TY - JOUR
AU - Yao, Wang
AB - Abstract Monolayer group-VIB transition-metal dichalcogenides have recently emerged as a new class of semiconductors in the two-dimensional limit. The attractive properties include the visible range direct band gap ideal for exploring optoelectronic applications; the intriguing physics associated with spin and valley pseudospin of carriers which implies potentials for novel electronics based on these internal degrees of freedom; the exceptionally strong Coulomb interaction due to the two-dimensional geometry and the large effective masses. The physics of excitons, the bound states of electrons and holes, has been one of the most actively studied topics on these two-dimensional semiconductors, where the excitons exhibit remarkably new features due to the strong Coulomb binding, the valley degeneracy of the band edges and the valley-dependent optical selection rules for interband transitions. Here, we give a brief overview of the experimental and theoretical findings on excitons in two-dimensional transition-metal dichalcogenides, with focus on the novel properties associated with their valley degrees of freedom. exciton, valley physics, two-dimensional semiconductor, transition metal dichalcogenides INTRODUCTION Atomically thin group-VIB transition-metal dichalcogenides (TMDs) have recently attracted vast interest as a new class of gapped semiconductors in the two-dimensional (2D) limit [1–3]. The compounds have the chemical composition of MX2, where M stands for the metal atom W or Mo and X is S or Se. The stable phases of bulk crystals are of a layered structure where the elementary unit, the monolayer, is an X-M-X covalently bonded 2D hexagonal lattice. The monolayers are stacked and bounded by the weak van der Waals interaction. Monolayers can be extracted from bulk crystals by mechanical exfoliation [4–6], or synthesized using chemical vapor deposition or molecular beam epitaxy [7–12], similar to the preparation of graphene. When TMDs are thinned down from bulk to monolayers, a striking change in their electronic structure is the crossover from indirect to a direct band gap at the degenerate but inequivalent K and −K valleys at the corners of the hexagonal Brillouin zone [4,5,8]. The direct band gap of monolayer TMDs is in the visible frequency range, ideal for the exploration of optoelectronic applications and semiconductor optics. 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. The attractive Coulomb interaction between the conduction band electron and the valence band hole can bound them into a hydrogen-like state, known as exciton, which is an elementary excitation that plays key role in optoelectronic phenomena. The bound electron–hole pair can also capture an extra electron or hole to form a negatively or positively charged exciton, also known as trion. Through the optical interband transition described above, excitons can be interconverted with photons. Neutral and charged excitons have been discovered in monolayer TMDs from the reflection and photoluminescence (PL) spectra [13–15], where the 2D geometry makes possible the remarkable electrostatic tunability between the neutral and the two charged configurations through gated controlled doping. The measured energy differences between the charged and neutral excitons point to an exceptionally large binding energy [13–15], which is also predicted by first-principles calculations [16,17], and jointly revealed by various measurements including the reflection spectra [18], two-photon absorption [19–22], and scanning tunneling spectroscopy [23,24]. The strong Coulomb binding arises from the reduced screening in the 2D geometry as well as the large effective masses of both the electron and the hole [17,20]. Another unique and interesting aspect of excitons in 2D TMDs is their valley configurations. As both the conduction and valence band edges are at the degenerate K and −K valleys, the lowest energy exciton states can be classified by the valley configurations as well as the spin configurations (Fig. 1). Those configurations with an electron and a hole in the same valley with opposite spin are bright excitons that can emit photon, as the spin and momentum conservation can be satisfied in the electron–hole recombination process. Interestingly, the interband optical transition in monolayer TMDs is associated with a valley selection rule originated from the 3-fold rotational symmetry of the 2D lattice [25,26], such that the valley configurations of the bright excitons directly correspond to the circular polarization of the emitted/absorbed photon. This makes possible the optical addressability of the excitonic valley pseudospin [13,27–29], a property unique to 2D TMDs which have attracted remarkable interest. Figure 1. Open in new tabDownload slide Valley configurations of neutral (X0), negatively charged (X−) and positively charged (X+) excitons in monolayer MoX2 (a) and WX2 (b) (X = S or Se). Solid (empty) dots denote electrons (holes), and red (blue) curves denote spin-up (spin-down) conduction and valence bands. For X0 and X−, only those with the hole in the K valley are shown, and for X+ only those with the electron in the K valley are shown. The remaining configurations are the time reversal of those shown in the figure. Valley configurations with white (gray) background are bright (dark) excitons. In the bright X±, the electron–hole pair that can recombine is shown in black color while the excess electron or hole is shown in orange color. Note that the two spin-split conduction bands have different effective mass due to the renormalization of the SOC, leading to conduction band crossings in MoX2 but not for WX2. Figure 1. Open in new tabDownload slide Valley configurations of neutral (X0), negatively charged (X−) and positively charged (X+) excitons in monolayer MoX2 (a) and WX2 (b) (X = S or Se). Solid (empty) dots denote electrons (holes), and red (blue) curves denote spin-up (spin-down) conduction and valence bands. For X0 and X−, only those with the hole in the K valley are shown, and for X+ only those with the electron in the K valley are shown. The remaining configurations are the time reversal of those shown in the figure. Valley configurations with white (gray) background are bright (dark) excitons. In the bright X±, the electron–hole pair that can recombine is shown in black color while the excess electron or hole is shown in orange color. Note that the two spin-split conduction bands have different effective mass due to the renormalization of the SOC, leading to conduction band crossings in MoX2 but not for WX2. Apart from the discovery of ultrastrong Coulomb binding and the optical addressability of valley pseudospin, other unique properties have been observed or predicted for valley excitons in 2D TMDs. These include the spin-layer locking effect and spectrally resolvable intra- and interlayer trions in bilayer [30,31], the valley–orbital coupling and excitonic fine structure from the electron–hole exchange [32], the trion valley-Hall effect [32], anomalous Rydberg series of excitonic excited states [17–20, 22, 24], valley Zeeman splitting and magnetic tuning of polarization and coherence of the excitonic valley pseudospin [33–36], and valley-selective optical Stark effect [37,38]. Besides, functional optoelectronic devices based on valley excitons in 2D TMDs are being demonstrated. In lateral p-i-n junctions electrostatically formed in monolayer WSe2, electroluminescence has been observed when the electrons and holes are injected into the intrinsic region from the n- and p-doped regions, respectively, under the forward bias [39–41]. Because of the strong Coulomb interaction, the electrons and holes form valley excitons before the radiative recombination [39–41]. A unique feature of these excitonic light-emitting p-n junctions is that the spectral weight of electroluminescence from the neutral and charged excitons is tunable by the bias current [39–41]. In some WSe2 p-n junctions, the electroluminescence is found to be circularly polarized, where the polarization changes sign when the p-n junction is flipped [42–44]. According to the valley optical selection rule, the polarization implies that the emission from excitons in the two valleys is unbalanced and is electrically controllable, which realized a prototype valley light-emitting transistor [42–44]. Moreover, valley-selective second-harmonic generation at the ground-state exciton resonance has been demonstrated in monolayer WSe2 field-effect transistors. The second-harmonic signal generated from such devices can be tuned over an order of magnitude by electrostatic doping, and the tunability arises from the electrostatic charging effect which transfers excitonic oscillator strength between the neutral and the charged excitons [45]. The present article is motivated by these exciting physics and potential applications from the extraordinary properties of valley excitons in 2D TMDs. Below we give an overview of the theoretical understanding and experimental findings on the various aspects of valley excitons. THE EXCITON SPIN AND VALLEY CONFIGURATIONS In monolayer TMDs, the conduction band minima and valence band maxima are both located at the degenerate K and −K points at the corners of the hexagonal Brillouin zone. The K and −K valleys are time reversals of each other. The conduction (valence) states at ±K valley mainly consist of transition-metal |${\rm d}_{z^2 } $| (⁠|${\rm d}_{x^2 - y^2 } \pm i{\rm d}_{xy} $|⁠) orbitals [46] with the magnetic quantum m = 0 (m = ±2). The strong spin–orbit coupling (SOC) from the metal d orbitals then leads to a large spin splitting for the valence band. The splitting value ranges from ∼150 meV for MoX2 to ∼450 meV for WX2 (X = S, Se). The mirror symmetry in the out-of-plane (z) direction dictates that the splitting has to be in the z-direction [30,31], while the time-reversal symmetry (with inversion symmetry breaking) dictates that the spin splitting has opposite signs at valley K and −K (Fig. 1). For low-energy configurations of excitons, because of the large spin splitting, we only need to consider the top spin subband for the valence states. For