Optical properties of materials¶
Solcore has several ways of accessing the optical properties of materials: databases and parametric dielectric functions.
Dielectric constants models and Adachi parametrization¶
Understanding the optical response of both established and novel materials is crucial to effective solar cell design. To efficiently model the complex dielectric function of a material Solcore incorporates an optical constant calculator based on the wellknown CriticalPoint ParabolicBand (CPPB) formalism popularised by Adachi ([1], [2], [3]). In this model, contributions to :math:`epsilon_2(omega)` from critical points in the Brillouin Zone at which the probability for optical transitions is large (van Hove singularities) are considered. The transition probability for such transitions is proportional to the joint density of states (JDOS) , which describes the number of available electronic states between the valence and conduction bands at given photon energy. The imaginary part of the complex dielectric function is related to the JDOS by:
Where is the momentum matrix element for transitions from the valence band () to the conduction band (). Critical point transitions are considered at the following points of symmetry in the band structure: :math:`E_0` corresponds to the optical transition at the :math:`Gamma` point and :math:`E_0 + Delta_0` to the transition from the spinorbit split off band to the conduction band at the :math:`Gamma` point. :math:`E_1` and :math:`E_1 + Delta_1` denote the transitions from the valence heavyhole (HH) band and the valence lighthole (LH) band respectively to the conduction band at the L point. The :math:`E’_0` triplet and :math:`E_2` transitions occur at higher energies, between the HH band and the split conduction bands at the point as well as across the wide gap X valley. The model also includes contributions from the lowest energy indirect bandgap transition and the exciton absorption at the :math:`E_0` critical point. The contributions listed above are summed to compute the overall value of . The real and imaginary components of the overall complex dielectric function are then related via the KramersKronig relations;
The CPPB model included with Solcore also incorporates a modification to the critical point broadening present in Adachi’s description, which is shown to produce a poor fit to experimental data in the vicinity of the and critical points ([4]). To give a more accurate description of the broadening of the optical dielectric function, Kim et al. proposed that a frequencydependent damping parameter be used to replace the damping constant given by Adachi at each critical point ([5], [6]);
Where is the damping constant used by Adachi and describes the shape of the lineshape broadening with producing purely Lorentzian character and producing a good approximation to Gaussian broadening.
The Solcore module absorption_calculator
contains the CPPB model
within the Custom_CPPB
class. The class offers a flexible way to
build up the optical constant model by adding individual critical point
contributions through the Oscillator structure type within Solcore. In
addition to the oscillator functions described by Adachi the
Custom_CPPB
class also provides additional oscilator models and the
Sellmeier equation for describing the real part of the dielectric
function for nonabsorbing materials ([7]).
Description of functions in this module
This module contains a collection of mathematical models to calculate the dielectric constant of a material. The modelling is largely based in the equations and naming used by:
 Woollam, Guide to using WVASE: Spectroscopic Ellipsometry Data Acquisition and Analysis Software. 2012, pp. 1–696.

class
solcore.absorption_calculator.dielectric_constant_models.
Poles
(A=0.0, Ec=5.0)[source]¶ Basic oscillator model for fitting ellipsometry and RAT data. It a pole without broadening and therefore can only represent nonabsorbing materials. Usually, the central energy must lie outside the spectral range of interest.
with:
 An = the amplitud of the oscillator (dimensionless)
 En = the central energy of the pole (in eV)
It is equivalent to Pol.2 in WVASE

name
= 'poles'¶

var
= 2¶

class
solcore.absorption_calculator.dielectric_constant_models.
Lorentz
(An=0.0, En=2.0, Brn=0.1)[source]¶ Classic Lorentz oscillator involving a central energy and a broadening responsible for the absorption.
with:
 An = the amplitud of the oscillator (dimensionless)
 Brn = the broadening (eV)
 En = the central energy (eV)
It is equivalent to Lor.2 in WVASE.

name
= 'lorentz'¶

var
= 3¶

class
solcore.absorption_calculator.dielectric_constant_models.
Gauss
(A=0.0, Ec=2.0, Br=0.1)[source]¶ Gauss oscillator involving a central energy and a broadening responsible for the absorption. It is defined as:
Epsilon_1 is calculated using the KramersKronig relationships
with:
 A = the amplitud of the oscillator (dimensionless)
 Br = the broadening (eV)
 Ec = the central energy (eV)
It is equivalent to Gau.0 in WVASE.

