Shape Ellipticity Dependence of Exciton Fine Levels and Optical Nonlinearities in CdSe and CdTe Nanocrystal Quantum Dots

Abstract

Shape ellipticity dependence of the exciton fine energy levels in CdTe and CdSe nanocrystal quantum dots were compared theoretically by considering the crystal structure and the Coulomb interaction of an electron and a hole. While quantum dot ellipticity changes from an oblate to prolate quantum dot via spherical shape, both the fine energy levels and the dipole moment in wurtzite structure of a CdSe quantum dot change linearly for ellipticity. In contrast, CdTe quantum dots were found to show a level crossing between the bright and dark exciton states with a significant change of the dipole moment due to the cubic structure. Shape ellipticity dependence of the optical nonlinearities in CdTe and CdSe nanocrystal quantum dots was also calculated by using semiconductor Bloch equations. For a spherical shape quantum dot, only $1^L$ dominates the optical nonlinearities in a CdSe quantum dot, but both $1^U$ and $0^U$ contribute in a CdTe quantum dot. As excitation pulse area becomes strong (${\sim}{\pi}$), the optical nonlinearities of both CdSe and CdTe quantum dots are mainly governed by absorption saturation. However, in the case of a prolate CdTe quantum dot, the real part of the nonlinear refractive index becomes relatively significant.

INTRODUCTION

Currently, quantum dots are of great interest for both fundamental research and industrial applications due to their strong three-dimensional quantum confinement effects [1]. For instance, a two-level system of the confinement states can be used as a qubit, and tunable fluorescence of nanocrystal quantum dots (NQDs) are used for bio-probe and display devices by controlling the size [2, 3]. Although the confinement energy levels of NQDs can be observed by absorption and photoluminescence (PL) spectrum [4-6], the presence of fine (~10 meV) energy levels were proposed theoretically. It was known that the fine energy levels of NQDs depend on not only the radius but also the shape ellipticity and the crystal structures of NQDs [7, 8], whereby the shape dependence of oscillator strength can also be obtained [9-11]. In order to observe the fine energy levels, high spectral-resolution photo-excitation luminescence (PLE) can be used for the ensemble NQDs. However, regarding randomly oriented NQDs dispersed on a substrate, micro-PL of a single NQD is necessary for accurate comparison with theory. Although much research has been reported on single NQD micro-PL, the fine energy levels have rarely been considered.

Additionally, optical nonlinearity of quantum dots can be applicable to various photonics technology such as nonlinear optical imaging and optical bio-sensors. While multi-photon processes of NQDs are often investigated, exciton optical nonlinearities were rarely reported. In this case, the shape ellipticity and fine energy levels should be considered. However, the shape dependence of optical nonlinearities of NQDs has never yet been reported. In this work, we have calculated theoretically the fine energy levels and dipole moments of CdSe and CdTe by considering the shape ellipticity and the crystal structure, and the nonlinear refractive index of each fine level was obtained by using the semiconductor Bloch equations.

THEORY

Suppose a spherical quantum dot (QD) with a radius a is surrounded by an infinite potential barrier, the ground state energy of an electron and a hole in the conduction and the fourfold degenerate valence band are given by \(E_{a}=\frac{h^{2} \pi^{2}}{2 m_{c}^{*} a^{2}}\)and  \(E_{\mathrm{h}}=\frac{h^{2} k^{2}(\beta)}{2 m_{\mathrm{hh}} a^{2}}\)  respectively, where \(\beta=m_{i j}^{*} / m_{h h}^{*}\) is the ratio of the light to heavy hole effective mass, and  \(k(\beta)\) is the first root of the equation\(j_{0}(k) j_{2}(\sqrt{\beta} k)+j_{2}(k) j_{0}(\sqrt{\beta} k)=0\)  . Thus, the exciton state of a spherical quantum dot is eightfold degenerate by two electron spin states (\(1 S_{3 / 2}^{\alpha}\)) and four hole spin states \(\left(1 S_{3 / 2}^{b}\right)\). However, the eightfold degeneracy of the spherical band-edge exciton is split into spin-degenerate five levels \(\left(0^{\mathrm{U}}, 0^{\mathrm{L}}, \pm 1^{\mathrm{U}}, \pm 1^{\mathrm{L}} \text { and }\pm 2\right)\) as a consequence of crystal structure, shape ellipticity, and electron-hole exchange interaction.

