Thermodynamics-based modeling of semiconductor lasers

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Thermodynamics-based modeling of semiconductor lasers

Collaborator: U. Bandelow, H. Gajewski, R. Hünlich

Cooperation with: H. Wenzel (Ferdinand-Braun-Institut für Höchstfrequenztechnik (FBH) Berlin)

Description:

In modern semiconductor devices such as high power transistors or lasers, thermal effects caused by strong electric and optical fields and by strong recombination play an important role and have to be included in the mathematical models. Indeed, there is a large variety of energy models for semiconductor devices. Typically, these models base on the usual state equations and continuity equations for the carrier densities, and on the balance of the total energy expressed by the equation

$\displaystyle \partial_{t}^{}$ u + $\displaystyle \nabla$ j_u = - $\displaystyle \gamma$

(1)

for the density u and the current density j_u of this total energy, where $\gamma$ counts for the radiation which is emitted from the device. Furthermore, differential relations for u and general thermodynamic relations for j_u are used to transform the energy balance equation (1) into a heat flow equation

c_h $\displaystyle \partial_{t}^{}$ T - $\displaystyle \nabla$ ^.( $\displaystyle \kappa_{L}^{}$ $\displaystyle \nabla$ T) = H,

(2)

where c_h is the heat capacity and $\kappa_{L}^{}$ the heat conductivity. While the heat flow equation (2) with the description of the source term H is well established, the discussion about its relation to the conservation law of energy is still ongoing.

In our model, the transport of electrons and holes is ruled by the drift-diffusion equations

- $\displaystyle \nabla$ ( $\displaystyle \varepsilon$ $\displaystyle \nabla$ $\displaystyle \varphi$ )	=	C + p - n ,	(3)
$\displaystyle \partial_{t}^{}$ n - $\displaystyle \nabla$ ^. $\displaystyle \bf { j}_{n}^{}$ = - R ,		$\displaystyle \partial_{t}^{}$ p + $\displaystyle \nabla$ ^. $\displaystyle \bf { j}_{p}^{}$ = - R	(4)

for the densities of electrons n and holes p, and the electrostatic potential $\varphi$ in the transverse cross section $\Omega$ of the laser. The recombination rate R in (4) involves all non-radiative and radiative recombination processes. The current densities $\bf { j}_{n}^{}$ and $\bf { j}_{p}^{}$ are driven by the gradients of the quasi-Fermi potentials f_n and f_p, which are linked with the carrier concentrations by means of Fermi-Dirac statistics:

n = N_c $\displaystyle \cal {F}$ _1/2 $\displaystyle \left(\vphantom{\frac{\varphi-f_n-e_c}{T}}\right.$ $\displaystyle {\frac{{\varphi-f_n-e_c}}{{T}}}$ $\displaystyle \left.\vphantom{\frac{\varphi-f_n-e_c}{T}}\right)$ , p = N_v $\displaystyle \cal {F}$ _1/2 $\displaystyle \left(\vphantom{\frac{e_v+f_p-\varphi}{T}}\right.$ $\displaystyle {\frac{{e_v+f_p-\varphi}}{{T}}}$ $\displaystyle \left.\vphantom{\frac{e_v+f_p-\varphi}{T}}\right)$ .

(5)

The optical field distribution $\chi$ ( $\bf { r}$ ) within $\Omega$ is governed by the waveguide equation

$\displaystyle \left[\vphantom{ \nabla ^2 +\frac{\omega^2}{c^2}\varepsilon_{opt}(\omega,{\bf { r}} ) -\beta^2 }\right.$ $\displaystyle \nabla^{2}_{}$ + $\displaystyle {\frac{{\omega^2}}{{c^2}}}$ $\displaystyle \varepsilon_{{opt}}^{}$ ( $\displaystyle \omega$ , $\displaystyle \bf { r}$ ) - $\displaystyle \beta^{2}_{}$ $\displaystyle \left.\vphantom{ \nabla ^2 +\frac{\omega^2}{c^2}\varepsilon_{opt}(\omega,{\bf { r}} ) -\beta^2 }\right]$ $\displaystyle \chi$ ( $\displaystyle \bf { r}$ ) = 0

(6)

where the optical response function $\varepsilon_{{opt}}^{}$ ( $\omega$ , $\bf { r}$ ) contains the refractive index and the material gain, which depends on almost all properties of the device and its operating state, in particular on n, p, and T. The respective (complex) eigenvalue $\beta$ readily counts the number N_s of photons in the laser by a corresponding photon rate equation

$\displaystyle \dot{{N_s}}$ = v_g(2 $\displaystyle \Im$ m $\displaystyle \beta$ - $\displaystyle \alpha_{0}^{}$ - $\displaystyle \alpha_{{m}}^{}$ )N_s + r^sp.

