Nonreciprocal Giant Magneto-Optic Effects in ... - ACS Publications

Letter pubs.acs.org/JPCL

Nonreciprocal Giant Magneto-Optic Effects in Transition-Metal Dichalcogenides without Magnetic Field Haixia Da,*,†,‡ Lei Gao,†,‡ Weiqiang Ding,∥ and Xiaohong Yan*,†,‡,§ †

College of Electronic Science and Engineering, Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu 210046, China ‡ Key Laboratory of Radio Frequency and Micro-Nano Electronics of Jiangsu Province, Department of Human Resources, Nanjing 210023, China § School of Material Science and Engineering, Jiangsu University, Zhenjiang 212013, China ∥ Physics Department, Harbin Institute of Technology, Harbin 150001, China ABSTRACT: Magnetic exchange field has been demonstrated to be effective in enhancing the valley splitting of monolayer transition-metal dichalcogenides experimentally. However, how magnetic exchange coupling affects the magnetooptical behaviors in massive Dirac systems ⃗ model and Kubo formula, we theoretically remains elusive. Using k·p⃗ report that optical Hall conductivity and giant magnetooptical effects can be induced in monolayer transition-metal dichalcogenides even if there is no any magnetic field involved when considering magnetic exchange interaction. Such an unusual result originates from the fact that the existence of magnetic exchange coupling effectively enables the breaking of time reversal symmetry, which grants the removal of valley degeneracy and unveils the possibility of generation and manipulation of magnetooptical effects in monolayer transition-metal dichalcogenides with no need for magnetic field. Our results suggest that the presence of magnetic exchange coupling of transition-metal dichalcogenides represents an alternative strategy capable of inducing magnetoopitcal effects, which can be extended to other monolayer massive Dirac systems.

T

other 2D materials beyond graphene. Silicene, TMDC, and even black phosphorus upon magnetic field have been discussed as possible MO candidates.18−21 Spin- and valleydependent MO properties have been investigated when monolayer MX2 (such as MoS2 and WSe2) is submitted to an external magnetic field.19,22 The valley-splitting and valleydependent optical selection rules have been revealed in monolayer MoS2 upon magnetic field, thus permitting the manipulation of valley polarization through control of magnetic field.19 The valley pseudospin in TMDC enables valley Zeeman splitting and valley-selective circular dichroism in the presence of magnetic field, which have been observed to be controllable by magnetic field.23−26 In the above works, the application of external magnetic field is always necessary because it, in principle, allows the possibility of giving rise to MO effects. However, the related MO behaviors of monolayer MX2 suffer from the requirement of unrealistic high magnetic field out of the experimental reach. Therefore, it is highly desirable to explore an alternative platform to generate the MO effects in 2D materials without the need of external magnetic field. It will endow 2D materials

ransition-metal dichalcogenide (TMDC) materials are the recent advent of 2D materials with the type of MX2, where M is a transition-metal atom (Mo, W, V, etc.) and X is a chalcogen atom (S, Se, or Te). Unlike the planar configuration of monolayer graphene, monolayer MX2 is a trigonal prismatic arrangement consisting of a single layer of M atoms and two hexagonal layers of X atoms, indicating its unique broken inversion symmetry.1,2 In addition, monolayer MX2 generally has strong spin−orbit coupling (SOC) due to the presence of heavy metal atoms.3,4 The broken inversion symmetry together with its strong SOC in MX2 enable fantastic physical properties as well as great opportunities in electronic and optoelectronic applications, such as superconductor, photoluminescence, ultrafast nonlinear absorption, opto-valleytronic imaging, and valley polarization.5−11 Conventional bulky magnetic materials with strong magnetooptical (MO) effects, that is, Faraday rotation (FR) and Kerr rotation (KR) angles, are always in demand for polarization controllers, switches, optical isolations, unidirectional transmission, as well as high-density data-storage devices.12−16 The recent explosive interest in MO effects is driven by the experimental demonstrations of FR effects in monolayer and multilayer graphene, where giant FR angle has been reported within such atomic layers under the application of magnetic field.17 Along this direction, much effort has been extended to © XXXX American Chemical Society

Received: July 11, 2017 Accepted: August 2, 2017 Published: August 2, 2017 3805

DOI: 10.1021/acs.jpclett.7b01786 J. Phys. Chem. Lett. 2017, 8, 3805−3812

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The Journal of Physical Chemistry Letters

⃗ Hamiltonian model for monolayer MX2 at two The k·p⃗ inequivalent K+ and K− valleys can be written as36,38

