Documentation

ParabBand(mstar::Vector{Float64},ϵ₀::Union{Float64,Vector{Float64}},type::Int64,deg::Int64)

Julia type to represent a band. Parameters:

• mstar: effective mass tensor
• ϵ₀: minimum/maximum of the band
• type: cconduction +1 or valence -1 band
• deg: degeneracy

Example

julia> mstar = [1.0, 1.0, 1.0, 0.0, 0.0, 0.0];
julia> ϵ₀ = 0.0;
julia> type = 1;
julia> deg = 1;
julia> band = ParabBand(mstar,ϵ₀,type,deg);  # band creation

BandStructure(n::Int64, bands::Vector{ParabBand}, μ::Union{Float64,Vector{Float64}})

Julia type to represent a band full band structure. Parameters:

• n: number of bands
• bands: vector of bands
• μ: Fermi level position

Example

julia> m_1 = [1., 2., 3., 0.0, 0.0, 0.0];
julia> m_2 = [.5, .7, .9, 0.0, 0.0, 0.0];
julia> band_1 = ParabBand(m_1,0.0,1,1);     # create the conduction band
julia> band_2 = ParabBand(m_2,-0.1,-1,1); 
julia> μ = collect(-.1:.01:.1);     
julia> model = BandStructure(2,[band_1,band_2],μ);    # band creation

electrical_conductivity(bandstructure::BandStructure, Ts::Union{Vector{Float64},Float64,Vector{Int64},Int64}, τ::Union{ScModel,Matthiessen}; exportasdf::Bool=false, fulltensor::Bool=false)

Function that by default computes the trace of the tensorial version of the electical conductivity in units of $(\Omega m)^{-1}$ for a given bandstructure and a choice of the relaxation time $\tau$, for given values of chemical potential $\mu$ and temperature Ts. The transport coefficient is returned as a matrix of dimensions (length(T),length(μ)). The boolean variable exportasdf allows to return the calculations as a DataFrame with all the parameters included. If fulltensor is set to true the full tensor is returned in place of the trace. Reference: "Theory of band warping and its effects on thermoelectronic transport properties", PHYSICAL REVIEW B89.

Example

julia> m = [1.0, 1.0, 1.0, 0.0, 0.0, 0.0];  # effectuve mass
julia> band = ParabBand(m,0.0,1,1); # build the band
julia> μ = collect(-.1:.01:.1); # fermi level position
julia> model = BandStructure(1,band_1,μ)    # build the band structure
julia> T = collect(50:10:650);  # temperature range
julia> τ_form = Scattering.constant()   # relaxation time

julia> σ = electrical_conductivity(model,T,τ_form); # compute the electrical conductivity

seebeck_coefficient(bandstructure::BandStructure, Ts::Union{Vector{Float64},Float64,Vector{Int64},Int64},τ::Union{ScModel,Matthiessen}; exportasdf::Bool=false, fulltensor::Bool=false)

Function that by default computes the Seebeck coefficient in units of $V/K$ for a given bandstructure and a choice of the relaxation time $\tau$, for given values of chemical potential $\mu$ and temperature Ts. The transport coefficient is returned as a matrix of dimensions (length(T),length(μ)). The boolean variable exportasdf allows to return the calculations as a DataFrame with all the parameters included. If fulltensor is set to true, the full tensor is returned in place of the trace. Reference: "Theory of band warping and its effects on thermoelectronic transport properties", PHYSICAL REVIEW B89.

Example

julia> m = [1.0, 1.0, 1.0, 0.0, 0.0, 0.0];  # effectuve mass
julia> band = ParabBand(m,0.0,1,1); # build the band
julia> μ = collect(-.1:.01:.1); # fermi level position
julia> model = BandStructure(1,band_1,μ)    # build the band structure
julia> T = collect(50:10:650);  # temperature range
julia> τ_form = Scattering.constant()   # relaxation time

julia> S = seebeck_coefficient(model,T,τ_form); # compute the Seebeck coefficient

carrier_concentration(bandstructure::BandStructure, Ts::Union{Vector{Float64},Float64,Vector{Int64},Int64}, τ::Union{ScModel,Matthiessen}; exportasdf::Bool=false)

Function that computes the carrier concentration for a given bandstructure and a choice of the relaxation time $\tau$, for given values of chemical potential $\mu$ and temperature Ts. The transport coefficient is returned as a matrix of dimensions (length(T),length(μ)). The boolean variable exportasdf allows to return the calculations as a DataFrame with all the parameters included.

