Quantitative first-principles theory of interface absorption in multilayer
heterostructures
Jordan A. Hachtel,1,2,* Ritesh Sachan,2 Rohan Mishra,1,2,3 and Sokrates T. Pantelides1,2
1
Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
3
Department of Mechanical Engineering and Materials Science, Washington University in St. Louis, St. Louis,
Missouri, 63130 USA
2
*Corresponding Author e-mail: jordan.a.hachtel@vanderbilt.edu
Supplemental Materials
Calculation of  i : The Complex Dielectric Function
To calculate  i , we need the absorption coefficient, which can be calculated from the complex
dielectric function of the supercell. The imaginary part of the dielectric function is determined using a
weighted sum over direct transitions at each k-point in the assigned k-point mesh. We calculate the real
and imaginary parts of the dielectric following the formulation put forth by Gajdos et al. using the VASP
code and the LOPTICS method where the imaginary part of the dielectric function is taken to be a
weighted sum over direct transitions from the occupied valence band states to unoccupied conduction
band states, and the real part is determined from a Kramers-Kronig Transformation [1]. For the bulk
absorption coefficients in isotropic materials such as NiSi2 and Si (cubic space group) all three
components are the same. However for our heterostructure calculations, the dielectric response for
electric field normal to the interface (z-direction) differs from that of the in-plane directions (x-, ydirections). To use Beer’s law to subtract  i , we only consider light travelling normal to the interface.
Since the photon itself is propagating in the z-direction that means for both linearly or circularly polarized
light the electric field will be in the x-y plane. Our calculations of the dielectric function reflect this, and
have identical x and y components for the dielectric tensor, with a different spectrum in the z component.
For all our calculations we use the x-component of the dielectric tensor, but using the y-component would
produce an identical result.
1
These calculations require a large number of empty conduction bands to converge. For each
heterostructure we use approximately 1.5 times the number of electrons in the system as the total number
of bands, and find that for this number of bands all heterostructure dielectric spectra have converged up to
4.2 eV (the maximum energy value we consider in our calculations). A dense k-point mesh is also
required to effectively sample the Brillouin zone and accurately determine the optical properties of a
system. For all our calculations we use 22 x 22 x 2 k-point mesh is used. Both increasing the in-plane kpoint mesh to 24 x 24 and increasing the out of plane k-points from 2 to 4 are found to have no effect on
the total energy or dielectric spectrum. An energy cutoff of 350 eV is found to be sufficient for these
calculations as well, and no noticeable change is seen when a cutoff of 400 eV is used instead.
Calculation of  i : The Absorption Coefficient
Once we have dielectric function we now need the absorption coefficient. The absorption coefficient
is defined as the decay constant for the electric field of a plane wave propagating through matter [2].
  ) 
2   4  

c
  
(Eq. S1)
Here κ is the extinction coefficient or the imaginary component of the complex refractive index. The
real and imaginary components of the complex refractive index (η and κ) can be calculated directly from
the real (ε1) and imaginary (ε2) parts of the dielectric function.
1 (   1 ( 2   2 ( 2
 (   (  
2
(Eq. S2)
Our output from our calculations of the dielectric function is the real and imaginary parts of the
dielectric function for specific values of energy, not frequency of wavelength. So we finally calculate the
absorption coefficient in terms of the energy (E).
  E) 
2  E )  E  4  E ) 4  E ) E 4 E



1 ( E )  1 ( E ) 2   2 ( E ) 2
c
 E)
hc
hc 2
(Eq. S3)
This gives us our absorption coefficient in 1/nm. The absorbance for a material which has arbitrary
units, is the absorption coefficient as defined in Eq. S1 times a linear thickness in nm.
2
Calculation of  i
From here we need to calculate the absorbance of the macroscopic and atomistic supercells. The
absorbance is the product of the thickness and the absorption coefficient. To calculate the atomistic
absorbance we multiply the DFT calculated supercell absorption coefficient with the thickness of the DFT
supercell (shown in FIG. S1). For consistency we require that the thicknesses used to calculate the
macroscopic absorbance must combine to the thickness of the atomistic supercell.
To divide the supercell thickness and determine the macroscopic layers we take the half way point
between the final unit cells of Si and NiSi2 on either side of the NiSi2 layer (shown in FIG. S1) to
determine the NiSi2 layer thickness, and subtract that value from the total supercell thickness to determine
the Si layer thickness.
