Valley Polarized Single Photon Source Based on Transition Metal Dichalcogenides Quantum Dots

Photonic quantum computer, quantum communication, quantum metrology, and optical quantum information processing require a development of efficient solid-state single photon sources. However, it still remains a challenge. We report theoretical framework and experimental development on a novel kind of valley-polarized single-photon emitter (SPE) based on two-dimensional transition metal dichalcogenides (TMDCs) quantum dots. In order to reveal the principle of the SPE, we make a brief review on the electronic structure of the TMDCs and excitonic behavior in photoluminescence (PL) and in magneto-PL of these materials. We also discuss coupled spin and valley physics, valleypolarized optical absorption, and magneto-optical absorption in TMDC quantum dots. We demonstrate that the valley-polarization is robust against dot size and magnetic field, but optical transition energies show sizable size-effect. Three versatile models, including density functional theory, tight-binding and effective k p method, have been adopted in our calculations and the corresponding results have been presented.


Introduction
Traditional semiconductors have been used for decades for making all sorts of devices like diodes, transistors, light emitting diodes, and lasers [1]. Due to the advances of technology in fabrication, it is possible not only to make ever pure semiconductor crystals, but also to study heterostructures, in which carriers (electrons or holes) are confined in thin sheets, narrow lines, or even a point [1,2]. Quantum dots (QDs) are zero-dimensional objects where all the three characteristics related to spin-valley degree of freedom inherited from its 2D bulk materials. Very recently, graphene QDs (GQDs) have attracted intensive research interest due to their high transparency and high surface area. Many remarkable applications ranging from energy conversion to display to biomedicine are prospected [11]. Nevertheless, from quantum nano-devices point of view, the TMDCs have advantages over graphene. For instance, the semiconducting TMDCs have a band gap large enough to form a QD using the electric field, as shown in Figure 1, unlike etched GQDs made on semi-metallic graphene.
In this chapter, we show the optical and magneto-optical properties of the TMDC QD's. We choose MoS 2, which has been widely studied in the literature as our example. Three versatile models including density functional theory, tight-binding, and effective k Á p approach have been adopted in our calculations and the corresponding results have been presented. We show that the valley-polarization is robust against dot size and magnetic field, but the optical transition energies show sizable size effect. Based on the computed optical absorption spectra, a novel kind of valley-polarized single-photon source based on TMDC quantum dot is proposed [15].

Physical properties of transition metal dichalcogenides 2.1. Electronic band structure of transition metal dichalcogenides
Layered TMDCs have the generic formula MX 2 , where M stands for a metal and X represents a chalcogen. The monolayer crystal structure of the MoS 2, which is one of the most studied TMDCs in the literature is shown in Figure 2. Notice that the monolayer MoS 2 has a trigonal prism crystal structure. An inversion asymmetry results in a large direct gap semiconductor with the gap lying at the two inequivalent K-points of the hexagonal Brillouin zone.
The major orbital contribution at the edge of the conduction band (CB) is from d 3z 2 Àr 2 orbital of the metal, plus minor contributions from p x , and p y orbitals of chalcogens. On the other hand, at the edge of the valence band (VB) at the K-point, the most important orbital contribution is due to a combination of d xy and d x 2 Ày 2 of the metal, which hybridize to p x and p y orbitals of the chalcogen atoms [24]. In addition, there is a strong spin-orbit interaction (SOI), especially in the valence band. The spatial inversion asymmetry along with strong SOI leads to a spin-valley coupling, which results in the spin of electron being locked to the valley [20,21,36], as demonstrated in Figure 3(a). Furthermore, the band structure in the K-valley presents time reversal symmetry with that in K 0 -valley. And, it changes dramatically as the number of the layers of the TMDC increases. When the thickness increases, the band gap in MoS 2 and other group VI TMDCs decreases, and more importantly, the material becomes an indirect gap semiconductor [37], as shown in Figure 3(b).
