# nextnano^{3} - Tutorial
## next generation 3D nano device simulator
## 1D Tutorial
## n-i-n Si resistor
Author:
Stefan Birner
If you want to obtain the input file that is used within this tutorial, please
submit a support ticket.
`-> Si_nin_resistor_classical_1D_nn3.in / *_nnp.in - ` input file for the next**nano**^{3}
and next**nano**++ software
Si_nin_resistor_quantum_1D_nn3.in / *_nnp.in - input file for the next**nano**^{3}
and next**nano**++ software
This tutorial is based on the example presented on p. 43 in Stefan Hackenbuchner's PhD thesis, TU Munich
(2002) and on the following paper:
Towards
fully quantum mechanical 3D device simulation
M. Sabathil, S. Hackenbuchner, J.A. Majewski, G. Zandler, P.
Vogl
Journal of Computational Electronics 1, 81 (2002)
## n-i-n Si resistor
`-> Si_nin_resistor_classical_1D_nn3.in`
Here, we illustrate our method for calculating the current by studying simple
one-dimensional examples that we can compare to full Pauli master equation
results. Our method amounts to calculating the electronic structure of a device
fully quantum mechanically, yet employing a semi-classical scheme for the
evaluation of the current. As we shall see, the results are close to those
obtained by the full Pauli master equation provided we limit ourselves to
situations not too far from equilibrium.
We consider a one-dimensional 300 nm Si-based n-i-n resistor at room
temperature, where "n-i-n"
stands for "n-doped / intrinsic
/ n-doped". The intrinsic region and the n-doped
regions are each 100 nm wide.
*The figure shows the geometry of the n-i-n Si
resistor.*
The n-doped regions at the left and right sides are doped with a doping
concentration of N_{D} = 1 x 10^{17} cm^{-3}.
The intrinsic region in the center of the device has a background concentration
of n_{i} = 1 x 10^{13} cm^{-3} (according to PhD thesis
of S. Hackenbuchner, p. 43) (why? compare with value
of n_{i} some lines below! Because of Fermi-Dirac statistics or because
of other valleys' contribution?). This value is calculated by next**nano**³
automatically and does not have to be entered in the input file. Assuming
Maxwell-Boltzmann statistics, the intrinsic carrier concentration n_{i}
is given by:
n_{i} = (N_{c} N_{v})^{1/2} exp (- E_{gap}
/ (2 k_{B} T))
T = 300 K; T: temperature
k_{B} T = eV (at room temperature)
E_{gap} = 1.095 eV; E_{gap}: band gap energy of Si at 300 K
N_{c} = 2.738*10^{19} cm^{-3} (see Tutorial "I-V characteristics of an n-doped Si structure")
N_{v} = 1.138*10^{19} cm^{-3} (see Tutorial "I-V characteristics of an n-doped Si structure")
Note: Using this equation, one obtains n_{i} = 1.12*10^{10}
cm^{-3}.
For a more detailed discussion of this equation (including Fermi-Dirac
statistics), please read the description in Tutorial "I-V characteristics of an n-doped Si structure".
At both ends of the device there are ohmic contacts.
The conductivity electron mass is given by
m_{e}*_{cond }= 3 / (1/0.916^{ }+ 2/0.19) m_{0 }=
0.258 m_{0}
whereas the DOS electron effective mass is given by
m_{e}*_{DOS} = (0.916·0.19^{2})^{1/3 }m_{0
}= 0.321m_{0}.
The static dielectric constant is given by epsilon = 11.7.
For the donors we assumed an ionization energy of 0.015 eV as well as a
degeneracy factor of 2.
For the mobility which should depend on the electric field and on the
concentration of ionized impurities we assumed```
mobility-model-simba-2
``` and used the following parameters:
```
$mobility-model-simba
!
```
!
material-name =
Si !
taken from the
SIMBA
manual
!
n-alpha-doping =
0.73 ! []
n-N-ref-doping =
1.072e17 ! [1/cm^{3}]
n-mu-min
= 55.2 ! [cm^{2}/Vs]
n-mu-doping =
1374.0 ! [cm^{2}/Vs]
n-gamma-temp
= 1.5 ! []
n-E0-saturation = 8.0e3 !
[V/cm]
n-T0-E-saturation = 300.0 ! [K]
n-temp-dependence-E = 0.0 ! [V/Kcm]
n-alpha-E
= 2.0 ! []
n-beta-E
= 0.5 ! []
n-v0-saturation =
1.03e7 ! [cm/s]
n-temp-dependence-v = 0.0
! [cm/Ks]
n-kappa-v
= 2.0 ! []
n-ET-perpendicular = 64.970e3 ! [V/cm]
The electron density in next**nano**³ can be calculated in two ways:
- classical density (Thomas-Fermi approximation)
- quantum mechanical density (local quasi-Fermi levels): The charge density
is calculated for a given applied voltage by assuming the carriers to be in
*
local* equilibrium that is characterized by energy-band dependent local
quasi-Fermi levels E_{F}(**x**) (i.e. in the simplest case, one for
holes and one for electrons)
These local quasi-Fermi levels are determined by global current conservation
div **j** = 0, where the current is assumed to be given by the
semi-classical relation **j** = µ(**x**) n(**x**) grad E_{F}(**x**)
where µ(**x**) is the electron mobility which is determined in our example
from```
mobility-model-simba-2
``` .
The carrier wave functions and energies are calculated by solving the
single-band Schrödinger-Poisson equation self-consistently.
