The correlation between geometry and electronic properties in two-dimensional semiconductor structures has recently become an active area of theoretical and experimental research due to its potential application for electronic purposes. The interplying between electronic properties and geometry appears as an efficient tool for manipulation of band structure or transport properties what allows for the design of electronic nano-devices with a desired functionality. In graphene systems, the coupling between geometry and the electronic band structure is the most visible for graphene nanoribbons (GNRs) and carbon nanotubes (CNTs)1,2. In the case of GNRs, the electronic states strongly depend on the width and on the edge structure of the ribbon. In the case of CNTs the electrical properties essentially depend on the diameter and on the shape of the edges. One of the most recognized examples of ballistic transport devices exhibiting correlation between geometry and electronic properties is the geometric diode. In the geomeric diode the electric current rectification is obtained due to geometrical asymmetry3,4. In many materials, energy bands posses a discrete number of inequivalent local minima or maxima for specific values of momenta. The minima usually known as valleys seem to be promising candidates for components of pseudospin or a binary variable5,6. The separation of charge current composed of electron states belonging to only one valley can be regarded to as valley polarization. For graphene in particular, several schemes have been proposed to achieve valley-current filtering depending on geometrical deformation. At the field of geometry-induced effects also strain-induced effects in graphene has been the topic of a large number of theoretical works aimed at understanding the impact of controlled geometrical deformations on electronic properties7,8,9,10,11,12,13.
Recent advances in the epitaxial growth of 2D crystals have opened up new opportunities towards novel devices based on van der Waals heterostructures in which TMDCs play a major role. Two-dimensional TMDC materials, such as MoS\(_2\), WS\(_2\), MoSe\(_2\), and WSe\(_2\), have received extensive attention in the past decade due to their extraordinary electronic, optical and thermal properties14,15,16. They are considered as ideal materials for next-generation electronics, photonic and opto-electronic devices, relying on ultimate atomic thicknesses17,18. The bandgap of semiconducting TMDCs can be sized by varying the number of layers, and it can be changed from indirect to direct approaching the single layer. This tunable bandgap in TMDCs is accompanied by a strong photoluminescence and large exciton binding energy, making them promising candidate for a variety of opto-electronic devices, including solar cells, photo-detectors, light-emitting diodes, and photo-transistors19,20,21,22. Similarly to traditional semiconductor heterostructures quantum dots (QDs) can be formed on atomically thin ML TMDCs materials by applying a gate voltage to tune the local band structure23. Gate-defined QDs, in addition provide an efficient tool to tune electrically the confinement geometry and the strength. The realization of a quantum dot device has been obtained from the nanosheet on a Si/SiO\(_2\) substrate, and quantum dot confinement has been achieved by the top gate technique24. The fabrication of single quantum dots defined by gates has been reported recently for bilayer and monolayer WSe\(_2\) and MoS\(_2\) materials and discrete levels at temperatures up to 10 K have been observed25,26. The Coulomb blockade in single and coupled dots in high quality single layer MoS\(_2\) and the Shubnikov-de Haas oscillations occurring at magnetic fields as low as 3.3T have been observed24. Intrinsic exciton-state mixing and nonlinear optical properties, particulary the mechanism of second harmonic generation in TMDCs monolayers have been recently investigated on the basis of symmetry analysis27. Many other investigations based on several approaches including tight binding method, ab initio calculations, a many-body Bethe-Salpeter equation, \(\varvec{k}\cdot \varvec{p}\) perturbative method and effective mass approximation have been performed on properties of excitons formed in 2D semiconductor materials, also in the presence of confinement28,29,30,31,32,33,34,35. One should be also added that the Coulomb-exchange interaction, having an important influence on the energy structure of the 2D excitons, can be separated into the long-range and the short-range parts. In particular, the long-range exchange interaction has quantum electrodynamic nature, by the analogy to the exchange interaction in a positronium36. A theoretical study on the long-range exchange interaction in excitons has been performed treating the bright exciton as a microscopic dipole which produces an electric field and the backaction of this field on the exciton leads to the long-range electron-hole exchange interaction. Formally this treatment corresponds to the decomposition of the Coulomb interaction up to the dipole term and calculation of matrix element of the dipole term using the antisymmetrized Bloch functions28.