Example

julia> m = [1.0, 1.0, 1.0, 0.0, 0.0, 0.0];  # effectuve mass
julia> band = ParabBand(m,0.0,1,1); # build the band
julia> μ = collect(-.1:.01:.1); # fermi level position
julia> model = BandStructure(1,band_1,μ)    # build the band structure
julia> T = collect(50:10:650);  # temperature range
julia> τ_form = Scattering.constant()   # relaxation time

julia> n = carrier_concentration(model,T,τ_form); # compute the carrier concentration

thermal_conductivity(bandstructure::BandStructure, Ts::Union{Vector{Float64},Float64,Vector{Int64},Int64},τ::Union{ScModel,Matthiessen}; exportasdf::Bool=false, fulltensor::Bool=false)

Function that computes the thermal conductivity in units of $W/K$ for a given bandstructure and a choice of the relaxation time $\tau$, for given values of chemical potential $\mu$ and temperature Ts. The transport coefficient is returned as a matrix of dimensions (length(T),length(μ)). The boolean variable exportasdf allows to return the calculations as a DataFrame with all the parameters included. If fulltensor is set to true, the full tensor is returned in place of the trace.. Reference: "Theory of band warping and its effects on thermoelectronic transport properties", PHYSICAL REVIEW B89.

Example

julia> m = [1.0, 1.0, 1.0, 0.0, 0.0, 0.0];  # effectuve mass
julia> band = ParabBand(m,0.0,1,1); # build the band
julia> μ = collect(-.1:.01:.1); # fermi level position
julia> model = BandStructure(1,band_1,μ)    # build the band structure
julia> T = collect(50:10:650);  # temperature range
julia> τ_form = Scattering.constant()   # relaxation time

julia> Κ = thermal_conductivity(model,T,τ_form); # compute the thermal conductivity

lorenz_tensor(bandstructure::BandStructure, Ts::Union{Vector{Float64},Float64,Vector{Int64},Int64},τ::Union{ScModel,Matthiessen}; exportasdf::Bool=false, fulltensor::Bool=false)

Function that computes the Lorentz tensor for a given bandstructure and a choice of the relaxation time $\tau$, for given values of chemical potential $\mu$ and temperature Ts. The transport coefficient is returned as a matrix of dimensions (length(T),length(μ)). The boolean variable exportasdf allows to return the calculations as a DataFrame with all the parameters included. If fulltensor is set to true, the full tensor is returned in place of the trace. The Lorentz tensor is defined as the ratio between thermal and electrical conductivity multiplied by temperature. Ref.: https://en.wikipedia.org/wiki/Wiedemann-Franz_law

Example

julia> m = [1.0, 1.0, 1.0, 0.0, 0.0, 0.0];  # effectuve mass
julia> band = ParabBand(m,0.0,1,1); # build the band
julia> μ = collect(-.1:.01:.1); # fermi level position
julia> model = BandStructure(1,band_1,μ)    # build the band structure
julia> T = collect(50:10:650);  # temperature range
julia> τ_form = Scattering.constant()   # relaxation time

julia> L = lorenz_tensor(model,T,τ_form); # compute the lorenz tensor

Ln(ϵ₀::Float64, μ::Float64, bandtype::Int64, T::Float64, β::Float64, n::Int64, idx::Int64, scm::ScModel)

This function is called when an integration is required. $n=0,1,2$ is the index of the kinetic coefficient. Reference: "Theory of band warping and its effects on thermoelectronic transport properties", PHYSICAL REVIEW B89.

constant(A::T=1.0) where {T<:Real}

Model for the relaxation time is set to the constant relaxation time approximation (CRTA), $\tau = A\cdot 10^{-14}$. Tje default value for A is 1.0.

Parameter:

• A: value for the A constant above.

Example

julia> τ_form = Scattering.constant()   # relaxation time

T_fun(τ::Function)

Set the temperature dependence of the relaxation time from a function given by the user.

Example

julia> f(T) = sqrt(T)  # τ ∝ √T
julia> τ_form = T_fun(f)   # relaxation time

ET_fun(τ::Function)

Set the energy and temperature dependence of the relaxation time from a function given by the user.

Example

julia> f(ϵ,T) = sqrt(ϵ)/T  # τ ∝ √ϵ/T
julia> τ_form = ET_fun(f)  # relaxation time

impurity(ϵ_im::Real, A_im::Real=1;γ::Real=1.)

Include impurity scattering in the simulations.

Parameters:

• ϵ_im: energy of the impurity in eV
• A: multiplicative constant in front of the functional expression
• γ: γ-parameter three-parameter Lorentzian function (Ref: https://en.wikipedia.org/wiki/Cauchy_distribution)

Example

julia> ϵ_im = 0.2  # eV
julia> τ_form = impurity(ϵ_im, A=1e-1)  # relaxation time

acoustic(bands_min::Union{BandStructure,Real}, A_sm::Real=1., τm_max::Real=1.; T₀::Real=50., μ_min::Real=2, μ_max::Real=2, s=2)

Include acoustic scattering in the simulations. Implementation of the functional expression in Wilson, Alan Herries. 1953, The theory of metals by A. H. Wilson Cambridge Uni. Press Cambridge, England.

