Finite flux solutions of the quantum Boltzmann equation can enhance semiconductor lasing
Phys. Rev. Lett. 84, 1894 - 1897 (2000)

Yuri V. Lvov[], Alan C. Newell []

[]   Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87544
[]   Department of Mathematics, University of Warwick, Coventry, CV4 7AL, UK

Abstract:

In this Letter we propose and illustrate in the context of the semiconductor laser that, in nonequilibrium fermionic systems with sources and sinks, the family of finite flux stationary solutions of the quantum Boltzmann equation are central and more important then the zero flux Fermi-Dirac spectrum. The results we present are the quantum analogue of the finite-flux Kolmogorov spectra which are central to understanding nonequilibrium classical systems such as high Reynolds number hydrodynamics and the wave turbulence encountered in water waves, plasmas and optics [1,2]. In particular, we show how semiconductor laser efficiency can be improved by pumping the system in such a way so as to maximize the flux of carriers (electrons and holes) towards the lasing frequencies.

Introduction and Main Ideas. -- The main goal (and principal novel feature) of this Letter is to point out the importance of a richer class of stationary spectra of fermionic systems. Unlike the Fermi-Dirac spectrum, which is a special case, these spectra allow for a finite transport of carriers between source energies at which the system is pumped and sink energies at which the system loses particles and energy to other states. Such finite flux distributions are the quantum analogue of the finite-flux Kolmogorov spectra of classical wave turbulence [2,3]. These exact solutions of the kinetic equations describe how, for example, energy, momentum and particle number inserted by the wind into ocean surface waves is transported by four wave resonances (for wavelengths $\lambda> 2 cm$., gravity waves dominate) and three wave resonances (for $\lambda< 2 cm$., surface tension is important) throughout the spectrum. Indeed such spectra have been observed experimentally [4]. The inverse cascade of predominantly particle number is responsible for ``old'' waves, namely the appearance of very long and fast traveling waves which outrun the storm and which are not directly driven by the wind but can only be indirectly generated by nonlinear processes. We suggest the same scenario is also vital to understanding the behavior of the quantum fermionic systems. We illustrate the idea in the context of the semiconductor laser (ubiquitous and advantageous in a wide range of applications) and demonstrate how, by using finite-flux solutions, the laser output can be enhanced. This means that the optical properties of a semiconductor material can be effectively altered without actually changing the material itself.

For the most part the semiconductor laser operates in a manner similar to the ``text book'' two level laser. Optical feedback is organized by a cavity and the coherent light output is generated by in-phase transitions of an electron from a higher to lower energy state. In semiconductors, the lower state is a valence band, from which electrons are excited by pumping into a conduction band, leaving behind positively charged holes with opposite momenta and spins. But they also differ from two-level lasers. First, since light emission is a recombination process, there must be both an electron and hole at the same absolute momentum and spin values. Second, there is a continuum of transition energies parametrized by the electron momentum $\bf k$ and the laser output is a weighted sum of contributions from polarizations corresponding to each momentum value. (Spin is included as part of the momentum vector.) In this regard, the semiconductor laser resembles an inhomogeneosly broadened two-level laser. Third, and most important of all, electrons and holes interact with each other via screened Coulomb forces [5]. This leads to a mechanism for the redistribution of carriers between different momentum states, a process well described by coupled quantum Boltzmann or kinetic equations (QKE). The electron and hole probability densities $n_k^-=<a_k^\dagger a_k>$ , $n_k^+=<b_k^\dagger b_k>$ evolve according to [5,6]:

$$\frac{\partial} {\partial t}n_{k}^s = 4\pi \int \vert T_{k k_1 k_2 k_3}\vert^2 \delta ({\bf k}+{\bf k_1}-{\bf {k_2}}-{\bf k_3})$$

$$\times \left( n_{\bf k_2}^s n_{\bf k_3}^s(1-n_{\bf k_1}^s)(1-n_{\bf k}^s) \right.$$

$$\left. -n_{\bf k}^sn_{\bf k_1}^s(1-n_{\bf k_2}^s)(1-n_{\bf k_3}^s)\right)\cr$$

$$+ \left( n_{\bf k_2}^s n_{\bf k_3}^{-s}(1-n_{\bf k_1}^{-s})(1-n_{\bf k}^{s}) \right.$$

$$\left. - n_{\bf k}^s n_{\bf k_1}^{-s}(1-n_{\bf k_2}^s)(1-n_{\bf k_3}^{-s})\right)\cr$$

