1
Photo Absorption Calculation in Bulk Silicon
and Silicon Nanowires: A Comparative Study
Md Golam Rabbani and Jianqing Qi

Abstract—We calculate the photo absorption coefficient in bulk
silicon and silicon nanowires using tight-binding method. For
nanowires we use both fully numerical and semi-analytical
approaches and find that latter is both very fast and accurate for
energies about 1-2 eV above the bandgap, while the former is
computationally intensive. Comparison between ‘direct’ bulk and
nanowire absorptions shows that for nanowires of diameter of
about 1nm or higher the absorption exceeds the bulk value.
Index Terms—Silicon, nanowire, absorption
I. INTRODUCTION
S
ILICON is the most important semiconductor, but bulk
silicon is not a good optoelectronic material because of its
indirect bandgap, which renders its photo absorption capability
worse than the direct bandgap semiconductors, like GaAs.
Recently silicon nanowires have attracted much attention
because they may have direct bandgap and this property can be
modified with application of strain [1].
Photoabsorption is an important property for any
optoelectronic material and for silicon nanowires to be
considered a good candidate for next generation optoelectronic
devices their absorption quality has to show improvement over
bulk silicon. We can do photoluminescence measurement but
before doing experimental measurement we have to convince
ourselves through theoretical studies that the nanowires hold
good potential for this purpose.
To do this theoretical calculation we have to first find the
stable structure for a specific nanowire and then find the
Hamiltonian representation and solve for its bandstructure and
momentum matrix elements. We use tight-binding (TB) [2]
approach which has been applied to various solids.
For comparison purpose, we calculate direction transition in
both bulk silicon and silicon nanowires. The calculation steps
are these: (i) write down TB Hamiltonian for a unit cell and its
connection to the nearest neighboring unit cells. [For
nanowires we have to perform molecular dynamics first to get
an optimized (stable) structure of the wire since we start with
wires cut from the bulk.] (ii) Calculate eigen values and eigen
vectors of the material at each value of the wave vector
using Bloch’s theorem. (iii) Find the momentum matrix
element at each . (iv) Calculate imaginary part of the
dielectric constant and use Kramers-Kronig relation to
calculate the real part of the dielectric constant (for bulk real
dielectric constant is not needed). (v) Finally use values
calculated so far to find absorption as function of photon
energy (or, wavelength).
This paper is organized as follows: We describe our
calculation methodology in Section II, present some results in
Section III and make conclusions in Section IV.
II. METHOD
We first describe the bulk calculation method in subsections A
and B, and then nanowire methods are described in C.
A. Method for bulk silicon
The unit cell of bulk silicon is shown in Fig. 1, in which A1
and A2 together are considered as a unit cell and the rest are
nearest neighbors of the unit cell. The Hamiltonian of the bulk
silicon is constructed by using the semi-empirical 10 orbital
(sp3d5s*) TB method [2]. The s, p, d and s* orbitals of silicon
are assumed to be of Slater type [3] which has been widely
used in quantum calculation since it was first proposed. The
sampling Brillouin Zone (BZ) is performed using
-points. The Bloch Theorem is used to find the
total Hamiltonian of the bulk silicon at each point:
(1)
where
is the TB Hamiltonian of the central unit cell
constructed by A1A2,
(i=6, 7, 8) is the interaction between
A1 and one of its nearest neighbors and
(i=3, 4, 5) is the
Fig. 1 unit cell of bulk silicon along with nearest neighbor atoms.
2
interaction between A2 and one of its nearest neighbors. The
translation vector
(i=2, 3, 4) represents the relative
position between the neighborhood atoms and the unit cell
atoms. The size of the total Hamiltonian is
,
where
here.
To find the band structure, one first fills the total
Hamiltonian according to Eq. (1) and then solves the following
eigenvalue-problem,
HΨ=EΨ
(2)
where Ψ is the wave function or eigen state. Due to the
symmetry of BZ, the filling of
can be confined with an
irreducible wedge [4], as shown in Fig. 2. In this case, it’s not
necessary to include all the points. It is enough to calculate
