Giant Valley Splitting and Valley Polarized Plasmonics in Group V

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Giant Valley Splitting and Valley Polarized Plasmonics in Group-V Transition-Metal Dichalcogenide Monolayers Jian Zhou, and Puru Jena J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.7b02507 • Publication Date (Web): 12 Nov 2017 Downloaded from http://pubs.acs.org on November 13, 2017

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The Journal of Physical Chemistry Letters

Giant Valley Splitting and Valley Polarized Plasmonics in Group-V Transition-Metal Dichalcogenide Monolayers Jian Zhou*, Puru Jena Physics Department, Virginia Commonwealth University, Richmond, Virginia 23284, USA Abstract Two-dimensional (2D) group-VI transition-metal dichalcogenides (TMDs) provide a promising platform to encode and manipulate quantum information in the valleytronics. However, the two valleys are energetically degenerate, protected by time-reversal symmetry (TRS). To lift this degeneracy, one needs to break the TRS by either applying an external magnetic field or using a magnetic rare-earth oxide substrate. Here, we predict a different strategy to achieve this goal. We propose that the ferromagnetic groupV TMD monolayer, in which the TRS is intrinsically broken, can produce a larger valley and spin splitting. A polarized ZnS(0001) surface is also used as a substrate, which shifts the valleys to the low energy regime (near the Fermi level). Moreover, by calculating its collective electronic excitation behaviors, we show that such system hosts a giant valley polarized THz plasmonics. Our results demonstrate a new way to design and use valleytronic devices, which are both fundamentally and technologically significant. TOC Graphic

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In addition to conventional information storage and manipulation techniques which are based on the charge and/or spin of an electron, a novel electronic degree of freedom, valley, has received considerable attention in the past decade. Exploiting the valley degree of freedom at the microscopic level can have important implications in nextgeneration information technology. The most studied system in this field is a honeycomb lattice where the two sublattices are different.1-7 Such broken inversion symmetry triggers a pair of valleys at the corners of the first Brillouin zone, denoted as K and K' (= –K). The discovery of two-dimensional (2D) transition-metal dichalcogenide (TMD, e.g. MoS2) monolayer family has provided a convenient material platform to study valley polarized and valley contrasting physics. In this system, one views the Mo and the vertically aligned S pair as two different sublattices.8 For example, recent theoretical and experimental studies have proved that the carriers of different valleys in a 2H-MoS2 monolayer can be selectively excited using circularly polarized light.9-13 Although the two valleys are inequivalent, in the freestanding MoS2 (and other group-VI TMD) monolayer, they are energetically degenerate due to time-reversal symmetry (TRS). In order to control and manipulate the two valleys individually for future memory and logic applications, lifting of the valley degeneracy in these materials will be of great interest and fundamental importance. To achieve this goal, one needs to break the TRS, so that EK ≠ EK'. Previous studies have provided two strategies to break the TRS in a group-VI TMD (such as WSe2, MoSe2, or MoTe2) monolayer. One is to apply an external magnetic field, which can induce a small valley splitting at the band edge (∼0.2 meV/T);14-17 another approach is to use the interfacial magnetic exchange field from a ferromagnetic substrate.18 Theoretically, it has been proved that a MoTe2 monolayer, once deposited on a rare-earth oxide (EuO) substrate, shows a giant valley (spin) splitting of ~170 (500) meV.19,20 This scheme has been experimentally verified using a WSe2 monolayer on a EuS substrate, which showed a valley splitting of 2.5 meV/T.21 Unfortunately, the valence band, with much larger valley splitting than conduction band, is well below the Fermi level. Nevertheless, these works provide great incentive to study the nondegenerate valley nondegenerate TMD monolayers, supported on semiconducting surfaces. However, all of these studies are limited to group-VI TMDs and the TRS is

