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2.1 Limitations of the Channel Material InGaAs
To obtain a high conduction band offset DEC of the channel/barrier interface the In content has to be maximized as shown in  Figure 2.2. A second benefit of a high In mole fraction is an improved transport behavior as shown in  Figure 2.4. The dark shaded area refers to data from [7-9] obtained from bulk GaAs. The light shaded area comprises data for In0.53Ga0.47As on InP substrates from different authors compiled in [10]. It shows much higher velocity over the whole electric field range. Unfortunately, the In content of InGaAs grown on GaAs without dislocations has tight limits which will be shown in the following.

Figure 2.4 Electron drift velocity in GaAs, InAs and InGaAs bulk material versus electric field. III-V semiconductors typically exhibit a local maximum in the v(E) characteristics. The velocity is increased with the In content in the whole range of electric field.

The lattice constants of GaAs, AlAs, InAs and their alloys are shown in  Figure 2.2 versus the band gap along with other important semiconductors. As shown the lattice constants of GaAs and AlAs differ only slightly. Therefore AlyGa1-yAs can be grown in almost arbitrary thicknesses on GaAs for all Al contents without introducing dislocations in the crystal (lattice matched). The lattice constant of InxGa1-xAs is significantly larger than that of GaAs for considerable In contents. Nevertheless, InxGa1-xAs can be grown with moderate In contents in thin strained layers on GaAs substrates without introducing dislocations which is called pseudomorphic growth.

The limitations of the In content and the corresponding thickness of the layer is given by the critical thickness (dC cr). If the layers thickness is increased above dC cr an increased dislocation density has to be expected. dC cr can be calculated according to the theory of Matthews and Blakeslee [11-13] by:
where aS is the lattice constant of the unstrained substrate. The inplane strain  is given by
and  is given by
The elastic constants C11 and C12 of InGaAs can be calculated by a linear interpolation of the binary values of GaAs and InAs which can be found in Table 2.1.

Table 2.1 Parameters taken from [16]
d1 GaAs [eV]
d2 GaAs [eV]
d1 InAs [eV]
d2 InAs [eV]
aGaAs [nm]
C11 GaAs [GPa]
C12 GaAs [GPa]
C11 InAs [GPa]
C12 InAs [GPa]
aInAs [nm]

In  Figure 2.5 the solutions for the critical thickness dC cr of (1) are shown. The theory of Matthews and Blakeslee is confirmed by various data of the literature. Samples below the critical thickness exhibit a low dislocation density whereas samples above the critical thickness show a high dislocation density, i. e. the strained layer starts to relax. The asterisk symbols in  Figure 2.5 represent samples with moderate dislocation density. This shows that the transition from pseudomorphic growth to relaxation is not abrupt. Typical channels of pseudomorphic HEMTs have a thickness of 12 nm and an In content of 20% which is below the critical thickness. But also HEMTs with other channel thicknesses and compositions have been reported [17, 18].

Figure 2.5 Critical thickness (bold line) of a GaAs/InxGa1-xAs/GaAs single quantum well according to the theory of Matthews and Blakeslee. Empty symbols represent samples with low dislocation density and filled symbols samples with high dislocation density. Samples with moderate dislocation density are indicated by the asterisks.

next up previous contents
Next: 2.2 Electron Energy Levels in Strained Quantum Wells Up: 2 The Principles of a HEMT Previous: 2 The Principles of a HEMT

Helmut Brech