707–8714 (2016).
[Crossref]

R. Schmidt, I. Niehues, R. Schneider, M. Drüppel, T. Deilmann, Michael Rohlfing, S. M. d. Vasconcellos, A. Castellanos-Gomez, and R. Bratschitsch, “Reversible uniaxial strain tuning in atomically thin WSe2,” 2D Mater. 3, 021011 (2016).
[Crossref]

J. O. Island, A. Kuc, E. H. Diependaal, R. Bratschitsch, H. S. J. v. d. Zant, T. Heine, and A. Castellanos-Gomez, “Precise and reversible band gap tuning in single-layer MoSe2 by uniaxial strain,” Nanoscale 8, 2589–2593 (2016).
[Crossref] [PubMed]

A. E. Maniadaki, G. Kopidakis, and I. N. Remediakis, “Strain engineering of electronic properties of transition metal dichalcogenide monolayers,” Solid State Commun. 227, 33–39 (2016).
[Crossref]

2015 (8)

H. Rostami, R. Roldán, E. Cappelluti, R. Asgari, and F. Guinea, “Theory of strain in single-layer transition metal dichalcogenides,” Phy. Rev. B 92, 195402 (2015).
[Crossref]

S. Yang, C. Wang, H. Sahin, H. Chen, Y. Li, S.-S. Li, A. Suslu, F. M. Peeters, Q. Liu, J. Li, and S. Tongay, “Tuning the optical, magnetic, and electrical properties of ReSe2 by nanoscale strain engineering,” Nano Lett. 15, 1660–1666 (2015).
[Crossref] [PubMed]

Y. Wang, C. Cong, W. Yang, J. Shang, N. Peimyoo, Y. Chen, J. Kang, J. Wang, W. Huang, and T. Yu, “Strain-induced direct-indirect bandgap transition and phonon modulation in monolayer WS2,” Nano Res. 8, 2562–2572 (2015).
[Crossref]

M. M. Glazov, E. L. Ivchenko, G. Wang, T. Amand, X. Marie, B. Urbaszek, and B. L. Liu, “Spin and valley dynamics of excitons in transition metal dichalcogenide monolayers,” Phys. Status Solidi B 252, 2349–2362 (2015).
[Crossref]

G. Plechinger, A. Castellanos-Gomez, M. Buscema, H. S. J. v. d. Zant, G. A. Steele, A. Kuc, T. Heine, C. Schüller, and T. Korn, “Control of biaxial strain in single-layer molybdenite using local thermal expansion of the substrate,” 2D Mater. 2, 015006 (2015).
[Crossref]

H. Li, A. W. Contryman, X. Qian, S. M. Ardakani, Y. Gong, X. Wang, J. M. Weisse, C. H. Lee, J. Zhao, P. M. Ajayan, J. Li, H. C. Manoharan, and X. Zheng, “Optoelectronic crystal of artificial atoms in strain-textured molybdenum disulphide,” Nat. Commun. 6, 7381 (2015).
[Crossref] [PubMed]

R. Roldán, A. Castellanos-Gomez, E. Cappelluti, and F. Guinea, “Strain engineering in semiconducting two-dimensional crystals,” J. Phys. :Condens. Matter 27, 313201 (2015).

H. Dery and Y. Song, “Polarization analysis of excitons in monolayer and bilayer transition-metal dichalcogenides,” Phys. Rev. B 92, 125431 (2015).
[Crossref]

2014 (5)

M. M. Glazov, T. Amand, X. Marie, D. Lagarde, L. Bouet, and B. Urbaszek, “Exciton fine structure and spin decoherence in monolayers of transition metal dichalcogenides,” Phys. Rev. B 89, 201302 (2014).
[Crossref]

S. B. Desai, G. Seol, J. S. Kang, H. Fang, C. Battaglia, R. Kapadia, J. W. Ager, J. Guo, and A. Javey, “Strain-induced indirect to direct bandgap transition in multilayer WSe2,” Nano Lett. 14, 4592–4597 (2014).
[Crossref] [PubMed]

