InGaAsPElectroOpticalModel

An ElectroOpticalModel is a PhotonicPolygon-specific object that will calculate the change in optical permitivity, \(\Delta \epsilon\), locally, due to an applied electric field to your structure. These models are not meant to be used as a stand-alone package, but to be fed into a PhotonicPolygon, which in turn ElectroOpticalSimulator will make use of to calculate the full electro-optical response.

InGaAsP-based modulators are one of the most mature technologies in integrated photonics, which translates into well established models that predict the electro-optic response of these alloys. All of the models below are specific to alloys lattice matched to InP, where the \(x\) can be predicted as:

\[x = \frac{y}{2.2020 - 0.659y}\]

In order to use many of the models, we first need to be able to predict some physical properties for all alloys. We will go over them one my one.

Effective mass

The effective masses are calculated according to Fiedler and Schlachetzki [6]:

\[\begin{split}\begin{align} m_e &= \left( 0.07 - 0.0308y \right)m_0 \\ m_{hh} &= \left( 0.6 - 0.218y + 0.07y^2 \right)m_0 \\ m_{lh} &= \left( 0.12 - 0.078y + 0.002y^2 \right)m_0 \end{align}\end{split}\]

where \(m_0\) is the vacuum electron mass.

Carrier effective density of states

The effective density of states are calculated from Bennett et al. [4]. For electrons we have:

\[N_C = 2 \left( \frac{m_e k_b T}{2\pi \hbar^2}\right)^{3/2}\]

and for holes:

\[N_v = 2 \left( \frac{m_h k_b T}{2\pi \hbar^2}\right)^{3/2}\]

with:

\[m_h = \left(m_{hh}^{1.5} + m_{lh}^{1.5}\right)^{2/3}\]

where \(k_b\) is the boltzman constant, \(T\) is the temperature and \(\hbar\) is the planck constant.

Spin orbit splitting

The spin orbit splitting is taken from Fiedler and Schlachetzki [6]:

\[\Delta_{so} = 0.119 + 0.30y - 0.107y^2 \hspace{1cm} [eV]\]

Bandgap narrowing

The bandgap narrowing is a well known effect in III-V semiconductors, where it is observed that the bandgap of heavily doped materials shrinks. Here we follow the work of Jain et al. [7] where the band gap narrowing (BGN) is estimated via:

\[\Delta E_g^{BGN} = A \times N^{1/3} + B \times N^{1/4} + C \times N^{1/2}\]

where \(N\) is the carrier concentration and the constants \(A\), \(B\) and \(C\) have been empirically found for GaAs, GaP, InP and InAs, allowing for interpolation for InGaAsP alloys via Fiedler and Schlachetzki [6]:

\[Q(x,y) = xyQ_{GaAs} + x(1-y)Q_{GaP} + y(1-x)Q_{InAs} + (1-x)(1-y)Q_{InP}\]

The constants are:

P-doped material

Material	A \(\times 10^{-9}\)	B \(\times 10^{-7}\)	C \(\times 10^{-12}\)
GaAs	9.83	3.90	3.90
InAs	8.34	2.91	4.53
InP	10.3	4.43	3.38
GaP	12.7	5.85	3.90

N-doped material

Material	A \(\times 10^{-9}\)	B \(\times 10^{-7}\)	C \(\times 10^{-12}\)
GaAs	16.5	2.39	91.4
InAs	14.0	1.97	57.9
InP	17.2	2.62	98.4
GaP	10.7	3.45	9.97

Charge carrier mobility

Despite most commercial software for the simulation of charge transport in semiconductors have a good material database, III-V (in particular InGaAsP) materials seldom have a correct value for mobility, and in particular doping dependent mobility values. For that reason we will follow the empirical model of Sotoodeh et al. [1] for low field mobility:

\[\mu (N, T) = \mu_{min} + \frac{\mu_{max}(300K)\left(\frac{300K}{T}\right)^{\theta_1}-\mu_{min}}{1+\left(\frac{N}{N_{ref}(300K)\left(\frac{T}{300K}\right)^{\theta_2}}\right)^\lambda}\]

Where the values of \(\lambda_n\), \(\theta_{n2}\), \(\log_10{N_{n,ref}(300K)}\), \(\lambda_p\), \(\mu_{p, min}\), \(\theta_{p2}\) and \(\log_10{N_{p,ref}(300K)}\) for \(In_{1-x}Ga_{x}As_{y}P_{1-y}\) are found via:

\[Q(In_{1-x}Ga_{x}As_{y}P_{1-y}) = yQ(In_{1-x}Ga_{x}As) + (1-y)Q(In_{1-x}Ga_{x}P)\]

