Model

We present the results of a numerical simulation of a greatly simplified model of semiconductor lasing in which we use parameter values which are realistic but make fairly severe approximations in which we (a) assume that the densities of electrons and holes are the same (even though their masses differ considerably) (b) ignore carrier recombination losses and (c) model the collision integral by a differential approximation [1], [2], [3] in which the principal contributions to wavevector quartets satisfying (1) are assumed to come from nearby neighbors. Despite the brutality of the approximations, the results we obtain are qualitatively similar to what we obtain using more sophisticated and complicated descriptions.

The semiconductor Maxwell-Bloch equations are [4],[5],

$$\frac{\partial e}{\partial t} = i \frac{\Omega }{2\epsilon _0}\int\mu_k p_k d{\bf k} - \gamma _E e ,$$ (2)

$$\frac{\partial p_k}{\partial t} = (i\Omega -i\omega _k-\gamma _P)p_k - \frac{i \mu_k}{2\hbar}(2n_k-1)e,$$ (3)

$$\frac{\partial n_k}{\partial t} = \Lambda (1-n_k) -\gamma _k n_k +\left( \frac{\partial n_k}{\parti... ...)_{\rm collision} -\frac{i}{2\hbar} \left( \mu_k p_k e^*-\mu _k p_k^* e\right).$$ (4)

Here $e(t)$ and $p_k(t)$ are the electric field and polarization at momentum $\bf k$ envelopes of the carrier wave $\exp{(-i\Omega t + i K z)}$ where $\Omega$ is the cavity frequency (we assume single mode operation only) and $n(k)$ is the carrier density for electrons and holes. The constants $\gamma _E$, $\gamma _P$ model electric field and homogeneous broadening losses, $\epsilon _0$ is dielectric constant, $\mu_k$ is the weighting accorded to different $\bf k$ momentum (modeled by $\mu_k=\mu_{k=0}/(1+\epsilon _k/\epsilon _{\rm gap})$), $\Lambda _k$ and $\gamma _k$ represent carrier pumping and damping. In (4), the collision term is all important and is given by

$$\frac{\partial} {\partial t}n_{k}= 4\pi \int \vert T_{k k_1 k_2 k... ...}+\omega _{k_1}-\omega _{k_2}-\omega _{k_3}) d{\bf k_1}d{\bf k_2} d {\bf k_3} ,$$ (5)

where $T_{k k_1 k_2 k_3}$ is the coupling coefficient measuring mutual electron and hole interactions. We make the weak assumption that all fields are isotropic and make a convenient transformation from $k$ ($=\vert{\bf k}\vert$) to $\omega$ via the dispersion relation $\omega =\omega ({\bf k})$ defining the carrier density $N_\omega$ by $\int N_\omega d \omega = \int n({\bf k}) d{\bf k}$ or $N_\omega =4\pi k^2 d k /(d\omega ) n({\bf k})$. Then, in the differential approximation, (5) can be written as both,

$$\frac{\partial N_\omega }{\partial t} = \frac{\partial^2 K}{... ...rtial\omega } (K-\omega \frac{\partial K}{\partial\omega }),$$ (6)

with

$$\\ \ \ \ \ K=-I\omega ^s \left( n^4_o\left(n_\omega ^{-1}\rig... ...rtial }{\partial\omega }, \ \ \ n_\omega =n({\bf k}(\omega )),$$

where $s$ is the number computed from the dispersion relation, the dependence of $T_{k k_1 k_2 k_3}$ on ${\bf k}$ and dimensions ($s$ is of the order of $7$ for semiconductors.) The conservation forms of the equations for $N_\omega$ and $E_\omega =\omega N_\omega$ allow us identify $Q=\frac{\partial K}{\partial \omega }$ (positive if carriers flow from high to low momenta) and $P=K-\omega \frac{\partial K}{\partial\omega }$ (positive if energy flows from low to high momenta) as the fluxes of carriers and energy respectively. Moreover, the equilibrium solutions are now all transparent. The general stationary solution of (6) is the integral of $K=Q\omega +P$ which contains four parameters, two (chemical potential and temperature) associated with the fact that $K$ is a second derivative, and two constant fluxes $Q$ and $P$ of carriers and energy. The Fermi-Dirac solution $n_\omega =(\exp{(A\omega +B)}+1)^{-1}$, the solution of $K=0$, has zero flux. We will now solve (2), (3) and (4) after angle averaging (4) and replacing $4\pi k^2 \frac{\partial k}{\partial \omega }\left(\frac{\partial n_k}{\partial \omega }\right)_{\rm collision}$ by $\frac{\partial^2 K}{\partial\omega ^2}$. The value of the constant $I$ is chosen to ensure that solutions of (6) relax in a time of 100 fs.