Quantum provides a set of models for simulation of various effects of quantum confinement of carriers in semiconductor devices. A self consistent Schrodinger - Poisson solver allows calculation of bound state energies and associated carier wave function self consistently with electrostatic potential. A Quantum moment transport model allow simulation of confinement effects on carrier transport. The Van Dort and Hansch models provide semi-empirical simulation of confinement effects on MOS devices. A quantum wells for gain and spontaneous recombination in light emitting devices. Quantum also has non-local tunneling models which calculate tunneling current by solving the Schrodinger equation. These can optionally include the effects of quantum confinement on tunneling currents.

Schrodinger-Poisson

To model the effects of quantum confinement, Quantum allows the self-consistent solution of the Schrodinger and Poisson’s equations. Quantum gives you freedom to solve Schrodinger equations in either in 1D slices or full 2D of the device in order to find eigen bound state energies and wave functions.

The eigen bound state energies and wave functions obtained are used to find the electron density, which is plugged into a corresponding 2D or3D Poisson’s equation. In order to estimate a charge control behavior of a device under applied bias, Quantum will first compute a local quasi Fermi level from drift-diffusion equation and then use it to determine the quantum mechanical electron density.

The figure above shows the potential variation of the conduction band edge, as well as the first seven bound state energy levels under the gate of a GaAs/AlGaAs HEMT device. Here we can see that there is confinement due to the heterojunction as well as a potential well due to depletion in the AlGaAs top layer.

This figure gives a comparison between the electron concentrations under the gate of the GaAs/AlGaAs HEMT device as predicted by the self-consistent Schrodinger- Poisson solution and as predicted by classical solution.

The first seven electron wave functions are shown in the figure above. These wave functions correspond to the lowest seven bound state energies. These wave functions peak on both sides of the heterojunction due to the potential well created by depletion directly under the gate in the AlGaAs layer.

This figure shows the first five bound state energies under the gate for a thin gate oxide NMOS device in inversion.

This figure shows the first five electron wave functions under the gate of the NMOS device in inversion.

2D Schrodinger Solver

This figure shows a representative solution for an “O” shaped quantum well.

The figure above shows a ground state wave function of an “S” shaped quantum well. The Schrodinger solver can ruduce computation time by discretizing the 2D equation only inside the well and thin oxide layer.

Non-local Tunneling Model

ATLAS has a number of models for non-local tunneling which take into account the spatial transfer of carriers during the tunneling process. They also couple the tunneling current to the current continuity equations to ensure self-consistency. One such model is the non-local band to band tunneling model which can calculate the tunneling current across a p-n junction in both forward and reverse bias. This results in the current voltage characteristics typical of a tunnel diode. The tunneling current through an insulator can also be calculated using a non-local model (Transmission Matrix technique). Optionally this can take account of quantum confinement in the inversion/accumulation region of the semiconductor. Band to band tunneling through the insulator can also be modeled for polysilicon contacts.

Tunneling current versus gate voltage for a MOS structure with a 2 nm oxide layer and an N-doped channel. The Fowler-Nordheim current is shown for comparison. Results with and without quantum confinement being modeled in the channel are shown. The tunneling mass assumed in the oxide is 0.5 m0.

Current-Voltage curves for a tunnel diode comprised of a Hg0.78 Cd0.2 Te degenerately doped p-n structure. The material bandgap is 116 meV, the device area is 1 µm2 and the simulation temperature is 80 K.

Quantum Moment Transport Models

ATLAS has two models for including some of the effects of quantum confinement into the semi-classical drift-diffusion and hydrodynamic carrier transport calculations. The density gradient (DG) method calculates a position dependent potential energy term according to higher derivatives of the carrier densities. The Bohm Quantum Potential (BQP) model calculates a position dependent potential energy term using an auxiliary equation derived from the Bohm interpretation of Quantum mechanics. These extra potential energies then modify the electron or hole distributions. Whilst both methods are derived from pure physics they retain some empiricalism, the DG method has one fitting parameter and BQP has two. This flexibility allows the models to approximate the quantum behavior of different classes of devices as well as a range of materials. It is possible to get close agreement between Poisson-Schrodinger results and DG or BQP under conditions of negligible current flow. The effects of quantum confinement on the device performance, including I-V characteristics, will then be calculated to a good approximation.

Shows the electron concentration under the gate of an NMOS device with a 2 nm thick oxide. The applied bias of 2 V has put the device in strong inversion. The BQP curve can be made even closer to the S-P curve by a better choice of parameters for the BQP model.

Shows the electron concentration under the gate of an NMOS device with a 2 nm thick oxide. The applied bias of 2 V has put the device in strong inversion. The BQP curve can be made even closer to the S-P curve by a better choice of parameters for the BQP model.

Shows the quasi-static Gate capacitance versus gate voltage for the NMOS device. The threshold voltage for inversion due to quantum confinement is correctly predicted by the BQP and S-P models. Quantum effects are only included for electrons in this figure.

Shows the electron density near the AlGaAs/GaAs interface in a HEMT structure as calculated by Classical, BQP and S-P

Shows the drain current versus gate bias for a HEMT structure with a drain bias of 0.5 V. The quantum confinement results in a reduced drain current in this case, although this effect will depend on the particular mobility model used.

Shows the lateral current density (i.e. parallel to the channel) for a HEMT structure along a line perpendicular to the channel. The drain bias is 0.5 V and the gate bias is 0.5 V. One can see a parallel conduction path in the AlGaAs as well as the quantum model smoothing out the current density in the channel.

The electron density on a cross-section of the 3 nm wide channel, halfway between source and drain. The gate bias is 0.5 V, the drain bias is 0.5 V and the carrier temperature at the position of the cross section is approximately 630 K. Quantum confinement effects are pronounced.

The conduction current density for the Double gate MOSFET at a gate bias of 0.5 V, a Drain bias of 0.5 V. The model used was Bohm Quantum Potential with energy balance. The concentration of the current near the centre of the channel is due to quantum confinement effects.

The electron temperature as obtained using the Energy Balance model for the Double Gate MOSFET structure. The profile is taken along the device between source and drain, that is perpendicular to the section in previous figure. Results for both classical and Bohm Quantum Potential show only a subtle difference in temperature distribution despite the large change in carrier distribution.

Rev. 120806_03