Electro-Optical Properties of Excitons in CdSe Nanoplatelets

Abstract

Applying a constant external electric field to a semiconductor nanostructure with Wannier–Mott excitons, in which the electron and hole interact via a centrally symmetric Coulomb potential, alters the symmetry of the system. When the electric field is applied parallel to the z-axis, the system exhibits cylindrical symmetry; when the field lies in the $x-y$ plane, the symmetry is broken. These symmetry changes affect the optical properties of the system. We present a theoretical calculation that yields analytical expressions for the optical functions of CdSe Nanoplatelets—reflectivity, transmissivity, and the absorption coefficient—in an external homogeneous electric field. From these, we focus on the absorption coefficient. We consider various configurations, with the external field oriented perpendicular and parallel to the platelet planes. Using the real density matrix approach, we calculate the linear electro-optical functions of CdSe nanoplatelets, taking into account the effect of dielectric confinement on excitonic states. We also discuss the impact of platelet geometry (thickness and lateral dimensions) and applied field strength on the spectrum.

1. Introduction

We focus on semiconductor nanostructures known as nanoplatelets (NPLs), which are quantum dots (QDs) but exhibit a different confinement mechanism than other QDs (e.g., GaAl/GaAlAs) and quantum wells (QWs), here referred to as standard QDs (QWs). In the standard QDs the electrons and the holes, created by the wave propagating in a QW, are confined in a nanostrukture of one type of semiconductors by an impenetrable barrier of a different semiconductor. In the considered below CdSe NPLs the confinement is of electrostatic origin and is made by a large dielectric contrast between the semiconductor (here CdSe) and its environment. Despite this difference, electrons and holes interact by a screened Coulomb potential, and bound electron-hole pairs, named excitons, are created. Colloidal group II–VI semiconductor nanostructures have emerged as one of the most studied inorganic systems. The quite uncomplicated synthesis and the ability to tune optical properties depending on their crystalline size due to the quantum confinement effect make them good candidates for highly efficient light emitters, bioimaging markers, and photovoltaic cells. Colloidal nanoplatelets (NPLs) are cuboid nanostructures with the thickness of several atomic layers and much larger lateral size so that the confinement is strong only in the thinnest dimension. In the two last decades, after the first synthesize of CdSe NPLs [1], a large number of works dedicated to this topic, appeared, see, for example, Refs [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18].

As in standard QDs, the optical properties of NPLs are dominated by excitons. So the major part of the initial research on NPLs was concentrated on the calculation of exciton characteristics, such as exciton binding energy, confinement eigenfunctions, and eigenvalues.

As in other nanostructures, subsequent research addressed electro- and magneto-optical properties. CdSe nanoplatelets (NPLs) have unique electro-optical properties due to strong 1D quantum confinement, featuring giant oscillator strength (GOST), narrow absorption/emission, large absorption cross-sections, high exciton binding energies, and excellent optical gain, making them ideal for LEDs, lasers, and sensors, with thickness dictating color and surface chemistry (ligands/doping) tuning efficiency and spectral features like their strong Stark effect under electric fields. Beginning with the pioneering work by Miller et al. [19], where the term ‘Quantum Confined Stark Effect’ (QCSE) was introduced, the optical response of quantum semiconductor nanostructures (wells, dots, wires) subjected to an interaction with the external constant electric field, has attracted many interest over the past decades, for example [20,21,22]). Besides the cognitive value, this attention is motivated by possible technological applications of QCSE, for example, Self-Electrooptic Effect Devices (SEEDs), fast optical switches and modulators, crucial for optical communications (for example [23]).

The optical properties of NPLs, including electro-absorption, were subject of extensive experimental and theoretical studies (for example [22]). However, it seems, that there are few theoretical articles on electro-optical effects. This makes an inspiration to the present work. We will discuss the effects of applied static electric field on NPL with dielectric confinement. The electron-hole Coulomb potential dielectrically screened with the static dielectric constant is adopted, and the valence band structure is considered in the cylindrical approximation, thus separating light- and heavy hole motions. In calculations we use the so-called Real Density Matrix Approach (RDMA) (see Ref. [24] for details about RDMA). This method enables to obtain analytic expressions for the NPLs electro-optical functions, where we have chosen the electro-absorption coefficient.

The paper is organized as follows. In Section 2 we present the calculation method, based on RDMA. In Section 3 and Section 4 we discuss the case of the electric field applied parallel to the NPL growth axis (z-axis). In the next section we analyze the case of the field applied parallel to the NPL plane, when the exciting electromagnetic wave energy is below the fundamental gap. In Section 6, we consider the case of a field applied parallel to the NPL plane when the exciting wave energy exceeds the fundamental gap, is discussed. Finally, Section 7 presents the conclusions. Appendix A and Appendix B provide detailed calculation procedures.

2. The Method

The real density matrix approach (RDMA) is particularly suitable for computing the effects of external fields because it includes both the relative motion of the carriers and the center-of-mass motion, where the interaction with the radiation field occurs.