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broken extrinsically. It is important to extend this study to other group TMD systems with new strategies. In this work, we propose an “inverted” approach to break the TRS and lift the valley degeneracy. Instead of using group-VI TMD monolayers, we focus on the magnetic group-V TMD monolayer, which intrinsically breaks the TRS. This allows us to use a nonmagnetic substrate to support the TMD monolayer, which further enhances the Rashba effect and the valley/spin splitting. We show that the substrate is important as it adjusts the band energy and moves the valleys close to the Fermi level. To demonstrate it, we choose a 2H-TaTe2 as the exemplary TMD monolayer, and use the (0001) surface of ZnS as the substrate with a small lattice mismatch (~2%). Note that group-V TMD monolayers can be stable in their 2H phase22-25 and layered transition metal ditelluride have also been synthesized.26,27 Thus, 2H-TaTe2 is also realizable. By performing full relativistic first-principles calculations, we find that this system possesses a giant valley splitting of over 320 meV. Most remarkable, by simulating its electrostatic screening behavior, we predict a remarkable valley polarized THz plasmonics in this system. Therefore, our discovery provides the valley degree of freedom a new dimension to the collective excitations in 2D TMD nanomaterials. The first-principles calculations are based on density functional theory (DFT) where the exchange correlation interaction is treated within the framework of generalized gradient approximation (GGA). We used the Perdew-Burke-Ernzerhof (PBE) functional28 as implemented in the Vienna Ab initio Simulation Package (VASP).29 The core and valence electrons are treated using projector-augmented wave (PAW) method30 and a plane wave basis set with a cutoff energy of 400 eV, respectively. In order to incorporate the strong correlation effect, we add an effective Hubbard U of 2 eV for the Ta-d orbitals.31,32 According to our test calculations, the specific value of U does not change our main results. The ZnS(0001) surface is modelled by a six-layer Zn-terminated slab. The bottom dangling bonds are saturated by adding pseudo-hydrogen atoms with a fractional charge of 0.5 electrons. We test the effect of slab thickness and it does not change our main conclusion. A vacuum space of 15 Å along the z-direction is used to avoid artificial interactions between neighboring images. The first Brillouin zone is

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represented by a Γ-centered Monkhorst-Pack k-mesh33 of (15×15×1). All the coordinates except the bottom layer of the ZnS slab are fully optimized without symmetry constraints using conjugated gradient method. The convergence criteria of total energy and force component are set to 1×10–4 eV and 0.01 eV/Å, respectively. The optimized geometric structure of a TaTe2 monolayer deposited on the ZnS(0001) surface is shown in Figure 1a. The ground state corresponds to an AA stacking with the Ta and Te atoms reside on top of Zn and S sites, respectively. The equilibrium distance between the Te and Zn is 2.7 Å. We use density functional perturbation theory34 to calculate its phonon dispersion (Supporting Information, Figure S1). No imaginary frequencies have been found, confirming its dynamical stability. We also performed Bader’s charge analysis35 which reveals that 0.13 electron per unit cell is transferred from the ZnS surface to TaTe2 (corresponding to a 1.02×1014 cm–2 electron doping). The Ta-d electrons show a clear spin polarization, which breaks the TRS intrinsically (Figure 1b). Consistent with previous studies on freestanding TaTe2 monolayer,32,36 we find that the ferromagnetic spin order of TaTe2 is energetically lower than antiferromagnetic coupling. Thus, in the following, we will focus on its ferromagnetic state.

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Figure 1. (a) Geometric structure (left panel, side view; right panel, top view) of a TaTe2 monolayer deposited on a Zn-terminated ZnS(0001) surface. (b) Iso-surface of spin density which is mainly located on the Ta atoms. (c) First Brillouin zone with high symmetry k-points.

The direct band gap of a 2H-TMD monolayer opens at the K and K' valleys (Figure 1c), which are well separated in the momentum space with suppression of intervalley scattering. Before performing DFT calculations, we briefly analyze its energy spectra using low-energy k·p method.3,18,19 Using the basis set functions | = | and

=

√

+ (each composed of two spin components, η = ±1 denotes

the valley index, and subscripts v and c indicate valence and conduction band, respectively), the four-band Hamiltonian in the vicinity of valleys reads (to first order in k)

δ 1 Hˆ 0 = vF (ησˆ x k x + σˆ y k y ) sˆ0 + σˆ z sˆ0 −  mc (σˆ 0 + σˆ z ) + mv (σˆ 0 − σˆ z )  sˆ z . 2 2

(1)

Here vF is the Fermi velocity, and and ̂ (i = 0, x, y, z) are Pauli matrices for the two basis set and spin degrees of freedom, respectively. The first term is the graphene-like band dispersion.37 The energy gap between the two basis sets is mimicked by δ, and the magnitude of spin splitting in valence (conduction) band is mv (mc). Here, we assume that both mv and mc are positive (ferromagnetic spin-up ground state) and satisfies δ > (mv + mc). In Figure 2a, we give a schematic plot of the ferromagnetic low energy band structure. The K and K' valleys show the same band spectra. The valence (conduction) band corresponds to σz = –1 (+1). The energy gap between spin-down valence and spinup conduction band is δ – mv – mc. In order to lift the degeneracy of the two valleys, we include an ad hoc intrinsic intraatomic (L·S) and Rashba form of spin-orbit coupling (SOC)38