In order to get insight into the physical origins of the band gap variations with the number of layers, Figure 4 shows evolutions of the band gaps (a) and band edges (b) of MoS 2 as a function of the number of layers. Notice that with increasing layer thickness, the indirect band gap (ΓÀK, ΓÀΛ) becomes smaller, while the direct excitonic transition ðKÀKÞ only slightly changes, as illustrated in Figure 4(a). Note also that the monolayer n ¼ 1 ð ÞMoS 2 is a direct band gap semiconductor, but it becomes an indirect band gap semiconductor when the number of layers is larger than one. This phase transition is also clearly demonstrated in Figure 4(b). In addition, as the number of layers increases, both the VB and CB edges at K-valley exhibit only slight variation while the degeneracy at Γ is lifted and a splitting of the bands occurs, which pushes the VB maximum to higher energy. The variation of the band gap is largely driven by the variation of the VB at Γ point. Going from a monolayer to a bilayer significantly raises the VB at Γ, resulting in a transition from the direct KÀK gap to an indirect ΓÀK gap. This dramatic change of electronic structure in monolayer MoS 2 results in the jump in monolayer photoluminescence efficiency.

Massive Dirac fermions
To gain insight of physics around the K-and K 0 -points, one can reduce multi-band tightbinding model to a two band k.p model, using Löwdin partitioning method [38]. For the monolayer TMDCs, one gets the Hamiltonian in the first order of k approximation, The solid curves were obtained using the QUANTUM ESPRESSO package with fully relativistic pseudopotentials under the Perdew-Burke-Ernzerhof generalized-gradient approximation, and a 16 Â 16 Â 1 k grid. The dashed curves were calculated from the tight-binding model, with cyan (red) representing states that are even (odd) under mirror operation with respect to the Mo plane. v 1;2 and c 1;2 label the bands close to the valence and conduction band edges near the K-and K 0 -points. The inset shows the hexagonal Brillouin zone (pink) associated with the triangular Bravais lattice of MoS 2 and an alternate rhombohedral primitive zone (black), and labels the principle high-symmetry points in reciprocal space [36] where Δ ¼ 1:6 eV is the band gap, t ¼ 1:1 eV is the effective hopping parameter, λ ¼ 0:075 eV is the SOI parameter, a ¼ 3:19 Å is the lattice parameter [39], τ ¼ AE1 is the valley index. The eigenvalues and eigenvectors can be derived straightforwardly as follows: and The energy dispersion around the K-and K 0 -points, described by Eq. 2, is shown in Figure 5.
In order to see the reliability of the k Á p approach, Figure 6 plots the energy spectrum of  monolayer MoS 2 calculated by the first principle and k Á p model. It can be found that in the vicinity of the K-(K 0 )-point, they have very good agreement.

Landau levels of monolayer MoS 2
For a perpendicular magnetic field applied to the MoS 2 sheet, we use Peierls substitution where e is the elementary charge. In the Landau Gauge A ! ¼ 0; Bx ð Þ, we define the operators Π AE ¼ τΠ x AE iΠ y , which have the following properties: where the magnetic length is . Using these operators, the destruction and creation operators are introduced in the following way: in the K-(τ ¼ 1) valley, andb in the K 0 (τ ¼ À1) valley. Then, the Hamiltonian in the presence of a perpendicular magnetic field can be well described by It is worth to mention that since the Zeeman effect is vanishingly small (< 5 meV), it is neglected. After some algebra calculations, the Landau levels are obtained as follows: where n is an integer and n ≥ 1, ω c ¼ ffiffi ffi 2 p =l B . The corresponding Landau fan diagrams of the monolayer MoS 2 are shown in Figure 7. The corresponding eigenfunctions are given by and where H n ðxÞ ¼ ðÀ1Þ n e x 2 d n dx n e Àx 2 : The eigenfunctions can be written in a compact form as, For the special case in which n ¼ 0, the eigenvalues become and corresponding eigenfunctions turn out to be

Optical selection rules
In splitting, as shown in Figure 8.
