Bandgap Variation and Miscibility Gaps of Thallium-Based Pseudo-Binary Alloys
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297 Mat. Res. Soc. Symp. Proc. Vol. 607 © 2000 Materials Research Society
BAND GAP VERSUS LATTICE CONSTANT Fig. 1 shows the plots of the calculated direct band gaps (at F) versus the lattice constants of III-V zincblende semiconductors consisting of (Ga, In, T1)(P, As, Sb). The band structures were calculated from a hybrid empirical pseudo-potential and tight-binding method [5]. This method was shown to be effective in incorporating the empirical information to produce accurate band structures for the III-V zincblende compounds. The Hamiltonians of these pure crystals were then used as input in the calculation of the alloy band structures using the so-called molecular coherent potential approximations (MCPA) [5, 6]. The input parameters for TIP, T1As, and TlSb were those deduced in Ref. [1]. The concentration (x) dependence of the calculated band gap Eg of a pseudo-binary alloy, e.g., Gal.xInxAs, is well described by a quadratic form: Eg=-x(1-x)b, where is the concentration weighed average of the pure crystal values. The values of the bowing
parameter b are given in Table I. These bowing parameters and the Vegard's law [7], i.e., the alloy lattice constant being the concentration weighed average, are then used to generate the band gap versus the lattice constant plots in Fig. 1. These results will be used along with the miscibility curves in the next section to discuss the selection of the Tl-based alloys and possible substrates for the growth of these alloys. Table L The calculatedbowing parameters b (in eV)for the energy gap at Ffor several Tlbased pseudo-binaryalloys A 11xBC. The calculatedand experimentalb values for other non-Tlbearing alloys are tabulatedin Table 7.102 of Ref 5. Also listed are the band gaps at Tfor the constituent compounds. (A, B)C
Eg (AC)
Eg (BC)
b
(Ga,TI)P (In,TI)P (In,TI)As (Ga,TI)As (Ga,TI)Sb (In,T1)Sb
2.866 1.420 0.422 1.520 0.822 0.237
-0.240 -0.240 -1.340 -1.340 -1.600 -1.600
Calc. 0.279 0.840 0.401 0.150 0.696 1.521
(Ga,In)As (Ga,In)Sb (Ga, In)P
1.520 0.822 2.866
0.422 0.237 1,420
0.348 0.406 0.517
Exp.
0.40 0.413 0.758, 0.7, 0.5
ALLOY MISCIBILITY GAP The two essential ingredients for the calculation of the phase diagram of pseudo-binary semiconductor alloys are the alloy mixing enthalpy AE and mixing entropy AS. A review of different approaches to AE and AS was given in Chapter 4 of Ref. 5. The principal methods for AE include the delta lattice parameter model [8], the valence force field (VFF) model [9], a
298
3 2
GaSb
1 -GaAs z Go
-1 5.4
lsTb 5.6
5.8
6.2
6.0
6.4
6.6
6.8
7.0
LATTICE CONSTANT (A)
Figure 1. Band gaps versus lattice constants
0) Cu 0
1200 800
Q)
I-
400 1200
800 400 800 600 400 0.0
K
0.2
I
I
0.4
0.6
a J XIXSb 0.8
x Figure 2. Miscibility gaps for three TI-based alloys
299
1.0
combined VFF and tight-binding method [10], and a total energy model based on the LDA calculation [ 11]. The entropy theory includes the regular-solution model, the generalized quasichemical approximation (GQCA) [5,12], the cluster variational met
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