1 Hˆ SO = − η λc (σˆ 0 + σˆ z ) + λv (σˆ 0 − σˆ z )  sˆz 2 , ˆ H R = λR (ησˆ x sˆ y − σˆ y sˆx )

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(2)


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where λc,v are the intrinsic SOC coefficients for valence and conduction bands, respectively, and λR is Rashba coefficient. The low-energy band profile of Ĥ0 + ĤSO + ĤR is shown in Figure 2b. Note that the out-of-plane spin component sz is no longer a good quantum number. Since the conduction band is mainly contributed by dz2, the intrinsic SOC coefficient of λc is nearly zero.8 Roughly speaking, the SOC term decreases (increases) the spin-up (down) valence band energy at the K valley, while the opposite effect occurs at the K' valley. Therefore, the SOC lifts the valley degeneracy with ∆VB (∆CB) (see Figure 2b) of ∆ VB = EVB,↓ ( K ) − EVB,↓ ( K ′ ) ≈ 2λv + ∆ CB

4λR2 δ − mc − mv + λc + λv

4λR2 = ECB,↑ ( K ′ ) − ECB,↑ ( K ) ≈ 2λc + δ − mc − mv + λc + λv

6


.

(3)

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Figure 2. (a) and (b), schematic low-energy band spectra in the vicinity of K and K' valleys. (c) and (d), DFT calculated band structure along the high symmetry k-path. Blue and red colors represent spin-up and spin-down channels, respectively. (a) and (c) are band structures without including SOC, and (b) and (d) are based on fully relativistic Hamiltonian.

Since the valley splitting occurs much more in the valence band than that in the conduction band, one needs to shift the valence band close to the Fermi level. The freestanding group-V TMD monolayer only contributes one d-electron in the low-energy bands, similar to that in ferrovalley VSe2 monolayer.39 Thus, the Fermi level lies below the valley gap in the system. In order to move the valley energy down close to the Fermi level, the polarized substrate ZnS(0001) surface provides an effective electric field which adds an additional term in the Hamiltonian

Hˆ sub = −U subσˆ 0 sˆ0 ,

(4)

where Usub > 0 is a constant. Thus, the Fermi level lies across the VB band, giving the opportunity to study the valley polarization and valley splitting at low energy. In Figures 2c and 2d we plot the DFT calculated band structure of TaTe2 monolayer on the ZnS(00001) surface, without and with SOC interactions, respectively. The semiconducting ZnS surface opens a large band gap around the Fermi level. In contrast to previous MoTe2 on EuO(111) case where the VB edges are deeply below the Fermi level (around –0.9 eV),19,20 here, the VB edges are above the Fermi level and lie in lower energy range (around 0.1 eV). This is because in both cases the polarized substrate adds the Ĥsub to the Hamiltonian so that the bands move downward. We find that the valley splitting in our system is also giant with ∆VB = 324 meV and ∆CB = 15 meV. The former value is around twice of that in the MoTe2@EuO(111) case (174 meV), while the latter value is half of that in MoTe2@EuO(111) system (30 meV). The spin splitting parameters are ∆v,K = 752 meV, ∆v,K' = 73 meV, ∆c,K = 642 meV, and ∆c,K' = 651 meV. The fitted parameters are δ = 0.747 eV, mv = 0.177 eV, mc = 0.337 eV, λv = 0.199 eV, and λc = 0.001 eV, and λR = 0.024 eV. We also calculated effective mass, m* of VB and CB at the two 7



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∗ valleys. Regardless of small anisotropy in the xy plane, the fitted values are , = 0.37

∗ m0, ,

"

∗ ∗ = 0.18 m0, , = 0.39 m0, and ,

"

= 0.16 m0. Here m0 is the electron rest

mass. The effective masses at the K valley are heavier than those at the K' valley. Since the valence band valley splitting ∆VB is much larger than the conduction band valley splitting ∆CB (λv >> λc ~ 0), we will focus on the valence band splitting which is close to the Fermi level. This makes our predicted system more applicable than the MoTe2 on EuO where the valence band lies ~ 0.9 eV below the Fermi level. Next, we explore the effect of biaxial strain on the valence band related valley splitting, as shown in Figure 3. The negative value is used to represent compressive strain, which is easier to achieve than a tensile strain on a surface. We find that the band structure (with SOC) remains similar under an intermediate compressive strain. One clearly observes that the valley splitting ∆VB lies around 300 meV. The spin splitting at the K valley ∆v,K is always larger than 600 meV, in large contrast with that at the K' valley ∆v,K' (lower than 150 meV). The ratio ∆v,K'/∆v,K is always below 20%, indicating a large valley polarized spin splitting. These facts demonstrate that the giant valley splitting in ZnS(0001) supported TaTe2 monolayer is robust against intermediate strain, different from previous MoTe2 on EuO.20

Figure 3. Biaxial strain effect on the valley splitting energy scales ∆VB, ∆v,K, and ∆v,K'.

