nextnano³ - Tutorial

next generation 3D nano device simulator

1D Tutorial

Electron concentration in doped semiconductors (Si, Ge, GaAs) - Compensated semiconductors

Author: Stefan Birner

If you want to obtain the input files that are used within this tutorial, please check if you can find them in the installation directory.
If you cannot find them, please submit a Support Ticket.
1) -> 1Dn_doped_bulk_Si.in
2) -> 1Dn_doped_bulk_GaAs.in
3) -> 1Dn_doped_bulk_Ge_a.in
3) -> 1Dn_doped_bulk_Ge_b.in
4) -> 1Dn_doped_bulk_Ge_a_compensated.in
5) -> 1Dn_doped_bulk_Ge_a_compensated_two_donors.in
 

1) Effective density of states N_c(T) of the conduction band in Si and GaAs

The effective density of states N_c(T) of a conduction band is defined as
 N_c(T) = 2 (2 pi m_e k_BT / h² )^3/2

            2: spin degeneracy factor

For an effective mass of m_e = 1 at a temperature of 300 K, the value of N_c = 25.09*10¹⁸ cm^-3.

The same equation holds for the effective density of states N_v(T) of a valence band where the valence band mass m_h has to be used.

Note that some of the conduction bands are degenerate (valley degeneracy), e.g.
- the X       conduction band minimum in a zinc blende semiconductor is three-fold degenerate, so N'_c(T) = 3 N_c(T). (Note: Each valley counts only 1/2 because they are at the boundary of the Brillouin zone.)
- the Delta conduction band minimum in silicon                                  is    six-fold degenerate, so N'_c(T) = 6 N_c(T). (Note: Each valley counts because they are not at the boundary of the Brillouin zone.)
- the L       conduction band minimum in germanium                           is   four-fold degenerate, so N'_c(T) = 4 N_c(T). (Note: Each valley counts only 1/2 because they are at the boundary of the Brillouin zone.)

1a) Si

In silicon, the electron mass at the X valley is an ellipsoid with a longitudinal and two transverse masses:
  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
The density of states mass is calculated as follows:
  m_e^X = m_e*_DOS = (m_l·m_t·m_t)^1/3 = (0.916 · 0.190²)^1/3 m₀ = 0.321 m₀
 

The value of k_BT at different temperatures reads:

• T =  10 K:       k_BT =  0.86173 meV
• T = 100 K:      k_BT =  8.6173   meV
• T = 300 K:      k_BT = 25.852    meV

Thus for Si we obtain the following values for the effective density of states N_c(T) at the sixfold X (i.e. Delta) conduction band minimum:

• T = 10K:       N_c = 6 * 2 (2 pi m_e k_BT / h² )^3/2 = 2 (0.321  * 0.8617*10^-3 * 2.0886*10¹⁴)^3/2 =   0.166626 * 10¹⁸ cm^-3
• T = 100K:      N_c = 6 * 2 (2 pi m_e k_BT / h² )^3/2 = 2 (0.321 * 8.617*10^-3   * 2.0886*10¹⁴)^3/2 =   5.26919   * 10¹⁸ cm^-3
• T = 300K:      N_c = 6 * 2 (2 pi m_e k_BT / h² )^3/2 = 2 (0.321 * 25.85*10^-3   * 2.0886*10¹⁴)^3/2 = 27.3795     * 10¹⁸ cm^-3

The output for N_c is contained in the following file: Nc_cb1D.dat
The units are 1 * 10¹⁸ cm^-3.
The first column refers to N_c of the conduction band at the Gamma point, the second at the L point, and the third at the X (or Delta) point.

