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\title{
Semiconductor Lasers and Kolmogorov Spectra.\\
{\bf Physics Letters A, 235, 499 (1997).}
}
\author{ Yuri V. Lvov$^{1,2}$ and Alan C. Newell$^{1,3}$\\
$^1$ Dep. of Mathematics, The University of Arizona, Tucson AZ
85721 USA \\
$^2$ Dep. of Physics, The University of Arizona, Tucson AZ 85721 USA \\
$^3$ Mathematical Institute, The Univ. of Warwick, Coventry CV47AL UK\\
{\bf Physics Letters A, {\bf 235}, pp. 499-503 (1997).}
}
\maketitle
\abstract{
In this letter, we make a prima facie case that there could be distinct
advantages to exploiting a new class of finite flux equilibrium
solutions of the Quantum Boltzmann equation in semiconductor lasers.}
\section{Introduction} At first sight, it may very well seem that the two
subjects linked in the title have little in common. What do semiconductor
lasers have to do with behavior normally associated with fully developed
hydrodynamic turbulence? In order to make the connection, we begin by
reviewing the salient features of semiconductor lasers. In many ways, they
are like two level lasers in that the coherent light output is associated
with the in phase transitions of an electron from a higher to lower energy
state. In semiconductors, the lower energy state is the valence band from
which sea electrons are removed leaving behind
positively charged holes. The higher energy state is the conduction band.
The quantum of energy released corresponds to an excited electron in the
conduction band combining with a hole in the lower band below the
bandgap. Bandgaps, or forbidden energy zones are features of the energy
spectrum of an electron in periodic potentials introduced in this case
by the periodic nature of the semiconductor lattice.
However, there are two important ways in which the semiconductor laser
differs from and is more complicated than the traditional two-level laser
model. First, there is a continuum of bandgaps parameterized by the
electron momentum ${\bbox k}$ and the laser output is a weighted sum of
contributions from polarizations corresponding to electron-hole pairs at
each momentum value. In this feature, the semiconductor laser resembles
an inhomogeneosly broadened two level laser. Second, electrons and holes
interact with each other via Coulomb forces. Although this interaction is
screened by the presence of many electrons and holes, it is nonetheless
sufficiently strong to lead to a nonlinear coupling between electrons and
holes at different momenta. The net effect of these collisions is a
redistribution of carriers (the common name for both electrons and holes)
across the momentum spectrum. In fact it is the fastest ($\approx 100$
fs.) process (for electric field pulses of duration greater than
picoseconds) and because of this, the gas of carriers essentially relaxes
to a distribution corresponding to an equilibrium of this collision
process. This equilibrium state is commonly taken to be that of
thermodynamic equilibrium for fermion gases, the Fermi-Dirac
distribution characterized by two parameters, the chemical potential
$\mu$ and temperature $T$, slightly modified by the presence of broadband
pumping and damping.
But the Fermi-Dirac distribution is not the only equilibrium of the
collision process. There are other stationary solutions, called finite flux
equilibria, for which there is a finite and constant flux of carriers and
energy across a given spectral window. The Fermi-Dirac solution has zero
flux of both quantities. It is the aim of this letter to suggest that
these finite flux equilibria are more relevant to situations in which
energy and carriers are added in one region of the spectrum, redistributed
via collision processes to another region where they are absorbed.
Moreover, it may be advantageous to pump the laser in this way because
such a strategy may partially overcome the deleterious effects of Pauli
blocking. The Pauli exclusion principle means that two electrons with the
same energy and spin cannot occupy the same state at a given momentum.
This leads to inefficiency because the pumping is effectively multiplied
by a factor $(1-n_s({\bbox k})), s=e,h$ for electrons and holes
respectively, denoting the probability of not finding electron (hole) in a
certain ${\bbox k}$ (used to denote both momentum and spin) state. But,
near the momentum value corresponding to the lasing frequency $\o_L$,
$n_s({\bbox k})$ is large ($n_e({\bbox k})+n_h({\bbox k})$ must exceed
unity) and Pauli blocking significant. Therefore, pumping the laser in a
window about $\o_0>\o_L$ in such a way that one balances the savings
gained by lessening the Pauli blocking (because the carriers density
$n_s({\bbox k})$ decreases with $k=|{\bbox k}|$) with the extra input
energy required (because $k$ is larger), and then using the finite flux
solution to transport carriers (and energy) back to lasing frequency,
seems an option worth considering. The aim of this letter is to
demonstrate, using the simplest possible model, that this alternative is
viable. More detailed results using more sophisticated (but far more
complicated) models will be given later.