In the case of CdSe, the spin-orbit coupling should be considered due to the wurtzite structure, where the heavy hole (\(j=\frac{3}{2}, m_{j}=\pm \frac{3}{2}\)) and light-hole (\(j=\frac{3}{2}, m,=\pm \frac{1}{2}\)) bands become separate with an energy of crystal field (\(\Delta_{\sigma}\) ). However, the diamond-like structure gives \(\Delta_{\sigma}=0\) in CdTe [12, 13]. Additionally, the non-sphericity energy (\(\Delta_{s h}\) ) is associated with the degree of ellipticity (\(\mu\)) and  \(\Delta_{\sigma}\)as

\(\Delta_{s h}=\Delta_{c r} v(\beta)+2 \mu E_{h}(a, \beta) u(\beta)\) ,(1)

\(v(\beta)=\int d r r^{2}\left[R_{0}^{2}(a, r)-\frac{3}{5} R_{2}^{2}\left(a_{4} r\right)\right]\) ,(2)

\(\int d r r^{2}\left[R_{0}^{2}(a, r)+R_{2}^{2}(a, r)\right]=1\), (3)

where \(R_{0}(a, r)\) and \(R_{2}(a, r)\) are the normalized radial functions, and the dimensionless functions  \(u(\beta)\) and  \(v(\beta)\) are associated with the shape ellipticity and crystal structure, respectively [15]. When the NQD shape becomes nonspherical, the radii of circular (b) and non-circular (c) cross-section can be defined as shown in Fig. 1. Compared to the radius of spherical NQD (a), prolate and oblate NQDs show  and      with      respectively. Therefore, while the NQD shape changes from oblate to prolate, the degree of ellipticity (     ) varies from negative to positive.

The electron-hole exchange energy  gives rise to an additional splitting ∆ as

\(\Delta_{c x}=\left(\frac{a_{e x}}{a}\right)^{3} E_{e x} w(\beta)\) ,(4)

\(w(\beta)=\frac{a^{2}}{6} \int_{0}^{a} d r \sin ^{2}\left(\frac{\pi r}{a}\right)\left[R_{0}^{2}(a, r)+\frac{1}{5} R_{2}^{2}(a, r)\right]\) , (5)

where the dimensionless function  describes the electronhole exchange interaction, and  is the exciton Bohr radius. Using the parameters of CdTe and CdSe summarized in Table 1 with a 3 nm-radius, the spin-degenerate five energy levels (0U , 0L , ±1U, ±1L and ±2) of an exciton is given by [14]

(6)

(7)

(8)

(9)

(10)

In order to obtain the refractive index spectrum of the exciton fine energy levels, the transition oscillator strengths of five levels were calculated as

(11)

(12)

(13)

where the sum of the five oscillator strength was normalized (±  ±      ). It is noticeable that   and ± are zero because they are optically inactive dark states [15, 16].

Given the oscillator strength (α) of a k-fine energy level (), the corresponding polarization  and occupancy  can be defined and they depend on time (t).