Once we have the two absorbances we subtract the macroscopic value from the atomistic, and divide
by two. The division by two is important because there are two interfaces in every supercell, and we wish
for  i to correspond only to a single interface.
FIG. S1.
Thicknesses for absorbance. We use the DFT
supercell pictured here to determine the thicknesses for both
macroscopic and atomistic absorbance. The atomistic
absorbance is calculated
Inaccuracy of the GGA Approximation
It is known that the absorption spectrum for semiconductors is not described adequately by the GGA
approximation. More sophisticated calculations such the Heyd, Scuseria and Ernzerhof HSE06 rangeseparated hybrid functional [3,4], one body Green’s function approaches based on the GW
3
approximation [5], and the Bethe-Salpeter Equation [6] are needed for accurate calculations that can be
compared with experimental data. However, such calculations are currently impractical for the large
supercells that must be used. The calculations in this paper are a proof-of-concept experiment for
extracting and applying interface effects on absorption, as opposed to a rigorous description of NiSi2/Si
heterostructure absorption, and are thus left at the GGA level.
Validation of DFT Absorption Coefficients
To validate our absorption coefficient calculations, we validate the imaginary part of our dielectric
functions, ε2. In this method, ε2 is the fundamental quantity, and the real part, ε1, is determined through a
Kramers-Kronig transformation of ε2, and hence if ε2 is valid, so is ε1. FIG. S2a shows our calculation for
the ε2 for NiSi2 which we compare against valence electron energy loss spectroscopy experiments from
Cheynet et al [7] and reflectivity measurements Amiotti et al [8]. The two main peaks (at 2.6 eV and 4.8
eV) both coincide with the experimental data, indicating our theoretical calculations for the optical
properties of NiSi2 are accurate.
For Si it is known that GGA underestimates the band-gap, so rather than comparing our ε2 for Si
against experiment we compare it to the first-principles Si dielectric spectrum in the PBE approximation
reported by Gajdos et al [1]. We make a comparison to theoretical values to validate that our calculations
on Si are well-converged to the known values of GGA, rather than a comparison to experimental values
of ε2 which would not match the GGA spectrum. FIG. S2b shows our theoretical ε2 spectrum for Si, which
also has peaks in the locations and with magnitudes corresponding to the known PBE peak locations and
magnitudes. Indicating that, while the Si in our calculations has some dissimilarities to experimental Si, it
is accurate within the PBE approximation.
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FIG. S2. Imaginary component of the dielectric function. (a)
NiSi2. The main features are the peak at 2.7 eV and the peak
at 4.5 eV which are both present at the same values in
experiments. (b) Si. Si is calculated within the PBE
approximation hence the underestimation of the band gap.
However all other peak values in the Si dielectric spectrum
match with the accepted values for PBE-Si.
(NiSi2)m/(Si2)n <111> 7-Fold Type A Heterostructure Supercells
Our atomistic heterostructure supercells are composed of NiSi 2 and crystalline Si. NiSi2, which has a
fluorite (CaF2) crystal structure with a lattice parameter of 5.47 Å, has a similar crystal structure and
lattice parameter to crystalline silicon (diamond cubic and 5.43 Å), which allows for the formation of
stable epitaxial interfaces [9–11]. To accommodate for this epitaxial strain on the atomic structure of the
interface, we fix the in-plane lattice parameters to the in-plane lattice parameters of NiSi2 and allow the
lattice parameter normal to the interface to relax. The final supercell is thus determined to be the in-plane
lattice parameters of NiSi2 with the perpendicular lattice parameter being the value that minimizes the
total energy [12]. An energy cutoff of 350 eV is used for the relaxations with a Monkhorst-Pack k-point
mesh of 8 x 8 x 2 centered around the Γ point [13].
All calculations for this paper are presented for a <111> interface between NiSi 2 with 7-Fold
coordination. 7-Fold refers to the number of nearest neighbor Si atoms for the interface Ni atoms. In bulk5
NiSi2 each Ni atom has eight nearest neighbors, however maintaining this coordination at the interface
results in the interface Si atoms being over-coordinated which generates a larger energy penalty [14]. All
supercells contain only one unit cell of either NiSi2 or Si2 per layer, as in for a (m=8, n=6) superlattice
there will be 8 Ni atoms and 28 Si atoms, the actual supercell is shown in FIG. S3.
