Direct-to-indirect bandgap transitions in 110 silicon nanowires
I. J. T. Jensen, A. G. Ulyashin, and O. M. Løvvik
Citation: Journal of Applied Physics 119, 015702 (2016); doi: 10.1063/1.4938063
View online: http://dx.doi.org/10.1063/1.4938063
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JOURNAL OF APPLIED PHYSICS 119, 015702 (2016)
Direct-to-indirect bandgap transitions in h110i silicon nanowires
I. J. T. Jensen,1,a) A. G. Ulyashin,1 and O. M. Løvvik1,2
1
SINTEF Materials and chemistry, P/O box 124 Blindern, 0314 Oslo, Norway
Department of Physics, University of Oslo, P/O box 1048 Blindern, 0316 Oslo, Norway
2
(Received 14 October 2015; accepted 4 December 2015; published online 7 January 2016)
The bandstructure of h110i silicon nano wires (SiNWs) with diameters (d) up to 6.1 nm were studied
using density functional theory. Three types of surface termination were investigated: H, F, and OH;
all giving quantum confinement induced direct bandgaps in the investigated size range. Comparison
of the calculated results to reported experimental values showed that trends in the bandstructure
behaviour were well reproduced. By studying the relative decrease of global and local minima in
the conduction band minimum with increasing d, it was possible to predict a direct-to-indirect
bandgap transition at d ¼ 9.2, 9.5, and 11.4 nm for H, F, and OH terminated NWs, respectively.
C 2016 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4938063]
V
I. INTRODUCTION
As the quest to fulfil Moore’s law continues to push the
physical limits of conventional electronic devices, semiconductor nanowires have attracted attention due to their potential
for high-performance nanoscale devices. Among the advantages of silicon nanowires (SiNWs) are the high surface-tovolume ratio and the compatibility with established Si-based
technology, in particular. The most intriguing property of
SiNWs, however, is related to their 1D nature, which results
in a quantum confinement induced direct bandgap.
The quantum confinement aspect of SiNWs has been
studied theoretically for more than twenty years,1 while
experimentally, research on SiNWs dates back to the late
1950s.2 However, SiNWs have seen a rapid increase in interest during the last decade. At present, single-crystal SiNWs
with diameters approaching molecular dimensions can be
produced in a controlled manner,3 while improved computational power has made it possible to calculate the properties
of SiNWs with increasingly large diameters.
The advantage of using computational methods is the
possibility to study a large selection of growth directions,
cross-section geometries, and terminating species for
SiNWs. In particular, the h001i and h110i growth directions
have received attention.4–9 Nolan et al. compared the bandstructure of H and OH-terminated h001i SiNWs with diameters below 2 nm and found that the functional group used to
terminate the Si dangling bonds significantly modifies the
band gap, with a decrease in band gap up to 1 eV when
replacing H with OH. Gao et al. compared h001i and h110i
SiNWs with F and SiO2-like termination for diameters up to
about 3 nm. For the h110i SiNWs, the bandgap vs. 1/d
behaviour of the H- and F-terminated models were similar,
approaching the bandgap of bulk Si for d approaching infinity. The SiO2-like termination, however, was found to induce
strain, which was drastically reducing the bandgap relative
to 1/d. The effect of surface passivation and orientation on
the electric properties of SiNWs were also explored by Zhuo
et al.,8 who investigated the band structure of h001i and
a)
IngvildThue.Jensen@sintef.no
0021-8979/2016/119(1)/015702/5/$30.00
h110i SiNWs with H, F, and CH3 passivation for diameters
below 3.4 nm. The h110i SiNWs were found to be more
influenced by their surface termination than h001i SiNWs.
In this work, the effects of surface passivation and diameter on the bandstructure of h110i Si NWs are studied using
density functional theory (DFT). Comparisons of bandgaps
and bandstructure are made for H, F, and OH terminated
h110i SiNWs with diameters up to 6.1 nm, and the calculated
results are used to predict the SiNW diameter for directto-indirect transition.
II. METHODS
DFT calculations at the PBE-GGA level, i.e., using the
Perdew-Burke-Ernzerhof generalized gradient approximation for the exchange-correlation energy,10 were performed
using the Vienna ab-initio simulation package (VASP).11,12
Silicon nanowires with diameters ranging from 1.2 to 5.8 nm
(without terminating species) were modelled in the Si h110i
growth direction. The h110i orientation was chosen as this
has been reported in the literature to be the preferred growth
direction experimentally for silicon NWs in this size
range.3,13 A hexagonal shape with f111g and f001g facets
was used in close compliance with experimental reports.3,14
The NWs were modelled in the middle of a supercell with
dimensions a b c, where b ¼ 3.86 Å and a and c were
large enough to allow 20 Å vacuum between the NWs.