holes at such valence band edge, the spin and valley indices are then locked together, i.e. valley K (−K) only have spin-up (-down) states. For the conduction states at the ±K valleys, the dominant |${\rm d}_{z^2 } $| component do not contribute SOC, but the small components from the transition metal dxz, dyz and the chalcogen px, py orbitals give rise to a finite spin splitting [47,48]. The form of the SOC is the same as the valence band, but the magnitude is much smaller (Ec, K, ↑ − Ec, K, ↓ ≈ −3 meV, −21, 29 and 36 meV for monolayer MoS2, MoSe2, WS2, and WSe2 respectively, from first-principle calculations [47]). It should be noted that this conduction band spin splitting has an overall sign change between MoX2 and WX2. On the other hand, SOC slightly renormalizes the effective mass of the two spin-split conduction bands, leading to conduction band crossings in MoX2 but not for WX2 [47] (the upper conduction band at ±K has larger/smaller effective mass than the lower one in MoX2/WX2, see Fig. 1). Both spin-split conduction bands are relevant for the low-energy excitons. Consider first the neutral exciton X0. When the electron and hole constituents are at different valleys, their direct recombination is forbidden as momentum conservation cannot be satisfied and the exciton is therefore dark (Fig. 1). When the electron and hole constituents are at the same valley with opposite (same) spin, the exciton is a bright (dark) one. Here we use the convention that a spin-down (-up) hole describes the absence of a spin-up (-down) valence electron. Optical interband transitions always conserve the spin as well as the valley index. Therefore, only the bright exciton can radiatively recombine and result in a photon emission, as the name itself implies. Bright excitons are directly observable both from the PL spectrum and the absorption (reflection) spectrum [4,5]. In addition to the lowest energy bright valley excitons (often referred as A excitons), the PL and absorption/reflection spectra also show the presence of a high-energy bright valley exciton configuration, i.e. B exciton, where the hole is in the higher energy spin-split valence band. The energy separation of the A and B excitons has been used to extract the valence band spin splitting [4,5,27,49]. The lifetime of B excitons is much shorter, as it will relax to the lower energy configurations through fast non-radiative channels. Below we will focus only on the A excitons that play more important roles in the optoelectronic phenomena. The creation operator for the K valley bright exciton can be written as |$\hat B_{{\bf k}, + }^{\dagger} \equiv \mathop \sum_{\bf q} \psi _{\bf k} \left( {\bf q} \right){\rm \hat e}_{{\bf K} + {\bf q} + \frac{{\bf k}}{2}, \uparrow }^{\dagger} \hat h_{ - {\bf K} - {\bf q} + \frac{{\bf k}}{2}, \Downarrow }^{\dagger} ,$| where k is the in-plane center-of-mass wavevector and ψk (q) describes the momentum space electron–hole relative motion, and |${\rm \hat e}_{{\bf K} + {\bf q} + \frac{{\bf k}}{2}, \uparrow }^{\dagger} $| (⁠|$\hat h_{ - {\bf K} - {\bf q} + \frac{{\bf k}}{2}, \Downarrow }^{\dagger} $|⁠) is the creation operator for an electron (hole) with the subscript denoting its momentum and spin. Considering the time-reversal symmetry between the two valleys, the −K valley bright X0 is |$\hat B_{{\bf k}, - }^{\dagger} \equiv \mathop \sum_{\bf q} \psi _{ - {\bf k}}^* \left( {\bf q} \right)\hat {\rm e}_{ - {\bf K} - {\bf q} + \frac{{\bf k}}{2}, \downarrow }^{\dagger} \hat h_{{\bf K} + {\bf q} + \frac{{\bf k}}{2}, \Uparrow }^{\dagger} $|⁠. The photon emission needs to satisfy both the energy and momentum conservation. Thus, even in the bright exciton branch, only those states that lie within the light cone (the conical region defined by E > ℏc|k|) can directly recombine and emit photons. For states outside the light cone, the radiative recombination needs the assistance of phonon scattering to satisfy the energy momentum conservation. When the TMDs sample is negatively (positively) doped, X0 can further bind an excess electron (hole) to lower its energy. The formed three-body bound quasiparticle is called negatively (positively) charged exciton or negative (positive) trion, denoted by X− (X+). Just like bright exciton X0, the bright trion is the one which can radiatively recombine, emitting a photon and leaving an electron/hole. In principle, a bright trion with any center-of-mass wavevector can recombine since the wavevector can be transferred to the resulted electron/hole. But detailed analysis shows that the radiatively emission rate (or the trion brightness) exponentially decays with the increase of the center-of-mass wavevector [50], which gives rise to a low-energy tail in the trion PL spectrum [14]. In X− (X+), to lower the energy, the two electrons (holes) shall have either opposite spin or valley index. Fig. 1 shows the various spin and valley configurations of trions. We only list those X0 and X− with their hole component in the K valley, and those X+ with electron in the K valley, while all other configurations are the time-reversal counter parts of the ones shown. The energy splitting between the different spin and valley configurations (λ0, λ± and |${\lambda_{-}^{\prime}} $| in Fig. 1) is estimated to be in the order of a few tens of meV, mainly due to the SOC splitting of the conduction bands, as well as the effective mass differences of the spin-split subbands. The latter effect comes in through the exciton binding energy. In MoX2, the contribution to λ0, λ± and |${\lambda_{-}^{\prime}} $| from the binding energy differences is expected to have an opposite sign to the contribution from the conduction band spin splitting. The lack of accurate information on the exciton binding energies of the two spin subbands has resulted in the uncertainty in the magnitude and even sign of λ, which need to be determined in future experiments. EXCITON BINDING ENERGY The binding energy of a neutral or charged exciton characterizes how stable the bound state is, and is determined by the strength of the Coulomb interaction and the effective masses of the electron and hole. The binding energy Eb of X0 is defined as its lowered energy compared to the free electron–hole pair, while the trion binding energy (also called charging energy) Ec is the difference between X± and the unbound state of an X0 plus a free electron or hole. In the exciton luminescence, Ec then corresponds to the spectra separation between the X0 emission and the X± emission. PL experiments clearly show well-separated X0 and X± emission peaks in 2D TMDs. From the energy splitting, the trion charging energy Ec is determined to be 18 meV [15], 30 meV [14], 20–40 meV [51] and 30 meV [13] in monolayer MoS2, MoSe2, WS2 and WSe2, respectively. For the X0 binding energy Eb, ab initio calculations give extremely large values in the order of ∼0.5–1 eV [16,17,52–55]. The direct determination of Eb from the PL spectra or absorption spectra has not been possible, as the edge of the band-to-band transition has not been unambiguously determined. While the one-photon process measures the exciton ground state (1s) energy, in two-photon process, it is the exciton excited state (e.g. 2p) that becomes bright. This makes possible the determination of the energy separation between the 1s and 2p states, which provide a lower bound of several hundred meV on the X0 binding energy Eb [19–22]. Also one-photon reflectance contrast spectrum has been used to extract the exciton Rydberg series from 1s to 5s in monolayer WS2 [18], a fitting then gives Eg ∼ 2.41 eV and Eb ∼ 0.32 eV. Various methods have been used to extract the band-to-band transition energy, and its difference from the X0 resonance measured by PL or absorption/reflection spectra then directly gives the exciton binding energy Eb. Two-photon-excitation-induced PL measurements in monolayer WS2 [21] and WSe2 [19] have claimed the observation of features of band-to-band transition at the energies 2.73 and 2.02 eV which then lead to the binding energy Eb ∼ 0.71 and 0.37 eV, respectively. On the other hand, scanning tunneling microscopy/spectroscopy has been carried out to directly measure the quasiparticle band gap (electronic band gap) [23,24]. Scanning tunneling spectroscopy (STS) shows a 2.2 eV band gap for monolayer MoSe2 grown on bilayer graphene [24], resulting in Eb ∼ 0.55 eV. Combined with ab initio calculation, Eb is found to be ∼0.65 eV without the screening of graphene [24]. Monolayer MoSe2 grown on highly oriented pyrolytic graphite (HOPG) shows a similar quasiparticle band gap of 2.1 eV in the STS measurement [12], which implies Eb ∼ 0.5 eV. Monolayer MoS2 grown on HOPG is shown to have a quasiparticle band gap of 2.15 or 2.35 eV depending on the threshold tunneling current [23,56], and the measured exciton PL peak is at 1.93 eV, which then leads to Eb ∼ 0.22 or 0.42 eV. STS also shows an quasiparticle band gap of 2.51 eV (2.59 eV) at 77 K for monolayer WSe2 (WS2) on HOPG [56]. The exciton resonance in monolayer WSe2 is measured to be ∼1.65 eV at room temperature [56], which then leads to Eb ∼ 0.86 eV. These measured X0 binding energy are in the same order of magnitude to those ab initio results. Such large values of Eb (one order of magnitude stronger than in bulk TMDs [57] and conventional GaAs-type quantum wells) come from the large effective masses of both the electron and hole, the spatial confinement in the out-of-plane direction and the reduced screening of the dielectric environment. The 2D nature of monolayer TMDs enhances the binding energy to four times of that in 3D case. The dielectric mismatch between the TMDs and the substrates/air enhances Eb even further, because the electric field line between the electron and the hole can penetrate into the air and the weakly screened substrates. The spatial-dependent effective dielectric constant (weaker at larger electron–hole separation) results in a significant deviation from a 2D hydrogen model, as indicated by the measured exciton Rydberg series [18,20,22]. The actual number on the exciton binding energy varies between measurements. The effects of substrates, sample fabrications on the exciton binding energy