[ Home ] [ News ] [ Contact ] [ Search ] [ nextnano.com ] 2D n / nin Si

 Download Software
 Copyright notice
 Publications

 GUI: nextnanomat
 Tool: nextnano
 Tool: nextnano++
 Tool: nextnano.QCL
 * password protected

 

 
Up
 

nextnano3 - Tutorial

next generation 3D nano device simulator

2D Tutorial

I-V characteristics of an n-doped Si structure

1) n-doped Si

2) n-i-n-doped Si (n-doped, intrinsic, n-doped)

Note: This tutorial's copyright is owned by Stefan Birner, www.nextnano.com.

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


I-V characteristics of an n-doped Si structure

1) n-doped 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:

    region-number = 1:  0 -  1 nm (blue)  : left contact
    region-number = 2:  1 - 19 nm (red)   :
    n-doped Si
    region-number = 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 half-way between the physical grid lines. For more information on the definition of the grid confer this page: Grids and Geometry
  • The Si is n-type doped with a donor concentration of ND = 1*1020 cm-3. The energy level is 0.044 eV below the conduction band edge.
    This leads to an electron density of n = 13.48 * 1018 cm-3. This is the concentration of the ionized donors.

    The Fermi level is taken to be at 0 eV in an equilibrium nextnano simulation (EF = 0).

    The distance of the conduction band from the Fermi level can be calculated in the following way:

    electron mass:
      conduction-band-masses = 0.156d0  0.156d0  0.156d0 ! GAMMA: m,m,m
                               1.420d0  0.130d0  0.130d0 ! L: ml,mt,mt
                               0.916d0  0.190d0  0.190d0 ! X: ml,mt,mt
     
    me = me*DOS = (mlmtmt)1/3 = (0.9160.192)1/3 m0 = 0.321 m0

      conduction-band-degeneracies = 2 8 12 ! valley degeneracy including spin degeneracy
        Degeneracy of "X" valley ("X" = Delta in Si): 6
        Spin degeneracy: 2

    Nc = 12 (2 pi me kBT / h )3/2 = 12 (0.321 * 0.026 * 2.0886*1014)3/2 = 12 * 2.2816*1018 cm-3 = 2.737916*1019 cm-3 (identical to nextnano result)
    (Gamma and L valley: NcGamma = 1.54615994*1018 cm-3   NcL = 1.55494502*1019 cm-3)
    Holes: Nvhh =  9.87481457*1018 cm-3    Nvlh = 1.50177430*1018 cm-3   Nvso = 2.84047719*1018 cm-3
    Note that heavy and light holes are degenerate for k = 0. => Nv = Nvhh + Nvlh = 1.137658887*1019 cm-3

    The Semiconductor equation: np = ni2 = Nc Nv exp( - Egap/kBT) = Nc 1.1377*1019 cm-3 exp( - 1.095/0.026) = 1.23792*1020 cm-6

    Egap = 1.095 eV

    ni = 1.11262*1010 cm-3

    p = ni/n = 9.1845 cm-3
     

    Maxwell-Boltzmann Fermi-Dirac
    n (T) = Nc(T) exp( (EF - Ec) / (kBT) ) n (T) = Nc(T) F1/2( (EF - Ec) / (kBT) )
    p (T) = Nv(T) exp( (Ev - EF) / (kBT) ) p (T) = Nv(T) F1/2( (Ev - EF) / (kBT) )
                         F1/2 = Fermi-Dirac integral of order 1/2 multiplied by 2/SQRT(pi) (i.e. F1/2 includes the Gamma prefactor)

    When using the Maxwell-Boltzmann statistics as an approximation we obtain for Ec:
    Ec = kBT ln (Nc / n) = 0.026 eV * ln ( 2.737916*1019 cm-3 / (13.47836*1018 cm-3)) = 0.026 eV * ln 2.031 = 0.0183 eV
    => Note: Inside the code we make use of the Fermi-Dirac integrals (Fermi-Dirac statistics). nextnano result: 0.01385 eV

    The distance of the valence band from the Fermi level can be calculated in the following way:
    Ev = - kBT ln (Nv / p) = - 0.026 * 42.538 = - 1.099     (Maxwell-Boltzmann statistics approximation)
    => Note: Inside the code we make use of the Fermi-Dirac integrals (Fermi-Dirac statistics). nextnano result: -1.08148 eV

  • The mobility model that is applied is called mobility-model-simba-2. It is described here: $mobility-model-simba
    The calculated electron mobility is:
    - at 0.00 V:  64.5 cm/Vs
    - at 0.20 V:  52.9 cm/Vs