We analyze the weak field limit, where the set of basic RDMA equations (‘constitutive equations’) reduces to a set of linearized equations which are the inter-band equations with only the linear source on the right-hand-side retained. The resulting linearized equations for the coherent amplitudes for the electron-hole pair of coordinates ${\mathbf{r}}_1={\mathbf{r}}_h$ and ${\mathbf{r}}_2={\mathbf{r}}_e$ between any pair of bands $\alpha$ (valence band) and b (conduction band) read:

$$-i(\hslash {\partial }_t+{\mathsf{\Gamma }}_{\alpha b})Y_{12}^{\alpha b}+H_{eh\alpha b}{Y}_{12}^{\alpha b}={\mathbf{M}}_{\alpha b}\mathbf{E},$$

(1)

where ${\mathsf{\Gamma }}_{\alpha b}=\hslash /T_2^{\alpha b}$ is a phenomenological damping coefficient, E is the electric vector of the impinging electromagnetic wave, and ${\mathbf{M}}_{\alpha b}$ is the inter-band transition dipole density. The notation $Y_{12}^{\alpha b}=Y^{\alpha b}({\mathbf{r}}_1,{\mathbf{r}}_2)$ is used. When external static fields F (electric field), and B (magnetic field), are applied, the effective-mass two-band Hamiltonian $H_{eh\alpha b}$ with gap $E_{g\alpha b}$ for any pair of bands is

$$\begin{array}{ccc}& & H_{eh\alpha ,b}=E_{g\alpha b}+{\displaystyle \frac{1}{2}}{\underline{\underline{m_e}}}^{-1}{\left({\mathbf{p}}_e-e{\displaystyle \frac{{\mathbf{r}}_e\times \mathbf{B}}{2}}\right)}^2\hfill \\ & & +{\displaystyle \frac{1}{2}}{\underline{\underline{m_h}}}^{-1}{\left({\mathbf{p}}_h+e{\displaystyle \frac{{\mathbf{r}}_h\times \mathbf{B}}{2}}\right)}^2+e\mathbf{F}·({\mathbf{r}}_e-{\mathbf{r}}_h)\hfill \\ & & +V_{\mathrm{conf}}({\mathbf{r}}_e,{\mathbf{r}}_h)-{\displaystyle \frac{e^2}{4\pi {\epsilon }_0{\epsilon }_1|{\mathbf{r}}_e-{\mathbf{r}}_h|}},\hfill \end{array}$$

(2)

where appropriate values of $E_{g\alpha b}$ and the effective mass tensors ${\underline{\underline{m}}}_{e,h}$, and $V_{\mathrm{conf}}$ are the surface potentials for electrons and holes. The coherent amplitudes $Y_{12}^{\alpha b}$ determine the total NPL polarization

$$\mathbf{P}\left(\mathbf{R}\right)=2\sum _{\alpha ,\dots ,b,\dots }\mathrm{Re}{\mathbf{M}}_{cw}\left(\mathbf{r}\right)Y^{cw}(\mathbf{R},\mathbf{r}),$$

(3)

where the summation includes all allowed excitonic transitions between the valence $(w=\alpha ,\beta \dots )$ and conduction $(c=a,b,\dots )$ bands. The vectors R and r denote the electron–hole center-of-mass and relative coordinates, respectively. The Equations (1)–(3) connect the polarization to the electric field. In addition, the electric field must obey the Maxwell equation

$$-c^2{\epsilon }_0\nabla \times \nabla \times \mathbf{E}-{\epsilon }_0{\epsilon }_1\ddot{\mathbf{E}}=\ddot{\mathbf{P}},$$

(4)

where the polarization is given in Equation (3), and ${\epsilon }_1$ is the dielectric constant of the material contained inside the NPL.

The RDMA scheme, described by Equations (1)–(4), is solved in the following steps.

1.

We solve Equation (1), with the Hamiltonian (2), to obtain the excitonic amplitudes $Y_{12}^{\alpha b}$.

2.

Having calculated the amplitudes, we use them in Equation (3) and, in the long wave approximation, determine the NPL susceptiblity.

3.

The so obtained susceptibility enables to calculate the optical functions (electro-reflectivity, transmissivity, absorption).

We consider dipole-allowed transitions at the $\mathsf{\Gamma }$ point of the Brillouin zone within a simple two-band model. Further, we assume that the electric field E is linearly polarized with a component $E_x$ and that the vector M has a non-vanishing component $M_x\left(\mathbf{r}\right)$ in the same direction. For the quantities $Y,\mathbf{E}\mathrm{and}\mathbf{P}$ the center-of-mass dependence is of the plane-wave form $\exp (i{k}_{z}Z-i\omega t)$. As in previous works, in the NPL internal region we separate the exciton center-of-mass and relative motion, and consider the case $\mathbf{B},\mathbf{F}\Vert z$. This assumption forces the cylindrical symmetry, and the Hamiltonian (2) takes the form

$$\begin{array}{ccc}\hfill H& =& E_g+{\displaystyle \frac{p_{hz}^2}{2m_{hz}}}+V_h\left(z_h\right)+{\displaystyle \frac{p_{ez}^2}{2m_{ez}}}+V_e\left(z_e\right)\hfill \\ & +& {\displaystyle \frac{1}{2{\mu }_{\Vert }}}({\mathbf{p}}_{e\rho }^2+{\mathbf{p}}_{h\rho }^2)+V_e\left({\rho }_e\right)+V_h\left({\rho }_h\right)\hfill \\ & +& {\displaystyle \frac{1}{8}}{\mu }_{\Vert }{\omega }_c^2{\rho }^2+{\displaystyle \frac{e}{2{\mu }_{\Vert }^{\prime }}}B{\mathcal{L}}_z+eF(z_e-z_h)\hfill \\ & -& {\displaystyle \frac{e}{M_{\Vert }}}{\mathbf{P}}_{\Vert }·(\mathit{\rho }\times \mathbf{B})-{\displaystyle \frac{e^2}{\sqrt{{\rho }^2+{(z_e-z_h)}^2}}},\hfill \end{array}$$

