# How to estimate the threshold current density of a semiconductor laser?

This is a continuation from the previous tutorial - ** experimental results of radiative and Auger recombination coefficients of semiconductor lasers**.

In this tutorial we show how the calculations presented in the radiative recombination in semiconductors tutorial and the nonradiative recombination in semiconductors tutorial can be used for estimating the threshold current density \(J_\text{th}\) of a semiconductor laser.

Because of the different facet reflectivities for the TE and TM modes, the TE mode generally has a lower threshold.

In the emission characteristics of semiconductor lasers tutorial we obtained \(J_\text{th}\) using the linear-gain model based on Equation (2-4-3) [refer to the gain and stimulated emission in semiconductor lasers tutorial].

From Equations (2-6-3) and (2-6-4) [refer to the emission characteristics of semiconductor lasers tutorial], \(J_\text{th}\) is given by

\[\tag{3-5-1}J_\text{th}=qd(A_\text{nr}n_\text{th}+Bn_\text{th}^2+Cn_\text{th}^3)+J_\text{L}\]

where \(d\) is the active-layer thickness and \(n_\text{th}\) is the injected carrier density at threshold. To account for the carrier leakage from the active region, we have introduced \(J_\text{L}\) into Equation (3-5-1).

In general, the carrier leakage is structure-dependent. For simplicity and generality, we consider a broad-area laser for which the leakage current density \(J_\text{L}\) is only due to carrier leakage over the ** heterojunctions** (interfaces between the active and cladding layers).

The threshold carrier density \(n_\text{th}\) is obtained using the threshold condition (2-3-9) [refer to the threshold condition and longitudinal modes of semiconductor lasers tutorial].

Using Equations (2-3-10) [refer to the threshold condition and longitudinal modes of semiconductor lasers tutorial] and (2-5-50) [refer to the waveguide modes in semiconductor lasers tutorial] we obtain

\[\tag{3-5-2}\Gamma{g}_\text{th}=\frac{1}{2L}\ln\left(\frac{1}{R_1R_2}\right)+\Gamma\alpha_\text{a}+(1-\Gamma)\alpha_\text{c}+\alpha_\text{scat}\]

where the first term represents facet loss, while the remaining terms account for internal losses.

Typically, \(L\approx250\) μm, \(R_1=R_2\approx0.32\), and the facet loss is about \(45\text{ cm}^{-1}\).

If we consider a 1.3-μm InGaAsP laser with InP cladding layers, the confinement factor \(\Gamma\) for the TE mode is about 0.47 for a 0.2-μm-thick active layer [see Figure 2-5 in the waveguide modes in semiconductor lasers tutorial]. We calculate internal losses using \(\alpha_\text{scat}\approx0\) and \(\alpha_\text{a}=\alpha_\text{c}=30\text{ cm}^{-1}\).

With these parameters, the threshold gain \(g_\text{th}\approx166\text{ cm}^{-1}\). The injected carrier density at room temperature (\(T=300\text{ K}\)) required to achieve this gain can be obtained using Figure 3-8 [refer to the radiative recombination in semiconductor lasers tutorial]; \(n_\text{th}\approx1.7\times10^{18}\text{ cm}^{-3}\).

The threshold current density \(J_\text{th}\) can now be obtained using Equation (3-5-1) provided the parameters \(A_\text{nr}\), \(B\), and \(C\) have been estimated from the results presented in the radiative recombination in semiconductors tutorial and the nonradiative recombination in semiconductors tutorial.

For good-quality InGaAsP lasers, the contribution of \(A_\text{nr}\) (trap and surface recombinations) is often negligible compared with the other terms in Equation (3-5-1), and \(J_\text{th}\) can be rewritten as

\[\tag{3-5-3}J_\text{th}\approx{J_\text{r}}+J_\text{nr}+J_\text{L}\]

where

\[\tag{3-5-4}J_\text{r}=qd(Bn_\text{th}^2)=qdR\]

\[\tag{3-5-5}J_\text{nr}=qd(Cn_\text{th}^3)=qdR_\text{a}\]

are the radiative and nonradiative components of the current density, respectively.

The radiative spontaneous-emission rate \(R\) and the Auger recombination rate \(R_\text{a}\) can be obtained from the results presented in the radiative recombination in semiconductors tutorial and the nonradiative recombination in semiconductors tutorial.

For the example under consideration, using \(d=0.2\) μm and \(g_\text{th}=166\text{ cm}^{-1}\), we obtain \(J_\text{r}\approx1\text{ kA}/\text{cm}^2\) from Figure 3-9 [refer to the radiative recombination in semiconductors tutorial].