    In our example the mobility does neither depend on the temperature (T = T0 = 300 K) nor on the perpendicular electric field (E_|_ = 0). If E_|_ /= 0 we would have to use mobility-model-simba-2e 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: = min                 + dop                      / (1 + ((ND + NA)       / Nref                        ) a_dop) =
                      = 55.2 cm/Vs + 1374.0 cm/Vs /  (1 + (1*1020 cm-3)  / 1.072*1017 cm-3) 0.73   ) = 64.47 cm/Vs

      $mobility-model-simba
       material-name  = Si       !
    taken from SIMBA manual

       n-alpha-doping = 0.73d0   ! [---]      
    a_dop
       n-N-ref-doping = 1.072d17 ! [1/cm^3]   
    Nref
       n-mu-min       = 55.2d0   ! [cm^2/Vs]  
    min
       n-mu-doping    = 1374.0d0 ! [cm^2/Vs]  
    dop

      $end_mobility-model-simba
  • 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).
      $voltage-sweep
       sweep-number            = 1
       sweep-active            = yes       ! 'yes'/'no'
       poisson-cluster-number  = 2         !
    right contact
       step-size               = 0.02d0    ! < 0.1 V
       number-of-steps         = 10
       data-out-every-nth-step = 1
      $end_voltage-sweep

 

Results

  • The current-voltage (I-V) characteristic can be found in the following file: IV_characteristics2D.dat (2D)

    The nextnano results match perfectly to the I-V characteristics obtained with a commercial software package.
    The units for the current in a 2D simulation are [A/m].
    Dividing this two-dimensional current value by the width of the device (in our case 5 nm) we obtain the current in units of [A/m] 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 three-dimensional current are [A]. Dividing by the area of the device of 25 nm, we obtain the 1D units of [A/m].
    The 1D and 3D input files are: 1DSi_n_doped.in, 3DSi_n_doped.in
     
    voltage current [A/m]
    (nextnano 2D)
    current [A/m]
    (nextnano 2D results
    divided by the width 5 nm)
    current [A/m]
    (nextnano 1D results)
    current [A]
    (nextnano 3D results)
    current [A/m]
    (nextnano 3D results
    divided by the square 5x5 nm)
    0 0 0 0 0 0
    0.04 153.3 3.066 * 1010 3.064 * 1010 0.766 * 10-6 3.064 * 1010
    0.08 298.1 5.962 * 1010 5.961 * 1010 1.490 * 10-6 5.961 * 1010
    0.12 428.1 8.562 * 1010 8.566 * 1010 2.141 * 10-6 8.566 * 1010
    0.16 540.5 1.081 * 1011 1.081 * 1011 2.704 * 10-6 1.081 * 1011
    0.20 634.7 1.269 * 1011 1.270 * 1011 3.175 * 10-6 1.270 * 1011

 

2) n-i-n-doped Si (n-doped, intrinsic, n-doped)

  • This is a similar structure as in 1).



    from left to right:
    region-number = 1:  0 -  1 nm (dark blue)  : left contact
    region-number = 2:  1 -  3 nm (bright blue):
    n-doped Si
    region-number = 3:  3 - 17 nm (green)      :
    intrinsic Si
    region-number = 4: 17 - 19 nm (yellow)     :
    n-doped Si
    region-number = 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 half-way 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 n-doped regions. (For more details on how to set accurate grid lines for a doping profile, confer $doping-function).
    Most of the Si region is now undoped (intrinsic) (x = 3 - 17 nm).
    Only the Si region next to the two contacts is n-type 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:
    - mobility-model-simba-0 (no dependence on electric field)
    - mobility-model-simba-2
    (mobility depends on parallel electric field)
    More information on the mobility models can be found here: $mobility-model-simba

 

Results

  • The current-voltage (I-V) characteristic can be found in the following file: IV_characteristics2D.dat (2D)



    The nextnano results match perfectly to the I-V characteristics obtained with a commercial software package.
    mobility-model-simba-0 does not include a dependence of the mobility on the parallel electric field, thus the current is proportional to the applied voltage.
    mobility-model-simba-2 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 two-dimensional current value by the width of the device (in our case 5 nm) we obtain the current in units of [A/m] 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 three-dimensional current are [A]. Dividing by the area of the device of 25 nm, we obtain the 1D units of [A/m].
    The 1D and 3D input files are: 1DSi_nin_doped.in, 3DSi_nin_doped.in
    Again, the nextnano 1D and 3D results are in agreement with the nextnano 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.