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For the ternary semiconductors the low field mobility is composed according to the Matthiesen rule with a bowing correction.

    $\displaystyle \frac{1}{\mu^{AB}} = \frac{1-x}{\mu^{A}}+\frac{x}{\mu^{B}}+ \frac{(1-x)\cdot x}{C_{\mu}}$ (3.40)

This concept is useful, if the two mobilities to combine are of similar magnitude. For InAlAs, however, in a one-valley modeling approach the binary mobilities to combine differ by two orders of magnitude. In this case,  (3.42) can supply negative mobility values for certain parameter combinations of bowing parameters, doping, temperature, and material composition. In Fig. 3.4 a one-valley approach is also given, which shows the significant disagreement with the available data at $ x$= 0.52. Fig. 3.3 shows the comparison of measurements, MC data and the composition dependent analytical low field mobility model for Al$ _{x}$Ga$ _{1-x}$As. The figures illustrate the improvement of the two valley modeling concept, since any one-valley bowing approach for electrons will either neglect certain compositions, where the mobility gets negative, to allow a precise fit of other compositions, or will deliver completely wrong values. The data are taken from [304]. For holes it was found, that a one-valley approach can fit the available data with sufficient accuracy. Fig. 3.4 shows the carrier mobility for In$ _{x}$Al$ _{1-x}$As versus material composition $ x$ for background concentration of 10$ ^{15}$ cm$ ^{-3}$. Although InAlAs is a very important material for both InP based HEMTs and HBTs, the understanding of the properties of InAlAs is still very poor, as can be seen in Fig. 3.4. Table 3.15 supplies the bowing parameters for (3.42).

Table 3.15: Bowing parameters for ternary alloys for harmonic bowing in (3.42).
Material C $ _{n,one valley}$ C $ _{n,\Gamma}$ C$ _{n,X}$ C$ _{\mu,p}$
  [cm$ ^2$/Vs] [cm$ ^2$/Vs] [cm$ ^2$/Vs] [cm$ ^2$/Vs]
Al$ _{x}$Ga$ _{1-x}$As 180 -250 1e6 1e6
In$ _{x}$Ga$ _{1-x}$As 1e6 1e6 - 1e6
In$ _{x}$Al$ _{1-x}$As 1e6$ ^1$ 1.5e8 1e6 1e6
Al$ _x$Ga$ _{1-x}$N 1e6 1e6 - 1e6

Figure 3.5: Carrier mobility as a function of material composition for In$ _x$Ga$ _{1-x}$As at $ {\it T}_\mathrm{L}$= 300 K.

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Fig. 3.5 shows the material dependence of the mobility in In$ _{x}$Ga$ _{1-x}$As. The simple quadratic bowing approach applies for the low field mobility shown for the whole material composition. The experimental data to compare are Hall data taken from HEMTs from the AlGaAs/InGaAs and the InAlAs/InGaAs materials system. The data for the pseudomorphic HEMTs are taken from [77] and [79]. For the metamorphic HEMTs grown on GaAs the data are taken from [314] and the compilation of [113], the data for InP based devices are taken from [322]. If a double channel concept is applied, a weighted average is determined for $ x$. It can be seen that the metamorphic devices reach similar mobility values as the corresponding InP devices. In Fig. 3.6 the comparison of MC data, measurements, and the analytical model for In$ _{0.48}$Al$ _{0.52}$As as a function of doping is shown. The data are taken from the compilation of Littlejohn et al. in [163] and selectively from Goto et al. in  [105]. By using a special value for $ \Gamma$ AlAs, given in Table 3.13 suitable for InAlAs only, very good agreement is achieved in Fig. 3.6. The problem of possibly negative mobility values in (3.42) can be overcome with the bowing parameter in Table 3.15, without having to apply a second material composition concept for the ternary alloys in the simulator, as e.g. suggested by Sotoodeh et al. in [273].

Figure 3.6: Comparison of the analytical model, measurement, and MC data for In$ _{0.52}$Al$ _{0.48}$As versus doping concentration at $ {\it T}_\mathrm{L}$= 300 K [105,163].

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Figure 3.7: Comparison of the analytical model, measurements, and MC data for In$ _{0.53}$Ga$ _{0.47}$As versus doping concentration at $ {\it T}_\mathrm{L}$= 300 K.

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In Fig. 3.7 a similar comparison for In$ _{0.53}$Ga$ _{0.47}$As is given. The figure stresses the importance of the residual background concentration for the low field properties [191]. To allow for the simulation of metamorphic and pseudomorphic HEMTs on InP substrate, MC simulations have been performed for $ x$ $ \neq$ 0.52 to support the available MC investigations in [120] and data compiled in [163].
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