Further, if we use \(C=2.3\times10^{-29}\text{ cm}^6/\text{s}\) as the experimentally deduced value [refer to the gain and stimulated emission of semiconductor laser tutorial], \(J_\text{nr}=0.36\text{ kA}/\text{cm}^2\) from Equation (3-5-5).

Thus the calculated \(J_\text{th}\) in our example is \(1.36\text{ kA}/\text{cm}^2\) in the absence of carrier leakage. This compares well with the measured values, which are typically in the range \(1-1.5\text{ kA}/\text{cm}^2\).

In the next part we calculate the carrier leakage over the heterojunctions in order to estimate \(J_\text{L}\).

### Carrier Leakage over the Heterojunctions

Heterojunction carrier leakage is caused by diffusion and drift of electrons and holes from the edges of the active region to the cladding layers, and is schematically shown in Figure 3-24(a).

The heterojunction leakage in a double heterostructure has been extensively studied. In thermal equilibrium, at the boundary between the active and cladding layers, a certain number of electrons and holes are present.

Figure 3-24(b) shows an energy-level diagram for the heterojunction of active and p-cladding layers; the electron and hole quasi-Fermi levels \(E_\text{fc}\) and \(E_\text{fv}\) are assumed to be continuous at the boundary.

The number of electrons \(n_\text{b}\) at the boundary of the p-cladding layer is given by [from Equation (3-2-27) from the radiative recombination in semiconductors tutorial]

\[\tag{3-5-6}n_\text{b}=N_\text{cc}\frac{2}{\pi^{1/2}}\displaystyle\int\limits_{\epsilon_\text{c}}^\infty\frac{\epsilon^{1/2}\text{d}\epsilon}{1+\exp(\epsilon-\epsilon_\text{fc})}\]

where

\[N_\text{cc}=2\left(\frac{2\pi{m_\text{c}k_\text{B}T}}{h^2}\right)^{3/2}\qquad\epsilon_\text{c}=\Delta{E_\text{c}}/k_\text{B}T\]

and \(m_\text{c}\) is the conduction-band effective mass of the p-cladding layer.

Note that \(n_\text{b}\) is the number of electrons with energy greater than the conduction-band barrier height \(\Delta{E}_\text{c}\).

Using the Boltzmann approximation for the non-degenerate case, Equation (3-5-6) may be simplified to yield

\[\tag{3-5-7}n_\text{b}=N_\text{cc}\frac{2}{\pi^{1/2}}\displaystyle\int\limits_{\epsilon_\text{c}}^\infty\epsilon^{1/2}\text{d}\epsilon\exp(-\epsilon+\epsilon_\text{fc})=N_\text{cc}\exp\left(\frac{-E_1}{k_\text{B}T}\right)\]

The quantity \(E_1=\Delta{E}_\text{c}-E_\text{fc}\) is shown in Figure 3-24.

In deriving the above, we have used the relation \(\epsilon_\text{fc}=E_\text{fc}/k_\text{B}T\). We now show that \(E_1\) is related to the band-gap-difference \(\Delta{E}_\text{g}\) between the active and cladding layers.

From Figure 3-24 we note that

\[\tag{3-5-8}E_1=\Delta{E_\text{c}}-E_\text{fc}=\Delta{E_\text{g}}-\Delta{E_\text{v}}-E_\text{fc}\]

Further, \(\Delta{E}_\text{v}\) is given by

\[\tag{3-5-9}\Delta{E_\text{v}}=E_\text{fv}'+E_\text{fv}\]

where \(E_\text{fv}'\) and \(E_\text{fv}\) are the hole quasi-Fermi levels in the p-cladding layer and active layer respectively.

From Equations (3-5-7), (3-5-8), and (3-5-9) it follows that

\[\tag{3-5-10}n_\text{b}=\frac{N_\text{cc}N_\text{vc}}{P}\exp\left(-\frac{\Delta{E_\text{g}+E_\text{fc}+E_\text{fv}}}{k_\text{B}T}\right)\]

where we have used the relation

\[\tag{3-5-11}P=N_\text{vc}\exp\left(\frac{-E_\text{fv}'}{k_\text{B}T}\right)\]

for the majority (hole) carrier density and

\[\tag{3-5-12}N_\text{vc}=2\left(\frac{2\pi{k_\text{B}T}}{h^2}\right)^{3/2}(m_\text{hh}^{3/2}+m_\text{lh}^{3/2})\]

is the valence-band density of states for the p-cladding layer.