The open system is
mimicked by using mixed Dirichlet and Neumann boundary conditions
(Fischetti (1998), Lent and Kirkner (1990), Frensley (1992)) at ohmic
contacts. *(Note: The feature of using "mixed" boundary conditions for the
Schrödinger equation is not supported any more.)* The Schrödinger, Poisson and current continuity equations are solved
iteratively. As a preparatory step, the built-in potential is calculated for
zero applied bias by solving the Schrödinger-Poisson equation
self-consistently employing a predictor-corrector approach and setting to zero
the electric field at the ohmic contacts. For applied bias, the Fermi level
and the potential at the contacts are then shifted according to the applied
potential which fixes the boundary conditions.
The main iteration scheme itself consists of two parts:
- In the first part, the wave functions and potential are kept fixed and the
quasi-Fermi are calculated self-consistently from the current continuity
equation.
- In the second part, the quasi-Fermi levels are kept constant, and the
density and the potential are calculated self-consistently from the
Schrödinger and Poisson equations.
### Classical and quantum mechanical electron densities at equilibrium, i.e.
applied bias = 0 V
Now let us first have a look at the electron densities for the cases of
a) classical and
b) quantum mechanical calculations.
The electron density is the sum over all three valleys (Gamma point, L point
and X point (or Delta for Si) in the Brillouin zone) whereas for Si the dominant
valley is the X valley which is sixfold degenerate (or twelvefold degenerate
including spin degeneracy). Thus we solve Schrödinger's equation only in the X
valley and take for the other valleys the classical density only.
For b) we have to choose appropriate boundary conditions where we choose
"mixed". *(Note: The feature of using "mixed" boundary conditions
for the Schrödinger equation is not supported any more.)* For comparison we also show the results of Dirichlet and Neumann
boundary conditions although we do not recommend the latter for current
calculations.
Dirichlet boundary conditions lead to a problem, namely the density at the
contacts is then zero (`-> ` no current).
` $quantum-model-electrons`
...
boundary-condition-100 = mixed
! mixed = Dirichlet + Neumann (i.e. the Schrödinger equation has to
be solved twice)* (not supported any more)*
boundary-condition-100 = Dirichlet !
Dirichlet
boundary-condition-100 = Neumann
! Neumann
To switch off the current calculation, we have to choose:
` $current-cluster`
...
deactivate-cluster = yes ! flag
to switch off current clusters
In the following figure we compare for 0 V applied bias, i.e. for equilibrium
where no current flows, the classical and the quantum mechanical electron
densities.
*Classical and quantum mechanical electron
densities for the n-i-n resistor. Dirichlet boundary conditions force the wave
function to be zero at the boundaries and thus the electron density is zero as
well. Thus to get
a meaningful and physical electron density at the boundaries we have to choose
"mixed" boundary conditions.**
(Note: The feature of using "mixed" boundary conditions for the Schrödinger
equation is not supported any more.)*
### Current-voltage characteristics of the Si n-i-n resistor
Now we vary the applied bias in steps of 0.05 V (```
->
$voltage-sweep
``` )
and solve the drift-diffusion equations without taking into account quantum
mechanical densities, i.e. for a classical simulation.
In addition to the above mentioned```
mobility-model-simba-2
``` which depends on doping and on the electric
field, we compare our results to the ones obtained with a constant mobility
model (µ = 1417 cm²/Vs).
```
$mobility-model-constant
```
material-name = Si
n-mu-lattice-temp = 1417.0 ! [cm²/Vs]
*IV characteristics* *of the 300 nm Si n-i-n resistor for
two different mobility models (classical simulations).*
The conduction band edges and the Fermi levels (i.e. chemical potentials) for
the electrons at different applied voltages are plotted in this figure:
### Quantum mechanical calculations
`-> Si_nin_resistor_quantum_1D_nn3.in`
As one may expect, true quantum mechanical effects play little role in this
case and both the next**nano**³ (i.e. the semi-classical drift-diffusion) and
the Pauli master equation approach yield practially identical results for the
density and conduction band edge energies (i.e. for the electrostatic
potential). We would like to point out that this good agreement is a nontrivial
finding, as we calculate the density quantum mechanically with self-consistently
computed local quasi-Fermi levels rather than semi-classically.
The following figure shows for an applied bias of 0.25 V the conduction band edge energies and the electron
densities. One can see that our results agree very well with the solution of the
Pauli master equation (M.V. Fischetti, J. Appl. Phys. **83**, 270 (1998)).
Fischetti obtains for the current density 6.8 * 10^{4} A/cm² whereas we
obtain 3.65 * 10^{4} A/cm² by using a (semi-)classical drift-diffusion model.
However, we note that the current is directly proportional to the mobility in
our model, i.e. changing the mobility therefore changes the value of the current
but does not affect the electron density or potential profile (*is this really
true?*). If we had chosen a constant mobility of µ = 1417 cm²/Vs then the current
at 0.25 V applied bias had been 7.67 * 10^{4} A/cm² (compare with I-V
characteristics above).
*The figure shows the calculated conduction band edges E*_{c}
and the electron densities n of the n-i-n structure as a function of position
inside the structure. The results obtained from
the Pauli master equation (Fischetti (1998), dashed lines) are compared to our quantum mechanical results (full
lines).
First one could think that the good agreement between the two approaches is
trivial, as quantum mechanical effects do not play any role in this example.
However, we want to stress that the electron density was calculated fully
quantum mechanically with self-consistent local quasi-Fermi levels.
**Conclusion**
Here, we demonstrated our approach to calculate the electronic structure in
nonequilibrium where we combine the stationary solutions of the Schrödinger
equation with a semi-classical drift-diffusion model. For the electrostatic
potential and the charge carrier density, the method leads to a very good
agreement with the more rigorous Pauli master equation approach. In addition,
the current can also be described accurately. |