name
= 'gauss'¶

var
= 3¶

class
solcore.absorption_calculator.dielectric_constant_models.
Drude
(An=0.0, Brn=0.1)[source]¶ The classic Drude oscillator describes free carrier effects on the dielectric response. Its form is a Lorentz oscillator with zero center energy.
with:
 An = the amplitud of the oscillator (eV)
 Brn = the broadening (eV)
It is equivalent to Drd.0 in WVASE.

name
= 'drude'¶

var
= 2¶

class
solcore.absorption_calculator.dielectric_constant_models.
Cauchy
(An=0.0, Bn=0.0, Cn=0.0, Ak=0.0, Bk=0.0, Ck=0)[source]¶ Cauchy oscillator model for fitting ellipsometry data. It is generally used to fit non absorbing materials, although an Urbach tail is included to indicate the onset of absorption.
with:
 An, Bn and Cn = the Cauchy coeficients (dimensionless, µm2 and µm4, respectively)
 Ak = the amplitud of the Urbach tail (dimensionless)
 Bk = the multiplicative factor of the Urbach tail (eV1)
 Ck = the energy offset of the Urbach tail (eV)  it is user selected, but redundant with Ak.
with Ak.
It is equivalent to Chy.0 in WVASE.

name
= 'cauchy'¶

var
= 5¶

class
solcore.absorption_calculator.dielectric_constant_models.
PolySegment
(energy, e2)[source]¶ The PolySegment model aims to approximate the dielectric function with a finite number of linear segments. It is similar to a pointbypoint fitting but likely using much less free variables. The segments are created for Epsilon_2 and Epsilon_1 is calculated using the KramersKronig relationships.
It allows for a maximum freedom for fitting data while reducing noise, but does not provide any physically relevant information.
The input parameters are the energies of the endpoints of the segments and the corresponding Epsilon_2 values.

name
= 'polysegment'¶


class
solcore.absorption_calculator.dielectric_constant_models.
Oscillator
(oscillator_type, material_parameters=None, **kwargs)[source]¶

class
solcore.absorption_calculator.dielectric_constant_models.
DielectricConstantModel
(e_inf=1.0, oscillators=())[source]¶ This class groups together several models for the dielectric constant applicable to a given layer. Based on all of them, it calculates the dielectric constant and the complex refractive index at any energy. It requires as input an offset, offen refered as epsilon_infinity.

add_oscillator
(name, **kwargs)[source]¶ Adds an oscillator to the dielectric constant model.
Parameters:  name – The name of the oscillator
 kwargs – The arguments needed to create that oscillator.
Returns:

remove_oscillator
(idx)[source]¶ Removes an oscillator from the model.
Parameters: idx – The number of the oscillator (starting in 1) Returns: None


solcore.absorption_calculator.adachi_alpha.
create_adachi_alpha
(material, Esteps=(1.42, 6, 3000), T=300, wl=None)[source]¶ Calculates the n, k and absorption coefficient of a material using Adachi’s formalism of critical points.
Parameters:  material – A Solcore material
 Esteps – (1.42, 6, 3000) A tuple with the start, end and step energies in which calculating the optical data
 T –
 Temperature in kelvin
 wl – (None) Optional array indicating the wavelengths in which calculating the data
Returns: A tuple containing 4 arrays: (Energy, n, k, alpha)
SOPRA database¶
n order to calculate and model the optical response of potential solar cell architectures and material systems, access to a library of accurate optical constant data is essential. Therefore, Solcore incorporates a resource of freely available optical constant data measured by Sopra S. A. and provided by Software Spectra Inc. The refractive index and extinction coefficient are provided for over 200 materials, including many IIIV, IIVI and group IV compounds in addition to a range of common metals, glasses and dielectrics.
Any material within the Sopra S.A. optical constant database can be
used with the “material” function, but they will have
only the optical parameters and . In the case of
materials that are in both databases, the keyword “sopra” will need to
be set to “True” when creating the material. Once a material is loaded
its , and absorption coefficient data is returned by
calling the appropriate method, for example SiO2.n(wavelength)
and
SiO2.k(wavelength)
. For certain materials in the database, the
optical constants are provided for a range of alloy compositions. In
these cases, any desired composition within the range can be specified
and the interpolated and data is returned.
This module gives access to the vast library of optical constant data, made freely available by the SOPRASA optoelectronics company founded in 1948. For further detail on the data and SOPRASA see the legacy website: http://www.sspectra.com/sopra.html

class
solcore.absorption_calculator.sopra_db.
sopra_database
(Material)[source]¶ Import the SOPRA_DB module from the solcore.material_system package and get started by selecting a material from the extensive list that SOPRASA compiled;
>>> GaAs = sopra_database('GaAs')
Once imported a number of useful methods can be called to return n, k and alpha data for the desired material.

