# Exciton Spectroscopy of Spatially Separated Electrons and Holes in the Dielectric Quantum Dots

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Energy of Ground State Exciton in Nanosystems

_{2}, surrounded by a dielectric matrix with ε

_{1}permittivity. In the QD volume h hole moves with the effective mass m

_{h}, and e electron with the effective mass m

_{е}

^{(1)}is situated in the matrix (r

_{е}and r

_{h}—the distance of electron and hole from the QD center). Let us assume that in QD the valence band has a parabolic form. We also assume that on the spherical surface boundary of QD/matrix infinite high potential barrier exists. Therefore, in the studied model h hole cannot go out from the QD volume, and e electron can not penetrate into the QD volume.

_{2}− ε

_{1})/(ε

_{2}+ ε

_{1}).

_{h}, is located in the center of the QD (wherein r

_{h}= 0), and the e electron with an effective mass m

_{е}

^{(1)}is localized in the matrix of the spherical surface QD (r

_{е}= r—the distance of e electron from the QD center). Such assumption is justified, as the ratio of the effective masses of electrons and holes in nanosystem $\left(({m}_{e}^{\left(1\right)}/{m}_{h})\ll 1\right)$. In the observed quasi-zero nanosystems model, in the frame of the above mentioned approximations, as well as in the effective mass approximation and the center of mass of nanosystem, the Hamiltonian of exciton (from spatially separated hole, moving in QD volume, and the electron situated in the dielectric matrix) takes the form:

_{0}= m

_{е}

^{(1)}m

_{h}/(m

_{е}

^{(1)}+ m

_{h})—the reduced mass of the exciton from spatially separated electrons and holes), the second describes the centrifugal energy of the exciton

_{2}permitivity. In the Hamiltonian (6), the energy of the Coulomb interaction between the electron and hole is described by the equation [5]:

_{h}= 0, takes the form:

_{ех}, where а

_{ех}= (ε

_{2}ћ

^{2}/µ е

^{2})—exciton Bohr radius, µ = m

_{е}

^{(2)}m

_{h}/(m

_{е}

^{(2)}+ m

_{h}) is the reduced exciton mass and m

_{е}

^{(2)}—the effective mass of electron in the semiconductor with permittivity ε

_{2}), the spherical surface between boundary of two media goes into the flat boundary surface. Wherein the exciton from the spatially separated electrons and holes (the hole moves in the semiconductor, and the electron is situated in the matrix) becomes two-dimensional. In the Hamiltonian (6) potential energy, describing the motion of the exciton in the nanosystem, containing QD with large radius (а >> а

_{ех}) the main contribution is the energy of the Coulomb interaction ${V}_{eh}\left(r\right)$ (8) between electron and hole. Interaction energy between electron and hole with their (9), (11) and “foreign” (11), (10) images gives a much smaller contribution to the potential energy (14) of the Hamiltonian (6). By the first approximation this contribution may be neglected. In the potential energy (14) of the Hamiltonian (6) remains only the energy of the Coulomb interaction (8) between electron and hole. The Schrodinger equation with this Hamiltonian describes a two-dimensional exciton (2D exciton) from spatially separated electron and hole energy spectrum, which takes the form [14]:

_{0}= 13.606 eV—the Rydberg constant. Bohr radius of such 2D exciton is described by the following equation:

_{h}/m

_{0}) = 6.2; the value of the effective mass of the electron in the matrix (m

_{е}

^{(1)}/m

_{0}) = 0.537 [17], situated in the matrix (vacuum oil), which was studied in experimental studies [2,3,4] (see Figure 1). Figure 1 shows the dependence of ${E}_{1,l}$(a) (19) for the states with l = 0, 1, 2, 3, 4, 5, 6, 7, 8 with quasi-stationary states spectrum border ${E}_{1,l}^{max}\left(a\right)$. The obtained results (see Figure 1) clearly illustrate the abovementioned qualitative features of considered dependences ${E}_{1,l}$(a) (22). The critical radii of QD for$l\le 8$ have are respectively values:

_{1,l}(а) (22) of the ground state of exciton (of spatially separated electrons and holes) in nanosystem containing aluminum oxide QD with average radius a from the interval (25), it follows that with the increase of QD radius Е

_{1,l}(а) (19) total energy of the ground state of the exciton increases. Herewith, the energy (22) of the ground state of the exciton significantly exceeds (3–49 times) the value of the binding energy of the exciton ${\tilde{E}}_{ex}^{2D}\approx (-$51.16 meV) in aluminum oxide single crystal. Starting from the radius $a\ge \tilde{{a}_{c}}\left(1,l\right)=19.1\mathrm{nm}$, the value of the total energy (22) of the ground state of the exciton asymptotically tends to the, accordingly, values of ${E}_{ex}^{2D}=(-$2.5038 eV) (17) (characterizing the binding energy of the ground state of the two-dimensional exciton (from spatially separated electrons and holes)) (see Figure 1).

