# Material Parameters of InGaAsP Quaternary Alloy Grown on InP

This is a continuation from the previous tutorial - lattice-mismatch effects on semiconductor epitaxial growth.

This tutorial describes the numerical values of various material parameters of InGaAsP quaternary alloy grown lattice-matched on InP.

A knowledge of the band-structure parameters, such as the band gap and the effective masses of the conduction and valence bands, is necessary to calculate the radiative and nonradiative Auger recombination rates.

The low-field minority carrier mobilities are also useful for calculating the diffusion coefficient that plays an important role in device performance.

Tables 4-1 to 4-3 list the band gap, the lattice constant, the effective masses, and the dielectric constant of the four binaries InP, GaAs, InAs, GaP that constitute the InGaAsP alloy.

## Band-Structure Parameters

The band structure of InGaAsP is similar to that of InP or GaAs with a direct gap at the $$\Gamma$$-point in $$\mathbf{k}$$-space.

A simplified model of the band structure is the four-band model of Kane shown in Figure 3-1 [refer to the radiative recombination in semiconductors tutorial].

Under photoexcitation, electrons and holes are created in the conduction and valence bands respectively; when they recombine, light at the band-gap energy $$E_\text{g}$$ is emitted.

This technique is called photoluminescence, and has been used to measure the direct band gap of $$\text{In}_{1-x}\text{Ga}_x\text{As}_y\text{P}_{1-y}$$ lattice-matched to InP.

Figure 4-17 shows the measured data.

From the data, the room-temperature band gap of $$\text{In}_{1-x}\text{Ga}_x\text{As}_y\text{P}_{1-y}$$ lattice-matched to InP is represented by the following expression:

$\tag{4-7-1}E_\text{g}(\text{in eV})=1.35-0.72y+0.12y^2\qquad\text{at 300 K}$

On the other hand, Nakajima et al. have suggested the following linear interpolations of the band gap from their measured data:

$\tag{4-7-2}E_\text{g}(\text{in Ev})=\begin{cases}0.74+0.61(1-y)\qquad\text{at 300 K}\\0.80+0.61(1-y)\qquad\text{at 77 K}\end{cases}$

The measurement of the spin-orbit splitting ($$\Delta$$) at the zone center has been carried out using electroreflectance. Figure 4-18 shows the measured data.

The electroreflectance technique has also been used to measure the direct band gap, and the results agree well with those obtained using the photoluminescence technique.

The temperature dependence of the band gap ($$\lambda\approx1.3$$ μm) of InGaAsP lattice-matched to InP has been obtained from measurements of the wavelength of a double-heterostructure injection laser as a function of the ambient temperature.

The result is shown in Figure 4-19. As temperature increases, the band gap decreases at a rate of 0.325 meV/K.

The conduction-band edge effective mass of $$\text{In}_{1-x}\text{Ga}_x\text{As}_y\text{P}_{1-y}$$ lattice-matched to InP has been measured usign techniques based on cyclotron resonance, magneto-absorption, and Shubnikov-de Hass effect, and the magneto-photon effect. Figure 4-20 shows the results of various measurements as compiled by Pearsall.

The results of these independent measurements agree very well. The following analytic expression has been proposed for the conduction-band-edge effective mass $$m_\text{c0}$$ based on cyclotron measurements:

$\tag{4-7-3}m_\text{c0}/m_0=0.080-0.039y$

where $$m_0$$ is the free-electron mass and $$y$$ is the $$\text{As}$$ mole fraction.

The valence-band effective masses of the InGaAsP alloy are not so well known because of difficulties in interpreting the experimental data.

The heavy-hole mass and the split-off-band mass are generally estimated from an interpolation of the measured binary values.

The light-hole effective mass of $$\text{In}_{1-x}\text{Ga}_x\text{As}_y\text{P}_{1-y}$$ has been obtained using the optical pumping technique. The measured data shown in Figure 4-21 agree well with the value obtained from an interpolation of the measured binary values.

The valence-band effective masses obtained from an interpolation of the binary values were used to calculate the radiative and nonradiative Auger recombination rates.

## Mobility

The low-field carrier mobility is an important parameter in the performance of semiconductor injection lasers. The diffusion coefficient $$D$$ is related to mobility $$\mu$$ by the relation $$D=\mu{k}_\text{B}T/q$$.

The diffusion of carriers can influence spatial-hole burning, multitransverse mode operation, and waveguiding in many semiconductor laser structures.

The low-field carrier mobility in InGaAsP alloys has been extensively studied. The phenomena related to hot-electron effects and high-field transport have also been discussed in several review articles.

In this tutorial we limit ourselves to a discussion of the measured low-field carrier mobility.

The room-temperature electron mobility of $$\text{In}_{1-x}\text{Ga}_x\text{As}_y\text{P}_{1-y}$$ has been extensively studied. Figure 4-22 shows the measured electron mobility as a function of the alloy composition in samples with a free-carrier concentration in the range of $$1\times10^{16}-4\times10^{16}\text{ cm}^{-3}$$.

The measurements were done on single layers of InGaAsP grown by LPE on (100)-oriented semi-insulating InP.

The mobility increases with decreasing temperature. The influence of various scattering mechanisms (such as ionized impurity scattering, optical phonon scattering, and alloy scattering) on the temperature dependence of the mobility has also been studied.

Hayes et al. have measured the hole mobility as a function of the mole fractions $$x$$ and $$y$$ in $$\text{In}_{1-x}\text{Ga}_x\text{As}_y\text{P}_{1-y}$$. Their experimental data are shown in Figure 4-23.

The samples used in the experiment were grown by LPE on (100)-oriented semi-insulating InP substrates. The shape of the mobility curve as a function of the alloy composition is principally governed by alloy scattering, which is large near the middle range of $$y$$ ($$y\approx0.5$$).

## Refractive Index

A knowledge of the refractive index is important for understanding the waveguiding properties of semiconductor lasers.

The InGaAsP double-heterostructure used for the fabrication of laser diodes generally have p-type InP and n-type InP as cladding layers. The wavelength dependence of the refractive index of InP has been studied by Petit and Turner.

For InGaAsP, several interpolation schemes using the measured binary values have been proposed and compared. Nahory and Pollack have suggested the following analytic form for the refractive index (at a wavelength corresponding to the band gap) of $$\text{In}_{1-x}\text{Ga}_x\text{As}_y\text{P}_{1-y}$$ lattice-matched to InP (for which $$x\approx0.45y$$):

$\tag{4-7-4}\mu(y)=3.4+0.256y-0.095y^2$

Values of the refractive indices have also been estimated using indirect methods such as far-field measurements and mode spacing of injection lasers. All of these values agree quite well.

Chandra et al. reported direct measurement of the refractive index of $$\text{In}_{1-x}\text{Ga}_x\text{As}_y\text{P}_{1-y}$$ lattice-matched to InP.

About 1-μm thick layers of InGaAsP were grown on InP by vapor-phase epitaxy. Transmission and reflection measurements were done on each sample as a function of the incident wavelength.

The spectral line width ($$\sim5\text{ Å}$$) of the light source was large enough to average out forward-backward wave interference effects occurring in thick substrates.

The refractive index was obtained from the observed modulation in reflection and transmission caused by interference of the light reflecting between the two surfaces of the epitaxial layer.

Figure 4-24 shows the measured refractive index as a function of wavelength for four lattice-matched compositions. The measured values agree well with the calculated values obtained using the interpolation method of Afromowitz.

The next tutorial introduces what is strained-layer epitaxy.