## 1. 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 spatial dimensions are quantized with sizes smaller than some specific characteristic lengths, e.g., the exciton Bohr radius [1, 2]. Because of confinement, electrons in the QDs occupy discrete energy levels, in a similar way as they do in atoms [2, 3]. For these reasons, QDs are also referred to as artificial atoms [1, 2]. In spite of some similarities between the QD and the real atom, the former demonstrates several special characteristics. For instance, its size can vary from a few to hundreds of nanometers, and it can trap from a very small number of electrons (

Various techniques have been developed to produce the QDs such as etching, regrowth from quantum well structures, beam epitaxy, lithography, holograph patterning, chemical synthesis, etc [1, 2]. Consequently, many kinds of QDs emerge. According to the electrical property of their parent material, they can be classified into metal, semiconductor, or super‐conducting dots. From geometry point of view, the QDs form two groups: two‐dimensional [2, 4, 5, 8] or three‐dimensional (3D) [1] dots. The former can be further divided into conventional 2D semiconductor QDs, such as self‐assembled‐ and gated‐QDs based on traditional semiconductor quantum wells [4–8, 10], and the novel QDs made from two‐dimensional‐layered materials (2DLMs) [11–15].

Atomically thin 2DLMs have revolutionized nanoscale materials science [16]. The interatomic interaction within layers is covalent in nature, while the layers are held together by weak van der Waals (vdW) forces. The family of 2D materials, which started with graphene [16], has expanded rapidly over the past few years and now includes insulators, semiconductors, semimetals, metals, and superconductors [17–19]. The most well studied 2D systems beyond graphene, are the silicene, germanene, stanine, and borophene, organic‐inorganic hybrid perovskites, insulator hexagonal boron‐nitride [17, 18], the anisotropic semiconductor phosphorene, transition metal‐carbides, ‐nitrides, ‐oxides, and ‐halides, as well as the transition metal dichalcogenides (TMDCs) [20–28]. Compared with traditional semiconducting materials, the 2DLMs take advantage of inherent flexibility and an atomically‐thin geometry. Moreover, because of their free dangling bonds at interfaces [25, 29, 30], two‐dimensional‐layered materials can easily be integrated with various substrates [17]. They can also be fabricated in complex‐sandwiched structures or even suspended to avoid the influence of the substrate [31]. The monolayer TMDCs with infinite geometry exhibit strong carrier confinement in one dimension but preserve the bulk‐like dispersion in the 2D plane. In contrast, electrons in a TMDC QD are restricted in three dimensions, which present size tunable electronic and optical properties in addition to the remarkable 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.

The applications of quantum dots are still mostly restricted to research laboratories, but they are remarkable due to the fact that QDs provide access to the quantum mechanical degrees of freedom of few carriers. Single electron transistors [1–6], the manipulation of one [4–7] or two [1–3] electron spins, manipulation of a single spin in a single magnetic ion‐doped QD [4–7] are only some examples. Optically active quantum dots can also be used in both quantum communication and quantum computation [4–7, 12–15]. The emerging field of quantum information technology, as unconditionally quantum cryptography, quantum‐photonic communication and computation, needs the development of individual photon sources [12–15, 32, 33]. Recently, individual photon emitters based on defects in TMDC monolayers with different sample types (

In this chapter, we show the optical and magneto‐optical properties of the TMDC QD's. We choose _{,} which has been widely studied in the literature as our example. Three versatile models including density functional theory, tight‐binding, and effective

## 2. Physical properties of transition metal dichalcogenides

### 2.1. Electronic band structure of transition metal dichalcogenides

Layered TMDCs have the generic formula _{,} which is one of the most studied TMDCs in the literature is shown in **Figure 2**. Notice that the monolayer

The major orbital contribution at the edge of the conduction band (CB) is from d_{,} and **Figure 3(a)**. Furthermore, the band structure in the **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**Figure 4(a)**. Note also that the monolayer **Figure 4(b)**. In addition, as the number of layers increases, both the VB and CB edges at

#### 2.1.1. Massive Dirac fermions

To gain insight of physics around the **k**.**p** model, using Löwdin partitioning method [38]. For the monolayer TMDCs, one gets the Hamiltonian in the first order of

where

and

The energy dispersion around the **Figure 5**. In order to see the reliability of the **Figure 6** plots the energy spectrum of monolayer MoS

#### 2.1.2. Landau levels of monolayer M o S 2

For a perpendicular magnetic field applied to the

where the magnetic length is

in the

in the

It is worth to mention that since the Zeeman effect is vanishingly small (

where **Figure 7**. The corresponding eigenfunctions are given by

and

where

The eigenfunctions can be written in a compact form as,

For the special case in which

and corresponding eigenfunctions turn out to be

### 2.2. Optical selection rules

In monolayer TMDCs, both the top of valence bands and the bottom of conduction bands are constructed primarily by the **Figure 8**.

We assume that the monolayer TMDCs are exposed to light fields with the energy _{,} which is orthogonal to the monolayer plane and much smaller than

with the light field

where _{,} and

For a circularly polarized (CP) light,

with

and

The optical transition rate for **Figure 8**. In contrast, linearly polarized light does not present valley‐selected emission and absorption spectra because both

### 2.3. 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 **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

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 **Figure 10** and also **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

The polarization of the photoluminescence from the TMDCs, which is defined by **Figure 11** illustrates photoluminescence spectra of monolayer WSe_{,} and

### 2.4. 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 **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.

### 2.5. 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 valence–band Landau levels can be calculated using the eigenfunctions in Eq. 21.

where

In the presence of magnetic field, both **Figure 12** [41]. In addition, the transition occurs from **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.

### 2.6. TMDC quantum dots and valley polarized single‐photon source

The Hamiltonian of the TMDC QDs in polar coordinates is given by [15]

As a matter of convenience, we get rid of the angular part by using the following *ansatz* for the eigenfunctions

where the quantum number **Figure 14**, becomes

After some algebra calculations, we get two decoupled equations. They are

and

where

and

where _{,} we obtains the secular equation

where **Figure 14**.

From **Figures 15** and **16**, we see that the bound states formed in a single valley, and **Figure 15(a, b)**, we can found that

### 2.7. 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 gauge, i.e.,

Then, the total Hamiltonian becomes _{,} which leads to the following two‐coupled differential equations:

In order to solve this eigenvalue problem, let us first decouple these two equations into

and

where

Solving these equations, we obtain the following two components of the eigenfunctions

where

With the eigenfunctions at hand, we can derive the secular equation for the eigenvalues by applying infinite mass boundary condition, i.e.,

**Figure 17(a)** illustrates the energy spectrum of the lowest four spinup conduction bands in the

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 **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

Let us turn to the energy spectrum of the valence band in the dot of **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 **Figure 17(a, b, e, f)**. Besides that, we also observe the emergent locked energy mode around **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 **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, the LLs of the bulk TMDC scale as

### 2.8. 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

In the 2D bulk **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 **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 **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 **Figure 21(c–f)**. Eventually, several individual absorption peaks merge together to yield a single composite‐peak 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

### 2.9. 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

### 2.10. 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

with the superscript

which involves the wave function

and the exchange interaction, i.e., Eq. 55 at

To calculate Coulomb interaction defined in Eq. 55, we expand

which is widely used in many‐body calculations [45]. Here

where

In our numerical calculations, we have used five modified single‐particle basis functions with angular momentum ranging from −2.5 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 (