# Lead-Salt Semiconductor Lasers

This is a continuation from the previous tutorial - ** optoelectronic integrated circuits (OEICs)**.

## 1. Lead-Salt Lasers

In previous tutorials we paid particular attention to GaAs and InGaAsP semiconductor lasers that emit light in the wavelength range of 0.8-1.6 μm in view of their important applications in optical data storage and optical fiber communications systems.

However, longer-wavelength semiconductor lasers have also been of considerable interest and have found applications in molecular spectroscopy among other things, because of their large spectral tuning range.

Figure 1-3 [refer to the the history of semiconductor lasers tutorial] showed the wavelength range covered by various material systems. Of particular interest are the lead-salt semiconductor lasers covering a large spectral range (2.5-34 μm) in the far-infrared region.

The interest in lead-salt semiconductor lasers is generated by the wide range of emission wavelengths that can be obtained by varying the composition of the various constituents. Injection lasers of the ternary materials Pb_{1-x}Sn_{x}Te and PbS_{1-x}Se_{x} can provide emission wavelengths in the range of 3 μm to ~ 34 μm.

The band gap of these materials also varies considerably with temperature; as a result, a large degree of spectral tuning can be obtained simply by varying the operating temperature.

Lead-salt lasers have been used for various spectroscopic studies, including high-resolution spectroscopy and detection of low concentrations of pollutants.

## 2. Materials and Physical Properties

Lead-salt lasers, as they are commonly called, usually refer to the ternary alloys Pb_{1-x}Sn_{x}Te, PbS_{1-x}Se_{x}, and Pb_{1-x}Sn_{x}Se and the quaternary material system of Pb_{1-x}Sn_{x}Se_{y}Te_{1-y}.

The various II-VI compounds that make up these ternaries and quaternary are PbS, PbSe, PbTe, SnTe, and SnSe. Table 13-1 lists the band gaps, lattice constants, and effective masses of the first four of these binaries.

These four compounds crystallize in the cubic rock-salt structure (similar to NaCl) or in the orthorhombic structure. This allows the formation of Pb_{1-x}Sn_{x}Te and PbS_{1-x}Se_{x} for all values of the molar fraction x, whereas Pb_{1-x}Sn_{x}Se can be fabricated only for lead-rich compositions (x < 0.4).

Figure 13-1 shows the variation in band gap and lattice constant of four ternaries with respect to their composition. The figure assumes a linear extrapolation between the binary endpoints. The solid line is for the band gap, and the dashed line is for the lattice constant.

Most of the lead-salt lasers use Pb_{1-x}Sn_{x}Te, PbS_{1-x}Sn_{x} or Pb_{1-x}Sn_{x}Se as the active region. The ternary PbTe_{1-x}Se_{x}, which can also be grown without a large lattice mismatch, is generally used in cladding layers. Also, quaternary PbSnSeTe injection lasers on PbTe substrates have been grown using PbTeSe cladding layers.

Figure 13-1 shows that the band gap of Pb_{1-x}Sn_{x}Te and Pb_{1-x}Sn_{x}Se approaches zero at a certain composition. In principle, this should allow the emission of very long wavelengths using these materials.

Figure 13-2 shows the measured band gap of Pb_{1-x}Sn_{x}Te obtained from the laser-emission wavelength as a function of the composition x. For x < 0.35, the band gap at 12 K may be represented by

\[\tag{13-2-1}E_\text{g}(\text{in eV})=0.19-0.543x\]

Coherent emission from Pb_{1-x}Sn_{x}Te was first observed by Dimmock et al. Laser emission has been observed in Pb_{1-x}Sn_{x}Te up to 34 μm at 12 K.

The compositional dependence of the band gap of Pb_{1-x}Sn_{x}Se can be obtained from a measurement of the laser-emission wavelength. Figure 13-3 shows the measured values at 12 K.

Another quantity of interest for the fabrication of double-heterostructure laser diodes is the refractive index of the material. Figure 13-4 shows the data for the refractive indices of PbTe, PbSe, and PbS at 77 K, 300 K, and 373 K. Similar values have been reported for the refractive indices of the binaries.

The following analytic form for the high-frequency dielectric constant (\(\boldsymbol{\epsilon}_\infty\)) of Pb_{1-x}Sn_{x}Te and Pb_{1-x}Sn_{x}Se (for \(0\le{x}\le0.2\)) has been suggested:

\[\tag{13-2-2}\boldsymbol{\epsilon}_\infty=\begin{cases}35+50x\qquad\qquad(T=4.2\text{K}, \text{Pb}_{1-x}\text{Sn}_x\text{Te})\\(5.5+0.8x)^2\qquad(T=100\text{K}, \text{Pb}_{1-x}\text{Sn}_x\text{Se})\end{cases}\]

By varying the composition x, an index difference sufficient for dielectric waveguiding can be obtained between active and cladding layers.

## 3. Band Structure

The band structure of the lead-salt narrow-gap semiconductors is considerably different from that of III-V semiconductors, where the conduction-band minimum and the valence-band maximum occur at the \(\gamma\)-point and the bands are isotropic with spherical constant energy surfaces in \(\mathbf{k}\) space.