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We see from Figure 2d that the valence and conduction bands at K are close to each other. Thus, one may wonder if they can be inverted by perturbation. Indeed, we find that under a biaxial tensile strain of 5%, band inversion happens at the K valley. This might lead to interesting valley-polarized topological feature (with a spin skyrmion around K), as we have discussed in a ferrimagnetic honeycomb lattice.7 However, by calculating its Berry curvature, we find that the inverted band still gives a topologically trivial state. This is because the valence and conduction bands in TMD monolayers arise from different orbitals, as discussed previously. Thus, the band inversion also leads to a nontrivial pseudo-spin (orbital-type) skyrmion, and the two skyrmions cancel each other, giving a zero topological index. In addition to the valley polarized (quantum) anomalous Hall effect7,19 and magnetooptic effects,40 as discussed in previous studies, we explore another potential application of this system, taking advantages of its Fermi level crossing the valence band. Note that recent experiments have characterized the plasmonic response of MoS2 using electron energy loss spectroscopy41 and angle-resolved reflectance spectroscopy.42 Using a compressed (–4%) state as a typical example (others yield similar results), we calculate its collective plasmon excitation and the screening spectrum of the two valleys separately.43-47 To do this, the valley-dependent dielectric function εη(q,ω) can be evaluated within the random phase approximation (RPA) as

ε K , K ′ ( q, ω ) = 1 − V ( q ) χ K , K ′ ( q, ω ) ,

(5)

where V(q) = e2/2ϵ0ϵbq is the Fourier transform of 2D Coulomb interaction term, and the background relative dielectric constant ϵb is set to 8 in the numerical simulation.48 The non-interacting polarization function χη(q,ω) is calculated using the Lindhard function49

χ K , K ′ ( q, ω ) = −

1

( 2π ) ∫ 2

K ,K ′

d 2k ∑ ψ m ,k +q eiq⋅r ψ n ,k n,m

2

×

f n ,k − f m , k + q En,k − Em ,k +q + hω + iξ

,

(6)

where the integral is performed in the triangular regions centered at K and K' valleys whose area is equal to half of the first Brillouin zone. The En,k is the eigenvalue of firstprinciples wave function #$,% of band-n at the k point in the momentum space. In our numerical simulation, a temperature of T = 300 K is used in the Fermi-Dirac distribution 9



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fn,k = {exp[(En,k – µ)/kBT] + 1}–1 for energy En,k, and the phenomenological broadening ξ = 1 meV is adopted. To obtain more accurate results, we take the first-principles wave function to calculate these terms, instead of using low energy Hamiltonian wave function. The zeros of the dynamical dielectric function ε(q,ω) give the excitation spectrum of the plasmon modes of the electron liquid. The real and complex parts of the dielectric function quantify the dispersion and loss of plasmon modes, respectively. The electron energy loss spectrum (EELS), defined as ℒ

, " '(, )*

= −Im.1/1

, " '(, )*2 ,

is the

spectral weight of the plasmon mode. In the zero broadening limit (ξ→0), the EELS ℒ

, " '(, )*

shows delta characters.

Figure 4 shows the EELS ℒ

, " '(, )* with

chemical potential µ = EF + 0.2 eV. The

calculated iso-energy surface of such chemical potential in the 2D momentum space is plotted as inset of Figure 4a, which shows a large ring around the K valley and a small ring around the K' valley, indicating the valley polarized behavior. In the low energy regime, marginal in-plane anisotropy is expected. Indeed, our test calculations of EELS along ±qx and ±qy directions show very small difference. Thus, we only discuss the positive qx situation. Valley polarized acoustic EELS feature can be clearly observed. At the K valley, due to large spin splitting (∆v,K), the plasmon is mainly contributed by intraband (VB) transition. Thus, the EELS is weakly damped in the momentum space (Figure 4a). On the contrary, the spin splitting ∆v,K' is much smaller, so that in the larger momentum region the plasmon has more interband scattering contribution (between VB and VB–1). Hence, the EELS in the K' valley is more damped (Figure 4b).