The \(\mu_{n,max}(300K)\), \(\mu_{n,min}(300K)\), \(\mu_{p,max}(300K)\), \(\theta_{n1}\), \(\theta_{p1}\) are found via:

\[Q(In_{1-x}Ga_{x}As_{y}P_{1-y}) = \frac{yQ(In_{1-x}Ga_{x}As) + (1-y)Q(In_{1-x}Ga_{x}P)}{1+my(1-y)}\]

with:

	\(\mu_{n,max}(300K)\)	\(\mu_{n,min}(300K)\)	\(\mu_{p,max}(300K)\)	\(\theta_{n1}\)	\(\theta_{p1}\)
m	6	6	6	1	1

Therefore, we just need to find the values for InGaAs and InGaP. To do so, we do a quadratic interpolation between each of the parameters, unless there are only two data points, in which case we do a linear interpolation. The values used are layed down in the tables below:

InGaP parameter list

x	0	0.51	1
\(\mu_{n, max}\)	5200	4300	152
\(\mu_{n, min}\)	400	400	10
\(N_{n, ref}\)	log10(3e17)	log10(2e16)	log10(4.4e18)
\(\lambda_{n}\)	0.47	0.70	0.80
\(\theta_{n,1}\)	2.0	1.66	1.60
\(\theta_{n, 2}\)	3.25		0.71
\(\mu_{p, max}\)	170	150	147
\(\mu_{p, min}\)	10	15	10
\(N_{p, ref}\)	log10(4.87e17)	log10(1.5e17)	log10(1.0e18)
\(\lambda_{p}\)	0.62	0.80	0.85
\(\theta_{p,1}\)	2.0	2.0	1.98
\(\theta_{p, 2}\)	3.0		0.0

InGaAs parameter list

x	0	0.47	1
\(\mu_{n, max}\)	34000	14000	9400
\(\mu_{n, min}\)	1000	300	500
\(N_{n, ref}\)	log10(1.1e18)	log10(1.3e17)	log10(6.0e16)
\(\lambda_{n}\)	0.32	0.48	0.394
\(\theta_{n,1}\)	1.57	1.59	2.1
\(\theta_{n, 2}\)	3.0	3.68	3.0
\(\mu_{p, max}\)	530	320	491.5
\(\mu_{p, min}\)	20	10	20
\(N_{p, ref}\)	log10(1.1e17)	log10(4.9e17)	log10(1.48e17)
\(\lambda_{p}\)	0.46	0.403	0.38
\(\theta_{p,1}\)	2.3	1.59	2.2
\(\theta_{p, 2}\)	3.0	3.0	3.0

Refractive index

The optical refractive index above the bandgap absorption edge is calculated via the modified single oscillator model through Fiedler and Schlachetzki [6]:

\[n^2 = 1+\frac{E_d}{E_0} + E_d \frac{E_d (\hbar \omega)^2}{E_0^3} + \frac{E_d}{2E_0^3(E_0^2 - E_g^2)} (\hbar \omega)^4ln\left[\frac{2E_0^2 - E_g^2 - (\hbar \omega)^2}{E_g^2 - (\hbar \omega)^2}\right]\]

where \(E_0\) and \(E_d\) are given by:

\[E_0 = 3.391 - 1.652y + 0.863y^2 - 0.123y^3\]

\[E_d = 28.91 - 9.278y + 5.626y^2\]

Whereas if any information above the bandgap is needed, we resort to the following formula from Seifert and Runge [10]:

\[\begin{split}\begin{aligned} n^* - 1 &\approx \frac{a}{b - (E + i\Gamma)^*} + \frac{A\sqrt{R}}{(E + i\Gamma)^*} \Bigg\{ \ln \frac{E_z^*}{E_z^* - (E + i\Gamma)^*} + \pi \Bigg[ 2 \cot \left( \pi \sqrt{\frac{R}{E_z^*}} \right) \\ &\quad - \cot \left( \pi \sqrt{\frac{R}{E_z^* - (E + i\Gamma)^*}} \right) - \cot \left( \pi \sqrt{\frac{R}{E_z^* + (E + i\Gamma)}} \right) \Bigg] \Bigg\} \end{aligned}\end{split}\]

where:

\[\begin{split}\begin{aligned} R &= -0.00115 + 0.0191E_g \\ \Gamma &= -0.000691 + 0.00433 E_g \\ A &= -0.0453 + 2.1103 E_g \\ a &= 72.32 + 12.78 E_g \\ b &= 4.84 + 4.66 E_g \\ c &= -0.015 + 0.02 E_g \\ d &= -0.178 + 1.042 E_g \end{aligned}\end{split}\]

Charge carrier electro-optic effects

Now that we have defined all the necessary physical properties needed to do our calculations, we can finally dive deeper into the electro-optic effects that take place. We will start with the effects that are governed by the electrons and holes. We will not consider in extreme details the physics behind each effect, for that we reccomend you follow the cited references. Instead, we will focus on the models that are employed and their possible shortcomings.

