Two-dimensional electron gas in a Si MOSFET¶

Introduction¶

In this tutorial, you can learn how to obtain carrier sheet densities in the inversion layer of MOSFET.

Layer sequence¶

The table below shows the materials, their widths, and their dopant concentrations for this tutorial.

material

width (nm)

doping

contact

$$10$$

p-Si

$$99$$

$$5\times10^{17}\;\mathrm{cm^{-3}}$$

SiO2

$$5$$

n-Si (poly-Si)

$$54$$

$$3\times10^{19}\;\mathrm{cm^{-3}}$$

Gate contact

$$1$$

The applied gate voltage leads to confined electron states at the p-Si/SiO2 interface (n-type inversion layer) whereas the holes are repelled from the p-Si/SiO2 surface towards the interior of the device (i.e. to the left side).

An applied source-drain voltage in the plane of the inversion layer will lead to a flow of current which depends on the sheet density in the inversion layer. The magnitude of the current is governed by the applied gate voltage, i.e. the gate controls the sheet density and thus switches the current on or off (MOSFET, metal-oxide-semiconductor field effect transistor).

Calculations¶

The temperature was set to $$300\;\mathrm{K}$$. Self-consistent solution of the 1D-Schrödinger-Poisson equation within single-band effective-mass approximation (using ellipsoidal effective mass tensors) for the (Delta) conduction band edges.

We vary the gate voltage from $$0\;\mathrm{V}$$ to $$2.5\;\mathrm{V}$$ in steps of $$0.1\;\mathrm{eV}$$.

Results¶

The following two figures show the band profiles and the electron density for two different gate voltages:

Figure 2.4.496: 0.7 V (The electron ground state is above the electron Fermi level $$E_{F,n}$$)

Figure 2.4.497: 2.5 V (The electron ground state is below the electron Fermi level $$E_{F,n}$$ and thus occupied, leading to a large quantum mechanical density)

The amplitude of the ground state $$\psi_{1}^2$$ is above $$E_{F,n}$$ as you can see.

The amplitude of the ground state $$\psi_{1}^2$$ is below $$E_{F,n}$$ as you can see.

In the figures above, cb and vb represent the conduction band and the valence band, respectively.

In the poly-silicon on the right side of the SiO2 barrier, the electrons get depleted from the oxide interface.

Due to the fact that the quasi Fermi level is nearly constant outside the SiO2 barrier, almost no current is flowing. Inside the SiO2 barrier, the quasi Fermi level has a step-like feature. However, as the electron density is close to zero inside the barrier, almost no current is eventually flowing.

The ground state electron level is associated with the longitudinal electron mass ($$m\mathrm{_{longitudinal}} = 0.916\;m_{0}$$). On the other hand, the second and the third eigenstate (which are degenerate) are associated with the transversal electron mass ($$m\mathrm{_{transversal}} = 0.190\;m_{0}$$). Due to this degeneracy, only two rather than three Schrödinger equations have to be solved: (a) $$V(z), m = m\mathrm{_{transversal}} = 0.190\;m_{0}$$ (b) $$V(z), m = m\mathrm{_{longitudinal}} = 0.916\;m_{0}$$ The potential $$V(z)$$ that enters into the Schrödinger equation is the same in these two cases.

The eigenvalues for $$m\mathrm{_{longitudinal}}$$ are contained in bias_*\Quantum\probabilities_shift_Quantum_region_X1.dat. The eigenvalues for $$m\mathrm{_{transversal}}$$ are contained in bias_*\Quantum\probabilities_shift_Quantum_region_X2.dat.

At $$2.5\;\mathrm{eV}$$, the energy spacing between the two lowest electron states is of the order $$100\;\mathrm{meV}$$ (in the case of the longitudinal effective mass). At $$2.5\;\mathrm{eV}$$, the energy spacing between the two lowest electron states is of the order $$130\;\mathrm{meV}$$ (in the case of the transversal effective mass). At $$2.5\;\mathrm{eV}$$, the energy spacing between the electron ground state of the longitudinal effective mass and the ground state of the transversal effective mass is of the order $$70\;\mathrm{meV}$$. Thus, in this case, one can safely assume that only first subband is occupied, i.e. the electron ground state with the longitudinal mass.

(to be fixed)

Electron sheet density in the inversion channel as a function of applied gate voltage¶

The file bias_*\Quantum\density_electron.dat contains the electron density across the MOSFET. Since the p-Si region, where the inversion channel is located, extends from $$x = 0\;\mathrm{nm}$$ to $$x = 99\;\mathrm{nm}$$, you have to integrate the electron density over the region to obtain the sheet density. To do it on nextnano++, structure{ region { integrate } } is used as following (structure{ region{ integrate{ } } }).

109region{
110    line{ x = [ $itf_start_contact,$itf_p_Si_SiO2 ] }
111    binary{ name = "Si" }
112    doping{
113        constant{
114            name = "B_acceptor"
115            conc = \$acceptor_conc
116        }
117    }
118    integrate{ electron_density{} }
119}
`

The output is in the file integrated_density_electron.dat.

Figure 2.4.498 shows the electron sheet density of the p-Si inversion layer.

To obtain the capacitance-voltage characteristics, you have to calculate the derivative of the sheet density.

Last update: 17/07/2024