In lead-salt compounds there are four equivalent conduction-band minima and valence-band maxima at the L-points of \(\mathbf{k}\) space. The minimum of each conduction band occurs at the same point in \(\mathbf{k}\) space as the maximum of the corresponding valence band so that direct transitions (electron-hole recombinations that conserve momentum) are allowed.

However, the constant energy surfaces in \(\mathbf{k}\) space are prolate spheroids characterized by the longitudinal and transverse effective masses \(m_\text{l}\) and \(m_\text{t}\), respectively. The ratio \(m_\text{l}/m_\text{t}\) is ~ 10 for PbTe and Pb_{1-x}Sn_{x}Te and ~ 1.5-2 for Pb_{1-x}Sn_{x}Se and PbS_{1-x}Se_{x}. Furthermore, conduction and valence bands are mirror images of each other with almost identical effective masses.

We noted previously that with increasing Sn concentration the band gap of Pb_{1-x}Sn_{x}Te, as measured optically, approaches zero (seed Figure 13-2). However, tunneling experiments have shown that SnTe is a semiconductor with a band gap of 0.3 eV at 4.2 K.

These observations suggest the following model for the band structure of Pb_{1-x}Sn_{x}Te as its composition varies. The valence band edge of PbTe is an \(\text{L}_6^+\) state and the conduction-band edge is an \(\text{L}_6^-\) state. With increasing Sn composition, the band gap initially decreases as the states \(\text{L}_6^+\) and \(\text{L}_6^-\) approach each other, reaching zero at some composition; the gap then increases with the \(\text{L}_6^+\) state now forming the conduction band edge and the \(\text{L}_6^-\) state forming the valence-band edge. This phenomenon is referred to as ** band inversion**.

Figure 13-5 shows schematically the band inversion in Pb_{1-x}Sn_{x}Te.

The energy-wave-vector relation of the conduction and valence bands of PbTe-type narrow-gap semiconductors in the two-band approximation of Kane is given by

\[\tag{13-3-1}E=\pm(\frac{1}{4}E_\text{g}^2+p_\text{t}^2k_\text{t}^2+p_\text{l}^2k_\text{l}^2)^{1/2}-\frac{1}{2}E_\text{g}\]

where the \(+\) sign is for the conduction band and the \(-\) sign is for the valence band. The band gap is \(E_\text{g}\), \(k_\text{t}\) and \(k_\text{l}\) are the transverse and longitudinal components of the wave vector, and \(p_\text{t}^2\) and \(p_\text{l}^2\) are related to the effective masses \(m_\text{t}\) and \(m_\text{l}\).

Note that the effective masses of the valence and conduction bands are the same and that the two bands are mirror images of each other.

The following approximate dispersion relations near the band edge can be obtained from Equation (13-3-1) by expanding the square root in a binomial series:

\[\tag{13-3-2a}E_\text{c}(k)=\frac{\hbar^2}{2m_\text{l}}k_\text{l}^2+\frac{\hbar^2}{2m_\text{t}}k_\text{t}^2\]

\[\tag{13-3-2b}E_\text{v}(k)=\frac{\hbar^2}{2m_\text{l}}k_\text{l}^2+\frac{\hbar^2}{2m_\text{t}}k_\text{t}^2\]

where

\[m_\text{l,t}=\frac{\hbar^2E_\text{g}}{2p_\text{l,t}^2}\]

and \(E_\text{c}(k)\) and \(E_\text{v}(k)\) are measured from the respective conduction- and valence-band edges into the band.

The number of electrons per unit volume at a temperature \(T\) is given by

\[\tag{13-3-3}n=\frac{2\eta}{(2\pi)^3}\displaystyle\int\text{d}^3\mathbf{k}f_\text{c}(E_\text{c})\]

where

\[f_\text{c}(E_\text{c})=\frac{1}{1+\exp[(E_\text{c}-E_\text{fc})/k_\text{B}T]}\]

is the Fermi factor, \(\eta\) is the number of conduction-band valleys (\(\eta=4\) for PbTe), and \(E_\text{fc}\) is the Fermi energy. A similar equation holds for holes.

Using the approximate dispersion relation of Equation (13-3-2), the carrier density becomes

\[\tag{13-3-4}n=N_\text{c}\displaystyle\int{E^{1/2}}f_\text{c}(E_\text{c})\text{d}E_\text{c}\]

where

\[\tag{13-3-5}N_\text{c}=\eta{m_\text{l}}^{1/2}m_\text{t}(2\pi{k_\text{B}T/h^2})^{3/2}\]

Under the Boltzmann or nondegenerate approximation, an equation similar to Equation (3-2-30) [refer to the radiative recombination in semiconductors tutorial] is obtained for both electrons and holes, and the Fermi factors are

\[\tag{13-3-6}f_\text{c}(E_\text{c})=\frac{n}{N_\text{c}}\exp\left(\frac{-E_\text{c}}{k_\text{B}T}\right)\]

\[\tag{13-3-7}f_\text{v}(E_\text{v})=\frac{p}{N_\text{v}}\exp\left(\frac{-E_\text{v}}{k_\text{B}T}\right)\]

where \(N_\text{c}=N_\text{v}\) and is given by Equation (13-3-5).

## 4. Optical Gain

The optical gain can be calculated following the radiative recombination in semiconductors tutorial. However, the analysis should be modified to take into account the anisotropic nature of the energy bands governed by the different effective masses \(m_\text{l}\) and \(m_\text{t}\), in the dispersion relations (13-3-2).