D. Akinwande, N. Petrone, and J. Hone, “Two-dimensional flexible nanoelectronics,” Nat. Commun. 5, 5678 (2014).
[Crossref] [PubMed]

M. Cazalilla, H. Ochoa, and F. Guinea, “Quantum spin hall effect in two-dimensional crystals of transition-metal dichalcogenides,” Phys. Rev. Lett. 113, 077201 (2014).
[Crossref] [PubMed]

X. Duan, C. Wang, J. C. Shaw, R. Cheng, Y. Chen, H. Li, X. Wu, Y. Tang, Q. Zhang, A. Pan, and et al., “Lateral epitaxial growth of two-dimensional layered semiconductor heterojunctions,” Nat. Nanotech. 9, 1024 (2014).
[Crossref]

2013 (10)

A. Kormányos, V. Zólyomi, N. D. Drummond, P. Rakyta, G. Burkard, and V. I. Falko, “Monolayer MoS2: Trigonal warping, the Γ valley, and spin-orbit coupling effects,” Phys. Rev. B 88, 045416 (2013).
[Crossref]

T. C. Berkelbach, M. S. Hybertsen, and D. R. Reichman, “Theory of neutral and charged excitons in monolayer transition metal dichalcogenides,” Phys. Rev. B 88, 045318 (2013).
[Crossref]

K. He, C. Poole, K. F. Mak, and J. Shan, “Experimental demonstration of continuous electronic structure tuning via strain in atomically thin MoS2,” Nano Lett. 13, 2931–2936 (2013).
[Crossref] [PubMed]

H. J. Conley, B. Wang, J. I. Ziegler, R. F. Haglund, S. T. Pantelides, and K. I. Bolotin, “Bandgap engineering of strained monolayer and bilayer MoS2,” Nano Lett. 13, 3626–3630 (2013).
[Crossref] [PubMed]

C. R. Zhu, G. Wang, B. L. Liu, X. Marie, X. F. Qiao, X. Zhang, X. X. Wu, H. Fan, P. H. Tan, T. Amand, and B. Urbaszek, “Strain tuning of optical emission energy and polarization in monolayer and bilayer MoS2,” Phys. Rev. B 88, 121301 (2013).
[Crossref]

A. Castellanos-Gomez, R. Roldán, E. Cappelluti, M. Buscema, F. Guinea, H. S. J. van der Zant, and G. A. Steele, “Local strain engineering in atomically thin MoS2,” Nano Lett. 13, 5361–5366 (2013).
[Crossref]

P. Tonndorf, R. Schmidt, P. Böttger, X. Zhang, J. Börner, A. Liebig, M. Albrecht, C. Kloc, O. Gordan, D. R. T. Zahn, S. M. de Vasconcellos, and R. Bratschitsch, “Photoluminescence emission and raman response of monolayer MoS2, MoSe2, and WSe2,” Opt. Express 21, 4908–4916 (2013).
[Crossref] [PubMed]

Y. Y. Hui, X. Liu, W. Jie, N. Y. Chan, J. Hao, Y.-T. Hsu, L.-J. Li, W. Guo, and S. P. Lau, “Exceptional tunability of band energy in a compressively strained trilayer MoS2 sheet,” ACS Nano 7, 7126–7131 (2013).
[Crossref] [PubMed]

D. Sercombe, S. Schwarz, O. Del Pozo-Zamudio, F. Liu, B. J. Robinson, E. A. Chekhovich, I. I. Tartakovskii, O. Kolosov, and A. I. Tartakovskii, “Optical investigation of the natural electron doping in thin MoS2 films deposited on dielectric substrates,” Sci. Rep. 3, 3489 (2013).
[Crossref]

T. Cheiwchanchamnangij, W. R. L. Lambrecht, Y. Song, and H. Dery, “Strain effects on the spin-orbit-induced band structure splittings in monolayer MoS2 and graphene,” Phys. Rev. B 88, 155404 (2013).
[Crossref]