In (1), $T_{k k_1, k_2 k_3}$ is the coupling coefficient due to Coulomb forces and is proportional to $1/(k^2+\kappa^2)$, with $\kappa$ being the inverse screening length. The kinetic energies are $\hbar \omega_k^- = \frac{\hbar^2 k^2}{2 m^-}$ , $\hbar \omega_k^+ = \frac{\hbar^2 k^2}{2 m^+}$ with $k=\vert {\bf k} \vert$, and $m^-$ and $m^+$ are the electron and hole masses respectively. For electric field pulses exceeding 1 ps. the carrier redistribution is the fastest process ( $\simeq 100 fs.$). Therefore, carriers relax adiabatically to attracting manifolds $n^s_{0{\bf k}}$ which are stable stationary solutions of the QKE consistent with the presence of sources (pumping) and sinks (laser output at lower momenta and dissipation via nonradiative recombination). It is commonly assumed [5,6], that this equilibrium state corresponds to the thermodynamic equilibrium of fermion gases, which is described by a Fermi-Dirac (FD) distribution. However, because of the presence of sources and sinks, a semiconductor laser is far from equilibrium. In the case where the sources and sinks are located at different parts of the momentum spectrum, carriers and energy flow between them. But the Fermi-Dirac (FD) solutions do not capture the relevant physics because they carry no flux. Whereas it has been appreciated by several authors (e.g. F. Jahnke and S.W.Koch [7]) that a finite flux of carriers is essential to compensate for the action of a sink (e.g. hole burning in semiconductor lasers), it has not been generally recognized that there are exact stationary solutions of the QKE which describe the constant flux of carriers between isolated sources and sinks. We will now demonstrate, in the semiconductor laser context, that such solutions are not only relevant but are realized and can be used to enhance the laser output.

Assuming isotropy, the most general steady state solutions of the QKE (1) belong to a six parameter family. The six parameters are the total numbers of electrons and holes

$$N^s= \int n_k^s d {\bf k} = \int N_\omega^s d\omega, s=+,-,\ \ \hbar\omega =({\hbar^2 k^2 })/({2 m}),$$

$$1/m=1/m^-+1/m^+, \ \ \ \ \ N_\omega^s=\Omega_0\left(\frac{d k}{d\omega}\right)k^{d-1}n_k^s({\bf k}(\omega)),$$

(where $\Omega_0$ is the surface area of a d-dimensional unit sphere), the total energy

$$E=\int E_\omega d\omega,\ \ \ E_\omega = (\epsilon_{gap}+{\hbar\omega m}/{m^-}) N^- -{\hbar\omega m}/{m^+} N^+,$$

and the three fluxes $Q^-$ , $Q^+$ and $P$ of the conserved densities $N^-_\omega$, $N^+_\omega$ and $E_\omega$. It is convenient to write (1) in conservation law form

$$\frac{\partial N_\omega^s}{\partial t} = \frac{\partial Q^s}{... ...l E_\omega}{\partial t} = -\frac{\partial P}{\partial \omega},$$ (1)

where $Q^s$ ($P$) is taken positive when the flux of $N_\omega^s$ ($E_\omega$) is towards lower (higher) energies. Stationary solutions occur when $Q^s$ and $P$ are constants. The FD states $(n^s_{k})^{-1}=\exp(\frac{\hbar}{T}(\frac{\omega m}{m^s}-\mu_s))+1,$ belong to a special zero flux three parameter sub-manifold of solutions for which $Q^s=P=0$. Here $T$ can be interpreted as the common electron and hole temperature, $\mu_-$ and $\mu_+$ as chemical potentials. FD states with zero flux are most relevant in situations where the damping and pumping are broadband and locally in balance in $\omega$ space.