The absorption coefficient between two discrete levels given by Eq. (3-2-21) [refer to the radiative recombination in semiconductors tutorial] is

\[\tag{13-4-1}\alpha(E)=\frac{q^2h}{2\epsilon_0m_0^2c\bar{\mu}E}|M_\text{if}|^2\delta(E_\text{i}-E_\text{f}-E)\]

where \(|M_\text{if}|^2\) is the momentum-matrix element and \(E\) is the photon energy. We assume that the band-to-band transitions conserve momentum (\(\mathbf{k}\)-conserving transitions). The matrix element \(|M_\text{if}|^2\) is then given by [see Equation (3-2-34) in the radiative recombination in semiconductors tutorial]

\[\tag{13-4-2}|M_\text{if}|^2=|M_0|^2\frac{(2\pi)^3}{V}\delta(\mathbf{k}_\text{c}-\mathbf{k}_\text{v})\]

where the \(\delta\) function assures conservation of momentum and \(V\) is the volume of the semiconductor. The matrix element \(|M_0|^2\) is related to the interband matrix elements \(p_\text{l}\) and \(p_\text{t}\) in Eq. (13-3-1).

We assume that the radiation is polarized along the x direction that makes an angle \(\theta\) with the crystallographic [111] direction. Then

\[\tag{13-4-3}|M_0|^2=(p_\text{l}^2\cos^2\theta+p_\text{t}^2\sin^2\theta)(m_0^2/\hbar^2)\]

When contributions from all four prolate spheroids are summed and the result is averaged over all directions, we obtain

\[\tag{13-4-4}|M_0|^2=4(\frac{1}{3}p_\text{l}^2+\frac{2}{3}p_\text{t}^2)\frac{m_0^2}{\hbar^2}\]

which is independent of the polarization as required by the symmetry of a cubic crystal.

The expression for \(\alpha(E)\) after summing over the available states is [see Eq. (3-2-40) in the radiative recombination in semiconductors tutorial]

\[\tag{13-4-5}\begin{align}\alpha(E)=&\frac{q^2h}{2\epsilon_0m_0^2c\bar{\mu}E}|M_0|^2\frac{(2\pi)^3}{V}2\left(\frac{V}{(2\pi)^3}\right)^2\frac{1}{V}\\&\times\displaystyle\iint(1-f_\text{c}-f_\text{v})\text{d}^3\mathbf{k}_\text{c}\text{d}^3\mathbf{k}_\text{v}\delta(\mathbf{k}_\text{c}-\mathbf{k}_\text{v})\delta(E_\text{i}-E_\text{f}-E)\end{align}\]

It can be simplified to become

\[\tag{13-4-6}\alpha(E)=\frac{q^2h|M_0|^2}{\epsilon_0m_0^2c\bar{\mu}E(2\pi)^3}[1-f_\text{c}(E_\text{c})-f_\text{v}(E_\text{v})]I\]

where

\[E_\text{c}=E_\text{v}=(E-E_\text{g})/2\]

\[\tag{13-4-7}I=\displaystyle\int\delta(E_\text{i}-E_\text{f}-E)\text{d}^3\mathbf{k}_\text{c}\]

The evaluation of the integral \(I\) is carried out using the dispersion relations given by Eq. (13-3-2).

The explicit dependence of \(E_\text{i}-E_\text{f}\) on \(\mathbf{k}_\text{c}=(k_x,k_y,k_z)\) is given by

\[\tag{13-4-8}E_\text{i}-E_\text{f}=(E_\text{g}-E)+2\left[\frac{\hbar^2}{2m_\text{l}}k_x^2+\frac{\hbar^2}{2m_\text{t}}(k_y^2+k_z^2)\right]\]

The integration in Eq. (13-4-7) should take into account the effective mass anisotropy arising from the prolate-spheroid shape of the constant energy surface in the \(\mathbf{k}_\text{c}\) space. A simple way is to define a new wave vector

\[\tag{13-4-9}\mathbf{k}_\text{c}'=(k_x',k_y,k_z)\qquad{k_x'}=k_x(m_\text{t}/m_\text{l})^{1/2}\]

so that the constant energy surface becomes spherical in the new integration variable:

\[\tag{13-4-10}E_\text{i}-E_\text{f}=(E_\text{g}-E)+2(\hbar^2/2m_\text{t})k_\text{c}'^2\]

The evaluation of \(I\) in Eq. (13-4-7) is now straightforward and the result is

\[\tag{13-4-11}\begin{align}I&=\left(\frac{m_\text{l}}{m_\text{t}}\right)^{1/2}\displaystyle\int\delta(E_\text{i}-E_\text{f})\text{d}^3\mathbf{k}_\text{c}'\\&=2\pi\left(\frac{m_\text{l}}{m_\text{t}}\right)^{1/2}\left(\frac{2m_\text{t}}{\hbar^2}\right)^{3/2}(E-E_\text{g})^{1/2}\end{align}\]

Equations (13-4-6) and (13-4-11) provide the expression for the absorption coefficient in lead-salt semiconductors.

Expressions for the total spontaneous emission rate can be derived in a manner similar to that discussed in the radiative recombination in semiconductors tutorial.