Band filling effect

When doping a semiconductor with additional donors, the fermi level gets closer to the conduction band, which ultimately can cause the occupation of energy levels above the minimum of the conduction band to be occupied. This means that the excitation of electrons will only occur at higher energy levels than the bandgap energy. This change in the absorption spectrum will translate into a change in the refractive index via the Kramers-Kronig relations. Here we follow the work of Bennett et al. [4], with the difference that the quasi-fermi levels will be taken from numerical calculations of the charge transport simulator.

The absorption due to the bandfilling effect is:

Warning

In the models below, we do not consider the bandgap narrowing as a separate effect. Instead we consider it in-built into every model that is dependent on the bandgap value by considering \(E_g \to E_g - \Delta E_{BGN}\)

\[\begin{split}\begin{aligned} \alpha(N,P,E) &= \frac{C_{hh}}{E} \sqrt{E - E_g - \Delta E_{BGN}} \left[ f_v(E_{ah}) - f_c(E_{bh}) - 1 \right] \\ &\quad + \frac{C_{lh}}{E} \sqrt{E - E_g - \Delta E_{BGN}} \left[ f_v(E_{al}) - f_c(E_{bl}) - 1 \right] \end{aligned}\end{split}\]

where \(f_v\) and \(f_c\) are the fermi-dirac distributions considering the fermi level as the quasi-fermi level for holes and electrons, respectively. \(E\) is the energy of the incoming photon. The change in absorption is then given by:

\[\Delta \alpha(N,P,E) = \alpha(N,P,E) - \alpha_0\]

where:

\[\alpha_0(N,P,E) = \frac{C_{hh}}{E}\sqrt{E - E_g - \Delta E_{BGN}} + \frac{C_{lh}}{E}\sqrt{E - E_g - \Delta E_{BGN}}\]

The constants \(C_{hh}\) and \(C_{lh}\) are adapted from a constant \(C = 4.4e12 cm^{-1} s^{-0.5}\) in Bennett et al. [4].

Warning

To determine the constants \(C_{hh}\) and \(C_{lh}\) for arbitrary concentrations of InGaAsP, we have adopted a new approach. We have noticed that in the work of Bennett et al. [4], the quaternary interpolation is done via the charge carrier masses alone. However, the parabolic absorption formula given by

\[\alpha_0(N,P,E) = \frac{C}{E}\sqrt{E - E_g}\]

tells us that the constant \(C\) can be written as Moss et al. [9]

\[C = \frac{2\pi e^2 (2m_r)^{3/2} |p_{m0}|^2}{3m_0^2 n_0 \epsilon_0 c h^3 \nu}\]

we see that the constant is dependent on the refractive index as well, which will have an impact and is not accounted by Bennet. At the same time, Vinchant et al. [12] states that a scaling factor is applied to the absorption spectrum so as to fit measurements at \(E_g + 0.2eV\), however, such scaling factor is not disclosed. For these reasons we have decided to employ a new model.

The reduced mass of the electron-heavy/light hole pairs is:

\[\begin{split}\begin{aligned} m_{r, ehh} &= \left(\frac{1}{m_e} + \frac{1}{m_hh}\right) \\ m_{r, elh} &= \left(\frac{1}{m_e} + \frac{1}{m_lh}\right) \end{aligned}\end{split}\]

The constants \(C_{hh}\) and \(C_{lh}\) are now calculated as:

\[C_{hh} = C \frac{m_{r,hh, InP}^{3/2}}{m_{r,hh, InP}^{3/2} + m_{r,lh, InP}^{3/2}} \left(\frac{m_{r, hh}}{m_{r,hh,InP}}\right)^{3/2} \frac{n_{0,InP}}{n_0}\]

\[C_{lh} = C \frac{m_{r,lh, InP}^{3/2}}{m_{r,hh, InP}^{3/2} + m_{r,lh, InP}^{3/2}} \left(\frac{m_{r, lh}}{m_{r,lh,InP}}\right)^{3/2} \frac{n_{0,InP}}{n_0}\]

The change in refractive index can now be calculated via the Kramers-Kronig integral. We have found, in accordance to the literature [12], that for all relevant alloys, the dependence with carrier concentration is linear. Therefore, we now emply the following slopes from \(\Delta n = m_n N + m_p P\):

Our model with bandgap narrowing

x	\(m_n\)	\(m_p\)
0.0	-4.935e-21	-1.339e-21
0.1	-6.053e-21	-1.582e-21
0.2	-7.589e-21	-1.884e-21
0.3	-9.738e-21	-2.266e-21
0.4	-12.96e-21	-2.758e-21
0.53	-20.48e-21	-3.6560e-21
0.6	-28.27e-21	-4.340e-21
0.7	-58.43e-21	-5.789e-21