We assume that the monolayer TMDCs are exposed to light fields with the energy ℏω and wave vector k l, which is orthogonal to the monolayer plane and much smaller than 1=a. Up to the first order approximation, the light-matter interaction Hamiltonian is described by, with the light field A in ¼ A 0α cos ðk l Á r À ωtÞ and v ¼ ð1=ℏÞ∇ k H being propagation velocity [15]. Here, A 0 andα stand for the amplitude and orientation of the polarization field, respectively. For an electron being excited by an incident photon from its initial state ji〉 to a final state jf 〉, the transition probability is given by the Fermi's golden rule as, where nðEÞ is the density of states available for the final state. Since the absorption intensity I is proportional to the transition rate, it can be evaluated by, where Ψ c ðΨ v Þ is the conduction (valence) band wave function, For a circularly polarized (CP) light,α ¼ ð1; cos ðωt À σπ=2Þ; 0Þ T , with σ ¼ AE1 denoting the corresponding positive and negative helicities and T stands for the transpose of a matrix, then the perturbed Hamiltonian becomes, . For example, for τ ¼ 1; s z ¼ AE1, the transition rate for the monolayer TMDCs is determined by, and 〈c, k The optical transition rate for τ ¼ À1 and s z ¼ AE1 can be obtained by replacing þ with Àin the H LÀM . From Eqs. (29) and (30), notice that under CP light excitation, a valley and spin polarized emission or absorption light is expected in monolayer MoS 2 , as shown in Figure 8. In contrast, linearly polarized light does not present valley-selected emission and absorption spectra because both K-and K 0 -valleys absorb light simultaneously.

Valley polarized photoluminescence and excitonic effects of the monolayer TMDCs
In monolayer TMDCs, strong Coulomb interactions due to reduced screening and strong 2D confinement lead to exceptionally high binding energies for excitons [23,24,36], which allow them be able to survive even at room temperature. Hence, the typical absorption spectra are usually characterized by strong excitonic peaks marked by A and B, located at 670 and 627 nm, respectively. The strong spin-orbit interaction in the valence band gives rise to a separation between them, as shown in Figure 9. In addition, an injection of electrons into the conduction band of MoS 2 , which can be realized by gate-doping [26], photoionization of impurities [28], substrates [25] or functionalization layers [22,27], leads to the formation of negatively charged excitons ðX À Þ. The peak of the X À is positioned at a lower energy side of neutral exciton with a binding energy about 36 meV for MoS 2 , see the peak indicated by X À in Figure 10. In addition, the emergence of the charged exciton is accompanied by a transfer of spectral weight from the exciton. Therefore, the intensity ratio between a neutral and charged exciton can be tuned externally. Besides, with increasing the nonequilibrium excess electron density, a red-shift of the excitonic ground-state absorption due to Coulomb-induced band gap shrinkage occurs. It is also worth to point out that on the one hand the trion can provide a novel channel for exciton relaxation, and on the other hand, it can also be excited by an optical phonon into an excitonic state to realize an upconversion process in monolayer WSe 2 .
In the regime of high exciton density, the exciton-exciton collision leads to exciton annihilation through Auger process or formation of biexciton in the monolayer TMDCs. The biexciton is identified as a sharply defined state in the PL, see P 0 in Figure 10 and also XX-peak in Figure 11. The nature of the biexcitonic state is supported by the dependence of its PL intensity on the excitation laser power. At low excitation laser intensity, the peaks P 0 and X grow superlinearly and linearly with incident laser power, whereas they increase sub-quadratically and sublinearly with the laser power at sufficiently high laser fluence. The large circular polarization of P 0 emission provides a further support for this assignment.
The polarization of the photoluminescence from the TMDCs, which is defined by η ¼ðI σ þ À I σ À Þ=ðI σ þ þ I σ À Þ, inherits that of the excitation source, where I σ þ (I σ À ) is PL intensity of right (left) hand circularly polarized light. Figure 11 illustrates photoluminescence spectra of monolayer WSe 2 excited by near-resonant circularly polarized radiation at 15 K. Notice that the peaks for X, X À , and XX all exhibit significant circular polarization. In addition to biexciton emission, the X and X À emission bands also exhibit strong valley polarization.