Anderson has calculated the gain-current relation in Pb_{1-x}Sn_{x}Te lasers using the nonparabolic terms in the energy-wave-vector relation for the conduction and valence bands. This model can be used to calculate the optical gain as a function of injected carrier density. Figure 13-6 shows the results for Pb_{0.82}Sn_{0.18}Te (\(\lambda\approx10\) μm) at 77 K.

The material is \(p\) type, and the four curves correspond to different doping levels. A lower injected-carrier concentration is needed to obtain the same gain if the active layer is heavily doped. This behavior is similar to that for InGaAsP shown in Fig. 3-10 [refer to the radiative recombination in semiconductors tutorial].

Figure 13-6 shows that the optical gain \(g\) varies approximately linearly with the injected carrier density; i.e.,

\[\tag{13-4-12}g=a(n-n_0)\]

where \(a\) is the gain coefficient and \(n_0\) is the injected carrier density at transparency.

The approximate validity of Eq. (13-4-12) for lead-salt lasers suggests that the results presented in the emission characteristics of semiconductor lasers tutorial can be applied to these lasers. In particular, the threshold current density can be obtained using Eqs. (2-6-2) and (2-6-3).

Note, however, that although the injected carrier density \(n_0\) needed for transparency (\(g=0\) in Figure 13-6) is lower for higher doping, the threshold current may not be lower. The latter is affected by free-carrier absorption as well as by nonradiative Auger recombination, both of which increase with increased doping.

## 5. Auger Recombination

The various Auger recombination mechanisms in a direct-gap semiconductor are discussed in the nonradiative recombination in semiconductors tutorial. In this section we calculate the Auger lifetime for the CCCH Auger process (see Figure 3-13 in the nonradiative recombination in semiconductors tutorial) using the band structure appropriate for lead-salt semiconductors.

In the nondegenerate approximation, the Auger rate \(R_\text{a}\) for spherical parabolic bands varies as [Equations (3-3-26) and (3-3-29) with \(m_\text{c0}=m_\text{c}\) in the nonradiative recombination in semiconductors tutorial]

\[\tag{13-5-1}R_\text{a}\propto{n^2p}\exp\left[\frac{-\mu{E_\text{g}}}{(1+\mu)k_\text{B}T}\right]\]

where \(\mu=m_\text{c}/m_\text{v}\), the ratio of the effective masses of the conduction and valence bands.

In PbTe semiconductors. four conduction and four valence bands exist at the center of the (111) zone faces. For Auger recombination involving electrons in the same pair of bands. Eq. (13-5-1) still holds for spheroidal energy surfaces so that the Auger rate for such a process is given by

\[\tag{13-5-2}R_\text{a}\propto{n^2p}\exp\left(\frac{-{E_\text{g}}}{2k_\text{B}T}\right)\]

In writing Eq. (13.5.2), we have assumed that p, = 1 because the conduction- and valence-band effective masses are nearly equal for PbTe semiconductors, as shown by Eq. (13-3-2).

Emtage showed that if the interacting electrons in the Auger process belong to different conduction-band valleys, the Auger rate is significantly enhanced and Eq. (13-5-2) takes the form

\[\tag{13-5-3}R_\text{a}\propto{n^2p}\exp\left(\frac{-r{E_\text{g}}}{2k_\text{B}T}\right)\]

where \(r=m_\text{t}/m_\text{l}\), the ratio of the transverse to longitudinal effective masses. The parameter \(r\approx0.1\) for PbTe and Pb_{1-x}Sn_{x}Te semiconductors.

We now generalize this result to take into account nonparabolicity.

Figure 13-7 diagrams the CCCH Auger process with electrons in different conduction-band valleys.

Electron 1 is in conduction band a and electron 2 is in conduction band b. After collision, electron 1 transfers to a state 1' in valence band a and electron 2 transfers to the excited state 2' in conduction band b.

When \(r\approx0.1\), the principal axes of bands a and b can be taken at right angles to each other, as shown in Fig. 13-7(ii). The electron energies in bands a and b in the parabolic approximation are

\[\tag{13-5-4a}E_\text{a}=\frac{\hbar^2}{2m_\text{l}}k_x^2+\frac{\hbar^2}{2m_\text{t}}(k_y^2+k_z^2)\]

\[\tag{13-5-4b}E_\text{b}=\frac{\hbar^2}{2m_\text{l}}k_y^2+\frac{\hbar^2}{2m_\text{t}}(k_x^2+k_z^2)\]

The holes are assumed to have the same effective mass as electrons.

The expression for the Auger rate per unit volume is given by [from Eq. (3-3-12) in the nonradiative recombination in semiconductors tutorial]

\[\tag{13-5-5}\begin{align}R_\text{a}=&\eta(\eta-1)\left(\frac{1}{(2\pi)^3}\right)^4\frac{8\pi}{\hbar}\displaystyle\iiiint|M_\text{if}|^2P(1,1',2,2')\\&\times\delta(E_\text{i}-E_\text{f})\delta(\mathbf{k}_1+\mathbf{k}_2-\mathbf{k}_1'-\mathbf{k}_2')\text{d}^3\mathbf{k}_1\text{d}^3\mathbf{k}_1'\text{d}^3\mathbf{k}_2\text{d}^3\mathbf{k}_2'\end{align}\]

where \(\eta\) is the number of equivalent conduction-band and valence-band valleys (\(\eta=4\) for Pb_{1-x}Sn_{x}Te) and \(|M_\text{if}|^2\) is the transition matrix element.

