Theoretical analysis of optical gain in uniaxial tensile strained and n<sup>+</sup>-doped Ge/GeSi quantum well

Jialin Jiang; Junqiang Sun

doi:10.1364/OE.24.014525

1. Introduction

Tremendous efforts have been made to achieve photonic integrated circuits (PICs) on silicon platform, since the low-cost and mass manufacturing for photonics device can be achieved by using mature Complementary Metal-Oxide-Semiconductor (CMOS) techniques [1]. The scope of applications for silicon photonics covers high-capacity and energy-efficient optical interconnect, and recently has been extended to spectroscopy, chemical and biological sensing [2]. A crucial and challenging issue for Si photonics is the electrically pumped light source because Si itself is unsuitable for light emission due to the indirect gap property. Recent years, the hybrid silicon laser which has shown desirable performance is an exciting success [3]. SiGeSn alloy also shows its potential since the material is expected to become direct gap at specific Sn concentration [4]. Another parallel approach which has attracted great interest is GeSi system. Germanium is compatible with CMOS process and has a unique band structure with its conduction band minima (CBM) at the L-points are only lower than the Γ-point by 136 meV. Furthermore, this energy difference can be reduced by tensile strain and the remaining empty states between the Γ- and L-valleys can be filled by extrinsic electrons from n-type doping [5]. As a result, more electrons can occupy the Γ-valley so as to enhance the direct gap optical gain. The groundbreaking works which realized an optically and an electrically pumped Ge laser were conducted by a MIT group [6, 7]. However, these devices suffer from high threshold current since the 0.2% tensile strain is insufficient to turn Ge into a direct gap material. Thus, subsequent research interest has focused on the strain engineering of Ge. Several methods such as using silicon nitride as an external stressor [8] or defining microstructures to enhance the internal strain [9] have been demonstrated. Until now, a 2.3% biaxial tensile strain [10] and a 5.7% uniaxial tensile strain [11] have been achieved in bulk Ge and the enhancement of Photoluminescence (PL) is observed.

Apart from bulk material, quantum structure materials also offer a possibility to realize an efficient Si-based laser considering its ability to reduce the threshold current in III-V materials system [12, 13]. However, it has been discussed that although reduced density of state is obtained, the transparency carrier density of the Ge/GeSi QW is still higher than that of bulk Ge since the larger band offset of Γ-valley makes the energy difference between the direct and indirect gap become larger [14]. In that paper, the QW is under weak tensile strain. While the QW with higher tensile strain is still likely to be fabricated by similar approaches used in bulk Ge [8, 9], which means a theoretical prediction for its gain characteristics is in demand. Recently, Fan has calculated the direct gap optical gain of the biaxial tensile strained Ge/GeSi QW [15]. However, the case of uniaxial tensile strained has rarely been reported.

In this paper, we calculate the direct gap optical gain of [100] uniaxial tensile strained and n⁺-doped Ge/Ge_0.85Si_0.15 QW. Free carrier absorption (FCA) is considered since it is the main loss in highly doped material. The strained band structures near the Γ- and L-point are obtained by the 8-Band k∙p method and single band effective mass approximation, respectively. The influences of uniaxial tensile strain and n-type doping on the band structure, carrier occupation, polarization dependent gain and FCA loss are discussed. Furthermore, net peak gain at different tensile strain value and n-type doping concentration is analyzed to consider the competition between gain and loss. Finally, we predict a significant net peak gain for TE-polarized light when the tensile strain and doping concentration are set at reasonable values.

2. Theoretical Model

2.1 Strained band structure

The k∙p method is a semiempirical theoretical model based on perturbation theory which can calculate the band structure near the high-symmetry point with adequate accuracy so that it serves as an efficient approach to study the optical and electrical properties of semiconductor. Different forms of the k∙p method have been developed depending on the number of band considered and the treatment of spin-orbit interaction. Furthermore, the model can also handle the situation of quantum structure, including quantum wells, wires and dots, by combined with the envelop function approximation (EFA) [16].

Considering the direct band gap of Ge is only 0.8 eV and can be even smaller at high strain level, we adopt an 8-Band k∙p method which includes the coupling between conduction band (CB) and valence band (VB). The basis functions, also called Bloch functions, are chosen as

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

Then the wavefuntion of electron and hole for quantum well can be written as

ψ_{m} (r, k_{t}) = \frac{e^{i k_{t} \cdot ρ}}{\sqrt{A}} \sum_{i = 1}^{8} g_{m}^{(i)} (z) | u_{i} 〉

where

r

is the position vector and

k_{t}

is the transverse wavevector,

ρ

is the transverse position vector, A is the area of quantum well in the x-y plane.

g_{m}^{(i)} (z)

denotes the envelop function and

| u_{i} 〉

is the Bloch function.

