— EDU — Electron transport in n-type Silicon¶

Problem¶

An n-type silicon layer of thickness $d = 1\; \mathrm{\mu m}$ is grown on a $1 \times 1\, \mathrm{cm}^2$ insulating substrate. It is doped with phosphorous (P) donors with a doping concentration of $N_\mathrm{D} = 1 \cdot 10^{16} \,\mathrm{cm^{-3}}$. Two ohmic contacts are located on the opposite sides of the sample, therefore, distanced by $l = 1\,\mathrm{cm}$ from each other.

Calculate:

mean drift velocity of charge carriers in the sample,
mean free path for the charge carriers in the sample by considering the effective scattering time and the mean drift velocity,
resistance and conductivity

at room temperature when 1 V of bias is applied to the contacts. Assume electron mobility $\mu_\mathrm{e} = 1222.58~\mathrm{cm^2/Vs}$ and hole mobility $\mu_\mathrm{h} = 425.54~\mathrm{cm^2/Vs}$.

Input file¶

The input file 1D_el_transport_Si_n_dop_nnp.in contains a 1D definition of 1 cm long n-doped Si at 300 K as stated in the problem. Assumed mobilities of carriers in Si are overwritten in the group database{}.

database{
    binary_zb{
        name = "Si"
        mobility_constant{
            electrons{ mumax = 1222.58 } # (cm2/Vs)
            holes{ mumax = 425.54} # (cm2/Vs)
        }
    }
}

The complete structure is n-doped with an impurity concentration of $N_\mathrm{D}=10^{16}~\mathrm{cm^{-3}}$. Activation energy of the dopants is taken from this table. Degeneracy is chosen 2 as typical for donors.

$doping_concentration = 1e16  # (cm^3)
$width = 1e7                  # (nm)


structure{
    region{ # Doping layer
        line{ x = [ -1.0, $width + 1.0 ] }
        doping{
            constant{
                name = "Phosphorus"
                conc = $doping_concentration
            }
        }
    }
}

impurities{
    donor{
        name = "Phosphorus"
        energy = 0.045      # (eV)
        degeneracy = 2
    }
}

The structure is biased with a voltage of 1 V and 0 V applied to the left and right contact, respectively.

contacts{ # this group is required in every input file
     ohmic{
         name = contact_right
         bias = 0.0 # (V)
     }
     ohmic{
         name = contact_left
         bias = 1.0 # (V)
     }
 }

The simulation of current inside the material is done based on the Drift-Diffusion model solved self-consistently with the Poisson equation. Therefore poisson{}, currents{}, and run{ current_poisson{} } groups are present in the input file. Constant mobility model is chosen for this simulation. Among multiple interesting outputs, the ones useful for solving the problem are also added: electron velocity, mobility and currents.

$mobility_model = constant
currents{
    mobility_model = $mobility_model
    recombination_model{}

    output_mobilities{}
    output_currents{}
    output_velocities{}
}

These can be found in output files: IV_characteristics.dat, velocity_electron.dat, and mobility_electron.dat. Computed values are used later in the tutorial to determine the scattering time, mean free path and resistance of the material.

Note

Scattering time of bulk crystal, mean free path and resistance cannot be outputted by the nextnano++ software.

Solutions¶

Mean drift velocity¶

The mean drift velocity $v_\mathrm{d,e}$ of the electrons at an applied electric field $E = \frac{U}{d} = \frac{1 ~\mathrm{V}}{1~\mathrm{cm}} = 1 ~\mathrm{V/cm}$ is given as follows:

\[v_\mathrm{d,e} = \mu \cdot E = \mu \cdot \frac{U}{d} = 1222.58 ~\mathrm{cm}^2/\mathrm{Vs} \cdot \frac{1 \mathrm{V}}{1~\mathrm{cm}} = 1222.58 ~\mathrm{cm/s} = 12.23~\mathrm{m/s}\]

The drift velocities of electrons and holes at each grid point (in units of $\mathrm{cm/s}$) can be found in the files bias_XXXXX/velocity_electron and bias_XXXXX/velocity_hole, respectively. From the simulation 1D_el_transport_Si_n_dop_nnp.in one can read the drift velocity for electrons $v_\mathrm{d,e} = 1222.5797\,\mathrm{cm/s}$.