Defect induced photoluminescence and single photon source
As known, vacancy defects, impurities, potential wells created by structural defects or local strain or other disorders might be introduced in the growth process of the TMDC materials [12,13,32,33]. They can produce localized states to participate the optical emission and absorption as manifested by P 1 to P 3 in Figure 10, and the emission bands on the lower energy side of the peak XX in Figure 11. Since the point defects can induce intervalley coupling, the defect-related emission peaks show no measurable circular polarization character. Besides, the excitons, trions, and even biexcitons can be trapped by theses crystal structure imperfections to form corresponding bound quasiparticles. Therefore, delocalized excition, charged exciton and biexciton emissions, and localized ones can coexist in the TMDCs. Interestingly, these carrier trapping centers can act as single-photon emitters to emit stable and sharp emission line [12,13,32,33]. For this kind of single quantum emitter, since the maximum number of emitted single photons is limited by the lifetime of the excited state, a saturation of the PL intensity at high excitation laser power is expected.

Magneto-optical properties of the monolayer TMDCs
The presence of a magnetic field induces a quantization of the energy levels. At high magnetic field, the Landau levels (LLs) form. The transition rate between the conduction-and valenceband Landau levels can be calculated using the eigenfunctions in Eq. 21.
〈Ψ τ, þ n jH CP, AE LÀM jΨ τ, À m 〉 ¼ M δ szc, szv ½ðτ þ σÞ c τ, þ 1;n c τ, À 2;m δ n, mþτ þ ðτ À σÞ c τ, þ 2;n c τ, À 1;m δ nþτ, m where In the presence of magnetic field, both σ þ and σ À absorptions take place in each valley. However, there is a great difference in their intensity. For instance, the absorption spectrum intensity of σ þ -light is 10 4 times larger than σ À -light in the K-valley. And, the lowest transition energy absorption spectrum demonstrates a valley polarization, i.e., σ þ in the K-valley and σ À in the K 0 -valley, as showed in Figure 12 [41]. In addition, the transition occurs from n ¼ 0 to n ¼ 1 LLs in the K 0 -valley, whereas n ¼ 1 to n ¼ 0 LLs in the K-valley. Therefore, the valley polarization remains in the magneto-optical absorption, as showed in Figure 13. It is worth to argue that the higher-order terms in the effective k.p model only induces about 0.1% correction to absorption spectrum intensity [41], which allows us neglect them safely.

TMDC quantum dots and valley polarized single-photon source
The Hamiltonian of the TMDC QDs in polar coordinates is given by [15] H ¼ As a matter of convenience, we get rid of the angular part by using the following ansatz for the eigenfunctions where the quantum number m ¼ j À τ=2, j is the quantum number related to the effective angular momenta J τ ef f ¼ L z þ ℏτσ z =2 with L z being the orbital angular momenta in theẑ direction. Then the Schrödinger equation for TMDC QDs, showed in Figure 14, becomes After some algebra calculations, we get two decoupled equations. They are a 00 ðrÞ þ a 0 ðrÞ r þ aðrÞ χ À m 2 r 2 ¼ 0 and b 00 ðrÞ þ b 0 ðrÞ r þ bðrÞ χ À ðm þ τÞ 2 r 2 ! ¼ 0 where χ ¼ ð2EÀΔÞðΔþ2EÀ2λsτÞ . These two second order differential equations can be straightforwardly resolved. Finally we obtain, and where N is the normalization constant and J n ðxÞ is Bessel Function of the first kind. Applying the infinite mass boundary condition ψ 2 ðR, θÞ ψ 1 ðR, θÞ ¼ iτe iτθ , we obtains the secular equation where R is the QD radius, see Figure 14.