The factor \(\eta(\eta-1)\) arises from the number of combinations of two dissimilar pairs of alleys out of \(\eta\) pairs.

\(P(1,1',2,2')\) is the product of the occupancy probabilities of various states, as in Eq. (3-3-13) [refer to the nonradiative recombination in semiconductors tutorial]. Under the nondegenerate approximation, it can be approximated by Eq. (3-3-16); i.e.,

\[\tag{13-5-6}P(1,1',2,2')=\frac{n^2p}{N_\text{c}^2N_\text{v}}\exp\left(\frac{-E_2'+E_\text{g}}{k_\text{B}T}\right)\]

where we have used the conservation of energy (\(E_1+E_2=E_2'-E_\text{g}-E_1'\)); where \(N_\text{c}=N_\text{v}\) and is given by Eq. (13-3-5); and where \(n\) and \(p\) are the total number of electrons and holes, respectively.

As discussed in the nonradiative recombination in semiconductors tutorial, there is a minimum value \(E_\text{T}\) for which the Auger transition takes place. Since \(P(1,1',2,2')\) decreases rapidly for values of \(E_2'\) higher than this minimum value, most of the contribution to the integral in Eq. (13-5-5) arises from \(E_2'\approx{E}_\text{T}\).

The minimum value of \(E_2'\) occurs when all the momentum vectors are aligned in the same direction (for example, along the \(x\) axis) in such a way that \(\mathbf{k}_1\), \(\mathbf{k}_2\), and \(\mathbf{k}_1'\) are directed in the positive direction and \(\mathbf{k}_2'\) is directed in the negative direction, satisfying the following momentum conservation law:

\[\tag{13-5-7}k_{2x}'=k_{1x}+k_{2x}+k_{1x}'\]

Since states 1 and 1' are in band a and states 2 and 2' are in band b, the energy conservation requires that

\[\tag{13-5-8}\frac{\hbar^2{k_{1x}^2}}{2m_\text{l}}+\frac{\hbar^2k_{2x}^2}{2m_\text{t}}=\frac{\hbar^2}{2m_\text{t}(E_2')}k_{2x}'^2-\frac{\hbar^2}{2m_\text{l}}k_{1x}'^2-E_\text{g}\]

where \(m_\text{t}(E_2')\) is the effective mass at \(E_2'\) and is introduced to take into account nonparabolicity of the bands.

Dropping the suffix \(x\) and writing \(k_1=ak_1'\), \(k_2=bk_1'\), \(r=m_\text{t}/m_\text{l}\), and \(\mu_\text{c}=m_\text{t}(E_2')/m_\text{t}\), Equations (13-5-7) and (13-5-8) may be written as

\[\tag{13-5-9}k_2'=(a+b+1)k_1'\]

\[\tag{13-5-10}rk_1^2+k_2^2=\frac{1}{\mu_\text{c}}k_2'^2-rk_1'^2-k_\text{g}^2\]

with \(k_g^2=2m_\text{t}E_\text{g}/\hbar^2\).

Eliminating \(k_1'\), we obtain

\[\tag{13-5-11}k_2'^2=k_\text{g}^2\frac{(a+b+1)^2}{(a+b+1)^2\mu_\text{c}^{-1}-(ra^2+b^2+r)}\]

The magnitude \(k_2'\) can now be minimized with respect to \(a\) and \(b\). The values of \(a\) and \(b\) for minimum \(k_2'\) are

\[\tag{13-5-12}a=1\qquad{b=r}\]

and the minimum value is given by

\[\tag{13-5-13}k_2'^2=k_\text{g}^2\frac{(2+r)}{(2+r)\mu_\text{c}^{-1}-r}\]

Hence the minimum value of \(E_\text{T}\) or \(E_2'\), which is also the energy of electron 2' for the most probable transition, is given by

\[\tag{13-5-14}E_2'=E_\text{T}=\frac{1}{\mu_\text{c}}\frac{\hbar^2k_2'^2}{2m_\text{t}}=\frac{(2+r)E_\text{g}}{(2+r)-r\mu_\text{c}}\]

We can define the energy difference \(\Delta{E}\) as

\[\tag{13-5-15}\Delta{E}=E_2'-E_\text{g}=\frac{r\mu_\text{c}}{(2+r)-r\mu_\text{c}}E_\text{g}\]

From Eqs. (13-5-5) and (13-5-6), it follows that the Auger rate varies approximately as

\[\tag{13-5-16}R_\text{a}\propto{n^2p}\exp\left(\frac{-\Delta{E}}{k_\text{B}T}\right)\]

with \(\Delta{E}\) given by (13-5-15).

For parabolic bands, \(\mu_\text{c}=1\), and Eq. (13-5-16) reduces to Eq. (13-5-3) such that \(\Delta{E}=rE_\text{g}/2\). For spherical parabolic bands, \(\mu_\text{c}=1\), \(r=1\), and \(\Delta{E}=E_\text{g}/2\).

These results are valid when \(r\ll1\), which is the case for Pb_{1-x}Sn_{x}Te (\(r\approx0.1\)).

Equation (13-5-16) shows that, under the nondegenerate approximation, the nonradiative Auger recombination rate increases with increasing temperature and is larger for low band-gap semiconductors.