more unique practical propositions in the MO devices from the perspective of applications. Some successful realizations of MO effects have been identified in 2D materials without magnetic field.27−29 Strain engineering is a possible way to achieve nonreciprocal MO effects of graphene in the absence of magnetic field.27 The opportunity of creating nonzero KR effect in bilayer graphene has been proposed by Nandkishore et.al when there is no magnetic field, which can be ascribed to the broken time-reversal symmetry of ground state caused by the electron−electron interactions.28 Hole-doped group-IIIA metalmonochalcogenide monolayers have been predicted to allow MO effects owing to their identified ferromagnetism, whose magnitudes can be tuned by the carrier concentrations.29 For TMDC family, circularly polarized light is an available way to produce the valley pseudomagnetic field in monolayer MX2, guaranteeing the selective control of valley degree of freedom free of magnetic field.30,31 In this regard, it still remains largely unexplored to induce the MO effects in monolayer MX2 if no magnetic field is involved theoretically as well as experimentally. Valley polarization has been reported to be induced in magnetically doped monolayer MX2 (such as, Mn-doped MoS2) and controlled by the doping site and dopant concentration theoretically.32 Recent experimental discoveries regarding the magnetic proximity effects on 2D materials hold potential for dramatically affecting electronic and magnetic behaviors, such as the controlled spin−orbit torques.33,34 Single-layer graphene has been demonstrated to be ferromagnetic owing to the proximity effects from an atomically flat yttrium iron garnet ferromagnetic thin film.35 It has recently been theoretically proposed that the enhanced valley splitting and large spin-valley polarization of monolayer MX2 (MoTe2, WSe2) supported by ferromagnetic substrates can be achieved by extrinsic magnetic exchange field36,37 and experimentally confirmed (unpublished results). In addition, it has been predicted that monolayer MX2 (VSe2) enable being ferrovalley materials when the intrinsic exchange interaction of localized electrons is considered.38 In light of the rapid progress of this direction, one may wonder whether monolayer MX2 with intrinsic or extrinsic magnetic exchange coupling (MEC) provides a new strategy of generating and engineering the MO properties without the need of magnetic field. To clarify this question, we study the optical Hall conductivity and MO effects of monolayer MX2 in the presence ⃗ model and Kubo formula.39 It is shown of MEC by using k·p⃗ that the presence of MEC is able to induce nonzero optical Hall conductivity in monolayer MX2, which is dramatically different from that of monolayer MX2 without MEC. Such a striking difference of optical Hall conductivity in monolayer MX2 between without and with MEC can be attributed to the fact that the presence of MEC in monolayer MX2 is able to break time reversal symmetry and thus generate nonzero optical Hall conductivity, clearly elucidating its role and effectively overcoming the limitations of requiring applied magnetic fields. In particular, following this principle, we show that this strategy can be extended to other massive Dirac systems. In addition, the FR angles obtained in monolayer MX2 with MEC appearing at the visible energy regime outperform those of conventional bulky magnetic materials, which can be manipulated by chemical potential as well as the MEC values. Our results unveil an alternative possibility of generation and manipulation of MO effects in 2D TMDC, allowing the possible MO candidates for the future MO devices.

Δ Ĥ = ℏvF(τkxσx̂ + k yσŷ ) ⊗ σ0 + σẑ ⊗ σ0 2 1 − σẑ + τλ ⊗ sẑ + [(M1σ+̂ + M 2σ ̂−] ⊗ sẑ 2

(1)

at the corners of the hexagonal Brillouin zone in the presence of MEC, where vF, k,⃗ and ŝz are Fermi velocity, wave vector, and spin index, respectively. The label of τ = ± 1 corresponds to K+ and K− valleys, respectively. σx, σy, and σz are 2 × 2 Pauli matrices for the pseudospin indices and σ0 is a unit matrix and 1 σ± = 2 (σ0 ± σz). λ is the SOC splitting and Δ is the band gap. M1 and M2 represent the respective effective exchange splitting values for the conduction and valence bands, which can be generated by the magnetic proximity effects or intrinsic exchange interaction. The AC-conductivity tensor of monolayer MX2 with MEC can be obtained by using the Kubo formula39

∫

e2 2ω

+∞

dω [f (ω − μ) − f (ω + Ω − μ)] 2π dk 2 ⇀ ⇀ Tr[vαA(ω + Ω, k )vβA(ω , k )], (α , β = x , y) (2π )2

σαβ(Ω) =

∫−∞

(2)

where f (x) =

1 exp[x / T ] + 1

is the Fermi−Dirac function. T and

μ are the temperature and chemical potential, respectively. Tr is the trace and A(ω) is the spectral function, connected to +∞ dω A ij (ω ′)

. electronic Green’s function Gij(z) by Gij(z) = ∫ −∞ 2π z − ω ′ vα,β is the velocity matrix that can be obtained by the derivatives of the Hamiltonian via Peierls substitution. The MO effects can be obtained by transfer matrix method when a linearly polarized wave normally impinges on monolayer MX2 with MEC, which is placed on a substrate, as shown in Figure 1a. The 4 × 4 transfer matrix method is a powerful tool to study wave propagations in multilayer structures and provides accurate access to evaluate optical and MO effects. The field vector (Ψ(z)) is obtained using the relation for a monochrome light upon the material40 d Ψ(z) iω = Δ(z)Ψ(z) dz c

