 
nextnano^{3}  Tutorial
next generation 3D nano device simulator
2D Tutorial
IV characteristics of an ndoped Si structure
1) ndoped Si
2) nindoped Si (ndoped, intrinsic, ndoped)
Author:
Stefan Birner
If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
1) > Si_n_doped_1D_nn3.in
> Si_n_doped_2D_nn3.in
> Si_n_doped_3D_nn3.in
2) > Si_nin_doped_1D_nn3.in
> Si_nin_doped_2D_nn3.in
> Si_nin_doped_3D_nn3.in
IV characteristics of an ndoped Si structure
1) ndoped Si
 The structure we are dealing with consists of bulk Si that is sandwiched
between two contacts.
The whole structure has the following dimensions:
 along x axis: 20 nm (1 nm contact, 18 nm Si, 1 nm contact)
 along y axis: 5 nm
from left to right:
regionnumber = 1: 0  1 nm (blue)
: left contact
regionnumber = 2: 1  19 nm (red)
: ndoped Si
regionnumber = 2: 19  21 nm (blue)
: right contact
Note: The grid lines that are shown in the figure are the
material grid lines. The grid lines that one specifies in the input file
are the physical grid lines. The material grid lines are placed
halfway between the physical grid lines. For more information on the
definition of the grid confer this page:
Grids and Geometry
 The Si is ntype doped with a donor concentration of N_{D }= 1*10^{20} cm^{3}.
The energy level is 0.044 eV below the conduction band edge.
This leads to an electron density of n = 13.48 * 10^{18} cm^{3}.
This is the concentration of the ionized donors.
The Fermi level is taken to be at 0 eV in an equilibrium nextnano³ simulation
(E_{F} = 0).
The distance of the conduction band from the Fermi level can be calculated
in the following way:
electron mass:
conductionbandmasses = 0.156d0
0.156d0 0.156d0 ! GAMMA: m,m,m
1.420d0 0.130d0 0.130d0 ! L: m_{l},m_{t},m_{t}
0.916d0 0.190d0 0.190d0 ! X: m_{l},m_{t},m_{t}
m_{e} = m_{e}*_{DOS} = (m_{l}·m_{t}·m_{t})^{1/3}
= (0.916·0.19^{2})^{1/3} m_{0 }= 0.321 m_{0}
conductionbanddegeneracies = 2 8 12
! valley degeneracy including spin degeneracy
Degeneracy of "X" valley ("X" = Delta in Si): 6
Spin degeneracy: 2
N_{c} = 12 (2 pi m_{e}
k_{B}T / h² )^{3/2} = 12 (0.321 * 0.026 * 2.0886*10^{14})^{3/2}
= 12 * 2.2816*10^{18} cm^{3} = 2.737916*10^{19} cm^{3}
(identical to nextnano³ result)
(Gamma and L valley: N_{c}^{Gamma} = 1.54615994*10^{18}
cm^{3} N_{c}^{L} = 1.55494502*10^{19}
cm^{3})
Holes: N_{v}^{hh} = 9.87481457*10^{18} cm^{3}
N_{v}^{lh} = 1.50177430*10^{18} cm^{3}
N_{v}^{so} = 2.84047719*10^{18} cm^{3}
Note that heavy and light holes are degenerate for k = 0.