FIG. S3. Supercell for (NiSi2)5/(Si2)6 superlattice. Atoms are present to show the bonding, while the
dark lines show the actual region of the supercell. FIG. 2 shows (m=5,n=6) but all other superlattices
contain the same a and b vectors, but have a different number of unit cells of NiSi2 and Si2 along the c
lattice vector.
The Case of Multiple Reflections
We have stated in the manuscript that NiSi2/Si2 was a low reflectivity interface. We will now justify
that statement. The reflectivity of the interface is calculated from the complex refractive index defined in
Eq. S3 as follows.
R(  
( NiSi2 ( )   Si ( )) 2  ( NiSi2 ( )   Si ( )) 2
( NiSi2 ( )   Si ( )) 2  ( NiSi2 ( )   Si ( )) 2
(Eq. S4)
Using this expression we plot the reflectivity of the interface in the optical regime, shown in Fig. S4.
As can be seen the reflectivity is less than 10% throughout the visible regime, indicating that the
reflectivity for this interface is low.
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FIG. S4. Reflectivity of the NiSi2/Si2 interface. From this
plot it can be seen that the reflectivity is less than 10%
throughout the optical regime
As stated in the main text of the paper, our method for accounting for interface absorption
atomistically cannot be applied to multiple reflections. To illustrate why, we will first examine the
equation for transmittivity through an absorbing medium, including multiple reflections [2].
T  1  R  e  l (1  R 2 e 2 l  R 4 e 2 l  ...)
2
(Eq. S5)
Where T is the fraction of light transmitted through the layer, R is the reflectivity of the front and
back surface which we assume is symmetric, α is the absorption coefficient of the material within the
layer, and l is the width of the layer. The first term in Eq. S5 is the ‘first-pass’ transmittivity, or fraction of
light that transmits through the medium without reflection, while the subsequent terms refer to increasing
orders of multiple reflections. Now let us extend the analysis to heterostructures with an interface. We
consider a two layer heterostructure so the light passes from vacuum (V) into material A, then into
material B, then back into vacuum. The first-pass transmittivity is relatively simple to define.
Tfp  1  RVA   e A tA  (1  RAB )  eB tB (1  RBV )  (1  RVA )  (1  RAB )  (1  RBV )  e( A tA B tB ) (Eq. S6)
In Eq. S6 we can recognize the macroscopic absorbance defined in the paper, this term can be
modified by the interface absorbance difference,  i , to make it atomistic.
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Tfp Atom  (1  RVA )  (1  RAB )  (1  RBV )  e( A tA B tB i )
(Eq. S7)
However, if we wish to consider multiple reflections things are not so simple. All possible
permutations of reflections between the surfaces and the interface must be considered. However, with
Taylor expansions all these reflections can be simplified to the following expression.
T
T
1  RVA  RAB  e
2 A t A
 RAB  RBV  e
fp
2 B tB
 RVA  RBV  (1  2RAB )  e 2( A t A  B tB )
(Eq. S8)
There are four terms in the denominator of Eq. S8. The first term represents the first pass, the
subsequent two terms refer to multiple reflections with each layer, and the last term refers to the
reflections between the two layers. Here we can correct the final term corresponding to light reflecting
back and forth between the two layers, because just as in the first pass, light interacts with the entire
interface, allowing us to use  Int . However, for the two terms representing multiple reflections within
the layer, such a correction is not possible. That is because when the light reflects within one layer it
interacts with part of the interface region, but not the whole interface region. Hence, in order to treat it
atomistically we would have to know a position dependent interface absorption coefficient, which is
beyond the capacity of our method. As a result, using this treatment it is not possible to consider multiple
reflections with atomistic accuracy.
We ignore the case of multiple reflections in our study, because for low-reflection/highabsorption materials such as NiSi2/Si or many of the interfaces considered in Ref. [24], the ‘first pass’
transmittivity, is dominant, and multiple reflections play a negligible role. However, for highreflection/low-absorption materials, the multiple reflections become dominant in a complex superlattice,
and the ‘first-pass’ transmittivity becomes only a fraction of the total transmitted light. This analysis is
not suited for such systems.
However, even in the case of high-reflection/low-absorption, the treatment establishes a lower
bound for the transmittivity. The only mechanism through which multiple reflections could decrease
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transmittivity would be through negative interference. However, such phase effects can only occur at the
scale of the wavelength of the light being considered. For wavelengths in the near the optical range (200
nm+) the size scales would be large enough where interface absorption effects would be negligible
compared to bulk effects anyway. Hence, even in the case of high-reflection/low-absorption materials a
lower bound on transmittivity is established.
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