2D Materials with Strong Spin-orbit Coupling: Topological and Electronic Transport Properties Artem Pulkin California Institute of Technology (Caltech), Pasadena, CA 91125, US Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland pulkin@caltech.edu November 1, 2017 Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 1 / 32
1 Topological insulators Spin-orbit coupling 2 Two-dimensional transition metal dichalcogenides Quantum spin Hall phase in 2D TMDs Edges and topological edge modes Structural phase boundaries Line defects and transport of electronic spin 3 Summary Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 1 / 32
Topological insulators
Timeline 1975-1981 (Nobel prize 1985) Quantum Hall effect 2005 Quantum spin Hall (QSH) effect, topological order 2006 First realization in HgTe quantum wells Materials? Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 2 / 32
The QSH effect in 2D Kane-Mele model H = [ t NN c c + ] iνt NNN c c spin NN NNN Tight-binding model of graphene E NNN E k k Spin-orbit coupling Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 3 / 32
Spin-orbit coupling Increases with atomic number Z: SO Z 2 n 3 l(l + 1) Valence shells in carbon: SO < mev Too small for spectroscopic and transport measurements Compare: HgTe quatnum wells: 10-100 mev (Bernevig et al. Science 314, 1757 (2006)) Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 4 / 32
Are there any 2D materials with a large spin-orbit coupling?
Two-dimensional transition metal dichalcogenides TMD = MX 2, M = {Mo, W}, X = {S, Se, Te} 2H phase K Γ K' = = =, stable phase (except for WTe 2 ), semiconductor in a hexagonal lattice; large spin-orbit splitting in the valence band (150 mev in MoS 2, up to 460 mev in WSe 2 ): spin-polarized states; spin-valley coupling Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 5 / 32
2D TMDs M. Chhowalla, et al., Nat Chem 5, 263275 (2013) Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 6 / 32
Applications Transistors LEDs Radisavljevic et al., Nat Nano 6 no. 3 147-150 (2011) Solar cells Bernardi el al., Nano Lett. 13 no. 8 3664-3670 (2013) Artem Pulkin (Caltech, EPFL) Amani et al., Science 350 no. 6264 1065-1068 (2015) Transport in 2D materials November 1, 2017 7 / 32
Is it possible to drive 2D TMDs into the QSH phase?
QSH effect in 2D TMDs 1T phase same material in a metastable structure hexagonal symmetry breaking rectangular unit cell formation of dimerization chains Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 8 / 32
Electronic properties of the 1T structural phase 0.3 Energy (ev) MoS 2 0.0 48 mev MoSe 2 35 mev MoTe 2 0.3 0.3 Energy (ev) WS 2 0.0 WSe 2 29 mev WTe 2 0.3 0.5 0.0 0.5 k y ( 2 b ) 0.5 0.0 0.5 k y ( 2 b ) 0.5 0.0 0.5 k y ( 2 b ) spin-degenerate bands (inversion + time reversal symmetries) semimetals or semiconductors with a 10 mev-order band gap topological band inversion Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 9 / 32
QSH phase in 1T 2D TMDs band inversion at Γ quantum spin Hall (QSH) topological phase Qian et al., Science 346, 1344-1347 (2014) Choe et al., Phys. Rev. B 93, 125109 (2016) Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 10 / 32
Is the QSH phase in 1T TMDs robust against lattice deformations?