(5)

where the cyclotron frequency is

$${\omega }_c={\displaystyle \frac{eB}{{\mu }_{\Vert }}},$$

(6)

and the reduced mass ${\mu }_{\Vert }^{\prime }$ is defined as

$${\displaystyle \frac{1}{{\mu }_{\Vert }^{\prime }}}={\displaystyle \frac{1}{m_{e\Vert }}}-{\displaystyle \frac{1}{m_{h\Vert }}}.$$

(7)

The operator ${\mathcal{L}}_z$ is the z-component of the angular-momentum operator. To analyze the electro-optical effects, we set all terms in the Hamiltonian (5) related to the magnetic field equal to zero.

3. The Electric Field Parallel to the Z-Axis

We analyze the changes in the NPL optical response when a constant external electric field F is applied along the z direction. We consider a CdSe nanoplatelet of cuboid shape located at the $z=0$ plane, with barriers at $x=\pm L_x/2,y=\pm L_y/2,z=\pm L_z/2$. For NPLs, the vertical dimension (‘thickness’) $L_z$ is typically on the order of a few monolayers, which, for the CdSe NPLs considered below, corresponds to 1–2 nm. The lateral dimensions are much larger, typically several dozen nm. A schematic representation of the nanoplatelet is shown in Figure 1.

As previously shown, the small vertical dimension modifies the electron and hole effective masses, which increase with decreasing thickness. This effect is illustrated in

Table 1. Masses, reduced masses, Rydberg energies, Luttinger parameters, and coherence radii, from Ref. [14]. Lengths in nm, masses in free electron mass $m_0$, energies in meV, ML means ‘monolayers’.

Parameter	3ML	4ML	5ML
$L_z$	1	1.33	1.67
$m_{ez}$	0.2567	0.2015	0.1635
$m_{e\Vert }$	0.3208	0.2519	0.2044
$a_{ez}^{*}$	1.236	1.575	1.94
$m_{hzH}$	1.1925	0.9754	0.8153
$m_{h\Vert H}$	0.4957	0.4337	0.3879
$m_{hzL}$	0.4149	0.3659	0.3302
$m_{h\Vert L}$	0.8121	0.6887	0.5963
${\mu }_{zH}$	0.2112	0.167	0.1362
${\mu }_{zL}$	0.1586	0.13	0.1094
$a_{ez}^{*}$	1.236	1.575	1.94
$a_{e\Vert }^{*}$	0.989	1.26	1.55
$a_{hzH}^{*}$	0.266	0.325	0.389
$a_{hzL}^{*}$	0.765	0.867	0.961
$E_{ez}$	918.13	686.81	530
$E_{hzH}$	221.13	162.1	130
$E_{hzL}$	579.53	352.26	277.58
$E_{zH}$	1139.4	816.8	636
$E_{zL}$	1497.4	1036.5	783.1
$R_{ez}^{*}$	96.98	76.12	61.77
$R_{\Vert H}^{*}$	73.58	60.20	51.28
$R_{hzH}^{*}$	450.5	368.5	308
$R_{hzL}^{*}$	156.74	138.23	124,66
$R_{\Vert L}^{*}$	86.88	69.67	57.49
${\gamma }_1$	1.6243	1.8789	2.1062
${\gamma }_2$	0.3929	0.4269	0.4488
${\rho }_{0H}$	0.20	0.18	0.17
${\rho }_{0L}$	0.22	0.19	0.18

. We consider the NPL response to a normally incident electromagnetic wave, linearly polarized along the x direction:

$${\mathbf{E}}_i(z,t)={\mathbf{E}}_{i0}\exp (i{k}_{0}z-i\omega t),k_0={\displaystyle \frac{\omega }{c}}.$$

(8)

The motion of electrons and holes along the z direction is affected by the dielectric confinement potential, which we assume to have the form

$$V_{e,h}\left(z\right)={\displaystyle \frac{{\gamma }_{e,h}}{(L/2)-z}}.$$

(9)

The coefficient $\gamma$ is proportional to the dielectric permittivities [25,26,27]:

$$\gamma \propto {\displaystyle \frac{{ϵ}_1-{ϵ}_2}{{ϵ}_1({ϵ}_1+{ϵ}_2)}},$$

(10)

where ${ϵ}_2\left({ϵ}_{out}\right)$ and ${ϵ}_1\left({ϵ}_{in}\right)$ denote the permittivities of the surrounding environment (ligands) and the internal medium, respectively, with ${ϵ}_2<<{ϵ}_1$. Using the potential (9), we solve the corresponding Schrödinger equation [14]. The resulting eigenvalues, which demonstrate the impact of the permittivities, are given in Appendix A, Equation (A1). In the calculations, we use the values ${ϵ}_1=6,{ϵ}_2=2$ [5].