Surface passivation with different terminating species was
investigated, namely, H, F, and OH. Figure 1 shows the input
model for the largest calculated NW, and details of all the
models can be found in Table I.
For the calculations, the plane-wave energy cut-off was
400 eV and a 1 9 1 Gamma sampling of k-points was
used to model the Brillouin zone. The chosen energy cut-off
and k-point values gave a convergence in the calculated band
gaps better than 0.01 eV. Structural optimizations of the
atomic positions were performed with the break condition for
the relaxation loop set to forces <0.01 eV/Å. Due to the
vacuum included in the unit cell, this meant that the structure
was allowed to expand or contract in the a and c directions,
while the growth direction (b) was kept constant. Relaxation
119, 015702-1
C 2016 AIP Publishing LLC
V
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Jensen, Ulyashin, and Løvvik
J. Appl. Phys. 119, 015702 (2016)
double, and triple in the b-direction. For H termination, the
results did not show any structural irregularities on a scale
larger than b, thus the single a b c supercell was judged
sufficiently large to give a representative picture of surface
passivation of silicon NWs using H and F. The latter was
based on the very similar behaviour of H and F as passivating
species, as will be shown. The case of OH-termination is
more complicated and will be discussed later.
III. RESULTS AND DISCUSSION
A. Bandgaps as a function of diameter
FIG. 1. Cross-sectional view of the largest h110i NW model before relaxation. Large atoms represent Si, while small atoms represent the termination,
i.e., H, F, or OH. The black atoms in the middle show the cross section of
the smallest NW. The NW has two different types of surfaces, f001g and
f111g. The diameter (d) is defined as the largest distance across the NW,
excluding the termination.
of the b axis was also tested for a selection of models but did
not seem to impact the bandgaps to any significant extent.
After structural relaxation, the electron density was calculated
to an electronic convergence less than 105 eV change in the
total energy. Finally, band structure calculations were performed along the direction in k-space corresponding to the
planes in the NW growth direction, keeping the electron density from the previous step constant. 25 k-points were used
for the 1.2–4.3 nm diameter models and 15 k-points for the 5
and 5.8 nm diameter models. Diameters were defined as the
largest distance across the NW cross section before structural
relaxation, not including terminating species. This was chosen for the sake of comparison between models with different
terminating species. When terminating species are included,
which is often the case in the literature, the largest models
calculated in this work have diameters of 6 and 6.1 nm for H
and F termination, respectively.
First principles based molecular dynamics (MD) as
implemented in VASP was performed on H and OH terminated NWs with 1.2 nm diameter to investigate how the
periodicity in the growth direction affects the arrangement of
the terminating species. The MD calculations were done at
300 K for 1000 cycles of 1 fs. Both MD and DFT calculations
were done for 1.2 nm diameter supercells which were single,
Table I lists the bandgaps calculated for each NW model.
Figure 2 shows the calculated bandgaps Eg as a function of
the inverse diameter (1/d) for H, F, and OH surface termination. The corresponding calculation for bulk Si is included as
a horizontal line to offer a point of reference, as DFT tends to
underestimate bandgaps. In general, the bandgap decreases
with the increase in diameter. The largest bandgaps are seen
for H terminated NWs with small diameters. Both H and
F terminations give a linear dependence of the inverse
diameter. H termination results in a steeper slope than Ftermination, both reaching a value about 0.1 eV below bulk
Si for 1/d ¼ 0. Although commonly used, the bulk bandgap of
Si might not be the most relevant value for comparison with
h110i NWs. This will be discussed in Section III C.
The behaviour of OH-terminated NWs is more irregular.
On average, the bandgaps follow a line with slope similar to
that of H terminated NWs, about 0.5 eV below. However, for
OH larger deviations are found from the line, in a seemingly
random manner. The bandgaps for diameters approaching
infinity drop well below the value of bulk Si. This was seen
previously for SiO2-like terminated NWs, where it was taken
as an indication of strain.5 This and the irregular behaviour
of OH terminated NWs raise questions about whether the
ideal OH positions have been found.