For the valence bands, the effective density of states of the heavy (hh), light (lh) and split-off (so) hole bands are contained in the file:
Nv_vb1D.dat: N_v(hh), N_v(lh), N_v(so)

To vary the temperature, the following specifier has to be varied:

 $global-parameters
 !lattice-temperature =  10.0d0  !  10 Kelvin
 !lattice-temperature = 100.0d0  ! 100 Kelvin
  lattice-temperature = 300.0d0  ! 300 Kelvin
 $end_global-parameters

1b) GaAs

For GaAs where the electron mass at the Gamma point is m_e = 0.067 m₀ we obtain:

• T = 10K:       N_c = 2 (2 pi m_e k_BT / h² )^3/2 = 2 (0.067  * 0.8617*10^-3 * 2.0886*10¹⁴)^3/2 = 0.00264852 * 10¹⁸ cm^-3
• T = 100K:      N_c =2 (2 pi m_e k_BT / h² )^3/2 = 2 (0.067  * 8.617*10^-3    * 2.0886*10¹⁴)^3/2 = 0.0837535   * 10¹⁸ cm^-3
• T = 300K:      N_c = 2 (2 pi m_e k_BT / h² )^3/2 = 2 (0.067 * 25.85*10^-3   * 2.0886*10¹⁴)^3/2 = 0.435196     * 10¹⁸ cm^-3

2) Electron concentration in doped semiconductors and position of Fermi level

The electron concentration is equivalent to the concentration of the ionized donors.

The Fermi level is assumed to be constant and equal to 0 eV in an equilibrium nextnano³ simulation (E_F = 0).

2a) n-doped Si

The Si layer is n-type doped with a donor concentration of N_D = 1*10¹⁷ cm^-3.
The energy level of the donor is set to 100 meV below the conduction band edge.

 $doping-function
  doping-function-number = 1
  impurity-number        = 1        ! properties of this impurity type have to be specified below
  doping-concentration   = 0.1d0    ! 0.1 * 10¹⁸ cm^-3 = 1 * 10¹⁷ cm^-3
  only-region            = 0d0 20d0 ! xmin = 0 nm, xmax = 20 nm
 $end_doping-function

 $impurity-parameters
  impurity-number             = 1        ! impurity numbers labelled in doping-function
  impurity-type               = n-type   ! n-type, p-type
  number-of-energy-levels     = 1        ! number of energy levels of this impurity (only 1 is currently allowed)
  energy-levels-relative      = 0.100d0  ! 100 meV below conduction band
 !energy-levels-relative      = 0.049d0  ! n-As-in-Si, energy relative to 'nearest' band edge (n-type -> conduction band
                                                                                                                                                           p-type -> valence band)
  degeneracy-of-energy-levels = 2        ! degeneracy of energy levels, 2 for n-type, 4 for p-type
 $end_impurity-parameters

At T = 0 K, the donors are neutral: N_D⁺ = N_D⁰

For T > 0 K, the electrons that are bound to the donors can be thermally activated into the conduction band.
It holds:

                      N_D⁺ / N_D⁰ = 1/g exp[ (E_D-E_F) / k_BT] =  1/g exp[ (E_c-E_D'-E_F) / k_BT]

where E_D' is the donor level relative to the conduction band edge and g is the degeneracy factor (g = 2 for donors).
Note that the degeneracy of neutral donors is 2, whereas the degeneracy of the ionized donors is 1.

If the position of the Fermi level relative to the conduction band edge is known, one can calculate the electron density analytically using:

        N_D = N_D⁰ + N_D⁺

        n = p + N_D⁺ ~= N_D⁺     (charge neutrality)

==> n ~= N_D / ( 1 + 2 exp[ (E_F-E_D) / (k_BT) ] ) =
             = N_D / ( 1 + 2 exp[ (E_F-(E_c-E_D')) / (k_BT) ] )                                 (Eq. 1)

The calculated electron density can be found in this file: densities/density1Del.dat
This is the total electron density, i.e. the sum over all conduction band minima at the Gamma, L and X (or Delta) points.

The file densities/density1DGamma_L_X.dat
contains the electron density for each valley (Gamma, L and X).