The purpose of this paper is a theoretical investigation of anisotropic quantum confinement effect on the linear response on external electric field of a neutral Mott-Wannier exciton formed inside the gate-defined quantum dot, for different geometries of the dot. The dot shape can be controlled electrostatically by appropriate system of metallic electrodes25,26. The electrodes generate the confining potential trapping additional electron added into conduction band (CB) as well as the hole in the valence band (VB). We model the confinement using non-centrosymmetric parabolic potential with the deformation of an elliptic type. In particular, we study the effect of the dot geometry on quantum properties of the Mott-Wannier exciton in terms of static dipole polarizability tensor and correlated probability distribution function of the electron-hole pair confined to the dot. Although obtained results can be directly linked with any ML TMDC system, due to relevant scalling relations derived in this paper, we explicitly demonstrate the dependence of static dipole polarizabilities on the dot geometry using material parameters appropriate for MoS\(_2\) ML structure. One should be noted that the static dipole polarizability is the property of atomic or molecular system that determines the behavior of neutral particles in the interaction with other particles such as in collision phenomena. The polarizability, in particular, allows for effective description of a long-range van der Waals exciton-exciton interaction or exciton-electron scattering in terms of the dipol-dipol and dipol-monopol interaction, respectively32. Moreover, the dipole polarizability is related to the dielectric constant and the reflection coefficient. Thus, in order to determine these parameters theoretically, the excitonic contribution to the total dielectric polarization should also be taken into account37.
Model
The system under study is sketched in Fig. 1. Due to anisotropy of confining potential the circular symmetry of the system, supposed within the effective mass approximation, is broken and in a consequence the angular momentum is not conserved quantity. The anisotropy leads to the mixing of states with different angular momentum quantum numbers. However, in the absence of external electric field the parity quantum number is still a good quantum number and only the states with even or odd angular momentum quantum numbers are mixed. If the external electric field is applied then the inversion symmetry is also broken and all the angular momentum eigenstates are mixed. For this reason the use of a more sophisticate method such as tight binding atomistic approach or Bethe-Salpeter equation based treatment including the noncentrosymmetric confinement is highly limited. We can see in Fig. 2 that parabolic-like behavior of bands is strongly pronounced close to the K-point. This is true for every ML TMDCs and the effective mass approximation used in this work based on this observation is optimal.
The confining potential acting on a particle with the mass m is supposed in the form of anisotropic parabolic potential,
$$\begin{aligned} \mathsf V_{\text {conf}}(\varvec{r})=m\mathsf U_{\text {conf}}(\varvec{r}), \end{aligned}$$
(1)
where \(\varvec{r}=\big [x,y\big ]\) is the coordinates vector of the electron and
$$\begin{aligned} \mathsf U_{\text {conf}}(\varvec{r})=\frac{1}{2}\left( \omega _x^{2}x^{2}+\omega _y^{2}y^{2}\right) , \end{aligned}$$
(2)
where oscillator frequencies \(\omega _x\) and \(\omega _y\) are different, in a general case. The dynamics of the electron in the CB and VB in the presence of confining potential, given by Eq. (1) and in the presence of in-plane external electric field, \(\varvec{F}=F\big [\cos \Phi ,\sin \Phi \big ]\), oriented along the direction given by the angle \(\Phi\), is described by the Hamiltonian \(\mathsf H_{\text {e}}^{\text {c}}\) and \(\mathsf H_{\text {e}}^{\text {v}}\), respectively. The Hamiltonians in the effective mass approximation are defined as