The calculation of electro-optical properties becomes significantly simpler when we consider an NPL of thickness $L_z$ with parabolic confinement potentials of harmonic-oscillator form:

$$V_{\mathrm{conf}}={\displaystyle \frac{1}{2}}{m}_{ez}{\omega }_{ez}^2{z}_e^2+{\displaystyle \frac{1}{2}}{m}_{hz}{\omega }_{hz}^2{z}_h^2,$$

(11)

where the energies $\hslash {\omega }_{ez},\hslash {\omega }_{hz}$ correspond to the electron and hole barrier heights. For the in-plane electron and hole motion, we retain the assumptions of Ref. [14], in which the cuboidal NPL is replaced with a cylinder of height $L_z$ and radius $r_{eff}=\sqrt{L_x{L}_y/\pi }.$ We neglect the hole motion and assume that the electron moves under the Coulomb attraction to a hole fixed at $\rho =0,z=0$, and under the confinement potential

$$\begin{array}{ccc}& & V_e\left({\rho }_e\right)=\left\{\begin{array}{c}0\mathrm{for}{\rho }_e\le R,\hfill \\ \infty \mathrm{for}{\rho }_e>R,\hfill \end{array}\right.\hfill \end{array}$$

(12)

where $R=r_{eff}$. In this case, with a constant applied electric field, the NPL Hamiltonian takes the form

$$\begin{array}{ccc}\hfill H& =& E_g+H_{m_{ez},{\omega }_{ez}}^{\left(1D\right)}\left(z_e\right)+H_{m_{hz},{\omega }_{hz}}^{\left(1D\right)}\left(z_h\right)\hfill \\ & +& H_{\mathrm{Coul}}^{\left(2D\right)}\left(\mathit{\rho }\right)+eF(z_e-z_h)\hfill \\ & +& {\displaystyle \frac{e^2}{4\pi {ϵ}_0{ϵ}_b{\rho }_e}}-{\displaystyle \frac{e^2}{\sqrt{{\rho }_e^2+{(z_e-z_h)}^2}}},\hfill \end{array}$$

(13)

which contains the one-dimensional oscillator Hamiltonians

$$H_{m,\omega }^{\left(1D\right)}\left(z\right)={\displaystyle \frac{p_z^2}{2m}}+{\displaystyle \frac{1}{2}}{m}_z{\omega }^2{z}^2,$$

(14)

and the two-dimensional Coulomb Hamiltonian

$$H_{\mathrm{Coul}}^{\left(2D\right)}\left(\mathit{\rho }\right)={\displaystyle \frac{{\mathbf{p}}_{\Vert }^2}{2m_{e\Vert }}}-{\displaystyle \frac{e^2}{4\pi {ϵ}_0{ϵ}_1{\rho }_e}}+V_e\left({\rho }_e\right).$$

(15)

Using the substitution

$$\begin{array}{ccc}& & {\zeta }_e=z_e+z_{0e},z_{0e}={\displaystyle \frac{eF}{m_{ez}{\omega }_{ez}^2}},\hfill \end{array}$$

(16)

$$\begin{array}{ccc}& & {\zeta }_h=z_h-z_{0h},z_{0h}={\displaystyle \frac{eF}{m_{hz}{\omega }_{hz}^2}},\hfill \end{array}$$

(17)

we obtain the QW Hamiltonians in the form

$$\begin{array}{ccc}\hfill H& =& E_g+H_{m_{ez},{\omega }_{ez}}^{\left(1D\right)}\left({\zeta }_e\right)+H_{m_{hz},{\omega }_{hz}}^{\left(1D\right)}\left({\zeta }_h\right)\hfill \\ & -& {\displaystyle \frac{{\left(eF\right)}^2}{2m_{ez}{\omega }_{ez}^2}}-{\displaystyle \frac{{\left(eF\right)}^2}{2m_{hz}{\omega }_{hz}^2}}+H_{\mathrm{Coul}}^{\left(2D\right)}\left({\rho }_e\right)\hfill \\ & +& {\displaystyle \frac{e^2}{4\pi {ϵ}_0{ϵ}_1{\rho }_e}}-{\displaystyle \frac{e^2}{\sqrt{{\rho }_e^2+{(z_e-z_h)}^2}}}.\hfill \end{array}$$

(18)

With this Hamiltonian, we solve the constitutive equation

$$(H-\hslash \omega -i\mathsf{\Gamma })Y=\mathbf{M}\mathbf{E}.$$

(19)

We use the long-wave approximation and seek solutions of the form

$$\begin{array}{ccc}\hfill Y({\rho }_e,{\zeta }_e,{\zeta }_h)& =& E\left(Z\right)\sum _{jm}{Y}_{0,jm}{\psi }_{jm}\left({\rho }_e\right)\hfill \\ & \times & {\psi }_{{\alpha }_{ez}}^{\left(1D\right)}\left({\zeta }_e\right){\psi }_{{\alpha }_{hz}}^{\left(1D\right)}\left({\zeta }_h\right),\hfill \end{array}$$

(20)

where ${\psi }_{jm}$ are the eigenfunctions of the Hamiltonian (15),

$$\begin{array}{ccc}\hfill {\psi }_{jm}(\xi ,\varphi )& =& C{\xi }^{\left|m\right|}{e}^{-\xi /2}\hfill \\ & \times & M\left(m+{\displaystyle \frac{1}{2}}-\eta ,2\left|m\right|+1;\xi \right){\displaystyle \frac{e^{im\varphi }}{\sqrt{2\pi }}}.\hfill \end{array}$$

(21)