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Figure 4. EELS ℒ

, " '(, )*

of (a) K and (b) K' valleys in the x-direction when

chemical potential µ = EF + 0.2 eV. Inset in (a) shows the k-resolved iso-energy plot of such chemical potential.

In the long wavelength limit (q→0), the plasmon follows a ℏ)~67 relation.44-47 According to our simulation, this scaling rule also holds in our system. Comparing Figures 4a and 4b, one observes that for the same excitation momentum q (small value), the plasmon corresponds to a higher energy at the K valley than that at the K' valley. Quantitatively speaking, we fit the coefficient αK,K' in the relation ℏ) = 8

, " 67

in the

small momentum region. We find that the αK = 0.66 eV·Å1/2 and αK' = 0.41 eV·Å1/2 = 0.62 αK, which again suggests a giant valley polarized plasmonics. We also note that at large momentum q region (> ~0.05 Å–1), the plasmon follows a linear relationship, ℏ)~7,

instead of ℏ)~67.

In order to further explore the valley polarized plasmonics in our proposed system, we adjust the chemical potential µ and examine the variations of long wavelength plasmon coefficients αK and αK'. The results are shown as open circles and open diamonds in Figures 5a and 5b, respectively. Throughout the large energy range, the valley polarized plasmon behavior is clearly seen. The plasmon coefficient at the K valley αK is always larger than that at the K' valley, αK'. We try to fit the coefficients α at each valley with respect to the chemical potential µ according to

9: ; 9:

=>?@,: A

=
?@,:

C:

B

(η = K, K'),

where EVBM,η is the VB maximum at each valley. The characteristic 8 D = E FGHI,

C:

is the plasmon coefficient value when the chemical potential µ lies at the Fermi level. Note that in previous studies, the scaling power γ is 1/4 in graphene monolayer (with linear band dispersion)50 and 1/2 in black phosphorus monolayer (with parabolic band dispersion).51 In our system, the best fit to scaling power is γK = 0.31 and γK' = 0.36 (solid curves in Figures 5a and 5b), due to the strong nonlinear or non-parabolic nature of the valence bands, under giant valley and spin splitting. The coefficients β (in unit of eV and Å) at the two valleys are found to be βK = 0.92 and βK' = 0.93.

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Figure 5. Variation of long wavelength plasmon coefficient α (ℏ) = 8 67) as a function of chemical potential µ (relative to the Fermi level) of (a) K valley and (b) K' valley. Solid curves are fitted scaling relation.

In conclusion, we provide a new strategy to lift the valley degeneracy of 2D TMD monolayer. Instead of using a magnetic proximity effect to break the TRS of MoTe2 monolayer, we use group-V TMD monolayer (TaTe2 as a typical example) where the TRS is broken intrinsically. A polarized ZnS(0001) surface is used to support the TaTe2 monolayer and shift the band energy. The valley splitting ∆VB in our proposed TaTe2 on ZnS(0001) system can be as large as ~ 300 meV. The spin splitting at the K valley is ∆v,K ~ 700 meV, while at the K' valley it is much smaller, ∆v,K' ~ 100 meV. These values are larger than previous results. In contrast to the deeply embedded valleys in previous studies, in our system the valley valence band lies near the Fermi level, which allows 12


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easier control and manipulation of valleys and spins in experiments and future applications. Furthermore, we predict a possible application of this low-energy valley splitting system, namely, valley polarized plasmonics. Our calculations reveal a large valley dependent plasmon behavior, whose resonance scales with chemical potential as ~ µγ (1/4 < γ < 1/2). This discovery gives valley a new dimension to the collective excitations at the surface. Note that similar results can be obtained when other elements in the same group of Ta and/or Te are used. Thus, our predictions can be extended and verified in different ferromagnetic group-V based TMD monolayers (See Supporting Information, Figure S2).32,36,52,53

Acknowledgments. This work is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No. DE-FG02-96ER45579. Resources of the National Energy Research Scientific Computing Center supported by the Office of Science of the U.S. Department of Energy under Contract No.DE-AC02-05CH11231 are also acknowledged. Supporting Information Available. Phonon dispersion along the high symmetry kpath, band structure (without and with SOC) of a 2H-TaSe2 deposited on ZnS(0001) surface. The authors declare no conflict of interest.

