Theoretical study of polarization dependence of carrier-induced refractive index change of quantum dot

Qingyuan Miao; Ziyi Yang; Jianji Dong; Ping-An He; Dexiu Huang

doi:10.1364/OE.26.002252

1. Introduction

Since quantum dot has a confined effect on the carriers in three dimensions, it has a completely discrete energy level structure. The unique energy level structure makes quantum dot have excellent optical properties. Optical devices such as all-optical logic gates [1,2], wavelength converters [3,4], all-optical switches [5-7] are the key devices to develop optical packet-switched networks characterized by high speed and flexibility. Carrier-induced refractive index change is the operating mechanism of these devices. In order to acquire good operating performance in actual applications, these devices must have low polarization dependence of refractive index change under carrier injection. Carrier-induced refractive index changes of bulk material [8–10] and quantum well material [11–13] have been analyzed, but there is a lack of the study on the refractive index change and its polarization dependence of quantum dot material. The polarization dependence of refractive index change is not only affected by interband transition from conduction band to valence band, but also by the intraband free-carrier absorption [14], which is different from the emission affected only by interband transition [15,16].

In this paper, a theoretical model to study the polarization dependence of carrier-induced refractive index change of quantum dot is constructed firstly. Then the influences of dot material component, barrier material component, aspect ratio and carrier density on the refractive index changes of TE mode and TM mode of InGaAs/InGaAsP columnar quantum dot are analyzed, and the physical mechanisms are dissected. Based on the above analysis, a multiparameter adjustment method is proposed and used to design quantum dots with low polarization dependence of refractive index change in C-band (1530 nm - 1565 nm).

2. Model

Traditional self-assembled quantum dot tends to be flat shape, and there is a strong biaxial compressive strain inside it, showing strong effect on transverse-electric (TE) mode and slight effect on transverse-magnetic (TM) mode, resulting in an obvious polarization dependence, which markedly reduces devices performance. Through cycled submonolayer deposition method, the quantum dots are vertically stacked with no interdot spacing, forming a columnar quantum dot structure whose aspect ratio can be varied in a wide range. The confinement potential and the strain distribution of columnar quantum dot vary with the change of aspect ratio, thus affecting the energy level structure and optical properties [17], which brings the possibility to achieve low polarization dependence of refractive index change. The schematic of columnar quantum dot used in the paper is shown in Fig. 1, in which the height is l and the bottom diameter is d. The dot material is InGaAs, the barrier material is InGaAsP.

Fig. 1 Schematic of columnar quantum dot.

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Coupling effect among the conduction and valence band has a strong impact on the band structure of quantum dot. Therefore, the coupling of the conduction and valence band needs to be taken into account to ensure the accuracy of the band structure calculation. The eight-band k·p model which contains strain,and the plane-wave expansion method are used to calculate the band structure [18]. The wave functions of conduction band and valence band are expressed as

ψ_{c} = \sum_{i = 1}^{8} F_{c}^{i} (r) u_{i} (r)

ψ_{v} = \sum_{i = 1}^{8} F_{v}^{i} (r) u_{i} (r)

where u_i (r) is basic wave function, and F_c, F_v are envelop functions corresponding to conduction and valence band respectively.

The calculations of momentum matrix elements are based on the band structure. The transition rate is derived from the Fermi's Golden rule. The optical absorption coefficient α which takes into account the scattering broadening mechanisms is given by [19]

α (ℏ ω) = C_{0} \frac{1}{V} \sum_{n_{c}} \sum_{n_{v}} {| \hat{e} \cdot {\hat{p}}_{c v} |}^{2} \frac{γ / π}{{(E_{c} - E_{v} - ℏ ω)}^{2} + γ^{2}} (f_{v} - f_{c})

in which ћ is reduced Planck’s constant; ω is the angular frequency of light; V is the volume of considered structure;

