Threebody systems in novel twodimensional materials Roman Ya

Three-body systems in novel two-dimensional materials Roman Ya. Kezerashvili New York City College of Technology City University of New York, USA EFB 24, Surrey University, UK 1 -6 September 2019 1

Zoo of 2 D Materials “What could we do with layered structures with just the right layers? ” asked Richard Feynman in his famous 1959 lecture, “There’s plenty of room at the bottom. ” Physics Today 69, 9, 38 (2016) Graphene Phosphorene TMDC Mo. S 2, Mo. Se 2 , Mo. Te 2 WS 2, WSe 2, WTe 2 Germanene 2 D MATERIALS Stanene TMTC MX 3 (M = Ti, Zr; X = S, Se) Hexagonal boron nitride h-BN Silicene These materials are the two-dimensional allotrope of the corresponding chemical element Layered materials are characterized by extended crystalline planar structures held together by strong in-plane covalent bonds and weak out-of -plane van der Waals forces. EFB 24, Surrey University, UK 1 -6 September 2019

Few-body Systems in Condensed Matter Versus in Nuclear Physics While the excitonic complexes like excitons, trions, biexcitons in condensed matter physics are very similar to the two- three- and fourbody bound systems in nuclear physics, there are major differences: (i) Excitonic complexes are excited in bulk materials as 3 D systems, in novel atomically thin materials they are 2 D systems and in nanowires, nanorods and nanotubes these complexes are considered as 1 D systems; (ii) The reduction of dimensionality itself necessitates a change to the formalism, and requires the modification of the bare Coulomb potential to account for non-local screening effects. The screening effects, resulting from the host lattice, make the Coulomb force between charge carriers much weaker than in atomic systems or nuclear systems; (iii) Band effects make the effective masses of the electrons and holes smaller than the bare electron mass. EFB 24, Surrey University, UK 1 -6 September 2019 3

Two Dimensional Transition Metal Dichalcogenide Atomic layers of hexagonal Transition Metal Top view on the hexagonal lattice Dichalcogenides (TMDC) represent a new of lying in the xy plane class of systems y x Positive and Negative Trion Complexes EFB 24, Surrey University, UK 1 -6 September 2019

A new cousin of graphene Upon the absorption of a photon from an optical field, a valence band electron can be excited to the conduction band, and the vacancy left behind in the valence band is usually described as a hole which is a quasiparticle carrying positive charge. Kormányos et al. , 2 D Mater. 2, 2015. a new cousin of graphene EFB 24, Surrey University, UK 1 -6 September 2019 5

Trions The special interest presents Trion which is a bound state of an interacting electrons and holes produced in semiconductor monolayer or heterstructure via, for example, the laser pumping. Positively charged Trion e Negatively charged Trion h h e Trion can be formed in bulk materials (3 D trions), monolayer semiconductor (2 D trions) and nanowires or nanotubes (1 D trions) EFB 24, Surrey University, UK 1 -6 September 2019

3 D 2 D 1 D Kezerashvili, Few-Body Syst. 60, (2019) EFB 24, Surrey University, UK 1 -6 September 2019

The 2 D screened electrostatic interaction potential Ritova-Keldysh potential Rytova, Proc. MSU Phys. , Astron. 3, 30 (1967). Keldysh, JETP Lett 29, 658 (1979). EFB 24, Surrey University, UK 1 -6 September 2019

Effect of Reduction of Dimensionality e Coulomb potential h in 3 D space in 2 D space As a result of reduction of dimensionality degrees of freedom decrease and the ground state energy increases by factor 4 EFB 24, Surrey University, UK 1 -6 September 2019

Trion e- h h e- The nonrelativisic trion Hamiltonian is given by e- h To separate out the center-of-mass and the relative motion of three particles let us intoroduce a set of mass-scaled Jacobi coordinates for the partition i as follows: e- e- h EFB 24, Surrey University, UK 1 -6 September 2019

