Screening effect on the exciton mediated nonlinear optical susceptibility of semiconductor quantum dots

Jessica E. Q. Bautista; Marcelo L. Lyra; R. P. A. Lima

doi:10.1364/OE.22.028270

1. Introduction

New devices operating in the optical and infrared regions have been recently proposed based on the unique properties of semiconductor quantum wells and quantum dots [1–6]. These structures present enhanced nonlinear optical susceptibilities as compared with those of bulk semiconductors, due to the electron confinement potential, which is particularly strong in quantum dots [7–17]. The optical response has been shown to be enhanced in the presence of a DC field in two dimensional triangular quantum dots [18, 19]. In particular, semiconductor quantum well and quantum dots with a semiparabolic confinement potential present very large nonlinear susceptibilities which results from a multiple resonance due to the uniform spacement of the sub-bands [12–14]. Excitonic transitions are expected to play a significant role in the low-temperature nonlinear optical properties of these semiconductor systems, such as the optical rectification, second and third-order harmonic generation and their corresponding contributions to the refraction index and absorption coefficient [7–15].

In the regime of weak exciton confinement, the electron-hole Coulombian interaction predominates over the confinement potential. Recently, a density matrix perturbation theory was used to compute the nonlinear optical susceptibilities due to third-harmonic generation and the corresponding nonlinear corrections to the refractive index and absorption coefficient resulting from the transitions of an electron-hole pair confined by a semi-parabolic potential and interacting with each other via a Coulomb potential [20]. These quantities were analyzed as a function of ratio between the confinement length L and the exciton Bohr radius a₀. It was shown that, associated with the degradation of the energy level separation caused by the Coulomb potential, there emerges multiple resonance peaks in the third-harmonic generation spectrum. Further, the third-order susceptibility was shown to decay as 1/β⁸ in the weak confinement regime of β = L/a₀ ≫ 1 due to the prevalence of the hydrogenoid character of the exciton eigenstates. However, the screening of the Coulombian electron-hole interaction shall be relevant in the weak confinement regime. It introduces a third length scale, namely the screening length ℓ, whose impact on the nonlinear optical response has to be properly evaluated.

Here, we will provide a detailed study of the exciton contribution to the nonlinear susceptibility of a one-dimensional quantum dot by incorporating the screening of the electron-hole coupling. We will compute the eigenenergies of the lowest levels as a function of the ratio between the screening length and the characteristic Bohr radius. We will show that the screening of the Coulomb potential acts by restoring the uniformity of the energy levels separation. In addition, a perturbative density matrix formulation will be used to estimate the screening effect on the third-order susceptibility and its corresponding contribution to the refraction index and absorption coefficient. In particular, we will show that the nonlinear optical susceptibility increases while recovering a single peaked spectrum when the screening length decreases.

2. Model and formalism

In nano-sized quantum wires, the electron motion is mainly parallel to the wire length. The presence of walls can confine the electron in a finite segment, characterizing a 1D quantum dot. Within the effective-mass approximation, the screening of the electron-hole interaction can be incorporated in the excitonic Hamiltonian of a one-dimensional quantum dot as [12–14]

ℋ = \frac{p_{e}^{2}}{2 m_{e}^{*}} + \frac{p_{h}^{2}}{2 m_{h}^{*}} + V_{0}^{e} (x_{e}) + V_{0}^{h} (x_{h}) - \frac{q^{2} e^{- | x_{e} - x_{h} | / ℓ}}{4 π ε | x_{e} - x_{h} |},

where the subindices e and h refer to the electron and to the hole, respectively. The above model captures the essential exciton contribution to the nonlinear optical properties of quantum dots and can be extended to 2D and 3D systems.

