Theoretical analysis of electro-refractive index variation in asymmetric Ge/SiGe coupled quantum wells

Yi Zhang; Junqiang Sun; Jianfeng Gao

doi:10.1364/OE.25.030032

1. Introduction

With the rapid development of optical communication and interconnect, electronic photonic integrated circuits (EPICs) play a more and more important role in data transfer. Silicon photonics is now considered as the most promising platform for the integration of electronic and photonic devices owing to the compatibility with mature complementary metal-oxide-semiconductor (CMOS) technology. However, the development of compact and efficient active components is still challenging. SiGe material system is an appealing solution to achieve CMOS compatible active components with competitive advantages in aspects of low working voltage, high speed operation and low power consumption. Although silicon and germanium are both indirect gap semiconductors, germanium has a useful direct Γ-valley band gap only 136 meV higher than the indirect L-valley band gap. In recent years, Ge/SiGe quantum wells (QW) have been intensely studied since the groundbreaking works for the first demonstration [1] of the quantum-confined Stark effect (QCSE) in SiGe material in 2005. Many significant breakthroughs [2–5] have been made for the Ge/SiGe quantum wells electro-absorption modulators based on the QCSE.

However, fewer efforts have been made to research the electro-refractive index variation in the Ge/SiGe QWs due to the change of the absorption coefficient. Only recently a preliminary demonstration [6] of the electro-refractive index variation has been reported with an effective refractive index variation of 1.3 × 10⁻³ under an electric field of 88 kV/cm and a figure of merit V_πL_π of 0.46 V cm for a Ge/SiGe QW. Obviously, the bias voltage of 8 V is too high for the CMOS circuits and the figure of merit V_πL_π is not so competitive with the III-V based multiple QWs. Shortly afterwards, coupled quantum wells were exploited to enhance the electro-optical performances for the optical modulators because of its amazing physical behavior of the coupling between the two wave functions through the thin barriers [7]. An outstanding demonstration of a refractive index variation up to 2.3 × 10⁻³ under a bias voltage of 1.5 V was reported for a symmetric Ge/SiGe CQW structure with an associated modulation efficiency V_πL_π of 0.046 V cm [8].

In this paper, a novel asymmetric Ge/SiGe CQW structure is proposed and analyzed theoretically. By designing two different width quantum wells for the CQW, we can tailor the electro-optical properties of the CQW through controlling the degree of the coupling between the wave functions. An 8-band k $\cdot$ p model is employed to calculate the eigenstates and absorption spectra of the CQWs. The simulation results show unique physical characteristics which strongly differ from the standard uncoupled QW. And the modulation performance is far better than the symmetric CQW previously demonstrated [8, 9].

2. Theory model

Multiband k $\cdot$ p method has been proved to be an efficient approach to calculate the band structure of semiconductors near the high symmetry point with sufficient accuracy [10]. Here, we use an 8-band k $\cdot$ p model to simulate the Γ point band structure of Ge/SiGe CQWs. The 8-band k $\cdot$ p model is in agreement with the experimental results even though the excitonic effects are ignored. The eight basis functions which are used to expand the Schrödinger equation are chosen including spin degeneracy as [11]

| S ↑ 〉, | X ↑ 〉, | Y ↑ 〉, | Z ↑ 〉, | S ↓ 〉, | X ↓ 〉, | Y ↓ 〉, | Z ↓ 〉

According to Burt’s k

\cdot

p theory, the Hamiltonian of the strain band can be expressed as [12]

