Upscaling of microscopically calculated characteristics to

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Upscaling of microscopically calculated characteristics to macroscopic state equations

Collaborator: U. Bandelow (FG 1), Th. Koprucki (FG 3)

Supported by: DFG: Priority Program ,,Analysis, Modellbildung und Simulation von Mehrskalenproblemen`` (Analysis, modeling and simulation of multiscale problems)

Description:

Eight-band kp-Hamiltonians [1] are detailed models for the calculation of quantum-confined states in semiconductor quantum-well structures, because they consistently include band mixing, spin-orbit interaction and strain effects. Quantities such as carrier densities or the optical response can be efficiently obtained in terms of the states ([2]). To utilize such microscopically calculated data for device simulation, e.g., with drift-diffusion-type models [2], [3], a suitable upscaling to macroscopic state equations is required. We have demonstrated upscaling schemes for the carrier densities, the peak gain characteristics and the spontaneous radiative recombination rate calculated with WIAS-QW for an example quantum-well structure [3].

The investigations have been carried out for an InGaAsP-based single quantum-well structure which is designed for emission at 1.55 $\mu m$ consisting of a compressively strained quantum well of width d_QW=7 nm sandwiched between tensile strained barriers.

As a first example for upscaling we consider the carrier densities. kp calculations [2] generically provide local carrier densities n(z) and p(z) depending on the Fermi levels F_e and F_h and the temperature T. Since the carriers localize very well in the quantum well [2] we can introduce an average carrier density per quantum well $\bar{n}=\int n(z)dz / d_{QW}$ as a natural quantity for upscaling. Thus we obtain from the microscopical model the relations $\bar{n}(F_e, k_B T)$ and $\bar{p}(F_h, k_B T)$ for the averaged carrier densities as depicted in Figure 1. The upscaling to a macroscopic level can then be performed by fitting the calculated relations $\bar{n}(F_e, k_B T)$ and $\bar{p}(F_h, k_B T)$ to macroscopic state equations such as the Fermi distribution used in drift-diffusion models. As shown in Figure 1 we observed that both quantities can be reasonably fitted to the relations

$\begin{displaymath} \bar{n}=N_c {\cal F}_{1/2}\Big(\frac{F_e-E_c}{k_BT}\Big), \quad \bar{p}=N_v {\cal F}_{1/2}\Big(\frac{E_v-F_h}{k_BT}\Big)\end{displaymath}$ (1)

by adjusting the effective band edges E_c and E_v and the band-edge densities of state N_c and N_v. ${\cal F}_{1/2}$ is the Fermi integral of order 1/2. The fit for the valence bands is very close to the calculated curve for a wide range of densities and temperatures whereas for the conduction bands the macroscopic state equation yields a good approximation only for a certain range of parameters. Thus, an upscaling of band-structure information by the dependence of the average density per quantum well on the Fermi level is possible. This allows us to treat the quantum well as a classical material with microscopically defined band-edge densities of state N_c and N_v and net band edges E_c and E_v which differ from the band-structure parameters of the well material and cannot be estimated elsewhere.

Fig. 1: Relation between the average carrier densities $\bar{n},\bar{p}$ and the Fermi levels F_e,F_h relative to the net band edge E_c, E_v for the temperatures T=290 K, 315 K, 340 K. The dashed lines indicate the fit to the macroscopic state equation (1). Left: electrons occupying the conduction subbands, right: holes occupying the valence subbands.
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Furthermore, the spectra of the optical material gain can be calculated in terms of the band-structure and transition matrix elements given by the wavefunctions [2], [3]. This defines a function $g(\hbar\omega, F_e, F_h, k_B T)$ depending on the transition energy $\,\hbar\omega$ , the Fermi levels F_e and F_h and the temperature T. Most lasers are designed to emit at the spectral gain maximum. Therefore the peak gain g_p is an important quantity.

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As a second example we have calculated the peak gain $g(\bar{n})$ (assuming charge neutrality $\bar n$ = $\bar p$ ) as a function of the average carrier density introduced above. The result is depicted in Figure 2. For sufficiently high carrier densities (above $2.5\cdot 10^{18} cm^{-3}$ ) this peak gain characteristics can be upscaled to a usually used state equation given by the logarithmic gain model

$\begin{displaymath} g_p(\bar{n})=g_0 \ln(\bar{n}/n_{tr})\end{displaymath}$

(2)

as shown in Figure 2. However, for low densities this way of upscaling completely fails. The constant g₀ appears to be nearly independent of the temperature whereas the transparency carrier density n_tr should roughly linearly increase with the temperature.

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As a third example we analyzed the upscaling of the spontaneous radiative recombination rate [3] which is a further important quantity for the analysis of semiconductor lasers, because it essentially determines the laser threshold. The spontaneous radiative recombination rate R_rad(F_e, F_h, T) obtained by the kp calculations depends on the Fermi levels F_e and F_h and the temperature. Again, for local charge neutrality $\bar{n}=\bar{p}$ we found out that this relation can be suitably upscaled to the power law

$\begin{displaymath} R_{rad}=B \cdot \bar{n}^\alpha\end{displaymath}$

as depicted in Figure 3. The exponent for the fit was approximately $\alpha=1.5$ , which differs from the commonly used models corresponding to $\alpha=2$ .

References:

U. BANDELOW, H.-CHR. KAISER, TH. KOPRUCKI, J. REHBERG, Spectral properties of $k\cdot p$ Schrödinger operators in one space dimension, Numer. Funct. Anal. Optim., 21 (2000), pp. 379-409.
$\dito$ , Modeling and simulation of strained quantum wells in semiconductor lasers, to appear in: Mathematics -- Key Technology for the Future II, K.-H. Hoffmann, W. Jäger, T. Lohmann, H. Schunck, eds., Springer, Berlin, Heidelberg.
U. BANDELOW, R. HÜNLICH, TH. KOPRUCKI, Simulation of static and dynamic properties of edge-emitting multi quantum well lasers, WIAS Preprint no. 799 , 2002.