newAD2 - optical characterization software

Universal dispersion model

Universal dispersion model (UDM) is basic model used in newAD2 for calculation of optical constants of condensed matter [7,8]. Model have no constant number of parameters. Number of the parameters depends on how many terms model contains. Individual terms represent types of elementary excitations in condensed matter. Number of terms is modified by model attributes and model contains at least three parametric contribution representing interband electronic transitions. The attributes and values of parameters can be saved to the file Universal-id.par in the command line mode of newAD2 by: par save. This is way how to distribute optical constants between individual fits of users. In the model (in modelfile) the name of parfile can be introduced instead of attributes:

modelfile

Universal[:attributes|parfile]

parfile

attributes
Nvc = value
Eg = value
Eh = value
...

newAD2 command

parameters save Universal-id

(save attributes and parameter values of medium id in the Universal-id.par)

Note that attributes and parameters are order insensitive.

udm program

The results of optical characterizations based on UDM can be easily exported by simple program udm. This utility generate table of optical constants as function of photon energy (eV), wavelength (nm), wavenumber (1/cm) or frequency (THz).

Usage:
udm - display help
udm parfile - print table of optical constants of material defined in parfile (4001 lines for energy in range 0.01-100 eV with exponential distribution)
udm parfile lambda=190~850:661 - print table of optical constants of material defined in parfile (661 lines for wavelength in range 190-850 nm withe linear distribution, i.e. with step 1 nm).
Fore more see help.

Download:
source code in C++
for Linux
for Windows

Electron excitations

Figure 1. Schematic diagram of density of states (DOS) of valence electrons electrons: σ - valence band; σ^* - conduction band; ξ^* - band of higher excited states; λ - occupied localized states; λ^* - unoccupied localized states.

Using of individual attributes corresponding to the individual types of elementary excitations will be demonstrated in the following sections. For description of electronic excitations the basic schema introduced in Fig. 1 will be used. Therefore, electronic excitations can be classified by all possible combinations of transitions between occupied bands σ and λ end empty bands σ^*, λ^* and ξ^*.

Interband transitions

Basic three-parametric model includes electronic transitions from valence to conduction bands. In the basic schema this transitions are represented by σ → σ^*. The parameters of this model are Nvc (transition strength, eV²), Eg (minimum energy of transitions - band gap, eV) and Eh (maximum energy of transitions, eV).

Figure 2. Schematic diagram of joint density of states of interband transitions.

modelfile

Universal

parfile


Nvc = 400
Eg = 5
Eh = 30

Note that in this case the parfile begin by obligatory empty line.

Figure 3. Optical constants calculated on the basis of three-parametric dispersion model of interband transitions. The gray lines represents absorption thresholds for given thickness of ambient.

Dielectric response exhibit nonzero absorption only for photon energy in the range from Eg and Eh. Dielectric response is represented by the symmetric broad band in the joint density of states (JDOS) representation having quadratic shape in the vicinity of Eg and Eh (see Figure 2). However, absorption peak is asymmetrical in representation of dielectric response by optical constants (see Figure 3). The optical constants are calculated using parameters introduced in parfile for demonstration. Such simple 3 parametric formula is usable only in the region of transparency of dielectric materials, i.e. for energy smaller than Eg or for values slightly above this value because usually absorption band have more complicated shape. Moreover, condensed matter absorb light above energy Eh thanks the existence of non-occupied ξ^* states. Therefore, in the practice more parametric models are necessary to use in the region of interband transitions. In the case of 3 parametric formula this model is equivalent to empirical models as is Cauchy or Sellmeier formula.

Modification of interband transitions by excitons

In the condensed matter the redistribution of transition strength from higher energies to the spectral parts with lower energies is presented due to multi-particle effects. Therefore, course of absorption is is deformed in comparison to the course of absorption corresponding to the one-particle approximation. Especially, for sufficiently low temperatures the crystalline matter exhibits sharp structures generally called excitons which disappear for high temperatures. Simultaneously joint density of states can exhibits also relatively sharp structures called Van Hove singularities, however temperature practically independent. Finally, the absorption bands of condensed matter exhibit various form given by forms of density of states (DOS) of valence and conduction electrons and probabilities of transitions of electrons between these occupied and unoccupied bands. All these effect even though they are or they are not true excitons are modeled by `excitonic' contributions described by 3 parameters: Aex (amplitude), Eex (characteristic energy, eV) and Bex (peak broadening, eV). Moreover, the parameter A0 is introduced in the set of parameters. This parameter represents amplitude of symmetrical contribution representing basic 3 parametric model. Using this modification it is possible to model any dielectric response of interband transitions of condensed matter.