According to theory of Kane and Bir-Pikus, the Hamiltonian for strained bulk material can be expressed as [17]

H = H_{0} + H_{s o} + H_{s t}

where

H_{s o}

and

H_{s t}

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

H_{0}

and

H_{s t}

are block diagonalized as

(\begin{matrix} H_{0}^{4 \times 4} \\ H_{0}^{4 \times 4} \end{matrix}), (\begin{matrix} H_{s t}^{4 \times 4} \\ H_{s t}^{4 \times 4} \end{matrix})

with

H_{0}^{4 \times 4} = (\begin{matrix} E_{c} + S k^{2} & i P k_{x} & i P k_{y} & i P k_{z} \\ - i P k_{x} & \begin{array}{l} E_{v, av} + ℏ^{2} k^{2} / 2 m_{0} \\ L' k_{x}^{2} + M (k_{y}^{2} + k_{z}^{2}) \end{array} & N' k_{x} k_{y} & N' k_{x} k_{z} \\ - i P k_{y} & N' k_{y} k_{x} & \begin{array}{l} E_{v, a v} + ℏ^{2} k^{2} / 2 m_{0} \\ L' k_{y}^{2} + M (k_{x}^{2} + k_{z}^{2}) \end{array} & N' k_{y} k_{z} \\ - i P k_{z} & N' k_{z} k_{x} & N' k_{z} k_{y} & \begin{array}{l} E_{v, a v} + ℏ^{2} k^{2} / 2 m_{0} \\ L' k_{z}^{2} + M (k_{x}^{2} + k_{y}^{2}) \end{array} \end{matrix})

where E_c and E_v,av are the unstrained band edge of CB and average energy of VB, respectively. It is convenient to determine those values and then deduce the band offset of the quantum well by the “model-solid” theory [18]. m₀ is the free electron mass. S, P,

L'

,

M

and

N'

are the Kane’s parameters and P is related to the energy parameter E_p by

P = \sqrt{ℏ^{2} E_{p} / 2 m_{0}}

. S,

L'

,

M

and

N'

are in units of

ℏ^{2} / 2 m_{0}

and are given by [19]

\begin{array}{l} S = \frac{1}{m_{c}} - E_{p} \frac{E_{g} + 2 Δ / 3}{E_{g} (E_{g} + Δ)} \\ L' = L + \frac{P^{2}}{E_{g}} \\ N' = N + \frac{P^{2}}{E_{g}} \end{array}

where

m_{c}

is the effective mass of conduction band, E_g is the unstrained band gap,

Δ

is the spin-orbit splitting energy. Note that the Dresselhaus parameters L and N for 6-band k∙p method should be corrected to exclude the coupling between the CB and VB since it has been considered in the off-diagonal matrix element. A challenging issue in 8-Band k∙p method under the EFA is the unphysical spurious solutions [19]. To eliminate the spurious solutions, we set

S = 1

and rescale E_p to fit the effective mass [20]

E_{p} = (\frac{1}{m_{c}} - 1) \frac{E_{g} (E_{g} + Δ)}{E_{g} + 2 Δ / 3}

The strain-induced

4 \times 4

Hamiltonian is

H_{s t}^{4 \times 4} = (\begin{matrix} a_{c} T r (ε) & 0 & 0 & 0 \\ 0 & l ε_{x x} + m (ε_{y y} + ε_{z z}) & n ε_{x y} & n ε_{x z} \\ 0 & n ε_{x y} & l ε_{y y} + m (ε_{x x} + ε_{z z}) & n ε_{y z} \\ 0 & n ε_{x z} & n ε_{y z} & l ε_{z z} + m (ε_{x x} + ε_{y y}) \end{matrix})

where

ε

is the strain tensor. In the case of uniaxial strain with the principal stress direction along x,

ε

only has three diagonal components

ε_{x x}

,

ε_{y y}

and

ε_{z z}

whose relationship can be described by

ε_{y y} = ε_{z z} = \frac{- C_{12}}{C_{12} + C_{11}} ε_{x x}

where C₁₂ and C₁₁ are the elastic stiffness constants. a_c is the deformation potential of conduction band. l, m and n are related to the Bir-Pikus deformation potential constants a, b and d by [17]

\begin{array}{l} l = a + 2 b \\ m = a - b \\ n = \sqrt{3} d \end{array}

The spin-orbit coupling Hamiltonian

H_{s o}

is expressed as

H_{s o}^{8 \times 8} = \frac{Δ}{3} (\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - i & 0 & 0 & 0 & 0 & 1 \\ 0 & i & 0 & 0 & 0 & 0 & 0 & - i \\ 0 & 0 & 0 & 0 & 0 & - 1 & i & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & i & 0 \\ 0 & 0 & 0 & - i & 0 & - i & 0 & 0 \\ 0 & 1 & i & 0 & 0 & 0 & 0 & 0 \end{matrix})

According to the EFA, the Schrodinger equation can be transformed into

\sum_{j = 1}^{8} H_{i, j} ({\hat{k}}_{z} = - i \frac{\partial}{\partial z}) g_{m}^{(j)} (z, k_{t}) = E_{m} g_{m}^{(i)} (z, k_{t})

Solving the above eigenvalue problem for each k-point in the x-y plane, one can obtain the band structure and envelop function. Owing to the z-dependence of material parameters,

k_{z}

here turns into an operator. Notice that the operator ordering is crucial. To ensure the hermiticity of Hamiltonian, the diagonal terms involving S,