static
material_list
()[source]¶ SOPRA_DB.material_list() :: Loads a list (.pdf file) of all available SOPRA materials.

load_n
(Lambda=None)[source]¶ SOPRA_DB.load_n(Lambda) :: Load refractive index (n) data of the requested material. Optional argument Lambda allows user to specify a custom wavelength range. data will be interpolated into
this range before output.Returns: Tuple of (Wavelength, n)

load_k
(Lambda=None)[source]¶ SOPRA_DB.load_k(Lambda) :: Load refractive index (n) data of the requested material. Optional argument Lambda allows user to specify a custom wavelength range. data will be interpolated into
this range before output.Returns: Tuple of (Wavelength, k)

load_alpha
(Lambda=None)[source]¶ SOPRA_DB.load_alpha(Lambda) :: Load refractive index (n) data of the requested material. Optional argument Lambda allows user to specify a custom wavelength range. data will be interpolated into
this range before output.Returns: Tuple of (Wavelength, alpha)

load_temperature
(Lambda, T=300)[source]¶  SOPRA_DB.load_temperature(T, Lambda) :: Loads n and k data for a set of materials with temperature dependent
 data sets
Optional argument T defaults to 300K Required argument Lambda specifies a wavelength range and the data is interpolated to fit. This is a
required argument here as not all data sets in a group are the same length (will be fixed in a subsequent update).Returns: Tuple of (Wavelength, n, k)

load_composition
(Lambda, **kwargs)[source]¶ SOPRA_DB.load_temperature(T, Lambda) :: Loads n and k data for a set of materials with varying composition. Required argument Lambda specifies a wavelength range and the data is interpolated to fit. This is a
required argument here as not all data sets in a group are the same length (will be fixed in a subsequent update).Keyword argument :: Specify the factional material and fraction of the desired alloy.
Returns: Tuple of (Wavelength, n, k)

static
Manually changing optical constants of a material¶
If you would like to define a material with optical constant data from a file, you can do this by telling Solcore the path to the optical constant data, e.g.:
this_dir = os.path.split(__file__)[0]
SiGeSn = material('Ge')(T=T, electron_mobility=0.05, hole_mobility=3.4e3)
SiGeSn.n_path = this_dir + '/SiGeSn_n.txt'
SiGeSn.k_path = this_dir + '/SiGeSn_k.txt'
In this case, we have defined a material which is like the builtin Solcore germanium material, but with new data for the refractive index and extinction coefficient from the files SiGeSn_n.txt and SiGeSn_k.txt, respectively, which are in the same folder as the Python script. The format of these files is tabseparated, with the first column being wavelength (in nm) and the second column n or k.
References¶
[1]  Adachi, S.: Model dielectric constants of GaP, GaAs, GaSb, InP, InAs, and InSb. Phys. Rev. B 35(14), 7454–7463 (1987) 
[2]  Adachi, S.: Optical dispersion relations for GaP, GaAs, GaSb, InP, InAs, InSb, Alx Ga1−x As, and In1−x Gax Asy P1−y . J. Appl. Phys. 66(12), 6030–6040 (1989) 
[3]  Adachi, S.: Optical dispersion relations for Si and Ge. J. Appl. Phys. 66(7), 3224–3231 (1989) 
[4]  Rakic ́, A.D., Majewski, M.L.: Modeling the optical dielectric func tion of GaAs and AlAs: extension of Adachi’s model. J. Appl. Phys. 80(10), 5909–5914 (1996) 
[5]  Kim, C.C., Garland, J.W., Abad, H., Raccah, P.M.: Modeling the optical dielectric function of semiconductors: extension of the criticalpoint parabolicband approximation. Phys. Rev. B 45(20), 11 749–11 767 (1992) 
[6]  Kim, C.C., Garland, J.W., Raccah, P.M.: Modeling the optical dielectric function of the alloy system AlxGa1xAs. Phys. Rev. B 47(4), 1876–1888 (1993) 
[7]  Woollam, J.A.: Guide to using WVASE 32: spectroscopic ellipsometry data acquisition and analysis software. J. A. Woollam Company (2008). `https://books.google.co.uk/books? id=xOupYgEACAAJ`_ 