## 3. Comparison of the Theory with Experiment

_{1,l}(а) (22) of the exciton in nanosystem containing aluminum oxide QD, varying radii a in the range of (25), (see Figure 1 and Figure 2 and Table 1) the band of surface exciton states appears (consisting of the stationary states band with ${E}_{1,l}={E}_{ex}^{2D}=$ 2.5038 eV width, located in the band gap of QD, and from a band from quasi-stationary states ${E}^{max}\text{}$ = 0.526 eV, located in the conduction band of QD. In [2,3,4] works it is revealed the formation of donor type additional band with (0.3$\u2013$0.4) eV width inside the bandgap aluminum oxide QD at the depth (2.3 eV and 1.96 eV) from the bottom of the conduction band of the QD. Let us suppose that such a band can be described by the band of surface exciton states. Then in the nanosystem levels E

_{d}= $-$2.3 eV and ($-$1.96 eV) correspond to the QDs with average radii: (14.0 nm, 14.9 nm, 16.1 nm, 17.0 nm, 18.1 nm) and (5 nm, 5.2 nm, 12.5 nm, 13 nm, 15.2 nm, 16.3 nm, 17.4 nm), respectively (see Figure 1), and besides the values of the radii lie in the range of average radii of QD studied in experimental conditions [2,3,4].

_{d}= $-$2.3 eV and ($-$1.96 eV)) [2,3,4]) excite the dipole moments of the transitions, the values of which are proportional to the a radii of QD and exceed by an order typical values for aluminum oxide single crystal [18,19]. Such mechanism of excitation of the dipole moments of transitions causes QD polarization in the field of the light wave, which creates an additional polarization of nanosystem, and as a result, the nonlinear increment to the refractive index of the nanosystem [2,3,4]. The increment in the range of frequencies below the resonance is positive. In the case of sufficient large length of interaction of light waves with the nanosystem the increase in the refractive index due to self-focusing of the beam can cause a waveguide channel. As a result, with an increase of the radiation intensity [2,3,4] “enlightenment” of nanosystem occurs.

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The dependence of the energy spectrum (${E}_{1,l}\left(a\right)-{E}_{g})$ of the exciton (from spatially separated electrons and holes) in the states (n = 1, l = 0, 1, 2, 3, 4, 5, 6, 7, 8) (where n and l—the principal and orbital quantum numbers of the electron) (solid line) on the a radius of the aluminum oxide QD, situated in the dielectric matrix (vacuum oil). The numbers at the curves indicate the value of l. The dashed lines indicate the boundaries of the spectrum of quasi-stationary states ${E}_{1,l}^{max}\left(a\right)$ of the exciton. Here ${E}_{g}$—bandgap of the aluminum oxide QD, ${E}_{ex}^{2D}=2.5038\mathrm{eV}$—the binding energy of the ground state of the two-dimensional exciton (from spatially separated electrons and holes).

**Figure 2.**Schematic representation of the energy levels ${E}_{1,l}\left(a\right)$ of the exciton (from spatially separated electron and hole) in the state (n = 1, l$\text{}\le $8) in the nanosystem consisting of Al2O3 QD with a radius situated the dielectric matrix (vacuum oil). The stationary states of the exciton (area 1) are located in the bandgap of Al2O3 QD (below the bottom of the conduction band ${E}_{\u0441}$ of Al2O3 QD). They are bounded below by level ${E}_{ex}^{2D}$, which is the binding energy of the ground state of the exciton (from spatially separated electrons and holes) in the nanosystems. Quasi-stationary states of the exciton (area 2) are in the ${E}_{\u0441}$ conduction band of Al2O3 QD (above the bottom of the conduction band ${E}_{\u0441}$). They are limited to the top boundary of the spectrum of quasi-stationary states ${E}_{1,l}^{max}\left(a\right)$ (see Figure 1). In this case, the hole moves in the valence band of QD, and the electron is located on energy levels (in area 1 and 2).

