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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 contact stefan.birner@nextnano.de.
-> 1Dbulk_n_Si_current.in
 


Resistance of a bulk n-type doped silicon sample

-> 1Dbulk_n_Si_current.in

Experiment

- Si sample of 1 cm x 1 cm x 1 m
- 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 P 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 0.1V from 0 V to 1 V (i.e. 10 voltage sweeps: $voltage-sweep).

 

Electron mobility

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      = 0.73d0   ! []
   n-N-ref-doping      = 1.072d17 ! [1/cm3]
   n-mu-min            = 55.2d0   ! [cm2/Vs]
   n-mu-doping         = 1374.0d0 ! [cm2/Vs]
 

This leads to an electron mobilty of (for details, see $mobility-model-simba)

      e = 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/mobility1D_ind010.dat (ind010 corresponds to an applied voltage of 1 V)
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.

 

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 = * F = * 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_velocity1D_ind010.dat (ind010 corresponds to an applied voltage of 1 V)
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 = * 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 m

 

Resistance / Conductivity

The calculated current density j (in units of [A/m2] for a 1D simulation) can be found in this file: current/IV_characteristics1D.dat

The calculated value for an applied voltage of 1 V is j = 19507 A/m2.
Taking into account the dimensions of the Si sample, this corresponds to a total current I of
       I = 19507 A/m2 * 1 cm * 1 m = 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 = e n e = 19507 A/m2 / 1 V/cm = 195 Ohm-1m-1 and can be found in this file: current/conductivity1D.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 m.)

 

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