Our model without bandgap narrowing

x	\(m_n\)	\(m_p\)
0.0	-3.520e-21	-1.258e-21
0.1	-4.284e-21	-1.479e-21
0.2	-5.276e-21	-1.752e-21
0.3	-6.597e-21	-2.092e-21
0.4	-8.410e-21	-2.523e-21
0.53	-12.01e-21	-3.289e-21
0.6	-21.67e-21	-4.982e-21
0.7	-35.52e-21	-6.936e-21

Vinchant model

x	\(m_n\)	\(m_p\)
-0.003	-5.625e-21	0
0.047	-6.192e-21	0
0.089	-6.637e-21	0
0.127	-7.122e-21	0
0.166	-7.446e-21	0
0.2	-7.932e-21	0
0.235	-8.255e-21	0
0.27	-8.984e-21	0
0.307	-9.469e-21	0
0.352	-10.85e-21	0
0.398	-12.06e-21	0
0.44	-13.35e-21	0
0.473	-14.65e-21	0
0.515	-16.43e-21	0
0.557	-18.86e-21	0
0.603	-21.93e-21	0
0.636	-25.33e-21	0
0.669	-28.25e-21	0
0.692	-31.48e-21	0
0.72	-36.34e-21	0
0.742	-40.87e-21	0
0.762	-47.35e-21	0
0.778	-52.85e-21	0
0.793	-58.35e-21	0

Plasma effect

In this model we only consider the plasma effect coming from n-dopants. The reason for this is that, in p-dopants, the inter-valence absorption mechanism is so much stronger that scattering effects in holes are negligible. We follow the model from Walukiewicz et al. [13], which is in accordance with Dumke et al. [5]. The change in absorption is given by:

\[\Delta \alpha(N, \lambda) = A_{\text{imp}}(N)\left(\frac{\lambda}{\lambda_0}\right)^{3.5} + A_{\text{op}}(N)\left(\frac{\lambda}{\lambda_0}\right)^{2.5} + A_{\text{ac}}(N)\left(\frac{\lambda}{\lambda_0}\right)^{1.5}\]

where \(\lambda_0 = 10\mu m\), and the constants \(A_{imp}\), \(A_{op}\), \(A_{ac}\) are interpolated from:

\(N \ (\text{cm}^{-3})\)	\(A_{\text{imp}}\)	\(A_{\text{op}}\)	\(A_{\text{ac}}\)
1.0e16	0.004	0.623	0.034
1.5e16	0.008	0.932	0.052
2.0e16	0.014	1.239	0.069
3.0e16	0.031	1.850	0.104
4.0e16	0.056	2.456	0.139
5.0e16	0.086	3.051	0.173
6.0e16	0.123	3.646	0.208
7.0e16	0.167	4.240	0.243
8.0e16	0.217	4.815	0.278
9.0e16	0.273	5.397	0.313
1.0e17	0.314	5.578	0.325
1.5e17	0.690	8.227	0.491
2.0e17	1.201	10.79	0.660
3.0e17	2.602	15.75	1.005
4.0e17	4.474	20.52	1.360
5.0e17	6.790	25.16	1.726
6.0e17	9.510	29.65	2.100
7.0e17	12.64	34.11	2.488
8.0e17	16.13	38.44	2.879
9.0e17	20.00	42.75	3.285
1.0e18	24.22	47.01	3.699
1.5e18	50.28	67.80	5.912
2.0e18	93.91	88.02	8.354
3.0e18	170.3	127.0	13.87
4.0e18	276.7	164.1	20.12
5.0e18	396.1	199.6	26.98
6.0e18	522.8	233.4	34.34

As for the change in refractive index, we consider the simple Lorentz model:

\[\Delta n(E) = -\frac{1}{2} \frac{N e^2}{m_e \epsilon_0 (E/\hbar)^2 n_0}\]

Intervalence absorption

The bandstructure of InGaAsP has two valence bands (light and heavy holes) and a third band due to the spin-orbit coupling. When a photon hits such a material which is hevily P-doped, it can cause an electron in the spin-orbit valence band to jump to one of the hole bands. This effect can be quite intense in III-V at 1550nm because the \(\Delta_{so} < 0.8eV\). In IngaAsP materials, this effect has been thorougly characterized, and here we employ the model from Weber [14]:

\[\Delta \alpha(E,N) = 4.252\times 10^{-20} \exp \left(-3.657 E\right) P\]

\[\Delta n(E,N) = -\frac{\hbar c \alpha_0}{\pi} \frac{1}{2E}\left(e^{-bE}E_i(bE) + e^{bE}E_1(bE)\right) P\]

where \(\alpha_0 = 4.252\times 10^-20 m^2\) and \(b = 3.657 eV^{-1}\). \(E_i\) and \(E_1\) are the exponential integrals.

Field effects

In InGaAsP materials you have a linear and a quadratic field effect.

