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Journal ArticleDOI

Band structure calculations of Si Ge Sn alloys: achieving direct band gap materials

01 Jul 2007-Semiconductor Science and Technology (IOP Publishing)-Vol. 22, Iss: 7, pp 742-748
TL;DR: In this article, the electronic structure of relaxed or strained Ge1−xSnx, and of strained Ge grown on relaxed Ge 1−x−ySixSny, was calculated by the self-consistent pseudo-potential plane wave method, within the mixed-atom supercell model of alloys, which was found to offer a much better accuracy than the virtual crystal approximation.
Abstract: Alloys of silicon (Si), germanium (Ge) and tin (Sn) are continuously attracting research attention as possible direct band gap semiconductors with prospective applications in optoelectronics. The direct gap property may be brought about by the alloy composition alone or combined with the influence of strain, when an alloy layer is grown on a virtual substrate of different compositions. In search for direct gap materials, the electronic structure of relaxed or strained Ge1−xSnx and Si1−xSnx alloys, and of strained Ge grown on relaxed Ge1−x−ySixSny, was calculated by the self-consistent pseudo-potential plane wave method, within the mixed-atom supercell model of alloys, which was found to offer a much better accuracy than the virtual crystal approximation. Expressions are given for the direct and indirect band gaps in relaxed Ge1−xSnx, strained Ge grown on relaxed SixGe1−x−ySny and strained Ge1−xSnx grown on a relaxed Ge1−ySny substrate, and these constitute the criteria for achieving a (finite) direct band gap semiconductor. Roughly speaking, good-size (up to ~0.5 eV) direct gap materials are achievable by subjecting Ge or Ge1−xSnx alloy layers to an intermediately large tensile strain, but not excessive because this would result in a small or zero direct gap (detailed criteria are given in the text). Unstrained Ge1−xSnx bulk becomes a direct gap material for Sn content of >17%, but offers only smaller values of the direct gap, typically ≤0.2 eV. On the other hand, relaxed SnxSi1−x alloys do not show a finite direct band gap.

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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Universities of Leeds, Sheffield and York
http://eprints.whiterose.ac.uk/
This is an author produced version of a paper published in Semiconductor
Science and Technology.
White Rose Research Online URL for this paper:
http://eprints.whiterose.ac.uk/4976/
Published paper
Moontragoon, P., Ikonic, Z. and Harrison, P. (2007) Band structure calculations
of Si–Ge–Sn alloys: achieving direct band gap materials.
Semiconductor Science
and Technology, 22 (7). pp. 742-748.
http://dx.doi.org/10.1088/0268-1242/22/7/012
White Rose Research Online
eprints@whiterose.ac.uk

Band Structure Calculations of Si-Ge-Sn Alloys:
Achieving direct band gap materials
Pairot Moontragoon, Zoran Ikoni´c, and Paul Harrison
Institute of Microwave and Photonics, School of Electronic and Electrical
Engineering, University of Leeds, Leeds, LS2 9JT, United Kingdom
E-mail: eenpm@leeds.ac.uk
Abstract. Alloys of silicon (Si), germanium (Ge) and tin (Sn) are continuously
attracting research attention as possible direct band gap semiconductors with
prospective applications in optoelectronics. The direct gap property may be brought
about by the alloy composition alone or combined with the influence of strain, when an
alloy layer is grown on a virtual substrate of different composition. In search for direct
gap materials, the electronic structure of relaxed or strained Ge
1x
Sn
x
and Si
1x
Sn
x
alloys, and of strained Ge grown on relaxed Ge
1xy
Si
x
Sn
y
, was calculated by the
self-consistent pseudo-potential plane wave method, within the mixed-atom supercell
model of alloys, which was found to offer a much better accuracy than the virtual
crystal approximation. Expressions are given for the direct and indirect band gaps
in relaxed Ge
1x
Sn
x
, strained Ge grown on relaxed Si
x
Ge
1xy
Sn
y
, and for strained
Ge
1x
Sn
x
grown on a relaxed Ge
1y
Sn
y
substrate, and these constitute the criteria for
achieving a (finite) direct band gap semiconductor. Roughly speaking, good-size (up
to 0.5 eV) direct gap materials are achievable by subjecting Ge or Ge
1x
Sn
x
alloy
layers to an intermediately large tensile strain, but not excessive because this would
result in a small or zero direct gap (detailed criteria are given in the text). Unstrained
Ge
1x
Sn
x
bulk becomes a direct gap material for Sn content of > 17%, but offers
only smaller values of the direct gap, typically 0.2 eV. On the other hand, relaxed
Sn
x
Si
1x
alloys do not show a finite direct band gap.
PACS numbers: 71.20.Mq, 71.15.-m, 71.22.+i