These finite flux equilibria are the analogies of the Kolmogorov spectra
associated with fully developed, high Reynolds number hydrodynamic
turbulence and the wave turbulence of surface gravity waves on the sea. In
the former context, energy is essentially added at large scales (by
stirring or some instability mechanism), is dissipated at small
(Kolmogorov and smaller) scales of the order of less than the inverse
three quarter power of the Reynolds number. It cascades via nonlinear
interactions from the large scales to the small scales through a window of
transparency (the inertial range in which neither forcing nor damping is
important) by the constant energy flux Kolmogorov solution. Indeed, for
hydrodynamic turbulence, the analogue to the Fermi-Dirac distribution,
the Rayleigh-Jeans spectrum of
equipatitions, is irrelevant altogether. The weak turbulence of surface
gravity waves is the classical analogue of the case of weakly interacting
fermions. The mechanism for energy and carrier density (particle number)
transfer is "energy" and "momentum" conserving binary collisions
satisfying the "four wave resonance" conditions \BE {\bbox k}+{\bbox
k_1}={\bbox k_2}+{\bbox k_3}, \ \ \ \ \o({\bbox k})+\o({\bbox
k_1})=\o({\bbox k_2})+\o({\bbox k_3}). \label{one} \EE In the
semiconductor context, $\hbar \o({\bbox k})=\hbar \o_{\rm gap}
+\e_e({\bbox k})+\e_h({\bbox k})$ (which can be well approximated by
$\a+\b k^2$) where $\hbar \o_{\rm gap}=\e_{\rm gap}$ corresponds to
the minimum bandgap and $\e_e({\bbox k})$, $\e_h({\bbox k})$ are electron
and hole energies. In each case, there is also a simple relation $E({\bbox
k})=\o n({\bbox k})$ between the spectral energy density $E({\bbox k})$
and carrier (particle number) density $n({\bbox k})$. As a consequence of
conservation of both energy and carriers, it can be argued (schematically
shown in Figure 1 and described in its caption), that the flux energy (and
some carriers) from intermediate momentum scales (around $k_0$ say) at
which it is injected, to higher momenta (where it is converted into heat)
must be accompanied by the flux of carriers and some energy from $k_0$ to
lower momenta at which it will be absorbed by the laser. It is the latter
solution that we plan to exploit.
\section{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
\cite{Optical}, \cite{HASS}, \cite{ZLF}
in
which the principal contributions to wavevector quartets satisfying
(\ref{one}) 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 \cite{Koch},\cite{Koch1},
\BEA \frac{\partial e}{\partial t}&=& i \frac{\O}{2\e_0}\int\mu_k p_k
d{\bbox k} - \g_E e ,\label{two}\\
\frac{\partial p_k}{\partial t}&=&(i\O-i\o_k-\g_P)p_k - \frac{i
\mu_k}{2\hbar}(2n_k-1)e,\label{three}\\
\frac{\partial n_k}{\partial t}&=&\L(1-n_k) -\g_k n_k +\left(
\frac{\partial n_k}{\partial t} \right)_{\rm collision} -\frac{i}{2\hbar}
\left( \mu_k p_k e^*-\m_k p_k^* e\right). \label{four}\EEA
Here $e(t)$ and $p_k(t)$ are the electric field and polarization at
momentum $\bbox k$ envelopes of the carrier wave $\exp{(-i\O t + i K z)}$
where $\O$ is the cavity frequency (we assume single mode operation only)
and $n(k)$ is the carrier density for electrons and holes. The constants
$\g_E$, $\g_P$ model electric field and homogeneous broadening losses,
$\e_0$ is dielectric constant, $\mu_k$ is the weighting accorded to
different $\bbox k$ momentum (modeled by
$\mu_k=\mu_{k=0}/(1+\e_k/\e_{\rm gap})$), $\L_k$ and $\g_k$
represent carrier pumping and damping. In (\ref{four}), the collision term
is all important and is given by
\BEA
\frac{\partial} {\partial t}n_{k}=
4\pi \int |T_{k k_1 k_2 k_3}|^2
\left( n_{\bbox k_2}n_{\bbox k_3}(1-n_{\bbox k_1}-n_{\bbox k})+
n_{\bbox k}n_{\bbox k_1}(n_{\bbox k_2}+n_{\bbox
k_3}-1)\right)\cr
\times\d({\bbox k}+{\bbox k_1}-\bbox{k_2}-{\bbox k_3})
\d(\o_{k}+\o_{k_1}-\o_{k_2}-\o_{k_3}) d{\bbox k_1}d{\bbox
k_2} d {\bbox k_3} , \label{five}\EEA
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 $k$ ($=|{\bbox k}|$) to $\o$ via the dispersion
relation $\o=\o({\bbox k})$ defining the carrier density $N_\o$ by $\int
N_\o d \o = \int n({\bbox k}) d{\bbox k}$ or $N_\o=4\p k^2 d k /(d\o)
n({\bbox k})$.