• At T =  10 K, the electron density is 0.0000372311 * 10¹⁸ cm^-3 = 3.72311 * 10¹³ cm^-3, i.e. it is practially zero.
• At T = 100 K, the electron density is    0.00154008 * 10¹⁸ cm^-3 = 1.54008 * 10¹⁵ cm^-3.
• At T = 300 K, the electron density is      0.0784660 * 10¹⁸ cm^-3 = 7.84660 * 10¹⁶ cm^-3.

This can be approximated analytically for small temperatures, leading to a formula which is independent of the position of the Fermi level:

n ~= ( 0.5 * N_c * N_D)^1/2 exp[ (E_D-E_c) / (2k_BT) ] = ...

• T =  100 K: n ~= ( 0.5 * 5.26919*10¹⁸ cm^-3 * 1*10¹⁷ cm^-3)^1/2 exp[ (-100 meV) / (2 * 8.6173 meV) ] =
                        =  1.550 * 10¹⁵ cm^-3.

2b) n-doped GaAs

For GaAs the electron densities at the Gamma conduction band minimum are as follows:

• At T =  10 K, the electron density is 0.0000323902 * 10¹⁸ cm^-3 = 3.23902 * 10¹³ cm^-3, i.e. it is practially zero.
• At T = 100 K, the electron density is  0.000217719 * 10¹⁸ cm^-3 = 2.17719 * 10¹⁴ cm^-3.
• At T = 300 K, the electron density is     0.0190349 * 10¹⁸ cm^-3 =  1.90349 * 10¹⁶ cm^-3.

This can be approximated analytically for small temperatures, leading to a formula which is independent of the position of the Fermi level:

n ~= ( 0.5 * N_c * N_D)^1/2 exp[ (E_D-E_c) / (2k_BT) ] = ...

• T =  100 K: n ~= ( 0.5 * 0.0837535*10¹⁸ cm^-3 * 1*10¹⁷ cm^-3)^1/2 exp[ (-100 meV) / (2 * 8.6173 meV) ] =
                        =  1.955 * 10¹⁴ cm^-3.

Position of the Fermi level

For small temperatures, the following relationship holds for the Fermi level:

      E_F(T) = 1/2 ( E_D + E_c )   + 1/2 k_BT ln[ 1/2 N_D/N_c ] =
              = 1/2 ( 2 E_c - E_D' ) + 1/2 k_BT ln[ 1/2 N_D/N_c ]

where E_D' is the donor level relative to the conduction band edge.

Thus for small temperatures, the distance of the Fermi level to the conduction band edge reads:

 E_c - E_F(T) = 1/2 E_D' - 1/2 k_BT ln[ 1/2 N_D/N_c ]

3a) The conduction band edge in Si is thus given by:

• T =    10 K, E_F= 0 eV:  E_c (10 K)   = 100 / 2 meV - 0.5 * 0.86173 meV  ln[ 1/2 1*10¹⁷ cm^-3 /  0.166626*10¹⁸ cm^-3 ] =
                                                    =        50 meV + 0.52 meV = 0.05052 eV
  nextnano³ result: E_c (X,  10 K) = 0.00724398 eV
  (The result differs from the analytical formuala because the electron density is practically zero, and thus of similar magnitude than the hole density!)
• T =  100 K, E_F= 0 eV:  E_c (100 K) = 100 / 2 meV - 0.5 * 8.6173 meV    ln[ 1/2 1*10¹⁷ cm^-3 / 5.26919*10¹⁸ cm^-3 ] =
                                                    =       50 meV + 20.07 meV = 0.07007 eV
  nextnano³ result: E_c (X, 100 K) = 0.0701253 eV

3b) The conduction band edge in GaAs is thus given by:

