2.2 Electron Energy Levels in Strained Quantum Wells
 
When InGaAs is grown pseudomorphically on GaAs the InGaAs layer is biaxially compressively strained because the lattice constant of unstrained InGaAs is larger than that of GaAs. Compressive strain causes an increase in E_g which acts contrary to the reduction in E_g with increasing In mole fraction, i. e. the reduction of E_g is partially canceled by the strain effect. The strained band gap energy  can be calculated by [19]:
 

(4)

where the hydrostatic part DE_h of the energy shift reads
 

(5)

 
and the uniaxial contribution DE_s of the energy shift reads
 

.

(6)

 
  is given by (2).

The values of the lattice constants a, the deformation potentials d₁ and d₂ as well as the elastic constants C₁₁ and C₁₂ for GaAs and InAs, respectively, can be found in Table 2.1.  The values for In_xGa_1-xAs are obtained by linear interpolation between the numbers for the binary materials.

Due to the uniaxial strain the degeneracy of the valence band is canceled. The positive sign in (6) holds for the shift of the heavy hole band and the negative sign for the shift of the light hole band. The investigations on strain dependence of the band discontinuities in the literature are contradictory [20, 21, 22]. Because no consistent results could be found in the literature 60 % of the energy difference between the conduction band and 40 % difference in the heavy hole valence band is assumed for DE_C and DE_V between GaAs and InGaAs as well as for the heterojunction between AlGaAs and InGaAs as described in Section 2.1.

In a quantum well the continuous energy levels of the conduction band split up into discrete levels. The energy levels E_e can be calculated by solving the Schrödinger equation for a single potential well with finite barrier height. The symmetrical part of the solution is given by
 

(7)

 
which reveals the lowest energy E_e, i. e. the ground state of the quantum levels in the potential well. In (7) d_C is the thickness of the quantum well, and h is the Planck constant, and m_AlGaAs and m_InGaAs are the effective masses of the barrier and channel materials, respectively.

For the current transport over a potential barrier the difference between the energy level of the electron and the energy maximum of the barrier has to be taken into account. Within MINIMOS­NT only properties of bulk material are used for the simulation models in the channel. To account for strain and quantum effects an effective band gap E_{g eff} and therefore an effective conduction band offset DE_{C eff} as shown in  Figure 2.6 is calculated separately according to (4) and (7) and used for the channel material.

The unstrained conduction band offset DE_{C unstrained} is reduced by the bandgap shift due to strain and the quantization of the electrons in the channel. Due to the critical thickness a trade off between the In content x and the channel thickness d_{C cr}(x) has to be made as described in Section 2.1. Therefore the following considerations are based on In_xGa_1­xAs layers and the critical layer thickness corresponding to the In content x. Given this limit for the thickness d_{C cr} of a quantum well for each In content the quantum level of electrons E_e can be calculated "along" the critical thickness d_{C cr}.

In  Figure 2.7 E_e is shown for an Al_0.2Ga_0.8As/In_xGa_1-xAs/Al_0.2Ga_0.8As quantum well versus the Indium content. Also DE_{C strained} and DE_{C unstrained} are shown. For a strained Al_0.2Ga_0.8As/In_0.2Ga_0.8As/Al_0.2Ga_0.8As channel with a critical thickness of 14 nm the energy level of electrons is 20.5 meV above the band edge (E_e= 26.5 meV for a 12 nm thick channel which is typically used). The shift in E_g due to strain (DE_h + DE_s) accounts for 45.5 meV. According to Figure 2.6 both effects reduce the unstrained DE_C from 331 meV to 265 meV.  Figure 2.7 shows that DE_{C eff} could be increased to about 320 meV for an In content of more than 50 %. But problems in epitaxial growth and effects such as surface roughness and alloy scattering severely degrade the crystal quality and thus the transport properties [23, 24, 25, 26].
 

Several attempts have been made to overcome the tight limitations of pseudomorphic growth on GaAs substrates. One possibility is to change the lattice constant for the growth conditions of the HEMT layers. This can be achieved by growing thick relaxed InGaAs layers on the GaAs substrate which is called metamorphic growth [27, 28, 29] or by using InP as substrate.  Figure 2.2 reveal the ternary systems InGaAs and InAlAs lattice matched to InP. The two materials yield an DE_C of about 700 meV. In addition the electron velocity is substantially increased compared to pseudomorphic InGaAs on GaAs as shown in  Figure 2.4. For HEMTs on InP extrinsic f_T's exceeding 300 GHz are reported [30]. The drawback concerning devices performance is low power capability. Due to the small E_g in the channel breakdown occurs for relatively low electric fields compared to GaAs based pseudomorphic HEMTs. But practically the most severe drawback is the high cost of InP wafers, the small wafer size (mostly 3'' today) and the incompatibility of the process technology with other cheaper standard processes that are used for instance in a MESFET production line.
 

• 2.3 Limitations of the Barrier Material AlGaAs

   
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