m_{e}^{*}

and

m_{h}^{*}

stand for the effective masses and ε is the permittivity of the semiconductor material. The exponential decay of the electron-hole interaction accounts for the screening of the Coulomb interaction with typical length ℓ. Such screening is caused by the presence of quasifree carriers in semiconductors that shield the free charges thus resulting in an effective short-range electrostatic coupling. The potential V₀ will be considered to be a semi-parabolic one with the same oscillation frequency for both electron and hole. Therefore, the electron-hole pair is restricted to the x > 0 region and are harmonically trapped. The eigenstates have all the same parity for such potential, which opens the possibility of achieving large dipole matrix elements. The above Hamiltonian can be rewritten using the center of mass

X = (m_{e}^{*} x_{e} + m_{h}^{*} x_{h}) / (m_{e}^{*} + m_{h}^{*})

and relative x = x_e − x_h coordinates. While the eigenfunctions and eigenstates of the center of mass Hamiltonian ℋ_X are analytically known, those associated with the relative electron-hole coordinate Hamiltonian ℋ_x have to the determined numerically. In terms of the typical length scale of the oscillations

L = \sqrt{ℏ / m ω_{0}}

(used as the characteristic confinement length), the adimensional screening parameter λ = a₀/ℓ, and the effective Bohr radius of the electron-hole pair a₀ = 4πεħ²/(μq²), the time-independent Schroedinger equation obeyed by the stationary states of ℋ_x can be written as

[- \frac{\partial^{2}}{\partial u^{2}} + u^{2} - \frac{2 β e^{- λ β | u |}}{| u |}] Φ (u) = \frac{2 E}{ℏ ω_{0}} Φ (u)

where u = x/L, and β = L/a₀ is the confinement parameter. The above equation can be solved numerically by employing a standard discretization procedure followed by a finite-difference approach which gives very accurate results for the lowest energy states [20–22].

In the following, we will investigate how the screening of the electron-hole coupling affects the contribution of the excitonic transitions to the nonlinear response to an optical radiation field of frequency ω. We will focus on the third-harmonic generation process on which the system absorbs three photons from the radiation field and emits a single photon of frequency 3ω. Within a density matrix perturbation formalism and considering the first four energy levels, the third-harmonic generation nonlinear susceptibility is given by χ⁽³⁾(3ω) = Aχ̃⁽³⁾(3ω), where $A = \frac{q^{4} {NL}^{4}}{ε_{0} {(ℏ ω_{0})}^{3}}$ and χ̃⁽³⁾(3ω) is a dimensionless susceptibility given by [23, 24]

\begin{array}{l} {\tilde{χ}}^{(3)} (3 ω) & = & M_{01} M_{12} M_{23} M_{30} [f_{1}^{-} (ω) f_{2}^{-} (ω) f_{3}^{-} (ω) + f_{1}^{-} (ω) f_{2}^{-} (ω) g_{3} (ω) \\ + & f_{1}^{-} (ω) f_{2}^{+} (ω) g_{3} (ω) + g_{1} (ω) f_{2}^{+} (ω) g_{3} (ω)] \end{array}

with M_nm = 〈Φ_m|u|Φ_n〉,

f_{n}^{\pm} (ω) = ω_{0} / (ω_{n 0} \pm n ω - i Γ_{n 0})

, and g_n(ω) = ω₀/(ω_n0 + 3ω/n − iΓ_n0). Here

| Φ_{n} 〉 = \sum_{j} Φ_{j}^{n} | j 〉

is the eigenvector corresponding to the n-th eigenstate. Γ_n0 is the line width associated with the transition between the n-th and the ground states and ω_n0 = E_n0/ħ = (E_n − E₀)/ħ. The third-harmonic generation contribution to the nonlinear refractive index and absorption coefficient can be readily obtained from the nonlinear susceptibility as [23, 24]

n_{2} (ω) = \frac{3 ℛ e χ^{(3)} (3 ω)}{4 ε_{0} n_{0}^{2} c}, α_{2} (ω) = \frac{3 ω ℐ m χ^{(3)} (3 ω)}{2 ε_{0} n_{0}^{2} c^{2}}

Here ε₀ is the vacuum permittivity, n₀ is the linear refraction index of the material, and c is the light velocity in vacuum. These nonlinear coefficients can also be written in terms of dimensionless quantities as

n_{2} (ω) = \frac{A}{ε_{0} n_{0}^{2} c} {\tilde{n}}_{2}

and

α_{2} (ω) = \frac{A ω_{0}}{ε_{0} n_{0}^{2} c^{2}} {\tilde{α}}_{2}

.