To investigate the position of the OH molecule, both
MD and DFT were performed on 1.2 nm diameter supercells
which were single, double, and triple in the b-direction. MD
TABLE I. NW models with calculated bandgaps for different diameters and
terminating species (X). The bandgaps are all direct. The listed diameters do
not include the terminating species.
Model
Diameter (nm)
1.2
1.9
2.7
3.5
4.3
5.0
5.8
Calculated bandgaps (eV)
#Si atoms
#X atoms
X¼H
X¼F
X ¼ OH
24
54
96
150
216
294
384
16
24
32
40
48
56
64
1.73
1.25
1.03
0.92
0.87
0.78
0.77
1.24
0.95
0.84
0.78
0.73
0.70
0.68
1.27
0.74
0.84
0.51
0.44
0.53
0.27
FIG. 2. Calculated bandgaps as a function of the inverse diameter (1/d). The
calculated bandgap for bulk Si is included as a horizontal line to offer a
point of reference, as DFT tends to underestimate bandgaps. Also included
are experimental results for H terminated f110g SiNWs from the work of
Ma et al.14
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Jensen, Ulyashin, and Løvvik
was conducted under the same conditions as described previously for H termination. The resulting structures were rather
chaotic, giving no clear indications as to what would be the
preferred positions for the OH molecule. The DFT calculations, on the other hand, gave highly ordered structures,
showing no irregularities on a scale larger than b. The deviations from linear dependence of Eg on 1/d for OH terminated
NW models point to an increase in complexity when using
molecules rather than single atoms to terminate SiNW
surfaces.
From the particle-in-a-box analogy, the band gap is
expected to scale linearly with 1/d2, rather than 1/d as seen
above. A limitation of the particle-in-a-box model, however,
is the disregard of surface and surroundings. Previously, it
has been argued that so-called self-energy corrections, which
J. Appl. Phys. 119, 015702 (2016)
arise from dielectric mismatch between the NWs and their
surroundings, will add an additional 1/d dependence.15
Others have reported a phenomenological dependence of
Eg upon the surface-to-volume ratio, highlighting the
importance of surface in semiconducting NW systems.6
Experimental results for H terminated f110g SiNWs have
been added to Figure 2 for comparisons. The experimental
band gaps are taken from the work of Ma et al. and were
measured using scanning tunneling spectroscopy.14 For
SiNWs with d 2.5 nm, the calculated and experimental
bandgaps as a function of inverse diameter are in good correspondence, following similar slopes. The underestimation of
the bandgaps caused by DFT turns out to be near constant in
this range, which makes it possible to reproduce trends in the
system even if the absolute values of the calculated bandgaps
FIG. 3. Examples of bandstructure for
NW with different diameters (d) and
termination. All terminated NWs calculated in this work exhibit a direct
bandgap in C, which decreases in magnitude with increasing d. In addition,
all structures have a local minimum in
the conduction band (Rmin) just below
k ¼ 0.7 Å1.
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Jensen, Ulyashin, and Løvvik
FIG. 4. Calculated bandstructure along the Si h110i direction for H-terminated
NWs with 1.2 and 5 nm diameters (red and black, respectively). C marks the
CBM and Rmin marks the local CB minimum along the R direction.
J. Appl. Phys. 119, 015702 (2016)
this evolvement towards indirect bandgap. In Figure 5, the
direct bandgap has been plotted as a function of 1/d for H, F,
and OH termination. Also included is the energy difference
between the local CB minimum Cmin and the valence band
maximum (VBM), labelled indirect bandgap. For all terminating species, both direct and indirect bandgaps are found
to vary linearly with 1/d. For OH, the linearity becomes
clearer if looking at the relative energy difference between
the global and the local CB minima, see Figure 5(d). From
extrapolation, a transition from direct to indirect bandgap is
found to occur at d ¼ 9.2, 9.5, and 11.4 nm for H, F, and OH
termination, respectively. This indicates that OH termination
preserves a direct bandgap for slightly larger diameters than
H and F.
C. The effect of NW surfaces
are too small compared to experimentally obtained results.
For SiNWs with d 2.5 nm, the limitations of the computational method are becoming apparent as the exchangecorrelation functionals will struggle to capture the confinement at this scale.