L'

and M are treated in a conventional symmetry way

S k_{z}^{2} \to {\hat{k}}_{z} S {\hat{k}}_{z}

. However, the off-diagonal terms involving

N'

should be treated in an asymmetry way [19]:

\begin{array}{l} k_{x} N' k_{z} = k_{x} N_{+} k_{z} + k_{z} N_{-} k_{x} \\ N_{+} = N' - M, N_{-} = M \end{array}

For the L-conduction band, we employ a simple effective mass approximation (EMA) to calculate the band structure. Ge has eight equivalent L-valleys, while only four of them are independent due to the periodicity of the Brillouin zone. Since the uniaxial tensile strain along x-direction does not break the degeneracy of them, we only need to consider the [111] valley. The Hamiltonian $H_{L}^{[111]}$ under the EMA is given in [21]. Then the band structure of L-valley can be determined by solving the eigenvalue equation as follow

H_{L}^{[111]} χ_{l, k_{t}} (z) = E_{l, k_{t}} χ_{l, k_{t}} (z)

2.2 Optical gain and FCA

According to the Fermi’s golden rule, the optical gain coefficient containing the Lorentzian lineshape function can be derived as [22]

g (ℏ ω) = \frac{π q^{2}}{n_{r} c ε_{0} ω L_{z} m_{0}^{2}} \sum_{n_{c}, n_{v}} \int \frac{d k_{t}}{4 π^{2}} {| \hat{e} \cdot M_{n_{c}, n_{v}} (k_{t}) |}^{2} (f_{n_{c}} - f_{n_{v}}) \frac{ℏ / π τ}{{[E_{n_{c}} (k_{t}) - E_{n_{v}} (k_{t}) - ℏ ω]}^{2} + {(ℏ / τ)}^{2}}

with

f_{n_{c}} = \frac{1}{1 + \exp [(E_{n_{c}} (k_{t}) - F_{c}) / k_{B} T]}, f_{n_{v}} = \frac{1}{1 + \exp [(E_{n_{v}} (k_{t}) - F_{v}) / k_{B} T]}

where

f_{n_{c}}

and

f_{n_{v}}

are the Fermi-Dirac function.

F_{c}

and

F_{v}

are the quasi-Fermi levels of conduction band and valence band, respectively.

k_{B}

is the Boltzmann’s constant and T is the temperature. q is the electron charge, c and

ε_{0}

are the velocity of light and permittivity in free space.

n_{r}

and L_z are the refractive index and well width of the quantum well, respectively.

τ

is the intraband relaxation time.

\hat{e}

is the unit vector of the optical electric field. It corresponds to transverse electric (TE) and transverse magnetic (TM) transitions when

\hat{e}

is parallel and perpendicular to the quantum well plane, respectively.

{| \hat{e} \cdot M_{n_{c}, n_{v}} (k_{t}) |}^{2}

represents the squared momentum matrix element for transition between the

n_{c} -th

conduction band and

n_{v} -th

valence band with

\begin{array}{l} | \hat{x} \cdot M_{m, n} (k_{t}) | = 〈 ψ_{m} (r, k_{t}) | p_{x} | ψ_{n} (r, k_{t}) 〉 \\ = \sqrt{\frac{E_{p} m_{0}}{2}} \int d z [g_{m}^{(1)}^{*} (z) g_{n}^{(2)} (z) + g_{m}^{(5)}^{*} (z) g_{n}^{(6)} (z) - g_{m}^{(2)}^{*} (z) g_{n}^{(1)} (z) - g_{m}^{(6)}^{*} (z) g_{n}^{(5)} (z)] \end{array}

\begin{array}{l} | \hat{y} \cdot M_{m, n} (k_{t}) | = 〈 ψ_{m} (r, k_{t}) | p_{y} | ψ_{n} (r, k_{t}) 〉 \\ = \sqrt{\frac{E_{p} m_{0}}{2}} \int d z [g_{m}^{(1)}^{*} (z) g_{n}^{(3)} (z) + g_{m}^{(5)}^{*} (z) g_{n}^{(7)} (z) - g_{m}^{(3)}^{*} (z) g_{n}^{(1)} (z) - g_{m}^{(7)}^{*} (z) g_{n}^{(5)} (z)] \end{array}

\begin{array}{l} | \hat{z} \cdot M_{m, n} (k_{t}) | = 〈 ψ_{m} (r, k_{t}) | p_{z} | ψ_{n} (r, k_{t}) 〉 \\ = \sqrt{\frac{E_{p} m_{0}}{2}} \int d z [g_{m}^{(1)}^{*} (z) g_{n}^{(4)} (z) + g_{m}^{(5)}^{*} (z) g_{n}^{(8)} (z) - g_{m}^{(4)}^{*} (z) g_{n}^{(1)} (z) - g_{m}^{(8)}^{*} (z) g_{n}^{(5)} (z)] \end{array}

When population inversion is achieved, the enormous free carrier density will lead to significant free carrier absorption (FCA). A classical Drude-Lorentz model is adopted to evaluate the FCA loss [23]:

α_{F C A} = \frac{λ^{2} q^{2}}{4 π^{2} c^{3} ε_{0} n_{r}} [\sum_{i} \frac{n_{Γ}^{i}}{η_{Γ} \times m_{Γ}^{i}} + \sum_{j} \frac{n_{h}^{j}}{η_{h} \times m_{h}^{j}} + \sum_{k} \frac{n_{L}^{k}}{η_{L} \times m_{L}^{k}}]

where

λ

is the free space wavelength,

n_{X}^{Y}

and

m_{X}^{Y}

(X = Γ, h, L;Y = i, j, k) are the carrier volume density and effective mass of conductivity, respectively. The superscripts denote the subband index. The subscripts Γ, h and L denote Γ-Valley, hole and L-Valley, respectively. In Eq. (18), we treat the subbands separately since the carrier density and effective mass of them are different. This is more preferable than using a total carrier density and effective mass, as [15] and [23] did.