Parameters:

• bands_min`: energy of the lowest band. If a BandStructure type is passed, the value is automatically derived.
• A_sm: multiplicative constant in front of the functional expression for the acoustic τ in semiconductors in units of $5\cdot 10^{-20}$ s.
• τm_max: free parameter to constraint the functional form for the acoustic τ in metals. $τ(ϵ₀+μ\_max,T) = τm\_max$ in units of $2\cdot 10^{-12}$ s, where $ϵ₀$ is the minimum/maximum of the band.
• T₀: minimum of temperature range in which the τ is defined (note: τ ∝ 1/(T-T₀), where T is the temperature input defined by the user.
• μ_min: left shift in energy from the energy of the lowest band (ϵ_min). It defines the first point at which τ is computed (i.e., $τ \propto 1/\sqrt{μ-(ϵ\_min-μ\_min)}$).
• μ_max: right shift in energy from the energy of the band $\ϵ₀$. It defines the last point at which τ is computed (i.e., $τ(ϵ₀+μ\_max,T) = τm\_max$).

Example

julia> band_1 = ParabBand([5.0, 5.0, 5.0, 0.0, 0.0, 0.0],1.0, 1,1);    # conduction band
julia> band_2 = ParabBand([0.1, 0.5, 3.0, 0.0, 0.0, 0.0],0.5,-1,1);    # valence band
julia> model = BandStructure(2,[band_1,band_2],0.8);   # build the two-band structure
julia> τ_form = Scattering.acoustic(model,T₀=180,μ_min=5,μ_max=5);  # relaxation time

matthiessen(τ_models::Array{ScModel};γ::Float64=-1.)

This function applies Matthiessen's rule to sum up different scattering mechanisms.

Parameters:

• τ_models: vector of relaxation time models
• γ: exponent in the generalized Matthiessen's rule:

\[\tau = (\sum_i τ_i^γ)^{1/γ}, \]

where each $\tau_i$ can be a function of $T$ and/or $\mu$.

Example

julia> ϵ_im = 0.2  # eV
julia> τ1 = constant()  # CRTA
julia> τ2 = impurity(ϵ_im,A=1e-1)   # impurity scattering
julia> τ_model =  matthiessen([τ1,τ2]);  # matthiessen's rule

plot(num_plots, x_axis, data, z; vlines=Float64[], annotations=[], color=ColorTypes.RGBA{Float64}(Colors.JULIA_LOGO_COLORS.blue), colorscheme=:viridis, titles=L"", xlabels=L"", ylabels=L"", zlabel=L"")

Plot the transport coefficients. y must be a vector if plotting a single line, or a matrix of shape (length(x_axis),length(z_axis)) if plotting more than one line.

Parameters:

- num_plots::Int64 : number of transport coefficients to plot in the same figure
- x::Union{Vector{Float64},Float64,Vector{Int64},Int64} # x axis vector
- y::Union{Vector{Float64},Matrix{Float64},Vector{Vector{Float64}},Vector{Matrix{Float64}}} # y axis vector/matrix
- z::Union{Vector{Float64},Vector{Int64}}   # z axis vector
- titles::Union{LaTeXString,Vector{LaTeXString}}    # titles for each plot
- xlabels::Union{LaTeXString,Vector{LaTeXString}}   # x labels for each plot
- ylabels::Union{LaTeXString,Vector{LaTeXString}}   # x labels for each plot
- zlabel::LaTeXString   # z labels for the shared colorbar
- color::ColorTypes.RGBA{Float64}   # line color
- colorscheme   # colormap
- vlines::Union{Float64,Array{Float64}} # add vertical lines to the plot
- annotations::Vector{Any}  # add text annotation to the plot

Example

1. Single line (z is empty). Plot of electrical conductivity, Seebeck and carrier concentration as functions of the band gap.

julia> titles  = [L"$σ$ vs gap, $μ$ = %$μ, $τ = const$", L"$S$ vs gap, $μ$ = %$μ, $τ = const$", L"$n$ vs gap, $μ$ = %$μ, $τ = const$"]
julia> xlabels = [L"$Gap$ $[eV]$" for i in 1:3];
julia> ylabels = [L"$\sigma$ $[(\Omega m)^{-1}]$", L"$S$ $[\mu VK^{-1}]$", L"n"];
julia> annotations = Dict(L"v1" => [(0.11,6e5),(0.11,0),(0.11,5e6)], L"v2" => [(0.01,6e5-5e3),(0.01,0),(0.01,5e6-5e4)]);
julia> fig = plot(3, gap, [σ,S,n], titles=titles, xlabels=xlabels, ylabels=ylabels, vlines=[0.0,0.1], annotations=annotations);

