## Abstract

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, valley‐polarized 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.

### Keywords

- single-photon source
- quantum dots
- transition metal dichalcogenides

## 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

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

#### 2.1.1. Massive Dirac fermions

To gain insight of physics around the ** k**.

**model, using Löwdin partitioning method [38]. For the monolayer TMDCs, one gets the Hamiltonian in the first order of**p

where

and

The energy dispersion around the

#### 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

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

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

### 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

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

The polarization of the photoluminescence from the TMDCs, which is defined by _{,} 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

### 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 ** k**.

**model only induces about 0.1% correction to absorption spectrum intensity [41], which allows us neglect them safely.**p

### 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

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

and

where

and

where _{,} we obtains the secular equation

where

From Figures 15 and 16, we see that the bound states formed in a single valley, and

### 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(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

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

### 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 (

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