Since \(\Delta{E}\) appears in the exponent of Eq. (13-5-16), nonparabolicity of the bands [determined by \(\mu_\text{c}\) in Eq. (13-5-15)] can significantly alter the calculated Auger rate. Ziep et al. have calculated the Auger rate in PbSnTe using the Fermi statistics.

An exact calculation of the Auger rates in lead-salt semiconductors is not possible because the band structure away from the band edge is not known accurately. Moreover, only order-of-magnitude estimates of the matrix element are available.

Rosman and Katzir have estimated the CCCH Auger lifetime in PbSnTe using Kane's two-band model [Eq. (13-3-1)]. Figure 13.8 shows the calculated Auger lifetime at 77 K as a function of the injected carrier density for p-type Pb_{0.82}Sn_{0.18}Te.

For an optical gain of \(\sim100\text{ cm}^{-1}\), which is the typical threshold gain for a semiconductor laser, the injected carrier density is \(\sim0.8\times10^{17}\text{ cm}^{-3}\) for low-doped material (see curve I in Fig. 13-6).

At this injected carrier density, the current lost to Auger recombination (\(qn/\tau_\text{A}\)) at 77 K is \(25\text{ A}/(\text{cm}^2\cdot\mu\text{m})\) using \(\tau_\text{A}=5\text{ ns}\) from Fig. 13-8. This is a significant fraction of the generally observed threshold current densities which are in the range of \(500-1000\text{ A}/(\text{cm}^2\cdot\mu\text{m})\) at 77 K.

Since the Auger rate increases with increasing temperature, the current lost to Auger recombination also increases with increasing temperature; hence the Auger effect may be responsible for the observed high temperature dependence of the threshold current of PbSnTe laser diodes.

Lead-salt lasers generally cannot operate near room temperature since almost all injected carriers combine nonradiatively and are not available for gain.

## 6. Laser Diode Fabrication

Single crystals of Pb_{1-x}Sn_{x}Te and Pb_{1-x}Sn_{x}Se have been prepared by conventional crystal growth techniques such as the Bridgman method and the Czochralski method.

Vapor-growth techniques, with or without a seed crystal, have also been used to grow single crystals of Pb_{1-x}Sn_{x}Te and Pb_{1-x}Sn_{x}Se. Thallium and bismuth are generally used as \(p\) and \(n\) dopants, respectively.

Single-heterostructure laser diodes have been fabricated from these single crystals using compositional interdiffusion (CID). In the CID technique, PbS_{1-x}Se_{x} or Pb_{1-x}Sn_{x}Se is annealed in the presence of PbS or PbSe powder, respectively, resulting in the out-diffusion of Se or Sn from the surface. Single-heterostructure lasers showed excellent tuning characteristics.

Double-heterostructure lasers have been fabricated using liquid-phase epitaxy (LPE), vapor-phase epitaxy (VPE), and the molecular-beam epitaxial (MBE) techniques. PbS_{1-x}Se_{x} double-heterostructure lasers have also been fabricated using CID.

Several groups have reported on the growth of Pb_{1-x}Sn_{x}Te layers by LPE on PbTe substrates. The growth occurs from a supersaturated solution consisting of polycrystalline PbTe, Pb, and Sn. The growth temperature is generally between 550 and 600°C. A conventional LPE multibin boat with the substrate on a slider can be used (see Fig. 4.3 in the what is liquid phase epitaxy tutorial).

In the Pb_{0.88}Sn_{0.12}Te-PbTe lasers fabricated by Groves et al., (100)-oriented p-PbTe, grown by the Bridgman method, was used as the substrate, and stripe-geometry lasers with ~ 50-μm-wide stripes were made using MgF_{2} dielectric mask on the p-PbTe substrate. The threshold current density at 77 K was ~ 4.2 kA/cm^{2} at \(\lambda\approx8.35\) μm.

Tomasetta and Fonstad also made double-heterostructure lasers of Pb_{0.82}Sn_{0.18}Te with PbTe cladding layers using LPE on p-PbTe substrates. For an active-layer thickness of 6 μm, they report a threshold current density of ~ 1.2 kA/cm^{2} at 77 K.

Vapor-phase epitaxy using H_{2} transport in an open tube has been used to grow epitaxial layers of PbSnTe on PbTe substrates. At a substrate temperature of ~ 700°C, the growth rate is typically ~0.3 μm/h. Fabrication of single-heterostructure lasers of Pb_{0.83}Sn_{0.17}Te-PbTe (\(\lambda\approx12.2\) μm at 77 K) by VPE has been reported.

Walpole et al. have fabricated Pb_{0.78}Sn_{0.22}Te double-heterostructure lasers using MBE. Thallium-doped p-PbTe was used as the substrate. In MBE, lead-salt alloys in powdered form are commonly used as sources, and the composition of the grown layer is varied by controlling with shutters the rate at which the constituents arrive at a heated substrate.

The walls of the growth chamber are cooler than the substrate, which is in direct line with the molecular beam originating from the sources. A variant of this technique is hot-wall vapor deposition (HWVD) which has been extensively used for the growth of PbS_{1-x}Se_{x} double-heterostructure lasers on n-type PbS substrates.

In HWVD the walls of the growth chamber are hot, so they reflect or diffuse molecules; the substrate is heated independently. The furnace can be designed so that either (i) only the diffused beam from the walls gets deposited or (ii) both the direct and the diffused beams get deposited.