(3)

where Ψ(z) = (Ex, Hy, Ey, −Hx)T is the field vector and Δ(z) is the Berreman matrix. The field vectors at two different positions z and z0 are connected by the transfer matrix T(z, z0), that is, Ψ(z) = T(z, z0)Ψ(z0). On the basis of the Cauchy and Hamilton theorem,40 the transfer matrix can be expressed by T (z , z 0) = β0I + β1Δ(z) + β2Δ(z)2 + β3Δ(z)3

coefficients

The

(4) f

4

β0 = −∑i = 1 λjλk λl λ λ i λ ,

are

ij ik il

f

4

β1 = ∑i = 1 (λjλk + λjλl + λk λl) λ λ i λ , ij ik il

β2 =

4 −∑i = 1 (λj

fi

+ λk + λl) λ λ

ij ik λil

4

f

, and β3 = ∑i = 1 λ λ i λ , where ij ik il

λij = λi − λj and fi = e−iω / cλi(z − z 0), with λi being the eigenvalues of the Berreman matrix, i, j, k, l = 1, 2, 3, 4. Assuming that (Eip, Eis) and (Etp, Ets) are the respective electric-field components of 3806


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(Re[σxx]) as a function of energy are shown in Figure 1c at different μ = 1, 0.5, −0.8, −1 eV, respectively. As observed, Re[σxx] are chemical-potential-dependent and are always positive values due to their non-negative sign for all optical transitions between conduction and valence bands for various μ. When μ = 1 eV, the curve is close to zero at low energies and then Re[σxx] starts to increase around the energy of 1.85 eV. It keeps on increasing with an increased energy and reaches the maximum value of 1.05 e2/ℏ at the energy of 2.18 eV. Beyond this maximum value, Re[σxx] keeps on slightly decreasing with the further increase in energy. For different μ, the profiles of Re[σxx] remain the similar but with different starting points for nonzero Re[σxx]. At μ = 0.5 eV (0.8 eV), the starting point for nonzero Re[σxx] appears at 1.51 eV (1.58 eV) and the maximum value of 1.27 e2/ℏ stays at almost the same energy of 1.85 eV. When μ = −1 eV, the curve of Re[σxx] is quite similar to that at μ = 1 eV. In Figure 1d, we plot optical Hall conductivity (Re[σxy]) as a function of energy with the same parameters as Figure 1c. In contrast with the cases of Re[σxx], it is found that the response of Re[σxy] as a function of energy vanishes in monolayer MoS2 and remains zero for various chemical potentials, which confirms that there is no anomalous charge Hall effect for monolayer MoS2 in the absence of MEC.19 Figure 1e,f shows Re[σxx] and Re[σxy] of monolayer MoS2 with MEC as a function of energy at μ = 1, 0.5, −0.8, and −1 eV, respectively, whose exchange splitting values in the conduction and valence bands are set to be M1, M2 = 50, 20 meV. We can see that the general profiles of Re[σxx] are similar to those in Figure 1 c with some small distortions, whose change are governed by the shifts of band structures induced by MEC. Remarkably, the trends in Re[σxy] are dramatically distinct from that of monolayer MoS2 without MEC. Nonzero values of Re[σxy] are observed and their values can be positive as well as negative. It is our main result, which is a direct manifestation of the role of MEC and demonstrates its ability to generate optical Hall conductivity without the need of magnetic field. In addition, the values of Re[σxy] can be around one tenth order of e2/ℏ, which is competitive with those of bilayer graphene in the absence of magnetic field and allows the possible observation of optical Hall effect in experiments.28 Chemical potential determines the distributions of massive Dirac carriers and dominates both interband and intraband transitions between conduction and valence bands, thus affecting the profiles of optical conductivities. When μ = 1 eV, Re[σxy] increases and reaches a positive peak with the value of 0.1 e2/ℏ around the energy of 1.85 eV. Beyond this inflection point, it is decreased until down to a negative valley with the value of −0.33 e2/ℏ at the energy of 2.0 eV, which is followed by a rapid increase, and the second peak appears at the energy of 2.16 eV with the value of 0.09 e2/ℏ. With the further increase in energy, Re[σxy] is vanishing and tends to be zero. When μ = 0.5 eV, the curve of Re[σxy] changes shape and shows one positive peak with a relatively large bandwidth ranging from 1.62 to 1.7 eV, with the value of 0.17 e2/ℏ along with two negative dips with the value of −0.06 e2/ℏ and −0.07 e2/ℏ at the energies of 1.54 and 1.78 eV, respectively. Clearly, the value of positive peak (negative valley) is slightly larger (much smaller) than those at μ = 1 eV. In contrast, there is only positive peak and one negative valley in the spectrum of Re[σxy] at μ − 0.8 eV, which appear at the energies of 1.7 and 1.6 eV with the values of 0.2 e2/ℏ and −0.13 e2/ℏ, respectively. When μ = 1 eV, the trend of Re[σxy] is still similar to that at μ = 1 eV.

Figure 1. (a) Schematic setup of Faraday rotation in the configuration composed of monolayer MX2 and the surrounding media, respectively. The direction of extrinsic or intrinsic exchange field is shown by an arrow. (b) Top view and side view of lattice structure of monolayer MX2 with M and X atoms indicated in different color balls. The structure consists of one sheet with M atoms in between and two sheets with X atoms in the respective top and bottom layers. Real parts of (c) longitudinal optical conductivity Re[σxx] and (d) optical Hall conductivity Re[σxy] of monolayer MoS2 without MEC (i.e., M1 = M2 = 0 meV) versus energy at μ = 1, 0.5, −0.8, −1 eV. The unit of optical conductivity is e2/ℏ. Zero optical Hall conductivity of monolayer MoS2 without MEC indicates the symmetry of band structures at two valleys owing to time reversal symmetry. (e) Re[σxx] and (f) Re[σxy] of monolayer MoS2 with MEC versus energy. The exchange splitting values for the conduction and valence bands are set to be M1, M2 = 50, 20 meV.