are not well understood. And in extraction of binding energy from the difference of exciton resonance from the quasiparticle band gap measured in STS measurements, care shall be taken in the attribution of conduction and valence band edges in the STS as the higher energy critical points Γ and Q can contribute much larger weight in the scanning tunneling spectra than the K points [3,24]. These all need further experiments to clarify. Usually excitons are classified into Frenkel excitons and Wannier–Mott excitons. In Frenkel excitons, the separation between the electron and hole is in the order of the unit cell and the binding energy is typically ∼ 0.1 − 1 eV. For typical Wannier–Mott exciton, the electron–hole separation is much larger than lattice constant, and the binding energy is much weaker than that of Frenkel excitons. Here in 2D TMDs, the wavefunction is still largely of the Wannier–Mott type, while the binding energy is comparable to the typical Frenkel exciton. The calculated 2D Bohr radius is about aB ∼ 1 nm [17], and the wavefunction for the electron–hole relative motion extends over several tens unit cells [17,20]. For trions in TMDs, a variational method shows that the average electron–hole distances are ∼1 and ∼2.5 nm [58]. EXCITON RADIATIVE AND NON-RADIATIVE DECAY Excitons can recombine radiatively and non-radiatively. The radiative lifetime of the bright X0 inside the light cone is determined by its oscillator strength and thus is also called intrinsic lifetime. A measurement using optical 2D coherent spectroscopy gives an exciton homogeneous linewidth of ∼meV in WSe2 [59]. This homogeneous linewidth can be attributed to the overall effect from excitonic radiative, non-radiative decay and pure dephasing, it then gives a lower bound of ∼0.2 ps to the exciton intrinsic radiative lifetime. There still lacks direct measurement on the intrinsic lifetime. For an ensemble of X0, those with momentums outside the light cone are unable to radiatively recombine but these excitons can be scattered into the light cone, e.g. by impurity or phonon scattering, and thus they act as a reservoir, so the ensemble averaged radiative lifetime could be significantly increased [60]. The reported averaged radiative lifetime can reach ∼ns scale [61–64]. The non-radiative decay rates sensitively depend on experimental parameters like exciton density, temperature or sample quality. The observed time-resolved PL and absorption signals in monolayer TMDs exhibit multi-exponential decays [62,63,65,66], indicating complex exciton dynamics. An important intrinsic non-radiative decay channel is the exciton–exciton annihilation. When two excitons collide, one exciton can recombine and transfer the excess energy to the second exciton, which is then ionized and becomes a free electron–hole pair. The annihilation rate increases with ratio Eb/Eg between the binding energy and the band gap, and is proportional to the exciton density. In monolayer TMDs, Eb/Eg is large because of the strong Coulomb binding, and thus it is expected that this non-radiative process dominates when the exciton density is high. An exciton–exciton annihilation rate of 0.2 − 0.5 cm2s− 1 has been estimated for MoSe2 [66], WS2 [21] and WSe2 [62], which gives an exciton–exciton annihilation lifetime of 2–5 ps for a density ∼1012 cm−2. In monolayer MoS2, the rate of exciton–exciton annihilation was reported to be ∼0.04 cm2s–1 [61]. Other non-radiative decay channels include interband carrier-phonon scattering [63], exciton captured by defect states [67,68] and relaxing to dark states [64,67]. BRIGHT EXCITON VALLEY POLARIZATION AND VALLEY COHERENCE For X0, the bright A exciton has the valley degeneracy that can be described as a pseudospin σ. |$\hat \sigma _z = + 1$| and |$\hat \sigma _z = - 1$| correspond respectively to the exciton being in the K and −K valley (Fig. 2). The valley optical selection rule for interband transition correlates the excitonic valley pseudospin and the polarization of the photon: K (−K) valley bright X0 can be interconverted with a σ+ (σ−) circularly polarized photon [25] (Fig. 2). This has been observed in polarization-resolved PL experiments in different monolayer TMDs [13,27–29,69]. Figure 2. Open in new tabDownload slide Optical addressability of valley pseudospin of bright X0. A bright X0 in the K valley corresponds to valley pseudospin up (⁠|$\hat \sigma _z = + 1$|⁠, north pole on the Bloch sphere), which emits σ+ circularly polarized photon, while bright X0 in the −K valley corresponds to valley pseudospin down (⁠|$\hat \sigma _z = - 1$|⁠, south pole on the Bloch sphere), which emits σ− circular polarized photon. The Bloch sphere equator corresponds to bright X0 with an in-plane valley pseudospin (equal superposition of the two valleys), which emits linearly polarized photon with the polarization indicated by green double arrows. Figure 2. Open in new tabDownload slide Optical addressability of valley pseudospin of bright X0. A bright X0 in the K valley corresponds to valley pseudospin up (⁠|$\hat \sigma _z = + 1$|⁠, north pole on the Bloch sphere), which emits σ+ circularly polarized photon, while bright X0 in the −K valley corresponds to valley pseudospin down (⁠|$\hat \sigma _z = - 1$|⁠, south pole on the Bloch sphere), which emits σ− circular polarized photon. The Bloch sphere equator corresponds to bright X0 with an in-plane valley pseudospin (equal superposition of the two valleys), which emits linearly polarized photon with the polarization indicated by green double arrows. The X0 valley polarization is defined as |$P_{+-} \equiv \frac{{\langle\hat B_{{\bf k}, + }^{\dagger} \hat B_{{\bf k}, + }\rangle -\langle \hat B_{{\bf k}, - }^{\dagger} \hat B_{{\bf k}, - }\rangle }}{{\langle\hat B_{{\bf k}, + }^{\dagger} \hat B_{{\bf k}, + }\rangle +\langle \hat B_{{\bf k}, - }^{\dagger} \hat B_{{\bf k}, - }\rangle }}$|⁠, which is directly reflected in the circular polarization of the luminescence. In the polarization-resolved PL measurements, circularly polarized laser excites electron–hole pairs selectively in a valley, which then form valley-polarized excitons. Intervalley relaxation leads to the decay of the valley polarization P+−. The observed large PL circular polarization indicates that the valley relaxation time is larger than the exciton radiative lifetime [27,70–72]. However, as the exciton decay dynamics has not been observed, the direct extraction of valley relaxation timescale from the PL polarization may not be reliable. A linearly polarized photon is in a coherent superposition of σ+ and σ− polarization state. Thus, by the valley optical selection rule, linearly polarized optical field can generate X0 in a coherent superposition of the two valley configurations, transferring the optical coherence to valley quantum coherence [13,25] (Fig. 2). The bright X0 with an in-plane valley pseudospin |$\hat B_{{\bf k},{\rm V}}^{\dagger} \equiv \frac{{{\rm e}^{ - {\rm i\theta }} \hat B_{{\bf k}, + }^{\dagger} + {\rm e}^{{\rm i\theta }} \hat B_{{\bf k}, - }^{\dagger} }}{{\sqrt 2 }}$| (⁠|$\hat B_{{\bf k},{\rm H}}^{\dagger} \equiv \frac{{{\rm e}^{ - {\rm i\theta }} \hat B_{{\bf k}, + }^{\dagger} - {\rm e}^{{\rm i\theta }} \hat B_{{\bf k}, - }^{\dagger} }}{{\sqrt 2 }}$|⁠) couples to the photon with linear polarization along |$\hat x\cos {\rm \theta } + \hat y\sin {\rm \theta }$| (⁠|$\hat x\sin {\rm \theta } - \hat y\cos {\rm \theta }$|⁠). The PL linear polarization is then |$P_{\rm VH} \equiv \frac{\langle \hat B_{{\bf k},{\rm V}}^{\dagger} \hat B_{{\bf k},{\rm V}}\rangle\,{-}\,\langle\hat B_{{\bf k},{\rm H}}^{\dagger} \hat B_{{\bf k},{\rm H}}\rangle}{\langle\hat B_{{\bf k},{\rm V}}^{\dagger}\hat B_{{\bf k},{\rm V}}\rangle\,{+}\,\langle\hat B_{{\bf k},{\rm H}}^{\dagger}\hat B_{{\bf k},{\rm H}}\rangle}\le \frac{2| \langle{\hat B_{{\bf k}, + }^{\dagger} \hat B_{{\bf k}, - } } \rangle |}{\langle\hat B_{{\bf k}, + }^{\dagger} \hat B_{{\bf k}, + }\rangle + \langle\hat B_{{\bf k}, - }^{\dagger} \hat B_{{\bf k}, - }\rangle}$|⁠. The maximum value of PVH as a function of θ direction gives the exciton valley coherence |$\big| \langle{\hat B_{{\bf k}, + }^{\dagger} \hat B_{{\bf k}, - } } \rangle\big|$|⁠. Under the excitation by linearly polarized laser, it is observed that the PL of X0 always carries the same linear polarization with the incident laser regardless of the crystalline orientation, an evidence of the optically generated valley coherence [13]. Since both the valley relaxation and valley pure dephasing lead to the decay of valley coherence, it is expected the valley pure dephasing time is also longer than the X0 radiative lifetime. In these polarization-resolved PL measurements, the optically generated excitons will experience various scattering processes including carrier–carrier Coulomb interaction, carrier-phonon and carrier-impurities scattering. They are dominated by intravalley scattering owing to the large k-space separation between the two valleys. The intravalley scattering does not couple to the valley pseudospin of excitons, thus preserving both the valley polarization and coherence. The intervalley scattering induces both valley depolarization and valley pure dephasing, and it involves the short wavelength component of Coulomb potential (V (K), which is typically very weak), or short wavelength phonons, or atomically sharp impurities. On the other hand, the electron–hole exchange interaction behaves like an in-plane effective magnetic field depending on the exciton center-of-mass motion, and gives rise to valley depolarization and decoherence [73]. Such effective magnetic field vanishes for k = 0 excitons generated by perpendicularly incident lasers. Nevertheless, combined with momentum scattering it can still lead to the decay of PL circular and linear polarizations (see detail in the next section). The bright trions also have the optical selection rule determined by its recombining electron–hole pair (Fig. 1). The observed trion PL shows strong circular polarization just like X0. But the linear polarization for X− is absent owing to the exchange-interaction-induced energy splitting [13] (see details in the next section). X+ cannot emit linear polarized photon