$$\begin{aligned} \mathsf H_{\text {e}}^{\text {c}}-E_c= & {} -\frac{\hbar ^{2}}{2m_{e}^{c}}\nabla _{ec}^{2} +m_{e}^{c}\mathsf U_{\text {conf}}(\varvec{r}_{ec})-q_e\varvec{F}\cdot \varvec{r}_{ec}, \end{aligned}$$
(3)
$$\begin{aligned} \mathsf H_{\text {e}}^{\text {v}}-E_v= & {} -\frac{\hbar ^{2}}{2m_{e}^{v}}\nabla _{ev}^{2} +m_{e}^{v}\mathsf U_{\text {conf}}(\varvec{r}_{ev})-q_e\varvec{F}\cdot \varvec{r}_{ev}, \end{aligned}$$
(4)
where \(E_{c(v)}\) is the edge energy of CB (VB), \(q_e\) is the electron charge \((q_e<0)\), \(m_{e}^{c(v)}\) is the electron effective mass in the CB (VB), where \(m_e^{c}>0\) \((m_e^{v}<0)\). In accordance with the commonly known interpretation, the absence of the electron in filled VB may be treated as a quasi-particle (hole) in the VB with the effective mass \(m_h\equiv -m_e^{v}>0\) and the charge \(q_h=-q_e>0\). Introducing the notation \(e=\mid q_e\mid\), \(m_e^{c}\equiv m_e, r_{ec}\equiv r_e, r_{ev}\equiv r_h\), we can rewrite above equations in the forms
$$\begin{aligned} \mathsf H_{\text {e}}^{\text {c}}-E_c=\mathsf h_{\text {e}}, \mathsf H_{\text {e}}^{\text {v}}-E_v=-\mathsf h_{\text {h}}, \end{aligned}$$
(5)
where the electron and the hole hamiltonians read
$$\begin{aligned} \mathsf h_{\text {e}}= & {} -\frac{\hbar ^{2}}{2m_{e}}\nabla _{e}^{2} +m_e\mathsf U_{\text {conf}}(\varvec{r}_{e})+e\varvec{F}\cdot \varvec{r}_{e}, \end{aligned}$$
(6)
$$\begin{aligned} \mathsf h_{\text {h}}= & {} -\frac{\hbar ^{2}}{2m_{h}}\nabla _{h}^{2} +m_h\mathsf U_{\text {conf}}(\varvec{r}_{h})-e\varvec{F}\cdot \varvec{r}_{h}. \end{aligned}$$
(7)
At this point we can note that observed experimentally excitonic spectra are a result of excitation of the electron from VB to CB, by photons. Thus, the excitonic energy levels correspond to the energy difference between the states in VB from which the electron is excited and the states in CB to which the electron is excited. This means that the spectrum, in the first approximation, is related to the eigenvalues of the operator \(\mathsf H_{\text {e}}^{\text {c}} -\mathsf H_{\text {e}}^{\text {v}}\). Taking into account electrostatic Coulomb interaction between the electron and the hole,
$$\begin{aligned} \mathsf V_{\text {eh}}=-\frac{e^{2}}{\varepsilon \mid \varvec{r}_e-\varvec{r}_h \mid }, \end{aligned}$$
(8)
where \(\varepsilon\) is the electric permittivity, we can define the excitonic Hamiltonian within the effective mass approximation as
$$\begin{aligned} \mathsf H_{\text {exc}}=\mathsf H_{\text {e}}^{\text {c}} -\mathsf H_{\text {e}}^{\text {v}}+\mathsf V_{\text {eh}}=E_c-E_v +\mathsf h_{\text {e}}+\mathsf h_{\text {h}}+\mathsf V_{\text {eh}}. \end{aligned}$$
(9)
We note that the above Hamiltonian is independent of the position of the zero-point energy in the energy scale, relatively to which the energy bands edges \(E_c, E_v\) are defined. Finally, the two-particle Hamiltonian describing the exciton as a pair of two interacting particles (electron-hole), in a more explicit form reads
$$\begin{aligned} \mathsf H_{\text {exc}}(\varvec{r}_e,\varvec{r}_h)=E_g-\frac{\hbar ^{2}}{2m_{e}}\nabla _{e}^{2} -\frac{\hbar ^{2}}{2m_{h}}\nabla _{h}^{2}+\mathsf V_{\text {conf}}(\varvec{r}_{e}) +\mathsf V_{\text {conf}}(\varvec{r}_{h})+e\varvec{F}\cdot (\varvec{r}_{e} -\varvec{r}_{h})+\mathsf V_{\text {eh}}, \end{aligned}$$
(10)
where \(E_g=E_c-E_v\) is the free-particle bandgap. Note that the free-particle bandgap becomes the optical bandgap, when the exciton is formed. The optical bangap is the energy distance from \(E_v\) to the lowest excitonic bound state (1s), due to the Coulomb interaction between the electron and a hole. The eigenvalue problem,
