Fp.dat:Hole quasi Fermi-level position
x-axis:Position(nm)
y-axis:Electron Energy(eV )
nt.dat:Trapped electron carrier density - position
x-axis:Position(nm)
y-axis:Carrier density(m-3)
Jn_plus_Jp.dat:Total current density (Jn+Jp) - position
x-axis:Position(nm)
y-axis:Total current density (Jn+Jp)(Am-2)
fsrhh.dat:Trap fermi level - position
x-axis:Position(nm)
y-axis:Electron Fermi level(eV )
phi.dat:Potential
x-axis:Position(nm)
y-axis:Potential(V )
p.dat:Total hole density - position
x-axis:Position(nm)
y-axis:Carrier density(m-3)
pf.dat:Free hole carrier density - position
x-axis:Position(nm)
y-axis:Carrier density(m-3)
mu_p.dat:Hole mobility - position
x-axis:Position(nm)
y-axis:Hole mobility(m2V -1s-1)
nt_map.dat:Charge carrier density - position
x-axis:Position(nm)
y-axis:Carrier density(m-3eV -1)
Rn.dat:Electron recombination rate - position
x-axis:Position(nm)
y-axis:Recombination rate(m-3s-1)
Eg.dat:Band gap-position
x-axis:Position(nm)
y-axis:Electron Energy(eV )
prelax.dat:Hole relaxation rate - position
x-axis:Position(nm)
y-axis:Rate(m-3s-1)
nrelax.dat:Electron relaxation rate - position
x-axis:Position(nm)
y-axis:Rate(m-3s-1)
n.dat:Total hole density - position
x-axis:Position(nm)
y-axis:Carrier density(m-3)
imat.dat:Material number - position
x-axis:Position(nm)
y-axis:Number(au)
mu_n.dat:Electron mobility - position
x-axis:Position(nm)
y-axis:Electron mobility(m2V -1s-1)
mu_n_ft.dat:Electron mobility free/all- position
x-axis:Position(nm)
y-axis:Mobility(m2V -1s-1)
fsrhn.dat:Trap fermi level - position
x-axis:Position(nm)
y-axis:Electron Fermi level(eV )
mu_p_ft.dat:Hole mobility free/all- position
x-axis:Position(nm)
y-axis:Mobility(m2V -1s-1)
charge.dat:Charge density - Appleid voltage
x-axis:Applied Voltage(V olts)
y-axis:Charge density(m-3)
Fi.dat:Equlibrium Fermi-level - position
x-axis:Position(nm)
y-axis:Energy(eV )
Jn_diffusion.dat:Diffusion current density - position
x-axis:Position(nm)
y-axis:Electron current density (diffusion)(Am-2)
Jn.dat:Current density - position
x-axis:Position(nm)
y-axis:Electron current density(Am-2)
Jp_drift.dat:Drift current density - position
x-axis:Position(nm)
y-axis:Hole current density (drift)(Am-2)
nf.dat:Free electron carrier density - position
x-axis:Position(nm)
y-axis:Carrier density(m-3)
Gp.dat:Free hole generation rate - position
x-axis:Position(nm)
y-axis:Generation rate(m-3s-1)
Ev.dat:HOMO-position
x-axis:Position(nm)
y-axis:Electron Energy(eV )
dnt.dat:Excess electron density - position
x-axis:Position(nm)
y-axis:Electron density(m-3)
pf_to_nt.dat:Free hole to trapped electron - position
x-axis:Position(nm)
y-axis:Rate(m-3s-1)
ivexternal.dat:Current - Appleid voltage
x-axis:Applied Voltage(V olts)
y-axis:Current(A)
Efield.dat:Material number - position
x-axis:Position(nm)
y-axis:Number(au)
Jp_diffusion.dat:Diffusion current density - position
x-axis:Position(nm)
y-axis:Hole current density (diffusion)(Am-2)
Rp.dat:Hole recombination rate - position
x-axis:Position(nm)
y-axis:Recombination rate(m-3s-1)
nf_to_pt.dat:Free electron to trapped hole - position
x-axis:Position(nm)
y-axis:Rate(m-3s-1)
Jn_drift.dat:Drift current density - position
x-axis:Position(nm)
y-axis:Electron current density (drift)(Am-2)
dpt.dat:Excess electron density - position
x-axis:Position(nm)
y-axis:Hole density(m-3)
Gn.dat:Free electron generation rate - position
x-axis:Position(nm)
y-axis:Generation rate(m-3s-1)
epsilon_r.dat:Relative permittivity - position
x-axis:Position(nm)
y-axis:Relative permittivity(au)
Nad.dat:Doping - position
x-axis:Position(nm)
y-axis:Doping density(m-3)
dn.dat:Change in free electron population - position
x-axis:Position(nm)
y-axis:Carrier density(m-3)
Jp.dat:Current density - position