• T =    10 K, E_F= 0 eV:  E_c (10 K)   = 100 / 2 meV - 0.5 * 0.86173 meV  ln[ 1/2 1*10¹⁷ cm^-3 /  0.00264852*10¹⁸ cm^-3 ] =
                                                    =        50 meV - 1.27 meV = 0.04873 eV
  nextnano³ result: E_c (Gamma,  10 K) = 0.00379127 eV
  (The result differs from the analytical formuala because the electron density is practically zero, and thus of similar magnitude than the hole density!)
• T =  100 K, E_F= 0 eV:  E_c (100 K) = 100 / 2 meV - 0.5 * 8.6173 meV    ln[ 1/2 1*10¹⁷ cm^-3 / 0.0837535*10¹⁸ cm^-3 ] =
                                                    =       50 meV + 2.22 meV = 0.05222 eV
  nextnano³ result: E_c (Gamma, 100 K) = 0.0512862 eV

2a) n-doped Si

Using Eq. (1), one can calculate the electron density analytically:

• At T =  10 K, the electron density is
  
  n ~= 1*10¹⁷ cm^-3 / ( 1 + 2 exp[ 49.48 meV / 0.86173 meV ] ) ~ 0
  
  nextnano³ result: 3.72311 * 10¹³ cm^-3, i.e. it is practially zero.
• At T =  100 K, the electron density is
  
  n ~= 1*10¹⁷ cm^-3 / ( 1 + 2 exp[ 29.93 meV / 8.6173 meV ] ) = 1.53 * 10¹⁵ cm^-3
  
  nextnano³ result: 1.54008 * 10¹⁵ cm^-3

2b) n-doped GaAs

• At T =  10 K, the electron density is
  
  n ~= 1*10¹⁷ cm^-3 / ( 1 + 2 exp[ 51.27 meV / 0.86173 meV ] ) ~ 0
  
  nextnano³ result: 3.23902 * 10¹³ cm^-3, i.e. it is practially zero.
• At T =  100 K, the electron density is
  
  n ~= 1*10¹⁷ cm^-3 / ( 1 + 2 exp[ 47.78 meV / 8.6173 meV ] ) = 1.95 * 10¹⁴ cm^-3
  
  nextnano³ result: 2.17719 * 10¹⁴ cm^-3

   

Numerical details

The distance of the conduction band from the Fermi level is calculated in the following way inside nextnano³:

For details, see also this tutorial: I-V characteristics of an n-doped Si structure

The Semiconductor equation: np = n_i² = N_c N_v exp( - E_gap/k_BT)

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 E_c:
E_c = k_BT ln (N_c / n)
=> Note: Inside the code we make use of the Fermi-Dirac integrals (Fermi-Dirac statistics).

The distance of the valence band from the Fermi level can be calculated in the following way:
E_v = - k_BT ln (N_v / p)         (Maxwell-Boltzmann statistics approximation)
=> Note: Inside the code we make use of the Fermi-Dirac integrals (Fermi-Dirac statistics).

3) Electron and hole concentration in doped Ge

This part of the tutorial is based on Fig. 5.5 and Fig. 5.6 in

      Holger T. Grahn, Introduction to Semiconductor Physics, World Scientific (2001)

3a) Electron and hole concentration in doped Ge as a function of temperature T

   ==> 1Dn_doped_bulk_Ge_a.in

   flow-scheme = 13: vary temperature as T

The Ge sample is n-type doped with a concentration of N_D = 1 * 10¹⁷ cm^-3. The ionization energy of the donor E_D is set to 10 meV.
(Note that the melting point of germanium is at 1211.40 K.)

The following figure shows the electron density as a function of temperature for a Ge sample.

In the saturation regime, the charge carrier density is equal to the doping density (between 150 K and 600 K), i.e. n =~ 1 * 10¹⁷ cm^-3.

For large temperatures, the intrinsic carrier density dominates, i.e. here the electron density is equal to the hole density.

The figure also shows the band gap of the Gamma, L and Delta conduction band minima as a function of temperature. The valence band edges are also shown (heavy, light and split-off hole).
One can clearly see that the band gaps decrease with increasing temperature.
This temperature dependence can be described using the Varshni formula.

