— DEV — I–V characteristics of n-doped Si structure#

Input files:

  • I-V_n-doped-Si_1D_nnp.in

  • I-V_n-doped-Si_2D_nnp.in

  • I-V_n-doped-Si_3D_nnp.in

  • I-V_nin-doped-Si_1D_nnp.in

  • I-V_nin-doped-Si_2D_nnp.in

  • I-V_nin-doped-Si_3D_nnp.in

Scope:

This tutorial aims to simulate the I-V characteristics of n-doped and n-i-n doped Si structures.

Output files:

  • IV_characteristics.dat

  • bias_xxxxx/bandedges.dat

I-V characteristics of an n-doped Si structure#

Structure#

The structure we are dealing with consists of bulk Si that is sandwiched between two contacts. The whole structure has the following dimensions (see also):

  • along \(x\)-axis: \(20\,\mathrm{nm}\) (\(1\,\mathrm{nm}\) contact, \(18\,\mathrm{nm}\) Si, \(1\,\mathrm{nm}\) contact)

  • along \(y\)-axis: \(5\,\mathrm{nm}\)

../../_images/nnpp_tu_I-V_n-doped-Si_1D_2D_3D_n-structure.jpg

Figure 179 Simulated structure consisting of a left and right contact (blue) and n-doped Si layer (red).#

The Si is n-type doped with a donor concentration of \(N_\mathrm{D} = 1\cdot 10^{20}\,\mathrm{cm^{-3}}\). The energy level is 0.044 eV below the conduction band edge. This leads to an electron density of \(n = 13.48 \cdot 10^{18}\,\mathrm{cm^{-3}}\), which corresponds to the concentration of the ionized donors. The Fermi level \(E_\mathrm{F}\) is taken to be at 0 eV in an equilibrium simulation, i.e. \(V = 0\,\mathrm{V}\). The distance of the conduction band from the Fermi level can be calculated in the following way:

  • For the effective electron mass at the \(\Delta\)-point we have:

\[m_e = m^*_{e,\mathrm{DOS}} = (m_\mathrm{l}\cdot m_\mathrm{t}\cdot m_\mathrm{t})^{\frac{1}{3}} = (0.916\cdot 0.19^2)^{\frac{1}{3}} m_0 = 0.321 m_0,\]

where \(m_\mathrm{l}\) is the longitudinal and \(m_\mathrm{t}\) is the transversal mass of the effective mass tensor.

  • The effective density of states reads:

\[N_\mathrm{c} = 12\cdot \left(\frac{2 \pi m_e k_\mathrm{B}T}{h^2}\right)^{\frac{3}{2}} = 12\cdot (0.321 \cdot 0.026 \cdot 2.0886\cdot 10^{14})^{\frac{3}{2}} = 12 \cdot 2.282\cdot 10^{18}\,\mathrm{cm^{-3}} = 2.738\cdot 10^{19}\,\mathrm{cm^{-3}},\]

where the factor of 12 arises due to the six-fold degeneracy of Si at \(\Delta\) and the two-fold spin degeneracy. Similarly, we obtain the effective density of states for holes:

\[ \begin{align}\begin{aligned}\begin{aligned}\\N_{\mathrm{v,\, hh}} &= 9.875\cdot 10^{18}\,\mathrm{cm^{-3}},\\N_{\mathrm{v,\, lh}} &= 1.502\cdot 10^{18}\,\mathrm{cm^{-3}}.\\\end{aligned}\end{aligned}\end{align} \]

Note that heavy and light holes are degenerate for \(k = 0\), i.e. \(N_\mathrm{v} = N_\mathrm{v,\,hh} + N_\mathrm{v,\,lh} = 1.1377\cdot 10^{19}\,\mathrm{cm^{-3}}\).

  • The Semiconductor equation is given by

\[np = n_\mathrm{i}^2 = N_\mathrm{c} N_\mathrm{v} \exp\left( - \frac{E_\mathrm{gap}}{k_\mathrm{B}T}\right) = N_\mathrm{c}\cdot 1.138\cdot 10^{19}\,\mathrm{cm^{-3}}\exp\left(-\frac{1.095}{0.026}\right) = 1.238\cdot 10^{20}\,\mathrm{cm^{-6}},\]

with \(E_\mathrm{gap} = 1.095\,\mathrm{eV}\), \(n_\mathrm{i} = 1.113\cdot 10^{10}\mathrm{cm^{-3}}\) and \(p = n_\mathrm{i}^2/n = 9.185\,\mathrm{cm^{-3}}\).