the summation within the irreducible wedge, that is, a 1/48th of
the whole BZ. The final results for related quantities will be
multiplied by 48.
photon wave vector. The real part of dielectric constant can
be found from by using the Kramers-Kronig formula.
For bulk silicon, by confining points in the irreducible
wedge and using Fourier transformation and Lorentz
broadening, the imaginary part is now written as,
(4)
where is set to be 5 meV in the calculation.
And the absorption coefficient is,
(5)
where
is the refractive index of bulk silicon.
B. Calculation of Moment Matrix Element
For both bulk silicon and nanowires, the momentum matrix
element between a conduction band ( ) and a valence band
( ),
, is calculated as below.
(6)
Fig. 2 First BZ of the reciprocal lattice with emphasis on the first octant
which carries the first irreducible wedge. (a) BZ of bulk silicon; (b)
irreducible wedge.
The complex dielectric constant
and the complex
refractive index
are written respectively as the following
forms,
Using the sp3d5s* scheme, the eigen state at each
be expanded as,
point can
where spans one s, three p, five d and one extra s* orbital for
higher excitation. Thus,
The absorption coefficient is calculated from the imaginary
part of dielectric constant :
(7)
can be given by:
(3)
where the summation run over all conduction ( ) and valence
( ) bands at each point in BZ. , ,
and are charge of
electron, permittivity of vacuum, rest mass of electron and
volume of the material, respectively.
is the momentum
matrix element mentioned above. is the unit vector along the
where
stands for the orbital centered at . The
Krönecker’s delta function ensures the direct transitions
between two states in conduction and valence band. Set
, then
3
a [110] directed 0.5nm diameter nanowire, which is a direct
bandgap (Eg~3.6eV) wire.
(8)
The property of the delta function allows us to write the
momentum matrix element as,
(9)
Fig. 3 A hydrogen passivated silicon nanowire. Golden spheres are Si atoms
while small white spheres represent H atoms.
To obtain the absorption coefficient, the band structure is
calculated. By using Eqs.(6) through (8), the momentum
matrix elements are found. Finally, by using Eqs. (4) and (5),
the imaginary part of dielectric constant and the absorption
coefficient are obtained in succession.
C. Method for nanowires
Calculations of bandstructure and momentum matrix
elements are similar to those for the bulk, so they are only
described here briefly, but steps for specific nanowires are
presented in detail.
a. Getting stable nanowire structures
In this work, we study absorption in both [110] and [100]
directed nanowires, which were ‘cut’ from bulk silicon crystal.
Different cross-sectional geometry can be considered by
keeping different number of unit cells in different directions.
Thus, the nanowires obtained are not stable. Two ‘postprocessings’ are done on them to get stable nanowires. First,
hydrogen atoms are added to each unsatisfied bond of the
silicon atoms on the surfaces so that there are no dangling
bonds. This is called hydrogen passivation. Second, molecular
dynamics simulation is done using the passivated nanowire as
an initial structure. We got optimized nanowires from
Daryoush Shiri [5]. We show a picture of a H2 passivated
nanowire in Fig. 3.
b. Calculating nanowire bandstructure
We use TB [2] Hamiltonian and Bloch’s theorem to
calculate the nanowire bandstructure. Here the assumptions are
that the nanowires are infinitely long. H2 passivation does not
affect the eigen values and the eigen vectors; we just consider
the hydrogen atoms as part of the nanowires. In the case of
bulk silicon, 1/48th of Brillouin zone is considered and the
result is multiplied by 48, while for nanowires wave vector k is
one dimensional and eigen values are eigen vectors are
symmetric with respect to k=0. Hence we only consider the
range k=0 to k=(pi/a) and multiply the result (imaginary part
of dielectric constant) by 2. This way we minimize memory
requirement for saving eigen energies and eigen vectors
needed for later calculations. Fig. 4 shows the bandstructure of
Fig. 4 Bandstructue of a [110] 0.5nm diameter nanowire.
 2 ( )
equation for  2
c. Calculating
The
given in Eq. 3 is valid for both bulk
and nanowires. However, the one dimensional wavevector for
nanowire permits further simplification as follows
 2 ( ) 


