nextnano^{3}  Tutorial
next generation 3D nano device simulator
1D Tutorial
SchrödingerPoisson  A comparison to the tutorial file of Greg Snider's
code
Author:
Stefan Birner
> 1DGreg_Snider_MANUAL.in
> 1DGreg_Snider_MANUAL_analytic_doping_function.in
> 2DGreg_Snider_MANUAL.in
==>
Download these input files
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We appreciate that Greg Snider (University of Notre Dame) provided his code,
the manual and the input files free of charge, so that we could use it here as a
test case.
His 1D Poisson/Schrödinger code can be obtained from here:
http://www.nd.edu/~gsnider/
This tutorial is based on his manual ('1D Poisson Manual.pdf ',
MANUAL.EX ).
SchrödingerPoisson  A comparison to the tutorial file of Greg Snider's
code
We simulate a structure consisting of


surface 
Schottky barrier of 0.6 V 

15 nm 
GaAs 
n type doped (10^{18} cm^{3}) 

20 nm 
Al_{0.3}Ga_{0.7}As 
n type doped (10^{18} cm^{3}) 

5 nm 
Al_{0.3}Ga_{0.7}As 


15 nm 
GaAs 
quantum well 

50 nm 
Al_{0.3}Ga_{0.7}As 


250 nm 
Al_{0.3}Ga_{0.7}As 
p type doped (10^{17} cm^{3}) 


substrate 

The grid resolution is 1 nm with the exception of the 250 nm layer which has
a resolution of 5 nm.
The dopants are assumed to be fully ionized.
The temperature is 300 K.
The Schrödinger equation will be solved between 5 nm and 195 nm.
Doping profile
We consider two further impurity profiles resulting from ion implantation
using LSS theory.
The donor and
acceptor profiles are written out to the file "doping_concentration1D.dat "
and look as follows:
The relevant parameters are:

implant 
dose [1/cm^{2}] 
projected range R_{p} [nm] 
projected straggle Delta R_{p} [nm] 

donor 
2 * 10^{12} 
86 
44 

acceptor 
1 * 10^{11} 
75 
20 
For further details on the LSS theory (ion implantation) and on the doping
profiles, please check the relevant keyword
$dopingfunction .
Conduction and valence band edges
The following figure shows the conduction and valence band edges as well as
the Fermi level (which is constant and has the value of 0 eV) for the structure
specified above. These bands are the solutions of the the selfconsistent
SchrödingerPoisson equation.
Both codes, nextnano³ and Greg Snider's "1D Poisson" lead to the same
results.
Electron eigenstates and eigenfunctions
Inside the GaAs quantum well there are three confined electron states. The
ground state is below the Fermi level and thus occupied.
The following figure shows a zoom of the GaAs Quantum well.
The wave functions as calculated with nextnano³ are nearly identical to
Greg Snider's "1D Poisson" code, as well as the energies.
However, there are tiny differences which is not too suprising as the conduction
band profile is not completely identical:
nextnano³ uses
multiple points at material interfaces which lead to sharp (abrupt)
interfaces.
nextnano³ interface: 39.5 nm (two grid points, one for GaAs and one for
AlGaAs)
54.5 nm (two grid points, one for GaAs and one for AlGaAs)
Greg Snider's interfaces: There is a grid point at 39 nm (AlGaAs) and the next
one at 40 nm (GaAs),
and at 54 (GaAs) nm and the next one at 55 nm (AlGaAs).
Thus at 39.5 and 54.5 nm, an interpolated value is used in the plot of the
conduction band diagram.
Electron states 
nextnano³ 
Greg Snider's "1D Poisson" code 
E_{1} [meV] 
 3.0 
 1.3 
E_{2} [meV] 
43.5 
44.0 
E_{3} [mV] 
117.5 
117.8 
Electron and hole densities
The electron and hole densities are depicted in this figure. Again, there is
also nice agreement between the two codes.
The integrated electron density in the
GaAs quantum well region is 0.664 * 10^{12}
cm^{2}. (Greg Snider's result: 0.636 * 10^{12}
cm^{2})
The integrated hole density in the right
most Al_{0.3}Ga_{0.7}As region is 1.033
* 10^{12} cm^{2}. (Greg Snider's result: 1.085
* 10^{12} cm^{2})
The relevant output files are:
 densities/int_el_dens.dat
 densities/int_hl_dens.dat
This tutorial shows very nicely that both codes, nextnano³ and Greg
Snider's "1D Poisson" lead to the same results.
Greg Snider's 1D Poisson/Schrödinger code can be obtained from here:
http://www.nd.edu/~gsnider/
2D and 3D simulations
> 2DGreg_Snider_MANUAL.in
> 3DGreg_Snider_MANUAL.in
The following figure shows the conduction band profile and the ground state
wave function (probability density psi²) of the same structure in a 2D simulation
where the y direction has been assumed to be of length 100 nm with periodic
boundary conditions.
 Please help us to improve our tutorial! Send comments to
support
[at] nextnano.com .