(7)

The field intensity N_s^.| $\chi$ ( $\bf { r}$ )|² times the material gain enters the radiative recombination rate in R in (4), which self-consistently completes our model ([1]). Based on an expression for the density of the free energy

f	=	$\displaystyle {\frac{{\varepsilon}}{{2}}}$ \| $\displaystyle \nabla$ $\displaystyle \varphi$ \|² + c_LT(1 - logT) + u_rad - Ts_rad
		+ n[T(log $\displaystyle {\frac{{n}}{{N_c}}}$ -1) + e_c] + p[T(log $\displaystyle {\frac{{p}}{{N_v}}}$ -1) - e_v],	(8)

we extended (3)-(7) to a thermodynamics-based system (2)-(7) of evolution equations for semiconductor lasers in a deductive way. Here, we only apply first principles like the entropy maximum principle and the principle of partial local equilibrium as well as the Onsager symmetry relations ([2]). In particular we could show that the heat source term H in (2) is

H	=	T $\displaystyle \nabla$ P_n^. $\displaystyle \bf { j}_{n}^{}$ - T $\displaystyle \nabla$ P_p^. $\displaystyle \bf { j}_{p}^{}$ + T $\displaystyle \nabla$ ^.( $\displaystyle \bf { j}_{n}^{}$ - $\displaystyle \bf { j}_{p}^{}$ )
		- $\displaystyle \bf { j}_{n}^{}$ ^.( $\displaystyle \nabla$ f_n - P_n $\displaystyle \nabla$ T) - $\displaystyle \bf { j}_{p}^{}$ ^.( $\displaystyle \nabla$ f_p + P_p $\displaystyle \nabla$ T)
		+ (u_n + u_p)R - $\displaystyle \partial_{t}^{}$ u_rad - $\displaystyle \gamma$ ,	(9)

with the thermoelectric powers P_n, P_p, and the energy density of the optical field u_rad = $\hbar$ $\omega$ | $\chi$ |²N_s. The current densities $\bf { j}_{n}^{}$ and $\bf { j}_{p}^{}$ are now driven by the gradients of the temperature T, too. Moreover, a continuity equation for the entropy density s

$\displaystyle {\frac{{\partial s}}{{\partial t}}}$ + $\displaystyle \nabla$ ^. $\displaystyle \bf { j}_{s}^{}$ = d /T,

(10)

with the entropy current density $\bf { j}_{s}^{}$ could be derived. The dissipation rate d

d	=	$\displaystyle {\frac{{\kappa_L}}{{T}}}$ \| $\displaystyle \nabla$ T\|² + $\displaystyle \sigma_{n}^{}$ \| $\displaystyle \nabla$ f_n - P_n $\displaystyle \nabla$ T\|² + $\displaystyle \sigma_{p}^{}$ \| $\displaystyle \nabla$ f_p + P_p $\displaystyle \nabla$ T\|²
		+ $\displaystyle \sigma_{{np}}^{}$ \| $\displaystyle \nabla$ (f_n - f_p) - (P_n + P_p) $\displaystyle \nabla$ T\|² + (f_p - f_n)R - $\displaystyle \gamma$	(11)

appears to be always positive for a device which is isolated from the outside world ( $\gamma$ = 0). Therefore, by partial integration of (10) and supposing no-flux boundary conditions, and $\gamma$ = 0, it follows, according to the second law of thermodynamics,

$\displaystyle {\frac{{dS}}{{dt}}}$ = $\displaystyle \int_{{\Omega}}^{}$ $\displaystyle {\frac{{ds}}{{dt}}}$ d $\displaystyle \Omega$ = $\displaystyle \int_{{\Omega}}^{}$ $\displaystyle {\frac{{d}}{{T}}}$ d $\displaystyle \Omega$ $\displaystyle \ge$ 0 .

(12)

In conclusion, as a feature, we are able to prove the thermodynamic correctness of our model in view of the second law of thermodynamics (12).

The complete energy transport model has been implemented in WIAS-TeSCA ([3]), a numerical code for the simulation of semiconductor devices. On this base, we have demonstrated the simulation of long-wavelength edge-emitting quantum well lasers, with a special focus on the self-heating of the device and the modulation response ([2]).

References:

U. BANDELOW, TH. KOPRUCKI, R. HÜNLICH,
Simulation of static and dynamic properties of edge-emitting multi quantum well lasers,
IEEE J. Select. Topics Quantum Electron., 9 (2003), pp. 798-806.
U. BANDELOW, H. GAJEWSKI, R. HÜNLICH, Thermodynamics-based modeling of edge-emitting quantum well lasers, in preparation.
WIAS-TeSCA.
http://www.wias-berlin.de/software/tesca, 2003.