 
nextnano^{3}  Tutorial
next generation 3D nano device simulator
1D Tutorial
pnjunction
Author:
Stefan Birner
If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
> 1DGaAs_pn_junction_nn3.in / *_nnp.in 
input file for the nextnano^{3} and nextnano++ software
> 1DGaAs_pn_junction_QM_nn3.in
> 2DGaAs_pn_junction_nn3.in / *_nnp.in 
input file for the nextnano^{3} and nextnano++ software
> 3DGaAs_pn_junction_nn3.in / *_nnp.in 
input file for the nextnano^{3} and nextnano++ software
> 1DGaAs_pn_junction_nn3_ForwardBias.in 
input file for the nextnano³ software
> 1DGaAs_pn_junction_nnp_ForwardBias.in 
input file for the nextnano++ software
pnjunction
> 1DGaAs_pn_junction_nn3.in / *_nnp.in  input file for the nextnano^{3} and nextnano++ software
This tutorial aims to reproduce figure 3.1 (p. 51) of
Joachim Piprek's
book "Semiconductor
Optoelectronic Devices  Introduction to Physics and Simulation" (Section 3.2 "pnJunctions").
Doping concentration
 The structure consists of 300 nm GaAs.
At the left and right boundaries, metal contacts are connected to the GaAs
semiconductor (i.e. from 0 nm to 10 nm, and from 310 nm to 320 nm).
The structure is ptype doped from 10 nm to 160 nm and ntype doped from 160
nm to 310 nm.
 The following figure shows the concentration of donors and acceptors of
the pnjunction.
In the ptype region between 10 nm and 160 nm, the number of acceptors N_{A}
is 0.5 x 10^{18} cm^{3}.
In the ntype region between 160 nm and 310 nm, the number of donors N_{D}
is 2.0 x 10^{18} cm^{3}.
Carrier concentrations
 The equilibrium condition for a pnjunction is achieved by a small
transfer of electrons from the n region to the p region, where they recombine
with holes. This leads to a depletion region (depletion width = w_{p}
+ w_{n}), i.e. the region around the pnjunction only has very few
free carriers left.
 The following figure shows the electron and hole densities and the
depletion region around the pnjunction at 160 nm. Here, we assumed that all
donors and acceptors are fully ionized.
Net charges (space charge)
 In the depletion region, a net charge results from the ionized donors N_{D}
and ionized acceptors N_{A}.
 The following figure shows the net charge density of the pnjunction.
Electric field
 The slope of the electric field is proportional to the net charge (Poisson
equation), thus the extremum of the electric field is expected to be at the
pnjunction.
 In regions without charges, the electric field is zero.
 The following figure shows the electric field of the pnjunction.
The extremum of the electric field F_{max} (at 160 nm) can be
approximated as follows:
F_{max} =  e N_{A} w_{p} / (epsilon epsilon_{0})
=  6.997 x 10^{14} V/m^{2} w_{p }= 387 kV/cm
=  e N_{D} w_{n} /
(epsilon epsilon_{0}) =  2.799 x 10^{15} V/m^{2} w_{n
}= 386 kV/cm
where
e = 1.6022 x 10^{19} As
epsilon = 12.93 (dielectric constant of GaAs)
epsilon_{0} = 8.854 x 10^{12} As/(Vm)
N_{A} = 0.5 x 10^{18} cm^{3
} N_{D} = 2.0 x 10^{18} cm^{3
} w_{p} = 55.3 nm^{
} w_{n} = 13.8 nm
Electrostatic potential, conduction and valence band edges
 In regions, where the electric field is zero, the electrostatic potential
is constant.
 The electrostatic potential phi determines the conduction and valence band
edges:
E_{c} = E_{c0}  e phi
E_{v} = E_{v0}  e phi
 The following figure shows the conduction and valence band edges, the
electrostatic potential and the Fermi level of the pnjunction.
Without external bias (i.e. equilibrium), the Fermi level E_{F} is
constant (E_{F} = 0 eV).
The builtin potential phi_{bi} was calculated by nextnano³ to
be equal to 1.426 V.
It can be approximated as follows:
phi_{bi} = F_{max} (w_{p} + w_{n})
/ 2
Assuming F_{max} = 387 kV/cm, this would indicate for the depletion
width: w_{p} + w_{n} = 73.7 nm.
To allow for a constant chemical potential (i.e. constant Fermi level E_{F}),
a total potential difference of e phi_{bi} is required.
Quantum mechanical calculation
> 1DGaAs_pn_junction_QM_nn3.in
 Here, instead of calculating the densities classically, we solve the
Schroedinger equation for the electrons, light and heavy holes in the
singleband approximation over the whole device. We calculate up to 300
eigenvalues for each band. Thus the electron and hole densities are calculated
purely quantum mechanically.
 The following figure shows the electron and hole concentrations for the
classical and quantum mechanical calculations. For the QM calculations,
different boundary conditions were used.
 Dirichlet boundary conditions force the wave functions to be zero at
the boundaries, thus the density goes to zero at the boundaries which is
unphysically.
 Neumann boundary conditions lead to unphysically large values at
the boundaries.
 Mixed boundary conditions are in between.
For the classical calculation, the densities at the boundaries are constant.
Nevertheless, in the interesting region around the pnjunction, all four
options lead to identical densities.
 The following figure shows the band edges of the pnjunction for the four
cases:
 classical calculation
 quantum mechanical calculation with Dirichlet boundary conditions
 quantum mechanical calculation with Neumann boundary conditions
 quantum mechanical calculation with mixed boundary conditions
For all cases the band edges are identical in the area around the pnjunction.
Tiny deviations exist at the boundaries of the device.
 This figure is a zoom into the right boundary of the conduction band edge.
On this scale, the tiny deviations for the different boundary conditions can
be clearly seen.
Nonequilibrium
 Socalled "quasiFermi levels" which are different for electrons (E_{F,n})
and holes (E_{F,p}) are used to describe nonequilibrium carrier
concentrations.
In equilibrium the quasiFermi levels are constant and have the same value for
both electrons and holes (E_{F,n} = E_{F,p} = 0 eV).
The current is proportional to the mobility and the gradient of the
quasiFermi level E_{F}.
> 1DGaAs_pn_junction_nn3_ForwardBias.in 
input file for the nextnano³ software
> 1DGaAs_pn_junction_nnp_ForwardBias.in 
input file for the nextnano++ software
2D/3D simulations
> 2DGaAs_pn_junction_nn3.in / *_nnp.in  input file for the nextnano^{3} and nextnano++ software
> 3DGaAs_pn_junction_nn3.in / *_nnp.in  input file for the nextnano^{3} and nextnano++ software
Input files for the same pn junction structure as in 1D, but this time for a
2D and 3D simulation are also available.
==> 2D: rectangle of dimension 320 nm x 200 nm
==> 3D: cuboid of dimension 320 nm x 200 nm x
100 nm
 Please help us to improve our tutorial! Send comments to
support
[at] nextnano.com .