\hat{e}

is unit vector in the polarization direction of incident light; γ is Lorentzian half-linewidth; E_c (E_v) is energy level in conduction (valence) band; f_c (f_v) is the Fermi Dirac distribution function of conduction (valence) band; C₀ = πe²/(n_rcε₀m₀²ω), where e, n_r, c, ε₀, m₀ are electron charge, the refractive index of material, the speed of light in free space, vacuum permittivity and free electron mass respectively;

{\hat{p}}_{c v}

is optical matrix element. In eight-band k·p model,

{\hat{p}}_{c v}

is given by [20]

{\hat{p}}_{c v} = 〈 ψ_{c} | \hat{p} | ψ_{v} 〉 = \sum_{i = 1}^{8} 〈 F_{c}^{i} | \hat{p} | F_{v}^{i} 〉 + \sum_{i, j = 1}^{8} 〈 F_{c}^{i} | F_{v}^{j} 〉 〈 u_{i} | \hat{p} | u_{j} 〉

The refractive index change due to interband transitions from conduction band to valence band is given by the Kramers-Kronig transformation [8]

△ n_{BF} = \frac{2 c ℏ}{e^{2}} P \int_{0}^{\infty} \frac{α (E^{'}, N) - α_{0} (E^{'}, 0)}{{E^{'}}^{2} - E^{2}} d E^{'}

where P indicates the principal value of the integral; N is the injected carrier density; α is the absorption coefficient with carrier injection; α₀ is the absorption coefficient in the absence of carrier injection.

The refractive index change due to intraband free-carrier absorption is given by [14,21]

△ n_{F C A} = - \frac{e^{2} λ^{2} N}{8 π^{2} c^{2} ε_{0} n_{r} m_{r}}

where λ is the photon wavelength, m_r is the reduced carrier mass.

△ n_{F C A}

is proportional to the carrier density and the square of the wavelength.

Total refractive index change is given by

△ n = △ n_{BF} + △ n_{FCA}

The relevant band structure parameters used in the calculations are shown in Table 1 [22].

Table 1. Band structure parameters.

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3. Results and Discussion

Figure 2 shows the refractive index change spectra of TE mode and TM mode corresponding to different dot material components of columnar quantum dot. The values of Ga component x in dot material In_1-xGa_xAs are taken in turn as 0, 0.1 and 0.2 following the arrow direction, the corresponding peak wavelength intervals of TE mode and TM mode are S₁, S₂ and S₃ respectively. The barrier material In_0.91Ga_0.0.9As_0.2P_0.8 (λ_g = 1.03 μm) is lattice-matched to the InP substrate. The dots are assumed uniformly distributed; dots density is 3.7 × 10²² m⁻³. The height l and the bottom diameter d of the columnar quantum dot are 7 nm and 10 nm respectively, and the aspect ratio η = l/d = 0.7. Carrier density is 1 × 10²⁴ m⁻³, the temperature is 300 K.

Fig. 2 The refractive index change spectra of TE mode and TM mode corresponding to different dot material components.

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It can be seen from Fig. 2 that with the addition of the Ga composition x, both the peak wavelengths of the refractive index changes of TE mode and TM mode move to the shorter wavelength direction (blue-shift), and the interval between the peak wavelengths of TE mode and TM mode decreases, namely, S₁>S₂>S₃. These results can be explained from Table 2.

Table 2. The parameters obtained under different dot material components.

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Table 2 lists the parameters obtained under different dot material components x, including bandgap E_{g_dot}, lattice constant a, initial lattice mismatch ratio ε₀, and biaxial strain ε_bia at the center of quantum dot. The initial lattice mismatch ratio ε₀ = (a_InP-a_dot)/a_dot, where a_InP and a_dot are the lattice constant of substrate and dot material respectively. As can be seen from Table 2, with the addition of Ga composition x, bandgap increases, which leads to the increasing of electron transition energy, therefore making the peak wavelength of refractive index change moves to the shorter wavelength direction. On the other hand, with the addition of x, lattice constant decreases. This leads to the reduction of the initial mismatch ratio, further causes the reduction of biaxial strain which is still positive, thus the energy difference between the light-hole and heavy-hole states of valence band is reduced. The transition from conduction band state to heavy-hole state only contributes to the refractive index change of TE mode, while the transition from conduction band state to light-hole state mainly contributes to the refractive index change of TM mode and only a little to TE mode. So that the reduction of the energy difference between the light-hole and heavy-hole states leads to the decreasing of the interval between the peak wavelengths of TE mode and TM mode.

