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Next: 3.2.9 Direct Generation/Recombination Up: 3.2 Material Models Previous: 3.2.7 Energy Relaxation Times

3.2.8 Shockley-Read-Hall Generation/Recombination

Although HEMTs are considered unipolar devices, phenomena such as frequency dispersion, gate currents, and electroluminescence experiments give a strong hint of the influence of holes, in particular, their generation and recombination in these devices. The generation/recombination rates are modeled according to the well known formulae:

    $\displaystyle R^{\mathrm{SRH}} =  \frac{n \cdot p - n_i^2}{\tau_p \cdot(n+n_1)+\tau_n \cdot (p+p_1)},$   and:$\displaystyle  n_i^2=
n_1 \cdot p_1$ (3.52)

    $\displaystyle n_1 = N_C({\it T}_\mathrm{L})\cdot \exp \bigg(\frac{-E_C + E_T}{{\it k}_{\mathrm{B}}\cdot {\it T}_\mathrm{L}}\bigg)$ (3.53)

    $\displaystyle p_1 = N_V({\it T}_\mathrm{L})\cdot \exp \bigg(\frac{ E_V - E_T}{{\it k}_{\mathrm{B}}\cdot {\it T}_\mathrm{L}}\bigg)$ (3.54)

The thermal recombination velocity is defined as:
    $\displaystyle v_{n,p} =  \sqrt{\frac{3\cdot {\it k}_{\mathrm{B}}\cdot {\it T}_\mathrm{L}}{m_{n,p}}}$ (3.55)

Corrections are applied to the lifetimes at $ {\it T}_\mathrm{L}$= 300 K according to:
    $\displaystyle \tau_{n, 300 K}  = \frac{1}{\sigma_{T,n}\cdot {\it N}_{\mathrm{T}}\cdot v_{n,300}+ S_n/y}$ (3.56)

    $\displaystyle \tau_{p, 300 K}  = \frac{1}{\sigma_{T,p}\cdot {\it N}_{\mathrm{T}}\cdot v_{p,300}+ S_p/y}$ (3.57)

where $ S_n$ and $ S_p$ are the surface recombination velocities to correct the influence of surface. y represents the distance to the surface. The temperature dependence is modeled according to:
    $\displaystyle \tau_{n}({\it T}_\mathrm{L})  = \bigg(\frac{300 K}{{\it T}_\mathrm{L}}\bigg)^{3/2} \cdot \frac{\tau_{n, 300}}{1+\Gamma_n}$ (3.58)

    $\displaystyle \tau_{p}({\it T}_\mathrm{L})  = \bigg(\frac{300 K}{{\it T}_\mathrm{L}}\bigg)^{3/2} \cdot \frac{\tau_{p, 300}}{1+\Gamma_p}$ (3.59)

where $ \Gamma_\nu$ is the field enhancement factor only relevant for materials with an indirect band gap. Thus, the trap assisted band to band tunneling (TBB) is neglected for direct band gap materials. The parameters for the SRH recombination model are summarized in Table 3.23. The exact traps energies have to be checked with analysis of the growth process given in the references. In this work, for AlGaAs midgap traps of $ E_T$= 0.8 eV were used. For InGaAs $ E_T$= 0.3 eV was typically assumed. The lifetimes obtained by (3.59)-(3.62), i.e. the product of $ N_T$ and $ \sigma_{T \nu}$ can be controlled by measurements of the minority lifetime in material of the same growth type, e.g. given in [7]. In Al$ _x$Ga$ _{1-x}$As, though a direct semiconductor for x $ \leq$ 0.45, the lifetime is dominated by SRH recombination. For the surface recombination velocity typical values are given.

Table 3.23: Relative Shockley-Read-Hall recombination model parameter values.
Material $ N_T$ $ E_T$ $ \sigma_{T n}$ $ \sigma_{T p}$ S$ _n$ S$ _p$ References
  [cm$ ^{-3}$] [eV] [cm$ ^2$] [cm$ ^2$] [cm/s] [cm/s]  
GaAs 1e14 0.2-0.8 1e-16 (MBE) - - 0 [215]
InP 1e13 0.1-0.5 $ <$ 1.7e-13 - - - [143,214]
Si 1e13 0 1e-15 1e-15 100 1e2-1e5 [146,251]
Al$ _x$Ga$ _{1-x}$As 1e15-1e16 0.2-0.8 5e-14 5e-15 0 0 [216]
In$ _x$Ga$ _{1-x}$As $ <$ 1e15 0.17-0.5 1e-15 - 0 0 [41]

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Next: 3.2.9 Direct Generation/Recombination Up: 3.2 Material Models Previous: 3.2.7 Energy Relaxation Times