Additionally, the dipole moment of k-state can also be obtained as      ∗  , where  ∗ is the total exciton effective mass, and the nonlinear refractive index can be obtained by solving the semiconductor Bloch equations (SBEs). With the electric field of a pump   ⋅   and test pulse  ⋅  , the SBEs in an n-order spatial Fourier series expansion can be given as [19-21]

(14)

(15)

where the Rabi frequency of the pump and test pulse can also be defined as

(16)

, and the γ is the dephasing factor and  is the fine exciton levels determined by the confinement energy and the exchange energy of electron and hole. Those equations enable us to obtain both the real ( ) and imaginary ( ) nonlinear refractive index spectrum for an injected pump pulse area (  ∞ ∞  ) [19, 22] using Fourier transformation of electric susceptibility (      ). As the calculation considers complex number, the Kramers-Kronig relation is not necessary. It is also noticeable that high order nonlinear terms of the SBEs are sufficiently small. Thus, we calculated numerically the polarization and occupancy up to the third order (n = 3) For example, the polarization of the fifth-order effect is smaller by three-orders-of magnitude (~103 ) compared to that of the third-order effect. Therefore, the occupancydependent refractive index can also be described in terms of pulse area.

III. RESULTS AND DISCUSSION

Regarding ellipticity dependence of the exciton fine energy levels in CdTe (Fig. 1(a)) and CdSe (Fig. 1(c)) QDs, a significant difference was obtained. When a spherical QD of CdTe is formed, the degenerate heavy- and light-hole states become separated with ∆  . Because the cubic structure gives rise to ∆   in CdTe, the fine exciton energy levels of a spherical CdTe QD are determined only by ∆.

As shown in Fig. 1(a), the energy level difference between ±1U and ±1L (~2.5 meV) for a spherical CdTe QD (  ) mainly results from the exchange interaction (∆), which becomes increased for decreasing the confinement size (∆ ∼  ). Additionally, as the shape ellipticity becomes significant, ∆ becomes involved, i.e. ∆ becomes positive or negative for oblate and prolate structures, respectively. For a spherical CdTe QD (  ), the dark exciton states (0L and ±2) are degenerate with the same exchange interaction energy of ∆ ~1.0 meV. However, those become split as ∆ becomes significant for oblate and prolate CdTe QDs (≠). As the shape ellipticity become significant, the wavefunction difference of the heavy- and light-holes states gives rise to an energy difference, which can be obtained by perturbation method. It is noticeable that the energy levels of 0U, 0L , and ±2 states show a monotonic dependence for  due to the linear dependence of ∆ and ∆ (Eqs. (6), (9), and (10)), but those of ±1U and ±1L states for  show a novel curve due to the nonlinear dependence of ∆ and ∆

Shape ellipticity () dependence of the fine levels is very sensitive to crystal structure, and is dominated by the non-sphericity energy (∆). As described in Eq. (1), ∆ is associated with the crystal field energy (∆ ). Regarding ∆ = 25 meV in CdSe and ∆   in CdTe, the fine level change for  is relatively small in CdTe compared to that in CdSe. As shown in Fig. 1(c), the variation of the fine levels in CdTe is only a few meV when the ellipticity of the QD shape changes from oblate (-0.3) to prolate (0.3). On the other hand, both bright and dark levels of CdSe show a significant shift of ~20 meV for  due to the large crystal field energy (∆ = 25 meV). Therefore, the fine levels of CdSe QDs are very sensitive to shape ellipticity compared to those of CdTe QDs. Additionally, the electron-hole exchange energy of CdTe is comparable to the non-sphericity energy. Therefore, the fine level change for  becomes nonlinear. On the other hand, the fine level change of CdSe shows a linear dependence for  due to the relatively small electron-hole exchange energy.

In Figs. 1(a) and 1(c), both 0U and 0L states have the same slope ∆ for ellipticity (), and the slope of ±2 states for  also have the same magnitude but the sign is opposite (∆). Figs. 1(b) and 1(d) show the exciton dipole moment of the fine levels in CdTe and CdSe, respectively. The dipole moments of the fine levels in CdSe barely depend on shape ellipticity. On the other hand, interesting features are seen in the dipole moment of CdTe. When CdTe forms a perfect sphere (  ), ±1L is not optically accessible with zero dipole moment. Therefore, only three bright states of 0U and ±1U can be observed. Interestingly, the dipole moment of 0U of has no dependence on shape ellipticity both in CdTe and CdSe.