x-axis:Position(nm)
y-axis:Hole current density(Am-2)
Ec.dat:LUMO-position
x-axis:Position(nm)
y-axis:Electron Energy(eV )
dp.dat:Change in free hole population - position
x-axis:Position(nm)
y-axis:Carrier density(m-3)
Fn.dat:Electron quasi Fermi-level position
x-axis:Position(nm)
y-axis:Electron Energy(eV )
jvexternal.dat:Current density - Appleid voltage
x-axis:Applied Voltage(V olts)
y-axis:Current density(Am-2)
Jp_drift_plus_diffusion.dat:Total current density (Jn+Jp) - position
x-axis:Position(nm)
y-axis:Total current density (Jn+Jp)(Am-2)
pt.dat:Trapped hole carrier density - position
x-axis:Position(nm)
y-axis:Carrier density(m-3)
To calculate the built in potential of device we must know the following things:
Then infinite recombination velocity on the contacts is assumed.
The left hand side of the device is given a reference potential of 0 V. See figure 1. We can then write the energy of the LUMO and HOMO on the left hand side of the device as:
![]() | (1) |
![]() | (2) |
For the left hand side of the device, we can use Maxwell-Boltzmann statistics to calculate the equilibrium Fermi-level (Fi).
![]() | (3) |
We can then calculate the minority carrier concentration on the left hand side using Fi
![]() | (4) |
The Fermi-level must be flat across the entire device because it is in equilibrium. However we know there is a built in potential, we can therefore write the potential of the conduction and valance band on the right hand side of the device in terms of phi to take account of the built in potential.
![]() | (5) |
![]() | (6) |
we can now calculate the potential using
![]() | (7) |
equation 5.
The minority concentration on the right hand side can now also be calculated using.
![]() | (8) |
The result of this calculation is that we now know the built in potential and minority carrier concentrations on both sides of the device.
On the left of the interface the electric field is given by
![]() | (9) |
and on the right hand side of the interface the electric field is given by
![]() | (10) |
Maxwel’s equations give us the relationship between the electric and magnetic fields for a plane wave.
![]() | (11) |
which simplifies to:
![]() | (12) |
Applying equation 12 to equations 9-10, we can get the magnetic field on the left of the interface
![]() | (13) |
and on the right of the interface
![]() | (14) |
Tidying up gives,
![]() | (15) |
![]() | (16) |
We now apply the electric and magnetic boundary conditions[1]
![]() | (17) |
![]() | (18) |
We let the interface be at z=0, which gives,
![]() | (19) |
and
![]() | (20) |
. The wavevector is given by
![]() | (21) |
. We can therefore write the magnetic boundary condition as
![]() | (22) |
Rearrange equation, 22 to give,
![]() | (23) |
Inserting in equation 19, gives
![]() | (24) |
![]() | (25) |
![]() | (26) |
Rearrange equation, 22 to give,
![]() | (27) |
Inserting in equation 19, gives
![]() | (28) |
![]() | (29) |
![]() | (30) |
Which is the same result as obtained in [2].
These equations become:
![]() | (31) |
and
![]() | (32) |
Accounting for propagation we can write. Note the change in sign between [2] and this work, this is because of how I have defined my wave equation.
![]() | (33) |
and
![]() | (34) |
where
![]() | (35) |
Note that in [2] they use ζ1 instead of ζ2. This is a mistake and makes a difference at interfaces. For a device with non reflecting back contacts:
=
The boundary condition on the right hand side is given by:
![]() | (36) |
where we say that E5- is zero and r12 is zero. Therefore we end up with
![]() | (37) |
as the boundary condition.
![]() | (38) |
And because the intensity is proportional to the square of the electric field the absorption coefficient becomes
![]() | (39) |
![]() | (40) |