3. Results

The numerical results were taken through the direct diagonalization of ℋ_x with 0 < u = x/L < 10 discretized on 10³ segments. The singularity of the Coulomb potential at x = 0 is avoided in the discretization with no compromise to the accuracy due to the vanishing of the wave-functions at the origin. The obtained values of the eigen-energies and corresponding eigenfunctions for the first few eigenstates in the extreme confinement limit β = 0 were verified to agree with the exact results for the pure harmonic oscillator. In the following, we will unveil the influence of the screening of the electron-hole coupling on the third-order nonlinear response. We will consider a typical value for the line width of the transitions Γ_n0 = Γ₀ = 0.1ω₀.

In third-harmonic generation processes, resonances appear at radiation frequencies near ω₁ = E₁₀/ħ, ω₂ = E₂₀/(2ħ), and ω₃ = E₃₀/(3ħ). In Fig. 1(a), we plot ω_n as a function of the confinement parameter β for a fixed value of the screening parameter λ = a₀/ℓ = 1.0. In the regime of strong confinement (β ≪ 1), the electron-hole coupling plays no major role and the level spacing is uniform, as expected due to the semiparabolic form of the confinement potential. As the confinement potential weakens (increasing values of β) the uniformity of the level spacements is lost and multiple resonance frequencies appear. Due to the short range of the screened Coulomb potential, these frequencies depict a growth with the confinement parameter β slower than the quadratic one expected for hydrogenoid-like energy levels. In Fig. 1(b) we show the corresponding dependence of the resonance frequencies on the screening parameter λ for a specific value of the confinement length β = L/a₀ = 3.0. In the regime of weak screening on which the typical screening length is much larger than the confinement length, the resonance frequencies retain the character of those resulting for a purely Coulombian electron-hole coupling. For the specific value of the confinement length illustrated, these resonance frequencies are split. The decrease of the screening length (increasing λ) effectively turns off the electron-hole coupling thus recovering the physical scenario of multiple resonances. This feature potentializes the emergence of a very intense third-order nonlinear susceptibility.

Fig. 1: (a) Normalized resonance frequencies ω_n/(nω₀) versus the confinement coefficient β = L/a₀ for the first three excited states and λ = 1.0. The collapse of the curves for β ≪ 1 indicates the uniformity of the level spacing distribution in the strong confinement regime. The splitting of the curves as β increases signals the non-uniformity of the level spacings. (b) ω_n/(nω₀) versus the screening parameter λ = a₀/ℓ and β = 3.0. The uniformity of the energy levels separation is recovered in the strong screening regime λ ≫ 1.

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The non-linear susceptibility χ⁽³⁾ was determined by computing the corresponding dipole matrix elements from the numerically obtained eigenfunctions of the discretized Hamiltonian ℋ_x. In Fig. 2(a) we show the dimensionless third-harmonic generation susceptibility |χ̃⁽³⁾| versus the normalized frequency ω/ω₀ and the screening parameter λ for the representative case of a confinement parameter β = L/a₀ = 3.0. One clearly notice the presence of three peaks in the weak screening regime which result from the lack of uniformity of the energy levels spacement promoted by the Coulomb coupling of the electron-hole pair. As the screening effect increases, these three peaks are displaced to lower frequencies and ultimately coalesce signaling the recovery of the multiple resonance condition. Further, the susceptibility is substantially enhanced at resonance. To better quantify the enhancement of the susceptibility peak at resonance, we plot in Fig. 2(b) the peak values of |χ̃⁽³⁾| as a function of the screening parameter λ. Notice that there is a strong crossover when the peaks start to coalesce. After the triple resonance condition is achieved, the susceptibility at resonance is over two orders of magnitude larger than in the non-screened regime.