PbS-PbS_{0.6}Se_{0.4} (\(\lambda\approx5\) μm) double-heterostructure lasers fabricated using HWVD have threshold current densities of ~400 A/cm^{2} at 77 K for an active-layer thickness of 1μm.

The control of the active-layer thickness obtained by MBE and HWVD is considerably better than that obtained by LPE. Further processing of the grown wafer to obtain the injection laser diodes is completed by depositing p and n contact metals and then cleaving the wafer to produce mirror facets. Indium is generally used as the n contact material, and Au and Pt are used for p contacts.

## 7. Laser Properties

For a nominal emission wavelength of ~ 10 μm, the refractive index of a lead-salt compound is about 5, and hence the wavelength in the active layer is ~ 2 μm. Thus an active-layer thickness in the range of 1-2 μm is desirable for good mode confinement without excessive increase in threshold.

From Eqs. (2-6-2) and (2-6-3) in the emission characteristics of semiconductor laswers tutorial, the threshold current density \(J_\text{th}\) varies linearly with \(d/\Gamma\), where \(d\) is the active-layer thickness and \(\Gamma\) is the confinement factor of the TE mode.

Figure 13-9 shows a plot of \(d/\Gamma\) as a function of the active-layer thickness \(d\) for \(\lambda=10\) μm. The active-layer and cladding-layer indices are assumed to be 5.0 and 4.5, respectively.

Similar to the case of InGaAsP lasers (see Fig. 5-2 in the broad area lasers tutorial), \(J_\text{th}\) is minimum for an optimum thickness \(d\). However, note that the active-layer thickness needed is considerably larger for 10-μm lead-salt lasers than for InGaAsP lasers.

The technology for growing the lead-salt system is developing rapidly. As a result, a comparison of different material systems for laser performance is premature at this point.

However, the following two general conclusions may be drawn. The longer the emission wavelength, (i) the lower the maximum operating temperature, and (ii) the lower the external differential quantum efficiency and maximum output power.

The former may be due to the Auger effect, which is larger at longer wavelengths, and the latter may be due to increased free-carrier absorption (\(\alpha_\text{fc}\propto\lambda^2\)) at longer wavelengths.

Figure 13-10 shows the measured light-current characteristics of a PbSnTe diode laser at 2 K. The maximum power obtained without any facet coating (curve L1) is 5 mW. This power output can be increased to over 10 mW (curve L2-RC) by using an antireflection (AR) coating on the front facet and a total-reflection coating on the back facet.

The nonradiative recombination mechanisms discussed in the nonradiative recombination in semiconductor lasers tutorial determine both the threshold current and its temperature dependence.

In contrast to InGaAsP lasers, surface recombination plays an important role in lead-salt lasers because of their relatively poor heterojunction-interface quality. A temperature-independent surface-recombination velocity \(S\) increases the threshold current by

\[\tag{13-7-1}\Delta{J}_\text{th}=qn_\text{th}SA\]

where \(q\) is the electron charge, \(n_\text{th}\) is the carrier density at threshold, and \(A\) is the surface area.

For thick active layers, the relative contribution of interface recombination to the total current is smaller than that for thin layers. Figure 13-11 shows the measured threshold current density as a function of temperature for PbSnTe-PbTe double-heterostructure lasers of different active-layer thicknesses.

The low temperature dependence observed for the thin-active-Iayer device is probably due to interface recombination, which dominates the measured threshold current at low temperatures.

An interface recombination velocity of ~ \(1\times10^5\text{ cm/s}\) has been measured for PbSnTe-PbTe interfaces at 5 K. Such a high interface recombination velocity can significantly reduce the internal quantum efficiency.

The low internal quantum efficiency observed in many lead-salt diode lasers is probably related to poor interfaces. Since the quality of the interface depends on the material growth and processing techniques used, it is not surprising that differing temperature dependences of the threshold current are observed for different types of diodes.

Figure 13-12 shows the measured data for PbS_{1-x}Se_{x} diodes of different types.

Quaternary PbSnSeTe diode lasers with PbSeTe cladding layers have been fabricated by LPE. Both PbTe and Pb_{0.8}Sn_{0.2}Te were used as substrates. The PbSeTe layers are used to provide good lattice matching (which should have reduced interface defects) between the substrate and the active region.

Figure 13-13 shows the measured temperature dependence of the threshold current for lasers emitting near 10 μm. The active-region thickness of the lasers in Fig. 13-13 falls in the range of 1.0-1.5 μm.

The saturation of laser threshold current with a decrease in temperature, similar to that shown in Fig. 13-11 for lasers with a thin active region, is not observed; this suggests that these devices have a reduced interface recombination velocity.

The threshold current as a function of temperature may be written as

\[\tag{13-7-2}I_\text{th}\approx{I_0}\exp(T/T_0)\]

The measured \(T_0\) values in various temperature ranges are also shown in Fig. 13-13.

The output power of the lead-salt family of diode lasers depends on the crystalline quality of the substrate. The maximum output power is observed to be inversely proportional to the dislocation density.

Diffused-junction diodes generally emit more power than single- or double-heterostructure diodes. Also, the output powers are higher under pulsed operation than under CW operation.