 e2
 0 m02V  2
e
2
 0m V 
2
0
2e
2
2
 eˆ  P
2
cv
cv
(k )  ( Ecv (k )   )
k
L
2
 2  2  2  eˆ  P
cv
cv
1
2
 0 m0 Anw
(k )  ( Ecv (k )   )
2
k
 1

2
 ( )
  eˆ  P
cv
cv k

2
(k )  ( Ecv ( k )   ) 

(10)
Here, we have used,

k
L
  2  2   dk   , where the
2 k
first 2 takes care of spin degeneracy while the second 2 is
because of the bandstructure symmetry with respect to k=0
volume (V )
point.
Also
has
been
replaced
by area ( Anw )  length ( L) . Next we use the following
property of the Dirac delta function
4

f ( x) ( g ( x))dx  
xzp
x
f ( xzp )
dg ( x)
dx x  xzp
(11)
f. Semi-analytical method
So far we have described the fully numerical method of
calculating absorption in nanowires. One drawback of this
approach is the high computational requirement to
2
where xzp are the solutions of g ( x )  0 .
calculating Pcv (k ) . For nanowires with diameter about 1 nm
Using Eq.10, Eq. 9 is written as
this calculation takes about 6-8 hours while the rest of the
calculation takes a few minutes. So some approximate method
is highly welcome, especially for nanowires with larger
 2 ( ) 

2e2 2 1  1
2
eˆ  Pcv (k )  ( Ecv (k )   ) 

2
2 
 0 m0 Anw  (  ) cv k



2 

eˆ  Pcv (k ) 
2e
1  1



 0 m02 Anw  (  )2 cv kzp dEcv (k )


dk k kzp 

2 2
where k zp are the solutions of
d. Calculating
Ecv (k )    0 .
Fortunately we have found that a combination of analytical and
numerical (called semi-analytical) approach can avoid the high
computational burden and while still provide reasonable
estimate compared to the fully numerical approach. The basis
for this approximation is the well-known effective mass
approximation (EMA) and the assumption of the constancy of
Pcv (k ) over k values so that Pcv (k  0) can substitute
(12)
2
2
for all Pcv (k ) . Below we write down the relevant equations
 2 as value
for this approach. First the bottom of the conduction subband
and the top of the valence subband are approximated,
respectively, as
of refractive index is well known in this case. But for
nanowire, we do not have such experimental data and we are
left with calculating  1 , which is easily done using Kramers-
Ec (k )  Ecmin 
Kronig relations as given below
Ev (k )  Evmax 
1 ( )  1 
  ( )(  )d (  ) 
P 2
  0 (  )2  (  )2 

2
(13)
 2 ( ) and 1 ( ) ,
we write down the
equation for refractive index and extinction ratio as
 1  i 2   2   2  i 2
where
0 , f
0
and

(15)
4 ( ) 2 ( )

c
c
f
c are,
k
2mh
(18)
2
k2
2meh
Ecv 0  Ecmin  Evmax and
where
meh 
Finally, absorption coefficient is given by
4 ( )
(17)
2
(19)
1  1
1 
 
.
meh  me mh 
meh can also be calculated from
2
 ( ) 
2
k  0 while me and mh are electron and
hole effective masses, respectively. Then we define Ecv (k ) as
Ecv (k )  Ec (k )  Ev (k )
(14)
 ( ) , we get
12   22  1

k2
2me
where Ecmin and Evmax are the values of conduction and
 Ecv 0 
1  i 2    i
Solving for
2
valence subbands at
where P denotes principal value.
e. Calculating absorption,  ( )
Having found
2
Pcv (k ) is expected to take days.
2
1 ( )
For bulk absorption, we just have to calculate
diameters, for which
(16)
respectively, the light wavelength,
frequency and speed in free space.
2
d 2 Ecv (k )
dk 2
(20)
k 0
2
dEcv (k )
k