η

is the mean free time which is doping-dependent and can be deduced from the mobility of bulk Ge by

η = μ m_{}^{*} / q

, where

m_{}^{*}

is the effective mass of conductivity for bulk Ge. The experimental data of mobility [24–26] and effective mass of conductivity for electron in L-conduction band and hole are given by

\begin{array}{l} μ_{L} = \frac{3900}{1 + \sqrt{N_{D} \times 10^{- 17}}}, μ_{h} = \frac{1900}{1 + \sqrt{N_{A} \times 2.1 \times 10^{- 17}}} \\ m_{L}^{*} = \frac{3}{1/ m_{l, L}^{*} +2/ m_{t, L}^{*}}, m_{h}^{*} = \frac{m_{h h}^{*}^{3 / 2} + m_{l h}^{*}^{3 / 2}}{m_{h h}^{*}^{1 / 2} + m_{l h}^{*}^{1 / 2}} \end{array}

where

μ_{L}

and

μ_{h}

are in the unit of

c m^{2} V^{- 1} s^{- 1}

,

N_{D}

and

N_{A}

are the donor and acceptor concentration in the unit of

c m^{- 3}

, respectively.

m_{h h}^{*} = 0.33 m_{0}

and

m_{l h}^{*} = 0.043 m_{0}

. The mean free times of Γ- and L-valley are considered equal due to the lack of experimental data of electron mobility in Γ-valley. Actually, this would have little effect on the calculated FCA loss since the Γ-valley part contributes little to the total loss.

As far as the effective masses of QW are concerned, it is hard to define the “curvature” mass since the band structures of valence band are nonparabolic and anisotropic. To address this problem, we calculate the effective mass of conductivity indirectly. According to the Boltzmann transport equation and the relaxation time approximation, the conductivity tensor can be written as [27]

σ_{υ, ξ} = - \frac{q^{2}}{2 π^{2} ℏ^{2} L_{z}} \int τ \frac{\partial f_{0}}{\partial k_{υ}} \frac{\partial E}{\partial k_{ξ}} d k_{x} d k_{y}

where

f_{0}

is the Fermi-Dirac function. Then using

m_{}^{*} = n q^{2} τ / σ

, one can obtain the effective masses of conductivity of each subbands.

Note that the mean free time $η$ is inversely proportional to the doping concentration, which means doping will raise the FCA. In addition, according to the Eq. (18), a small effective mass of conductivity can also lead to significant FCA.

2.3 Fermi level and Carrier density

Prior to evaluating the optical gain and FCA, the quasi-Fermi levels and carrier densities in subbands under carrier injection must be determined. Without carrier injection, the CB and VB are in thermal equilibrium and the charge neutrality condition can be written as

n_{0} + N_{A}^{-} = p_{0} + N_{D}^{+}

where

n_{0}

and

p_{0}

are electron and hole densities at thermal equilibrium,

N_{A}^{-}

and

N_{D}^{+}

are ionized acceptor and donor concentrations. Commonly, we use the following relations for n-type doped semiconductor

n_{0} \approx N_{D}^{+} - N_{A}^{-} > > p_{0} = \frac{n_{i}^{2}}{n_{0}}

After the injection of excess carrier caused by optical or electrical pumping, the thermal equilibrium between CB and VB is disturbed and new thermal equilibrium is established in CB and VB, separately. The total carrier density comprises the thermal equilibrium and the injection part and can be expressed as

n = n_{0} + δ n = n_{Γ} + n_{L}

p = p_{0} + δ p

Where

δ n

and

δ p

are the excess electron and hole densities, respectively.

n_{Γ}

and

n_{L}

are electron densities in

Γ

- and L-valleys, respectively. p is the hole density. Since the dispersion relation of VB is severely nonparabolic and anisotropic, it is more appropriate to calculate the carrier density by numerically integrating over the 2-D k-space:

n_{Γ} = \sum_{n_{c}} \frac{1}{2 π^{2} L_{z}} \int \frac{1}{1 + \exp [(E_{n_{c}}^{Γ} (k_{t}) - F_{c}) / k_{B} T]} d k_{x} d k_{y}

n_{L} = \sum_{γ} \frac{2}{π^{2} L_{z}} \int \frac{1}{1 + \exp [(E_{γ}^{L} (k_{t}) - F_{c}) / k_{B} T]} d k_{x} d k_{y}

p = \sum_{n_{v}} \frac{1}{2 π^{2} L_{z}} \int \frac{1}{1 + \exp [(F_{v} - E_{n_{v}}^{h} (k_{t})) / k_{B} T]} d k_{x} d k_{y}

In Eqs. (24)a) and (24c), a factor of 2 has been considered due to the spin degeneracy of conduction band. Similarly, a factor of 8 is included in Eq. (24)b) on account of the degeneracy of spin and four equivalent L-valleys. The sum indices

n_{c}

,

γ

and

n_{v}

could just run over the several lowest subbands instead of all the subbands for the convenience of calculation. Then by combining Eqs. (22)-(24), the quasi-Fermi levels and the carrier densities can be obtained.