The dipole moment of ±1U state in CdTe (Fig. 1(b)) decreases as positive  increases up to 0.3, and a small change of the dipole moment is seen in the ±1U and ±1L states of CdSe near  ~ 0.3 (Fig. 1(d)). However, the dipole moment of the ±1L state in CdTe (Fig. 1(b)) looks sensitive to the shape change.

In a spherical CdTe QD, the ±1L state has a null dipole moment. Therefore, only the degenerate 0U and ±1U are optically active. When those three bright states are resonantly excited by a 100 fs laser, a change of the refractive index is induced. As shown in Figs. 2(a) and 2(b), the real part () of the nonlinear refractive index spectrum is shown for increasing pulse area (), where the spectral energy is given in terms of the energy difference between the lowest fine exciton levels and bulk band gap ( ). For the energy below the resonance level of 0U and ±1U states, the refractive index decreases for increasing . However, the energy higher than the resonance level shows the opposite feature. As a result, the real part of the refractive index at the resonance level decreases while  increases up to  × , but increases afterward (Fig. 2(c)). When this result is considered in a point of the Kerr effect, where the refractive index change shows a linear dependence for excitation intensity (∆∼∼  ), this approximation seems valid up to ∼ × . Likewise, the imaginary part () of the nonlinear refractive index spectrum is shown for increasing pulse area () in Figs. 2(d) and 2(e). As shown in Fig. 2(f), an absorption saturation can be seen at the resonance level of 0U and ±1U states. When ensemble CdSe quantum dots are dispersed with a density of ~1012 cm2 , roughly ~80 nJ energy is necessary to give a state bleaching under resonant pulse excitation. Therefore, the full saturation can be observed near 8 µJ of pulse excitation when single quantum dot is excited by a picosecond pulse.

While a single degenerate bright level consists of three bright states of 0U and ±1U in CdTe QDs, three separate bright levels are given in CdSe QDs, where 0U, ±1U, and ±1L are located in the highest, intermediate, and lowest levels, respectively. Because those states have similar dipole moments, nonlinear modulation of a laser pulse depends on spectral tuning. We calculated the central energy of our laser near the resonance energy of ±1L which has a spectral width of ~20 meV. Therefore, the nonlinearity is dominated by ±1L states, but the contribution of ±1U and 0U states are not negligible as shown in Fig. 3(a). As excitation dependence of the real refractive index spectrum is shown as a contour map in Fig. 3(b), a refractive index change at a certain spectral energy can also be plotted as shown in Fig. 3(c). It is noticeable that the refractive index change of ±1L states becomes maximized when ∼, but other high levels need stronger excitation to obtain a similar change.

Likewise, excitation dependence of the imaginary part were also shown in Figs. 3(d), 3(e), and 3(f). When ≥, extinction coefficient () near ±1L states becomes negative, i.e. gain. However, other states of ±1U and 0U are not saturated with ∼. When excitation is weak enough (≲ × ), those optical nonlinearities of ∆ and ∆ can also be approximated into the Kerr effect (∼  ). The slope of refractive index change for excitation can be decreased while the pulse duration becomes elongated from femtosecond to picosecond. For intuitive understanding, it is noticeable that this model is based on semi-classical light-matter interaction. While the excitation intensity is similar to classical nonlinear optics, the state filling plays an important role. Therefore, the state occupancy (f) gives rise to a unique feature compared to the classical nonlinear optics.

As shown in Fig. 4(a), the occupancy of the bright states in a spherical CdTe increases as pulse area () is increased up to . However, for   , the occupancy decreases gradually. This result implies a transient stimulated emission, where the negative extinction coefficient (  ) is seen in Fig. 2(d). It is noticeable that the occupancy of 0U becomes saturated when ∼ × , and this can be explained by the relatively small oscillator strength. The occupancy of ±1L remains zero as it is an optically forbidden dark state.