because it only has two valley configurations with the excess hole in opposite valleys. When a linear superposition of these two X+ configuration emit a photon, the photon polarization is entangled with the valley pseudospin of the remaining hole, which eliminates the coherence between the σ+ and σ− polarization states of the photon. EXCITON FINE STRUCTURE FROM COULOMB EXCHANGE The binding of the electrons and holes into neutral and charged excitons is primarily due to the direct part of the Coulomb interaction. Coulomb interaction also has the exchange part. In the context of valley excitons, the exchange Coulomb interaction results in diagonal energy shift and off-diagonal coupling on the valley configurations, as summarized in Fig. 3a and b. There are 10 exchange-coupling terms, five are shown in the figure and the rest can be obtained by time reversal. In Fig. 3a and b, the right columns are the diagrams for the Coulomb exchange between the conduction and valence electrons, and the left columns schematically illustrate the corresponding effect on the electron and hole in their valley configurations. Processes II–IV are electron–hole exchange. Processes I and V correspond to electron–electron exchange, where process V is usually suppressed due to the energy mismatch from the conduction band spin splitting ranging from a few to a few tens of meV. Figure 3. Open in new tabDownload slide The diagonal energy shift (a) and off-diagonal coupling (b) due to the Coulomb exchange interaction between electron and hole and between electrons. (c) The effect of Coulomb exchange on X0 and X− in WX2. Dashed thin double arrows denote a diagonal energy shift (I, II, III) of the valley excitons, while the solid thick double arrows denote the off-diagonal coupling (IV) between different valley configurations. Figure 3. Open in new tabDownload slide The diagonal energy shift (a) and off-diagonal coupling (b) due to the Coulomb exchange interaction between electron and hole and between electrons. (c) The effect of Coulomb exchange on X0 and X− in WX2. Dashed thin double arrows denote a diagonal energy shift (I, II, III) of the valley excitons, while the solid thick double arrows denote the off-diagonal coupling (IV) between different valley configurations. For the two bright X0 in opposite valleys, only processes III and IV are involved (Fig. 3c). Process III gives an overall energy shift independent of valley, while IV leads to a coupling between the two valleys. Using the valley pseudospin raising/lower operators |$\hat \sigma _ + \equiv \hat B_{{\bf k} + }^{\dagger} \hat B_{{\bf k} - } $| and |$\hat \sigma _ - \equiv \hat B_{{\bf k} - }^{\dagger} \hat B_{{\bf k} + } $|⁠, the effect of electron–hole exchange interaction can be written as \begin{equation} \hat H_{0,{\rm ex}} = J_{\bf k}^{{\rm intra}} + \left( {J_{\bf k}^{{\rm inter}} \hat \sigma _ + + {\rm h}.{\rm c}.} \right).\end{equation} (1) Here, |$J_{\bf k}^{{\rm inter}} $| (⁠|$J_{\bf k}^{{\rm intra}} $|⁠) is the intervalley coupling (intravalley energy shift) which comes from the process IV (III). |$\hat H_{0,{\rm ex}} $| splits the bright X0 into two branches. Each branch is the equal superposition of the two valleys, i.e. the valley pseudospin lies in the plane (⁠|$\langle\hat \sigma _z\rangle = 0$|⁠), and has a dispersion |$\hbar \omega _0 + \frac{{\hbar ^2 k^2 }}{{2M_0 }} + J_{\bf k}^{{\rm intra}} \pm \left| {J_{\bf k}^{{\rm inter}} } \right|$| (Fig. 4a and b). Figure 4. Open in new tabDownload slide (a) Valley–orbit splitting of bright X0 by the intervalley electron–hole exchange |$J_{\bf k}^{{\rm inter}} $| in the presence of the C3 rotational symmetry. (b) Valley–orbit coupled bright X0 dispersions when both the intervalley electron–hole exchange |$J_{\bf k}^{{\rm inter}} $| and the intravalley electron–hole exchange |$J_{\bf k}^{{\rm intra}} $| are taking into account. The latter is a valley-independent energy shift that renormalize the dispersion of both the upper and lower branches. The single-headed green arrows denote the excitonic valley pseudospin. The inset is the dispersion within the light cone, and the double-headed arrows denote the linear polarization of the photon emission. (c) and (d) Bright X0 dispersions in the presence of a tensile strain (breaking C3 rotational symmetry), without (c) and with (d) the intravalley exchange correction |$J_{\bf k}^{{\rm intra}} $|⁠, respectively. Part (a) and (c) are partly adapted from [32]. Copyright 2014, Nature Publishing Group. Figure 4. Open in new tabDownload slide (a) Valley–orbit splitting of bright X0 by the intervalley electron–hole exchange |$J_{\bf k}^{{\rm inter}} $| in the presence of the C3 rotational symmetry. (b) Valley–orbit coupled bright X0 dispersions when both the intervalley electron–hole exchange |$J_{\bf k}^{{\rm inter}} $| and the intravalley electron–hole exchange |$J_{\bf k}^{{\rm intra}} $| are taking into account. The latter is a valley-independent energy shift that renormalize the dispersion of both the upper and lower branches. The single-headed green arrows denote the excitonic valley pseudospin. The inset is the dispersion within the light cone, and the double-headed arrows denote the linear polarization of the photon emission. (c) and (d) Bright X0 dispersions in the presence of a tensile strain (breaking C3 rotational symmetry), without (c) and with (d) the intravalley exchange correction |$J_{\bf k}^{{\rm intra}} $|⁠, respectively. Part (a) and (c) are partly adapted from [32]. Copyright 2014, Nature Publishing Group. The exact forms of |$J_{\bf k}^{{\rm inter}} $| and |$J_{\bf k}^{{\rm intra}} $| depend on the X0 wavefunction. Symmetry analysis shows that up to the leading order of k, |$J_{\bf k}^{{\rm inter}} $| has a form of |$J_{\bf k}^{{\rm inter}} \propto - \left| {\psi \left( {{\bf r}_{{\rm eh}} = 0} \right)} \right|^2 V\left( {\bf k} \right)k^2 {\rm e}^{ - 2{\rm i\theta }} $| with ψ (reh) the real space wavefunction of the electron–hole relative motion, V (k) the k-space Coulomb potential, |$\theta \equiv {\rm atan}\frac{{k_x }}{{k_y }}$|⁠, and |$J_{\bf k}^{{\rm intra}} = \left| {J_{\bf k}^{{\rm inter}} } \right| + {\rm Const}.$| [32]. The factor e−2iθ is from the requirement of in-plane rotational symmetry, which further dictates that each split X0 branch has a chirality index 2. Inside the light cone (k ≤ ω0/c ∼ 10− 3K), the upper (lower) branch is coupled to linear polarized photons with polarization directions longitudinal (transverse) to the k direction, and the splitting |$2\left| {J_{\bf k}^{{\rm inter}} } \right|$| corresponds to the longitudinal-transverse (LT) splitting, which is well known in GaAs quantum wells. In monolayer TMDs, because of the large Coulomb interaction the LT splitting is greatly enhanced, about two orders of magnitude larger than in GaAs quantum wells [32]. For the unscreened Coulomb interaction of form |$V\!\left( {\bf k} \right) = \frac{{2\pi e^2 }}{{\varepsilon k}}$|⁠, it is found that |$J_{\bf k}^{{\rm inter}} = - \frac{k}{K}J\!{\rm e}^{ - 2{\rm i\theta }} $| with J ∼ 1 eV and K the distance between the Γ and ±K points of the Brillouin zone [32]. The dispersion is shown in Fig. 4a and b. Inside the light cone, the upper branch shows a close to linear dispersion, and thus can be viewed as a massless Dirac particle with chirality 2. The LT splitting near the edge of the light cone is estimated to be ∼meV, which could be much larger than the bright X0 intrinsic line width. The LT splitting can induce valley depolarization and decoherence [73–75]. Depending on the momentum scattering rate τ−1, the system can be divided into strong scattering (⁠|${\rm \bar \Omega }\tau \ll 1$|⁠) or weak scattering (⁠|${\rm \bar \Omega }\tau \gtrsim 1$|⁠) regime, where |${\rm \bar \Omega }$| is the ensemble averaged LT splitting value. In the strong scattering regime, the X0 dynamics can be described by the motional narrowing effect and the valley relaxation rate is given by |$\sim {\rm \bar \Omega }^2 \tau $| [73–75]. Since in monolayer TMDs the splitting is very large, it is possibly in the weak scattering regime. Further theoretical studies are needed to understand the role of LT splitting on the exciton valley relaxation and decoherence in the weak scattering regime. The 3-fold rotational (C3) symmetry of the lattice dictates that |$J_{\bf k}^{{\rm inter}} $| vanishes at k = 0, as X0 with zero center-of-mass wavevector inherits the C3 symmetry of ±K points. A finite |$J_{{\bf k} = 0}^{{\rm inter}} $| would lead to bright X0 eigenstates which can emit linearly polarized photons, thus violates the rotational symmetry. An in-plane uniaxial strain breaks the rotational symmetry and gives rise to a non-zero |$J_{{\bf k} = 0}^{{\rm inter}} $|⁠, which acts like an in-plane Zeeman field on the valley pseudospin and modifies the bright X0 dispersion [32] (see Fig. 4c and d). Exchange interaction also affects trions. For the low-energy configurations of X+, there is only the diagonal energy shift: process II for dark X+ and process III for bright X+. They induce different energy shifts for bright and dark X+. X− has more valley and spin configurations. The bright X− configurations are formed when a bright X0 with valley pseudospin |$\hat \sigma _z $| binds a low-energy excess electron with spin |$\hat s_z $|⁠, and thus they can be characterized by these two indices. Here we focus on the ground-state configurations of bright X−. In MoX2, the ground state of bright X− has 2-fold degeneracy (⁠|$\hat \sigma _z \hat s_z = - 1$|⁠, c.f. Fig. 1), similar to bright X+, and the exchange interaction only leads to a trivial energy shift. WX2 is different because of the opposite conduction band spin splitting from the MoX2 case (Fig. 1), and the ground states of bright X− can have four configurations. First, the exchange processes I and II exist only for the configurations with |$\hat \sigma _z \hat s_z = 1$|⁠, which leads to an energy splitting δ between |$\hat \sigma _z \hat s_z = 1$| and −1 (Fig. 3c). Second, process IV couples configurations |$\hat \sigma _z = 1$| and −1 with the same |$\hat s_z $|⁠. Third, process III induces a global energy shift independent of |$\hat \sigma _z $| and |$\hat s_z $|⁠. For bright X− with a center-of-mass wavevector |$ - \hat s_z {\bf K} + {\bf k}$|⁠, its total exchange Hamiltonian is \begin{eqnarray} \hat H_{- ,{\rm ex}} &=& J_{\bf k}^{{\rm intra}} \,{+}\, \frac{\delta }{2} ( {\hat\sigma_z \hat s_z \,{+}\, 1} ) \,{+}\, ( {J_{\bf k}^{{\rm inter}} \hat \sigma_+ \,{+}\, {\rm h.c.