Therefore, changing the dot material component can adjust not only the peak wavelengths of refractive index change, but also the interval between the peak wavelengths of TE mode and TM mode, which is useful for the design of quantum dot with low polarization dependence of carrier-induced refractive index change.

The refractive index change spectra of TE mode and TM mode corresponding to different barrier material components are shown in Fig. 3. The values of Ga component x in barrier material In_1-xGa_xAs_yP_1-y are taken in turn as 0.1, 0.15 and 0.2 following the arrow direction, and the bandgaps of the corresponding barrier material are 1.19 eV, 1.11 eV and 1.04 eV respectively. The lattice constant of the barrier is matched to the substrate when y = 0.4184x/(0.1894 + 0.013x). The dot material is In_0.9Ga_0.1As, and the other parameters are the same as those of Fig. 2. As can be seen from Fig. 3, with the addition of the Ga composition x, both the peak wavelengths of the refractive index changes of TE mode and TM mode move to the longer wavelength direction (red-shift). This is because the bandgap of barrier material reduces with the addition of Ga composition x. The bandgap reduction of barrier materia results in decreased confinement to the carrier of dot area, thus the electron transition energy reduces, so that the peak wavelength of refractive index change moves to the longer wavelength direction. Therefore, changing barrier material component is also a means to adjust the peak wavelengths of the refractive index changes of TE mode and TM mode.

Fig. 3 The refractive index change spectra of TE mode and TM mode corresponding to different barrier material components.

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The refractive index change spectra of TE mode and TM mode corresponding to different aspect ratios of quantum dot are shown in Fig. 4. Since the single-layer aspect ratio η of quantum dot is about 0.1, while the bottom diameter d maintains 10 nm, the height l is taken sequentially as 1 nm, 5 nm, 9 nm and 13 nm, and the corresponding aspect ratios η are 0.1, 0.5, 0.9 and 1.3 respectively. The dot material is In_0.9Ga_0.1As. The other parameters are the same as those of Fig. 2. As can be seen from Fig. 4, when the aspect ratio increases, the peak wavelengths of refractive index change move by a large margin to the longer wavelength direction (red-shift). This is arising from the rapid weakening of the confinement effect on the carrier due to the increasing height of quantum dot, which results in a decrease of the transition energy, thus the peak wavelength increases. It is also observed that as the aspect ratio increases, the relative magnitude of the refractive index changes of TM mode to TE mode is increasing, and finally the magnitude of the TM mode exceeds the TE mode in Fig. 4(d). This can be explained from Table 3.

Fig. 4 The refractive index change spectra of TE mode and TM mode corresponding to different aspect ratio η. (a) η = 0.1, (b) η = 0.5, (c) η = 0.9, (d) η = 1.3.

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Table 3. Biaxial strain at the center of quantum dot and the momentum matrix elements in the transition from conduction band ground state to valence band ground state.