$$\begin{aligned} \mathsf H_{\text {exc}}(\varvec{r}_e,\varvec{r}_h)\Psi _{\text {exc}}(\varvec{r}_e,\varvec{r}_h)= E_{\text {exc}}\Psi _{\text {exc}}(\varvec{r}_e,\varvec{r}_h) \end{aligned}$$
(11)
can be effectively solved after introducing the center-of-mass vector \(\varvec{R}=(m_e\varvec{r}_e+m_h\varvec{r}_h)/(m_e+m_h)\) and the relative motion vector \(\varvec{r}=\varvec{r}_e-\varvec{r}_h\), that separates the two-particle Hamiltonian (10) into the sum of two independent parts \(\mathsf H_{\text {exc}}(\varvec{R},\varvec{r})=\mathsf H_{\text {c.m.}}(\varvec{R})+ \mathsf H_{\text {rel}}(\varvec{r})\). In a consequence \(E_{\text {exc}}=E_{\text {c.m.}}+E_{\text {rel}}\) and \(\Psi _{\text {exc}}(\varvec{r}_e,\varvec{r}_h)=\Psi _{\text {c.m.}}(\varvec{R})\Psi (\varvec{r})\). Details of calculations are given in the Appendix. Since the external electric field does not affect the c.m. motion (due to opposite signs of the charges of two constituents), we consider only the relative motion part of the total Hamiltonian. It is convenient for further analysis to introduce atomic units: \(a_0=\frac{\hbar ^{2}}{m_0e^{2}}\simeq 0.529\text{\AA}\) as unit of length, \(E_0=\frac{m_0e^{4}}{\hbar ^{2}}\simeq 27.2\)eV as unit of energy and \(F_0=\frac{m_0^{2} e^{5}}{\hbar ^{4}}\simeq 5.14\times 10^{11}\)V/m as unit of electric field. Finally the relative motion Schrödinger equation, in polar coordinates (\(r,\vartheta\)) introduced in the plane of the system, takes the form:
$$\begin{aligned} \Big [-\frac{1}{2\mu }\nabla _r^{2}-\frac{1}{\varepsilon r}+\mu \Omega ^{2}r^{2}\Big (\frac{a^{2}}{1+a^{2}} +\frac{1-a^{2}}{1+a^{2}}\sin ^{2}\vartheta \Big )+\eta r(\cos \Phi \cos \vartheta +\sin \Phi \sin \vartheta )\Big ]\Psi =E\Psi , \end{aligned}$$
(12)
where we have introduced following dimensionless parameters: \(\mu =m_r/m_0\), where \(m_r=m_em_h/(m_e+m_h)\) is the reduced mass of the system, \(\Omega =\hbar \omega /E_0\), where \(\omega ^{2}=(\omega _x^{2}+\omega _y^{2})/2\) is average square of the oscillator frequency, \(a=\omega _x/\omega _y\) is the anisotropy parameter, \(\eta =F/F_0\) and \(E=E_{\text {rel}}/E_0\). Here \(r=\mid \varvec{r}_e-\varvec{r}_h\mid /a_0\). The energy eigenvalues are functions of the system parameters. Multiplying Eq. (12) by \(\mu\) and using appropriate scaling of radial variable \((r\varepsilon /\mu \rightarrow r)\), we can find usefull scaling relation for energies
$$\begin{aligned} E(\Omega ,\eta ,\mu ,\varepsilon )=\frac{\mu }{\varepsilon ^{2}} E\Bigg (\frac{\Omega \varepsilon ^{2}}{\mu },\frac{\eta \varepsilon ^{3}}{\mu ^{2}},1,1\Bigg ), \end{aligned}$$
(13)
that in the case (\(\Omega =0, \eta =0\)) reduces to the well known formula for energies of an ideal 2D hydrogen atom,
$$\begin{aligned} E_n(0,0,\mu ,\varepsilon )=-\frac{\mu }{\varepsilon ^{2}}\frac{1}{2N_n^{2}},N_n=n-\frac{1}{2}, \end{aligned}$$
(14)
where \(n=1,2,\ldots\) and energies are given in the units of \(E_0\). By expanding both sides of the scaling relation, given by Eq. (13) in powers of \(\eta ^{2}\) and equating coefficients with equal powers of \(\eta ^{2}\) we obtain, in particular, a scaling relation for the dipole polarizability (see following section)
$$\begin{aligned} \alpha (\Omega ,\mu ,\varepsilon )=\mu ^{-3}\varepsilon ^{4}\alpha \Big (\frac{\Omega \varepsilon ^{2}}{\mu },1,1\Big ). \end{aligned}$$
(15)
In the case of unconfined system (\(\Omega =0\)) we obtain \(\alpha =\mu ^{-3}\varepsilon ^{4}\alpha ^{2D}\), where \(\alpha ^{2D}=21/128\) is the polarizability of an ideal 2D H-like atom38. The scaling (15) gives in particular a direct relation between Mott-Wannier exciton polarizabilities for two arbitrary ML MX\(_2\) materials, \(\alpha _1/\alpha _2=(\mu _2/\mu _1)^{3}(\varepsilon _1/\varepsilon _2)^{4}\).