2012 (5)

K. F. Mak, K. He, J. Shan, and T. F. Heinz, “Control of valley polarization in monolayer MoS2 by optical helicity,” Nat. Nanotech. 7, 494 (2012).
[Crossref]

G. Kioseoglou, A. T. Hanbicki, M. Currie, A. L. Friedman, D. Gunlycke, and B. T. Jonker, “Valley polarization and intervalley scattering in monolayer MoS2,” Appl. Phys. Lett. 101, 221907 (2012).
[Crossref]

T. Cao, G. Wang, W. Han, H. Ye, C. Zhu, J. Shi, Q. Niu, P. Tan, E. Wang, B. Liu, and J. Feng, “Valley-selective circular dichroism of monolayer molybdenum disulphide,” Nat. Commun. 3, 887 (2012).
[Crossref] [PubMed]

D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, “Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides,” Phys. Rev. Lett. 108, 196802 (2012).
[Crossref]

H. Peelaers and C. G. Van de Walle, “Effects of strain on band structure and effective masses in MoS2,” Phys. Rev. B 86, 241401 (2012).
[Crossref]

2011 (1)

S. Bertolazzi, J. Brivio, and A. Kis, “Stretching and breaking of ultrathin MoS2,” ACS Nano 5, 9703–9709 (2011).
[Crossref] [PubMed]

2010 (1)

R. Winkler and U. Zülicke, “Invariant expansion for the trigonal band structure of graphene,” Phys. Rev. B 82, 245313 (2010).
[Crossref]

Ager, J. W.

S. B. Desai, G. Seol, J. S. Kang, H. Fang, C. Battaglia, R. Kapadia, J. W. Ager, J. Guo, and A. Javey, “Strain-induced indirect to direct bandgap transition in multilayer WSe2,” Nano Lett. 14, 4592–4597 (2014).
[Crossref] [PubMed]

Ajayan, P. M.

H. Li, A. W. Contryman, X. Qian, S. M. Ardakani, Y. Gong, X. Wang, J. M. Weisse, C. H. Lee, J. Zhao, P. M. Ajayan, J. Li, H. C. Manoharan, and X. Zheng, “Optoelectronic crystal of artificial atoms in strain-textured molybdenum disulphide,” Nat. Commun. 6, 7381 (2015).
[Crossref] [PubMed]

Akinwande, D.

D. Akinwande, N. Petrone, and J. Hone, “Two-dimensional flexible nanoelectronics,” Nat. Commun. 5, 5678 (2014).
[Crossref] [PubMed]

Albrecht, M.

Amand, T.

G. Wang, A. Chernikov, M. M. Glazov, T. F. Heinz, X. Marie, T. Amand, and B. Urbaszek, “Colloquium: Excitons in atomically thin transition metal dichalcogenides,” Rev. Mod. Phys. 90, 021001 (2018).
[Crossref]

M. M. Glazov, E. L. Ivchenko, G. Wang, T. Amand, X. Marie, B. Urbaszek, and B. L. Liu, “Spin and valley dynamics of excitons in transition metal dichalcogenide monolayers,” Phys. Status Solidi B 252, 2349–2362 (2015).
[Crossref]

M. M. Glazov, T. Amand, X. Marie, D. Lagarde, L. Bouet, and B. Urbaszek, “Exciton fine structure and spin decoherence in monolayers of transition metal dichalcogenides,” Phys. Rev. B 89, 201302 (2014).
[Crossref]