the incident and transmitted waves, the transmitted wave can be linked to the incident wave by the Jone transmission matrix ⎛ Et p ⎞ ⎛ t pp t ps ⎞⎛ Ei p ⎞ ⎜ ⎟=⎜ ⎟⎜ ⎟ ⎜ s ⎟ ⎜ t t ⎟⎜ s ⎟ ⎝ Et ⎠ ⎝ sp ss ⎠⎝ Ei ⎠

(5)

where tij is the ratio of the incident j and transmitted i polarized electric fields. Subscripts p and s denote electromagnetic waves that are parallel and perpendicular to the incident plane x − z, respectively. Thus we have the complex FR angles as41 tsp t ps , Θs = θs + iηs = Θp = θp + iηp = t pp t pp (6) where θp(s) and ηp(s) are FR angles and ellipticities for p (s) polarization waves, respectively. The atomic configuration of monolayer MX2 is shown in Figure 1b. We use monolayer MoS2 as a representative example in the numerical calculations, whose parameters include vF = 5.5 × 105 m/s, a = 3.86 Å, Δ = 1.66 eV, and λso = 0.15 eV.4 The temperature is set to be 10 K for better resolving the optical transitions appearing in a monolayer MoS2, and the broadening parameter is assumed to be 30 meV. We first investigate the case of monolayer MoS2 without MEC, whose real parts of longitudinal optical conductivity 3807


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MoS2 with MEC at μ = 1 eV, as shown in Figure 2d. The difference is that the depth of right negative valley of Re[σxy] is larger than that of left negative valley at M1, M2 = 200, 100 meV, which is kind of opposite to the case in Figure 2a. Therefore, it has been shown that optical Hall conductivity of monolayer MoS2 with MEC can be manipulated by chemical potential or the values of MEC. To elucidate the dramatic difference of Re[σxy] between without and with MEC, we plot the evolutions of energy spectra of monolayer MoS2 for both spin-up and spin-down states at K+ and K− valleys for different exchange splitting values in Figure 3, respectively. The energy spectra of

Our results show that the 2D nature of monolayer MX2 with MEC enables chemical-potential-controlled optical Hall conductivity. From the Hamiltonian in eq 1, we can understand that the spin degeneracy between spin-up and spin-down states at K+ and K− valleys can be further lifted by the introduction of MEC, whose magnitudes will determine the shifts of the spin-up and spin-down bands at the conduction and valence bands. Therefore, it allows another possible manipulation of optical Hall conductivity by MEC values. From a material perspective, MoTe2 (VSe2) has recently been reported to have the exchange splitting values of 206 meV (555 meV) and 170 meV (464 meV) in the conduction and valence bands, respectively.36,38 Therefore, the exchange splitting values in the conduction and valence bands are taken to be M1, M2 = 50, 20 meV, M1, M2 = 100, 50 meV, and M1, M2 = 200, 100 meV, respectively. Figure 2a shows the spectrum of Re[σxy] in monolayer MoS2 with

Figure 2. Re[σxy] of monolayer MoS2 with MEC versus energy at (a) μ = 1 eV, (b) μ = 0.5 eV, (c) μ = −0.8 eV, and (d) μ = −1 eV, respectively. The exchange splitting values in the conduction and valence bands are set to be M1, M2 = 50, 20 meV (black solid line), M1, M2 = 100, 50 meV (blue dashed line), and M1, M2 = 200, 10 meV (red dashed−dotted line), respectively.

MEC as a function of energy at μ = 1 eV. In contrast with the case of monolayer MoS2 with M1, M2 = 50, 20 meV, the original left positive peak at the energy of 1.85 eV moves toward low energy, and the right positive peak at the energy of 2.16 eV moves toward high energy with MEC values increasing. Besides, the original single valley at the energy of 2 eV splits into two valleys with the values of −0.26 e2/ℏ and −0.23 e2/ℏ, respectively. A small hump forms in between these two negative valleys, and the separation between the negative valleys gets larger with the further increase in MEC value to M1, M2 = 200, 100 meV. From Figure 2b, one can see that Re[σxy] has similar profiles along with the higher positive peak values and two deeper valleys with the increase in MEC value when μ = 0.5 eV. As shown in Figure 2c, there is a similar trend for Re[σxy] below the energy of 1.7 eV with μ being −0.8 eV, where the position of negative valley moves toward low-energy regime along with the increased absolute values of Re[σxy]. There is some difference in the spectrum of Re[σxy] beyond the energy of 1.7 eV for different MEC values. The original single positive peak at M1, M2 = 50, 20 meV splits into two peaks, and thus a new valley appears. With MEC value increasing, a new positive peak moves toward high energy with the larger Re[σxy] and a new valley gets deeper and even down below zero. When μ is down to −1 eV, Re[σxy] has very similar profiles as that of monolayer

