Cs2AgBiBr6 material model

1. Introduction

This page contains the OghmaNano material model for Cs2AgBiBr6 (Cs2AgBiBr6).

Caesium silver bismuth bromide (Cs2AgBiBr6), lead-free cubic double perovskite (indirect-gap absorber)

The model is written in Lua and provides simulation-ready material parameterisations for use within OghmaNano. For documentation, licensing, references, and information about the scope and accuracy of these models, see the material scripting documentation.

2. Lua material model

-- See end of file for copyright, licensing and documentation links.

local material = {}

function material.name()
	local enabled = true

	return "Cs2AgBiBr6", enabled
end


function material.description()
	local enabled = true

	return "Caesium silver bismuth bromide (Cs2AgBiBr6), lead-free cubic double perovskite (indirect-gap absorber)", enabled
end


function material.formula()
	local enabled = true

	return "Cs2AgBiBr6", enabled
end


function material.Eg(state)
	-- Units: eV
	--
	-- Reference:
	-- A. H. Slavney, T. Hu, A. M. Lindenberg, H. I. Karunadasa,
	-- "A Bismuth-Halide Double Perovskite with Long Carrier
	-- Recombination Lifetimes for Photovoltaic Applications",
	-- J. Am. Chem. Soc. 138, 2138-2141, 2016.
	--
	-- INDIRECT gap ~1.95-2.0 eV at 300 K (the lowest direct transition
	-- lies higher). A value of 2.0 eV is used.
	--
	-- Note: the indirect nature limits absorption near the edge and is
	-- one reason single-junction efficiencies are modest despite the
	-- long carrier lifetimes. Constant value; T-dependence not well
	-- characterised.

	local enabled = true
	local value = 2.0

	return value, enabled
end


function material.Xi(state)
	-- Electron affinity
	-- Units: eV
	--
	-- Reference:
	-- Photoemission / band-alignment studies of Cs2AgBiBr6.
	--
	-- Conduction-band minimum ~ -3.9 eV vs vacuum (electron affinity
	-- ~3.9 eV); with Eg ~2.0 eV the valence band is near ~ -5.9 eV.
	-- Reported band positions scatter by several tenths of an eV.
	-- Approximate.

	local enabled = true
	local value = 3.9

	return value, enabled
end


function material.Nc(state)
	-- Effective conduction-band density of states
	-- Units: m^-3
	--
	-- Note: heavier, more localised bands than the lead iodides
	-- (indirect, mixed Ag/Bi character). Order-of-magnitude
	-- placeholder from m* of order ~0.3-0.5 m0.

	local enabled = true
	local T = state.T
	local value = 5.0e24*(T/300.0)^1.5

	return value, enabled
end


function material.Nv(state)
	-- Effective valence-band density of states
	-- Units: m^-3
	--
	-- Note: order-of-magnitude placeholder; see Nc note.

	local enabled = true
	local T = state.T
	local value = 5.0e24*(T/300.0)^1.5

	return value, enabled
end


function material.mu_e(state)
	-- Low-field electron mobility
	-- Units: m^2 V^-1 s^-1
	--
	-- Reference:
	-- Transport studies of Cs2AgBiBr6 single crystals and films.
	--
	-- Mobility is modest (single-crystal ~0.5-12 cm^2/V/s; films
	-- lower). A representative 1 cm^2/V/s = 1e-4 m^2/V/s is used with a
	-- (300/T)^1.5 phonon form. Approximate.

	local enabled = true
	local T = state.T
	local value = 1.0e-4*(300.0/T)^1.5

	return value, enabled
end

function material.mue_x(state)
	return material.mu_e(state)
end

function material.mue_y(state)
	return material.mu_e(state)
end

function material.mue_z(state)
	return material.mu_e(state)
end


function material.mu_h(state)
	-- Low-field hole mobility
	-- Units: m^2 V^-1 s^-1
	--
	-- Note: comparable to or below the electron value; not tightly
	-- constrained. 1 cm^2/V/s = 1e-4 m^2/V/s with (300/T)^1.5.
	-- Approximate.

	local enabled = true
	local T = state.T
	local value = 1.0e-4*(300.0/T)^1.5

	return value, enabled
end

function material.muh_x(state)
	return material.mu_h(state)
end

function material.muh_y(state)
	return material.mu_h(state)
end

function material.muh_z(state)
	return material.mu_h(state)
end


function material.epsilonr(state)
	-- Relative static permittivity
	-- Dimensionless
	--
	-- Reference:
	-- Dielectric studies of Cs2AgBiBr6.
	--
	-- Frequency-dependent: high-frequency epsilon_inf ~4.5-5.5; the
	-- low-frequency (ionic) value is higher, ~30-60. A low-frequency
	-- value ~30 is used for drift-diffusion. Approximate.

	local enabled = true
	local value = 30.0

	return value, enabled
end


function material.free_to_free_recombination(state)
	-- Radiative (band-to-band) recombination coefficient
	-- Units: m^3 s^-1
	--
	-- Note: Cs2AgBiBr6 is an INDIRECT-gap semiconductor, so intrinsic
	-- band-to-band radiative recombination is weak (this is linked to
	-- its long carrier lifetimes). Weak, poorly constrained
	-- placeholder ~1e-18 m^3/s (well below the direct-gap lead
	-- halides). Approximate.

