# Experimental Results of Radiative and Auger Recombination Coefficients in Semiconductor Lasers

This is a continuation from the previous tutorial - ** nonradiative recombination in semiconductors**.

In this tutorial we present the experimental results on the measurement of radiative and Auger coefficients (the parameters \(B\) and \(C\)) [refer to Equation (2-4-9) in the gain and stimulated emission in semiconductor lasers tutorial] for InGaAsP material.

Many of these measurement were made in an effort to understand the observed higher temperature dependence of the threshold current of InGaAsP lasers (\(\lambda\approx1.3\) μm and 1.55 μm) compared with that of AlGaAs lasers (\(\lambda\approx0.82\) μm). Measurements under conditions of both optical and electrical excitations have been reported.

We first discuss the measurements under optical excitation of InGaAsP from a mode-locked Nd:YAG (\(\lambda\approx1.06\) μm) laser. Figure 3-20 shows the experimental setup.

The mode-locked user pulses (~ 120 ps long) are focused to a diameter of ~ 8 μm onto a 1.3-μm InGaAsP-InP double heterostructure. The optical radiation is absorbed in the InGaAsP layer (\(\lambda\approx1.3\) μm) and not in the InP cladding layers.

From the measured transmission of the laser pulse and the spot size, the carrier density generated in the double-heterostructure well (InGaAsP layer) can be estimated.

The optically generated carriers can recombine through various mechanisms including radiative and Auger recombinations. The carrier lifetime is measured by measuring the decay of the photoluminescence signal at 1.3 μm.

Figure 3-21 shows the experimental data for carrier lifetime as a function of the carrier density.

In the absence of recombination at defects and surfaces, the carrier lifetime \(\tau\) may be written as

\[\tag{3-4-1}\frac{1}{\tau}=B(n)n+Cn^2\]

where \(B(n)\) is the radiative recombination coefficient, \(C\) is the Auger coefficient, and \(n\) is the carrier density.

For undoped InGaAsP material, it is found that \(B\approx2\times10^{-10}\text{ cm}^3/\text{s}\) for \(n=1\times10^{18}\text{ cm}^{-3}\) and \(C\approx2.3\times10^{-29}\text{ cm}^6/\text{s}\) when \(\lambda=1.3\) μm and \(T\approx300\text{ K}\).

Henry et al. have reported similar measurements of the carrier lifetime in InGaAsP-InP double heterostructures at \(\lambda=1.3\) μm. However, they varied the carrier concentration by doping the active layer (n- or p-type).

A lower-power mode-locked dye laser was used to generate a small number of excess carriers, and the decay rate of these carriers was obtained by measuring the luminescence (\(\lambda\approx1.3\) μm) decay time. The lifetime \(\tau\) is given by Equation (3-4-1) as before. The estimated Auger coefficient is \(1\times10^{-29}\text{ cm}^6/\text{s}\) from these measurements.

The radiative and Auger coefficients in InGaAsP have also been estimated by the technique of absorption bleaching using 1.06-μm pump and probe pulses.

In this experiment, a train of high-power pules (85-ps width, 8-W power) from a mode-locked Nd:YAG laser was focused onto the InGaAsP double-heterostructure sample. This produced a high density of carriers in the InGaAsP layer.

Further absorption of a probe beam, also at 1.06 μm, is reduced (or bleached) because the available states are already occupied. This is known as the Burstein effect.

Thus, by delaying the probe beam with respect to the pump beam, one can measure the change in absorption, from which the carrier decay time can be obtained.

The estimated radiative and Auger coefficients for 1.3-μm InGaAsP are \(1.2\times10^{-10}\text{ cm}^3/\text{s}\) and \(1.5-2.8\times10^{-29}\text{ cm}^6/\text{s}\), respectively.

In the CHHS Auger process (Figure 3-13), two heavy holes interact and create an electron in the conduction band and a hole in the split-off band. The electron and the split-off-band hole can recombine with the emission of photons at an energy \(E_\text{g}+\Delta\).

The CHHS Auger coefficient has been estimated by measuring the intensity of this high-energy (\(E_\text{g}+\Delta\)) luminescence. For 1.3-μm InGaAsP, \(E_\text{g}+\Delta\approx1.31\text{ eV}\).