Pockels effect

The pockels effect follows the works of Adachi and Oe [1] and Adachi and Oe [2]. The electro-optic coefficient \(r_{41}\) is given by the sum of the free and piezoelectric contributions. Namely:

Note

For simplicity, here we will not write the \(\Delta E_{BGN}\) contribution, but consider it implied.

\[r_{41, free} = -\frac{1}{\epsilon_r^2} \left(E_0 g\left(\frac{E}{E_g}\right) + F_0\right)\]

\[\begin{split}\begin{aligned} r_{41, \text{piezo}} &= - \frac{1}{\epsilon_r^2} \Bigg( C \bigg( -g\left(\frac{E}{E_g}\right) + 4 \frac{E_g}{\Delta_{\text{so}}} \bigg[ f\left(\frac{E}{E_g}\right) \\ &\quad - \left(\frac{E_g}{E_g + \Delta_{\text{so}}}\right)^{1.5} f\left(\frac{E}{E_g + \Delta_{\text{so}}}\right) \bigg] \bigg) + D \Bigg) e_{14} \end{aligned}\end{split}\]

where

\[g(\chi) = \frac{1}{\chi^2} \left( 2 - (1 + \chi)^{-0.5} - (1 - \chi)^{-0.5} \right)\]

\[f(\chi) = \frac{1}{\chi^2} \left( 2 - (1 + \chi)^{0.5} - (1 - \chi)^{0.5} \right)\]

The constants \(E_0\), \(F_0\), \(C\), \(D\) are found via interpolation as:

\[Q(x,y) = xyQ_{GaAs} + x(1-y)Q_{GaP} + y(1-x)Q_{InAs} + (1-x)(1-y)Q_{InP}\]

and the constants for the binaries are found in the table below:

Constants for materials :align: center :header-rows: 1

Constant	InP	GaP	GaAs	InAs
E0 (m V^-1)	-42.06e-12	-83.31e-12	-71.48e-12	-30.23e-12
F0 (m V^-1)	91.32e-12	16.60e-12	123.16e-12	197.88e-12
C (m^2 N^-1)	-0.36e-10	-0.06e-10	-0.21e-10	-1.48e-10
D (m^2 N^-1)	2.60e-10	1.92e-10	2.12e-10	2.32e-10

Kerr effect

The kerr effect is due to the Franz-Keldysh effect, which causes a ripple in the absorption band-edge, which translates to a quadratic change in the refractive index. Here, we follow the work of Adachi and Oe [3] and Maat [8]. Contrary to other materials, like ferroelectric materials like lithium niobate, the quadratic response to an electric field is complex. This is a consequence of the Franz Keldysh effect. The absorption for TE and TM polarization is given by:

\[\Delta \alpha (E)_{TE,TM} = A_{TE,TM}\lambda \frac{|E_ext|}{E_g - E} 10^{-B_{TE,TM}\frac{(E_g-E)^{3/2}}{|E_{ext}|}}\]

As for the \(S_{11}\) and \(S_{12}\), they are given by:

\[\begin{split}\begin{aligned} S_{11} &= C_{TE} \frac{E^2}{\epsilon_r^2(E_g^2 - E^2)^2} \\ S_{12} &= C_{TM} \frac{E^2}{\epsilon_r^2(E_g^2 - E^2)^2} \end{aligned}\end{split}\]

and the constants \(C_{TE}\) and \(C_{TM}\):

\[\begin{split}\begin{aligned} C_{TE} &= -3.10\times 10^{-18} eV^2 m^2 V^{-2} \\ C_{TM} &= -5.60\times 10^{-18} eV^2 m^2 V^{-2} \end{aligned}\end{split}\]

class imodulator.ElectroOpticalModel.InGaAsPElectroOpticalModel(mup, mun, Ec, Ev, Efn, Efp, N, P, Efield, reg, T=300, y=0, bandgap_model='jain', BF_model='vinchant', growing_direction='z')

__init__(mup, mun, Ec, Ev, Efn, Efp, N, P, Efield, reg, T=300, y=0, bandgap_model='jain', BF_model='vinchant', growing_direction='z')

Base model for In_{1-x}Ga_{x}As_{y}P_{1-y}. It gives the changes in absorption and refractive index for charge carrier effects.

[!!] It expects Quantities for a single voltage The kwargs are to accomodate interoperability with other ElectroOpticalModels NOTE: the equations used for the modification of the bandstructure are applied assuming validity for compensated material, but that is not the case. It may be a source of error. Beware.