3 D Trion where P is the permutation operator for two identical particles. The Faddeev equations in 3 D conguration space can be written in the form of system of second order differential equations The Hamiltonian H 0 is the operator of kinetic energy written in terms of corresponding Jacobi coordinates, while VAA and VAB are the potentials of the pairwise interactions between the particles. EFB 24, Surrey University, UK 1 -6 September 2019

Binding Energy of Trion in Bulk Semiconductors Positively charged trion in 3 D bulk materials is unbound Filikhin, Kezerashvili, Vlahovic, PLA 382 (2018) 787 EFB 24, Surrey University, UK 1 -6 September 2019 e h h

The origin of binding energies difference for t trions . To analyze the origin of the difference of binding energies for the trions let us write the Schrödinger equation for the trion in the system of reference relative to the non-identical particle e e h Let's introduce the interaction between two identical particle as where the parameter a controls the strength of this interaction. e EFB 24, Surrey University, UK 1 -6 September 2019 h h

Eigen functions for eeh and hhe systems The negative trion has more extended distribution within about 300 x 400 A, than less extended distribution within 100 x 100 A for the positive trion. The contour plots of the Faddeev components U and W for eeh and hhe systems EFB 24, Surrey University, UK 1 -6 September 2019

Two Approaches The low-energy effective two-band single-electron Hamiltonian in the form of a spinor with a gapped spectrum for TMDCs in the k · p approximation: D. Xiao, et al. , PRL. 108, 196802 (2012). The energy gap Δ = 1. 6 -- 1. 8 e. V The spin splitting 2λ = 0. 1 -- 0. 5 e. V Qasi-Relativistic approach Berman and Kezerashvili, PRB 93, 245410 (2016) A nonrelativistic approach within the framework of a potential model using effective mass approximation Berman, Kezerashvili, PRB 96 , 094502 (2017) Mak, et al. , Nat. Materials 12 (2013) 207 Significant spin-orbit splitting in the valence band leads to the formation of TMDC layers two types of excitons: Type A excitons are formed by spin-down holes from the valence band spin-up electrons from the conduction band Type B excitons are formed by spin-up holes from the valence band spin-down electrons from the conduction band. EFB 24, Surrey University, UK 1 -6 September 2019 15

2 D Trions To obtain a solution of the Schrodinger equation for the 2 D trion we employ hyperspherical coordinates in 4 D configuration space. Details of the method will be given in Tsiklauri’s talk on Tuesday EFB 24, Surrey University, UK 1 -6 September 2019

Analytical Approach The effective potential energy can be evaluated by employing the Reynal-Revai unitary transformation Kezerashvili, Tsiklauri APS 2019 EFB 24, Surrey University, UK 1 -6 September 2019 17

By rescaling variable r and introducing a new function this equation can be reduced to the known Weber's equation. The solution of this equation in a first approximation is a parabolic cylinder functions Dn The eigenvalues that correspond to these eigenfunctions are EFB 24, Surrey University, UK 1 -6 September 2019

2 D Ritova-Keldysh potential EFB 24, Surrey University, UK 1 -6 September 2019 19

Binding Energies of 2 D Trionss Filikhin, Kezerashvili, Tsiklauri, Vlahovic, Nanotechnology 29 (2018) 124002 EFB 24, Surrey University, UK 1 -6 September 2019

Binding Energies of 2 D Trionss In monolayer TMDC material negatively and positively charged trions are bound In monolayer TMDC binding energy of positively charged trions is always greater than negatively charged trions Filikhin, Kezerashvili, Tsiklauri, Vlahovic, Nanotechnology 29 (2018) 124002 EFB 24, Surrey University, UK 1 -6 September 2019