B. Band structure—Direct versus indirect bandgap
Figure 3 shows the calculated band structure along the
growth direction for d ¼ 1.2, 2.7, and 4.3 nm. All the calculated models with H, F, and OH termination exhibit a direct
bandgap at C. In addition to the conduction band minimum
(CBM) at C, a local minimum (Rmin) is seen in the conduction band just below k ¼ 0.7 Å1. The energy difference
between this local minimum and the CBM decreases with
increasing d, i.e., the NW bandgap character goes towards
indirect as the diameter increases. Figure 4 shows an
enhanced view of the conduction band for the H-terminated
NWs with the smallest and the largest diameter, highlighting
Figure 6 shows parts of the band diagram for bulk Si
calculated using the same level of DFT as for the Si NW
investigation. The left part of Figure 6 shows the D direction
where the CBM is found, while the right part of the figure
shows the R direction, which corresponds to the growth
direction of the Si NWs. As can be seen, the R direction for
bulk Si has an indirect bandgap with the CBM in the region
corresponding to the local (indirect) CBM found for the
SiNWs. Thus, the transition from direct to indirect bandgap
predicted from Figure 5 might be a result of quantum confinement coming to an end. The size of the indirect bandgap
along the R direction in Figure 6, however, is far greater
than the values calculated for the largest SiNWs. Thus, the
evolvement of the bandgap from quantum confined NW
towards bulk Si is not quite as simple as one might lead to
believe from representations such as Figure 2, where the
direct bandgaps of the NWs are compared directly to the
indirect bandgap of bulk Si.
FIG. 5. Calculated difference between
VBM and CBM (labeled direct) and
between VBM and local CB minimum
Cmin (labeled indirect), as a function of
inverse diameter (1/d). (a) H termination, (b) F termination, and (c) OH termination. The data are extrapolated to
find the transition from direct to indirect bandgap. In (d) the difference
between CBM and local minimum
Cmin is plotted for the NWs with OH
termination, to better illustrate the
linearity.
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Jensen, Ulyashin, and Løvvik
J. Appl. Phys. 119, 015702 (2016)
FIG. 6. Parts of the band diagram for bulk Si calculated using the same level
of DFT as for the Si NW investigation: The D direction where the CBM is
found (left) and the R direction, which corresponds to the growth direction
of the h110i Si NWs in this work (right). In the latter, the k axis is reversed
compared to the convention, in order to compare directly to the NW bandstructure plots.
A complicating aspect that must be taken into account is
the effect of the terminating species on the surface of the
NWs. While quantum confinement facilitates a direct bandgap
which increases as the diameter of the NW decreases, the Si
dangling bonds on the NW surface act to destroy the bandgap
altogether. The unpaired valence electrons at the SiNW surface form electrically active traps with energies in the Si
bandgap. This is remedied to a certain extent by introducing
terminating species such as H, F, or OH. As seen above, however, the bandgaps of the surface terminated NWs are smaller
than the bandgaps of bulk Si. To investigate the effect of surface, two different models were constructed: a f001g surface
model and a f111g surface model, both terminated by H. For
each model, two Si layer thicknesses were calculated, corresponding to the distance between the planes in the smallest
and the largest NW. The bandstructure in the NW growth
direction is shown in Figure 7(a) for the f111g surface model
and (b) for the f001g surface model. As can be seen, the
f001g surface model exhibits a quantum confinement induced
direct bandgap, while the bandgap of the f111g surface model
was found to be indirect for both model thicknesses. This corresponds well to recent results for ultra-thin Si films from Lin
et al.16 and shows how different surface orientations give rise
to different quantum confinement effects in the same material.
IV. CONCLUSIONS
The bandstructure of h110i SiNWs with different diameters and terminating species was studied using DFT at the
PBE-GGA level. Comparison to existing experimental
values14 demonstrated that it is possible to use DFT at this
level to predict trends in the bandstructure behaviour. The
bandgaps were found to decrease linearly with 1/d, and F
and OH terminations were found to reduce the bandgap compared with H termination for NWs with equivalent d. A local
minimum in the CB was observed to decrease more rapidly
with increasing d than the CBM. By looking at the relative
difference between the CBM and this local minimum, it was
FIG. 7. Calculated bandstructure in the 110 direction for Si f001g and
f111g surface models ((a) and (b), respectively) with thicknesses corresponding to the largest and the smallest Si NWs studied in this work.
possible to predict a direct-to-indirect bandgap transition at
d ¼ 9.2, 9.5, and 11.4 nm for H, F, and OH termination,
respectively.
ACKNOWLEDGMENTS
The authors are grateful for the financial support of the
European project nanoPV in the framework of FP7-Energy2009-2.1.1, Grant No. 241281.
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