Theoretical study of polarization insensitivity of carrier-induced refractive index change of multiple quantum well

Qingyuan Miao; Qunjie Zhou; Jun Cui; Ping-An He; Dexiu Huang

doi:10.1364/OE.22.031893

1. Introduction

Carrier-induced refractive index change is not only the operating mechanism of some optical communication devices, such as optical switches [1,2], wavelength converters [3], but also a important parameter for semiconductor optical amplifiers [4] and semiconductor lasers [5], etc. The operating temperature of these devices is varied with carrier injection and ambient temperature. In order to acquire good operating performances in actual applications, it is necessary to make the refractive index change insensitive to the polarization state when active region temperature changes. Partial characteristics of carrier-induced refractive index change and its polarization dependent have been studied in previous models [4,6–8]. But in these models, temperature is supposed to be constant; the influence of temperature change on the polarization dependent of refractive index change is not considered. Thus the application range of these models is restricted.

In this paper, polarization insensitivity of carrier-induced refractive index change of 1.55 μm tensile-strained multiple quantum well (MQW) is investigated by using a comprehensive model. The model takes into account the temperature variation as well as the nonuniform distribution of injected carrier in MQW, which is much more in accord with the actual situation. The magnitude of refractive index change and its polarization insensitivity over a wide wavelength range at different temperatures are analyzed. The polarization insensitivity of refractive index change versus carrier concentration is also discussed. Furthermore, MQWs with similar polarization insensitivity of refractive index change, but with different material and structure parameters are given.

2. Model

The polarization sensitivity of refractive index change includes two factors: the polarization sensitivity of material refractive index change and the polarization sensitivity of optical confinement factor. For a MQW, the material refractive index change depends on material and structural parameters. The material refractive index change induced by carrier injection originates from three effects [8], the anomalous dispersion effect due to bandfilling through interband transition, the bandgap shrinkage effect due to carrier interaction and the plasma effect due to free-carrier absorption through intraband transition.

The contribution of anomalous dispersion on the refractive index change is obtained from the Kramers–Kronig transform of the change in the optical absorption coefficient

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

where c, ћ, e are the speed of light in free space, Planck’s constant divided by 2π, the electron charge respectively; α represents the absorption coefficient with the injection of electron concentration N and the hole concentration P; α₀ represents the absorption coefficient in the absence of injection.

The calculation of optical absorption coefficient α is based on the energy band structure. For quantum well structure, according to k·p theory and effective mass equations, parabolic-band model for conduction band and Luttinger–Kohn’s model for valence band are used. The optical absorption coefficient which takes into account the scattering broadening mechanisms is given by [9]

α^{p} (ℏ ω) = \frac{π \cdot e^{2}}{n_{e} c ε_{0} m_{0}^{2} ω} \cdot \frac{2}{L_{z}} \sum_{η, σ} \sum_{n, m} \int_{0}^{\infty} \frac{k_{t} d k_{t}}{2 π} M_{n m}^{p, σ} (k_{t}) \frac{γ / π}{{(E_{c v}^{n m σ} (k_{t}) - ℏ ω)}^{2} + γ^{2}} (f_{v}^{σ m} - f_{c}^{n})

in which n_e, ε₀, m₀ are the refractive index, the permittivity of free space and the electron rest mass respectively; L_z is the well width; η is the electron spin state; σ denotes the upper and lower blocks of the Hamiltonian; γ is the Lorentzian half-linewidth; f_cⁿ (f_v^m) is the Fermi occupation probability for electrons in the n_th (m_th) conduction (valence) subband and is related to carrier concentration;

E_{c v}^{n m}

is the transition energy between the n_th conduction and m_th valence subband;

M_{n m}^{p}

is the momentum elements.

When carriers inject inside the active region, their interaction leads to lowering the conduction band edge, while the top of the valence band edge moves up, causes the bandgap shrinkage. For certain carrier concentration, the magnitude of the reduction in bandgap energy is given by [8]

Δ E_{g} (χ) = - (\frac{e}{2 π ε_{0} ε_{s}}) {(\frac{3}{π})}^{1 / 3} χ^{1 / 3}

where ε_s is the relative static dielectric constant; χ is the concentration of free carrier.

The effect of bandgap shrinkage on refractive index change is taken into account by adding ΔE_g to the bulk bandgap in the calculation of energy band structure. Considering bandgap shrinkage effect, the expression of refractive index change in Eq. (1) can be rewritten as

Δ n_{B F + B G S} = \frac{2 c ℏ}{e^{2}} p \int_{0}^{\infty} \frac{α (χ, {E^{'}}_{g}, E) - α_{0} (E_{g}, E)}{{E^{'}}^{2} - E^{2}} d E^{'}

where α has taken into account the bandgap shrinkage, the bulk bandgap E_g turns into E′_g.