But broadband pumping, and the FD distributions generated by it, may be inefficient. Observe from Fig. (1(a)) that at room temperatures and typical operating conditions, the FD distributions are much broader than the gain band for lasing and therefore much energy is used in exciting transitions at momenta which, because there are no fluxes, do not directly contribute to lasing. Therefore, we are led to investigate what happens if, instead of pumping broadly, we pump the semiconductor in a relatively narrow spectral region around an energy value $\hbar \omega_0$ that is greater then the lasing energy $\hbar \omega_L$ (See Fig. (1(b))). Such local pumping is possible through optical pumping [8] or through resonant tunneling of electrons through multiple quantum well structures [9]. In this case, a significant portion of carriers and their associated energies will flow back from the source ( $\hbar \omega_0$) to sink (lasing at $\hbar \omega_L$) energies and thereby involve electrons and holes at all momenta and energies in lasing. At the same time, and because of conservation of carriers and energy, some carriers will flow to higher energy ($\hbar\omega_R$) values and their energies will be absorbed at various energy levels $\hbar\omega_R>\hbar\omega_0$ due to many processes e.g. (i) Absorption of the charge carriers with high kinetic energies that leave the optically active region and thus contribute to the electrical pumping current without contributing to the light amplification. (ii) Non-radiative recombination of electron-hole pairs mediated by impurities, dislocations, interface roughness. (iii) Auger processes.

Figure 1: (a) Stationary Fermi-Dirac electron and hole distributions $n_{\omega }^-$, $n_{\omega }^+$, the inversion $n_{\omega }^- + n_{\omega }^+ - 1$ and gain $\sqrt {\omega }(n_{\omega }^- + n_{\omega }^+ - 1)$ for the broadly pumped semiconductor laser. (b) Stationary (finite flux) distributions for the narrowband pumping case. Lasing only occurs over energies where the inversion is positive. Note the effect of the finite flux is to compress the original FD distribution.

Results.-- In the remainder of this Letter, we provide concrete evidence to support our idea. We numerically solve both the QKE (1) and the single mode semiconductor laser Maxwell-Bloch equations in the free carrier limit for the electric field envelope $e(t)$, polarization envelope $p_\omega$ and electron and hole distributions $n^-_\omega, n^+_\omega$ [5],[10] :

$$\frac{\partial e}{\partial t} = i \frac{\Omega}{2\epsilon_0} \frac{V}{(2\pi)^d} \int d_\omega p_\omega \frac{d{\bf k}}{d\omega}d\omega - \gamma_E e ,$$

$$\frac{\partial p_\omega}{\partial t} = (i\Omega-i\tilde\omega-\gamma_P)p_\omega - \frac{i d_\omega}{2\hbar}(n_\omega^++n_\omega^- -1)e,$$

$$\frac{\partial n_\omega^s}{\partial t} = \Lambda_\omega^s(1-n_\omega^s) -\gamma_\omega n_\omega^s +\left( ... ...l t} \right)_{\rm\tiny coll} -\frac{i}{ \hbar} {\rm Im}(d_\omega p_\omega e^*).$$

Here $V$ is a sample volume and we use typical semiconductor laser parameters suggested in [5]. $\Omega=1440 {\rm meV}/\hbar$ is the cavity frequency. The electric field damping $\gamma_E$ equals $6\times 10^{10} /sec$, the polarization decay (dephasing) $\gamma_P=10^{13}/sec$, $\epsilon_0$ is the permittivity of free space, $d_\omega$ is the dipole matrix element $d_{\omega=0}/(1+\epsilon_k/\epsilon_{\rm gap}),\ \ d_{\omega=0}=3\times 10^{-10} {\rm meters}\times {\rm the \ electron \ charge}$, and the nonradiative carrier damping $\gamma_\omega$ equals $10^{10}/sec$. In (3), ${\Lambda_\omega^s}$ is the pumping due to the injection current (taken to be between $0.001 ps^{-1}$ and $0.01 ps^{-1}$), $1-n_\omega^s$ is the Pauli blocking factor (because of Pauli exclusion principle, $n^s_\omega$ cannot be pumped over unity) and $\hbar\tilde\omega=\epsilon_{\rm gap}+\epsilon_{e,k}+\epsilon_{h,k}$. We further assume that all fields are isotropic and make a transformation from $k$ ($=\vert{\bf k}\vert$) to $\omega$ via the dispersion relation $\hbar\omega({\bf k})= \hbar^2 k^2/(2 m)$ and define the carrier density $n_\omega^s$ as $n^s({\bf k}(\omega))$. The collision terms in (3) are given in (1) and approximated by the differential approximation [1,2,10,11]. It is equivalent to the assumption that the spectral transfer of carriers is very local in $\omega$ space. The differential approximation reads

