Band structure calculations of Si Ge Sn alloys: achieving direct band gap materials
Summary (3 min read)
1. Introduction
- Recent years have witnessed a widespread use of optoelectronic devices.
- These are not expected to show an indirect-direct transition [1], because of their compressive strain: it is presently believed that a direct gap can only appear in tensilely strained or relaxed Ge1−xSnx, e.g. [5].
- The results for bGeSn are quite remote from each other (even in sign), and both grossly deviate from the experimental value, bGeSn= 2.8 eV [1, 5].
- This clearly indicates that VCA cannot explain the behaviour of disordered Ge1−xSnx alloys [14], although it is considered reasonably accurate for Si1−xGex.
- The aim of this work is to theoretically explore various possibilities of achieving tunable direct gap semiconductors based on group IV materials, and to investigate the composition dependence of their electronic and optical properties.
2. Computational method
- For band structure calculation of SiGeSn alloys the authors use the charge self-consistent pseudopotential Xα method.
- The calculation starts with the construction of the effective potential, including the pseudopotential, the Hartree potential and the exchange-correlation potential Veff (r) = Vps + VHartree(r) + Vxc(r), (1) where VHartree (r) = ∫ e2n(ŕ) |r−ŕ| dŕ, and n (r) is the real-space electron density.
- The calculation of this potential was similar to the calculation of n (q), except that the cube root of n (r) on the cubic grid was taken before transforming from [n (r)] 1 3 to [n (q)] 1 3 by fast Fourier transform.
- These are slightly readjusted values from those given in Ref. [21], as dictated by a larger energy cutoff, and a different method of integration over the Brillouin zone, used in this work.
- Including the spin-orbit coupling would bring slight quantitative corrections in the calculated values of band gaps (with the ionic pseudopotential formfunctions re-adjusted to reproduce the known experimental values for elemental Si, Ge and Sn in this case), but this would not affect the predicted direct-indirect crossover points.
3. Alloy models and their validity
- Alloy properties can be evaluated either within the virtual crystal approximation (VCA) with identical, average-composition atoms populating the lattice sites of the minimumvolume crystalline unit cell, or by populating individual lattice sites only with pure element atoms, in proportion to the alloy composition (“mixed atom method”), in which case one has to use a supercell, with increased volume.
- The former is simpler but may be grossly inaccurate in some cases.
- The lattice constant of an alloy can be estimated from Vegard’s law [25] a0(x) AB = (1− x)aA0 + xaB0 , (6) where aA0 and a B 0 are the lattice constants of elemental crystals of atoms A and B respectively, and a more accurate expression (with bowing) was taken where available.
- The calculated band gaps of Si1−xGex are shown in Fig. 2.
- For strained materials it is also worth comparing the deformation potentials.
4.1. Relaxed Ge1−xSnx alloys
- In studying of the composition dependence of the band structure of unstrained Ge1−xSnx alloy, it is important to note the strong bowing effect in the lattice constant of this alloy, that should be taken into account, even though the size of this effect is not very well known.
- Å, although its validity has been experimentally established only for the Sn content ≤0.2 [35].
- Previous calculations [15] have predicted slightly larger value of the bowing parameter, ≈0.3 Å, though still in reasonable agreement with the experiment.
- The optical (band gap) bowing parameters are clearly much smaller than the above experimental value [1, 5], again showing that the VCA cannot properly predict the composition dependence of the electronic structure of Ge1−xSnx alloys.
- It should also be noted, when comparing theoretical and experimental values for the direct band gap, that the optical measurements of this gap are quite difficult because of small total absorption of actual samples, and measurements necessarily contain a degree of uncertainty.
4.2. Strained Ge on relaxed Ge1−x−ySixSny alloys
- This system is currently believed to be of great practical interest, since it offers a direct band gap in Ge at a reasonable level of strain (>1.8%) as well as type-I heterostructure [11] (of importance for realisation of quantum well structures), together with a small thermal expansion mismatch between the two materials [6].
- This level of strain is considered acceptable for growth of good quality layers, provided they are below the critical thickness [6] (the same limitation applies to strained Ge1−xSnx grown on relaxed Ge1−ySny, considered in the next subsection).
- The line defined by EL = EΓ in the x-y plane is the boundary between regions where the band gap of strained Ge is direct or indirect.
- A direct band gap is achieved for sufficiently large tensile strain of Ge, achievable by growing it on appropriate Ge1−x−ySixSny alloy substrate, as given in Fig. 5.
4.3. Strained Ge1−xSnx on relaxed Ge1−ySny alloys
- For this calculation one needs the elastic constants for the GeSn alloy.
- This was estimated by linear interpolation, since no quadratic correction parameter for this alloy is known.
- Here again the authors find the region in the parameter space that corresponds to a direct band gap semiconductor, achieved by the combined influence of material composition and tensile strain, Fig. 6.