**Table 1.**Energy transitions $\Delta {E}_{l-1}^{l}\left(a\right)$ (expressed by meV), in which the orbital quantum number l changes by one (wherein l takes values from 1 to 8) between quasi-stationary and stationary states of the exciton (from the spatially separated electrons and holes) (see Figure 1), appearing in the nannosystem containing aluminum oxide QD with a radius (expressed by nm), situated in the dielectric matrix (vacuum oil).

а (nm) | $\Delta {\mathit{E}}_{0}^{1}\left(\mathit{a}\right)$ (meV) | $\Delta {\mathit{E}}_{1}^{2}\left(\mathit{a}\right)$ (meV) | $\Delta {\mathit{E}}_{2}^{3}\left(\mathit{a}\right)$ (meV) | $\Delta {\mathit{E}}_{3}^{4}\left(\mathit{a}\right)$ (meV) | $\Delta {\mathit{E}}_{4}^{5}\left(\mathit{a}\right)$ (meV) | $\Delta {\mathit{E}}_{5}^{6}\left(\mathit{a}\right)$ (meV) | $\Delta {\mathit{E}}_{6}^{7}\left(\mathit{a}\right)$ (meV) | $\Delta {\mathit{E}}_{7}^{8}\left(\mathit{a}\right)$ (meV) |
---|---|---|---|---|---|---|---|---|

5.2 | 501 | |||||||

5.3 | 426 | |||||||

5.6 | 401 | |||||||

5.8 | 351 | |||||||

6.0 | 325 | |||||||

6.5 | 250 | |||||||

6.9 | 200 | 826 | ||||||

7.2 | 180 | 801 | ||||||

7.5 | 150 | 751 | ||||||

8.0 | 75 | 676 | ||||||

8.6 | 24 | 570 | 1150 | |||||

9.0 | 0 | 451 | 926 | |||||

9.5 | 0 | 325 | 851 | |||||

10 | 0 | 225 | 751 | |||||

10.3 | 0 | 201 | 701 | 1172 | ||||

11 | 0 | 75 | 576 | 976 | ||||

11.9 | 0 | 0 | 350 | 750 | 1192 | |||

12.5 | 0 | 0 | 225 | 651 | 1080 | |||

13 | 0 | 0 | 124 | 502 | 976 | |||

13.6 | 0 | 0 | 52 | 388 | 851 | 1352 | ||

14 | 0 | 0 | 0 | 275 | 626 | 1302 | ||

15.2 | 0 | 0 | 0 | 63 | 376 | 1001 | 1480 | |

16 | 0 | 0 | 0 | 0 | 150 | 501 | 1281 | |

16.9 | 0 | 0 | 0 | 0 | 0 | 224 | 628 | 2000 |

17.5 | 0 | 0 | 0 | 0 | 0 | 63 | 300 | 1552 |

18 | 0 | 0 | 0 | 0 | 0 | 0 | 150 | 1052 |

18.5 | 0 | 0 | 0 | 0 | 0 | 0 | 34 | 401 |

18.8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |

19.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

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**MDPI and ACS Style**

Pokutnyi, S.I.; Kulchin, Y.N.; Dzyuba, V.P.; Hayrapetyan, D.B.
Exciton Spectroscopy of Spatially Separated Electrons and Holes in the Dielectric Quantum Dots. *Crystals* **2018**, *8*, 148.
https://doi.org/10.3390/cryst8040148

**AMA Style**

Pokutnyi SI, Kulchin YN, Dzyuba VP, Hayrapetyan DB.
Exciton Spectroscopy of Spatially Separated Electrons and Holes in the Dielectric Quantum Dots. *Crystals*. 2018; 8(4):148.
https://doi.org/10.3390/cryst8040148

**Chicago/Turabian Style**

Pokutnyi, Sergey I., Yuri N. Kulchin, Vladimir P. Dzyuba, and David B. Hayrapetyan.
2018. "Exciton Spectroscopy of Spatially Separated Electrons and Holes in the Dielectric Quantum Dots" *Crystals* 8, no. 4: 148.
https://doi.org/10.3390/cryst8040148