For the plasma effect due to free-carrier absorption, it has the following expressions [8]

Δ n_{F C A} = - \frac{e^{2} λ^{2}}{8 π^{2} c^{2} ε_{0} n_{e}} (\frac{N}{m_{e}^{}} + \frac{P}{m_{h}^{}})

where m_e, m_h are the effective mass for electron and hole respectively. Since the carriers are confined in the wells, they are not free to move in response to the electric field of the TM polarization, the plasma effect does not produce a contribution to the refractive index change of TM mode. But the movement of the confined carriers is not restricted in the direction of the electric field of the TE polarization, the plasma effect contributes to the refractive index change of TE mode [10].

Combination of three effects above and considering the optical confinement factor, the total refractive index change is given by

Δ n_{t o t a l} = Γ \cdot (Δ n_{B F + B G S} + Δ n_{F C A})

In the calculation of optical confinement factor for TE mode and TM mode, a MQW planar waveguide approximation is taken and a transfer matrix method is used to achieve effective refractive index, propagation constants and mode field distribution for SCH layer, barrier layers and well layers of MQW.

The influence of temperature on refractive index change is achieved by changing the bulk bandgap energy of material. The bulk bandgap energy is given by Varshni relationship [11]

E_{g} (T) = E_{g} (0) - \frac{α T^{2}}{T + β}

in which E_g(T) and E_g(0) represent the bandgap energies at temperature T and 0 respectively; α and β are fitting parameters.

For a MQW structure, carrier distribution is nonuniform. Without leakage, electron concentration in each well decreases from n-cladding to p-cladding of the MQW, while hole concentration in each well decreases from p-cladding to n-cladding. Since the effective mass of hole is larger than that of electron and the mobility of hole is lower than the mobility of electron, hole distribution is more nonuniform than electron distribution. In this paper, a carrier rate equation model has been employed to simulate the carrier distribution in MQW [12], the calculation of some relevant parameters is referred to [13].

3. Results and discussion

Either carrier injection or ambient temperature change will result in the variation of operating temperature. The temperature change will have an influence on band structure, quasi-Fermi level, and carrier distribution, which will further change the refractive index. The transition from the first conduction (C1) subband to the first heavy hole (HH1) subband only contributes to the refractive index change of TE mode, while the transition from C1 subband to the first light hole (LH1) subband mainly contributes to refractive index change of TM mode. For an unstrained quantum well, the transition speed of C1-HH1 is larger than that of C1-LH1. And the plasma effect only contributes to the refractive index change of TE mode. In addition, the optical confinement factors of TE mode is larger than that of TM mode when the cross section of active region is not a square. So tensile strain is introduced to make light hole band rise to the first subband of valance band to realize polarization insensitivity of refractive index change.

A 1.55 μm MQW with polarization insensitivity of refractive index change has been designed. The MQW structure is made up with four 0.35% tensile-strained In_0.481Ga_0.519As quantum wells, five In_0.91Ga_0.09As_0.2P_0.8 barrier layers which are lattice-matched to InP substrate, and two separate-confinement heterostructure (SCH) layers which have the same materials with barrier layers. The thicknesses of each well layer, barrier layer and SCH layer are 11 nm, 16 nm and 180 nm respectively. Figure 1(a) shows the spectrums of refractive index change at different temperatures, in which the injection carrier density is 2.0 × 10²⁴ m⁻³. The carrier concentration in each well at 20°C is listed in Table 1.

Fig. 1 (a) Refractive index change spectrum; (b) ρ versus wavelength at different temperatures.

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Table 1. Carrier concentration in each well

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As temperature increases, the bulk bandgap energy for well and barrier shrinks, the quasi-Fermi levels of conduction band and valence band are close to their band edge respectively. These effects cause the slight reduction of the magnitudes of peak refractive index change, as shown in Fig. 1(a). At the same time, due to the bulk bandgap shrinkage, the peak wavelength of refractive index change is red-shifted. The magnitudes of refractive index change keep large (more than 3 × 10⁻³) over 1.50 μm to 1.60 μm wavelength range as temperature varies from 0°C to 40°C. The refractive index change keeps polarization insensitivity at the mean time. A parameter ρ is introduced to describe the polarization sensitivity for refractive index change, which is defined as follows

ρ = | \frac{Δ n_{t o t a l}^{T E} - Δ n_{t o t a l}^{T M}}{Δ n_{t o t a l}^{T E} + Δ n_{t o t a l}^{T M}} | \times 100 %

According to the definition, the smaller ρ is, the better polarization insensitivity will be. Corresponding to Fig. 1(a), the ρ versus wavelength is shown in Fig. 1(b). It shows that the values of ρ keep smaller than 6% within 1.50 μm to 1.60 μm wavelength range as temperature varies from 0°C to 40°C, and the values of ρ are even less than 1.5% in the wavelength range of 1.530 μm to 1.565 μm (C band).