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Table 3 lists the parameters obtained under different aspect ratios, including the biaxial strain ε_bia at the center of quantum dot, and the momentum matrix element ${| \hat{e} \cdot {\hat{p}}_{c v} |}^{2}$ in the transition from conduction band ground state to valence band ground state. As can be seen from Table 3, with the increase of aspect ratio, the biaxial strain decreases gradually from a positive value, and becomes negative when aspect ratio equals 1.3, meanwhile the momentum matrix element of the TM mode increases continuously, and is ultimately greater than the TE mode. These are caused by the change of the internal strain with respect to aspect ratio of quantum dot. In the flat form, there is a strong compressive strain in quantum dot. At this time, the biaxial strain is positive and the valence band ground state is the heavy-hole state. The transition mainly occurs between the conduction band state and the heavy-hole state, which contributes to the TE mode momentum matrix elements. With the increase of the aspect ratio, the lattice mismatch on the side of quantum dot alleviates the bottom compressive strain, which makes the biaxial strain decrease. The light-hole state rises gradually with the decrease of biaxial strain. When the biaxial strain decreases to a negative value, the light-hole state surpasses the heavy-hole state, and becomes the ground state of the valence-band. At this time, the transition primarily occurs between the conduction band state and the light-hole state, which mainly contributes to the TM mode momentum matrix elements, and only a little to TE mode. Therefore, the refractive index change of TM mode gradually exceeds the TE mode to dominate.

Based on the above analysis, it can be seen that in the design of quantum dot with low polarization dependence of carrier-induced refractive index change, dot material component, barrier material component and aspect ratio are all important adjustment parameters. On the one hand, aspect ratio can be used to adjust the peak wavelengths of the refractive index changes of TE mode and TM mode with a wide wavelength range. By combining with the adjustments by dot material component and barrier material component, the peak wavelengths can be moved to a desired operating wavelength range, and the peak wavelengths of TE mode and TM mode tend to be the same. On the other hand, aspect ratio can be used to adjust the relative magnitude of the refractive index changes of TE mode to TM mode, so that their magnitudes tend to be the same. The above multiparameter adjustment method provides good flexibility for realizing low polarization dependence.

Figure 5 shows the refractive index change spectra of TE mode and TM mode corresponding to different carrier densities. Carrier density is taken in turn as 0.2 × 10²⁴ m⁻³, 0.4 × 10²⁴ m⁻³ and 1.2 × 10²⁴ m⁻³ following the arrow direction. The dot material is In_0.9Ga_0.1As, and the other parameters are the same as those in Fig. 2. As can be seen from Fig. 5, with the increasing of carrier density, the magnitudes of the refractive index changes of TE mode and TM mode increase, and the peak wavelengths of TE mode and TM mode slightly move to the longer wavelengths direction. The reason is that, along with the increase of the carrier density, the quasi-Fermi level of conduction band rises, the quasi-Fermi level of valence band decreases, the number of reversal particles increases, and the transition from the conduction band to the valence band is enhanced. in addition, the free-carrier absorption is proportional to the carrier density and the square of the wavelength, resulting in a slight red-shift of the peak wavelength. Therefore, a desired magnitude of refractive index change can be obtained by providing a certain carrier density.

Fig. 5 The refractive index change spectra of TE mode and TM mode corresponding to different carrier density.

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By adjusting dot material component, barrier material component and aspect ratio comprehensively, two sets of columnar quantum dot materials with low polarization dependence of refractive index change within C-band are obtained. The quantum dot material parameters are shown in Table 4, the aspect ratios are all 1.0 with taking the height and bottom diameter to be both 10 nm. When the carrier density is 1 × 10²⁴ m⁻³ and the temperature is 300 K, the corresponding refractive index change spectra are shown in Fig. 6. As can be seen from Fig. 6, the refractive index change spectra of the two columnar quantum dots are basically coincident within C-band.

Table 4. Two sets of material parameters of quantum dot.

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Fig. 6 Refractive index change spectra. (a) 1st group, (b) 2nd group.

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The parameter ρ is introduced to describe the polarization dependence of refractive index change, which is defined as $ρ = | △ n^{T E} - △ n^{T M} | / | △ n^{T E} + △ n^{T M} | \times 100 %$ . The smaller ρ is, the lower polarization dependence will be. Take Fig. 6(a) as an example, the spectrum of ρ is shown in Fig. 7. It shows that ρ keeps smaller than 1.5% within C-band, which reveals a low polarization dependence of refractive index change.