}} ).\!\!\!\!\nonumber\\ &&\\\nonumber \end{eqnarray} (2) The splitting δ is nearly independent on k because processes I and II correspond to Coulomb scattering with a large wavevector ∼K. Its strength is estimated to be δ ∼ 6 meV [32]. The values of |$J_{\bf k}^{{\rm inter}} $| and |$J_{\bf k}^{{\rm intra}} $| depend on the X– wavefunction. Nevertheless from symmetry analysis, |$J_{{\bf k} = 0}^{{\rm inter}} = 0$| because of the 3-fold rotational symmetry, and |$J_{\bf k}^{{\rm inter}} \approx - \frac{k}{K}J\!{\rm e}^{ - 2{\rm i\theta }} $| with comparable magnitude to that of X0. And |$J_{\bf k}^{{\rm intra}} \approx \left| {J_{\bf k}^{{\rm inter}} } \right| + {\rm Const}$|⁠. Processes I and II between the recombining electron–hole pair and the excess electron therefore act as an out-of-plane Zeeman field (with the sign dependent on |$\hat s_z $|⁠) on the valley pseudospin |$\hat \sigma _z $|⁠. The coherent superposition of the two configurations with |$\hat \sigma _z = 1$| and −1 with the same |$\hat s_z $| is then destroyed by this effective Zeeman field. Therefore, X– PL cannot be linearly polarized in monolayer TMDs [13]. The direct observation of the excitonic fine structure is still challenging with the existing sample qualities. The estimated energy splitting is ∼meV while the measured PL linewidth is at least ∼5 meV in MoSe2 and WSe2 [13,14] and much larger in MoS2 and WS2 [15,21]. Nevertheless, there have been indirect evidences supporting these predictions. As mentioned above, the splitting between X− configurations due to the exchange interaction (processes I and II, c.f. Fig. 3) explains the absence of linear polarization in the X− emission in contrast to the X0 emission [13]. It is also consistent with the observation that in bilayer WSe2 the interlayer trion can have linear polarization, because such exchange interaction is suppressed when the recombining electron–hole pair and the excess electron are located in opposite layers [31]. Moreover, in a perpendicular magnetic field, the polarizations of the X0 and X− PL peaks show distinct magnetic field dependence, which can be explained by the different exchange fine structures of X0 and X− (see the section entitled ‘Magneto response of valley excitons’) [33]. BERRY CURVATURE AND VALLEY HALL EFFECT OF TRION In an in-plane electric field E, a charged particle can acquire a transverse anomalous velocity due to the Berry phase effect, in addition to the normal velocity in the longitudinal direction. This may be described by the semiclassical equation of motion for a wavepacket: |$\hbar {\bf \dot r} = \frac{{\partial E_{\bf k} }}{{\partial {\bf k}}} - e{\bf E} \times {\bf \Omega }_{\bf k} $| [76], where Ek is the energy dispersion, e is the charge of the carrier (negative for electron), and r and k are the central coordinates of the wavepacket in real and momentum space, respectively. Ωk is the Berry curvature that characterizes the Berry phase effect [76]. It arises from the dependence of the internal structure of particle wavefunction on the center-of-mass wavevector. For Bloch electron, Ωk = i〈∇kuk| × |∇kuk〉, where uk is the periodic part of the Bloch wavefunction. In monolayer TMDs, because of the inversion symmetry breaking, the conduction and valence band both acquire finite Berry curvatures at the K and −K valleys [2], where the time-reversal symmetry dictates that the curvature must have opposite signs at K and −K in each band. Thus, in an in-plane electric field, carriers at the two valleys will flow in opposite transverse directions, i.e. a valley Hall effect [2,25,26,77,78]. Trion is also a charged particle. The dependence of the internal structure of trion wavefunction on the center-of-mass wavevector can give rise to a similar gauge structure to that of the electron [79]. An in-plane electric field can induce an anomalous transverse motion and give rise to the Hall effects of trion. In monolayer TMDs, the trions as composite particles can acquire valley-dependent Berry curvature from two distinct origins, from the inheritance of the Berry curvature from the Bloch band, and from the Coulomb exchange interaction between the electron and hole constituents. For the positively charged trion X+, its Berry curvature is mainly inherited from the Bloch bands [79]. The electrons and holes in ±K valley have nearly k-independent values of Berry curvature Ω± K + k ∼ ± 10 Å2 [25]. The bright X+ then acquires a Berry curvature given by the sum of the curvatures of its electron and hole constituents. As the Berry curvatures from the two holes in opposite valleys cancel, the X+ Berry curvature is determined by the electron constituent, and thus the two valley configurations of bright X+ have opposite Berry curvatures, giving rise to the valley Hall effect. The trion valley Hall effect can lead to valley polarization of trions with opposite signs at the two edges, which can be detected as the circularly polarized luminescence since the trion is associated with the valley optical selection rule. Also, the excess hole left behind upon the trion recombination is valley- and spin-polarized, and the trion valley Hall effect can therefore be exploited for the generation of spin and valley polarization of carriers. Similarly, bright X− will also inherit the Berry curvature from its electron and hole constituents. Moreover, in WX2, a much stronger Berry curvature can arise from the exchange interactions between the electrons and holes as shown in Equation (2). For bright X− with a center-of-mass wavevector |$ - \hat s_z {\bf K} + {\bf k}$|⁠, the out-of-plane Zeeman exchange term |$\frac{\delta }{2}\hat \sigma _z \hat s_z $| together with the intervalley exchange term |$J_{\bf k}^{{\rm inter}} $| gives rise to a Berry curvature [32]: \begin{equation} {\rm \Omega }_{ - \hat s_z {\bf K} + {\bf k}} = - \hat \sigma _z \frac{{2J^2 }}{{K^2 \delta ^2 }}\left( {1 + \frac{{4k^2 J^2 }}{{K^2 \delta ^2 }}} \right)^{ - \frac{3}{2}} . \end{equation} (3) The sign of |${\rm \Omega }_{ - \hat s_z {\bf K} + {\bf k}} $| is determined by |$\hat \sigma _z $|⁠, the valley index of the recombining electron–hole pair (Fig. 5), and it is independent of spin index |$\hat s_z $| of the excess electron. With the estimated exchange coupling strength J ∼ 1 eV and δ ∼ 6 meV, the peak value of |$\left| {{\rm \Omega }_{ - \hat s_z {\bf K} + {\bf k}} } \right|$| can be ∼ 104Å2 in the neighborhood of k = 0, which is several orders larger than the curvature of the electron or hole. The luminescence polarization of bright X− is determined by |$\hat \sigma _z $| through the valley optical selection rule, and thus the X− valley Hall effect may also be detected from a spatially dependent PL polarization pattern (Fig. 5), similar to the X+ valley Hall effect. Figure 5. Open in new tabDownload slide (a) The four valley configurations of bright X− characterized by the valley pseudospin |$\hat \sigma _z $| of the recombining electron–hole pair, and spin |$\hat s_z $| of the excess electron. (b) The valley Hall effect of the bright X−. Under an in-plane electric field E, X− with different |$\hat \sigma _z $| will move to opposite transverse directions, and therefore the trion luminescence can have a spatially dependent polarization pattern. Figure 5. Open in new tabDownload slide (a) The four valley configurations of bright X− characterized by the valley pseudospin |$\hat \sigma _z $| of the recombining electron–hole pair, and spin |$\hat s_z $| of the excess electron. (b) The valley Hall effect of the bright X−. Under an in-plane electric field E, X− with different |$\hat \sigma _z $| will move to opposite transverse directions, and therefore the trion luminescence can have a spatially dependent polarization pattern. Various relaxation mechanisms can inhibit the observation of the trion valley Hall effect. If the population decay is fast, the trions with opposite Berry curvatures cannot move sufficiently away from each other during their lifetime. A valley flip process (valley relaxation) changes the sign of the Berry curvature and the direction of the transverse motion, suppressing the valley Hall current. Also, we note that the upper and lower energy branches of X− dispersions have the opposite Berry curvature, and the two branches are separated only by a small energy gap of a few meV in the light cone. Non-adiabatic dynamics can then cause the transitions between the two branches, which will also diminish the valley Hall effect. The non-adiabaticity can come from the momentum scattering which changes the X− wavevector, so the scattering rate shall not be too large in order to observe the trion valley Hall effect. A room temperature X0 diffusion coefficient of D ∼ 12 − 15 cm2s− 1 has been measured in monolayer MoSe2 [80] and WSe2 [81], the relation D = τkBT/M0 then leads to the X0 scattering time τ ∼ 0.2 ps. Unlike X0, X± are charged particles so the interactions with charged impurities and piezoelectric types of phonons should be stronger. We expect the X± momentum scattering time to be shorter than X0. A low temperature and clean sample shall facilitate the observation of trion valley Hall effect. MAGNETO RESPONSE OF VALLEY EXCITONS In this section, we discuss the response of valley excitons to external magnetic fields. Compared to excitons in conventional semiconductors such as GaAs, a remarkably new feature in monolayer TMDs is that an exciton configuration and its time-reversal counterpart have different valley configurations. The momentum mismatch between these distinct valley configurations will suppress their off-diagonal coupling in various occasions. For example, in monolayer TMDs, the two valley configurations of bright X0 cannot be coupled by the in-plane magnetic field as it cannot supply the momentum mismatch. Experimentally, it has been shown that the exciton valley polarization in monolayer MoS2 has no response to