$$\frac{\partial N^s_\omega}{\partial t} = - \frac{m^s}{m}\frac{\pa... ...ial \omega^2} K_s - s \frac{m^s}{m} \frac{ \partial J}{\partial \omega}, s=+,-.$$

$$K_s(\omega)=I_1\omega^{\gamma+6} ( (n^s_\omega)^4 (1/n^s_\omega)''+ (n^s_\omega)^2 (\ln n^s_\omega)''),$$

$$J(\omega)=I_2 \omega^{\gamma+4} ( (n^-_\omega)'n^+_\omega(1-n_\omega^+)$$

$$- (m^{-1} m^-)(n^+_\omega)'n_\omega^-(1-n_\omega^-) ),\ \ '=d/(d\omega),$$ (2)

where $I_1, I_2$ are semiconductor relaxation time constants [10] and $\gamma$ equals unity for semiconductors. The fluxes are given by $Q^s = - m^s m^{-1} \frac{\partial K_s}{\partial \omega} - s m^s m^{-1} J$ and $P=-\epsilon_{gap}Q^- +\hbar(\omega\frac{\partial}{\partial\omega}(K_++K_-)-(K_++K_-)).$ We point out that $J$ is proportional to $-\frac{(m^{-1} m^-)}{\hbar}\frac{\partial}{\partial\omega}\ln\frac{n^{-}}{1-n^{-}} +\frac{(m^{-1} m^+)}{\hbar}\frac{\partial}{\partial\omega}\ln\frac{n^+}{1-n^+}$, or to the difference between the local inverse temperatures $(T^-)^{-1}(\omega)-(T^+)^{-1}(\omega)$ (for FD distributions, these are constant and equal) and thus the role of the cross term is to equalize electron and hole temperatures. We tested the validity of the model and of our numerical model by simulating the broadband pumping and reproducing the expected (but slightly modified due to hole burning) FD carrier distribution and laser turn on and output characteristics. Note that even for the broadband pumping case, there is some flux of carriers across the spectrum. Indeed, the maximum of the pumping $\Lambda_\omega^s(1-n_\omega^s)$ is located at the middle part of the spectrum, see Fig. (1(a)). The maximum of the carrier absorption is located at small $\omega$. Therefore one unavoidably has a flux of carriers across the spectrum!

We also observed, consistent with theoretical arguments one can make from (4) [10], that the net effect of the positive finite fluxes is to compress the FD spectra to smaller $\omega$ values. Such a compression, which is particularly strong for the electron distribution, can be interpreted as an effective temperature and chemical potential decrease. Therefore finite fluxes effectively increase inversion in the lasing part of the spectrum. In addition, the operation of a semiconductor laser may be optimized by choosing the pumping energy $\hbar \omega_0$ to (i) make $\hbar \omega_0$ big enough to minimize Pauli Blocking, because electrons are best pumped where $n_\omega^-$ is small, (ii) to make $\hbar \omega_0$ small enough (closer to the lasing energy $\hbar \omega_L$) to increases the flux of carriers towards the lasing energy. The flux formulae are given in the next paragraph.