Fig. 7 The spectrum of ρ within C-band

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4. Conclusion

In this work, we constructed a theoretical model to study the polarization dependence of carrier-induced refractive index change of quantum dot. The influences of dot material component, barrier material component, aspect ratio and carrier density on the refractive index changes of TE mode and TM mode of InGaAs/InGaAsP columnar quantum dot are analyzed and the physical mechanisms are dissected. A multiparameter adjustment method has been proposed and used to design quantum dot with low polarization dependence of refractive index change within C-band. The result shows that, quantum dots with different material parameters are anticipated to have similar characteristics of low polarization dependence, which demonstrates the design flexibility.

Funding

National Natural Science Foundation of China (NSFC) (60877039, 61622502, and 61475052).

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	Lattice constant	Bandgap	Deformation potential /eV			Elastic stiffness constants /10¹¹dyne·cm⁻²			Electron effective mass	Luttinger constant
	a₀ /Å	E_g /eV	a_c	a_v	b	C₁₁	C₁₂	C₄₄	m_eff /m₀	γ₁	γ₂	γ₃
GaAs	5.6533	1.424	−7.17	1.16	−1.7	11.88	5.376	5.94	0.067	6.8	1.9	2.73
InAs	6.0584	0.354	−5.08	1.00	−1.8	8.329	4.526	3.96	0.023	20.4	8.3	9.1
InP	5.8688	1.344	−5.04	1.27	−1.7	10.11	5.61	4.56	0.077	4.95	1.65	2.35
GaP	5.4505	2.780	−7.14	1.70	−1.8	14.05	6.203	7.033	0.25	4.05	0.49	1.25

x	E_{g_dot}/eV	a/Å	ε₀	ε_bia
0	0.324	6.058	−3.12%	0.82%
0.1	0.398	6.018	−2.47%	0.65%
0.2	0.480	5.977	−1.81%	0.48%

η	ε_bia	${\| \hat{e} \cdot {\hat{p}}_{c v} \|}^{2}$ /kg·eV
η	ε_bia	TE	TM
0.1	3.69%	4.34 × 10⁻⁵⁰	3.96 × 10⁻⁷⁹
0.5	1.46%	4.60 × 10⁻⁴⁹	2.51 × 10⁻⁷⁸
0.9	0.03%	4.78 × 10⁻⁴⁹	2.64 × 10⁻⁷⁵
1.3	−0.76%	1.06 × 10⁻⁴⁹	5.80 × 10⁻⁴⁹

	Dot material	Barrier material
1st group	In_0.95Ga_0.05As (λ_g = 3.45 μm)	In_0.82Ga_0.18As_0.40P_0.60 (λ_g = 1.17 μm)
2nd group	In_0.85Ga_0.15As (λ_g = 2.84 μm)	In_0.77Ga_0.23As_0.50P_0.50 (λ_g = 1.24 μm)

	Lattice constant	Bandgap	Deformation potential /eV			Elastic stiffness constants /10¹¹dyne·cm⁻²			Electron effective mass	Luttinger constant
	a₀ /Å	E_g /eV	a_c	a_v	b	C₁₁	C₁₂	C₄₄	m_eff /m₀	γ₁	γ₂	γ₃
GaAs	5.6533	1.424	−7.17	1.16	−1.7	11.88	5.376	5.94	0.067	6.8	1.9	2.73
InAs	6.0584	0.354	−5.08	1.00	−1.8	8.329	4.526	3.96	0.023	20.4	8.3	9.1
InP	5.8688	1.344	−5.04	1.27	−1.7	10.11	5.61	4.56	0.077	4.95	1.65	2.35
GaP	5.4505	2.780	−7.14	1.70	−1.8	14.05	6.203	7.033	0.25	4.05	0.49	1.25

Theoretical study of polarization dependence of carrier-induced refractive index change of quantum dot

Abstract

1. Introduction

2. Model

3. Results and Discussion

4. Conclusion

Funding

References and links

Cited By

Figures (7)

Tables (4)

Equations (7)

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