The finite-field method
In order to determine components of the dipole polarizability tensor we apply the finite-field method described below. The expectation value of the dipole moment of the system in the presence of external electric field is the sum of the permanent dipole moment and the contribution induced by the field \(\varvec{F}\),
$$\begin{aligned} P_i(\varvec{F})=P^{(0)}_i+\alpha _{ij}F_j+\frac{1}{2}\gamma _{ijk}F_jF_k+\cdots , \end{aligned}$$
(16)
where \(\alpha _{ij}\) is the polarizability tensor, \(\gamma _{ijk}\) is the first hyperpolarizability and the summation over j, k is supposed. On the other hand according to the Hellman-Feynman theorem we can write39
$$\begin{aligned} P_i(\varvec{F})=-\frac{\partial E}{\partial F_i}=-\Big (\frac{\partial E}{\partial F_i}\Big )_0-\Big (\frac{\partial ^{2} E}{\partial F_i\partial F_j}\Big )_0F_j-\frac{1}{2}\Big (\frac{\partial ^{3} E}{\partial F_i\partial F_j\partial F_k}\Big )_0F_jF_k-\cdots , \end{aligned}$$
(17)
where \(E(\varvec{F})\) is the energy of the system as function of electric field. Since the c.m. motion is not affected by the constant uniform electric field, E denotes effectively the relative motion energy. By comparing Eqs. (16) and (17) one obtains
$$\begin{aligned} \alpha _{ij}=-\Big (\frac{\partial ^{2} E}{\partial F_i\partial F_j}\Big )_0. \end{aligned}$$
(18)
The second-order derivatives of the energy with respect to the electric field can be defined by the Taylor expansion of the field-dependent energy \(E(\varvec{F})\),
$$\begin{aligned} E(\varvec{F})=E(0)+\frac{1}{2}\Big (\frac{\partial ^{2} E}{\partial F_i\partial F_j}\Big )_0F_iF_j+\cdots , \end{aligned}$$
(19)
that contains only even powers of the field magnitude, due to the parity conservation in the field-free system. The last expansion in the low-field limit is equivalent to the perturbation expansion and the components of the dipole polarizability tensor are conventionally obtained using the perturbative approach. For an ideal 2D hydrogen atom the perturbative expansion is known up to third-order in \(F^{2}\)40,
$$\begin{aligned} E^{2D}(F)=-2-\frac{21}{2^{8}}F^{2}-\frac{22947}{2^{20}}F^{4}-\frac{48653931}{2^{31}}F^{6}-\cdots \end{aligned}$$
(20)
and the scalar (due to circular symmetry) polarizability of 2D H-like atom is \(\alpha ^{2D}=21/128\)38. However, in the problem considered in this paper the use of the perturbation method is very inefficient since we do not have analytical eigen-solutions of the unperturbed problem, summation over which must be performed in the second-order perturbation theory. Instead, we perform exact diagonalization of the total Hamiltonian including an electric field for several different values of weak electric field \((F_{(i)}, i=1,2,\ldots )\). This gives corresponding energies (\(E_{(i)}\), i=1,2,…) In the next step we construct the system of linear equations
$$\begin{aligned} A_{0}+A_{1}F_{(i)}^{2}+A_{2}F_{(i)}^{4}+A_{3}F_{(i)}^{6}+\cdots =E_{(i)},\quad i=1,2,\ldots \end{aligned}$$
(21)
for unknown coefficients \(A_{i}\) and with given r.h.s. At this point we note that axes x, y are chosen according to the symmetry of the system. In a consequence the polarizability tensor is diagonal with respect to these axes. Thus, it is sufficient to calculate principal moments of the polarization tensor taking the electric field vector oriented along the axis x and y, separately. Taking the vector of electric field as \(\varvec{F}=(F,0)\) and solving the linear system (21) we equate the coefficents of the perturbation expansion given by Eq. (19) with the coefficients \(A_{i}\): \(E(0)=A_0, \alpha _{xx}=-2A_1,\ldots\) Similarly, taking \(\varvec{F}=(0,F)\) we obtain the component \(\alpha _{yy}\). In this manner we can reconstruct the perturbation series without summation over states. In particular, as the test of the method, the perturbative expansion corresponding to the 2D hydrogen problem, given in Eq. (20) has been reconstructed with high precision. Finally, the magnitude of the dipole polarizability for any direction is given by
$$\begin{aligned} \alpha (\Phi )=\alpha _{xx}\cos ^{2}\Phi +\alpha _{yy}\sin ^{2}\Phi , \end{aligned}$$
(22)
where the angel \(\Phi\) indicates the spatial direction. We note that the finite-field method described above has been succesfully employed for determination of the relativistic magnetic susceptibilities of 3D Dirac H-like atoms41.