3. Simulation results and discussion

The Ge fraction is set at 0.85 to obtain a type-I QW which enables confinement of both electrons and holes in the active layer [28]. Assuming that the quantum well grew along the z-direction, i.e., the [001] direction, with the well and barrier width fixed at 12 and 15 nm, respectively. A global homogeneous uniaxial tensile strain along the x-direction is set in the material. Namely, tensile strain exists in both well and barrier. It should be noted that uniaxial strain along different directions might give different results. For example, the uniaxial strain along [110] and [111] direction will cause the splittings of the equivalent L-conduction bands [18]. This effect is beyond the scope of this paper. The temperature is set to 300K in our simulation and the material parameters for bulk Ge and Si are listed in Table 1. The energy gap of Ge_xSi_1-x alloy at Γ-point is given by [15]

E_{g}^{Γ, G e S i} = 0.7985 x + 4.185 (1 - x) - 0.14 x (1 - x)

The Kane’s parameters L, M, N of Ge_xSi_1-x alloy are given by [28]

\begin{array}{l} Q (x) = Q (0) + α \ln (1 - W x^{β}) \\ W = 1 - \exp {[Q (1) - Q (0)] / α} \end{array}

where Q represents L, M, and N with

\begin{array}{l} α =6 .7064, β =1 .35 \\ L (0) = - 6.69, M (0) = - 4.62, N (0) = - 8.56 \\ L (1) = - 21.65, M (1) = - 5.02, N (1) = - 23.48 \end{array}

Other parameters for Ge_xSi_1-x alloy are obtained by simple linear interpolation between Ge and Si. All the simulations are performed for single quantum well. However, the calculated optical gain and FCA loss can be easily extended to a multiple quantum wells structure with

N_{w}

-wells by

G_{M Q W} = N_{w} G_{S Q W}

[23], as long as the barrier width is thick enough to make the bound states of different quantum wells decoupled.

Table 1. Material parameters for bulk Ge and Si at 300K in the calculation^a

View Table

The band structures near Γ-point for unstrained and 4% uniaxial tensile strained QW are shown in Fig. 1. Remarkable nonparabolicity and anisotropy are observed in the unstrained subband structure of hole due to the band mixing effects between heavy-hole (HH) and light-hole (LH) band [29]. When uniaxial tensile strain is applied, the E-k curves of hole become flat but still remain nonparabolic and anisotropic, indicating that a numerical integration over the 2-D k-space to calculate the optical gain, effective masses of conductivity and carrier densities is more preferable, as Eqs. (15), (20) and (24) show.

Fig. 1 Band structure of (a) unstrained and (b) 4% uniaxial tensile strained Ge/Ge_0.85Si_0.15 QW

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We also show the band lineups of the 4% uniaxial tensile strained QW together with the subband edges in Fig. 2. As can be seen, the energy separation of the CBM at Γ- and L-point for bulk Ge is close to zero, while the band edge of the lowest Γ-conduction subband $E_{Γ 1}$ still lies above that of the lowest L-conduction subband $E_{L 1}$ by an energy distance in consequence of the larger band offset of Γ-valley. For the valence band of bulk material, the LH band shifts above the HH band due to the uniaxial tensile strain. However, it is improper to denote the subbands of the QW by LH or HH since the effective mass is hard to be determined.

Fig. 2 Band lineups and subband edges of 4% uniaxial tensile strained Ge/Ge_0.85Si_0.15 QW (a) conduction band (b) valence band

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A prerequisite for direct gap transition is that electrons fill up the L-valleys and then occupy the Γ-valley. For bulk Ge and Ge/GeSi QW, the 8-fold degeneracy property and large density of state (DOS) of the L-valleys prevent efficient electron injection into Γ-valley. Thus tensile strain and n-type doping are critical to making more electrons populate the Γ-valley. Figure 3(a) illustrates the ratio of electron density in Γ-valley to the total electron density as a function of the injected carrier density at various strain value. The doping concentration is fixed at $1 \times 10^{19} {cm}^{- 3}$ . When the QW is under a weak uniaxial tensile strain of 1%, the ratio is below 0.1% and the slope of the curve is small, leading to a poor light-emitting efficiency and slope efficiency. As the strain value rises up to 4%, more than 0.4% electrons can occupy the Γ-valley and the curve becomes steeper implying an improved slope efficiency. We also perform similar simulation at various doping concentration with a strain value of 4%. An improvement on the ratio caused by n-type doping is observed, as shown in Fig. 3(b).