As a example the modification with two excitons will be introduced even when the number of excitons is unlimited.

Figure 4. Schematic diagram of joint density of states of interband transitions modified by two excitons.

modelfile

Universal:ex=2

parfile

ex=2
Nvc = 400
Eg = 5
Eh = 30
A0 = 1
Aex1 = 0.2
Eex1 = 6
Bex1 = 0.5
Aex2 = 0.3
Eex2 = 12
Bex2 = 1

Figure 5. Optical constants of the interband transitions modified by two excitonic terms. Doted lines represent model without excitonic modification. The gray lines represents absorption thresholds for given thickness of ambient.

Note that negative or invalid value of attribute ex call exception and ex=0 is valid but without effect. For fitting it is necessary one of the amplitudes fix in non zero value because the amplitudes are fully correlated with parameter Nvc (usually it is chosen A0=1). It is important that all `excitonic' contributions are limited to spectral interval between Eg and Eh. For prevent the correlations between the parameters values of characteristic energies Eex are limited to this interval.

Excitations of valence electrons into the higher energy states

In addition to excitation of valence electrons into the conduction band the electrons can also be excited into the higher energy states. As excited electron energy is higher as the final states losing band character and electrons are closer to free electrons, i.e. described by endless band without significant structures. Therefore, joint density of states approach to zero as ~1/E (see Fig. 6). In the basic schema this band is represented by transitions (σ+λ) → ξ^*. In the UDM this contribution is described by simple 2 parametric function with parameters: Ex (energy minimum of higher energy states measured owing to Fermi level, eV) and Nvx (transition strength, eV²).

Figure 6. Schematic diagram of joint density of states of transitions into the higher energy states.

modelfile

Universal:ex=2:he

parfile

ex=2:he
Nvc = 400
Eg = 5
Eh = 30
A0 = 1
Aex1 = 0.2
Eex1 = 6
Bex1 = 0.5
Aex2 = 0.3
Eex2 = 12
Bex2 = 1
Nvx = 1000
Ex = 10

Figure 7. Optical constants calculated on the basis of model of excitations of extended valence electrons. Doted lines represent model without transitions into higher energy states. The gray lines represents absorption thresholds for given thickness of ambient.

We note that absorption of this contribution begins at energy Ex+Eg/2. The Ex is fixed in the physically reasonable range between Eg a Eh Since the transitions into the higher energy states have usually energy outside the range of common laboratory instruments (e.g. > 10 eV), such model is over-parametrized for usual purposes and fixation of same parameters is needed. We usually fix parameters Ex and Eh at estimated values, or values predicted from theoretical calculations. In such cases the free parameter Nvx plays the role of offset in the real part of dielectric response, which is usually described by the ε_∞ parameter. Another option is to extend the experimental data with tabulated data of optical constants if available.

Urbach (exponential) tail

When characterizing films in a spectral range around band gap energy Eg, we have to include the effects concerning localized states excitations, since their influence can not be neglected in this spectral range. Localized states in solids exists mainly in amorphous structures and materials with defects and dopants. However, they can be found also in perfect crystals due to disorder caused by atomic vibrations at finite temperatures. Basic parametrization of the Urbach tail includes the λ → σ^* and σ → λ^* transitions. It is worth noting, that the localized states are sometimes described as a subgap absorption. This is misleading, since majority of transitions described by Urbach tail actually have energy bigger than the band gap. In the framework of UDM we use two parameters to represent this contributions: Eu (Urbach energy - determines the speed of exponential decay, eV) and Nut (transition strength of this contribution, eV²). The overall shape of this contribution also depends on the Eg a Eh parameters of the band structure (see figure 4). The Em parameter shown in the figure 8 is not independent but is rather defined as Em = (Eg+Eh)/2.