Ec: Conduction band energy in eV. Assumed to have shape (N) Ev: Valence band energy in eV. Assumed to have shape (N) Efn: Conduction quasi fermi level energy in eV. Assumed to have shape (N) Ep: Valence quasi fermi level energy in eV. Assumed to have shape (N) Efield: induced electric field. Assumed to have shape (N, 3). n: Electron concentration in cm^-3. Assumed to have shape (N) p: Hole concentration in cm^-3. Assumed to have shape (N) T: Temperature of operation in kelvin y: Concentration. Must be between 0 and 1. Assumed to have shape (Nx). growing direction: the axis of growth. Must be x,y or z. Default is z. BF_model: if ‘vinchant’ it will use the data calculated by Vinchant 1992 [7]. If ‘BGN’ it will use the result from the model while including BGN, and ‘no BGN’ will not include it.

References: 1) http://www.ioffe.ru/SVA/NSM/Semicond/InP/bandstr.html#Masses 2) Bennett, B.R., R.A. Soref, and J.A. Del Alamo. “Carrier-Induced Change in Refractive Index of InP, GaAs and InGaAsP.” IEEE Journal of Quantum Electronics 26, no. 1 (January 1990): 113–22. https://doi.org/10.1109/3.44924. 3) Adachi, Sadao. Properties of Group-IV, III-V and II-VI Semiconductors. Wiley Series in Materials for Electronic and Optoelectronic Applications. Chichester, West Sussex, England: John Wiley & Sons, Ltd, 2006. 4) Sze, S. M., and Kwok Kwok Ng. Physics of Semiconductor Devices. 3rd ed. Hoboken, N.J: Wiley-Interscience, 2007. 5) Fiedler, F., and A. Schlachetzki. “Optical Parameters of InP-Based Waveguides.” Solid-State Electronics 30, no. 1 (January 1987): 73–83. https://doi.org/10.1016/0038-1101(87)90032-3. 6) Moss, T. S., Geoffrey John Burrell, and Brian Ellis. Semiconductor Opto-Electronics. London: Butterworths, 1973. 7) Vinchant, J.-F., J.A. Cavailles, M. Erman, P. Jarry, and M. Renaud. “InP/GaInAsP Guided-Wave Phase Modulators Based on Carrier-Induced Effects: Theory and Experiment.” Journal of Lightwave Technology 10, no. 1 (January 1992): 63–70. https://doi.org/10.1109/50.108738.

get_BGN()

Return the bandgap narrowing energy of InGaAsP. Temperature dependence was taken from [1] page 121.

BGN is calculated for uncompensated InGaAsP.

if model=’jain’ it makes use of the model from [2]. The BGN adaptation for InGaAsP is based on the Vegard’s law taken from [5].

References: 1) Adachi, Sadao. Properties of Group-IV, III-V and II-VI Semiconductors. Wiley Series in Materials for Electronic and Optoelectronic Applications. Chichester, West Sussex, England: John Wiley & Sons, Ltd, 2006. 2) Jain, S. C., J. M. McGregor, and D. J. Roulston. “Band‐gap Narrowing in Novel III‐V Semiconductors.” Journal of Applied Physics 68, no. 7 (October 1990): 3747–49. https://doi.org/10.1063/1.346291. 3) Bennett, B.R., R.A. Soref, and J.A. Del Alamo. “Carrier-Induced Change in Refractive Index of InP, GaAs and InGaAsP.” IEEE Journal of Quantum Electronics 26, no. 1 (January 1990): 113–22. https://doi.org/10.1109/3.44924. 4) http://www.ioffe.ru/SVA/NSM/Semicond/InP 5) Fiedler, F., and A. Schlachetzki. “Optical Parameters of InP-Based Waveguides.” Solid-State Electronics 30, no. 1 (January 1987): 73–83. https://doi.org/10.1016/0038-1101(87)90032-3.

get_dalpha_BF()

Gives the change in absorption due to the bandfilling effect. The treatment if based on [1] but we neglect the quasi-fermi levels and use the fermi level numerically calculated instead.

Note that if the bandgap_model is not ‘none’ the BGN effect is taken into consideration within the bandfilling.

E must be a Pint quantity in energy using the parent object’s register!!

References: 1) Bennett, B.R., R.A. Soref, and J.A. Del Alamo. “Carrier-Induced Change in Refractive Index of InP, GaAs and InGaAsP.” IEEE Journal of Quantum Electronics 26, no. 1 (January 1990): 113–22. https://doi.org/10.1109/3.44924.

get_dalpha_iv(E=None)

Returns the intervalence absorption component. This is calculated from [1], eq. 16.

This formula may give worse results at low dopings and underpredict absorption. [2]

E: float. Pint quantity in energy using the parent object’s register.