From Figures 15 and 16, we see that the bound states formed in a single valley, and E τ ðjÞ 6 ¼ E τ ðÀjÞ for both conduction-and valence-bands. In addition, the electron-hole symmetry is broken. Due to the confinement potential, the effective time reversal symmetry (TRS) is broken within a single valley, even without the magnetic field, similarly to graphene QDs [42]. On the other hand, the inverse asymmetry of the crystalline structure, the terms of spin-orbit interaction and the confinement potential, do not commute with the effective inversion operator, defined in a single valley P e ¼ I τ ⊗ σ x . Consequently, the QDs do not preserve the electron-hole symmetry in the same valley. However, comparing Figure 15(a, b), we can found that E τ ðjÞ ¼ E Àτ ðÀjÞ was still true. This is attributed to the TRS, where THT À1 ¼ H, the TRS operator is defined as T ¼ iτ x ⊗ s y C, with C being the conjugate complex operator.

Landau levels in monolayer MoS 2 quantum dots
Similarly to what we did in the case of the monolayer TMDCs, for quantum dot subjected to a perpendicular magnetic field, we do the Peierls substitution and use now the symmetric Then, the total Hamiltonian becomes H þ H B, which leads to the following two-coupled differential equations: In order to solve this eigenvalue problem, let us first decouple these two equations into a 00 ðrÞ þ a 0 ðrÞ r þ aðrÞ À r 2 4l 4 and Figure 15.
Solving these equations, we obtain the following two components of the eigenfunctions bðrÞ ¼ Nr jmþτj e À r 2 4l 2 SðxÞ is the sign function, 1 F 1 a, b, x ð Þis the confluent hypergeometric function of the first kind, ε is an arbitrary constant (0 < ε < 1) used to avoid a singularity in the function SðxÞ.
With the eigenfunctions at hand, we can derive the secular equation for the eigenvalues by applying infinite mass boundary condition, i.e., However, the presence of magnetic field breaks down this symmetry and leads to splittings of the atomic orbitals of the dot. Moreover, in the regime of weak magnetic fields, unlike the monolayer MoS 2 in which the energy shows linear B response, valley dependent energy levels with nonlinear B response are observed. Such effect is attributed to the competition of the QD confinement with the magnetic field effect [14].
As B increases, an effective confinement induced by the magnetic field gradually becomes comparable to that of the dot. Hence, their contributions to the electronic energy are balanced. With a further increasing of B, magnetic field effect starts to dominate the features of the energy spectrum. Accordingly, the LLs which show a linear dependence on B, became of the heavily massive Dirac character, are formed just like in the pristine monolayer MoS 2 . The higher the energy level is, the stronger the magnetic field needed to form the corresponding LL. For instance, the lowest LL is formed around a critical value B ¼ B c ¼ 2T. Interestingly, in the Hall regime, an energy locked (energy independent on B) mode referring to the lowest spin-down conduction band in the K 0 -valley emerges, as shown in Figure 17(b). It is expected to be an analog to the zero energy mode in gapless graphene, associating with certain topological properties. These novel features of the QD energy spectrum are tunable by QD size [14]. Figure 17(c, d) are the corresponding analogs of Figure 17(a, b), but for a dot with R ¼ 40 nm. Comparing with the 70nm dot (B c ¼ 2T), here the energy locked mode takes place until B c ¼ 4T, which turns out to be larger than the one (2 T) for the 70 nm dot, indicating the locked energy modes arise from the competition between the dot confinement and the applied magnetic field [14].
Let us turn to the energy spectrum of the valence band in the dot of R ¼ 70nm. Figure 17(e, f) show the B field dependence of the energy spectrum of the highest four valence bands for several values of angular momentum ms with m ¼ 0, AE 1; AE 2 in the K-valley with spin-up (τ ¼ 1; s ¼ 1) and K 0 -valley with spindown (τ ¼ À1, s ¼ À1), respectively. In contrast to the conduction band, here we find EðmÞ 6 ¼ EðÀmÞ even at B ¼ 0 due to the spin-orbit coupling, see Figure 17(a, b, e, f). Besides that, we also observe the emergent locked energy mode around B c ¼ 2T, but referring to the highest valence band. We should emphasize that the locked energy modes for the conduction and valence bands appear in distinct valleys with opposite spin, see Figure 17(b, e). The corresponding valence band energy spectra for the 40 nm dot are shown in Figure 17(g-h), where the locked energy mode associated with a larger B c ¼ 4T, also arises from the combined effect of the dot confinement and the applied magnetic field, similar to the conduction band. Therefore, the flat band or energy locked modes appear only in the valence band of the K-valley and the conduction band of the K 0 -valley [14]. The effect of changing the magnetic field direction is showed in Figure 19.