Here, j and m are the principal and magnetic quantum numbers of the two-dimensional excitonic state, $M(a,b,z)$ is the confluent hypergeometric function (notation of Ref. [28]), $\rho ={\rho }_e/a_{e\Vert }^{*}$, and

$$\begin{array}{ccc}\hfill \eta & =& {\displaystyle \frac{2}{\kappa }},\xi =\kappa \rho ,a_{e\Vert }^{*}={\displaystyle \frac{m_0}{m_{e\Vert }}}{ϵ}_1{a}_B^{*},\hfill \\ \hfill {\kappa }^2& =& -4{\displaystyle \frac{2m_{e\Vert }}{{\hslash }^2}}{a}_{e\Vert }^{*2}E=4/ϵ,\hfill \end{array}$$

where $m_0$ is the free-electron mass and $a_B^{*}=0.0529\mathrm{nm}$ is the hydrogen Bohr radius. The functions ${\psi }_{{\alpha }_z,N}^{\left(1D\right)}\left(z\right)$ (N = 0, 1, …) are the quantum-oscillator eigenfunctions of the Hamiltonian (14),

$$\begin{array}{ccc}& & {\psi }_{{\alpha }_z,N}^{\left(1D\right)}\left(z\right)={\pi }^{-1/4}\sqrt{{\displaystyle \frac{{\alpha }_z}{2^{N}N!}}}{H}_N\left({\alpha }_{z}z\right)e^{-{\displaystyle \frac{{\alpha }_z^2}{2}}{z}^2},\hfill \\ & & {\alpha }_z=\sqrt{{\displaystyle \frac{m_z{\omega }_z}{\hslash }}},\hfill \end{array}$$

(22)

where $H_N\left(x\right)$ are Hermite polynomials $(N=0,1,\dots )$. We analyze the lowest confinement state, $N_e=N_h=0$.

For further calculations, we define the dipole density M. In view of the experiments reported in Refs. [6,22], in which resonances due to 1SH and 1SL excitons were observed, we choose

$${\mathbf{M}}_{S,\alpha }\left(\mathbf{r}\right)={\mathbf{M}}_{0,S\alpha }{N}_{S\alpha }\exp (-\rho /{\rho }_{0\alpha })\delta (z_e-z_h),$$

(23)

where ${\rho }_{0\alpha }$ are the coherence radii

$$\begin{array}{ccc}\hfill {\rho }_{0H}& =& \sqrt{{\displaystyle \frac{R_{\Vert H}^{*}}{E_g}}},{\rho }_{0L}=\sqrt{{\displaystyle \frac{R_{\Vert L}^{*}}{E_g}}},\hfill \end{array}$$

(24)

and $N_{S,\alpha }$ are normalization constants defined by

$$N_{S,\alpha }\underset{0}{\overset{\mathcal{R}}{\int }}\rho \exp (-\rho /{\rho }_{0\alpha })d\rho =1.$$

(25)

The integrated dipole strengths ${\mathbf{M}}_{0,S\alpha }$ for CdSe NPLs of various sizes are given in Ref. [14].

Substituting the series (20) into the constitutive equation (19), we calculate the expansion coefficients $Y_{0,jm}$. The resulting exciton amplitudes are inserted into Equation (3), yielding the NPL polarization and, through $\mathbf{P}={ϵ}_0\chi \mathbf{E}$, the susceptibility $\chi$

$$\begin{array}{ccc}\hfill \chi & =& {\displaystyle \frac{2}{{ϵ}_0}}\sum {\displaystyle \frac{|\langle M|{\psi }_{jm}\left({\rho }_e\right){\psi }_{{\alpha }_{ez}}^{\left(1D\right)}\left({\zeta }_e\right){\psi }_{{\alpha }_{hz}}^{\left(1D\right)}\left({\zeta }_h\right)\rangle {|}^2}{E_{res,jm}-\hslash \omega -i\mathsf{\Gamma }}},\hfill \end{array}$$

(26)

where the summation runs over the excitonic transitions considered. The exciton resonance energies are defined as

$$\begin{array}{ccc}\hfill E_{res,jm}& =& E_g+W_{0e}+W_{0h}+E_{jm}\hfill \\ & +& 〈{\displaystyle \frac{2}{\rho }}〉+\Delta E+E_{b,jm},\hfill \end{array}$$

(27)

where

$$\begin{array}{ccc}\hfill W_{0e,h}& =& {\displaystyle \frac{1}{2}}\hslash {\omega }_{e,hz},\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill 〈{\displaystyle \frac{2}{\rho }}〉& =& 2R_{e\Vert }^{*}\underset{0}{\overset{R}{\int }}d\rho {\psi }_{jm}^2\left(\rho \right),\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill \Delta E& =& -{\displaystyle \frac{{\left(eF\right)}^2}{2m_{ez}{\omega }_{e{z}^2}}}-{\displaystyle \frac{{\left(eF\right)}^2}{2m_{hz}{\omega }_{hz}^2}},\hfill \end{array}$$

(30)

$E_{jm}$ are the eigenvalues of the operator (15), and $\Delta E$ is the Stark shift. The excitonic binding energy $E_{b,jm}$ is

$$\begin{array}{ccc}\hfill E_{b,jm}& =& -2R_{e\Vert }^{*}\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}d{z}_{e}d{z}_h{\left({\psi }_{{\alpha }_{ez}}^{\left(1D\right)}\right)}^2\left(z_e\right){\left({\psi }_{{\alpha }_{hz}}^{\left(1D\right)}\right)}^2\left(z_h\right)\hfill \\ & \times & \underset{0}{\overset{R}{\int }}{\displaystyle \frac{{\psi }_{jm}^2\left(\rho \right)\rho d\rho }{\sqrt{{\rho }^2+{(z_e-z_h)}^2}}}.\hfill \end{array}$$