Figure 3. Energy spectra of monolayer MoS2 without and with MEC at K+ and K− valleys, where the exchange splitting values are (a) M1 = M2 = 0 meV (b) M1, M2 = 50, 20 meV (c) M1, M2 = 100, 50 meV, and (d) M1, M2 = 200, 100 meV, respectively. Red and blue lines are spinup and spin-down states, respectively. The spin-up (spin-down) bands are labeled as c1 (c2) for the conduction bands and ν1 (ν2) for valence bands at both K+ and K− valleys, respectively. Optical transitions of monolayer MoS2 with MEC at K+ and K− valleys with M1, M2 = 50, 20 meV for different chemical potentials (e) μ = 1 eV, (f) μ = 0.5 eV, (g) μ = −0.8 eV, and (h) μ = −1 eV, respectively. The straight dashed lines indicate the chosen positions of chemical potential and the vertical arrows indicate the optical transitions between conduction and valence bands at K+ and K− valleys.

monolayer MoS2 at M1 = M2 = 0 meV have been shown in Figure 3a for comparison. The band dispersion of monolayer MoS2 at K+ and K− valleys can be distinguishable because the intervalley scattering can be safely ignored. The spin degeneracy is partly lifted at a given valley due to large SOC interaction introduced by the heavy metal atoms. The 3808


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are not allowed due to the Pauli blocking principle if the incident energy is not large enough. Another optical transition (labeled as B1) from the ν2 band to c2 band can be excited by the incident energy of 2.16 eV. On the contrary, it is noted that the optical transition labeled as A2 (B2) from the ν1 (ν2) band to c1 (c2) band needs the excited energy of 2.09 eV (1.93 eV) at K− valley. The combined results of Re[σxy] with positive weights at K+ valley and negative weights at K− valleys give rise to two positive peaks (1.86 and 2.14 eV) and a negative valley at the energy of 2.0 eV. Moving from Figure 3f−h, we find that the optical transitions of A1(B1) correspond to the excited energy of 1.61 eV (1.71 eV), 1.72 eV (1.70 eV), and 2.15 eV (1.87 eV) with decreasing μ from 0.5 to −1 eV at K+ valley, respectively. The optical transitions react differently under the finite MEC at K− valley, whose excited energies for the optical transitions of A2(B2) are referred to as 1.67 eV (1.65 eV), 1.66 eV (1.75 eV), and 1.92 eV (2.01 eV). Therefore, these optical transitions accurately guarantee the positions of their positive peaks. One may notice that a few negative valleys cannot be confirmed from the above transitions. Here we like to emphasize that only spin-conserving optical transitions between the conduction and valence bands are listed in Figure 3 e−h because they dominate the major contributions to Re[σxy]. However, the spin-flip optical transitions are also included in our calculations due to SOC term, whose effects can be resolved in the spectrum of Re[σxy]. For example, two negative valleys in the spectrum of Re[σxy] at μ = 0.5 eV appear at the energy of 1.54 and 1.78 eV, which exactly coincide with the excited energies of spin-flip optical transitions from ν2 (ν1) band to c1 (c2) band. In addition, from these identified transitions, it is also found that the excited energies are almost the same for the optical transitions between conduction and valence bands, resulting in their similarity of Re[σxy] between μ = 1 eV and μ = −1 eV. Therefore, the positions of peaks and valleys in Re[σxy] are well explained by the combination of Pauli blocking, spin-conserving, and spin-flip optical transitions between the conduction and valence bands. Under this scheme, we can understand that the change of the band structures in monolayer MoS2 for different MEC values directly affects the magnitudes of incident excited energy inducing the spin-conserving and spin-flip optical transitions, leading to the MEC-modulated optical Hall conductivity, as shown in Figure 2. In addition, the sign of optical Hall conductivity can be reversed if we take the opposite signs for MEC values. Therefore, the response of optical Hall conductivity in monolayer MoS2 with MEC demonstrated here is an indication of the nature of asymmetry band structures in MoS2 between two valleys, therefore allowing the specific reflection of the change of band structures via the fingerprints of optical Hall conductivity. Before closing this section, we would like to extract the conditions for inducing optical Hall effects in the system of monolayer TMDCs. In Table 1, we list the existence of optical Hall conductivity for several different settings of Δ, λ, and MEC, whose values are artificially set to be zero for clarifying their roles in generating optical Hall conductivity. It can be found that when the SOC term is zero the optical Hall conductivity vanishes no matter whether there is the inversion asymmetry or not in the system. However, optical Hall conductivity appears for monolayer MX2 when the SOC term exists even if there is no inversion asymmetry, which indicates that the inversion asymmetry is a nonessential factor for generating optical Hall effects in monolayer MX2 system.