Band Structure Calculations of Si-Ge-Sn Alloys: Achieving direct band gap materials 2
1. Introduction
Recent years have witnessed a widespread use of optoelectronic devices. This technology
relies on direct band gap III-V materials like GaAs, which is expensive and highly toxic.
An interesting alternative would be a direct gap alloy based on group IV materials Si,
Ge, and Sn, which are generally compatible with silicon technology, and this has been
widely investigated. However, early studies of epitaxial SiGeSn alloys have revealed
the difficulties of their growth, with the exception of Si
1x
Ge
x
binary alloys. With
large lattice mismatch between α-Sn (6.489
˚
A) and Ge (5.646
˚
A) or Si (5.431
˚
A),
approximately 15% and 17% respectively, and the instability of cubic α-Sn above 13
o
C, bulk alloys with Sn cannot be readily grown [1]. Furthermore, because of a lower
surface free energy of alpha-tin and germanium, there can be segregation on the surface
[2]. These difficulties have been overcome by low temperature molecular beam epitaxy
(MBE), which has enabled epitaxial growth of e.g. strained Ge
1x
Sn
x
superlattices [3]
and random Ge
1x
Sn
x
alloys [4] on a Ge substrate. However, 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 app ear in tensilely strained or relaxed Ge
1x
Sn
x
,
e.g. [5]. Further advances were made by ultra-high-vacuum chemical vapor deposition
(UHV-CVD) and uniform homogeneous relaxed Ge
1x
Sn
x
alloys with x < 0.2 have
been grown on silicon[6, 7]. Experimental investigations revealed significant changes
in optical constants and redshifts in the interband transition energy as x varied [8],
indicating wide tunability of the band gap of these alloys.
The binary GeSn and ternary SiGeSn alloys are considered to b e very prosp ective
materials for infrared detectors, as pointed at by Soref and Perry [9], who used
linear interpolation scheme to calculate the electronic band structure and optical
properties of Ge
1xy
Si
x
Sn
y
alloys and concluded that these will be tunable direct
band gap semiconductors. Furthermore, both the direct and indirect band gap in
Ge decrease with tensile strain, but the former (initially 140 meV above) does so
faster, eventually delivering a direct gap material. Therefore, one can use strained
Ge, grown on ternary Ge
1xy
Si
x
Sn
y
alloys [10, 11]. There have since been a number of
theoretical investigations of the electronic structure of SiGeSn alloys and the influence
of composition fluctuations. For instance, using the tight-binding method within the
virtual crystal approximation (VCA), the bowing parameter b
GeSn
value of 0.30 eV [12]
for Ge
1x
Sn
x
was calculated, while another, pseudopotential based calculation [13] gave
the value of -0.40 eV. The latter also predicted that Ge
1x
Sn
x
alloys become direct
gap materials, with 0.55 > E
g
> 0 eV for 0.2 < x < 0.6. The results for b
GeSn
are quite remote from each other (even in sign), and both grossly deviate from the
experimental value, b
GeSn
= 2.8 eV [1, 5]. This clearly indicates that VCA cannot explain
the behaviour of disordered Ge
1x
Sn
x
alloys [14], although it is considered reasonably
accurate for Si
1x
Ge
x
. In order to take into account the alloy disorder effects, the
Coherent Potential Approximation (CPA) was employed for Si
1x
Ge
x
alloys. According
to Chibane et al. [15], who used the model developed by Zunger et al. [16], the