H = [\begin{matrix} H^{4 \times 4} & 0 \\ 0 & H^{4 \times 4} \end{matrix}] + H_{s o} + H_{s t r}

where

H_{s o}

,

H_{s t r}

and

H^{4 \times 4}

are spin-orbit coupling Hamiltonian, strain-induced Hamiltonian and

4 \times 4

Kane Hamiltonian respectively. When multiple thin film epitaxial layers are grown continuously, a pseudomorphic heterointerface is formed between them. All the films are distorted following the principle of minimizing the total elastic energy. The combining in-plane lattice constant and in-plane strains of each layer can be described as [13]

\begin{array}{l} a_{| |} = \frac{a_{1} G_{1} h_{1} + a_{2} G_{2} h_{2} + \dots + a_{n} G_{n} h_{n}}{G_{1} h_{1} + G_{2} h_{2} + \dots + G_{n} h_{n}} \\ ε_{| |}^{i} = \frac{a_{| |}}{a_{i}} - 1 \end{array}

where

a_{i}

denotes the unstrained lattice constant,

h_{i}

denotes the layer thickness and

G_{i}

is the shear modulus. The specific calculation procedures of band structure near the Γ point are demonstrated in detail in [14].

According to the K selection rules, the direct absorption coefficient can be derived as [15]

\begin{array}{l} α_{σ} (ℏ ω) = \frac{π q^{2}}{n_{r} c m_{0}^{2} ω L_{w}} \sum_{s, t} \iint \frac{2 d k_{x} d k_{y}}{{(2 π)}^{2}} {| 〈 Ψ_{s}^{c} | {\hat{p}}_{σ} | Ψ_{t}^{v} 〉 |}^{2} \cdot [f_{s}^{c} (k_{| |}) - f_{t}^{v} (k_{| |})] \\ \times \frac{γ / (2 π)}{{[E_{s}^{c} (k_{| |}) - E_{t}^{v} (k_{| |}) - ℏ ω]}^{2} + {(γ / 2)}^{2}} \end{array}

with the Fermi functions

\begin{array}{l} f_{s}^{c} (k_{| |}) = {1 + \exp [\frac{E_{s}^{c} (k_{| |}) - F_{c}}{k_{B} T}]}^{- 1} \\ f_{t}^{v} (k_{| |}) = {1 + \exp [\frac{E_{t}^{v} (k_{| |}) - F_{v}}{k_{B} T}]}^{- 1} \end{array}

where

ω

is the angular frequency,

q

is the electron charge,

n_{r}

is the refractive index of the CQW,

c

is the light speed in vacuum,

ε_{0}

is the free-space permittivity,

L_{w}

is the width of the CQW,

k_{| |}

is the in-plane wave vector of reciprocal space. The momentum matrix elements

〈 Ψ_{s}^{c} | {\hat{p}}_{σ} | Ψ_{t}^{v} 〉

are calculated between

s

conduction bands and

t

valence bands.

σ

denotes the polarization

x, y, z

.

F_{c}, F_{v}

are quasi-Fermi levels,

k_{B}

is the Boltzmann constant,

T

denotes the temperature.

γ

is the Lorentzian linewidth defined by Full-Width-at-Half-Maximum (FWHM).

As for the indirect bandgap absorption, we adopt the model described in [18, 26]. The indirect bandgap absorption coefficient can be expressed as

α_{i n d} = A [\frac{1}{1 - e^{- Θ / T}} {(\frac{ℏ ω - E_{g, i n d} - k_{B} Θ}{ℏ ω})}^{2} + \frac{1}{e^{Θ / T} - 1} {(\frac{ℏ ω - E_{g, i n d} + k_{B} Θ}{ℏ ω})}^{2}]

where

A

is a temperature dependent constant,

T

denotes the temperature,

Θ

is the equivalent temperature corresponding to the phonon energy and

E_{g, i n d}

is the indirect bandgap.

According to the Kramers-Kronig relation, the electro-refractive index variation is given by [16, 17]

Δ n (v) = \frac{c}{π} \int \frac{Δ α (v^{'})}{v^{' 2} - v^{2}} d v^{'}

where

v

is the frequency and

Δ α

is the direct absorption variation resulting from the applied bias. Here we neglect the effect of indirect absorption because the change of indirect absorption caused by the applied electric field is too small [18]. Thus we assume that the indirect absorption of the Ge/SiGe CQWs has little influence on the electro-absorption variation and electro-refractive index variation.