A comparison of the energy spectrum of the 70 nm dot with that of the bulk TMDC (i.e., infinite geometry) is shown in Figure 18. Because of the large effective mass at the band edges, þ t 2 a 2 ω 2 c n q in the low energy region, which resembles conventional 2D semiconductors more than Dirac fermions. The n ¼ 0LL appears only in the conduction band of K 0 -valley (and the valence band of K-valley, not shown), implying the lifting of valley degeneracy for the ground state. As magnetic field increases, there is an evolution of the energy spectrum from atomic energies to Landau levels. More specifically, in the regime of weak B field, the atomic structure emerges, where the energy is distinct from that of the bulk case. On the other hand, in the strong field regime, the energy in the bulk TMDC and quantum dot becomes identical because the effective confinement due to the magnetic field dominates physical behaviors of the dot. Note that although the two band model we used here is widely adopted in the literature [39], the model itself still has limitations, e.g., it cannot properly describe the spin splitting of the conduction band, the trigonal warping of the spectrum and the degeneracy breakdown by applied magnetic field for Landau levels with the same quantum number. However, for usual experimental setups, these effects in the vicinity of the K or K 0 valley in which we are interested play a minor role. Hence it can be safely neglected.

Optical selection rules in monolayer MoS 2 quantum dots
In the QDs as demonstrated in Figure 20, the optical transition matrix elements in the QDs are computed by, where s zv (s zc ) denote the valence (conduction) band spin state, R Àσ ¼ a Ã c b v rdr with a c=v and b c=v the radial components of the conduction/valence band spinor.
The selection rule for optical transitions in TMDC QDs is defined by m v À m c ¼ AEτ and s zv ¼ s zc , i.e., the angular momentum of the initial and final states differs by AE1, but the spin of these two states is the same. The magnitude of transition rates is determined by the integral R Àσ and R σ for the transitions taking place in the valley τ ¼ Àσ and σ, respectively. Since ja c ðrÞj > jb c ðrÞj and ja v ðrÞj < jb v ðrÞj, the integral R Àσ is much smaller than the R σ . As a Figure 19. Energy spectrum of the states with spin-up in the K-valley (a) and spindown in the K 0 -valley (b) of a monolayer MoS 2 QD with radius R ¼ 70 nm, as a function of the magnetic field along both positive and negativeẑ-direction, for angular momentum m ¼ 0 (red curves), À1 (pink curves), 1 (green curves), À2 (black curves), and 2 (blue curves).
consequence, the absorption in the valley τ ¼ σ is stronger than that in the τ ¼ Àσ, which leads to a valley selected absorption. In a special case in which one component of the wave function spinor is equal to zero such as n ll ¼ 0 LL, only photons with the helicity σ ¼ τ is absorbed. Then, one obtains a dichroism η ¼ 1. For the linear polarized light, however, there is no valley polarization in the absorption spectrum, in contrast to the case of CPL [15].