Electronic structure in equilibrium At the density functional theory level (GGA): MoS 2, MoSe 2, WSe 2 have a band gap; WS 2, MoSe 2, WTe 2 are semimetals; k x k x k x WS 2 MoTe 2 WTe 2 k y ky ky Hole, electron pockets in semimetallic 1T TMDs Close to semiconducting phase transition phase? Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 11 / 32
Band gap under strain in 1T 2D TMDs b (Å) b (Å) a 3.3 3.2 3.1 c 3.3 3.2 3.1 semiconductor semimetal PBE 57 mev QSH MoS 2 b 5.5 5.6 5.7 5.8 5.9 6.0 d semiconductor trivial 63 mev QSH PBE semimetal 5.5 5.6 5.7 5.8 5.9 6.0 a (Å) WS 2 3.4 3.3 3.2 3.4 3.3 3.2 semiconductor semimetal PBE 92 mev MoSe 2 5.7 5.8 5.9 6.0 6.1 6.2 semiconductor semimetal PBE 5.7 5.8 5.9 6.0 6.1 6.2 a (Å) 120 mev WSe 2 120 105 90 75 60 45 30 15 0 E g, mev Pulkin & Yazyev Journal of Electron Spectroscopy and Related Phenomena 219 72-76 (2017) Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 12 / 32
Conclusions 1T -TMDs posess a topological band inversion at the Gamma point; MoS 2, MoSe 2, WSe 2 also have a positive band gap topological insulators; The size of the band gap is sensitive to lattice deformations; Both semiconductor-to-semimetal and topological phase transitions can be induced by strain Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 13 / 32
How do topological edge states in 1T TMDs look like?
Recall: Kane-Mele model Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 14 / 32
Edges in 1T TMDs The zigzag edge 1,2 = neutral m1,m2 = metal-rich c1,c2 = chalcogen-rich Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 15 / 32
Edges in 1T TMDs m2 is always preferred for metal-rich conditions; 2 is usually preferred for chemically balanced conditions; c1, c2 are equally preferred for calcogen-rich conditions Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 15 / 32
Electronic properties of edges in 1T TMDs Energetically preferred terminations are considered; Method: DFT + Green s function (NEGF) M-rich Balanced -1 1 C-rich 0 Energy (ev) Energy (ev) Energy (ev) MoS 2 1 MoTe 2 MoSe 2 WS 2 WTe 2 WSe 2 0 1 2 2 2 2 2 m2 m2 m2 m2 m2 m2 c2 c1 c1 c2 c2 c1 0-1 1-1 0 10 k y ( /b) Artem Pulkin (Caltech, EPFL) 10 k y ( /b) 10 k y ( /b) 10 k y ( /b) Transport in 2D materials 10 k y ( /b) 1 k y ( /b) November 1, 2017 16 / 32
Is topological protection of the ballistic transport regime possible?
Topologically protected transport Protected Unprotected topological protection protection against back-scattering 0 A single spin channel is shown Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 17 / 32
Ballistic transport along 1T -WSe 2 edges 0.1 Energy (ev) Energy (ev) 0.0-0.1 0.1 0.0-0.1 Y WSe 2 (2) WSe 2 (c2) Y non-uniform dispersion of edge modes; protected transport is possible in a narrow energy region and only at specific edges of 1T -WSe 2 Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 18 / 32
What is available experimentally?
Experimental observations of the 1T phase in 2D WSe 2 Defect-free bulk Images are courtesy of Miguel M. Ugeda, nanogune, San Sebastian, Spain Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 19 / 32
Experimental observations of the 1T phase in 2D WSe 2 Defect-free bulk? Regular periodic edges Images are courtesy of Miguel M. Ugeda, nanogune, San Sebastian, Spain Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 19 / 32
Experimental observations of the 1T phase in 2D WSe 2 Defect-free bulk? Regular periodic edges! Structural phase boundaries Images are courtesy of Miguel M. Ugeda, nanogune, San Sebastian, Spain Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 19 / 32
Structural phase boundary is topologically non-trivial
1T -WSe 2 interface states 1T' 2H Images are courtesy of Miguel M. Ugeda, nanogune, San Sebastian, Spain Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 20 / 32
Can theory confirm the presence of interface modes? Are these modes topological?