Band Structure Calculations of Si-Ge-Sn Alloys: Achieving direct band gap materials 3
calculated optical band gap gap bowing is in good agreement with experiment for small
Sn contents. However, so far there is no theoretical model which properly describes
the optical properties of GeSiSn alloys in a wide range of compositions. 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 we use the charge self-consistent
pseudopotential Xα method. It finds the self-consistent solution of the Schr¨odinger
equation, with the lattice constituents described by ionic pseudopotential formfunctions.
Compared to the first-principles density functional theory in the local density
approximation [17, 18], which perform the total energy minimization, the Xα method
is able to reproduce the electronic structure (i.e. the band gaps, or optical properties
of semiconductors) with very good accuracy, without any additional schemes like GW
approximation or “scissors correction” as are employed in total energy approaches. On
the other hand, this method would not deliver the ground state properties (e.g. the
atomic coordinates relaxation, or lattice constant bowing) very accurately, though these
can be externally supplied to the calculation. Since our interest in this work are the
electronic properties, the Xα method was adopted, and experimentally obtained lattice
parameters were used where avilable. The calculation starts with the construction of
the effective potential, including the pseudopotential, the Hartree potential and the
exchange-correlation potential
V
eff
(r) = V
ps
+ V
Hartree
(r) + V
xc
(r), (1)
where V
Hartree
(r) =
R
e
2
n(´r)
|r´r|
d´r, and n (r) is the real-space electron density. The Hartree
potential is evaluated in momentum space
V
Hartree
(q) = 4πe
2
n (q)
q
2
, (2)
where q is the wave vector and n (q) are the Fourier coefficients of charge density. This
is evaluated in two steps. First, the density was computed on a 16 × 16 × 16 grid
of the simple unit cell in real space. Second, the fast Fourier transform was used to
transform from n (r) to n (q). The exchange-correlation potential was evaluated in a
similar manner. The local exchange-correlation potential of the Slater type has been
chosen, defined as
V
xc
(q) = α
3
2
e
2
µ
3
π
1
3
[n (q)]
1
3
, (3)
where [n (q)]
1
3
are the Fourier coefficients of the cub e root of charge density. Based on
the approximation of Slater [19], α is a constant which Schl¨uter et al. [20] have set to
0.79. The calculation of this potential was similar to the calculation of n (q), except

Band Structure Calculations of Si-Ge-Sn Alloys: Achieving direct band gap materials 4
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. The V
xc
(q) was then evaluated using Eq. (3). As
for the pseudopotential, we have used the same form as Srivastava [21],
V
ps
(q) =
µ
b
1
q
2
(cos(b
2
q) + b
3
) exp(b
4
q
4
), (4)
with the parameters for Si, Ge, and Sn given in Table 1. 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.
Parameter Si Ge Sn
b
1
(Ry) -1.213 -1.032 0.401
b
2
0.785 0.758 1.101
b
3
-0.335 -0.345 0.041
b
4
0.020 0.024 0.018
Table 1. Parameters of the pseudopotential of Si, Ge and α-Sn
The electronic structure is found by solving the Kohn-Sham equation (in atomic
units ~ = 2m
e
=
e
2
2
= 1):
£
−∇
2
+ V
eff
(r)
¤
ψ
n
(k; r) = ²
n
(k; r) ψ
n
(k; r) , (5)
which is done using the self-consistent pseudopotential plane waves method [22], with a
kinetic energy cutoff of 24 Ry, the value which gives good convergence of the calculation.
The improved linear tetrahedral method [23] was used for integration over the Brillouin
zone, with 34 k-points in the irreducible wedge. The convergence of the self-consistent
calculation was considered to be adequate when the total energy of the system was
stable to within 10
3
Ry. In these calculations we do not account for the spin-orbit
coupling, b ecause it would double the size of the problem while not being essential for
the aim of this work, which is to find whether the smallest band gap is direct or indirect
(and this is determined only by the behaviour of the conduction band). 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. Similar conclusion on a relatively
small influence of spin-orbit coupling on the topic of interest here has been drawn in
[24], based on empirical pseudopotential calculations in the Ge-Sn alloy.
The calculated band structure of bulk Si, Ge, and α-Sn are given in Fig. 1, with the
band gap of Si (Ge) being 1.2 eV (0.71 eV), while α-Sn has zero direct band gap. The
longitudinal (m
l
) and transverse effective mass (m
t
) for the X-valley in Si are 0.87m
0
and 0.22m
0
. Similarly, the L-valley in Ge has m
l
=1.68m
0
and m
t
=0.16m
0
. All these
are in very good agreement with the published values, indicating that the parameters
can be reliably used for further calculations.

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Frequently Asked Questions (10)
Q1. What are the contributions in "Band structure calculations of si-ge-sn alloys: achieving direct band gap materials" ?

In this paper, the authors used the pseudo-potential plane wave ( PWP ) method to calculate the direct and indirect band gap of the Si-Ge-Snx alloys. 

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. 

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]. 

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. 

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. 

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. 

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]. 

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. 

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 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.