Mean free path¶

The mean free path can be calculated by the simple formula $l_{\mathrm{mfp}} = v_\mathrm{d,e} \cdot t_\mathrm{eff,e}$. We already determined the drift velocity $v_\mathrm{d,e}$. We only have to find the effective scattering time $t_\mathrm{eff,e}$. The effective scattering time of the electrons $t_\mathrm{eff,e}$ can be calculated as follows:

\[t_\mathrm{eff,e} = \mu\cdot \frac{m_\mathrm{e,cond}}{e} = 1222.58 ~\mathrm{cm^2/Vs} \cdot 0.258 \frac{m_0}{e} = 1.79 \cdot 10^{-13}~ s = 0.18~ \mathrm{ps}\]

where the conduction electron mass is given by

\[m_\mathrm{e,cond} = \frac{1}{1/0.916 + 2/0.19} m_0 = 0.258\;m_0.\]

Therefore, the mean free path for bulk Si is given by

\[l_\mathrm{mfp} = v_{\mathrm{d},e} \cdot t_{\mathrm{eff},e} = 0.0022~\mathrm{nm}.\]

Resistance and conductivity¶

The calculated current density $j$ (in units of [A/cm$^2$] for a 1D simulation) can be found in the file: bias_xxxxx/IV_characteristics.dat. For an applied voltage of 1 V the calculated value reads

\[j = 19507~\mathrm{A/m^2} = 1.9507~\mathrm{A/cm^2}.\]

Taking into account the dimensions of the Si sample ($A=1~\mathrm{cm^2}$), this corresponds to a total current $I$ of

\[I = 19507~\mathrm{A/m^2} \cdot 1~\mathrm{cm} \cdot 1 ~\mu\mathrm{m} = 1.9507 \cdot 10^{-4}~\mathrm{A} = 0.2~\mathrm{mA}.\]

The ohmic resistance is thus given by

\[R = \frac{U}{I} = \frac{1~V}{1.9507\cdot 10^{-4}~\mathrm{A}} = 5105.2 ~\Omega = 5.1 ~\mathrm{k}\Omega.\]

The conductivity $\sigma$ is given by

\[\sigma = \frac{j}{E} = \mu_e\,n\, e = \frac{19507~\mathrm{A/m^2}}{1~ \mathrm{V/cm}} = 195 ~\Omega\mathrm{m}.\]

and is related to the resistance as follows:

\[\sigma = \frac{j}{E} = \frac{I / A}{U / d} = \frac{1}{w\, R},\]

where $w$ is the width of the sample. Here, $w = 1~\mathrm{\mu m}$.

Further Exercises¶

Repeat the calculations for InSb assuming electron mobility $\mu_{e,\mathrm{InSb}} = 4\cdot 10^5~\mathrm{cm^2/Vs}$ and compare your findings with the results you have obtained for Si.
Repeat the calculations for Two-dimensional electron gases (2DEGs) in AlGaAs/GaAs heterostructures assuming electron mobility $\mu_{e,\mathrm{2DEG}} = 10^7~\mathrm{cm^2/Vs}$ and compare your findings with the results you have obtained for Si.

Hint

You can change the material to, e.g., InSb by altering the variable $material. Custom material parameters, which should not be taken from the default, should be specified in the group database{}.

Answers¶

Drift velocity

Electrons in InSb in a field of 1 V/cm have mean drift velocities of 4 $\cdot$ 10⁵ cm/s = 4 km/s.

Two-dimensional electron gases (2DEGs) in a field of 1 V/cm in AlGaAs/GaAs heterostructures have mean drift velocities of the order ~100 km/s.

Scattering time

An effective scattering time for electrons in InSb ($m_e$ = 0.0135 $\cdot\,m_0$) is 3.1 ps.

An effective scattering time for two-dimensional electron gases (2DEGs) in AlGaAs/GaAs heterostructures ($m_e$ = 0.2 $m_0$) is of the order 1.1 ns.

Mean free path

$l_\mathrm{mfp} = 12.4~\mathrm{nm}$ for InSb.

$l_\mathrm{mfp} = 110~\mathrm{\mu m}$ for AlGaAs/GaAs (2DEG).

Last update: nn/nn/nnnn

— EDU — Electron transport in n-type Silicon¶

Header¶

Problem¶

Input file¶

Solutions¶

Mean drift velocity¶

Mean free path¶

Resistance and conductivity¶

Further Exercises¶

Answers¶