1. Multi-lines. Plot of electrical conductivity, Seebeck and carrier concentration as functions of the bandgapandtemperature` (z axis).

julia> titles  = [L"$σ$ vs gap, $μ$ = %$μ, $τ = const$", L"$S$ vs gap, $μ$ = %$μ, $τ = const$", L"$n$ vs gap, $μ$ = %$μ, $τ = const$"]
julia> xlabels = [L"$Gap$ $[eV]$" for i in 1:3];
julia> ylabels = [L"$\sigma$ $[(\Omega m)^{-1}]$", L"$S$ $[\mu VK^{-1}]$", L"n"];
julia> zlabel  = L"$T$ $[K]$";
julia> annotations = Dict(L"v1" => [(0.11,6e5),(0.11,0),(0.11,5e6)], L"v2" => [(0.01,6e5-5e3),(0.01,0),(0.01,5e6-5e4)]);
julia> fig = plot(3, gap, [σ,S,n], T, titles=titles, xlabels=xlabels, ylabels=ylabels, zlabel=zlabel, vlines=[0.0,0.1], annotations=annotations);

`

plot_bandstructure(bs::BandStructure,xaxis::AbstractArray=range(-1, 1, length=100); colors=nothing, label="")

Plot a given parabolic band structure.

Parameters:

• bs: band structure
• xaxis: x axis vector
• colors: tuple of three color from Colors. First: conduction bands. Second: valence band. Third: Fermi level.

Examples:

julia> band_1 = ParabBand([5.0, 5.0, 5.0, 0.0, 0.0, 0.0],1.0, 1,1);    # conduction band
julia> band_2 = ParabBand([0.1, 0.5, 3.0, 0.0, 0.0, 0.0],0.5,-1,1);    # valence band
julia> μ = 0.8;   # Fermi level
julia> model = BandStructure(2,[band_1,band_2],μ);   # build the two-band structure
julia> fig,ax = plot_bandstructure(model,colors=(:green,:red,:blue))

plot_τ(scm::Union{ScModel,Matthiessen}, t::Union{Vector{Float64},Vector{Int64}}, type="e-T",E_argmax::Int64=50; μ=nothing, ϵ₀::Float64=-42., bandtype::Int64=0)

3D plot of a given relaxation time as a function of temperature, energy and Fermi level.

Parameters:

• scm: relaxation time model
• t: temperature
• type: type of plot specificed in "xaxis-zaxis" format. Available types are "e-T","T-e","μ-T","T".
• μ: Fermi level vector. It's mandatory if type="μ-T".
• ϵ₀: band energy. It's mandatory for acoustic relaxation time.
• bandtype: band type (conduction, valence). It's mandatory for acoustic relaxation time.

Examples:

1. Constant relaxation time

julia> T = collect(300:10:650);  # temperature
julia> τ_form = Scattering.constant()
julia> fig = plot_τ(τ_form, T, "T");

1. Acoustic

julia> ϵ₀ = .1;  # band energy
julia> type = 1;  # band type (conduction)
julia> T = collect(300:10:650);  # temperature
julia> μ = collect(0.0:0.1:1.0);  # Fermi level
julia> τ_form = Scattering.acoustic(model,T₀=180,μ_min=5,μ_max=5);
julia> fig = plot_τ(τ_form, T, "μ-T", μ=μ, ϵ₀=ϵ₀, bandtype=type);

savefig(fullpath::String, fig::Figure)

Export to disk a given figure to fullpath.

Example

1. Save the plot for the relaxation time as a function of temperature and Fermi level.

julia> τ_form = Scattering.acoustic(model,T₀=180,μ_min=5,μ_max=5);
julia> fig = plot_τ(τ_form, T, "μ-T", μ=μ, ϵ₀=ϵ₀, bandtype=type);
julia> savefig("acoustic_tau.png", fig);

1. Save the plot for the electrical conductivity, Seebeck coefficient and carrier density as a function of temperature and Fermi level.

julia> fig = plot(3, T, [σ,S*1e6,n], μ, titles=titles, xlabels=xlabels, ylabels=ylabels, zlabel=zlabel);
julia> savefig("transport_coefficients.png", fig);

savedata(path::String, data::DataFrame)

Export to disk a given tranport coefficients simulation to a csv file.