(31)

4. Results of Specific Calculations for $\mathit{F}\Vert \mathit{z}$

We performed calculations for three CdSe nanoplatelets analyzed in Ref. [6], with the following dimensions:

• 3ML $1.0\times 56\times 41\mathrm{nm}$,
• 4ML $1.33\times 17\times 15\mathrm{nm}$,
• 5ML $1.67\times 30\times 11\mathrm{nm}$.

All the parameters used in the calculations are listed in Table 1 and

Table 2. Sizes and exciton states energies, transition matrix elements M, oscillator strengths, and damping parameters, for disks analyzed by Brumberg et al. [6], lengths in nm, matrix elements M in $e·\mathrm{nm}$, energies in meV, the energy gap at room temperature 1750 meV, notation: 1: $56\times 41,3\mathrm{ML}$, 2: $17\times 15,4\mathrm{ML}$ 3: $30\times 11,5\mathrm{ML}$, 4: $17\times 15,4\mathrm{ML}$, 5: $30\times 11,4\mathrm{ML}$, 6: $56\times 41,4\mathrm{ML}$.

lat. extension	1	2	3	4	5	6
$a_{e\Vert }^{*}$	1	1.26	1.553	1.26	1.26	1.26
$r_{\mathrm{eff}}$	27	9	10.25	9	10.25	27
$\mathcal{R}$	27	7.15	6.6	7.15	8.134	21.455
1SH	2640.44	2382.2	2242.2	2540	2537.7	2531
$\lambda$	469.2	520.5	553	488.6	489	490.4
1SL	3178.4	2745.25	2498.4	2761	2758.7	2752
$\lambda$	390	451.68	496.31	449.5	449.8	450.9
$M_{0SH}$	0.625	0.22	0.19	0.22	0.19	0.625
$f_{SH}$	4.16	4.77	5.47	4.77	4.6	4.96
$f_{SL}$	3.72	3.75	4.77	3.75	4.44	4.41
${\mathsf{\Gamma }}_{SH}$	4.63	2.53	2.86	2.53	2.86	2.86

. We calculated the Stark shift as a function of the lateral dimension $L_z$ and the applied field strength F, obtaining

$$\Delta E_{H,L}=-C{\left({\displaystyle \frac{F}{F_{IB}}}\right)}^2{M}_{zH,L}{L}_z^4,$$

(32)

where

$$C={\displaystyle \frac{R_B^{*3}}{{\left(556\right)}^2}}=8.137\times {10}^6,$$

(33)

and $M_{zH,L}=m_{ez}+m_{hzH,L}$ is the total exciton mass in the z direction. The quantity $F_{IB}$ denotes the ionization field strength

$$F_{IB}=2.57\times {10}^6{\displaystyle \frac{\mathrm{kV}}{\mathrm{cm}}}.$$

(34)

Details of the calculations are provided in Appendix A. The dependence of the Stark shift on the applied field strength is shown in Figure 2.

Using the eigenfunctions (22) and the dipole densities (23), we calculated the oscillator strengths associated with vertical motion, obtaining

$$f_{L_z,F}=\exp \left(-1.15M_z{L}_z^2·{10}^{-4}{x}^2\right).$$

(35)

The dependence of the oscillator strength on the applied field strength is depicted in Figure 3. The oscillator strength decreases with increasing field strength.

Finally, we calculated the absorption coefficient as the imaginary part of the susceptibility (26). This expression requires the resonance energies $E_{res,jm}$. An important component of the resonance energy is the exciton binding energy, defined in Equation (31). The dependence of the binding energy, resonance energies, and absorption coefficient on the applied field strength is shown in Figure 4, Figure 5 and Figure 6. In particular, the binding energy decreases with increasing field strength, as shown in Figure 4. Consequently, the resonance energies increase despite the Stark shift, which acts in the opposite direction. The absorption maxima shift toward higher energies (shorter wavelengths), in contrast to the behavior observed in quantum wells (Figure 5). Such a shift was observed experimentally by Baghdasaryan et al. [22]. Moreover, the heavy- and light-exciton absorption maxima approach each other and merge in the high-field limit (Figure 6).

5. The Electric Field Parallel to the NPL Plane, Excitation Below Gap

In this section, we analyze the electro-optical effects when the external electric field F is applied parallel to the NPL plane along the x-axis. This case is more complicated than the configuration $F\Vert z$, in which the electric field is decoupled from the two-dimensional Coulomb potential acting in the $x-y$ plane. Here, the field, parallel to the x-axis, acts in the same plane as the cylindrically symmetric Coulomb potential, which makes it impossible to find an analytic solution of the corresponding Schrödinger equation. To gain insight into the effect of the electric field, we adopt a two-dimensional model in which the electron moves along the x-axis in the interval $-L_x/2\le x\le L_x/2$, while the electron and hole motion in the z direction is retained, in the plane $x,y=0$. With these assumptions, the Hamiltonian takes the form