For very small temperatures, the position of the Fermi level (chemical potential) is close to the L conduction band edge, i.e. at E_F = E_gap^L - E_D/2 = E_c^L - E_D/2.
For temperatures between 600 and 900 K, the Fermi level is around the middle of the band gap.



In this example, the effective density of states at 300 K was calculated as follows:

    N_c (300 K) = 61.068 * 10¹⁸ cm^-3
    N_v (300 K) =   5.411 * 10¹⁸ cm^-3

This figure shows the same results as above, but this time the Fermi level is set to a constant value E_F = 0 eV.
Consequently, the conduction and valence band edges are plotted with respect to the constant Fermi level.
This figure shows more clearly how the band edges align with respect to the Fermi level.

The data that has been plotted in the above shown figures can be found in this file: sweep_temperature1D.dat

3b) Electron and hole concentration in doped Ge as a function of inverse temperature 1/T

   ==> 1Dn_doped_bulk_Ge_b.in

   flow-scheme = 14: vary temperature as 1000/T

The same data as above is plotted but this time as a function of inverse temperature in units of 1000/T.

The slope of the electron density approaches at high temperatures (i.e. left side of the figure) the value of E_gap/2 (if it holds N_c = N_v which is not the case in our example).
At 300 K,  E_gap/2 = 0.332 eV.
At 600 K,  E_gap/2 = 0.267 eV.
At 900 K,  E_gap/2 = 0.200 eV.
At low temperatures the slope is E_D/2 = 5 meV = 0.0005 eV.


 

This figure shows the same results as above, but this time the Fermi level is set to a constant value E_F = 0 eV.
Consequently, the conduction and valence band edges are plotted with respect to the constant Fermi level.
This figure shows more clearly how the band edges align with respect to the Fermi level.

4) Compensated semiconductors - one donor level and one acceptor level

   ==> 1Dn_doped_bulk_Ge_a_compensated.in

   flow-scheme = 13: vary temperature as T

• An interesting situation occurs in real semiconductors that have both, n-type and p-type doping. It is rather impossible to fabricate devices that are merely n- or p-type because unintentional doping through impurities is always present.
• We assume the same Ge sample as above, i.e. the Ge sample is n-type doped with a concentration of N_D = 1 * 10¹⁷ cm^-3.
  The ionization energy of the donor E_D is set to 10 meV.
  In contrast to the above shown figures, also acceptors are present with varying acceptor concentration ranging from
  - N_A = 1 * 10¹⁴ cm^-3    (N_A << N_D)  ==>  N_D - N_A =~ N_D
  - N_A = 1 * 10¹⁶ cm^-3    (N_A  <  N_D)  ==>  N_D - N_A = 9*10¹⁶ cm^-3
  - N_A = 5 * 10¹⁶ cm^-3    (N_A  <  N_D)  ==>  N_D - N_A = 5*10¹⁶ cm^-3
  - N_A = 7 * 10¹⁶ cm^-3    (N_A  <  N_D)  ==>  N_D - N_A = 3*10¹⁶ cm^-3
  - N_A = 9 * 10¹⁶ cm^-3    (N_A  <  N_D)  ==>  N_D - N_A = 1*10¹⁶ cm^-3
  - N_A = 1 * 10¹⁷ cm^-3    (N_A  =  N_D)  ==>  N_D - N_A = 0  ==>  intrinisc density is recovered
  The ionization energy of the acceptor E_A is set to 10 meV.
• The following figure shows the electron density vs. temperature for different acceptor concentrations N_A.
  Note that the donor concentration N_D is kept constant.
  
  
  
  If N_A = N_D, then the intrinsic behavior of the electron density is recovered. In any case, the intrinsic density dominates at high temperatures.
  In the saturation regime (between 100 K and 400 K), the nominally dopant density (N_D - N_A) determines the electron density.
  