  • The occupation of the different energy states can either be described by Maxwell-Boltzmann statistics:

\[ \begin{align}\begin{aligned}\begin{aligned}\\n(T) &= N_\mathrm{c}(T)\exp\left(\frac{E_\mathrm{F} - E_\mathrm{c}}{k_\mathrm{B}T}\right),\\p(T) &= N_\mathrm{v}(T)\exp\left(\frac{E_\mathrm{v} - E_\mathrm{F}}{k_\mathrm{B}T}\right),\\\end{aligned}\end{aligned}\end{align} \]

or Fermi-Dirac statistics:

\[ \begin{align}\begin{aligned}\begin{aligned}\\ n(T) &= N_\mathrm{c}(T) \mathcal{F}_{1/2}\left(\frac{E_\mathrm{F} - E_\mathrm{c}}{k_\mathrm{B}T}\right),\\p(T) &= N_\mathrm{v}(T) \mathcal{F}_{1/2}\left(\frac{E_\mathrm{v} - E_\mathrm{F}}{k_\mathrm{B}T}\right),\\\end{aligned}\end{aligned}\end{align} \]

where \(\mathcal{F}_{1/2}\) is the Fermi-Dirac integral of order \(1/2\) multiplied by the factor \(2/\sqrt{\pi}\) (i.e. \(\mathcal{F}_{1/2}\) includes the Gamma pre-factor)

When using the Maxwell-Boltzmann statistics as an approximation, we obtain:

\[ \begin{align}\begin{aligned}\begin{aligned}\\E_\mathrm{c} &= k_\mathrm{B}T \ln\left(\frac{N_\mathrm{c}}{n}\right) = 0.026\,\mathrm{eV}\cdot\ln\left(\frac{2.738\cdot 10^{19}\,\mathrm{cm^{-3}}}{13.478\cdot 10^{18}\,\mathrm{cm^{-3}}}\right) = 0.026\,\mathrm{eV}\cdot \ln(2.031) = 18.3\,\mathrm{meV},\\E_\mathrm{v} &= - k_\mathrm{B}T \ln\left(\frac{N_\mathrm{v}}{p}\right) = - 0.026\,\mathrm{eV} \cdot 42.538 = - 1.099\,\mathrm{eV}.\\\end{aligned}\end{aligned}\end{align} \]

Note that nextnano++ uses the Fermi-Dirac integrals (Fermi-Dirac statistics), where the following results are obtained: \(E_\mathrm{c}=13.85\,\mathrm{meV}\) and \(E_\mathrm{v} = -1.0815\,\mathrm{eV}\).

Results#

We sweep the voltage at the right contact from \(0.0\,\mathrm{V}\) to \(0.2\,\mathrm{V}\) in 10 steps. The input files used for the simulations are I-V_n-doped-Si_1D_nnp.in, I-V_n-doped-Si_2D_nnp.in I-V_n-doped-Si_3D_nnp.in. The calculated current density for each bias point can be found in IV_characteristics.dat. The resulting I-V characteristics is depicted in Figure 180.

../../_images/nnpp_tu_I-V_n-doped-Si_1D_2D_3D_n-IV.jpg

Figure 180 Simulated I-V characteristics of an n-doped Si structure using constant mobility model.#

The nextnano++ results are in agreement with the I-V characteristics obtained with nextnano³. The units for the current in a 2D simulation are [\(\mathrm{A/m}\)]. Dividing this two-dimensional current value by the width of the device (in our case 5 nm) we obtain the current in units of [\(\mathrm{A/cm^2}\)], 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 [\(\mathrm{A}\)]. Dividing by the area of the device of \(25\,\mathrm{nm^2}\), we obtain the 1D units of [\(\mathrm{A/cm^2}\)].

I-V characteristics of an n-i-n-doped Si structure#

Structure#

The second example is an n-i-n (n-doped, intrinsic, n-doped) Si structure, which is shown in Figure 181. The width of the intrinsic region is 14 nm, and the n-doped regions are both 2 nm wide.

../../_images/nnpp_tu_I-V_n-doped-Si_1D_2D_3D_nin-structure.jpg

Figure 181 Simulated n-i-n structure consisting of a left contact (dark blue), n-doped Si (light blue), intrinsic Si (green), n-doped Si (yellow) and right contact (red).#

Results#

In Figure 182 the current-voltage (I-V) characteristic is shown. The input files used for the simulations are I-V_nin-doped-Si_1D_nnp.in, I-V_nin-doped-Si_2D_nnp.in I-V_nin-doped-Si_3D_nnp.in. The data of the I-V curve can be found in the corresponding file IV_characteristics.dat.

../../_images/nnpp_tu_I-V_nin-doped-Si_1D_2D_3D_n-IV.jpg

Figure 182 Simulated I-V characteristics of the n-i-n doped Si structure using constant mobility model.#

In order to compare the results from 1D, 2D and 3D simulations, we have divided the 2D current by the width of the device (in our case 5 nm) and the 3D current by the cross-section area of the device of (in our case \(25\,\mathrm{nm^2}\)), to get the current density in units of [\(\mathrm{A/cm^2}\)]. The obtained results are in perfect agreement.

Figure 183 shows the conduction band profile (bias_xxxxx/bandedges.dat) for different voltages.

../../_images/nnpp_tu_I-V_n-doped-Si_1D_2D_3D_cb-profile.jpg

Figure 183 Simulated conduction band profile of the n-i-n Si structure for different voltages.#

Last update: 17/07/2024