MoS2 with moving defects and Schottky Barrier Lowering.
Memristor simulation with additional moving positively charged defects and Schottky barrier lowering at the contacts.
module Ex107_MoS2_withIons_BarrierLowering
using ChargeTransport
using ExtendableGrids
using PyPlotyou can also use other Plotters, if you add them to the example file
function main(;Plotter = PyPlot, plotting = false, verbose = false, test = false, barrierLowering = true)
if plotting
Plotter.close("all")
end
################################################################################
if test == false
println("Set up grid, regions and time mesh")
end
################################################################################
# region numbers
regionflake = 1
# boundary region numbers
bregionLeft = 1
bregionRight = 2
# grid
h_flake = 1.0 * μm # length of the conducting channelnon-uniform grid
coord1 = geomspace(0.0, h_flake/2, 5e-4 * h_flake, 2e-2 * h_flake)
coord2 = geomspace(h_flake/2, h_flake, 2e-2 * h_flake, 5e-4 * h_flake)
coord = glue(coord1, coord2)
grid = simplexgrid(coord)
# set region in grid
cellmask!(grid, [0.0], [h_flake], regionflake, tol = 1.0e-18)
if plotting
gridplot(grid, Plotter = Plotter)
Plotter.title("Grid")
end
if test == false
println("*** done\n")
end
################################################################################
if test == false
println("Define physical parameters and model")
end
################################################################################
# set indices of unknowns
iphin = 1 # electron quasi Fermi potential
iphip = 2 # hole quasi Fermi potential
iphix = 3
numberOfCarriers = 3 # electrons, holes and ionsWe define the physical data
T = 300.0 * K
εr = 9.0 * 1.0 # relative dielectric permittivity
εi = 1.0 * εr # image force dielectric permittivity
Ec = - 4.0 * eV
Ev = - 5.3 * eV
Ex = - 4.38 * eV
Nc = 2 * ( 2 * pi * 0.55 * mₑ * kB * T/(Planck_constant^2) )^(3/2) /m^3
Nv = 2 * ( 2 * pi * 0.71 * mₑ * kB * T/(Planck_constant^2) )^(3/2) /m^3
Nx = 1.0e28 / (m^3)
μn = 1e-4 * (m^2) / (V * s)
μp = 1e-4 * (m^2) / (V * s)
μx = 0.8e-13 * (m^2) / (V * s)
# Schottky contact
barrierLeft = 0.225 * eV
barrierRight = 0.215 * eV
An = 4 * pi * q * 0.55 * mₑ * kB^2 / Planck_constant^3
Ap = 4 * pi * q * 0.71 * mₑ * kB^2 / Planck_constant^3
vn = An * T^2 / (q*Nc)
vp = Ap * T^2 / (q*Nv)
Nd = 1.0e17 / (m^3) # doping
Area = 2.1e-11 * m^2 # Area of electrodeScan protocol information
endTime = 9.6 * s
amplitude = 12.0 * V
scanrate = 4*amplitude/endTime
# Define scan protocol function
function scanProtocol(t)
if 0.0 <= t && t <= endTime/4
biasVal = 0.0 + scanrate * t
elseif t >= endTime/4 && t <= 3*endTime/4
biasVal = amplitude .- scanrate *(t-endTime/4)
elseif t >= 3*endTime/4 && t <= endTime
biasVal = - amplitude .+ scanrate * (t-3*endTime/4)
else
biasVal = 0.0
end
return biasVal
endApply zero voltage on left boundary and a linear scan protocol on right boundary
contactVoltageFunction = [zeroVoltage, scanProtocol]
if test == false
println("*** done\n")
end
################################################################################
if test == false
println("Define System and fill in information about model")
end
################################################################################
# Initialize Data instance and fill in predefined data
data = Data(grid, numberOfCarriers, contactVoltageFunction = contactVoltageFunction)
data.modelType = Transient
data.F = [FermiDiracOneHalfTeSCA, FermiDiracOneHalfTeSCA, FermiDiracMinusOne]
data.bulkRecombination = set_bulk_recombination(;iphin = iphin, iphip = iphip,
bulk_recomb_Auger = false,
bulk_recomb_radiative = false,
bulk_recomb_SRH = false)
if barrierLowering
data.boundaryType[bregionLeft] = SchottkyBarrierLowering
data.boundaryType[bregionRight] = SchottkyBarrierLowering
else
data.boundaryType[bregionLeft] = SchottkyContact
data.boundaryType[bregionRight] = SchottkyContact
end
data.fluxApproximation .= ExcessChemicalPotential
enable_ionic_carrier!(data, ionicCarrier = iphix, regions = [regionflake])
if test == false
println("*** done\n")
end
################################################################################
if test == false
println("Define Params and fill in physical parameters")
end
################################################################################
params = Params(grid, numberOfCarriers)