4.4. Relaxed Si1−xSnx alloys
- In this calculation the lattice bowing parameter of the alloy lattice constants of SiSn alloys was set to zero.
- Its value has not been experimentally determined, and (although it may seem a bit surprising in view of a very large difference in atomic radii) the very recent LDA calculations [38] predict a negligible deviation of SiSn alloy lattice constant from the Vegard’s law.
- Therefore, the VCA predicts the indirect-to-direct band gap transition in the relaxed Si1−xSnx when the Sn content exceeds approximately 0.55, while the (more accurate) mixed atom method does not show any such transition.
- This finding may be contrasted to the indication given in the recent work [38] that the direct-indirect crossover in SiSn occurs at approx 25% Sn.
- This was reached using the LDA and (in contrast to their calculation) accounting for the atomic position relaxation, but in order to overcome the well-known LDA shortcomming in the bandgap prediction the “scissors” correction was employed, which itself brings in a degree of uncertainty.
5. Conclusion
- Using local density functional theory and the self-consistent pseudo-potential plane wave method the authors have explored some important properties of GeSiSn alloys, relevant for optoelectronic applications.
- In particular, the authors have studied relaxed Ge1−xSnx alloys, strained Ge grown on relaxed Ge1−x−ySixSny alloys, strained Ge1−xSnx grown on relaxed Ge1−ySny alloys and relaxed SnxSi1−x alloys.
- These were modelled by the mixed atom method, the accuracy of which proved to be far better than that of the virtual crystal approximation, using the available experimental data for comparison.
- Band structure calculations show that relaxed Ge1−xSnx alloys have an indirect-to-direct band gap crossover at a Sn content of ≈0.17, with the bowing parameter equal to 2.49 eV.
- In contrast, within the mixedatom approach the SnxSi1−x alloys never show a finite direct band gap (while the VCA calculation does predict it).
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Frequently Asked Questions (10)
Q2. What is the purpose of this work?
The aim of this work is to theoretically explore various possibilities of achieving tunable direct gap semiconductors based on group IV materials, and to investigate the composition dependence of their electronic and optical properties.
Q3. How can bulk alloys with Sn be grown?
With large lattice mismatch between α-Sn (6.489 Å) and Ge (5.646 Å) or Si (5.431 Å), approximately 15% and 17% respectively, and the instability of cubic α-Sn above 13o C, bulk alloys with Sn cannot be readily grown [1].
Q4. What is the corresponding band gap in the alloy?
Band structure calculations show that relaxed Ge1−xSnx alloys have an indirect-to-direct band gap crossover at a Sn content of ≈0.17, with the bowing parameter equal to 2.49 eV.
Q5. How can one evaluate the properties of Si, Ge, and -Sn?
Alloy properties can be evaluated either within the virtual crystal approximation (VCA) with identical, average-composition atoms populating the lattice sites of the minimumvolume crystalline unit cell, or by populating individual lattice sites only with pure element atoms, in proportion to the alloy composition (“mixed atom method”), in which case one has to use a supercell, with increased volume.
Q6. What is the lattice constant of Ge1xySny?
In the electronic structure calculations the lattice constant of Ge1−x−ySixSny alloys was taken to depend on the Si content (x) and Ge content (y) as [35]aGeSiSn(x, y) = aGe +4SiGex + θSiGe(1− x) +4SnGey + θSnGey(1− y), (12)where 4SiGe = aSi − aGe, 4SnGe = aSn − aGe, and θSiGe = −0.026 Å.
Q7. What is the optimum strain for the Ge1xSnx alloy?
This system is currently believed to be of great practical interest, since it offers a direct band gap in Ge at a reasonable level of strain (>1.8%) as well as type-I heterostructure [11] (of importance for realisation of quantum well structures), together with a small thermal expansion mismatch between the two materials [6].
Q8. What is the description of the optical band gap bowing?
According to Chibane et al. [15], who used the model developed by Zunger et al. [16], thecalculated optical band gap gap bowing is in good agreement with experiment for small Sn contents.
Q9. What is the lattice constant of an alloy?
The lattice constant of an alloy can be estimated from Vegard’s law [25]a0(x) AB = (1− x)aA0 + xaB0 , (6)where aA0 and a B 0 are the lattice constants of elemental crystals of atoms A and B respectively, and a more accurate expression (with bowing) was taken where available.
Q10. What is the valence band of the X valley?
The computational method used in this work, and the conditions (strain along the [001] axis), allow meaningful extraction of:: the uniaxial deformation potential b for the valence band at Γ, the sum of valence and conduction band (at Γ) hydrostatic deformation potentials av +ac, and the uniaxial deformation potential Ξ∆u of the X (i.e. ∆) valley of the conduction band.