Atomic structures of phase boundaries in WSe 2 Construct model: choose the zigzag edge of the 2H phase (2x); choose the zigzag edge of the 1T phase (4x, half discarded); concatenate Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 21 / 32
Electronic structure of phase boundaries in WSe 2 Method: DFT + NEGF E (ev) 1 0 1 2 3 4 E (ev) -1 1 0 5 6 7 8-1 Y K K' Y Y K K' Y Y K K' Y Y K K' Y 1, 3 and 6, 8 are similar Multiple spectroscopic signatures No topological protection of transport Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 22 / 32
Conclusions Edge modes of real topological materials require ab-initio description; DFT+NEGF calculations reveal multiple spin-polarized modes spanning a large energy region in 1T -TMDs; The dispersion of edge modes is consistent with the QSH phase; Specific 1T -WSe 2 edges are suitable for ballistic charge carrier transport protected against back-scattering; Regular topological phase boundary is accessible experimentally! Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 23 / 32
Topological edge states carry spin-polarized current in a non-magnetic media applications for spintronics and quantum computing. Other examples in 2D?
Idea 2H phase K Γ K' = = =, Discriminate valleys in some physical process spin-valley coupling induce spin polarization Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 24 / 32
Example: optical excitation of charge carriers in semiconducting 2D MoS 2 Cao et al. Nat Commun 3 887 (2012) Also: Mak et al. Nat Nano 7 no. 8 494-498 (2012); Zeng et al., Nat Nano 7 no. 8 490-493 (2012) Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 25 / 32
Is it possible to achieve valley polarization in an all-electric manner?
Idea: valley-polarized transport across a line defect D. Gunlycke and C.T. White PRL 106, 136806 (2011), graphene Symmetry: T ν (θ) = T ν ( θ) T ν (θ), Polarization: P(θ) = T ν=1 T ν= 1 T ν=1 + T ν= 1 sin θ T ν - transmission probability; ν = ±1 - valley; θ - group velocity angle In TMDs valley ν and spin σ are coupled! Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 26 / 32
Results of simulations: line defects in 2H-MoS2 Pulkin & Yazyev, Phys. Rev. B 93 041419(R) (2016) valley and spin filtering with strong energy dependence spin-orbit transport gap for holes in IDB1 Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 27 / 32
Poor/no transport across inversion domain boundaries: why?
Ballistic transport periodic Line defect MoS2 MoS2 semi-infinite Take into account: conservation of energy ( = ballistic transport); conservation of pseudo-momentum ( = periodic line defect); conservation of spin ( = planar non-magnetic defects) Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 28 / 32
Ballistic transport periodic Line defect MoS2 MoS2 semi-infinite Project the bands onto 1D Brillouin zone (BZ) of the defect (size 2π/d) d - periodicity of the defect a - TMD lattice constant (3-4 Å) Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 28 / 32
Transport gap E t Spin match (no gap) e.g. sulfur vacancy line Spin mismatch (transport gap) e.g. inversion domain boundary K K K K K K K K Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 29 / 32
Transport gap E t : criterion Other line defects with a transport gap? Defect periodicity vector d = n L a 1,L + m L a 2,L = n R a 1,R + m R a 2,R d = (1, 0) d = a = 0.3 nm d = (1, 0) L = ( 1, 0) R d = a = 0.3 nm Komsa et. al. PRB 88, 035301 (2013); Zhou et. al. Nano Lett. 13, 2615-2622 (2013) d = (3, 5) L = (5, 3) R d = 7a = 2.2 nm Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 30 / 32
Transport gap E t : criterion Spin match (no gap) e.g. sulfur vacancy line Spin mismatch (transport gap) e.g. inversion domain boundary K K K K K K (n L m L ) mod 3 = (n R m R ) mod 3 0 K K 0 (n L m L ) mod 3 (n R m R ) mod 3 0 Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 31 / 32
Summary 2D TMDs are prospective materials for modern electronics, spintronics and topological electronic structure community; There is a large experimental effort towards confirming the QSH phase in 1T -TMDs; Line defects found in these materials can be employed for spin-selective transport both along or across the defect, with or without relying on topological arguments; In either case, the spin polarization of charge carrier current exists without net magnetization and macroscopic magnetic fields: fewer spin relaxation channels and the increased spin lifetime in the material Artem Pulkin (Caltech, EPFL) Transport in 2D materials November 1, 2017 32 / 32
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