From Eq. 18, we also have
and
dk
meh
k
2meh
2
 Ecv (k )  Ecv 0 
so that
5
2
dEcv (k )

dk
2meh
2
 Ecv (k )  Ecv 0 
meh
Substituting Eq. 20 in Eq. 10 and using
(21)
Ecv (k )   ,
eˆ  Pcv (k  0) meh
1 2e2 2 1
1
 2 ( ) 
2
2 
2  0 m0 Anw (  ) cv
  Ecv 0
2
Equations for
1 ( )
(22)
and  ( ) remain unchanged in semi-
analytical approach.
III. RESULTS
The absorption coefficient versus photon energy for bulk
silicon is shown in Fig. 5. We observe a direct band gap at
3.24 eV, which is consistent with the previous theoretical [6]
and experimental results [7]. We also see that the magnitude of
absorption coefficient for bulk silicon is between 105 and 106
cm-1.
Fig. 5 Absorption in bulk silicon
Next in Fig. 6 absorption coefficients of 4 different
nanowires are plotted. Since we do not use any Lorentz
broadening the absorption is zero for energies below the
bandgap and the log scale of the y-axis does not show zero
absorption. Here we note that the narrow nanowires have
higher bandgap so that their absorptions do not start until a
higher energy. As the diameter increases we get absorption at
lower energies (because of decrease in bandgap). Another
important behavior to take notice of is that the absorption
actually increases with the diameter (at least, for the 4
nanowires that we have simulated). For comparable diameters,
the [110] directed wires show better absorption compared to
[100] directed ones. The absorption profile also maintains the
joint density of state characteristics. If we compare absorptions
in bulk silicon (Fig. 5) and silicon nanowires (Fig. 6), we see
that the narrowest nanowires have absorptions mostly less than
the bulk but the two wider nanowires have better absorption
performance and for this ‘direct only’ absorption calculation
the [110], 1.1nm nanowire have quite high absorption around
photon energy of about 3eV.
Fig. 6 Absorption in 4 different nanowires
Fig. 7 Comparison of absorptions calculated in numerical and semianalytical
approaches.
Fig. 7 compares the absorptions calculated in full numerical
method with those in semi-analytical approach for two
different nanowires. For the [110], 0.5nm nanowire the full
numerical approach takes less than an hour but it takes about 8
hours for the [110], 1.1nm nanowire. In comparison, the
semianalytical calculation for each takes less than 2 minutes.
But from Fig. 7 it is evident that the accuracy for the
semianalytical approach is quite good, especially for energy
below 4.5eV. The reason is that the Ecv(k) curves lend
themselves to effective mass approach (EMA) for this energy
range and beyond that the curves tend to bend down instead of
going up as in EMA. Many of the peak positions still match
though magnitude does not. Overall, the semi-analytical
approach can help find the close to bandgap absorption
property very quickly.
IV. CONCLUSION
We have presented detailed method on how to calculate the
photo absorption in both bulk silicon and silicon nanowires.
The full numerical method is computationally very demanding
6
for both bulk and nanowire, but here for bulk we used the
reduced wedge and for nanowires we implemented a semianalytical approach (along with full numerical) to reduce the
computational requirement to reasonable limits. Comparison
shows that for nanowires with diameter of about 1nm or above,
the absorption exceeds the bulk absorption. As we have not
included all the absorption phenomena (effect of electronphonon interaction, presence of excitons, etc.), further studies
on this will give us a better estimation
ACKNOWLEDGMENT
We thank Daryoush Shiri for helping us with notes, initial
codes and providing us with tight-binding Hamiltonians. We
also thank Professor Anantram for pointing us to this work.
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