Then, in the differential approximation, (\ref{five}) can be written as
both, \BE \frac{\partial N_\o}{\partial t} = \frac{\partial^2
K}{\partial\o^2} \ \ \ \ \ \ {\rm and \ \ \ \ } \frac{\partial \o
N_\o}{\partial t} = - \frac{\partial }{\partial\o}
(K-\o\frac{\partial K}{\partial\o}), \label{six}\EE
with
$$\\ \ \ \ \ K=-I\o^s \left( n^4_o\left(n_\o^{-1}\right)''
+n_\o^2 \left(\ln n_\o\right)''\right), \ \ \ ()'=\frac{\partial
}{\partial\o}, \ \ \ n_\o=n({\bbox k}(\o)),$$
where $s$ is the number computed from the dispersion relation, the
dependence of
$T_{k k_1 k_2 k_3}$ on ${\bbox k}$ and dimensions ($s$ is of the order of
$7$ for semiconductors.) The conservation forms of the equations for
$N_\o$ and $E_\o=\o N_\o$ allow us identify $Q=\frac{\partial K}{\partial
\o}$ (positive if carriers flow from high to low momenta) and
$P=K-\o\frac{\partial K}{\partial\o}$ (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 (\ref{six}) is the integral of $K=Q\o+P$ 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
$n_\o=(\exp{(A\o+B)}+1)^{-1}$, the solution of $K=0$, has zero flux. We
will now solve (\ref{two}), (\ref{three}) and (\ref{four}) after angle
averaging (\ref{four}) and replacing $4\pi k^2 \frac{\partial k}{\partial
\o}\left(\frac{\partial n_k}{\partial \o}\right)_{\rm collision}$ by
$\frac{\partial^2 K}{\partial\o^2}$. The value of the constant $I$ is
chosen to ensure that solutions of (\ref{six}) relax in a time of 100 fs.
\section{Results}
We show the results in figures 2, 3, and 4. First, to test accuracy,
we show, in Figure 2, the relaxation of (\ref{six}) to a pure Fermi-Dirac
spectrum in the window $\o_L=1<\o<\o_0=2$. The boundary conditions
correspond to $P=Q=0$ at both ends. We then modify boundary
conditions to read $Q=Q_0>0$ and $P=0$ at both ends. Next, in Figure 3 and 4, we show the
results of two experiments in which we compare the efficiencies of two
experiments in which we arrange to (i) pump broadly so that the effective
carrier distribution equilibrium has zero flux and (ii) pump carriers and
energy into a narrow band of frequencies about $\o_0$ and simulate this by
specifying carrier and energy flux rates $Q_L$ and $P_L=-\o_L Q_L$
($P_L$ chosen so that the energy absorbed by the laser is consistent with
the number of carriers absorbed there) at the boundary $\o=\o_0$.
$\o=\o_L$ is the frequency at which the system lases.
In both cases, the rate of addition of carriers and energy is
(approximately) the same. The results support the idea that it is
worth exploring the exploitation of the finite flux equilibrium. The
carrier density of the equilibrium solutions at $\o_0$ is small thus
making pumping more efficient there. The output of the laser is
greater by a factor of 10. While we do not claim that, when all
effects are taken account of, this advantage will necessary remain, we
do suggest that the strategy of using finite flux equilibrium
solutions of the Quantum Boltzmann equation is worth further
exploration.