Fig. 3 The ratio of electron density in Γ-valley to the total electron density versus the injected carrier density at various (a) uniaxial tensile strain with a doping concentration of $1 \times 10^{19} {cm}^{- 3}$ and (b) n-type doping concentration with a strain value of 4%.

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With the injection of excess carrier caused by optical or electrical pumping, the quasi-Fermi levels of electron and hole move towards conduction band and valence band, respectively. When the Bernard-Duraffourg relation $E_{F c} - E_{F v} > E_{g}^{Γ}$ is satisfied, population inversion can be obtained, leading to a positive optical gain. However, it is not sufficient to get positive net gain due to the FCA loss. We then perform simulations to clarify the influences of uniaxial tensile strain and n-type doping on the optical gain and FCA loss. It is interesting to find that the TE-polarized light has a larger gain in the uniaxial tensile strained QW which is in contrast to the case of biaxial tensile strained where the transverse-magnetic (TM)-polarized light dominates [15]. This can be easily understood when we look into the momentum matrix element, as described by Eq. (17). After solving the eigenvalue problem as described by Eq. (12), it is found that the uniaxial tensile strain modifies the wavefunction of the first valence band, making the $| Y 〉$ states dominate. As a result, the overlap integration between the envelop function of the $| S 〉$ state and $| Y 〉$ state becomes relatively larger which finally leads to a greater gain of y-electric-polarized light, i.e. the so-called TE-polarized light.

We now discuss the case of TE-polarized light since it dominates. Figures 4(a) and 4(b) show the gain spectra for TE-polarized light and FCA loss of the QW under various tensile strain value and n-type doping concentration, respectively. As can be seen in Fig. 4(a), with the increase of tensile strain from 1% to 4%, the optical gain is enhanced by a factor of 27. Meanwhile, the gain peak position shifts from 0.786 eV to 0.567 eV, which corresponds to the mid-infrared wavelengths, implying that it could be applied in chip-scale optical link, chemical and biological sensing. The dotted curves illustrate that the FCA loss is nearly independent to the uniaxial tensile strain which also implies the effective mass of conductivity is less sensitive to the uniaxial tensile strain. From the simulation results here, we can draw a conclusion that a higher uniaxial tensile strain is always desirable since it can improve the optical gain dramatically without bringing any additional FCA loss. As for the n-type doping, it both enhances the optical gain and FCA loss, as Fig. 4(b) shows. So the net peak gain under different doping concentration should be calculated to seek for an optimal doping concentration, as will be included in the following discussions.

Fig. 4 (a) Gain spectra for TE-polarized light and FCA loss of the QW under various tensile strain value. The n-type doping concentration is $1 \times 10^{19} {cm}^{- 3}$ . The injected carrier density is $4 \times 10^{19} {cm}^{- 3}$ . (b) Gain spectra for TE-polarized light and FCA loss of the QW under various n-type doping concentration. The tensile strain value is 4%. The injected carrier density is $4 \times 10^{19} {cm}^{- 3}$ .

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The net peak gain for TE-polarized light as a function of tensile strain and doping concentration with an injected carrier density of $4 \times 10^{19} {cm}^{- 3}$ is mapped out in Fig. 5(a). The white region indicates a negative net peak gain. At low strain level, the net peak gain is enhanced when n-type doping concentration increases from $5 \times 10^{18} {cm}^{- 3}$ to $5 \times 10^{19} {cm}^{- 3}$ . However, the enhancement becomes weak when the strain value approaches 5%. This is because the QW is nearly converted into a direct gap material at such high strain level, therefore the extrinsic electrons are no longer necessary for compensating the energy difference, and instead it can lead to large FCA loss.

Fig. 5 Neat peak gain for TE-polarized light as a function of (a) tensile strain and doping concentration with an injected carrier density of $4 \times 10^{19} {cm}^{- 3}$ , (b) tensile strain and injected carrier density with a doping concentration of $1 \times 10^{19} {cm}^{- 3}$ , (c) doping concentration and injected carrier density with a strain value of 4%.

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Figure 5(b) shows the net peak gain for TE-polarized light as a function of tensile strain and injected carrier density with a doping concentration of $1 \times 10^{19} {cm}^{- 3}$ . Figure 5(c) shows the net peak gain for TE-polarized light as a function of doping concentration and injected carrier density with a strain value of 4%. The edges of the white regions in both two contour maps represent the transparency carrier density where the optical gain and FCA loss cancel out. As can be seen, the transparency carrier density declines by 10x when the uniaxial tensile strain increases from 1.2% to 5%. While the effect of n-type doping is not as remarkable as that of the tensile strain but still can be utilized to reduce the transparency carrier density. It is also found that a net peak gain up to 2061 cm⁻¹ can be obtained if the uniaxial tensile stain, n-type doping concentration and injected carrier density reach 4%, $1 \times 10^{19} {cm}^{- 3}$ and $4 \times 10^{19} {cm}^{- 3}$ , respectively. Those values are realistic if we assume that the similar strain technique [30] and n-type doping [6] which have been experimentally demonstrated in bulk Ge can be applied in the Ge/GeSi QW.