$$\begin{array}{ccc}\hfill H& =& E_g+H_{m_{ez},{\omega }_{ez}}^{\left(1D\right)}\left(z_e\right)+H_{m_{hz},{\omega }_{hz}}^{\left(1D\right)}\left(z_h\right)\hfill \\ & +& V\left(x\right)+H_{\mathrm{Coul}}^{\left(1D\right)}\left(x\right)+eFx\hfill \\ & -& H_{\mathrm{Coul}}^{\left(1D\right)}\left(x\right)-{\displaystyle \frac{e^2}{\sqrt{x^2+{(z_e-z_h)}^2}}},\hfill \\ \hfill H_{\mathrm{Coul}}^{\left(1D\right)}\left(x\right)& =& -{\displaystyle \frac{{\hslash }^2}{2m}}{\displaystyle \frac{d^2}{d{x}^2}}-{\displaystyle \frac{e^2}{4\pi {\epsilon }_0{\epsilon }_b\left|x\right|}}+V\left(x\right),\hfill \end{array}$$

(36)

where

$$\begin{array}{ccc}& & V\left(x\right)=\left\{\begin{array}{c}0\mathrm{for}\left|x\right|\le L_x/2,\hfill \\ \infty \mathrm{for}\left|x\right|>L_x/2.\hfill \end{array}\right\}.\hfill \end{array}$$

(37)

The eigenfunctions of the operator $H_{\mathrm{Coul}}^{\left(1D\right)}\left(x\right)$ have the form

$$\begin{array}{ccc}\hfill \psi \left(\xi \right)& =& \left|\xi \right|e^{-\left|\xi \right|/2}M(1-\eta ;2;|\xi \left|\right),\hfill \\ \hfill \xi & =& \kappa {\displaystyle \frac{x}{a_{e\Vert }^{*}}},\kappa =2\sqrt{-\epsilon }={\displaystyle \frac{2}{\eta }},\hfill \end{array}$$

(38)

see Appendix B. Repeating the method used in the case $F\Vert z$, i.e., replacing the eigenfunction (38) with the function (22) using appropriate values of the coefficient $\alpha$, we obtain the Stark shift in the form

$$\Delta E=4.5\times {10}^{-4}{F}^2,$$

(39)

where the result is given in meV. The magnitude of $\Delta E$ depends not only on the field strength F but also on the NPL size, represented here by $L_x/2=R$ (see Equation (12)). The values of $\Delta E$ for NPLs with of sizes are shown in Figure 7. The presented values are approximately one order of magnitude larger than the Stark shift for a field applied in the z direction (Figure 2). The curves in Figure 7 should begin at $F=0$. However, because they are very close to one another, they are slightly offset for clarity. Despite the strong simplifications of the model, the calculated eigenvalues are in good agreement with the experimental values at $F=0$ reported in Ref. [6], see Figure 8. Using the data in

Table 3. Parameters for calculating the Stark shift, excitonic resonance energies vs applied field strength and lateral area, field strength in kV/cm, $X={10}^{2}F$, energies in meV, wavelengths in nm, oscillator strength to be multiplied by ${10}^{-3}$.

Param.	$17\times 15$	$30\times 11$	$56\times 41$
${\alpha }^{-2}$	3.283	3.22	3.03
${\alpha }^{-4}$	10.78	10.375	9.22
${10}^5·{\xi }_0/F^2$	1.89	1.82	1.62
${10}^6·\Delta \epsilon /F^2$	4.72	4.55	4.04
${10}^4\Delta E/F^2$	4.5	4.33	3.85
	$X=0$
$|E_b|$	227.9	226.38	281.76
$E_{resH}$	2450.166	2448	2441
$\lambda$	506	506.54	508
$E_{resL}$	2670.166	2668	2661
$\lambda$	464.39	464.77	466
	$X=0.5$
$|E_b|$	243.52	243.04	241.67
$E_{resH}$	2408.3	2410.5	2417.36
$\lambda$	514.88	514.41	512.95
$E_{resL}$	2628.3	2630.5	2637.6
$\lambda$	471.8	471.4	470.12
	$X=1$
$|E_b|$	288.15	287.46	278.72
$E_{resH}$	2360.2	2362.84	2377.42
$\lambda$	525.36	524.79	521.57
$E_{resL}$	2580	2582.84	2597.42
$\lambda$	480.6	480.1	477.4
	$X=2$
$|E_b|$	478.49	481.63	477.2
$E_{resH}$	2174.45	2173	2177.42
$\lambda$	570.25	570.63	569.48
$E_{resL}$	2394.45	2393.0	2397.4
$\lambda$	517.86	518.18	517.22

, we calculated the normalized absorption, which depends on both the NPL size and the applied field strength. The results are illustrated in Figure 9 and Figure 10.