3. Design of Ge/SiGe CQW

Figure 1 shows a schematic diagram of the proposed asymmetric Ge/Si_0.15Ge_0.85 coupled quantum wells. As you see, our design consists of eight pairs of CQWs: 8 × [6nm Ge QW + 1.6nm Si_0.1Ge_0.9 inner barrier + 12nm Ge QW + 24nm Si_0.15Ge_0.85 outer barrier]. Differing from previous works using symmetric CQW structures, we adopt asymmetric CQW design to achieve better modulation effects which are described in detail in the next section. Moreover, we set the wider Ge QW of the CQW at the same width as the outer Si_0.15Ge_0.85 barrier to realize a lower applied voltage for a given electric field and a higher density of CQWs per active region [5]. The outer barriers are wide enough to avoid coupling in neighboring CQWs and the simulation results of multiple CQWs hardly show any difference from the single CQW. The Ge fraction of inner barriers is higher than outer barriers so that we can obtain lower energy gap between the Ge quantum wells and the inner barriers which could enhance coupling between the narrow quantum well (QW1) and the wide quantum well (QW2).

Fig. 1 (a) Schematic of the asymmetric CQW. (b) Epitaxy design of the CQWs structure.

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Considering the lattice mismatch between silicon and germanium, we insert a buffer Ge heterolayers followed by an anneal process at temperature of 800°C usually to decrease dislocations and surface roughness [1, 3, 5, 19]. This process induces a tensile strain of 0.1%-0.3% and the typical value 0.2% is adopted in our simulation [10, 20].

4. Simulation results and discussion

The parameters of the materials used in the simulation are listed in Table 1. As for the Si_1-xG_x alloy parameter, the DKK parameters are described in [21] and the energy gap of Si_1-xG_x alloy at the Γ point is expressed as [22]

E g (S i_{1 - x} G e_{x}) = 0.7985 x + 4.185 (1 - x) - 0.14 x (1 - x) (e V)

Other parameters for Si_1-xG_x alloy are linearly extrapolated between Si and Ge. The temperature is set at 300K with the linewidth factor of

γ = 3 m e v

. All the simulations are carried out for single CQW due to the limited computational resources.

Table 1. The parameters at 300 K of Si and Ge used in the proposed scheme.

View Table

The in-plane energy dispersion near the Γ point of the asymmetric CQWs is shown in Fig. 2. While the electron dispersions are close to parabolic and isotropic, the hole states are significantly nonparabolic and anisotropic due to the band mixing effects between heavy-hole (HH), light-hole (LH) and spin-orbit splitting band [23]. The electron states are sparse while the hole states are denser. Figure 3 shows the energy eigenvalue of electron and hole states at the Γ point of the CQWs as a function of the electric field. With the increasing of the applied electric field, the band gap between HH1 and e1 is decreasing obviously which indicates the red shift of the absorption spectrum. Besides, it can be found that light-hole states move faster than heavy-hole states. At first the LH1 energy level is lower than HH3 level without applied electric field, but the LH1 energy level moves higher than HH2 level when the electric field adds up to 50 kV/cm.

Fig. 2 Energy dispersion near the Γ point of the CQW.

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Fig. 3 Energy eigenvalue of electron and hole states at the Γ point of the CQW as a function of electric field.