In the numerical experiments, we solve (3) for $t\ge 0$ on $\omega_1\le\omega\le\omega_0$ with $\omega_1$ just less than $\omega _L$, the lasing frequency. The boundary conditions and pumping rates are: (a) Broadband case; $Q^+=Q^-=0$ at $\omega=\omega_1,\omega_0$, $\Lambda_\omega=$FD, typical pump profile. (b) Narrowband case: $Q^+=Q^-=P=0$ at $\omega=\omega_1$, $Q^+=Q^-=Q_L$, $P=-(\epsilon_{gap}+\hbar\omega_L)Q_L$, $\omega=\omega_0$. $\Lambda^s_\omega=0$. In both cases the background dissipation $\gamma^s n^s_\omega$ is the same. $Q_L$ is calculated as follows. Consider Fig. (2). From the conservation of electrons, holes and total energy, the assumptions that $P_R=-\omega_R(Q_R^+ + Q_R^-)$, $P_L=-\omega_L(Q_L^+ + Q_L^-)$ (namely, that both carriers and energy are absorbed by the dissipation and laser states respectively) and charge neutrality $Q_L^+=Q_L^-$, we find after a little calculation that $Q_L=Q_0 (\omega_R-\omega_0)/(\omega_R-\omega_L)$, $\frac{-P_L}{-P_0}=\frac{e_{gap}+\hbar\omega_L}{\epsilon_{gap}+\hbar\omega_0}\fr... ...-\omega_0}{\omega_R-\omega_L}\simeq \frac{\omega_R-\omega_0}{\omega_R-\omega_L}$ since $\epsilon_{gap}\simeq 1400$ meV, $\hbar\omega_L\simeq 40$ meV and $\hbar\omega_0\simeq 200$ meV. To make the comparison with broadband pumping, we choose $Q_0 = \int \Lambda_\omega(1-n_\omega^-) \frac{d k}{d \omega}d\omega$ such that the net particle and energy input rates are the same (energy $\simeq e_{gap} \times$ particle number). Because of Pauli Blocking, the absorption of power for the pump may be slower in the broadband case than in the narrowband case.

Figure 2: This picture explains the setup for input-output fluxes. Carriers (electrons and holes) and energy are added at $\omega _0$ with rates $Q_0^{s}$,$s=+,-$ and $P_0$. Energy and some carriers are dissipated at $\omega _R>\omega _0$ (an idealization) and are absorbed by the laser at $\omega _L$. Finite flux stationary solutions are realized in the windows $(\omega _L,\omega _0)$ and $(\omega _0,\omega _R)$ although in practice there will be some losses in both of these regions. The electron and hole fluxes to the left (right) are $Q_L^{-}$, $Q_L^{+}$ ($-Q_R^{-}$, $-Q_R^{+}$). The rate of energy flowing to the left (right) is $-P_L$ ($P_R$). Note that fluxes are defined so that $Q_L^s>0$, $s=+,-$, $Q^s_R<0$, $P_L<0, P_R>0$.

In Fig. 3 we present the output power as a function of the pumping strength. We observe that the output power is consistently higher for narrowband pumping than for broadband pumping, especially for weak pumping. The lasing threshold value is less for narrowband pumping. The distribution functions are more compressed towards lower energies for narrowband pumping than they are for broadband pumping because of the higher leftward fluxes associated with the former. The results for different mass ratios $m^+=m^-$, $m^+=2 m^-$ are similar.

Figure 3: The output power (in arbitrary units) as a function of pumping strength for narrowband and broadband pumping for $m^+=6 m^-$. The narrowband output power is shown by thick points, the broadband output power is shown by thin points.

Summary.-- In this letter, we have shown that the Quantum Kinetic Equation (QKE) has a new and richer class of steady state solutions than the Fermi-Dirac (FD) distribution. These new solutions generalize FD to include finite fluxes of the conserved quantities. We demonstrated how these finite flux solutions of the QKE can be used in the semiconductor laser context and in particular how the finite fluxes of carriers and energy can be exploited to give improved laser efficiency. But semiconductor lasers are just one particular example. There is a huge number of other potential applications of these spectra in situations where the system is driven far from equilibrium by the presence of sources and sinks operating at different energies. Two other examples: the growth of (Bose) condensates in superfluids is a direct consequence of an inverse cascade of particles; likewise, the onset of intermittency in (classical) optical waves of diffraction in nonlinear media where the refractive index increases with intensity is a result of filaments triggered by instabilities of long waves which in turn are driven by an inverse cascade of particles [1]. We point out also that optoelectronics is a fast growing field and the idea of effectively changing optical properties without changing the materials themselves is certainly worth more attention.

We thank I.R. Gabitov, R. Indik, S.W. Koch, U. Leonhardt, J.White and V.E. Zakharov for many helpful discussions.