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A. Castellanos-Gomez, R. Roldán, E. Cappelluti, M. Buscema, F. Guinea, H. S. J. van der Zant, and G. A. Steele, “Local strain engineering in atomically thin MoS2,” Nano Lett. 13, 5361–5366 (2013).
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R. Schmidt, I. Niehues, R. Schneider, M. Drüppel, T. Deilmann, Michael Rohlfing, S. M. d. Vasconcellos, A. Castellanos-Gomez, and R. Bratschitsch, “Reversible uniaxial strain tuning in atomically thin WSe2,” 2D Mater. 3, 021011 (2016).
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J. O. Island, A. Kuc, E. H. Diependaal, R. Bratschitsch, H. S. J. v. d. Zant, T. Heine, and A. Castellanos-Gomez, “Precise and reversible band gap tuning in single-layer MoSe2 by uniaxial strain,” Nanoscale 8, 2589–2593 (2016).
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R. Schmidt, I. Niehues, R. Schneider, M. Drüppel, T. Deilmann, Michael Rohlfing, S. M. d. Vasconcellos, A. Castellanos-Gomez, and R. Bratschitsch, “Reversible uniaxial strain tuning in atomically thin WSe2,” 2D Mater. 3, 021011 (2016).
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M. Feierabend, A. Morlet, G. Berghäuser, and E. Malic, “Impact of strain on the optical fingerprint of monolayer transition-metal dichalcogenides,” Phys. Rev. B 96, 045425 (2017).
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D. Lloyd, X. Liu, J. W. Christopher, L. Cantley, A. Wadehra, B. L. Kim, B. B. Goldberg, A. K. Swan, and J. S. Bunch, “Band gap engineering with ultralarge biaxial strains in suspended monolayer MoS2,” Nano Lett. 16, 5836–5841 (2016).
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Q. Zhang, Z. Chang, G. Xu, Z. Wang, Y. Zhang, Z.-Q. Xu, S. Chen, Q. Bao, J. Z. Liu, Y.-W. Mai, and et al., “Strain relaxation of monolayer WS2 on plastic substrate,” Adv. Funct. Mater. 26, 8707–8714 (2016).
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C. R. Zhu, G. Wang, B. L. Liu, X. Marie, X. F. Qiao, X. Zhang, X. X. Wu, H. Fan, P. H. Tan, T. Amand, and B. Urbaszek, “Strain tuning of optical emission energy and polarization in monolayer and bilayer MoS2,” Phys. Rev. B 88, 121301 (2013).
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Nano Res. (1)

Y. Wang, C. Cong, W. Yang, J. Shang, N. Peimyoo, Y. Chen, J. Kang, J. Wang, W. Huang, and T. Yu, “Strain-induced direct-indirect bandgap transition and phonon modulation in monolayer WS2,” Nano Res. 8, 2562–2572 (2015).
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Nanoscale (1)

J. O. Island, A. Kuc, E. H. Diependaal, R. Bratschitsch, H. S. J. v. d. Zant, T. Heine, and A. Castellanos-Gomez, “Precise and reversible band gap tuning in single-layer MoSe2 by uniaxial strain,” Nanoscale 8, 2589–2593 (2016).
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Nat. Commun. (4)

T. Cao, G. Wang, W. Han, H. Ye, C. Zhu, J. Shi, Q. Niu, P. Tan, E. Wang, B. Liu, and J. Feng, “Valley-selective circular dichroism of monolayer molybdenum disulphide,” Nat. Commun. 3, 887 (2012).
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H. Li, A. W. Contryman, X. Qian, S. M. Ardakani, Y. Gong, X. Wang, J. M. Weisse, C. H. Lee, J. Zhao, P. M. Ajayan, J. Li, H. C. Manoharan, and X. Zheng, “Optoelectronic crystal of artificial atoms in strain-textured molybdenum disulphide,” Nat. Commun. 6, 7381 (2015).
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C. Palacios-Berraquero, D. M. Kara, A. R.-P. Montblanch, M. Barbone, P. Latawiec, D. Yoon, A. K. Ott, M. Loncar, A. C. Ferrari, and M. Atatüre, “Large-scale quantum-emitter arrays in atomically thin semiconductors,” Nat. Commun. 8, 15093 (2017).
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K. F. Mak, K. He, J. Shan, and T. F. Heinz, “Control of valley polarization in monolayer MoS2 by optical helicity,” Nat. Nanotech. 7, 494 (2012).
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Phy. Rev. B (1)