Material 33. 7 26 32. 0 33. 8; 32 32. 8 28. 2 21 27. 7 28. 4; 31 27. 6 33. 8 26 28; 33. 1 34. 0; 30 33. 1 29. 5 22 28. 5 29. 6; 27 28. 3 Experiment 1. Mak et al. , Nat. Mater. 12, 207 (2013) 2. Zhang, et al. , ACS Nano 9, 8514 (2015) 3. Christopher, at al. , Sci. Rep. 7, 14062 (2017) 4. Jones et al. , Nat. Nanotechnol. 8, 634 (2013) 5. Singh et al. , Phys. Rev. Lett. 112, 21680 (2014) 6. Plechinger et al. , Phys. Status Solidi RRL 9, 457 (2015) 7. Shang, et al. , ACS Nano 9, 647 (2015) 8. Zhu, et al. , Proc. Natl. Acad. Sci. USA 111, 11606 (2014) 9. Jones et al. , Nat. Nanotechnol. 8, 634 (2013) 10. Wang et al. , Phys. Rev. B 90, 075413 (2014 11. Courtade et al. , Phys. Rev. B 96, 085302 (2017) Theory 12. Zhang, Kidd, Varga, Nano Lett. 15, 7002 (2015), PRB 93, 125423 (2016) 13. Berkelbach, Hybertsen, Reichman, PRB 88, 045318 (2013) 14. Velizhanin, Saxena, PRB 92, 195305 (2015) 15. Kylanpaa, Komsa, PRB 92, 205418 (2015). 16. Mayers, et. al. , PRB 92, 161404 (2015) 17. Szyniszewski, et al. , PRB 95, 081301 (2017) 18. Kezerashvili, Tsiklauri, Few-Body Syst. 58, 18 (2017) EFB 24, Surrey University, UK 1 -6 September 2019 22

Intravalley and Intervalley Trions Schematic representation of the low-energy band structure for 2 D TMDC material and formation of the intravaley and intervaley trions In the negatively charged trion the extra electron can reside either in the same valley or in a different valley WSe₂ Freestanding Air, ε=1 Supported Si. O₂, ε=3. 9 Encapsulated h-BN, ε=4. 9 r₀, A X⁻ , L=0 38. 7 34. 6 29. 4 45. 11 X⁺ L=0 28. 3 23. 4 20. 6 45. 11 X⁻ , L=1 30. 3 25. 7 22. 8 45. 11 EFB 24, Surrey University, UK 1 -6 September 2019 23

Theory of Charge Carriers in Buckled 2 D Materials J. Zhao, H. Liu, et al. , Prog. Mat. Sci. 83, 24 -151 (2016). • Hamiltonian of charge carriers near the K/K’ points given as: • Dispersion relation: • Electric field-dependent charge carrier mass: • Excitons formed from the large gap are A excitons • Excitons formed from the small gap are B C. J. Tabert and E. J. Nicol, Phys. excitons Rev. B 89, 195410 (2014). EFB 24, Surrey University, UK 1 -6 September 2019

Excitons in Xene monolayers • Brunetti, Berman, and Kezerashvili, Phys. Rev. B 98, 125406 (2018). EFB 24, Surrey University, UK 1 -6 September 2019

The Schrödinger equation for the anisotropic trion Phosphorene monolayer has an anisotropic band structure. Consequently, the charge carriers, an electron and hole, have anisotropic effective mass. orine Trions in Phosph • EFB 24, Surrey University, UK 1 -6 September 2019

Summary Reduction of dimensionality affects on kinetic energy and modifies the Coulomb potential due to the screening The binding energies of the trions are calculated for different bulk materials based on the Faddeev equation for AAB system in configuration space. It was found that the binding energy of negative trion is relatively small, while the positive trion is unbound. Calculations within the method of hyperspherical harmonics show that in 2 D monolayer TMDC semiconductors due to the reduced dimensionality negatively and positively charged trions are bound and the binding energy of positive trion is always greater than for the negative trion. A study of trions in 2 D monolayers within the method of the Faddeev equation requires an extension of this formalism to 2 D configuration or momentum spaces. EFB 24, Surrey University, UK 1 -6 September 2019 27

I wish to thank my colaborators Igor Filikhin Shalva Tsiklauri Branislav Vlahovic