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Figure 4 shows the calculated wave functions at the Γ point of the asymmetric CQWs for e1, e2, HH1, HH2 and LH1 energy states under different electric fields. At first the e1 energy state and the HH1 state are mostly located in the wide quantum well (QW2) without applied field, which is different from those of symmetric CQWs distributed symmetrically. With the increasing of the applied electric field, the e1 state gradually moves towards the narrow quantum well (QW1). On the contrary, the hole states move to the wide quantum well (QW2) when the electric field augments. Because of the asymmetry of the CQWs, the original selected rules in standard uncoupled quantum wells are broken and the optical transitions: HH1-e2, LH1-e2 and HH2-e1 are no longer forbidden. In Fig. 5, we show the normalized momentum matrix elements (MME) at the Γ point for several optical transitions of TE and TM polarizations as a function of the electric field. As for TE polarization, the HH1-e1 and LH1-e1 transitions are gradually reducing while the HH1-e2 and LH1-e2 transitions are mainly increasing with the intensifying of the applied electric field. The situation for HH2-e2 and HH2-e1 transitions is different due to the obvious moving of HH2 energy state from QW1 to QW2. Therefore the HH2-e2 transition is first decreasing and then increasing as the electric field increases and the case of HH2-e1 transition is just the opposite. As for TM polarization, the optical transitions involved with heavy hole states are vanishingly small and the LH1-e1 transition is much more remarkable than that of TE polarization. Hence the optical absorption of TM polarization largely depends on transitions concerned with light hole state.

Fig. 4 Simulated wave functions at the Γ point of the CQW under the electric field of: (a) 0 kV/cm, (b) 10 kV/cm, (c) 20 kV/cm, (d) 30 kV/cm, (e) 40 kV/cm, (f) 50 kV/cm.

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Fig. 5 Normalized momentum matrix elements (MME) at the Γ point of several optical transitions as a function of electric field for: (a) TE polarization, (b) TM polarization.

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Figure 6 shows the absorption coefficient consisting of direct and indirect bandgap absorption for the asymmetric Ge/SiGe CQWs in TE and TM polarizations under different fields. The exciton absorption peaks are caused by the interband optical transitions between various conduction bands and valence bands. As you can see, the exciton peaks of TE polarization are much more than that of TM polarization. This is because the exciton peaks involved with heavy hole states are faint for TM polarization. As for TE polarization, the first absorption edge is about at 1445 nm without electric field. As the field gradually increases, the first band edge shifts towards long wavelength due to the reduction of the exciton binding energy of the HH1-e1 transition. Unlike typical uncoupled quantum wells, the red shift is more outstanding for the asymmetric CQWs, which can achieve a move of about 27 nm wavelength under an applied field of 40 kV/cm. For those previously demonstrated uncouple quantum wells [1], we can only gain a shift less than 10 nm wavelength under the same electric field. The numerous exciton transitions and the prominent red shift lead to a large electro-refractive index change.

Fig. 6 Absorption spectrum of the CQWs under different electric field for: (a) TE polarization, (b) TM polarization.

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Figure 7 shows the electro-refractive index variation of the asymmetric Ge/SiGe CQWs for TE and TM polarizations as a function of the wavelength under different field operations. As for TE polarization, the refractive index variation firstly increases then decreases when the applied field augments, which differs from that of classical uncoupled quantum wells increasing quadratically with the electric field. For practical application, the electro-refractive effect should be employed in the wavelength range of low optical absorption. In Fig. 7, the dash lines denote the wavelength range of high absorption under a certain electric field while the solid lines represent the wavelength range of low absorption. As you can see, the wavelength should be longer than 1461 nm to obtain low optical absorption when the applied field is 30 kV/cm. Figure 7 shows a local maximum about 9 × 10⁻³ of the electro-refractive index variation at the wavelength of about 1461 nm under the field of 30 kV/cm, which is much higher than that of standard uncoupled quantum wells. By contrast, an effective electro-refractive index variation of 1.3 × 10⁻³ was demonstrated at the wavelength of 1475 nm with an applied field of 88 kV/cm in the uncouple quantum wells [6]. The operating wavelength can be further shifted towards long wavelength by introducing uniaxial tensile stress [24]. Taking account of the built-in electric field of the p-i-n junction, which we assume about 16 kV/cm [10, 25], it needs only a bias voltage as low as 1 V to realize an electric field of 30 kV/cm. The bias voltage can be further decreased by reducing the total number of the CQWs at the cost of the optical confinement factor of the active region in the waveguide. Using the asymmetric CQW structure under a bias voltage of 1 V, a 100 μm long waveguide structure is enough to obtain a π phase shift at 1461 nm. The according absorption coefficient is about 1325 cm⁻¹ and the absorption length, which is defined as the distance where the intensity of the light is reduced by a factor 1/e, is about 7.5 μm at the wavelength of 1461 nm under the field of 30 kV/cm. To obtain lower absorption coefficient and longer absorption length, the wavelength should be longer than 1465 nm. The product of half-wave voltage and length of phase shift region, namely V_πL_π, is approximately 0.01 V cm, which is one order of magnitude smaller than that of uncoupled Ge/SiGe quantum wells and has competitive advantage over many silicon based phase modulators. Furthermore, the proposed CQWs structure works under reverse bias voltage, so the energy consumption is expected to be much lower than modulators of carrier injection. These characteristics pave the way for high speed, low voltage, low power consumption and compact optical phase modulators in silicon photonics devices.