in-plane magnetic field up to 9 T [28,69]. On the other hand, the valley excitons show interesting response to out-of-plane magnetic field. In monolayer TMDs, the mirror symmetry about the metal atom plane dictates that the magnetic moments of the carriers are in the out-of-plane (z) direction. And the time-reversal symmetry requires the magnetic moments of the ±K valleys to have the same magnitude but opposite signs. An out-of-plane magnetic field then lifts the valley degeneracy, i.e. a valley Zeeman effect. With the valley optical selection rule, the valley Zeeman effect may be detected from the polarization-resolved PL measurement, where the magnetic field is expected to split the σ+ and σ− PL peaks that correspond to the interband transition energies in valley K and −K, respectively [33–36]. The overall valley Zeeman splitting has three contributions [33]. The first is the spin magnetic moment. This contribution does not affect the optical resonances, because optical transitions conserve spin so that the shift of the initial and final states due to the spin magnetic moment is the same [25]. The second is the magnetic moment of atomic orbital or the intracellular component [33–36]. The conduction (valence) band in the ±K valley mainly consists of transition-metal |${\rm d}_{z^2 } $| (⁠|${\rm d}_{x^2 - y^2 } \pm i{\rm d}_{xy} $|⁠) orbitals with the magnetic quantum m = 0 (m = ±2). This contributes to a Zeeman shift of 0 and ± 2μBB for the conduction and valence band, respectively. Therefore, it is a major contribution to the magneto splitting of the σ+ and σ− PL peaks. The third is the valley magnetic moment associated with the Berry phase effect (or the lattice contribution) [33–36]. Note that within the minimum two-band k·p model for band edge electrons and holes in monolayer TMDs (i.e. massive Dirac fermion model [25]), because of the particle-hole symmetry, the valley magnetic moment (lattice contribution) contributes identical Zeeman shift for conduction and valence band. Nevertheless, corrections beyond this model result in a finite difference for the electron and hole valley magnetic moment, and it is this difference, as well as the atomic orbital contribution, that is measured by the splitting of the σ+ and σ− PL peaks [33–36]. It is also found that such a perpendicular magnetic field can be used for tuning the polarization and coherence of the excitonic valley pseudospin. Monolayer MoSe2 shows no PL polarization in the absence of the magnetic field. Applying a perpendicular magnetic field lifts the valley degeneracy and creates a population imbalance in the two valleys, which then leads to a field-dependent PL polarization [35,36]. In monolayer WSe2, the valley polarization is preserved during the exciton and trion lifetime, so the PL polarization sign is determined by the helicity of the excitation laser. The polarization of X0 either increase or decrease with magnetic field depending on the relationship between the pump light helicity and the magnetic field direction [33]. In contrast, the X− PL polarization always increases with magnetic field [33,34]. Such distinct behaviors of bright X0 and X− are explained by their qualitative different dispersion relations induced by the exchange interaction (see Equations (1) and (2) in the previous section), which result in different valley depolarization processes [33]. On the other hand, the valley coherence is always suppressed when applying an out-of-plane magnetic field, because the lifting of the valley degeneracy destroys the coherent pathways in the X0 formation process [33]. EXCITONS IN BILAYER TMDS HOMO- AND HETEROSTRUCTURES When monolayer TMDs are stacked on top of each other, the van der Waals interaction can bound them into homostructures as well as various heterostructures. These offer new possibility to study the physics of excitons, in particular the interlayer excitons where the composite particles reside in different layers. Interlayer trions have been observed in WSe2 homobilayers [31], and neutral interlayer exciton has been observed in TMDs heterobilayers [82–86]. These interlayer excitons and trions exhibit remarkable properties including the tunability by the interlayer bias. Below, we briefly introduce these interlayer excitons. In TMDs homo- or heterobilayers, the interlayer coupling between states at the K valleys is weak compared to the energy mismatch and hence the hybridization between the layers is negligible for these states [3]. Concerning the valley excitons with the electron and hole all from the K valleys of the two layers, each of these electron or hole constituents is largely localized either in the upper or lower layer. The layer separation (∼7 Å) is comparable to the intralayer exciton Bohr radius (∼1 nm), and thus the strong Coulomb interaction between the electrons and holes in different layers can bind them into interlayer neutral or charged excitons. The interlayer X0 has been observed in monolayer MoSe2-WSe2 vertical heterostructures through PL and PL excitation spectroscopy [82–86]. These heterostructures have the type-II band alignment, and therefore the lowest energy configuration of X0 is an interlayer one, with the electron and hole residing in opposite layers. Taking the MoSe2-WSe2 vertical heterostructures for example [82], the conduction band minimum (valence band maximum) is located in the MoSe2 (WSe2) layer, and thus the interlayer X0 has an energy much lower than the intralayer X0. While interlayer hopping is substantially quenched for both the electron and hole by the conduction band offset and valence band offset, respectively, its residue effect leads to small-layer hybridization that allows the direct radiative recombination of interlayer X0. The radiative recombination of interlayer X0 gives a distinct PL peak. The spatially indirect nature reduces the optical dipole of interlayer X0, thus substantially extends its lifetime, which is observed to exceed nanosecond [82]. Another unique aspect of the interlayer X0 is that it corresponds to a permanent electric dipole in the out-of-plane direction. This makes possible the tuning of its energy by the interlayer bias. Moreover, it also gives rise to strong dipole–dipole repulsive interaction, observed as a blue shift in the exciton resonance with increasing power [82]. The repulsive interaction and the ultralong lifetime of interlayer exciton makes it an ideal candidate to explore the exotic phenomenon of excitonic Bose Einstein condensate [87]. In exfoliated WSe2 homobilayer, interlayer trion has been observed [31]. TMDs bilayers exfoliated from the natural crystals are mostly 2H stacking, where the two layers are 180 degree rotation of each other. This 180 degree rotation switches the K and −K valleys. With the spin-valley coupling in each monolayer, the spin-splitting in 2H bilayer has a sign depending on both the valley and the layer index, which quenches the interlayer hopping at the K valleys and results in the spin-layer locking effect: the spin-up and -down states in each valley are localized in opposite layers [2,3]. An out-of-plane electric field can then induce a spin Zeeman splitting [30]. The electrically induced Zeeman splitting of the conduction band ΔEc is larger than the valence band one ΔEv owing to the fact that the electrons (holes) have a close to zero (weak but finite) interlayer hybridization. Considering the intralayer X0, the one localized in the upper layer has an energy difference ΔEc − ΔEv from that in the lower layer in presence of the out-of-plane electric field. In doped bilayer, the intralayer X0 can bind a low-energy excess electron or hole in the same layer (in the opposite layer) to form an intralayer (interlayer) trion. The interlayer trion has a smaller charging energy Δc,inter than the intralayer trion Δc,intra. The total energy difference between the inter- and intralayer trion is (ΔEc − ΔEv) + (Ec, intra − Ec, inter) for X−, and (ΔEc − ΔEv) − (Ec, intra − Ec, inter) for X+. The energy difference is larger for X− than X+. In bilayer WSe2, the splitting between the interlayer and interlayer X− has been observed in the PL, where the X− peak splits into a doublet with the increase of interlayer bias [31]. Large valley polarizations are observed for both the interlayer and the intralayer X− under circularly polarized pump [31,88]. Under linearly polarized pump, however, valley coherence is observed only for the interlayer X−, but not for the intralayer one [31]. This is because the intralayer X− with all three particles localized in the same layer is similar to the trion in a monolayer, where the exchange coupling with the excess electron destroys the valley coherence of the recombining electron–hole pair. In contrast, for interlayer X−, with the excess electron in the opposite layer, its exchange coupling with the recombining electron–hole pair is suppressed and the valley coherence can therefore be preserved. In summary, we provide here an overview of the properties of valley excitons in 2D group-VIB TMDs. For this emerging topic, materials issues and the lack of physical insights on many observed properties make a comprehensive review of various aspects of excitons in these new materials impractical at this stage. Instead, we have focused on the physics associated with valley degrees of freedom, which distinguish 2D TMDs excitons from those in conventional semiconductors. A lot of issues need to be addressed by future experimental and theoretical studies. For example, one of the most urgent issues that need to be thoroughly studied is the exciton relaxation and decoherence mechanisms. The exciton relaxation dynamics in monolayer TMDs are shown to be complicated, and the sample quality and the excitation power both affect the exciton decay timescales. Multiple timescales are observed in time-resolved spectroscopy, showing the relevance of non-radiative decay channels. Further experiments are needed for understanding the various mechanisms of exciton non-radiative decay, as well as the possible roles of dark excitons in the radiative decay. The observation of valley polarization and coherence of excitons in PL experiments imply that the T1 and T2 times of