In actual application, some devices are usually required to work for a wide range of carrier concentration variation which will cause refractive index change. For carrier distribution in MQW, temperature change will affect carrier concentration in each well by change the carrier escape time. Temperature increasing causes the reduction of escape time which leads to the weakening of nonuniform of carrier distribution. This may influence polarization insensitivity of refractive index change on carrier concentration. Figure 2 shows ρ versus carrier concentration at different temperatures at 1.55 μm wavelength. Since the carrier distribution is nonuniform in MQW, the carrier concentration indicated in abscissa of the figure represents the electron concentration in the first well which is closed to n-cladding. As shown in Fig. 2, when the carrier concentration varies from 1 × 10²⁴ m⁻³ to 3 × 10²⁴ m⁻³, the value of ρ are floating around 1%. It indicates that the polarization insensitivity of refractive index change can maintain for a wide range of carrier concentration when temperature changes from 0°C to 40°C.

Fig. 2 ρ versus carrier concentration at different temperatures at 1.55 μm wavelength.

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Through appropriate adjustments, other MQWs with different material and structure parameters can also realize polarization insensitivity of refractive index change. Two sets of material and structural parameters are listed in Table 2. The corresponding refractive index change spectrums are shown in Figs. 3(a) and 3(b) respectively.

Table 2. Two sets of material and structural parameters

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Fig. 3 (a) Refractive index change spectrum for 1st group; (b) Refractive index change spectrum for 2nd group.

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4. Conclusion

The characteristic analyses of polarization insensitivity of carrier-induced refractive index change of 1.55 μm tensile-strained MQW have been accomplished by using a comprehensive model. The model has considered the temperature variation as well as the nonuniform distribution of injected carrier in MQW, which greatly improves the actual application scope of previous models. The simulation results reveal that tensile-strained MQW can achieve polarization insensitivity of carrier-induced refractive index change over a wide wavelength range as temperature changes from 0°C to 40°C, while the magnitude of refractive index change keeps a large value (more than 3 × 10⁻³). Furthermore the polarization insensitivity of refractive index change can maintain for a wide range of carrier concentration when temperature changes from 0°C to 40°C. Most important, MQWs with different material and structure parameters can realize similar polarization insensitivity of refractive index change, which demonstrates the design flexibility.

The model presented here can be modified to investigate the electro refractive effect changes in new emerging Ge/SiGe MQW materials, the Γ, L and ∆-valley conduction band edges should be considered respectively.

Acknowledgments

This work was supported by the Chinese Natural Science Foundation under Grant No. 60877039.

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ith well from n-cladding to p-cladding	Electeron concentration	Hole concentration
1	2.01 × 10²⁴ m⁻³	1.77 × 10²⁴ m⁻³
2	1.97 × 10²⁴ m⁻³	1.83 × 10²⁴ m⁻³
3	1.93 × 10²⁴m⁻³	1.95 × 10²⁴ m⁻³
4	1.90 × 10²⁴ m⁻³	2.13 × 10²⁴ m⁻³

	Strain	Material of well layer	Well width	Well numbers	Material of barrier layer	Thickness of barrier layer	Thickness of SCH layer
1st group	0.4%	In_0.474Ga_0.526As	10 nm	4	In_0.9Ga_0.1As_0.22P_0.78	16 nm	146 nm
2nd group	0.3%	In_0.489Ga_0.511As	12 nm	4	In_0.918Ga_0.082As_0.18P_0.82	16 nm	146 nm

ith well from n-cladding to p-cladding	Electeron concentration	Hole concentration
1	2.01 × 10²⁴ m⁻³	1.77 × 10²⁴ m⁻³
2	1.97 × 10²⁴ m⁻³	1.83 × 10²⁴ m⁻³
3	1.93 × 10²⁴m⁻³	1.95 × 10²⁴ m⁻³
4	1.90 × 10²⁴ m⁻³	2.13 × 10²⁴ m⁻³

	Strain	Material of well layer	Well width	Well numbers	Material of barrier layer	Thickness of barrier layer	Thickness of SCH layer
1st group	0.4%	In_0.474Ga_0.526As	10 nm	4	In_0.9Ga_0.1As_0.22P_0.78	16 nm	146 nm
2nd group	0.3%	In_0.489Ga_0.511As	12 nm	4	In_0.918Ga_0.082As_0.18P_0.82	16 nm	146 nm

Theoretical study of polarization insensitivity of carrier-induced refractive index change of multiple quantum well

Abstract

1. Introduction

2. Model

3. Results and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Tables (2)

Equations (8)

Optics Express