In the 2D bulk MoS 2 , the bottom of the conduction band at the two valleys is characterized by the orbital angular momentum m ¼ 0. In contrast, at the top of the valence band, the orbitals with m ¼ 2 in the K-valley, whereas m is equal to À2 in the K 0 -valley. The valley dependent angular momentum in the valence band allows one to address different valleys by controlling the photon angular momentum, i.e., the helicity of the CP light. Such valley-specific circular dichroism of interband transitions in the 2D bulk has been confirmed [19,43,44]. The question is whether its counterpart QD also possesses this exotic optical property. Figure 21(a, b) depict zero-field band-edge optical absorption spectrum of a 70-nm dot pumped by the CP light field. Interestingly, one notices that (i) the polarization of the absorption spectrum is locked with the valley degree of freedom, manifested by the intensity of absorption spectrum with σ ¼ τ being about 106 times stronger than that with σ ¼ τ, and (ii) the spectrum is spin-polarized. Thus, the QDs indeed inherit the valley and spin dependent optical selection rule from their counterpart of 2D bulk MoS 2 . In spite of the distinction in the spin-and valley-polarization of absorption spectra in the distinct valleys, their patterns are the same required by the time reversal symmetry. In Figure 21(c, f) we show the zero-field optical absorption spectrum as a function of excitation energy for several values of dot-radius within R ¼ 20-80 nm. The involved transitions in Figure 21(c, f) lagged by the numbers have been schematically illustrated in Figure 21(g). Alike conventional semiconductor QDs, several peaks stemmed from discrete excitations of  the MoS 2 QD are observed. As the dot size decreases, the peaks of absorption spectrum undergo a blue shift. In other words, a reduction of the dot size pushes the electron excitations to take place between higher energy states, as a result of the enhancement of the confinement on carriers induced by a shrink of the dot. Therefore, the spin-coupled valley selective absorption with a tunable transition frequency can be achieved in QDs by varying dot geometry, in contrast to the 2D bulk where a fixed transition frequency is uniquely determined by the bulk band structure. In addition to the blue shift of the transition frequency, the absorption intensity can also be controlled by dot geometry due to size dependence of the peak interval. In fact, an increasing of dot size results in a reduction of the energy separation among confined states. Thus, as the dot size increases, the absorption peaks get closer and closer, see Figure 21(c-f).
Eventually, several individual absorption peaks merge together to yield a single compositepeak with an enhanced intensity. For instance, for the 20-nm dot (Figure 21c), the lowest energy peak is generated by only one transition, labeled by (1) (see also Figure 21g). However, for the dot with R ¼ 80nm, we observe a highly enhanced absorption intensity labeled by 1 þ 2 þ 3 ð Þ , which in the 20-nm dot refers to three separated peaks with weak absorption intensities indicated by (1), (2), (3), respectively.

Magneto-optical properties of monolayer MoS 2 quantum dots
With the knowledge of the energy spectrum and eigenfunctions of the QD, we are ready to study its magneto-optical properties. The optical transition matrix in Eq. 27 is applicable to the current case provided that we use newly obtained wavefunctions presented in Eq. 47 and Eq. 48. Figure 22 shows the magneto-optical absorption spectra for the spin-down states in the K 0valley of a monolayer MoS 2 QD with R ¼ 40 nm excited by left-hand circularly polarized  light σ À , for several values of magnetic field ranging from 0 to 15 T. A few interesting features are observed. Firstly, the magneto-optical absorption is also spin-and valley-dependent, as it does in optical absorption spectrum at zero field. In particular the lowest transition energy absorption peak related to the interband transition involving n ll ¼ 0 LL is totally valley polarized with dichroism equal to 1. Thus, the polarization of magneto-optical absorption locks with the valley. Secondly, for a fixed value of dot size, increasing (decreasing) the strength of the magnetic field results in a blue (red) shift in the absorption spectrum. This arises from the fact that the magnetic field induces an effective confinement characterized by the magnetic length l B ¼ ffiffiffiffiffiffiffiffiffiffi ℏ=eB p , which is more pronounced for a stronger magnetic field. In addition, the magnetic quantization induced by the magnetic field favors the QD to absorb photons with higher energies. The stronger the magnetic field, the greater the capacity of the QDs to absorb the photons of higher energy. Thirdly, in the high magnetic-field regime the absorption intensity can be highly enhanced by increasing the strength of magnetic field due to increased degeneracy of the LLs.