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Phys. Rev. B (10)

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Figures (3)

Fig. 1
Fig. 1 Effect of uniaxial/biaxial strain on the degree of optical polarization of TMDs for compressive/tensile strain at different excess energies ∆E, as measured from the conduction band minimum.
Fig. 2
Fig. 2 Uniaxial strain dependence of A-exciton PL peak energy shift for monolayer TMDs, comparing our calculations (in blue) with experimental data (symbols) along with their best fit line (red-dashed). References: Conley et al. [5], Island et al. [21], Wang et al. [18], Zhang et al. [19], Schmidt et al. [20], Maniadaki et al. [24].
Fig. 3
Fig. 3 Biaxial strain dependence of A-exciton PL peak energy shift for monolayer TMDs, comparing our calculations (blue-solid) with experimental data (symbols) along with their best fit line (red-dashed). References: Lloyd et al. [11], Frisenda et al. [45].

Tables (2)

Tables Icon

Table 1 k.p parameters fi (eV), lattice constant a (Å) [28], 2D polarizability χ2D (Å) [31] for different TMDs.

Tables Icon

Table 2 PL peak redshift under uniaxial or biaxial strain in comparison with results from literature. Our results (this work) have both uniaxial strain/stress (i.e., ν : 0/0.37) cases with the values in parentheses corresponding those without the excitonic contribution.

Equations (17)

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H = [ ( f 0 + f 1 2 ) + ( f 3 + f 4 ) ( ε x x + ε y y ) f 2 a ( k x i k y ) + f 5 ( ε x x ε y y + 2 i ε x y ) f 2 a ( k x + i k y ) + f 5 ( ε x x ε y y + 2 i ε x y ) ( f 0 f 1 2 ) + ( f 3 f 4 ) ( ε x x + ε y y ) ] ,
Biaxial strain : ε y y = ε x x , ε x y = ε y x = 0 , Uniaxial strain : ε x x 0 , ε y y = ε x y = ε y x = 0 , Shear strain : ε y y = ε x x , ε x y = ε y x 0 , Uniaxial stress : ε y y = ν ε x x , ε x y = ε y x = 0 ,
E c / ν ( q ) = f 0 + f 3 ε H ± E g 2 1 + 4 [ r ( q , ε H ) ] 2 ,
r ( q , ε H ) f 2 a q f 1 + 2 f 4 ε H ,
| U c = ( x 1 x 2 ) , | U v = ( x 2 x 1 * ) ,
x 1 = e i ϕ 1 + r 2 ,
x 2 = r 1 + r 2 .
P u ( k ) m 0 U c | H ^ k u | U v ,
P ± ( k ) = m 0 f 2 a 2 U c | σ ^ ± | U v .
η ( k ) | P + ( k ) | 2 | P ( k ) | 2 | P + ( k ) | 2 + | P ( k ) | 2 .
η ( q , ε H ) = 1 [ r ( q , ε H ) ] 4 1 + [ r ( q , ε H ) ] 4 ,
m c / v * = 2 ( 2 E c / v k 2 | k x 0 , k y 0 ) = ± 2 ( f 1 + 2 f 4 ε H ) 2 ( f 2 a ) 2 ,
H X = 2 D 2 2 μ + V 2 D ( ρ ) .
V 2 D ( ρ ) = π ( κ a + κ b ) ρ 0 [ H 0 ( ρ / ρ 0 ) Y 0 ( ρ / ρ 0 ) ] ,
Ψ X ( ρ ; λ ) = 2 π λ 2 exp ( ρ / λ ) .
V ( λ ) = 2 π ρ 0 λ 2 0 [ H 0 ( ρ / ρ 0 ) Y 0 ( ρ / ρ 0 ) ] exp ( 2 ρ / λ ) ρ d ρ .
k x k x + α ( ε y y ε x x ) , k y k y + α 2 ε x y ,

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