Opvdm manual

Roderick C. I. MacKenzie

December 17, 2014

roderick.mackenzie@nottingham.ac.uk

1 Output files.

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)

1.1 Calculating the built in potential

To calculate the built in potential of device we must know the following things:

The majority carrier concentrations on the contacts n and p.
The effective densities of states N_LUMO and N_HOMO.
The effective band gap E_g

Then infinite recombination velocity on the contacts is assumed.

Figure 1: Band structure of device in equilibrium.

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:

ELUMO = - χ

(1)

EHOMO = - χ - Eg

(2)

For the left hand side of the device, we can use Maxwell-Boltzmann statistics to calculate the equilibrium Fermi-level (F_i).

(E - F ) pl = Nvexp --HOMO------p kT

(3)

We can then calculate the minority carrier concentration on the left hand side using F_i

( ) Fn---ELUMO-- nl = Ncexp kT

(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.

Ev = - χ- qϕ

(5)

Ev = - χ - Eg - qϕ

(6)

we can now calculate the potential using

( ) nr = Ncexp Fn --ELUMO-- kT

(7)

equation 5.

The minority concentration on the right hand side can now also be calculated using.

(E - F ) pr = Nvexp --v----HOMO-- kT

(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.

2 Optical model

On the left of the interface the electric field is given by

E = E+ e-jk1z + E- ejk1z 1 1 1

(9)

and on the right hand side of the interface the electric field is given by

E2 = E+2 e-jk2z + E-2 ejk2z

(10)

Maxwel’s equations give us the relationship between the electric and magnetic fields for a plane wave.

∇ × E = - jωμH

(11)

which simplifies to:

∂E-= - jω μH ∂z

(12)

Applying equation 12 to equations 9-10, we can get the magnetic field on the left of the interface

- jμ ωHy = - jk E+ e-jk1z + jk E -ejk1z 1 1 1 1 1

(13)

and on the right of the interface

- jμωHy = - jk E+ e-jk2z + jk E -ejk2z. 2 2 2 2 2

(14)

Tidying up gives,

k k Hy1 = ---E+1 e-jk1z - ---E1-ejk1z ωμ ωμ

(15)

Hy = -k-E+ e-jk2z - -k-E -ejk2z 2 ωμ 2 ωμ 2

(16)

2.1 Boundary conditions

We now apply the electric and magnetic boundary conditions[1]

n× (E2 - E1 ) = 0

(17)

n × (H - H ) = 0 2 1

(18)

We let the interface be at z=0, which gives,

+ - + - (E2 + E 2 )- (E 1 + E1 ) = 0

(19)

and

k k -1-(E+2 - E -2 )- (E+1 - E1-)-2-= 0 ωμ ωμ

(20)

. The wavevector is given by

k = 2ω-= ωn- λ c

(21)

. We can therefore write the magnetic boundary condition as

n2(E+2 - E -2 )- n1 (E+1 - E -1 ) = 0

(22)

2.2 Forward propagating wave

Rearrange equation, 22 to give,

- + n2 + - E 1 = E1 - n-(E2 - E 2 ) 1

(23)

Inserting in equation 19, gives

+ - + + n2 + - E 2 + E 2 = E1 + E 1 - n--(E 2 - E2 ) 1

(24)

+ + - n2 + - 2E1 = E2 + E 2 + --(E 2 - E2 ) n1

(25)

n1 n1 - n2 2E+1------- = E+2 + E-2 ------- n1 + n2 n1 + n2

(26)

2.3 Backwards propagating wave

Rearrange equation, 22 to give,

+ - n2 + - E 1 = E1 + n-(E2 - E 2 ) 1

(27)

Inserting in equation 19, gives

+ - - n2- + - - E 2 + E 2 = E1 + n1(E2 - E 2 ) + E1

(28)

- + - n2- + - 2E1 = E2 + E 2 - n1(E 2 - E2 )

(29)

n n - n 2E -1----1-- = E+2--1---2 + E -2 n1 + n2 n1 + n2

(30)

Which is the same result as obtained in [2].

These equations become:

E -1 t12 = E+2 r12 + E2-

(31)

and

E+1 t12 = E+2 + E -2 r12

(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.

+ + ζ2d1 - -ζ2d1 E1 t12 = E2 e + E2 r12e

(33)

and

- + ζ2d1 - -ζ2d1 E1 t12 = E2 r12e + E 2 e

(34)

where

2π ζ = ---¯n λ

(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:

( eζd 0 0 0 r01e-ζd 0 0 0 )
| - t12 eζd 0 0 0 r12e- ζd 0 0 |
|| 0 - t eζd 0 0 0 r e-ζd 0 ||
|| 23 ζd 23 -ζd||
|| 0 0ζd - t34 e 0 -0ζd 0 r34e ||
| 0 r12e 1 0 0 - t12 e 0 0 |
|| 0 0 r23eζd 0 0 - t23 e-ζd 0 ||
( 0 0 0 r34eζd 0 0 - t34 e-ζd )
0 0 0 0 0 0 0 - t45 ( E+1 )
| E+ |
|| E2+ ||
|| 3+||
|| E4-||
| E1-|
|| E2 ||
( E3-)
E4- = ( t01Eexternal)
| 0 |
|| ||
|| 0 ||
| 0 |
( 0 )
0