2D Simulation Models for Quantum Mechanical Effects

Quantum provides a set of models for simulation of various effects of quantum confinement and quantum transport of carriers in semiconductor devices. A self consistent Schrodinger – Poisson solver allows calculation of bound state energies and associated carrier wave functions self consistently with electrostatic potential. Schrodinger solvers can be combined with Non-equilibrium Green’s Function (NEGF) Approach in order to model ballistic quantum transport in 2D or cylindrical devices with strong transverse confinement.

An alternative approach to modeling subband transport in nanoscale devices is given by Mode-Space Drift-Diffusion Model, which combines transverse Schrodinger with 1D drift-diffusion equations. A quantum moment transport model allows simulation of confinement effects on carrier transport and yet keeps the simplicity of a conventional drift diffusion approach. It also allows quantum confinement effects to be included in the energy balance/hydrodynamic transport model. A quantum well model takes confinement into account when computing 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 and can be used for band-to-band and oxide tunneling.

Schrodinger-Poisson

To model the effects of quantum confinement, Quantum allows the self-consistent solution of the Schrodinger and Poisson equations. The Schrodinger equation is solved in 1D, 2D or cylindrical geometry in order to find eigen energies and wave functions. The eigen energies and wave functions obtained are used to find the quantum electron density, which is plugged into a 2D Poisson equation. Quantum2D can solve Schrodinger and Poisson equations on the same rectangular or triangular mesh or create its own rectangular Schrodinger mesh and iterpolate quantum quantities to Poisson mesh. Fast convergence is achieved by utilizing a predictor-corrector scheme. In addition to a real space 2D Schrodinger solver, a fast product-space solver is available, which finds 2D wave functions as a linear combination of products of 1D solutions in two directions. A computation time for this method scales linearly with the number of nodes in one direction, which allows to simulate much larger devices with geometries close to rectangular.

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. 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.
Radial part of electron wave functions (left) and corresponding eigen energies (right) found by cylindrical Schrodinger solver. Eigen states of different radial and orbital quantum numbers are sorted with respect to the total eigen energy.



This figure shows self-consistent quantum electron density in rectangular , triangular and circular shaped channels under a gate bias of 2 Volts. Silicon channel with anisotropic effective mass, aluminum gate and 1nm silicon oxide were assumed in these examples. A triangular mesh and 2D real space Schrodinger solver were used.

Wave functions in the 10 nm diameter channel of surround gate transistor for the valley with in-plane effective masses mx=my=mt=0.19m0 and out of plane mass mz=me=0.91m0 Self-consistent quantum electron density in a 14X14 nm rectangular structure doped to 1020 cm-3 as found by fast product-space 2D Schrodinger solver on a mesh with 5041 grid points.

 

Ballistic Quantum Transport with Non Equilibrium Green’s Function Approach

As MOS field-effect transistors are scaled down to a nanometer regime, quantum effects in both transverse and transport directions start playing a major role in determining device characteristics. In order to address the new challenge, Silvaco has deployed new quantum mechanical models based on Non Equilibrium Green’s Function (NEGF) approach. This is a fully quantum mechanical approach, which treats such effects as source-to-drain tunneling, ballistic transport and quantum confinement on equal footing. The new NEGF solver is suitable to model ballistic quantum transport in such devices as double gate or surround gate MOSFETs, using rectangular or cylindrical geometries. The modeling starts with a solution of a 1D Schrodinger equation in the transverse slices of the device in order to find eigen functions and eigen energies. Then, NEGF quantum transport equations are solved for electron densities and current of electrons in various sub-bands (modes), propagating from source to drain. In general, Coupled Mode Space (CMS) method is used to account for mixing of electron modes. In a simpler case of uniform cross-section, Uncoupled Mode Space (UMS) method can be used. The NEGF simulation gives common current-voltage characteristics and self-consistent quantum electron and current density. It also provides an insight into device physics by storing energy dependent quantities such as transmission coefficient, local density of states, electron and current density per unit energy.

Quantum nature of electrons in transverse and transport directions is taken into account.

The flared-up channel access geometry causes spreading of current in source and drain extension regions.

2D contour of self-consistent electron density (left) and current flow (right)
in double gate transistor at Vg = 1V

 

Electron density (left) and conduction band (right) as functions of coordinate along the transport direction of a double gate transistor for various gate biases.
Id-Vg (left) and Id-Vd (right) characteristics of ballistic DGT computed with NEGF approach. Due to the assumption of ballistic transport, the computed current gives an upper limit for the particular device.