excitonic valley pseudospin are slow compared to the exciton radiative and non-radiative decay. Polarization-resolved pump-probe and spectral hole-burning measurements are providing useful information on the excitonic valley dynamics [64,71,89,90]. But a thorough understanding of the mechanisms and timescales for the relaxation and decoherence of excitonic valley pseudospin is still lacking, which is key to the exploration of the applications and new physics associated with valley excitons. FUNDING HY and WY acknowledge support by the Croucher Foundation (Croucher Innovation Award), the University of Hong Kong (OYRA), and the RGC of Hong Kong (HKU17305914P). XC acknowledges support by RGC (HKU9/CRF/13G) and UGC (AoE/P-04/08) of Hong Kong. XX acknowledges support from DoE, BES, Materials Science and Engineering Division (DE-SC0008145), the National Science Foundation (DMR-1150719), and the National Science Foundation, Office of Emerging Frontiers in Research and Innovation (EFRI-1433496). REFERENCES 1. Wang QH , Kalantar-Zadeh K , Kis A , et al. Electronics and optoelectronics of two-dimensional transition metal dichalcogenides , Nat Nanotechnol , 2012 , vol. 7 (pg. 699 - 712 ) Google Scholar Crossref Search ADS PubMed WorldCat 2. Xu X , Yao W , Xiao D , et al. Spin and pseudospins in layered transition metal dichalcogenides , Nat Phys , 2014 , vol. 10 (pg. 343 - 50 ) Google Scholar Crossref Search ADS WorldCat 3. Liu G-B , Xiao D , Yao Y , et al. Electronic structures and theoretical modelling of two-dimensional group-VIB transition metal dichalcogenides , Chem Soc Rev , 2015 doi: 10.1039/C4CS00301B Google Scholar OpenURL Placeholder Text WorldCat 4. Mak KF , Lee C , Hone J , et al. Atomically thin MoS2: a new direct-gap semiconductor , Phys Rev Lett , 2010 , vol. 105 pg. 136805 Google Scholar Crossref Search ADS PubMed WorldCat 5. Splendiani A , Sun L , Zhang Y , et al. Emerging photoluminescence in monolayer MoS2 , Nano Lett , 2010 , vol. 10 (pg. 1271 - 5 ) Google Scholar Crossref Search ADS PubMed WorldCat 6. Novoselov KS , Jiang D , Schedin F , et al. Two-dimensional atomic crystals , Proc Natl Acad Sci USA , 2005 , vol. 102 (pg. 10451 - 3 ) Google Scholar Crossref Search ADS PubMed WorldCat 7. Liu K-K , Zhang W , Lee Y-H , et al. Growth of large-area and highly crystalline MoS2 thin layers on insulating substrates , Nano Lett , 2012 , vol. 12 (pg. 1538 - 44 ) Google Scholar Crossref Search ADS PubMed WorldCat 8. Zhang Y , Chang T-R , Zhou B , et al. Direct observation of the transition from indirect to direct bandgap in atomically thin epitaxial MoSe2 , Nat Nanotechnol , 2014 , vol. 9 (pg. 111 - 5 ) Google Scholar Crossref Search ADS PubMed WorldCat 9. Zhan Y , Liu Z , Najmaei S , et al. Large-area vapor-phase growth and characterization of MoS2 atomic layers on a SiO2 substrate , Small , 2012 , vol. 8 (pg. 966 - 71 ) Google Scholar Crossref Search ADS PubMed WorldCat 10. van der Zande AM , Huang PY , Chenet DA , et al. Grains and grain boundaries in highly crystalline monolayer molybdenum disulphide , Nat Mater , 2013 , vol. 12 (pg. 554 - 61 ) Google Scholar Crossref Search ADS PubMed WorldCat 11. Najmaei S , Liu Z , Zhou W , et al. Vapour phase growth and grain boundary structure of molybdenum disulphide atomic layers , Nat Mater , 2013 , vol. 12 (pg. 754 - 9 ) Google Scholar Crossref Search ADS PubMed WorldCat 12. Liu H , Jiao L , Yang F , et al. Dense network of one-dimensional midgap metallic modes in monolayer MoSe2 and their spatial undulations , Phys Rev Lett , 2014 , vol. 113 pg. 066105 Google Scholar Crossref Search ADS PubMed WorldCat 13. Jones AM , Yu H , Ghimire NJ , et al. Optical generation of excitonic valley coherence in monolayer WSe2 , Nat Nanotechol , 2013 , vol. 8 (pg. 634 - 8 ) Google Scholar Crossref Search ADS WorldCat 14. Ross JS , Wu S , Yu H , et al. Electrical control of neutral and charged excitons in a monolayer semiconductor , Nat Commun , 2012 , vol. 4 pg. 1474 Google Scholar Crossref Search ADS WorldCat 15. Mak KF , He K , Lee C , et al. Tightly bound trions in monolayer MoS2 , Nat Mater , 2012 , vol. 12 (pg. 207 - 11 ) Google Scholar Crossref Search ADS PubMed WorldCat 16. Feng J , Qian X , Huang C-W , et al. Strain-engineered artificial atom as a broad-spectrum solar energy funnel , Nat Photon , 2012 , vol. 6 (pg. 866 - 72 ) Google Scholar Crossref Search ADS WorldCat 17. Qiu DY , da Jornada FH , Louie SG . Optical spectrum of MoS2: many-body effects and diversity of exciton states , Phys Rev Lett , 2013 , vol. 111 pg. 216805 Google Scholar Crossref Search ADS PubMed WorldCat 18. Chernikov A , Berkelbach TC , Hill HM , et al. Exciton binding energy and nonhydrogenic Rydberg series in monolayer WS2 , Phys Rev Lett , 2014 , vol. 113 pg. 076802 Google Scholar Crossref Search ADS PubMed WorldCat 19. He K , Kumar N , Zhao L , et al. Tightly bound excitons in monolayer WSe2 , Phys Rev Lett , 2014 , vol. 113 pg. 026803 Google Scholar Crossref Search ADS PubMed WorldCat 20. Ye Z , Cao T , O’Brien K , et al. Probing excitonic dark states in single-layer tungsten disulphide , Nature , 2014 , vol. 513 (pg. 214 - 8 ) Google Scholar Crossref Search ADS PubMed WorldCat 21. Zhu B , Chen X , Cui X . Exciton binding energy of monolayer WS2 , 2014 arXiv: 1403.5108 OpenURL Placeholder Text WorldCat 22. Wang G , Marie X , Gerber I , et al. Non-linear optical spectroscopy of excited exciton states for efficient valley coherence generation in WSe2 monolayers , 2014 arXiv:1404.0056 OpenURL Placeholder Text WorldCat 23. Zhang C , Johnson A , Hsu C-L , et al. Direct imaging of band profile in single layer MoS2 on graphite: quasiparticle energy gap, metallic edge states, and edge band bending , Nano Lett , 2014 , vol. 14 (pg. 2443 - 7 ) Google Scholar Crossref Search ADS PubMed WorldCat 24. Ugeda MM , Bradley AJ , Shi S-F , et al. Giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor , Nat Mater , 2014 , vol. 13 (pg. 1091 - 5 ) Google Scholar Crossref Search ADS PubMed WorldCat 25. Xiao D , Liu G-B , Feng W , et al. Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides , Phys Rev Lett , 2012 , vol. 108 pg. 196802 Google Scholar Crossref Search ADS PubMed WorldCat 26. Yao W , Xiao D , Niu Q . Valley-dependent optoelectronics from inversion symmetry breaking , Phys Rev B , 2008 , vol. 77 pg. 235406 Google Scholar Crossref Search ADS WorldCat 27. Mak KF , He K , Shan J , et al. Control of valley polarization in monolayer MoS2 by optical helicity , Nat Nanotechol , 2012 , vol. 7 (pg. 494 - 8 ) Google Scholar Crossref Search ADS WorldCat 28. Zeng H , Dai J , Yao W , et al. Valley polarization in MoS2 monolayers by optical pumping , Nat Nanotechol , 2012 , vol. 7 (pg. 490 - 3 ) Google Scholar Crossref Search ADS WorldCat 29. Cao T , Wang G , Han W , et al. Valley-selective circular dichroism of monolayer molybdenum disulphide , Nat Commun , 2012 , vol. 3 pg. 887 Google Scholar Crossref Search ADS PubMed WorldCat 30. Gong Z , Liu G-B , Yu H , et al. Magnetoelectric effects and valley-controlled spin quantum gates in transition metal dichalcogenide bilayers , Nat Commun , 2013 , vol. 4 pg. 2053 Google Scholar Crossref Search ADS PubMed WorldCat 31. Jones AM , Yu H , Ross JS , et al. Spin-layer locking effects in optical orientation of exciton spin in bilayer WSe2 , Nat Phys , 2014 , vol. 10 (pg. 130 - 4 ) Google Scholar Crossref Search ADS WorldCat 32. Yu H , Liu G-B , Gong P , et al. Dirac cones and Dirac saddle points of bright excitons in monolayer transition metal dichalcogenides , Nat Commun , 2014 , vol. 5 pg. 3876 Google Scholar Crossref Search ADS PubMed WorldCat 33. Aivazian G , Gong Z , Jones AM , et al. Magnetic control of valley pseudospin in monolayer WSe2 , Nat Phys , 2015 doi:10.1038/nphys3201 Google Scholar OpenURL Placeholder Text WorldCat 34. Srivastava A , Sidler M , Allain AV , et al. Valley Zeeman effect in elementary optical excitations of a monolayer WSe2 , 2014 arXiv:1407.2624 OpenURL Placeholder Text WorldCat 35. Li Y , Ludwig J , Low T , et al. Valley splitting and polarization by the Zeeman effect in monolayer MoSe2 , Phys Rev Lett , 2014 , vol. 113 pg. 266804 Google Scholar Crossref Search ADS PubMed WorldCat 36. MacNeill D , Heikes C , Mak KF , et al. Breaking of valley degeneracy by magnetic field in monolayer MoSe2 , 2014 arXiv:1407.0686 OpenURL Placeholder Text WorldCat 37. Sie EJ , McIver JW , Lee Y-H , et al. Valley-selective optical Stark effect in monolayer WS2 , Nat Mater , 2014 doi:10.1038/nmat4156 Google Scholar OpenURL Placeholder Text WorldCat 38. Kim J , Hong X , Jin C , et al. Ultrafast generation of pseudo-magnetic field for valley excitons in WSe2 monolayers , Science , 2014 , vol. 346 (pg. 1205 - 8 ) Google Scholar Crossref Search ADS PubMed WorldCat 39. Ross JS , Klement P , Jones AM , et al. Electrically tunable excitonic light-emitting diodes based on monolayer WSe2 p-n junctions , Nat Nanotechol , 2014 , vol. 9 (pg. 268 - 72 ) Google Scholar Crossref Search ADS WorldCat 40. Pospischil A , Furchi MM , Mueller T . Solar-energy conversion and light emission in an atomic monolayer p-n diode , Nat Nanotechol , 2014 , vol. 9 (pg. 257 - 61 ) Google Scholar Crossref Search ADS WorldCat 41. Baugher BWH , Churchill HOH , Yang Y , et al. Optoelectronic devices based on electrically tunable p-n diodes in a monolayer dichalcogenide , Nat Nanotechnol , 2014 , vol. 9 (pg. 262 - 7 ) Google Scholar Crossref Search ADS PubMed WorldCat 42. Zhang YJ , Oka T , Suzuki R , et al. Electrically switchable chiral light-emitting transistor , Science , 2014 , vol. 344 (pg. 725 - 8 ) Google Scholar Crossref Search ADS PubMed WorldCat 43. Yao W . Valley light-emitting transistor , NPG Asia Mater , 2014 , vol. 6 pg. e124 Google Scholar Crossref Search ADS WorldCat 44. Yu H , Wu Y , Liu G-B , et al. Nonlinear valley and spin currents from Fermi pocket anisotropy in 2D crystals , Phys Rev Lett , 2014 , vol. 113 pg. 156603 Google Scholar Crossref Search ADS PubMed WorldCat 45. Seyler KL , Schaibley JR , Gong P , et al. Valley-selective monolayer second-harmonic generation transistor , 2015 in press OpenURL Placeholder Text WorldCat 46. Mattheiss LF . Band structures of transition-metal-dichalcogenide layer compounds , Phys Rev B , 1973 , vol. 8 pg. 3719 Google Scholar Crossref Search ADS WorldCat 47. Liu G-B , Shan W-Y , Yao Y , et al. Three-band tight-binding model for monolayers of group-VIB transition metal dichalcogenides , Phys Rev B , 2013 , vol. 88 pg. 085433 Google Scholar Crossref Search ADS WorldCat 48. Kormányos A , Zólyomi V , Drummond ND , et al. Monolayer MoS2: trigonal warping, the Γ valley, and spin–orbit coupling effects , Phys Rev B , 2013 , vol. 88 pg. 045416 Google Scholar Crossref Search ADS WorldCat 49. Zeng H , Liu G-B , Dai J , et al. Optical signature of symmetry variations and spin-valley coupling in atomically thin tungsten dichalcogenides , Sci Rep , 2013 , vol. 3 pg. 1608 Google Scholar Crossref Search ADS PubMed WorldCat 50. Esser A , Runge E , Zimmermann R , et al. Photoluminescence and radiative lifetime of trions in GaAs quantum wells , Phys Rev B , 2000 , vol. 62 pg. 8232 Google Scholar Crossref Search ADS WorldCat 51. Mitioglu AA , Plochocka P , Jadczak JN , et al. Optical manipulation of the exciton charge state in single-layer tungsten disulfide , Phys Rev B , 2013 , vol. 88 pg. 245403 Google Scholar Crossref Search ADS WorldCat 52. Cheiwchanchamnangij T , Lambrecht WRL . Quasiparticle band structure calculation of monolayer, bilayer, and bulk MoS2 , Phys Rev B , 2012 , vol. 85 pg. 205302 Google Scholar Crossref Search ADS WorldCat 53. Ramasubramaniam A. . Large excitonic effects in monolayers of molybdenum and tungsten dichalcogenides , Phys Rev B , 2012 , vol. 86 pg. 115409 Google Scholar Crossref Search ADS WorldCat 54. Komsa H-P , Krasheninnikov AV . Effects of confinement and environment on the electronic structure and exciton binding energy of MoS2 from first principles , Phys Rev B , 2012 , vol. 86 pg. 241201 Google Scholar Crossref Search ADS WorldCat 55. Shi H , Pan H , Zhang Y-W , et al. Quasiparticle band structures and optical properties of strained monolayer MoS2 and WS2 , Phys Rev B , 2013 , vol. 87 pg. 155304 Google Scholar Crossref Search ADS WorldCat 56. Chiu M-H , Zhang C , Shiu HW , et al. Determination of band alignment in transition metal dichalcogenides heterojunctions , 2014 arXiv:1406.5137 OpenURL Placeholder Text WorldCat 57. Beal AR , Knights JC , Liang WY . Transmission spectra of some transition metal dichalcogenides. II. Group VIA: trigonal prismatic coordination , J Phys C: Solid State Phys , 1972 , vol. 5 pg. 3540 Google Scholar Crossref Search ADS WorldCat 58. Berkelbach TC , Hybertsen MS , Reichman DR . Theory of neutral and charged excitons in monolayer transition metal dichalcogenides , Phys Rev B , 2013 , vol. 88 pg. 045318 Google Scholar Crossref Search ADS WorldCat 59. Moody G , Dass CK , Hao K , et al. Intrinsic exciton linewidth in monolayer transition metal dichalcogenides , 2014 arXiv:1410.3143 OpenURL Placeholder Text WorldCat 60. Wang H , Zhang C , Chan W , et al. Radiative lifetimes of excitons and trions in monolayers of metal dichalcogenide MoS2 , 2014 arXiv:1409.3996 OpenURL Placeholder Text WorldCat 61. Sun D , Rao Y , Reider GA , et al. Observation of rapid exciton–exciton annihilation in monolayer molybdenum disulfide , Nano Lett , 2014 , vol. 14 (pg. 5625 - 9 ) Google Scholar Crossref Search ADS PubMed WorldCat 62. Mouri S , Miyauchi Y , Toh M , et al. Nonlinear photoluminescence in atomically thin layered WSe2 arising from diffusion-assisted exciton–exciton annihilation , Phys Rev B , 2014 , vol. 90 pg. 155449 Google Scholar Crossref Search ADS WorldCat 63. Shi H , Yan R , Bertolazzi S , et al. Exciton dynamics in suspended monolayer and few-layer MoS2 2D crystals , ACS Nano , 2013 , vol. 7 (pg. 1072 - 80 ) Google Scholar Crossref Search ADS PubMed WorldCat 64. Schaibley JR , Karin T , Yu H , et al. Nanosecond relaxation times of excitonic states in monolayer MoSe2 , 2015 in press OpenURL Placeholder Text WorldCat 65. Korn T , Heydrich S , Hirmer M , et al. Low-temperature photocarrier dynamics in monolayer MoS2 , Appl Phys Lett , 2011 , vol. 99 pg. 102109 Google Scholar Crossref Search ADS WorldCat 66. Kumar N , Cui Q , Ceballos F , et al. Exciton-exciton annihilation in MoSe2 monolayers , Phys Rev B , 2014 , vol. 89 pg. 125427 Google Scholar Crossref Search ADS WorldCat 67. Liu X , Ji Q , Gao Z , et al. Ultrafast terahertz probe of transient evolution of charged and neutral phase of photoexcited electron–hole gas in monolayer semiconductor , 2014 arXiv:1410.2939 OpenURL Placeholder Text WorldCat 68. Wang H , Strait JH , Zhang C , et al. Fast exciton annihilation by capture of electrons or holes by defects via auger scattering in monolayer metal dichalcogenides , 2014 arXiv:1410.3141 OpenURL Placeholder Text WorldCat 69. Sallen G , Bouet L , Marie X , et al. Robust optical emission polarization in MoS2 monolayers through selective valley excitation , Phys Rev B , 2012 , vol. 86 pg. 081301(R) Google Scholar Crossref Search ADS WorldCat 70. Wang G , Bouet L , Lagarde D , et al. Valley dynamics probed through charged and neutral exciton emission in monolayer WSe2 , Phys Rev B , 2014 , vol. 90 pg. 075413 Google Scholar Crossref Search ADS WorldCat 71. Mai C , Barrette A , Yu Y , et al. Many-body Effects in valleytronics: direct measurement of valley lifetimes in single-layer MoS2 , Nano Lett , 2014 , vol. 14 (pg. 202 - 6 ) Google Scholar Crossref Search ADS PubMed WorldCat 72. Wang Q , Ge S , Li X , et al. Valley carrier dynamics in monolayer molybdenum disulfide from helicity-resolved ultrafast pump-probe spectroscopy , ACS Nano , 2013 , vol. 7 (pg. 11087 - 93 ) Google Scholar Crossref Search ADS PubMed WorldCat 73. Maialle MZ , de Andrada E Silva EA , Sham LJ . Exciton spin dynamics in quantum wells , Phys Rev B , 1993 , vol. 47 pg. 15776 Google Scholar Crossref Search ADS WorldCat 74. Yu T , Wu MW . Valley depolarization due to intervalley and intravalley electron–hole exchange interactions in monolayer MoS2 , Phys Rev B , 2014 , vol. 89 pg. 205303 Google Scholar Crossref Search ADS WorldCat 75. Glazov MM , Amand T , Marie X , et al. Exciton fine structure and spin decoherence in monolayers of transition metal dichalcogenides , Phys Rev B , 2014 , vol. 89 pg. 201302 Google Scholar Crossref Search ADS WorldCat 76. Xiao D , Chang M-C , Niu Q . Berry phase effects on electronic properties , Rev Mod Phys , 2010 , vol. 82 pg. 1959 Google Scholar Crossref Search ADS WorldCat 77. Xiao D , Yao W , Niu Q . Valley-contrasting physics in graphene: magnetic moment and topological transport , Phys Rev Lett , 2007 , vol. 99 pg. 236809 Google Scholar Crossref Search ADS PubMed WorldCat 78. Mak KF , McGill KL , Park J , et al. The valley Hall effect in MoS2 transistors , Science , 2014 , vol. 344 (pg. 1489 - 92 ) Google Scholar Crossref Search ADS PubMed WorldCat 79. Yao W , Niu Q . Berry phase effect on the exciton transport and on the exciton bose-einstein condensate , Phys Rev Lett , 2008 , vol. 101 pg. 106401 Google Scholar Crossref Search ADS PubMed WorldCat 80. Kumar N , Cui Q , Ceballos F , et al. Exciton diffusion in monolayer and bulk MoSe2 , Nanoscale , 2014 , vol. 6 (pg. 4915 - 9 ) Google Scholar Crossref Search ADS PubMed WorldCat 81. Cui Q , Ceballos F , Kumar N , et al. Transient absorption microscopy of monolayer and bulk WSe2 , ACS Nano , 2014 , vol. 8 (pg. 2970 - 6 ) Google Scholar Crossref Search ADS PubMed WorldCat 82. Rivera P , Schaibley JR , Jones AM , et al. Observation of long-lived interlayer excitons in monolayer MoSe2-WSe2 heterostructures , Nat Commun , 2015 in press Google Scholar OpenURL Placeholder Text WorldCat 83. Lee C-H , Lee G-H , van der Zande AM , et al. Atomically thin p-n junctions with van der Waals heterointerfaces , Nat Nanotechol , 2014 , vol. 9 (pg. 676 - 81 ) Google Scholar Crossref Search ADS WorldCat 84. Furchi MM , Pospischil A , Libisch F , et al. Photovoltaic effect in an electrically tunable van der Waals heterojunction , Nano Lett , 2014 , vol. 14 (pg. 4785 - 91 ) Google Scholar Crossref Search ADS PubMed WorldCat 85. Cheng R , Li D , Zhou H , et al. Electroluminescence and photocurrent generation from atomically sharp WSe2/MoS2 heterojunction p-n diodes , Nano Lett , 2014 , vol. 14 (pg. 5590 - 7 ) Google Scholar Crossref Search ADS PubMed WorldCat 86. Fang H , Battaglia C , Carraroc C , et al. Strong interlayer coupling in van der Waals heterostructures built from single-layer chalcogenides , Proc Natl Acad Sci USA , 2014 , vol. 111 (pg. 6198 - 202 ) Google Scholar Crossref Search ADS PubMed WorldCat 87. Fogler MM , Butov LV , Novoselov KS . High-temperature superfluidity with indirect excitons in van der Waals heterostructures , Nat Commun , 2014 , vol. 5 pg. 4555 Google Scholar Crossref Search ADS PubMed WorldCat 88. Zhu B , Zeng H , Dai J , et al. Anomalously robust valley polarization and valley coherence in bilayer WS2 , Proc Natl Acad Sci USA , 2014 , vol. 111 (pg. 11606 - 11 ) Google Scholar Crossref Search ADS PubMed WorldCat 89. Mai C , Semenov YG , Barrette A , et al. Exciton valley relaxation in a single layer of WS2 measured by ultrafast spectroscopy , Phys Rev B , 2014 , vol. 90 pg. 041414 Google Scholar Crossref Search ADS WorldCat 90. Kumar N , He J , He D , et al. Valley and spin dynamics in MoSe2 two-dimensional crystals , Nanoscale , 2014 , vol. 6 (pg. 12690 - 5 ) Google Scholar Crossref Search ADS PubMed WorldCat © The Author(s) 2014. Published by Oxford University Press on behalf of China Science Publishing & Media Ltd. All rights reserved. For Permissions, please email: journals.permissions@oup.com This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits non-commercial reuse, distribution, and reproduction in any medium, provided the original work is properly cited. for commercial re-use, please contact journals.permissions@oup.com © The Author(s) 2014. Published by Oxford University Press on behalf of China Science Publishing & Media Ltd. All rights reserved. For Permissions, please email: journals.permissions@oup.com
TI - Valley excitons in two-dimensional semiconductors
JF - National Science Review
DO - 10.1093/nsr/nwu078
DA - 2015-03-01
UR - https://www.deepdyve.com/lp/oxford-university-press/valley-excitons-in-two-dimensional-semiconductors-dk83wgaaBW
SP - 57
EP - 70
VL - 2
IS - 1
DP - DeepDyve
ER -