Fig. 7 Electro-refractive index variation of the CQWs under different electric field operation for: (a) TE polarization, (b) TM polarization.

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5. Conclusion

We propose and analyze theoretically an asymmetric Ge/SiGe coupled quantum well for silicon based optical phase modulator. The band structure and electronic wave function of the CQWs are calculated by an 8-band k $\cdot$ p model. The simulation results show unique characteristics with respect to standard uncoupled quantum wells. The absorption coefficient and electro-refractive index variation of the asymmetric Ge/SiGe CQWs are calculated and discussed in detail. We can obtain an electro-refractive index variation as high as 9 × 10⁻³ at the wavelength of about 1461 nm under the electric field of 30 kV/cm. The product of half-wave voltage and length of phase shift region, namely V_πL_π, is deduced to be 0.01 V cm, which is one order of magnitude smaller than that of uncoupled Ge/SiGe quantum wells. In addition, the proposed Ge/SiGe CQWs modulator has much lower power consumption than modulators of carrier injection due to reverse working voltage. The proposed asymmetric Ge/SiGe CQWs structure provides a promising approach to realizing high speed, low voltage, low power consumption and compact optical phase modulators in silicon-based integrated optoelectronic devices.

Funding

National Natural Science Foundation of China (NSFC) (Grant No. 61435004).

References and links

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Parameters	Si	Ge
Lattice constants a (nm)	0.54304^a	0.56579^a
Elastic constants C₁₁ (GPa)	165.77^a	128.53^a
C₁₂ (GPa)	63.93^a	48.28^a
Spin-orbit split energies
Δ (eV)	0.044^a	0.289^a
Average valence band energies
E_V,av (eV)	−7.03^b	−6.35^b
Bandgaps
E_g^Γ (eV)	4.185^c	0.7985^c
E_g^L (eV)	1.65^c	0.664^c
Deformation potentials
a_c (eV)	−10.39^a	−10.41^a
a_v (eV)	1.8^a	1.24^a
b (eV)	−2.1^a	−2.86^a
d (eV)	−4.85^a	−5.28^a
Conduction band effective mass
m_c (m₀)	0.528^c	0.038^c
DKK parameters
L ( $ℏ$ ²/2m₀)	−6.69^d	−31.34^d
M ( $ℏ$ ²/2m₀)	−4.62^d	−5.9^d
N ( $ℏ$ ²/2m₀)	−8.56^d	−34.14^d
Indirect absorption parameters
Θ (K)		320^e
A (cm⁻¹)		1150^f

Theoretical analysis of electro-refractive index variation in asymmetric Ge/SiGe coupled quantum wells

Abstract

1. Introduction

2. Theory model

3. Design of Ge/SiGe CQW

4. Simulation results and discussion

5. Conclusion

Funding

References and links

Cited By

Figures (7)

Tables (1)

Equations (8)

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