conduction bands between spin-up and spin-down states remain degenerate, but the valence bands with different spin states are split by 150 meV. In particular, the symmetry of band dispersions between K+ and K− valleys indicates that monolayer MoS2 respects time reversal symmetry in the absence of MEC, which implies that the optical Hall conductivity of each valley shares the exact same value but with opposite sign and thus the net optical Hall conductivity is zero. In contrast with the case of M1 = M2 = 0 meV, the changes induced by MEC can be identified in energy spectra of monolayer MoS2 at K+ and K− valleys from Figure 3b,d. As observed, the splitting between spin-up and spin-down states can be engineered by the value of MEC for both conduction and valence bands. In the presence of MEC, the original spin degeneracy between spin-up (c1) and spin-down (c2) energy levels is lifted in the conduction bands of monolayer MoS2 without MEC, whose magnitudes of spin splitting between c1 and c2 bands are the respective 2|M1| and −2|M1| at K+ and K− valleys. Therefore, one can see that c1 band moves upward while c2 band moves downward in the conduction bands at K+ valley. On the contrary, it is noted that the c1 band moves downward but the c2 band moves upward in the conduction band at K− valley. Therefore, the conduction bands between K+ and K− valleys remain symmetrical because of its same absolute spin splitting values. The case is totally different for the valence bands, whose magnitudes of spin splitting between spin-up (ν1) and spin-down (ν2) energy levels are the respective 2(λ + |M2|) and 2(λ − |M2|) at K+ and K− valleys. Obviously, the spin splitting values in valence bands are strongly modified owing to the existence of MEC. As observed, ν1 band moves toward high energy and ν2 band moves toward low energy at K+ valley, which follows the same shifting trends as those of c1 and c2 bands. However, ν1 band also moves upward and ν2 band moves downward, resulting in the decreased spin splitting between spin-up and spin-down states in valence bands at K− valley. In particular, the original symmetry of band structures of monolayer MoS2 without MEC is broken between K+ and K− valleys. Thus the optical Hall conductivity at each valley has a different value despite their different sign, leading to nonzero values of Re[σxy] in the absence of magnetic field. Thus the existence of MEC is kind of an analogue of the magnetic field, effectively breaking time reversal symmetry of monolayer MoS2 now and producing nonvanishing optical Hall conductivity. Clearly, Re[σxy] of monolayer MoS2 with MEC hosts totally different responses from those of 2D materials under the application of magnetic field. The spectrum of Re[σxy] in 2D materials generally exhibits multiple peaks due to the allowed optical transitions between discrete Landau levels caused by magnetic field.42 Therefore, it determines the sensitivity of Re[σxy] to magnetic field and chemical potential. However, monolayer MoS2 with MEC does not support the Landau levels, and thus its optical conductivity is contributed by the intraband and interband optical transitions between the conduction and valence bands. To better clarify the behaviors of Re[σxy], Figure 3e−h shows the evolutions of optical transitions in monolayer MoS2 with MEC for different μ, where exchange splitting value is fixed to be M1, M2 = 50, 20 meV. The positions of chemical potential are shown using dashed horizontal lines, and the optical transitions are indicated as vertical arrows in Figure 3e−h. When μ = 1 eV in Figure 3e, the incident energy of 1.87 eV enables the optical transition (labeled as A1) from the ν1 band to c1 band at K+ valley because the interband optical transitions 3809


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The Journal of Physical Chemistry Letters Table 1. Possible Existence of Optical Hall Conductivity in Monolayer MoS2 for Various Different Settings of Δ, λ, and MECa Δ

λ

MEC

existence of optical Hall conductivity

≠0 0 ≠0 0 ≠0

≠0 0 0 ≠0 ≠0

0 ≠0 ≠0 ≠0 ≠0

no no no yes yes

a Values of Δ, λ, and MEC are artificially set to be zero to check their roles in generating optical Hall conductivity.

Therefore, the essential prerequisites for generating optical Hall effects in monolayer TMDCs are the coexistence of SOC term and MEC. It is different from those for generating valley Hall effects, where the presence of inversion asymmetry is also needed to induce nonzero Berry phase.19 Our results have confirmed that there is no charge Hall effect for monolayer MoS2 without MEC, but nonzero MEC can break reciprocity and induce optical Hall effect. In this regard, monolayer MoS2 with MEC might present itself as a suitable MO candidate without any need for external magnetic field. It has been demonstrated that optical Hall conductivity can be achieved in monolayer MoS2 with MEC, implying the appearance of nonreciprocal MO effects. Regard to MEC we introduce here, it can be originated from two kinds of mechanisms, that is, extrinsic magnetic exchange field due to magnetic proximity effects37 or intrinsic magnetic exchange interactions such as VSe2 or magnetically doped MX2.32,38 If the former is taken for generating MO effects in 2D materials experimentally, the contribution from the magnetic substrates to total MO effects is unavoidable and it thus complicates the MO signals. The intrinsic one might be more preferable for detecting the MO effects of monolayer MX2 without ruling out the possible effects from the magnetic substrates. Here we take the latter as an example for FR calculations, where a nonmagnetic dielectric material Si is used as a substrate. Figure 4a,b shows the FR angles (θF) and the ellipticities (ηF) as a function of energy in a simple configuration in which a monolayer MoS2 with MEC sits on a Si substrate with the thickness of 30 nm, respectively. The exchange splitting values are fixed at M1, M2 = 100, 50 meV and temperature is 10 K. The general trends of θF here are quite similar to those of Re[σxy] because of the simple configuration. For any μ, we can see that θF and ηF retain small values in the low-energy range, and they remain back to the close-zero values in the highenergy region. Clearly, θF and ηF exhibit the expected nonzero values over a certain energy range with the possible maximum rotation angles of ±0.3°. Most importantly, the MO responses of monolayer MoS2 with MEC occur in the visible-energy region, which is different from the case of monolayer graphene whose MO responses always stay in terahertz energy regions because of their different band gaps. In addition, there exists a transition between the positive and negative values of θF, whose transition position depends on the value of μ. Therefore, our results show that chemical potential can be employed to drive tunable FR angles. Such a dependence of θF on μ can be traced back to the nature of the Fermi−Dirac distribution of carriers and the chemical potential-dependent optical transitions. It is noted that the maximum of θF (around ±0.3°/nm) in monolayer MoS2 with MEC at M1, M2 = 100, 50 meV is smaller than that of monolayer graphene under the application of