Figure 8. Schematic diagram of joint density of states representing Urbach (exponential) tail.

modelfile

Universal:ex=2:he:ut

parfile

ex=2:he:ut
Nvc = 400
Eg = 5
Eh = 30
A0 = 1
Aex1 = 0.2
Eex1 = 6
Bex1 = 0.5
Aex2 = 0.3
Eex2 = 12
Bex2 = 1
Nvx = 1000
Ex = 10
Nut = 40 
Eu = 0.1

Figure 9. Optical constants calculated on the basis of the model included excitations from/into localized into/from extended states (Urbach tail). Doted lines represent model without localized states contributions. The gray lines represents absorption thresholds for given thickness of ambient.

We note that the usage of "Urbach tail" name should be restricted only to describe effects caused by localized states originated in disorder of amorphous materials. The typical value of Urbach energy at the room temperature is about 50 meV. The term "exponential tail" is more suitable in other cases. The origin of join density of states as used in UDM is given by the assumption of symmetry of valence and conduction band. However, there is actually no reason for this assumption, hence the model should in theory contain at least two Urbach energies corresponding to λ → σ^* and σ → λ^* absorption processes. Moreover, localized states have different origins and should have also different characteristic energies and transition strengths. In reality they are indistinguishable and according to our experience the two parameter model is sufficient.

Absorption of the localized states

The last electron excitations type missing in UDM are transitions between localized electron states represented in the basic schema as λ → λ^* transitions. They can be easily modeled by Gaussian-broadened discrete spectra (Gaussian peaks), where every characteristic energy corresponds to single type of localized states. This is not the case for broad absorption peaks which can not be connected to such processes (in this case they represent rather λ → σ^* a σ → λ^* transitions described in the previous paragraph). Each absorption peak in the model is characterized by three parameters: Eloc (characteristic excitation energy corresponding to the localized states type, eV), Bloc (broadening parameter of the Gaussian peak - rms value, eV) and Nloc (transition strength, eV²).

Figure 10. Schematic diagram of joint density of states of Gaussian broadened peaks. In the figure are introduced two extreme examples, i.e. when broadening energy is comparable or higher then characteristic energy and example when broadening energy is much smaller than characteristic energy.

modelfile

Universal:ex=2:he:ut:loc=2

parfile

ex=2:he:ut:loc=2
Nvc = 400
Eg = 5
Eh = 30
A0 = 1
Aex1 = 0.2
Eex1 = 6
Bex1 = 0.5
Aex2 = 0.3
Eex2 = 12
Bex2 = 1
Nvx = 1000
Ex = 10
Nut = 40 
Eu = 0.1
Nloc1 = 0.1
Eloc1 = 4.5
Bloc1 = 0.5
Nloc2 = 0.001
Eloc2 = 3
Bloc2 = 0.1

Figure 11. Optical constants calculated on the basis of the model included excitations between two types of localized states. Doted lines represent model without these excitations. The gray lines represents absorption thresholds for given thickness of ambient.

Free electrons contributions

It will be implemented.

Core electrons excitations

It will be implemented.

Phonon absorption

Phonon absorption is modeled in the UDM as Gaussian-broadened discrete spectra, similarly to absorption of the localized states. The only difference in the parametrization is the use of different units for characteristic energy and broadening parameter: nuph (characteristic wavenumber of phonon, 1/cm), betaph (broadening parameter - fwhm value, 1/cm) a Nph (phonon transition strength, eV²).

modelfile

Universal:ex=2:he:ut:loc=2:ph=6

parfile

ex=2:he:ut:loc=2:ph=6
Nvc = 400
Eg = 5
Eh = 30
A0 = 1
Aex1 = 0.2
Eex1 = 6
Bex1 = 0.5
Aex2 = 0.3
Eex2 = 12
Bex2 = 1
Nvx = 1000
Ex = 10
Nut = 40 
Eu = 0.1
Nloc1 = 0.1
Eloc1 = 4.5
Bloc1 = 0.5
Nloc2 = 0.001
Eloc2 = 3
Bloc2 = 0.1
Nph1 = 0.1
nuph1 = 1000
betaph1 = 100
Nph2 = 0.01
nuph2 = 750
betaph2 = 100
Nph3 = 0.05
nuph3 = 500
betaph3 = 100
Nph4 = 0.01
nuph4 = 1250
betaph4 = 100
Nph5 = 1e-5
nuph5 = 3600
betaph5 = 100
Nph6 = 1e-4
nuph6 = 100
betaph6 = 2500

Figure 12. Optical constants calculated on the basis of the model included six phonon excitations. Doted lines represent same model without these phonon excitations. The gray lines represents absorption thresholds for given thickness of ambient.