  The variation of the acceptor concentration is specified here:
  
    $doping-function
   doping-function-number = 2
   impurity-number        = 2         ! p-type impurity
   doping-concentration   = 0.0001d0  ! ND >> NA ! 0.0001 * 1018 / cm3 = 1 * 1014 / cm3
 ! doping-concentration   = 0.01d0    ! ND >  NA ! 0.01   * 1018 / cm3 = 1 * 1016 / cm3
 ! doping-concentration   = 0.05d0    ! ND >  NA ! 0.05   * 1018 / cm3 = 5 * 1016 / cm3
 ! doping-concentration   = 0.07d0    ! ND >  NA ! 0.07   * 1018 / cm3 = 7 * 1016 / cm3
 ! doping-concentration   = 0.09d0    ! ND >  NA ! 0.09   * 1018 / cm3 = 9 * 1016 / cm3
 ! doping-concentration   = 0.1d0     ! ND =  NA ! 0.1    * 1018 / cm3 = 1 * 1017 / cm3
 
• The following figure shows the Fermi level as a function of temperature for various acceptor densities N_A.
  Note that the indirect band gap at the L point decreases with temperature.
  As expected, for the "intrinisic case" (N_D - N_A = 0  ==>  intrinisc density is recovered), the Fermi level at 0 Kelvin lies in the middle of the band gap at E_F = 0.74 eV / 2 = 0.37 eV.
  
  

5) Compensated semiconductors - two donor levels and one acceptor level

   ==> 1Dn_doped_bulk_Ge_a_compensated_two_donors.in

   flow-scheme = 13: vary temperature as T

We assume a Ge sample which has two donor levels and one acceptor level.

The acceptor concentration is N_A = 4 * 10¹⁶ cm^-3.
The ionization energy of the acceptor E_A is set to 10 meV.

The donor concentrations N_D,1 and N_D,2 are varied.
The ionization energy of the upper donor level E_D,1 is set to 10 meV.
The ionization energy of the lower donor level E_D,2 is set to 100 meV.
 

We distinguish between three cases:

• 5a)
      N_D,1 = N_D,2 = 5 * 10¹⁶ cm^-3  >  N_A
   => N_D,1 + N_D,2  >  N_A
   
• 5b)
      N_D,1 = 3 * 10¹⁶ cm^-3  <  N_A
      N_D,2 = 7 * 10¹⁶ cm^-3  >  N_A
      N_D,1 + N_D,2  >  N_A
   
• 5c)   (one donor level and one acceptor level)
      N_D,1 = 10 * 10¹⁶ cm^-3  >  N_A
      N_D,2 = 0
   
• The following figure shows the result for these three cases. 5c) is similar to the figures that have been discussed already.
  At the dopant density of N_D,1 - N_A = 6 * 10¹⁶ cm^-3 the saturation regime forms.
  
  For case 5b) only the ionization of the second donor level E_D,2 is relevant at low temperatures.
  
  For case 5a) two plateaus form:
  - for the ionization of the upper donor level E_D,1: N_D,1               - N_A = 1 * 10¹⁶ cm^-3
  - for the ionization of the lower donor level E_D,2: N_D,1 + N_D,2 - N_A = 6 * 10¹⁶ cm^-3
  
  
  
  For comparison, for case 5a) the energetic position of the second donor level E_D,2 has been varied between 70 meV, 100 meV and 130 meV.
  The energetic position of the upper donor level is at 10 meV.
   
• The following figure shows the Fermi level for the cases 5a), 5b) and 5c).
  The valence band edge is fixed at 0 eV. The indirect band gap at the L point decreases with increasing temperature.
  
  For case 5b), where only the ionization of the second donor level E_D,2 is relevant at low temperatures, one can clearly see the position of the Fermi level at 100 meV below the conduction band edge: E_F (T = 0 K) = E_C - E_D,2
  
  For case 5a) the Fermi level moves from the upper donor level E_D,1 at low temperatures to the lower donor level E_D,2 at around 200 K and start to ionize the second donor level, eventually forming the second "plateau" in the electron density (see figure above for the electron density).
  
  At high temperatures, the Fermi levels moves approximately into the middle of the band gap.