=> N_{v} = N_{v}^{hh}
+ N_{v}^{lh} = 1.137658887*10^{19} cm^{3}
The Semiconductor equation: np = n_{i}^{2} = N_{c} N_{v} exp(
 E_{gap}/k_{B}T) = N_{c}
1.1377*10^{19} cm^{3} exp(  1.095/0.026) = 1.23792*10^{20}
cm^{6
}E_{gap} = 1.095 eV^{
}n_{i} = 1.11262*10^{10} cm^{3}
p = n_{i}²/n = 9.1845 cm^{3}
MaxwellBoltzmann 
FermiDirac 
n (T) = N_{c}(T) exp(
(E_{F}  E_{c}) / (k_{B}T) ) 
n (T) = N_{c}(T) F_{1/2}(
(E_{F}  E_{c}) / (k_{B}T) ) 
p (T) = N_{v}(T) exp(
(E_{v}  E_{F}) / (k_{B}T) ) 
p (T) = N_{v}(T) F_{1/2}(
(E_{v}  E_{F}) / (k_{B}T) ) 

F_{1/2} = FermiDirac integral of order 1/2 multiplied by
2/SQRT(pi) (i.e. F_{1/2} includes
the Gamma prefactor) 
When using the MaxwellBoltzmann statistics as an approximation we obtain
for E_{c}:
E_{c} = k_{B}T ln (N_{c }/ n) = 0.026 eV * ln (
2.737916*10^{19} cm^{3} / (13.47836*10^{18} cm^{3}))
= 0.026 eV * ln 2.031 = 0.0183 eV
=> Note: Inside the code we make use of the FermiDirac integrals (FermiDirac
statistics). nextnano³ result: 0.01385
eV
The distance of the valence band from the Fermi level can be calculated in
the following way:
E_{v} =  k_{B}T ln (N_{v} / p) =  0.026 * 42.538 =
 1.099
(MaxwellBoltzmann statistics approximation)
=> Note: Inside the code we make use of the FermiDirac integrals (FermiDirac
statistics). nextnano³ result: 1.08148
eV
 The mobility model that is applied is called
mobilitymodelsimba2 . It is described here:
$mobilitymodelsimba
The calculated electron mobility is:
 at 0.00 V: 64.5 cm²/Vs
 at 0.20 V: 52.9 cm²/Vs
In our example the mobility does neither depend on the temperature (T = T_{0
}= 300 K) nor on the perpendicular electric field (E_{__}
= 0). If E_{__} /= 0 we would have to use
mobilitymodelsimba2e instead, in order to take
into account the dependence on the perpendicular electric field.
Therefore the equation for the electron mobility reduces to:
at 0.00 V:
µ = µ_{min
}+ µ_{dop
}/ (1 + ((N_{D} + N_{A})
/ N_{ref
})^{ a_dop}) =
= 55.2 cm²/Vs + 1374.0 cm²/Vs / (1 + (1*10^{20} cm^{3})
/ 1.072*10^{17} cm^{3}) ^{0.73 }) = 64.47
cm²/Vs
$mobilitymodelsimba
materialname = Si
! taken from
SIMBA
manual
nalphadoping = 0.73d0 !
[] a_dop
nNrefdoping = 1.072d17 ! [1/cm^3]
N_{ref}
nmumin =
55.2d0 ! [cm^2/Vs]
µ_{min}
nmudoping = 1374.0d0
! [cm^2/Vs] µ_{dop
}
$end_mobilitymodelsimba
 We sweep the voltage at the right contact and calculate the current
density for 0.00 V, 0.02 V, 0.04 V, ..., 0.20 V (
10
steps).
$voltagesweep
sweepnumber
= 1
sweepactive
= yes ! 'yes'/'no'
poissonclusternumber = 2
! right contact
stepsize
= 0.02d0 ! < 0.1 V
numberofsteps =
10
dataouteverynthstep = 1
$end_voltagesweep
Results
 The currentvoltage (IV) characteristic can be found in the following
file:
IV_characteristics2D.dat (2D)
The nextnano³ results match perfectly to the IV characteristics obtained with a
commercial software package.
The units for the current in a 2D simulation are [A/m].
Dividing this twodimensional current value by the width of the device (in our
case 5 nm) we obtain the current in units of [A/m²] which is the usual unit of
a 1D simulation.
As our simple 2D example structure is basically equivalent to a 1D structure
we can easily compare our 2D results with the 1D results to check for
consistency. It is also possible to perform a 3D simulation. In this case, the
units for the threedimensional current are [A]. Dividing by the area of the
device of 25 nm², we obtain the 1D units of [A/m²].
