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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
1−x
Sn
x
and Si
1−x
Sn
x
alloys, and of strained Ge grown on relaxed Ge
1−x−y
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
1−x
Sn
x
, strained Ge grown on relaxed Si
x
Ge
1−x−y
Sn
y
, and for strained
Ge
1−x
Sn
x
grown on a relaxed Ge
1−y
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
1−x
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
1−x
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
1−x
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
1−x
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
1−x
Sn
x
superlattices [3]
and random Ge
1−x
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
1−x
Sn
x
,
e.g. [5]. Further advances were made by ultra-high-vacuum chemical vapor deposition
(UHV-CVD) and uniform homogeneous relaxed Ge
1−x
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
1−x−y
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
1−x−y
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
1−x
Sn
x
was calculated, while another, pseudopotential based calculation [13] gave
the value of -0.40 eV. The latter also predicted that Ge
1−x
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
1−x
Sn
x
alloys [14], although it is considered reasonably
accurate for Si
1−x
Ge
x
. In order to take into account the alloy disorder effects, the
Coherent Potential Approximation (CPA) was employed for Si
1−x
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.
![Figure 2. The band gaps of Si1−xGex at X and L points, calculated within the VCA and the mixed-atom method, together with data from the literature (a – Ref. [27], b – Ref. [28]).](/figures/figure-2-the-band-gaps-of-si1-xgex-at-x-and-l-points-221tcqvu.webp)
![Figure 4. The minimum band gap of relaxed Ge1−xSnx for Γ and L valleys, calculated within the VCA and within the mixed atom method. A couple of available experimental values are also displayed (a – Ref. [37], b – Ref. [5]).](/figures/figure-4-the-minimum-band-gap-of-relaxed-ge1-xsnx-for-g-and-1yajheqz.webp)