References: 1) Weber, J.-P. “Optimization of the Carrier-Induced Effective Index Change in InGaAsP Waveguides-Application to Tunable Bragg Filters.” IEEE Journal of Quantum Electronics 30, no. 8 (August 1994): 1801–16. https://doi.org/10.1109/3.301645. 2) Casey, H. C., and P. L. Carter. “Variation of Intervalence Band Absorption with Hole Concentration in p -Type InP.” Applied Physics Letters 44, no. 1 (January 1, 1984): 82–83. https://doi.org/10.1063/1.94561.

get_dalpha_plasma()

Gives the change in absorption due to the plasma effect as reported in [1]. Simply put, the model is based on a second order perturbation theory and considers 3 scattering mechanisms: electron - optical phonon, electron - acoustical phonon and electron - ionized impurity. It assumes room temperature of 300K. It is limited to concentrations below 6e18 cm^-3 and above 1116cm^-3

E must be a Pint quantity in energy using the parent object’s register!!

The values predicted by the below formula are consistent with the experimental results from [2].

References: 1) Walukiewicz, W., J. Lagowski, L. Jastrzebski, P. Rava, M. Lichtensteiger, C. H. Gatos, and H. C. Gatos. “Electron Mobility and Free-Carrier Absorption in InP; Determination of the Compensation Ratio.” Journal of Applied Physics 51, no. 5 (May 1, 1980): 2659–68. https://doi.org/10.1063/1.327925.

2. Dumke, W. P., M. R. Lorenz, and G. D. Pettit. “Intra- and Interband Free-Carrier Absorption and the Fundamental Absorption Edge in n -Type InP.” Physical Review B 1, no. 12 (June 15, 1970): 4668–73. https://doi.org/10.1103/PhysRevB.1.4668.

get_dn_BF()

Returns the change in refractive index based only on the band filling effect based on the kramers kronig relations.

get_dn_iv(E=None)

Returns the intervalence absorption component. This is calculated from [1], eq. 17 and 18.

E must be a Pint quantity in energy using the parent object’s register!!

References: 1) Weber, J.-P. “Optimization of the Carrier-Induced Effective Index Change in InGaAsP Waveguides-Application to Tunable Bragg Filters.” IEEE Journal of Quantum Electronics 30, no. 8 (August 1994): 1801–16. https://doi.org/10.1109/3.301645.

get_dn_plasma(E=None)

Returns the change in refractive index based on the plasma effect. It makes use of [1], page 79, eq.5.2.14

E: float. Pint quantity in energy using the parent object’s register.

References: 1) Hunsperger, Robert G. Integrated Optics: Theory and Technology. 6th ed. Advanced Texts in Physics. New York London: Springer, 2009.

get_dperm(fractions=False)

This function returns the change in permitivity tensor

Return type:

ndarray

get_dperm_kerr()

Returns the change in refractive index owing to the Kerr effect.

E: energy of the field’s photons. Pint quantity Efield: Electric field components. It is assumed to have dimensions (Nx, 3)

References: [1] - Maat, Derk Hendrik Pieter. InP-Based Integrated MZI Switches for Optical Communication, 2001.

get_dperm_pockels()

Returns the change in refractive index owing to the pockels effect.

E: energy of the field’s photons. Pint quantity Efield: Electric field components. It is assumed to have dimensions (Nx, 3)

References: 1) Adachi, Sadao, and Kunishige Oe. “Internal Strain and Photoelastic Effects in Ga 1− x Al x As/GaAs and In 1− x Ga x As y P 1− y /InP Crystals.” Journal of Applied Physics 54, no. 11 (November 1983): 6620–27. https://doi.org/10.1063/1.331898. 2) Adachi, Sadao, and Kunishige Oe. “Linear Electro‐optic Effects in Zincblende‐type Semiconductors: Key Properties of InGaAsP Relevant to Device Design.” Journal of Applied Physics 56, no. 1 (July 1984): 74–80. https://doi.org/10.1063/1.333731.

get_eps_s()

Returns the relative permeability as a function of wavelength and adjusted for the different concentrations. It uses the modified single oscillator model and the conversion formulas from the single oscilator model formula as outlined in [1] (Appendix).

References: 1) Fiedler, F., and A. Schlachetzki. “Optical Parameters of InP-Based Waveguides.” Solid-State Electronics 30, no. 1 (January 1987): 73–83. https://doi.org/10.1016/0038-1101(87)90032-3.

get_mobility()

Returns the electron and hole mobility for In_{1-x}Ga_{x}As_{y}P_{1-y} based on an interpolation scheme reported by [1]

References: 1- Sotoodeh, M., A. H. Khalid, and A. A. Rezazadeh. “Empirical Low-Field Mobility Model for III–V Compounds Applicable in Device Simulation Codes.” Journal of Applied Physics 87, no. 6 (March 15, 2000): 2890–2900. https://doi.org/10.1063/1.372274.