The emission characteristics of lead-salt lasers can be improved by adopting the fabrication techniques used in the GaAs and InGaAsP material systems.

For example, buried-heterostructure PbSnTe lasers have been developed. The PbSnTe active layer with a width of 4 μm and thickness of 1 μm was buried in the PbTeSe confining layers using a two-step LPE procedure. These lasers have a threshold current of ~ 60 mA at 50 K and can provide up to 1.7 mW of single-mode power at a wavelength of ~ 8 μm. Furthermore, the operating temperature can be increased up to 105 K with 300 μW of output power.

Cleaved-coupled-cavity lead-salt lasers have also been developed. Similar to the case of InGaAsP lasers discussed in the coupled cavity semiconductor lasers tutorial, these PbSnTe lasers provide an improvement in spectral purity together with a reduction in the threshold current. Furthermore, the laser diode can be tuned by adjusting the current in the controller section.

Another development in lead-salt lasers is adoption of the quantum-well structure (see the quantum well semiconductor lasers tutorial).

The motivation was to see if the operating temperature of lead-salt lasers could be increased through quantum-size effects. The stripe-geometry quantum-well lasers used a new quaternary compound, Pb_{1-x}Eu_{x}Se_{y}Te_{1-x} (which has previously been used to fabricate regular double-heterostructure lasers).

The single-quantum-well (SQW) structure used a PbTe quantum well as the active region and PbEuSeTe for the cladding layers. The quantum well ranged in thickness from 30 to 250 nm. The output power was highest for 30-nm quantum wells, ranging up to 5 mW for multimode emission.

The threshold current was found to be significantly reduced because of the quantum-size effects. These SQW lasers could be temperature tuned from \(\lambda\) = 6.45 μm (at 13 K) to 4.01 μm (at 241 K).

Under CW conditions, the highest operating temperature was 174 K (at 4.41 μm) while pulsed operation could be achieved up to 241 K.

These properties suggest that quantum-well lead-salt lasers may prove useful for spectroscopic and fiber optic applications.

## 8. Tuning Characteristics

One of the first applications of lead-salt diode lasers was in spectroscopy because of the large tuning range that can be obtained simply by varying the operating temperature. In addition to temperature tuning, pressure tuning and magnetic-field tuning can also be used. In this section we briefly discuss the tuning characteristics of lead-salt lasers.

The band gap of a semiconductor varies with temperature. Figure 13-14 shows the data for three lead-salt binaries. The band gap increases at a rate of ~ 0.5 meV/K, which corresponds to a wavelength change of 10-20 nm/K depending on the wavelength.

Clearly, by varying the temperature the emission wavelength can be varied over a large range.

Large-range tuning of this nature usually occurs in a discrete fashion; i.e., the laser-emission wavelength jumps from one longitudinal mode to another as the temperature varies.

Continuous tuning over a small range can also be obtained because the position of the cavity mode itself depends on the temperature through a temperature-dependent refractive index. The wavelengths of the longitudinal modes \(\lambda_m\) are determined by [refer to the threshold condition and longitudinal modes of semiconductor lasers tutorial]

\[\tag{13-8-1}m\lambda_m=2\mu{L}\]

where \(\mu\) is the refractive index, \(L\) is the length of the cavity, and \(m\) is an integer.

With a change in temperature \(\Delta{T}\), \(\mu\) changes by

\[\tag{13-8-2}\Delta\mu=(\text{d}\mu/\text{d}T)\Delta{T}\]

Hence the mode position changes by

\[\tag{13-8-3}\Delta\lambda_m=\frac{(\text{d}\mu/\text{d}T)}{\mu}\lambda_m\Delta{T}\]

The continuous-tuning rate for typical lead-salt lasers is about one-third the discrete tuning rate with temperature.

The active-region temperature of the diode laser that enters in Eqs. (13-8-2) and (13-8-3) depends on both the ambient temperature and the operating current (through ohmic heating). Thus a fine tuning (which may be sufficient for some spectroscopic applications) can be achieved simply by varying the operating current.

Figure 13-15 shows the measured data for Pb_{x}Sn_{1-x}Te diode lasers. Continuous tuning is observed over a small current range followed by successive mode jumps.

The measured change in emission wavelength with temperature of various lead-salt diode lasers is shown in Fig. 13-16. The large tuning range is primarily due to an increase in the band gap with increasing temperature.

Large changes in the band gap and hence the emission wavelength of these narrow-gap semiconductors can also be obtained by applying hydrostatic pressure. Figure 13-17 shows the band gap at 77 K as a function of hydrostatic pressure for three binary lead-salt semiconductors.

Pressure tuning has been used for spectroscopic applications. Generally a combination of temperature tuning (using current) and pressure tuning can be used. Band-gap tuning also occurs when a uniaxial pressure is applied. Although uniaxial pressure is considerably simpler to apply than hydrostatic pressure, there is a chance that the laser may get crushed.

Magnetic-field tuning can also be used as a laboratory spectroscopic tool. Tuning results from quantization of the valence and conduction bands into the Landau levels, the separation of which varies linearly with the magnetic field.

The measured tuning rate of PbS_{0.82}Se_{0.18} diode lasers is in the range of 0.4-2 MHz/G, depending on whether the tuning occurs within one cavity mode or between different cavity modes.

The next tutorial discusses about ** infrared and visible semiconductor lasers using other material systems**.