4. Conclusion

We have estimated the direct gap optical gain of the [100] uniaxial tensile strained and n⁺-doped Ge/GeSi QW on the basis of 8-band k∙p method. In the simulation, FCA is considered to be the main loss and is calculated by classical Drude-Lorentz model. The influences of uniaxial tensile strain and n-type doping on the optical gain, FCA and net peak gain are analyzed. Interestingly, simulation results show that the TE-polarized light dominates in the uniaxial tensile strained QW which is in contrast to the case of biaxial tensile strained. The significant enhancement of net peak gain and reduction of transparency carrier density induced by uniaxial tensile strain are observed. As for the impact of n-type doping, our results show it is not as powerful as that of uniaxial tensile strain and can get weak at high strain level, suggesting that the tensile strain is the top-priority factor. Finally, we predict a net peak gain of 2061 cm⁻¹ for TE-polarized light under an injection level of $4 \times 10^{19} {cm}^{- 3}$ if a uniaxial tensile strain of 4% and a n-type doping concentration of $1 \times 10^{19} {cm}^{- 3}$ that have been experimentally demonstrated in bulk Ge can be introduced in the Ge/GeSi QW. This result indicates that the uniaxial tensile strained and n⁺-doped Ge/GeSi QW can be a promising candidate for Si-based light source.

Acknowledgment

This work is supported by the National Natural Science Foundation of China under Grant No. 61435004

References and links

1. B. Jalali and S. Fathpour, “Silicon photonics,” J. Lightwave Technol. 24(12), 4600–4615 (2006). [CrossRef]

2. R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics 4(8), 495–497 (2010). [CrossRef]

3. A. W. Fang, H. Park, O. Cohen, R. Jones, M. J. Paniccia, and J. E. Bowers, “Electrically pumped hybrid AlGaInAs-silicon evanescent laser,” Opt. Express 14(20), 9203–9210 (2006). [CrossRef] [PubMed]

4. S. Wirths, R. Geiger, N. von den Driesch, G. Mussler, T. Stoica, S. Mantl, Z. Ikonic, M. Luysberg, S. Chiussi, J. M. Hartmann, H. Sigg, J. Faist, D. Buca, and D. Grützmacher, “Lasing in direct-bandgap GeSn alloy grown on Si,” Nat. Photonics 9(2), 88–92 (2015). [CrossRef]

5. J. Liu, X. Sun, D. Pan, X. Wang, L. C. Kimerling, T. L. Koch, and J. Michel, “Tensile-strained, n-type Ge as a gain medium for monolithic laser integration on Si,” Opt. Express 15(18), 11272–11277 (2007). [CrossRef] [PubMed]

6. J. Liu, X. Sun, R. Camacho-Aguilera, L. C. Kimerling, and J. Michel, “Ge-on-Si laser operating at room temperature,” Opt. Lett. 35(5), 679–681 (2010). [CrossRef] [PubMed]

7. R. E. Camacho-Aguilera, Y. Cai, N. Patel, J. T. Bessette, M. Romagnoli, L. C. Kimerling, and J. Michel, “An electrically pumped germanium laser,” Opt. Express 20(10), 11316–11320 (2012). [CrossRef] [PubMed]

8. A. Ghrib, M. de Kersauson, M. El Kurdi, R. Jakomin, G. Beaudoin, S. Sauvage, G. Fishman, G. Ndong, M. Chaigneau, R. Ossikovski, I. Sagnes, and P. Boucaud, “Control of tensile strain in germanium waveguides through silicon nitride layers,” Appl. Phys. Lett. 100(20), 201104 (2012). [CrossRef]

9. M. J. Süess, R. Geiger, R. A. Minamisawa, G. Schiefler, J. Frigerio, D. Chrastina, G. Isella, R. Spolenak, J. Faist, and H. Sigg, “Analysis of enhanced light emission from highly strained germanium microbridges,” Nat. Photonics 7(6), 466–472 (2013). [CrossRef]

10. R. W. Millar, K. Gallacher, J. Frigerio, A. Ballabio, A. Bashir, I. MacLaren, G. Isella, and D. J. Paul, “Analysis of Ge micro-cavities with in-plane tensile strains above 2%,” Opt. Express 24(5), 4365–4374 (2016). [CrossRef]

11. D. S. Sukhdeo, D. Nam, J.-H. Kang, M. L. Brongersma, and K. C. Saraswat, “Direct bandgap germanium-on-silicon inferred from 5.7% <100> uniaxial tensile strain [Invited],” Photonics Res. 2(3), A8–A13 (2014). [CrossRef]

12. A. Yariv, “Scaling laws and minimum threshold currents for quantum-confined semiconductor lasers,” Appl. Phys. Lett. 53(12), 1033–1035 (1988). [CrossRef]

13. H. K. Choi and C. A. Wang, “InGaAs/AlGaAs strained single quantum well diode lasers with extremely low threshold current density and high efficiency,” Appl. Phys. Lett. 57(4), 321–323 (1990). [CrossRef]

14. Y. Cai, Z. Han, X. Wang, R. Camacho-Aguilera, L. C. Kimerling, J. Michel, and J. Liu, “Analysis of threshold current behavior for bulk and quantum-well germanium laser structures,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1901009 (2013). [CrossRef]

15. W. J. Fan, “Tensile-strain and doping enhanced direct bandgap optical transition of n+ doped Ge/GeSi quantum wells,” J. Appl. Phys. 114(18), 183106 (2013).

