Monte Carlo methods for electron transport
Further, it physically represents the probability of particle occupation of energy k at position r and time t. In addition, due to being a seven-dimensional integro-differential equation (six dimensions in the phase space and one in time) the solution to the BTE is cumbersome and can be solved in closed analytical form under very special restrictions.
The Monte Carlo model in essence tracks the particle trajectory at each free flight and chooses a corresponding scattering mechanism stochastically.
The semiclassical equation describing the motion of an electron is where F is the electric field, E(k) is the energy dispersion relation, and k is the momentum wave vector.
[1] Both drift diffusion (DD) and the hydrodynamic (HD) models can be derived from the moments of the Boltzmann transport equation (BTE) using simplified approximation valid for long channel devices.
[2] On the other hand, the HD method solves the DD scheme with the energy balance equations obtained from the moments of BTE.
Needless to say, an accurate discretization method is required in HD simulation, since the governing equations are strongly coupled and one has to deal with larger number of variables compared to the DD scheme.
The accuracy of semiclassical models are compared based on the BTE by investigating how they treat the classical velocity overshoot problem, a key short channel effect (SCE) in transistor structures.
Essentially, velocity overshoot is a nonlocal effects of scaled devices, which is related to the experimentally observed increase in current drive and transconductance.
[6] The summary of simulation results (Illinois Tool: MOCA) with DD and HD model is shown in figure beside.
The velocity overshoot is observed only near the drain junction in the MC data and the HD model fits well in that region.
From the MC data, it can be noticed that the velocity overshoot effect is abrupt in the high-field region, which is not properly included in the HD model.
The band structure is used to compute the movement of carriers under the action of the electric field, scattering rate, and final state after the collision.
In case of silicon, conduction band minima does not lie at k = 0 and the effective mass depends on the crystallographic orientation of the minimum as where
For higher applied fields, carriers reside above the minimum and the dispersion relation, E(k), does not satisfy the simple parabolic expression described above.
Full band approach for Monte Carlo simulation was first used by Karl Hess at the University of Illinois at Urbana-Champaign.
This method couples the ensemble Monte Carlo procedure to Poisson's equation, and is the most suitable for device simulation.
Important charge transport properties of semiconductor devices such as the deviance from Ohm's law and the saturation of carriers mobility are a direct consequence of scattering mechanisms.
The semiconductor Monte Carlo simulation, in this scope, is a very powerful tool for the ease and the precision with which an almost exhaustive array of scattering mechanisms can be included.
Electronphonon interactions are essentially inelastic since a phonon of definite energy is either emitted or absorbed by the scattered particle.
Before characterizing scattering mechanisms in greater mathematical details, it is important to note that when running semiconductor Monte Carlo simulations, one has to deal mainly with the following types of scattering events:[9] Acoustic Phonon: The charge carrier exchanges energy with an acoustic mode of the vibration of atoms in the crystal lattice.
Because the mass of an electron is relatively small in comparison to the one of an impurity, the Coulomb cross section decreases rapidly with the difference of the modulus of momentum between the initial and final state.
As discussed in section II-I, the quantum many-body problem arising from the interaction of a carrier with its surrounding environment (phonons, electrons, holes, plasmons, impurities,...) can be reduced to a two-body problem using the quasiparticle approximation, which separates the carrier of interest from the rest of the crystal.
[9] Within these approximations, Fermi's Golden Rule gives, to the first order, the transition probability per unit time for a scattering mechanism from a state
: where H' is the perturbation Hamiltonian representing the collision and E and E′ are respectively the initial and final energies of the system constituted of both the carrier and the electron and phonon gas.
, generally referred to as the matrix element, mathematically represents an inner product of the initial and final wave functions of the carrier:[12] In a crystal lattice, the wavefunctions
Umklapp processes (or U-processes) change the momentum of the particle after scattering and are therefore limiting the conduction in semiconductor crystals.
, one finds The two spherical angles can then be chosen, in the uniform case, by generating two random numbers 0 < r1, r2 < 1 such that The current trend of scaling down semiconductor devices has forced physicists to incorporate quantum mechanical issues in order to acquire a thorough understanding of device behavior.
This complication, however, can be avoided in the case of practical devices like the modern day MOSFET, by employing quantum corrections within a semi-classical framework.
The standard Boltzmann Transport Equation is obtained when the non-local terms on the LHS disappear in the limit of slow spatial variations.
Even though the above-mentioned potentials for quantum correction differ in their method of calculation and their basic assumptions, yet when it comes to their inclusion into Monte Carlo simulation they are all incorporated the same way.
