nextnano3 - Tutorial
next generation 3D nano device simulator
1D Tutorial
Resistance of a bulk n-type doped silicon sample
Author:
Stefan Birner
If you want to obtain the input file that is used within this tutorial, please
submit a support ticket.
-> bulk_n_Si_current_1D_simba_nn3.in
- input file for the nextnano3
software
-> bulk_n_Si_current_1D_nn3.in /
*_nnp.in -
Resistance of a bulk n-type doped silicon sample
-> bulk_n_Si_current_1D_simba_nn3.in
Experiment
- Si sample of 1 cm x 1 cm x 1 µm
- ohmic resitance R = 5 kOhm at 300 K
- applied voltage 1 V (along d = 1 cm)
- n-type phosphorous doping with a concentration of 1 * 1016
cm-3
Simulation
We consider a one-dimensional n-type doped Si sample of length d = 1 cm at room
temperature (300 K).
The Si sample is n-type doped with phosphorous (P) donors with a doping
concentration of ND = 1 x 1016 cm-3.
At both ends of the device there are ohmic contacts.
We vary the applied voltage in steps of 0.1 V from 0 V to 1 V (i.e. 10 voltage
sweeps:
$voltage-sweep).
Electron mobility
a) bulk_n_Si_current_1D_simba_nn3.in - SIMBA mobility model
b) bulk_n_Si_current_1D_nn3.in - constant mobility model
a) For the mobility which depends on the concentration of ionized impurities we
assume
mobility-model-simba-0 and use 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/cm3]
n-mu-min
= 55.2 ! [cm2/Vs]
n-mu-doping =
1374.0 ! [cm2/Vs]
This leads to an electron mobilty of (for details, see
$mobility-model-simba)
µe = 55.2 cm2/Vs +
1374 cm2/Vs / [ 1 + ( 1*1016 cm-3
/ 1.072*1017 cm-3)0.73 ] =
1222.58 cm2/Vs
The mobility output can be found in this file:
current/mobility_V010.dat (V010 corresponds to an
applied voltage of 1 V in our example.)
The second column contains the electron mobility, the third column the hole
mobility (for each grid point) in units of [cm2/Vs].
Comparison:
- InSb has mobilities of 4 * 105 cm2/Vs.
- Two-dimensional electron gases (2DEGs) in AlGaAs/GaAs
heterostructures have mobilities of the order ~107 cm2/Vs.
b) bulk_n_Si_current_1D_nn3.in - constant mobility model
In this input file, a constant mobility is used.
µe =
1417 cm2/Vs
Mean drift velocity
The mean drift velocity of the electrons at an applied electric field F of F
= U / d = 1 V / 1 cm = 1 V/cm is given as follows:
vd,e = µ * F = µ * U / d = 1222.58 cm2/Vs *
1 V / 1cm = 1222.58 cm/s = 12.23 m/s
The drift velocity output can be found in this file:
current/drift_velocity_V010.dat (V010 corresponds to
an applied voltage of 1 V in our example.)
The second column contains the electron drift velocity, the third column the
hole drift velocity (for each grid point) in units of [cm/s].
Comparison:
- InSb has mean drift velocities of 4 * 105 cm/s = 4
km/s (at a field of 1 V/cm).
- Two-dimensional electron gases (2DEGs) in AlGaAs/GaAs
heterostructures have mean drift velocities of the order ~100 km/s (at a field
of 1 V/cm).
Scattering time
The effective scattering time of the electrons teff,e can be calculated as follows:
teff,e = µ * me,cond / e = 1222.58 cm2/Vs *
0.258 m0 / e = 1.79 * 10-13 s = 0.18
ps
where the conductivity electron mass is given by me,cond = 3 / (1/0.916 + 2/0.19) m0 =
0.258 m0.
Comparison:
- InSb (me = 0.0135 m0) has an effective
scattering time of 3.1 ps.
- Two-dimensional electron gases (2DEGs) in AlGaAs/GaAs
heterostructures (me = 0.2 m0) have an effective
scattering time of the order 1.1 ns.
Mean free path
The mean free path is the distance traveled between two collisions.
Assuming that the mean free path is given by lmfp = vd,e
* teff,e we obtain:
- Si:
lmfp = vd,e * teff,e = 0.0022 nm
- InSb:
lmfp = 12.4 nm
- AlGaAs/GaAs 2DEG: lmfp = 110 µm
Resistance / Conductivity
The calculated current density j (in units of [A/m2] for a 1D
simulation) can be found in this file:
current/IV_characteristics.dat
The calculated value for an applied voltage of 1 V is j = 19507 A/m2
= 1.9507 A/cm2.
Taking into account the dimensions of the Si sample, this corresponds to a total
current I of
I = 19507 A/m2 * 1 cm * 1
µm = 1.9507 * 10-4 A = 0.2 mA
The ohmic resistance is thus given by R = U / I = 1 V / 1.9507 * 10-4
A = 5105.2 Ohm = 5.1 kOhm
The conductivity sigma is given by sigma = j / F = µe n e = 19507 A/m2
/ 1 V/cm = 195 Ohm-1m-1
and can be found in this file:
current/conductivity_V010.dat
The conductivity is related to the resistance as follows: sigma = j / F = (I
/ A) / (U / d) = 1 / ( w R )
where w is the thickness of the sample. (Here, w = 1 µm.)
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