6. Quantum-Confined Franz–Keldysh Effect

When the excitation energy exceeds the total confinement energy (including the energy gap) and an external electric field is applied parallel to the NPL plane, oscillations appear in the optical functions. In unbounded media, these oscillations are known as the Franz–Keldysh effect (see, for example, [29]). In such media, they take the form of outgoing waves with variable periodicity and amplitude (see, for example, [30,31]). The situation changes when the medium, as considered here, is a finite NPL. Instead of outgoing waves, standing waves appear, with periodicity and amplitude depending on both the applied field strength and the NPL size. This phenomenon can be referred to as the quantum-confined Franz–Keldysh effect (QCFKE). The term has been used previously [32]; however, in that work, only partial confinement (a QW) was considered, the field was applied in the z direction, and the effect appeared in the limit $L_z\to \infty$. The situation examined here differs substantially: confinement is present in three directions. Therefore, the term quantum-confined Franz–Keldysh effect is justified. In the excitation region considered, the electron–hole Coulomb interaction can be neglected in the lowest approximation. The remaining terms in the Hamiltonian correspond to kinetic energies and constitute the total confinement. To analyze the NPL QCFKE, we use the constitutive equation

$$\begin{array}{ccc}& & \left(k^2-{\displaystyle \frac{d^2}{d{\xi }^2}}+f\xi \right)Y\left(\xi \right)={\displaystyle \frac{1}{a_{e\Vert }^{*}}}M\left(\xi \right)E,\hfill \\ & & k^2={\displaystyle \frac{E_{conf,tot}-\hslash \omega }{R_{e\Vert }^{*}}},\hfill \end{array}$$

(40)

where

$$\begin{array}{ccc}\hfill \xi & =& {\displaystyle \frac{x_e}{a_{e\Vert }^{*}}},f={\displaystyle \frac{F}{F_I}},F_I={\displaystyle \frac{R_{e\Vert }^{*}}{e·a_{e\Vert }^{*}}}.\hfill \end{array}$$

(41)

In the above equations, the electric field F is parallel to the x-axis, and we retain the assumptions that the hole is located at the center of the the NPL. As in the case of unbounded media, Equation (40) can be solved using an appropriate Green’s function, which takes the form

$$\begin{array}{ccc}\hfill G(\xi ,{\xi }^{\prime })& =& g^{<}·g^{>},\hfill \\ \hfill g^{<}& =& {\displaystyle \frac{\pi }{f^{1/3}}}\{\mathrm{Bi}\left[f^{1/3}\left({\xi }^{<}+{\displaystyle \frac{k^2}{f}}\right)\right]\hfill \\ & \times & \mathrm{Ai}\left[f^{1/3}\left({\displaystyle \frac{k^2}{f}}\right)\right]\hfill \\ & -& \mathrm{Bi}\left[f^{1/3}\left({\displaystyle \frac{k^2}{f}}\right)\right]\mathrm{Ai}\left[f^{1/3}\left({\xi }^{<}+{\displaystyle \frac{k^2}{f}}\right)\right]\},\hfill \\ \hfill g^{>}& =& \mathrm{Bi}\left[f^{1/3}\left({\xi }^{>}+{\displaystyle \frac{k^2}{f}}\right)\right]\hfill \\ & \times & \mathrm{Ai}\left[f^{1/3}\left(\mathcal{L}+{\displaystyle \frac{k^2}{f}}\right)\right]\hfill \\ & -& \mathrm{Bi}\left[f^{1/3}\left(\mathcal{L}+{\displaystyle \frac{k^2}{f}}\right)\right]\mathrm{Ai}\left[f^{1/3}\left({\xi }^{>}+{\displaystyle \frac{k^2}{f}}\right)\right].\hfill \end{array}$$

(42)

Here $\mathrm{Ai}\left(z\right),\mathrm{Bi}\left(z\right)$ are the Airy functions ([28]), and the notation ${\xi }^{<},{\xi }^{>}$ means ${\xi }^{<}=\min (\xi ,{\xi }^{\prime }),{\xi }^{>}=\max (\xi ,{\xi }^{\prime })$, and $\mathcal{L}=L_x/2a_{e\Vert }^{*}$. The confinement effect is included since

$$G(0,{\xi }^{\prime })=G(\xi ,\mathcal{L})=0.$$

(43)

When the dipole density $M\left(\xi \right)$ is defined, we obtain the NPL susceptibility in the form

$$\chi =MGM=\underset{0}{\overset{\mathcal{L}}{\int }}d\xi d{\xi }^{\prime }G(\xi ,{\xi }^{\prime })M\left(\xi \right)M\left({\xi }^{\prime }\right).$$

(44)

The absorption coefficient is proportional to the imaginary part of the susceptibility. Its profile, exhibiting Franz–Keldysh oscillations, is shown in Figure 11 for two values of the applied field and two lateral NPL dimensions. A strong dependence on both the field strength and the NPL size is observed.

7. Conclusions

In this study, we have investigated the electro-optical properties of CdSe NPLs with excitons for various orientations of an applied static electric field. These properties, compared with those of bulk materials and QWs, exhibit significant modifications due to the total confinement of electrons and holes, resulting in a pronounced dependence of the spectra on NPL size. In addition to different field orientations, we have separately analyzed two excitation-energy regimes: below and above the gap. For excitation energies below the gap and for both orientations of the field, exciton resonances are observed. They shift to lower energies (for an electric field parallel to the NPL plane) and to higher energies (for an electric field parallel to the z-axis). The latter effect arises from a reduction in the exciton binding energy, which outweighs the quadratic Stark shift that is also present. When the excitation energy exceeds the gap (including the confinement energy), Franz–Keldysh oscillations appear, with periodicity depending on both the applied field strength and the NPL size. These results demonstrate that CdSe NPLs provide an additional mechanism for controlling exciton states not only through thickness (z direction) and lateral size variation but also through the strength and orientation of an applied static electric field. This capability paves the way for the construction of high-sensitivity modulators based on NPLs. The obtained results agree well with the available experimental data.