params.temperature = T
params.UT = (kB * params.temperature) / q
params.chargeNumbers[iphin] = -1
params.chargeNumbers[iphip] = 1
params.chargeNumbers[iphix] = 2
for ireg in 1:length([regionflake]) # region data
params.dielectricConstant[ireg] = εr * ε0
params.dielectricConstantImageForce[ireg] = εi * ε0
# effective DOS, band-edge energy and mobilities
params.densityOfStates[iphin, ireg] = Nc
params.densityOfStates[iphip, ireg] = Nv
params.bandEdgeEnergy[iphin, ireg] = Ec
params.bandEdgeEnergy[iphip, ireg] = Ev
params.mobility[iphin, ireg] = μn
params.mobility[iphip, ireg] = μp
params.densityOfStates[iphix, ireg] = Nx
params.bandEdgeEnergy[iphix, ireg] = Ex
params.mobility[iphix, ireg] = μx
end
params.SchottkyBarrier[bregionLeft] = barrierLeft
params.SchottkyBarrier[bregionRight] = barrierRight
params.bVelocity[iphin, bregionLeft] = vn
params.bVelocity[iphin, bregionRight] = vn
params.bVelocity[iphip, bregionLeft] = vp
params.bVelocity[iphip, bregionRight] = vp
# interior doping
params.doping[iphin, regionflake] = Nd
data.params = params
ctsys = System(grid, data, unknown_storage=:sparse)
if test == false
println("*** done\n")
end
################################################################################
if test == false
println("Define control parameters for Solver")
end
################################################################################
control = SolverControl()
if verbose == true
control.verbose = verbose
else
control.verbose = "eda" # still print the time values
end
if test == true
control.verbose = false # do not show time values in testing case
end
control.damp_initial = 0.9
control.damp_growth = 1.61 # >= 1
control.max_round = 5
control.abstol = 1.0e-9
control.reltol = 1.0e-9
control.tol_round = 1.0e-9
control.Δu_opt = Inf
control.Δt = 1.0e-4
control.Δt_min = 1.0e-5
control.Δt_max = 5.0e-2
control.Δt_grow = 1.05
if test == false
println("*** done\n")
end
################################################################################
if test == false
println("Compute solution in thermodynamic equilibrium")
end
################################################################################
# initialize solution and starting vectors
solEQ = equilibrium_solve!(ctsys, control = control, nonlinear_steps = 0)
inival = solEQ
if plotting
label_solution, label_density, label_energy = set_plotting_labels(data)
label_energy[1, iphix] = "\$E_x-q\\psi\$"; label_energy[2, iphix] = "\$ - q \\varphi_x\$"
label_density[iphix] = "\$ n_x\$"; label_solution[iphix] = "\$ \\varphi_x\$"
Plotter.figure()
plot_densities(Plotter, ctsys, solEQ,"Equilibrium", label_density)
Plotter.legend()
Plotter.figure()
plot_solution(Plotter, ctsys, solEQ, "Equilibrium", label_solution)
end
if test == false
println("*** done\n")
end
################################################################################
if test == false
println("IV Measurement loop")
end
################################################################################
sol = solve(ctsys, inival = inival, times=(0.0, endTime), control = control)
if test == false
println("*** done\n")
end
################################################################################
######### IV curve calculation
################################################################################
IV = zeros(0) # for saving I-V data
tvalues = sol.t
number_tsteps = length(tvalues)
biasValues = scanProtocol.(tvalues)
factory = TestFunctionFactory(ctsys)
tf = testfunction(factory, [bregionLeft], [bregionRight])
push!(IV, 0.0)
for istep = 2:number_tsteps
Δt = tvalues[istep] - tvalues[istep-1] # Time step size
inival = sol[istep-1]
solution = sol[istep]
I = integrate(ctsys, tf, solution, inival, Δt)
current = 0.0
for ii = 1:numberOfCarriers+1
current = current + I[ii]
end
push!(IV, current)
end
if plotting
Plotter.figure()
Plotter.plot(tvalues, biasValues, marker = "x")
Plotter.xlabel("time [s]")
Plotter.ylabel("voltage [V]")
Plotter.grid()
Plotter.figure()
Plotter.semilogy(biasValues, abs.(Area .* IV), linewidth = 5, color = "black")
Plotter.grid()
Plotter.xlabel("applied bias [V]")
Plotter.ylabel("total current [A]")
end
testval = sum(filter(!isnan, solEQ))/length(solEQ)
return testval
end # main
function test()
main(test = true, barrierLowering = true) ≈ -1692.2303837883194
end
end # moduleThis page was generated using Literate.jl.