16. M. G. Burt, “Fundamentals of envelope function theory for electronic states and photonic modes in nanostructures,” J. Phys. Condens. Matter 11(9), R53–R83 (1999). [CrossRef]

17. T. B. Bahder, “Eight-band k∙p model of strained zinc-blende crystals,” Phys. Rev. B 41(17), 11992–12001 (1990). [CrossRef]

18. C. G. Van de Walle, and Van de Walle CG, “Band lineups and deformation potentials in the model-solid theory,” Phys. Rev. B Condens. Matter 39(3), 1871–1883 (1989). [CrossRef] [PubMed]

19. B. A. Foreman, “Elimination of spurious solutions from eight-band k∙p theory,” Phys. Rev. B 56(20), R12748 (1997). [CrossRef]

20. S. Birner, “Modeling of semiconductor nanostructures and semiconductor-electrolyte interfaces”, Ph.D. thesis, (Technical University Muenchen, Germany, 2011).

21. S.-W. Chang and S. L. Chuang, “Theory of optical gain of Ge-Si_xGe_ySn_1−x−y quantum-well lasers,” IEEE J. Quantum Electron. 43(3), 249–256 (2007). [CrossRef]

22. S. L. Chuang, Physics of Photonic Devices, 2nd ed., (Wiley, 2009).

23. G.-E. Chang, S.-W. Chang, and S. L. Chuang, “Theory for n-type doped, tensile-strained Ge-Si(_x)Ge(_y)Sn_1-x-y quantum-well lasers at telecom wavelength,” Opt. Express 17(14), 11246–11258 (2009). [CrossRef] [PubMed]

24. C. Hilsum, “Simple empirical relationship between mobility and carrier concentration,” Electron. Lett. 10(13), 259–260 (1974). [CrossRef]

25. B. G. Streetman, Solid State Electronic Devices, 4th ed. (Prentice-Hall, 1995).

26. S. M. Sze and J. C. Irvin, “Resistivity, mobility and impurity levels in GaAs, Ge, and Si at 300°K,” Solid-State Electron. 11(6), 599–602 (1968). [CrossRef]

27. M. Lundstrom, Fundamentals of Carrier Transport, 2nd ed. (Cambridge University, 2000).

28. M. M. Rieger and P. Vogl, “Electronic-band parameters in strained Si_1-xGe_x alloys on Si_1-yGe_y substrates,” Phys. Rev. B Condens. Matter 48(19), 14276–14287 (1993). [CrossRef] [PubMed]

29. S. W. Corzine, R. H. Yan, and L. A. Coldren, “Theoretical gain in strained InGaAs/AlGaAs quantum wells including valence-band mixing effects,” Appl. Phys. Lett. 57(26), 2835–2837 (1990). [CrossRef]

30. V. Reboud, A. Gassenq, G. Osvaldo Dias, K. Guilloy, J. M. Escalante, S. Tardif, N. Pauc, J.-M. Hartmann, J. Widiez, E. Gomez, E. Bellet Amalric, D. Fowler, D. Rouchon, I. Duchemin, Y. M. Niquet, F. Rieutord, J. Faist, R. Geiger, T. Zabel, E. Marin, H. Sigg, A. Chelnokov, and V. Calvo, “Ultra-high amplified strains in 200-mm optical germanium-on-insulator (GeOI) substrates: towards CMOS-compatible Ge lasers,” Proc. SPIE 9752, 97520F (2016). [CrossRef]

Parameters	Symbol	Ge	Si
Deformation potentials (eV)
	$a_{c}$	−8.24	1.98
	$a_{L}$	−1.54	−0.66
	$a_{v}$	1.24	2.46
	$b$	−2.9	−2.1
Kane parameters
	L	−31.34^b	−6.69^b
	M	−5.90^b	−4.62^b
	N	−34.14^b	−8.56^b
Energy parameter (eV)	$E_{p}$	26.3	21.6
Average valence band energy (eV)	$E_{a v}$	−6.35^c	−7.03^c
Band gaps (eV)
	$E_{g}^{Γ}$	0.7985	4.185
	$E_{g}^{L}$	0.664	1.65
Effective masses
	$m_{c} / m_{0}$	0.038	0.528
	$m_{t, L} / m_{0}$	0.0807	0.133
	$m_{l, L} / m_{0}$	1.57	1.659
Spin-orbit split off energy (eV)	$Δ$	0.29	0.044
Elastic stiffness constants (10Gpa)
	$C_{11}$	12.853	16.577
	$C_{12}$	4.826	6.393
Refractive index	$n_{r}$	4.02	3.4

Theoretical analysis of optical gain in uniaxial tensile strained and n⁺-doped Ge/GeSi quantum well

Abstract

1. Introduction

2. Theoretical Model

2.1 Strained band structure

2.2 Optical gain and FCA

2.3 Fermi level and Carrier density

3. Simulation results and discussion

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (5)

Tables (1)

Equations (32)

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