Note that \(m_\text{c}\), \(m_\text{hh}\), and \(m_\text{lh}\) are the effective masses for the p-cladding layer and can be obtained using Table 3-1 [refer to the radiative recombination in semiconductors tutorial].

The quantities \(E_\text{fc}\) and \(E_\text{fv}\) can be calculated from the known carrier density in the active region [using Equation (3-2-28) from the radiative recombination in semiconductors tutorial].

An equation similar to (3-5-10) can be derived for the density of holes (\(p_\text{b}\)) at the boundary between the n-cladding layer and the active layer.

Equation (3-5-10) shows that \(n_\text{b}\) increases rapidly with increasing temperature and suggest that the carrier leakage can be a major carrier-loss mechanism at high temperatures, especially for low heterojunction-barrier heights.

The electron leakage current density \(j_\text{n}\) at the p-cladding layer is given by

\[\tag{3-5-13}j_\text{n}=-qD_\text{n}\frac{\text{d}n}{\text{d}x}+qn\mu_\text{n}E\]

where the first term represents diffusive leakage and the second term represents drift leakage in the presence of an electric field \(E\).

\(D_\text{n}\) is the electron diffusivity, \(\mu_\text{n}\) is the minority carrier mobility and \(n(x)\) is the density of electrons at a distance \(x\) from the boundary between the active region and the P-cladding layer.

The current \(j_\text{n}\) also satisfies the continuity equation

\[\tag{3-5-14}\frac{1}{q}\frac{\text{d}j_\text{n}}{\text{d}x}+\frac{n}{\tau_\text{n}}=0\]

where \(\tau_\text{n}\) is the minority carrier lifetime.

Equation (3-5-13) can be solved using the boundary condition \(n(x=0)=n_\text{b}\) and \(n(x=h)=0\). The second condition assumes that the minority carrier density at the contact, which is at a distance \(h\) from the boundary, is \(0\).

The result for the electron leakage current at \(x=0\) is

\[\tag{3-5-15}j_\text{n}=qD_\text{n}n_\text{b}\frac{(Z-Z_1)\exp(Z_2h)+(Z_2-Z)\exp(Z_1h)}{\exp(Z_2h)-\exp(Z_1h)}\]

where

\[\tag{3-5-16}Z=qE/k_\text{B}T\]

\[\tag{3-5-17}Z_{2,1}=\frac{1}{2}Z\pm\left(\frac{1}{L_\text{n}^2}+\frac{1}{4}Z^2\right)^{1/2}\]

\[\tag{3-5-18}L_\text{n}=(D_\text{n}\tau_\text{n})^{1/2}\]

\(L_\text{n}\) is the electron diffusion length. In the above, the relation \(D_\text{n}=\mu_\text{n}k_\text{B}T/q\) has been used.

In the limit \(E=0\), Equation (3-5-15) reduces to the case of pure diffusive leakage, and for \(L_\text{n}\gg{h}\), it reduces to \(i_\text{n}=qn_\text{b}\mu_\text{n}E\), which holds if only drift leakage is present.

A similar equation can be derived for the hole leakage current \(j_\text{p}\) in the n-cladding layer. However, since the diffusion length and mobility of electrons are large compared to those of holes, the electron leakage is considerably larger than the hole leakage.

The total leakage is given by the sum

\[\tag{3-5-19}J_\text{L}=j_\text{n}+j_\text{p}\]

Figure 3-25 shows the calculated \(J_\text{L}\) for the diffusive heterojunction leakage in an InGaAsP-InP laser.

At higher temperatures, the leakage current is considerably higher. Also, the leakage current increases rapidly when the barrier height decreases.

At 1.3-μm and 1.55-μm wavelengths, however, the calculated \(J_\text{L}\ll100\text{ A}/\text{cm}^2\) near room temperature, implying that the hetero-barrier carrier leakage is not significant for such InGaAsP layers.

The calculation assumes that the electron and hole concentrations in the n- and p-cladding layers are \(3\times10^{17}\text{ cm}^{-3}\), respectively, and that the electron and hole diffusion lengths are 5 μm and 1 μm, respectively. The thickness \(h\) of the p-cladding layer is assumed to be 2 μm.

The drift leakage current increases rapidly with a decrease in p-cladding-layer doping. For a carrier concentration of \(3\times10^{17}\text{ cm}^{-3}\) or higher, which is normally the case for semiconductor lasers, the drift leakage is small compared to the diffusive leakage.

The next tutorial discusses about ** temperature dependence of threshold current of semiconductor lasers**.