Excitonic effect in monolayer MoS 2 quantum dots
The optical and magneto-optical absorptions that we have discussed so far are based on the independent electron-hole picture. In reality, there is a strong Coulomb interaction between the electron and hole in MoS 2 . The full treatment of the Coulomb interaction using manybody theory is beyond the scope of this chapter. However, the excitonic effects in absorption spectrum of the monolayer MoS 2 QDs can be properly addressed by an exact diagonalization, which is adopted in this work. The Hamiltonian used to describe the exciton is, Hðr e , r h Þ ¼ H e ðr e Þ þ H h ðr h Þ þ V eÀh ðr e À r h Þ, where H eðhÞ is the single electron (hole) Hamiltonian in QDs (see Eq. 1) and V eÀh is the electron-hole Coulomb interaction given by V eÀh ðjr e À r h jÞ ¼ ð1=4πE r E 0 Þðe 2 =jr e À r h jÞ. Here E 0 is the permittivity, E r is the dielectric constant, and r e and r h respectively stand for the position of electron and hole. An exciton can be understood as a coherent combination of electron-hole pairs. Thus, the wavefunctions of an exciton can be constructed based on a direct product of single-particle wave functions for the electron and hole. Since the electron-hole Coulomb interaction is larger than or at least in the same magnitude as the quantum confinement in the MoS 2 QDs, the excitonic effect play an very important role. Hence the single-particle wavefunctions should be modified due to the presence of the Coulomb interaction. Therefore, to get quick convergence of numerical calculation, instead of using simple single-particle wavefunction, we use the modified one to construct the exciton states, i.e., χ j ðr e, h Þ ¼ N exp ðÀr e, h =r b ÞΨ j ðr e, h Þ, where the exponential factor is a hydrogen-like s-wave state, N is the normalization constant, Ψ j is the wave function of the Hamiltonian H e;h , and r b is the exciton bohr radius, which has the value of r b $ 1 nm in MoS 2 [24]. Then, the exciton wave function Ψ exc can be straightforwardly written as, Ψ ν exc ðr e , r h Þ ¼ which involves the wave function χ i of the electron and hole in the absence of Coulomb interaction, with j ¼ p, r, s, q indicating the state index. We should emphasize that Eq. 55 contains both the direct interaction between the electron and hole, i.e., Eq. 55 at p ¼ q and r ¼ s, To calculate Coulomb interaction defined in Eq. 55, we expand 1=jr e À r h j in terms of halfinteger Legendre function of the second kind Q mÀ1=2 , i.e., which is widely used in many-body calculations [45]. Here θ eðhÞ is the polar angle of the position vector r eðhÞ in the 2D plane of MoS 2 , ξ ¼ ðr 2 e þ r 2 h Þ=2r e r h , E m ¼ 1 for m ¼ 0 and E m ¼ 2 for m 6 ¼ 0. The exciton energy and the corresponding wavefunction can be obtained by diagonalization of the matrix of many-particle Hamiltonian Hðr e , r h Þ. With the exciton state at hand (Eq. 54), we are ready to the determine the excitonic absorption involving a transition from the ground state j0〉 to exciton state jf 〉 ¼ jΨ exc 〉, where E ν exc and ℏω are the exciton-and photon-energy, respectively, and P ¼ X i, j δ sz, sz 0 〈χ i jH LÀM jχ j 〉 a i, sz h j, sz 0 is the polarization operator, with a i, sz and h j, sz 0 being the electron and hole annihilation operators. After straightforward calculation, we finally obtain In our numerical calculations, we have used five modified single-particle basis functions with angular momentum ranging from À2.5 to 1.5. Figure 23 shows that there is an exciton absorption peak located at around 550 meV (i.e., exciton binding energy) below the band-edge absorption. And, the excitonic absorption peak shifts monotonically to higher absorption energy as the dot size is increased. Above the band gap, however, the spectrum is similar to what we found in previous sections in the band-to-band transitions using the independent electron-hole model. Since the exciton absorption peak is far away from the band-edge absorption, one can in principle study them separately. And, the Coulomb interaction between electron-hole pair does not change the valley selectivity and our general conclusion. Finally, it is worth to remarking that one can shift the excitonic absorption peak to a higher absorption energy, by varying the band gap parameter (Δ) in our model Hamiltonian.