The 1D and 3D input files are:
1DSi_n_doped.in , 3DSi_n_doped.in
voltage 
current [A/m]
(nextnano³ 2D) 
current [A/m²]
(nextnano³ 2D results
divided by the width 5 nm) 
current [A/m²]
(nextnano³ 1D results) 
current [A]
(nextnano³ 3D results) 
current [A/m²]
(nextnano³ 3D results
divided by the square 5x5 nm²) 
0 
0 
0 
0 
0 
0 
0.04 
153.3 
3.066 * 10 ^{10} 
3.064
* 10 ^{10} 
0.766
* 10^{6} 
3.06 4
* 10 ^{10} 
0.08 
298.1 
5.962 * 10 ^{10} 
5.961
* 10 ^{10} 
1.490
* 10^{6} 
5.961
* 10 ^{10} 
0.12 
428.1 
8.562 * 10 ^{10} 
8.566
* 10 ^{10} 
2.141
* 10^{6} 
8.566
* 10 ^{10} 
0.16 
540.5 
1.081 * 10 ^{11} 
1.081 * 10 ^{11} 
2.704
* 10^{6} 
1.081
* 10 ^{11} 
0.20 
634.7 
1.269
* 10 ^{11} 
1.270
* 10 ^{11} 
3.175
* 10^{6} 
1.270
* 10 ^{11} 
2) nindoped Si (ndoped, intrinsic, ndoped)
 This is a similar structure as in 1).
from left to right:
regionnumber = 1: 0  1 nm
(dark blue) : left contact
regionnumber = 2: 1  3 nm (bright
blue): ndoped Si
regionnumber = 3: 3  17 nm (green)
: intrinsic Si
regionnumber = 4: 17  19 nm (yellow)
: ndoped Si
regionnumber = 5: 19  21 nm (red)
: right contact
Note: The grid lines that are shown in the figure are the
material grid lines. The grid lines that one specifies in the input file
are the physical grid lines. The material grid lines are placed
halfway between the physical grid lines. For more information on the
definition of the grid confer this page:
Grids and Geometry
To take into account the doping profile properly we define separate region
clusters for the ndoped regions. (For more details on how
to set accurate grid lines for a doping profile, confer
$dopingfunction ). Most of the Si region is now undoped (intrinsic) (x = 3  17 nm). Only the Si region next to the two contacts is ntype doped (x = 1  3 nm, x =
17  19 nm). The doping concentration is the same as in 1).
 We will compare two different mobility models:
 mobilitymodelsimba0
(no dependence on electric field)
 mobilitymodelsimba2
(mobility depends on parallel electric field)
More information on the mobility models can be found here:
$mobilitymodelsimba
Results
 The currentvoltage (IV) characteristic can be found in the following
file:
IV_characteristics2D.dat (2D)
The nextnano³ results match perfectly to the IV characteristics obtained with a
commercial software package.
mobilitymodelsimba0
does not include a dependence of the mobility on the parallel electric
field, thus the current is proportional to the applied voltage.
mobilitymodelsimba2
takes into account a dependence of the mobility on the parallel
electric field. In this case the current is smaller because the mobility
decreases when the applied voltage increases.
The units for the current in a 2D simulation are [A/m].
Dividing this twodimensional current value by the width of the device (in our
case 5 nm) we obtain the current in units of [A/m²] which is the usual unit of
a 1D simulation.
As our simple 2D example structure is basically equivalent to a 1D structure
we can easily compare our 2D results with the 1D results to check for
consistency. It is also possible to perform a 3D simulation. In this case, the
units for the threedimensional current are [A]. Dividing by the area of the
device of 25 nm², we obtain the 1D units of [A/m²].
The 1D and 3D input files are:
1DSi_nin_doped.in , 3DSi_nin_doped.in
Again, the nextnano³ 1D and 3D results are in agreement with the nextnano³
2D results.
 The following figure shows the conduction band profile (
band_structure/cb1D_003_ind*.dat )
for different voltages.
 Please help us to improve our tutorial. Send comments to
support
[at] nextnano.com .