Figure 4. (a) Faraday rotation (θF) and (b) ellipticities (ηF) of a monolayer MoS2 with MEC sitting on a nonmagnetic Si substrate whose thickness is 30 nm as a function of energy at μ = 1, 0.5, −0.8, and −1 eV. The exchange splitting values are fixed at M1, M2 = 100, 50 meV and temperature is 10 K. (c) Faraday rotation (θF) and (d) ellipticities (ηF) of a monolayer MoS2 with MEC, whose parameters are the same as those of panels a and b except for temperature being 300 K.

magnetic field, whose value has been experimentally demonstrated to be up to 6°.17 We know the allowed optical transitions between Landau levels determine the scale of Re[σxy] to be around 1−10 e2/ℏ in monolayer graphene upon magnetic field, but the order of Re[σxy] in monolayer MoS2 with MEC is ∼0.1 e2/ℏ, which is one or two orders lower than that in monolayer graphene. Therefore, θF of monolayer MoS2 with MEC is lower than that of monolayer graphene owing to their different intrinsic mechanisms behind. However, the FR angle of monolayer MoS2 with MEC still surpasses that of conventional magnetic bulky materials considering its atomic thickness. For example, bismuth-substituted yttrium iron garnet (Bi:YIG) has been reported to have an FR angle of −0.2°/μm at the wavelength of 720 nm. A well-designed periodic magnetophotonic structure enables the enhancement of FR angle up to −0.02°/nm with total Bi:YIG thickness of 150 nm.43 Therefore, the FR angles of monolayer MoS2 with MEC reported here are still an order of magnitude larger than that in Bi:YIG in the visible-energy region. Besides, in contrast with bulky magnetic materials with difficulty of modulating MO effects, the FR angles in monolayer MoS2 with MEC are strongly engineered by chemical potential as well as MEC values, and the elaborate designed structure might further boost MO performances of monolayer MoS2. Figure 4c,d shows θF and ηF for monolayer MoS2 with MEC as a function of energy at room temperature, where other parameters remian the same as those in Figure 4a,b. As observed, the general trend is that the performance of θF gets degraded with temperature increase. For example, the maximum positive θF decreases from 0.28 to 0.18° when temperature increases from 10 K to room temperature at μ = 1 eV. In addition, some details of FR angles tend to be smoothed out, which can be attributed to the smear of optical transitions owing to thermal broadening at high temperature. The similar degradation also happens in the case of ηF. However, despite the degradation of FR performance, it is found that the FR angle of monolayer MX2 with MEC at room temperature is still higher than that reported for Bi:YIG and comparable to that of monolayer metal-monochalcogenide.29 Finally, we like to note 3810