 

Transmission coefficient (left), current spectrum (center) and density of states (right) in the ballistic double gate transistor. (a) Each step of the transmission coefficient corresponds to an extra electron sub-band becoming available for transport. (b) The lowest sub-band gives the highest contribution to current. (c) Oscillatory nature of density of states in the source extension region is due to electron reflections at the injection barrier.

 

Drift Diffusion Mode-Space Model

The Drift Diffusion Mode-Space Model (DDMS) is a semi-classical approach to transport in devices with strong transverse confinement and is a simpler alternative to mode-space NEGF approach. Similarly to the mode-space NEGF, the solution is decoupled into 1D or cylindrical Schrodinger equation in transverse direction and 1D transport equations in each subband. In this model, however, a classical drift-diffusion equation is solved instead of a quantum transport equation. Thus, the model captures quantum effects in transverse direction and yet inherits all familiar ATLAS models for mobility, recombination, impact ionization and band-to band tunneling.

Here we show a current flow as computed by DDMS in a double gate FET with Lg=30nm and body thickness t=2nm. 1D drift-diffusion equations are solved in each subband in the presence of trap recombination (SRH), band-to-band tunneling (BBT) and impact ionization. Current density is increased in the end of the channel due to e-h pair generation.

 


This figure shows electron (top left) and hole (top right) carrier densities and the lowest electron subband energy with (green) and without (red) generation-recombination mechanisms. The e-h pair generation causes a slight increase in electron concentration (top left) and a tremendous increase in the concentration of holes (top right), which are accumulated in the channel. The charge accumulation resets the threshold voltage and decreases the source injection barrier for electrons (bottom).

 

Drain current- drain voltage characteristics, computed by DDMS, show a strong floating body effect and a poor saturation (right) when the mechanisms of BBT and impact ionization are present as compared to the case when generation-recombination is neglected. (left).

 

Band-to-Band Quantum Tunneling Models

Quantum has the capability to calculate band-to-band tunneling in semiconductors. Both the trap assisted and direct components can be calculated. The direct component can be calculated by using either a local or non-local model. In the local model, the electric field at each point is used to give a rate for the generation of electron-hole pairs at that point. The non-local model is more sophisticated in that it calculates the tunneling current for each energy at which tunneling is possible. Furthermore the sources (reverse bias) and sinks (forward bias) of carriers occur at the correctly spatially separated positions in the device. An example of forward current in a tunnel diode, calculated using the non-local model, is shown in the following figures.

Current-Voltage curves for a tunnel diode comprised of an Hg0.78 Cd0.22Te degenerately doped p-n structure. The material bandgap is 116 meV and the device temperature is 80 K. Conduction band (Ec) and valence band (Ev) energy profile versus position for the tunnel diode. The bias on the anode is 0.015 Volts, corresponding to maximum tunneling current.

 

The electron sink (in n-type material) and hole sink (in p-type material) for the tunnel diode. The bias on the anode is 0.015 Volts.

 

Oxide Tunneling Models

Quantum has a range of models for calculating tunneling through an oxide from a semiconducting channel. The most sophisticated of these solves the Schrodinger equation along the tunneling path using a transmission matrix technique and performs an integration over energy. It can optionally include quantization effects in the channel by coupling to solutions of the Schrodinger equation there. An example of tunneling current calculated using this sophisticated model is shown in the figure on the right. It is compared with the result from the Fowler-Nordheim model.

Tunneling current versus gate voltage for a MOS Capacitor with an acceptor doped channel and an oxide thickness of 2 nm. The Fowler Nordheim current agrees with the direct tunnel current at high bias. The direct tunnel current with channel quantization is also shown, the shift in the centroid in charge due to quantization is the major factor in enhancing it, most notably at higher bias.

 

Quantum Moment Transport Models

Quantum has a model for including some of the effects of quantum confinement in the semiclassical drift-diffusion and hydrodynamic carrier transport calculations. 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. This extra potential energy modifies the electron and/or hole distribution. The model is derived from pure physics, although it retains some empiricism, having two fitting parameters. This flexibility allows the model 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 Schrodinger-Poisson results and 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.

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. Total electron charge increasing with gate voltage as the NMOS goes into inversion. The BQP and S-P values are similar, and can be made closer to the S-P curve by a better choice of parameters for the BQP model.

 

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. Electron density near the AlGaAs/GaAs interface in a HEMT structure as calculated by Classical, BQP and S-P
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. 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 center 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. 110113_08