\section{Acknowledgments} We are grateful for support from AFOSR
Contract 94-1-0144-DEF and F49620-97-1-0002.
\section{Figure Captions.}
\begin{itemize}\item Figure 1 \\
Carriers and energy are added at $\o_0$ at rates $Q_0$ and $\o_0 Q_0$.
Energy and some carriers are dissipated at $\o_R>\o_0$ (an idealization)
and carriers and some energy are absorbed by the laser at $\o_L$. (The
carriers number will build until the laser switches on.) A little
calculation shows $Q_L=Q_0 (\o_R-\o_0)/(\o_R-\o_L)$,
$Q_R=Q_0(\o_L-\o_0)/(\o_R-\o_L)$, $P_R=Q_0 \o_R (\o_0-\o_L)/(\o_R-\o_L)$,
$P_L= \o_L Q_0 (\o_0-\o_R)/(\o_R-\o_L)$. Finite flux stationary solutions
are realized in the windows $(\o_L,\o_0)$ and $(\o_0,\o_R)$ although in
practice there will be some losses through both these regions. \item
{Figure 2.}\\ To test accuracy we take some initial distribution function
(thin line) and plot its time evolution as described by (\ref{six}) with
boundary conditions $P=Q=0$ at both ends. The distribution function
relaxes to Fermi-Dirac state (thick line). Several intermediate states
is shown by long-dashed and short-dashed lines (Figure 2a). We then
modify boundary conditions to $P=0,Q=Q_0>0$ at both ends. Then initial
distribution function (thick line) relaxes to finite-$Q$-equilibria as
shown by long-dashed line. We then change boundary conditions to $P=0,
Q=Q_1>Q_0$ at both ends, so that distribution function is shown by
short-dashed line. Increasing $Q$ at boundaries even further, so that
$P=0, Q=Q_2>Q_1$ at both ends, the distribution function is given by
dotted line (Figure 2b). \item Figure 3.\\ We now solve
(\ref{two}-\ref{four}) with the collision term given by (\ref{six}). We
pump broadly, so that the effective carrier distribution has zero flux.
The initial distribution function (thin line) builds up because of a
global pumping (dashed lines), until the laser switches on. The final
(steady) distribution function is shown by thick solid line (Figure 3a).
The output power (in arbitrary units) as a function of time (measured
in relaxation times $\simeq 100$fs) is also shown (Figure 3b). \item
Figure 4.\\ We pump in the narrow region around $\o_0\simeq 200 {\rm meV}$
and we model this by specifying carrier and energy flux rates $Q_L$ and
$P_L=-\o_L Q_L$. The initial distribution function (thin line) builds up
because of influx of particles and energy from right boundary (dashed
lines), until the laser switches on. The final (steady) distribution
function is shown by thick solid line and corresponds to a flux of
particles and energy from right boundary (where we add particles and
energy) to the left boundary, where the system lases (Figure 4a).
The output power as a function of time is also shown (Figure 4b).
\end{itemize}
\newpage
\begin{centerline}{\LARGE Figure 1}\end{centerline}
\
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\newpage
\begin{thebibliography}{100}
\bibitem{Optical} S. Dyachenko, A.C. Newell, A. Pushkarev, V.E.
Zakharov, Physica D {\bf 57}, 96-160, (1992)
\bibitem{HASS}
S. Hasselmann, K. Hasselmanm, J.H. Allender and T.P. Barnett
J.Phys.Oceonography, {\bf 15}, 1378, (1985).
\bibitem{ZLF}V.E. Zakharov, V.S. L'vov
and G.Falkovich, "Kolmogorov Spectra of Turbulence", Springer-Verlag, 1992\bibitem{Koch} W.W. Chow, S.W. Koch,
M. Sargent, "Semiconductor Laser Physics", Springer-Verlag, (1994).
\bibitem{Koch1} H.Haug and S.W. Koch {in} ``Quantum theory of the optical
and electronic properties of the semiconductors'' World Scientific,
(1990). \end{thebibliography}
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