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(5) Costanzo, D.; Jo, S.; Berger, H.; Morpurgo, A. F. Gate-induced superconductivity in atomically thin MoS2 crystals. Nat. Nanotechnol. 2016, 11, 339−344. (6) Splendiani, A.; Sun, L.; Zhang, Y.; Li, T.; Kim, J.; Chim, C. Y.; Galli, G.; Wang, F. Emerging Photoluminescence in Monolayer MoS2. Nano Lett. 2010, 10, 1271−1275. (7) Neumann, A.; Lindlau, J.; Colombier, L.; Nutz, M.; Najmaei, S.; Lou, J.; Mohite, A. D.; Yamaguchi, H.; Högele, A. Opto-valleytronic imaging of atomically thin Semiconductors. Nat. Nanotechnol. 2017, 12, 329−334. (8) Mak, K. F.; He, K.; Shan, J.; Heinz, T. F. Control of Valley Polarization in Monolayer MoS2 by Optical Helicity. Nat. Nanotechnol. 2012, 7, 494−498. (9) 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. (10) Cao, T.; Wang, G.; Han, W.; Ye, H.; Zhu, C.; Shi, J.; Niu, Q.; Tan, P.; Wang, E.; Liu, B.; et al. Valley-Selective Circular Dichroism of Monolayer Molybdenum Disulphide. Nat. Commun. 2012, 3, 887. (11) Jones, A. M.; Yu, H. Y.; Ghimire, N. J.; Wu, S. F.; Aivazian, G.; Ross, J. S.; Zhao, B.; Yan, J. Q.; Mandrus, D. G.; Xiao, D.; et al. Optical generation of excitonic valley coherence in monolayer WSe2. Nat. Nanotechnol. 2013, 8, 634−638. (12) Luo, X. G.; Zhou, M.; Liu, J. F.; Qiu, T.; Yu, Z. F. Magnetooptical metamaterials with extraordinarily strong magneto-optical effect. Appl. Phys. Lett. 2016, 108, 131104. (13) Chin, J. Y.; Steinle, T.; Wehlus, T.; Dregely, D.; Weiss, T.; Belotelov, V. I.; Stritzker, B.; Giessen, H. Nonreciprocal plasmonics enables giant enhancement of thin-film Faraday rotation. Nat. Commun. 2013, 4, 1599. (14) Ross, M. B.; Bourgeois, M. R.; Mirkin, C. A.; Schatz, G. C. Magneto-Optical Response of Cobalt Interacting with Plasmonic Nanoparticle Superlattices. J. Phys. Chem. Lett. 2016, 7, 4732−4738. (15) Bossini, D.; Belotelov, V. I.; Zvezdin, A. K.; Kalish, A. N.; Kimel, A. V. Magnetoplasmonics and Femtosecond Optomagnetism at the Nanoscale. ACS Photonics 2016, 3, 1385−1400. (16) Floess, D.; Chin, J. Y.; Kawatani, A.; Dregely, D.; Habermeier, H. U.; Weiss, T.; Giessen, H. Tunable and switchable polarization rotation with non-reciprocal plasmonic thin films at designated wavelengths. Light: Sci. Appl. 2015, 4, 284. (17) Crassee, I.; Levallois, J.; Walter, A. L.; Ostler, M.; Bostwick, A.; Rotenberg, E.; Seyller, T.; van der Marel, D.; Kuzmenko, A. B. Giant Faraday rotation in single- and multilayer graphene. Nat. Phys. 2011, 7, 48−51. (18) Tabert, C. J.; Nicol, E. J. Valley-Spin Polarization in the Magneto-Optical Response of Silicene and Other Similar 2D Crystals. Phys. Rev. Lett. 2013, 110, 197402. (19) Chu, R. L.; Li, X.; Wu, S. F.; Niu, Q.; Yao, W.; Xu, X. D.; Zhang, C. W. Valley-splitting and valley-dependent inter-Landau-level optical transitions in monolayer MoS2 quantum Hall systems. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90, 045427. (20) Tahir, M.; Vasilopoulos, P.; Peeters, F. M. Magneto-optical transport properties of monolayer phosphorene. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 92, 045420. (21) Pereira, J. M.; Katsnelson, M. I. Landau levels of single-layer and bilayer phosphorene. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 92, 075437. (22) Tahir, M.; Vasilopoulos, P. Magneto-optical transport properties of monolayer WSe2. Phys. Rev. B: Condens. Matter Mater. Phys. 2016, 94, 045415. (23) Srivastava, A.; Sidler, M.; Allain, A. V.; Lembke, D. S.; Kis, A.; Imamoğlu, A. Valley Zeeman effect in elementary optical excitations of monolayer WSe2. Nat. Phys. 2015, 11, 141−147. (24) MacNeill, D.; Heikes, C.; Mak, K. F.; Anderson, Z.; Kormányos, A.; Zólyomi, V.; Park, J.; Ralph, D. C. Breaking of Valley Degeneracy by Magnetic Field in Monolayer MoSe2. Phys. Rev. Lett. 2015, 114, 037401. (25) Aivazian, G.; Gong, Z. R.; Jones, A. M.; Chu, R. L.; Yan, J.; Mandrus, D. G.; Zhang, C. W.; Cobden, D.; Yao, W.; Xu, X. Magnetic

that the FR angle obtained here is not limited to monolayer MoS2 under MEC because our Hamiltonian also works for other 2D massive Dirac materials by making proper adjustments. We expect that this strategy might open an alternative way to foster MO effects in 2D materials without magnetic field. ⃗ model and Kubo formula, By the combination of k·p⃗ monolayer MX2 with MEC has been demonstrated to have nonzero optical Hall conductivity and thus produce nonreciprocal MO effects, which is totally different from that of monolayer MX2 without MEC. It is found that the FR angles of monolayer MX2 with MEC appear in the visible-energy region, whose angle dwarfs those of conventional bulky magnetic materials and can be controlled by chemical potential and MEC values. We argue that the induced MO effects are allowed by the effective time reversal asymmetry due to the presence of MEC, and this strategy provides a new direction in the quest for generating MO effects without the requirement of magnetic field. Our results highlight the emerging possibility to achieve giant MO effects in 2D massive Dirac materials, which enables new perspectives on MO effects and holds potential for establishing nonreciprocal devices.

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AUTHOR INFORMATION

Corresponding Authors

*Tel/Fax: +008683811210. E-mail: [email protected] (H.D.). *Tel/Fax: +008685866016. E-mail: [email protected] (X.Y.). ORCID

Haixia Da: 0000-0001-8504-2987 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by Jiangsu Specially Appointed Professor Plan (Grant No. RK033STP16002), Natural Science Foundation of Jiangsu Province (Grant No. BK20161513), Six Categories of Summit Talents of Jiangsu Province of China (Grant No. 2016-JNHB-060), NUPTSF (NY215027), National Natural Science Foundation of China (NSFC11374162 and NSFC51651202), and Major Program of Natural Science Foundation by the Ministry of Education of China (TJ215009). W.D. acknowledges the support of Harbin Science and Technology Innovation Talent Foundation (Grant No. RC2014QN001009).

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REFERENCES

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