We chose a model with six absorption peaks representing hypothetical material to demonstrate the phonon absorption. Narrow absorption peaks with well-defined wavenumber in this example represent single-phonon processes (first 5 absorption peaks). The first 4 represents the vibrations of majority atoms in atomic matrix. Wavenumbers of such vibrations are in the order of hundreds 1/cm up to approx. 1000 1/cm (heavier atoms have lower specific frequencies, lighter atoms have higher frequencies). The fifth absorption peak at thousands of 1/cm represents bond of light atom with the matrix of heavier atoms (the specific value of 3600 1/cm corresponds to the OH bond) The strength of this peak is proportional to the concentration of specific group (OH) in the material (this is valid only for low concentrations). The last peak can not be connected to single-phonon absorption. It represents a broad band corresponding to multi-phonon processes. The number of peaks chosen in the dispersion models is material specific and also depends on the sample thickness While for thin film it is usually sufficient to model the strongest peaks of single-phonon absorption, it is needed for thick films (tablets) to model even weaker processes of multi-phonon absorption (in this case the efficiency of the model decreases with number of absorption peaks).

Published

Material	Form	Sample	Characterization	Range	Published	Produced	Submitted	Download
Al₂O₃, alumina, aluminium oxide	amorphous film (120 nm)	X2890	ellipsometry & photometry	FIR (80 1/cm) - VUV (10.8 eV)	SPIE 9628 (2015) 96281U	Meopta	Daniel Franta	parfile table figure
Si, silicon	crystalline, Float zone		ellipsometry & photometry	FIR (80 1/cm) - VUV (8.5 eV)	unpublished		Daniel Franta	parfile table figure
HfO₂, hafnia, hafnium dioxide	amorphous film (120 nm)	X2194	ellipsometry & photometry	FIR (80 1/cm) - VUV (10.8 eV)	Applied Optics 54 (2015) 9108, SPIE 9628 (2015) 96281U	Meopta	Daniel Franta	parfile table figure
SiO₂, silica, silicon dioxide	amorphous slab (0.405 mm)	Lithosil Q2	ellipsometry & photometry & OC (Palik data & Schott prism)	FIR (80 1/cm) - VUV (8.5 eV)	SPIE 9890 (2016) 989014	Schott	Daniel Franta	parfile table figure
MgF₂, magnesium fluoride	polycrystalline film (120 nm)	X2935	ellipsometry & photometry	FIR (80 1/cm) - EUV (50 eV)	SPIE 9628 (2015) 96281U	Meopta	Daniel Franta	parfile table figure
SiO₂, silica, silicon dioxide	amorphous film (800 nm)	X2546	ellipsometry & photometry	FIR (80 1/cm) - EUV (45 eV)	SPIE 9890 (2016) 989014	Meopta	Daniel Franta	parfile table figure
SiO₂, silica, silicon dioxide	amorphous film (800 nm)	X2551	ellipsometry & photometry	FIR (80 1/cm) - EUV (50 eV)	SPIE 9628 (2015) 96281U	Meopta	Daniel Franta	parfile table figure
SiO₂, silica, silicon dioxide	amorphous film (800 nm)	X2551	ellipsometry & photometry	FIR (80 1/cm) - EUV (45 eV)	SPIE 9890 (2016) 989014	Meopta	Daniel Franta	parfile table figure
Ta₂O₅, tantalum pentoxide	amorphous film (200 nm)	X2656	ellipsometry & photometry	FIR (80 1/cm) - VUV (8.7 eV)	SPIE 9628 (2015) 96281U	Meopta	Daniel Franta	parfile table figure
TiO₂, titania, titanium dioxid	amorphous film (100 nm)	X2801	ellipsometry & photometry	FIR (80 1/cm) - VUV (10.8 eV)	SPIE 9628 (2015) 96281U	Meopta	Daniel Franta	parfile table figure