References

[1]

Sadao Adachi and Kunishige Oe. Internal strain and photoelastic effects in Ga $_\textrm 1− \textit x $ Al $_\textrm \textit x $ As/GaAs and In $_\textrm 1− \textit x $ Ga $_\textrm \textit x $ As $_\textrm \textit y $ P $_\textrm 1− \textit y $ /InP crystals. Journal of Applied Physics, 54(11):6620–6627, November 1983. URL: https://pubs.aip.org/aip/jap/article/54/11/6620-6627/13142 (visited on 2023-05-04), doi:10.1063/1.331898.

[2]

Sadao Adachi and Kunishige Oe. Linear electro‐optic effects in zincblende‐type semiconductors: Key properties of InGaAsP relevant to device design. Journal of Applied Physics, 56(1):74–80, July 1984. URL: http://aip.scitation.org/doi/10.1063/1.333731 (visited on 2023-04-19), doi:10.1063/1.333731.

[3]

Sadao Adachi and Kunishige Oe. Quadratic electro‐optic (Kerr) effects in zincblende‐type semiconductors: Key properties of InGaAsP relevant to device design. Journal of Applied Physics, 56(5):1499–1504, September 1984. URL: https://pubs.aip.org/aip/jap/article/56/5/1499-1504/170597 (visited on 2023-05-15), doi:10.1063/1.334105.

[4] (1,2,3,4)

B.R. Bennett, R.A. Soref, and J.A. Del Alamo. Carrier-induced change in refractive index of InP, GaAs and InGaAsP. IEEE Journal of Quantum Electronics, 26(1):113–122, January 1990. URL: http://ieeexplore.ieee.org/document/44924/ (visited on 2023-04-18), doi:10.1109/3.44924.

[5]

W. P. Dumke, M. R. Lorenz, and G. D. Pettit. Intra- and Interband Free-Carrier Absorption and the Fundamental Absorption Edge in n -Type InP. Physical Review B, 1(12):4668–4673, June 1970. URL: https://link.aps.org/doi/10.1103/PhysRevB.1.4668 (visited on 2023-08-10), doi:10.1103/PhysRevB.1.4668.

[6] (1,2,3,4)

F. Fiedler and A. Schlachetzki. Optical parameters of InP-based waveguides. Solid-State Electronics, 30(1):73–83, January 1987. URL: https://linkinghub.elsevier.com/retrieve/pii/0038110187900323 (visited on 2023-05-16), doi:10.1016/0038-1101(87)90032-3.

[7]

S. C. Jain, J. M. McGregor, and D. J. Roulston. Band‐gap narrowing in novel III‐V semiconductors. Journal of Applied Physics, 68(7):3747–3749, October 1990. URL: http://aip.scitation.org/doi/10.1063/1.346291 (visited on 2023-04-19), doi:10.1063/1.346291.

[8]

missing publisher in maat_inp-based_2001

[9]

T. S. Moss, Geoffrey John Burrell, and Brian Ellis. Semiconductor opto-electronics. Butterworths, London, 1973. ISBN 978-0-408-70326-0.

[10]

Sten Seifert and Patrick Runge. Revised refractive index and absorption of In_1-xGa_xas_yp_1-y lattice-matched to InP in transparent and absorption IR-region. Optical Materials Express, 6(2):629, February 2016. URL: https://opg.optica.org/abstract.cfm?URI=ome-6-2-629 (visited on 2023-08-29), doi:10.1364/OME.6.000629.

[11]

M. Sotoodeh, A. H. Khalid, and A. A. Rezazadeh. Empirical low-field mobility model for III–V compounds applicable in device simulation codes. Journal of Applied Physics, 87(6):2890–2900, March 2000. URL: https://pubs.aip.org/aip/jap/article/87/6/2890-2900/489121 (visited on 2023-05-24), doi:10.1063/1.372274.

[12] (1,2)

J.-F. Vinchant, J.A. Cavailles, M. Erman, P. Jarry, and M. Renaud. InP/GaInAsP guided-wave phase modulators based on carrier-induced effects: theory and experiment. Journal of Lightwave Technology, 10(1):63–70, January 1992. URL: http://ieeexplore.ieee.org/document/108738/ (visited on 2023-04-06), doi:10.1109/50.108738.

[13]

W. Walukiewicz, J. Lagowski, L. Jastrzebski, P. Rava, M. Lichtensteiger, C. H. Gatos, and H. C. Gatos. Electron mobility and free-carrier absorption in InP; determination of the compensation ratio. Journal of Applied Physics, 51(5):2659–2668, May 1980. URL: https://pubs.aip.org/jap/article/51/5/2659/498806/Electron-mobility-and-free-carrier-absorption-in (visited on 2023-08-14), doi:10.1063/1.327925.

[14]

J.-P. Weber. Optimization of the carrier-induced effective index change in InGaAsP waveguides-application to tunable Bragg filters. IEEE Journal of Quantum Electronics, 30(8):1801–1816, August 1994. URL: http://ieeexplore.ieee.org/document/301645/ (visited on 2023-05-03), doi:10.1109/3.301645.