Figure 3-22 shows the experimental results. Toward low energies, the onset of the band-to-band transition (at \(\sim0.96\text{ eV}\)) can be seen. The estimated CHHS Auger coefficient is \(\sim5\times10^{-29}\text{ cm}^6/\text{s}\) at \(300\text{ K}\).

Measurements of the radiative and Auger coefficients have also been reported using differential carrier-lifetime measurements. The differential carrier lifetime \(\tau_n\) is defined as

\[\tag{3-4-2}\frac{1}{\tau_n}=\frac{\partial{R}}{\partial{n}}\]

where \(R(n)\) is the total recombination rate [below threshold; see Equation (2-4-9) in the gain and stimulated emission in semiconductor lasers tutorial] given by

\[\tag{3-4-3}R(n)=A_\text{nr}n+B(n)n^2+Cn^3\]

The injected current \(I\) is related to the carrier density \(n\) by the relation

\[\tag{3-4-4}I=qVR(n)\]

where \(V\) is the active volume.

Since the spontaneous-emission rate is proportional to \(B(n)n^2\), Equations (3-4-3) and (3-4-4) show that in the presence of the Auger effect, the spontaneous-emission rate should vary sublinearly with increasing \(I\).

For \(C=0\) and a constant \(B\), the relation between \(\tau_n\) and \(I\) becomes

\[\tag{3-4-5}\frac{1}{\tau_n^2}=A_\text{nr}^2+\frac{4B}{qV}I\]

Thus for \(C=0\), \(\tau_n^{-2}\) varies linearly with the current \(I\).

In the presence of the third term in Equation (3-4-3), \(1/\tau_n^2\) versus \(I\) becomes superlinear with increasing \(I\).

Figure 3-23 shows an example each of the measured \(1/\tau_n^2\) versus \(I\) and the measured spontaneous-emission rate versus \(I\) for 1.3-μm InGaAsP lasers.

The coefficients \(A_\text{nr}\), \(B(n)\), and \(C\) in Equation (3-4-3) can be obtained by curve-fitting the data of Figure 3-23. An extensive analysis of such data shows that \(B(n)\) for 1.3-μm InGaAsP may be written as

\[\tag{3-4-6}B(n)=B_0-B_1n\]

with \(B_0=0.5-0.7\times10^{-10}\text{ cm}^3/\text{s}\) and \(B_1/B_0=1.7-2.2\times10^{-19}\text{ cm}^3\) at \(300\text{ K}\). The Auger coefficient \(C=1-2\times10^{-29}\text{ cm}^6/\text{s}\) for undoped InGaAsP and \(9\times10^{-29}\text{ cm}^6/\text{s}\) for p-type InGaAsP. Thompson arrived at an Auger coefficient of \(3\times10^{-29}\text{ cm}^6/\text{s}\) from an independent analysis of these data.

Uji et al. analyzed the output-power-saturation characteristics of 1.3-μm InGaAsP light-emitting diodes. They showed that by measuring the light-current characteristics of diodes of different active-region thicknesses, the effects of heterojunction leakage and Auger recombination can be separated.

They estimate an Auger coefficient in the range of \(3-8\times10^{-29}\text{ cm}^6/\text{s}\) and a radiative recombination coefficient in the range of \(0.7-1.5\times10^{-10}\text{ cm}^3/\text{s}\) at an injected carrier density of \(\sim1\times10^{18}\text{ cm}^{-3}\).

In an indirect method, the Auger coefficient in InGaAsP material is estimated by fitting the measured threshold current to stripe width for stripe-geometry lasers [see Figure 2-7 in the emission characteristics of semiconductor lasers tutorial]. The estimated values are \(3\times10^{-29}\text{ cm}^6/\text{s}\) and \(9\times10^{-29}\text{ cm}^6/\text{s}\) for 1.3-μm and 1.55-μm InGaAsP respectively.

The various experimental values for the radiative and Auger recombination rates in InGaAsP are listed in Table 3-2 and 3-3 respectively.

Note that different methods lead to considerable differences in the values. In spite of these large differences, it appears that the measured radiative coefficient is smaller (probably by 30%) than the calculated value and that the measured Auger coefficient is smaller (by a factor of 3 to 5) than the theoretical values obtained using the Kane model of the band structure. The Auger rate calculation using the Chelikowsky and Cohen model of the band structure agrees well with the experimental data.

The next tutorial discusses about ** how to estimate the threshold current density of a semiconductor laser**.