*

Corresponding author: [email protected]

References: (1) Pesin, D.; MacDonald, A. H. Spintronics and Pseudospintronics in Graphene and Topological Insulators. Nat. Mater. 2012, 11, 409-416. (2) Rycerz, A.; Tworzydlo, J.; Beenakker, C. W. J. Valley Filter and Valley Valve in Graphene. Nat. Phys. 2007, 3, 172-175.

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(3) Xiao, D.; Yao, W.; Niu, Q. Valley-Contrasting Physics in Graphene: Magnetic Moment and Topological Transport. Phys. Rev. Lett. 2007, 99, 236809. (4) Li, X.; Cao, T.; Niu, Q.; Shi, J.; Feng, J. Coupling the Valley Degree of Freedom to Antiferromagnetic Order. Proc. Natl. Acad. Sci. U.S.A. 2013, 110, 3738-3742. (5) Zhang, F.; MacDonald, A. H.; Mele, E. J. Valley Chern Numbers and Boundary Modes in Gapped Bilayer Graphene. Proc. Natl. Acad. Sci. U.S.A. 2013, 110, 1054610551. (6) Zhou, J.; Huang, C.; Kan, E.; Jena, P. Valley Contrasting in Epitaxial Growth of In/Tl Homoatomic Monolayer with Anomalous Nernst Conductance. Phys. Rev. B 2016, 94, 035151. (7) Zhou, J.; Sun, Q.; Jena, P. Valley-Polarized Quantum Anomalous Hall Effect in Ferrimagnetic Honeycomb Lattices. Phys. Rev. Lett. 2017, 119, 046403. (8) Xiao, D.; Liu, G.-B.; Feng, W.; Xu, X.; Yao, W. Coupled Spin and Valley Physics in Monolayers of MoS2 and Other Group-VI Dichalcogenides. Phys. Rev. Lett. 2012, 108, 196802. (9) Cao, T.; Wang, G.; Han, W.; Ye, H.; Zhu, C.; Shi, J.; Niu, Q.; Tan, P.; Wang, E.; Liu, B.; Feng, J. Valley-Selective Circular Dichroism of Monolayer Molybdenum Disulphide. Nat. Commun. 2012, 3, 887. (10) Mak, K.; He, K.; Shan, J.; Heinz, T. F. Control of Valley Polarization in Monolayer MoS2 by Optical Helicity. Nat. Nanotechnol. 2012, 7, 494-498. (11) Zeng, H. L.; Dai, J. F.; Yao, W.; Xiao, D.; Cui, X. D. Valley Polarization in MoS2 Monolayers by Optical Pumping. Nat. Nanotechnol. 2012, 7, 490-493. (12) Mai, C.; Barrette, A.; Yu, Y.; Semenov, Y. G.; Kim, K. W.; Cao, L.; Gundogdu, K. Many-Body Effects in Valleytronics: Direct Measurement of Valley Lifetimes in Single-Layer MoS2. Nano Lett. 2014, 14, 202-206.

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Figure 1. (a) Geometric structure (left panel, side view; right panel, top view) of a TaTe2 monolayer deposited on a Zn-terminated ZnS(0001) surface. (b) Iso-surface of spin density which is mainly located on the Ta atoms. (c) First Brillouin zone with high symmetry k-points. 80x98mm (160 x 160 DPI)



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Figure 2. (a) and (b), schematic low-energy band spectra in the vicinity of K and K' valleys. (c) and (d), DFT calculated band structure along the high symmetry k-path. Blue and red colors represent spin-up and spindown channels, respectively. (a) and (c) are band structures without including SOC, and (b) and (d) are based on fully relativistic Hamiltonian. 98x122mm (300 x 300 DPI)


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Figure 3. Biaxial strain effect on the valley splitting energy scales ∆VB, ∆v,K, and ∆v,K'. 50x32mm (300 x 300 DPI)



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Figure 4. EELS L_(K,K^' ) (q,ω) of (a) K and (b) K' valleys in the x-direction when chemical potential µ = EF + 0.2 eV. Inset in (a) shows the k-resolved iso-energy plot of such chemical potential. 33x14mm (300 x 300 DPI)


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Figure 5. Variation of long wavelength plasmon coefficient α (ℏω=α√q) as a function of chemical potential µ (relative to the Fermi level) of (a) K valley and (b) K' valley. Solid curves are fitted scaling relation. 93x110mm (300 x 300 DPI)


Giant Valley Splitting and Valley Polarized Plasmonics in Group V

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