Fig. 2: (a) Dimensionless third-order susceptibility (|χ̃⁽³⁾(3ω)|) as a function of the radiation frequency ω/ω₀ and screening parameter λ for a confinement parameter β = L/a₀ = 3.0. In the weak screening regime λ << 1 the susceptibility spectrum depicts three peaks signaling distinct resonance frequencies. In the extreme confinement regime, the uniform level spacing leads to a triple resonance condition with very large nonlinearity. (b) Peak values of the susceptibility as a function of the screening strength parameter λ. The three peaks coalesce as λ increases. The susceptibility is enhanced by over two orders of magnitude as compared to the non-screened regime.

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Finally, the corresponding contributions of the third-harmonic generation process to the nonlinear refractive index and optical absorption coefficient are shown in Figs. 3(a) and 3(b), respectively, as a function of the normalized frequency and screening strength. Both quantities depict the main trends exhibited by the non-linear susceptibility. Their values at resonance are substantially larger in the presence of a strong screening of the electron-hole interaction with the resonance frequency being displaced to a lower value (redshift) in the strong screening regime. We stress that the above reported enhancement of the exciton contribution to the non-linear absorption and refractive index shall be particularly important in 3D quantum dots where the screening of the electron-hole coupling is substantially stronger.

Fig. 3: (a) The nonlinear refractive index ñ₂ and (b) the nonlinear optical absorption coefficient α̃₂ as a function of the radiation frequency ω/ω₀ and screening parameter λ for a confinement parameter β = L/a₀ = 3.0. As the screening becomes stronger, the resonance is enhanced and displaced to lower frequencies.

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4. Summary and conclusions

We showed that the screening of the electron-hole interaction has a strong impact on the exciton contribution to the third-harmonic generation optical properties of a one-dimensional semiconductor quantum dot with a semi-parabolic confining potential. Within a density matrix approach, we computed the third-order nonlinear susceptibility and its corresponding contribution to the refractive index and optical absorption coefficient. A non-screened electron-hole Coulomb interaction breaks down the uniformity of the energy levels spacement thus leading to multiple peaks in the susceptibility spectrum. Taking into account the screening length of the electron-hole coupling, the uniformity of the energy level distribution is recovered when the screening length becomes much smaller than the size of the confinement region. As a consequence, the distinct peaks in the susceptibility spectrum coalesce as the screening becomes stronger and ultimately leads to a multiple resonance condition. The peak value of the susceptibility thus exhibits a crossover as a function of the screening strength being over two orders of magnitude larger in the regime of strong screening as compared to the resonance values in the non-screened regime. Also, the resonance frequency is displaced to a lower value in the presence of screening. The crossover between these regimes takes place when the screening length ℓ ≃ a₀ = 17.26nm for GaAs. Using the Thomas-Fermi theory for a one-dimensional electron gas, we estimate that such crossover will take place at an electron linear density of the order of 3 × 10⁵cm⁻¹. The reported effects shall be more evident in 3D quantum dots on which the screening of the electron-hole pair is stronger. These features are reflected in the spectra of the nonlinear refractive index and optical absorption coefficient. Therefore, all three length scales associated with the effective Bohr radius, the confinement length, and the screening length have to be carefully estimated to identify the relevant mechanisms associated with the excitonic contribution to the nonlinear third-order optical processes in semiconductor quantum dots.

Acknowledgments

We thank CAPES via project PPCP-Mercosul, CNPq, and FINEP (Brazilian Research Agencies) as well as FAPEAL (Alagoas State Research Agency) for partial financial support.

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Screening effect on the exciton mediated nonlinear optical susceptibility of semiconductor quantum dots

Abstract

1. Introduction

2. Model and formalism

3. Results

4. Summary and conclusions

Acknowledgments

References

Cited By

Figures (3)

Equations (4)

Optics Express