	local enabled = true
	local value = 1.0e-18

	return value, enabled
end


function material.auger_Cn(state)
	-- Electron Auger recombination coefficient
	-- Units: m^6 s^-1
	--
	-- Note: not well characterised. Order-of-magnitude placeholder.

	local enabled = true
	local value = 1.0e-42

	return value, enabled
end


function material.auger_Cp(state)
	-- Hole Auger recombination coefficient
	-- Units: m^6 s^-1
	--
	-- Note: order-of-magnitude placeholder; see Cn note.

	local enabled = true
	local value = 1.0e-42

	return value, enabled
end


function material.ss_srh_trap_energy(state)
	-- SRH trap energy relative to the middle of the band gap.
	-- Units: eV
	--
	-- Positive values are above mid-gap (towards the conduction band).
	-- Negative values are below mid-gap (towards the valence band).

	local enabled = true
	local value = 0.0

	return value, enabled
end


function material.ss_srh_Nt(state)
	-- SRH trap density
	-- Units: m^-3
	--
	-- Note: often the dominant recombination channel in this indirect
	-- absorber. Quality-dependent placeholder; set from measured
	-- lifetime.

	local enabled = true
	local value = 1.0e21

	return value, enabled
end


function material.ss_srh_sigma_n(state)
	-- Electron capture cross section
	-- Units: m^2

	local enabled = true
	local value = 1.0e-19

	return value, enabled
end


function material.ss_srh_sigma_p(state)
	-- Hole capture cross section
	-- Units: m^2

	local enabled = true
	local value = 1.0e-19

	return value, enabled
end


function material.thermal_conductivity(state)
	-- Thermal conductivity
	-- Units: W m^-1 K^-1
	--
	-- Note: ultralow, like other halide perovskites, ~0.3-0.5 W/m/K
	-- (strong anharmonicity). A value of 0.4 W/m/K is used.
	-- Approximate.

	local enabled = true
	local value = 0.4

	return value, enabled
end


function material.heat_capacity(state)
	-- Specific heat capacity
	-- Units: J kg^-1 K^-1
	--
	-- Note: ~300 J/kg/K used as a representative placeholder (limited
	-- calorimetric data). Approximate.

	local enabled = true
	local value = 300.0

	return value, enabled
end


function material.density(state)
	-- Mass density
	-- Units: kg m^-3
	--
	-- Reference:
	-- A. H. Slavney et al., J. Am. Chem. Soc. 138, 2138, 2016
	-- (structure). Cs2AgBiBr6 rho ~ 4.6 g/cm^3.

	local enabled = true
	local value = 4600.0

	return value, enabled
end


function material.lattice_constant(state)
	-- Cubic lattice constant
	-- Units: m
	--
	-- Reference:
	-- A. H. Slavney et al., J. Am. Chem. Soc. 138, 2138, 2016.
	--
	-- Cs2AgBiBr6 IS cubic (double-perovskite / elpasolite, Fm-3m) at
	-- room temperature, so this is a genuine cubic lattice constant:
	-- a = 11.25 A at 300 K. The large cell reflects the rock-salt
	-- ordering of Ag and Bi on the B-sites (doubled perovskite cell).

	local enabled = true
	local value = 11.25e-10

	return value, enabled
end


function material.print()
	local state = {
		T = 300.0,
		x = 0.0,
		y = 0.0,
		z = 0.0,
		photon_density = 0.0,
	}

	print(string.format("Material:               %s", material.name()))
	print(string.format("Description:            %s", material.description()))
	print(string.format("Formula:                %s", material.formula()))
	print(string.format("Temperature:            %.2f K", state.T))
	print(string.format("Position:               %.6e, %.6e, %.6e m", state.x, state.y, state.z))
	print(string.format("Photon density:         %.6e m^-3", state.photon_density))

	print(string.format("Band gap:               %.6f eV", material.Eg(state)))
	print(string.format("Electron affinity:      %.6f eV", material.Xi(state)))
	print(string.format("Electron mobility:      %.6e m^2/V/s", material.mu_e(state)))
	print(string.format("Hole mobility:          %.6e m^2/V/s", material.mu_h(state)))
	print(string.format("Nc:                     %.6e m^-3", material.Nc(state)))
	print(string.format("Nv:                     %.6e m^-3", material.Nv(state)))
	print(string.format("Relative permittivity:  %.6f", material.epsilonr(state)))

	print(string.format("Radiative coeff.:       %.6e m^3/s", material.free_to_free_recombination(state)))
	print(string.format("Electron Auger coeff.:  %.6e m^6/s", material.auger_Cn(state)))
	print(string.format("Hole Auger coeff.:      %.6e m^6/s", material.auger_Cp(state)))

	print(string.format("SRH trap energy:        %.6f eV", material.ss_srh_trap_energy(state)))
	print(string.format("SRH trap density:       %.6e m^-3", material.ss_srh_Nt(state)))
	print(string.format("SRH sigma n:            %.6e m^2", material.ss_srh_sigma_n(state)))
	print(string.format("SRH sigma p:            %.6e m^2", material.ss_srh_sigma_p(state)))

	print(string.format("Thermal conductivity:   %.6e W/m/K", material.thermal_conductivity(state)))
	print(string.format("Heat capacity:          %.6e J/kg/K", material.heat_capacity(state)))
	print(string.format("Mass density:           %.6e kg/m^3", material.density(state)))
end

return material

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-- This file is part of the OghmaNano Materials Model Library.
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-- https://www.oghma-nano.com
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-- https://www.oghma-nano.com/manual/material-scripts.html
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