The Monte Carlo method for electron transport is a semiclassical Monte Carlo (MC) approach of modeling semiconductor transport. Assuming the carrier motion consists of free flights interrupted by scattering mechanisms, a computer is utilized to simulate the trajectories of particles as they move across the device under the influence of an electric field using classical mechanics. The scattering events and the duration of particle flight is determined through the use of random numbers.
Background
Boltzmann transport equation The Boltzmann transport equation model has been the main tool used in the analysis of transport in semiconductors. The BTE equation is given by:
The distribution function, f, is a dimensionless function which is used to extract all observable of interest and gives a full depiction of electron distribution in both real and k-space. 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. Numerically, solution to the BTE is employed using either a deterministic method or a stochastic method. Deterministic method solution is based on a grid-based numerical method such as the spherical harmonics approach, whereas the Monte Carlo is the stochastic approach used to solve the BTE.
Monte Carlo method The semiclassical Monte Carlo method is a statistical method used to yield exact solution to the Boltzmann transport equation which includes complex band structure and scattering processes. This approach is semiclassical for the reason that scattering mechanisms are treated quantum mechanically using the Fermi's Golden Rule, whereas the transport between scattering events is treated using the classical particle notion. The Monte Carlo model in essence tracks the particle trajectory at each free flight and chooses a corresponding scattering mechanism stochastically. Two of the great advantages of semiclassical Monte Carlo are its capability to provide accurate quantum mechanical treatment of various distinct scattering mechanisms within the scattering terms, and the absence of assumption about the form of carrier distribution in energy or k-space. 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. To solve the above equation, one needs strong knowledge of the band structure (E(k)). The E(k) relation describes how the particle moves inside the device, in addition to depicting useful information necessary for transport such as the density of states (DOS) and the particle velocity. A Full-band E(K) relation can be obtained using the semi-empirical pseudopotential method.
Hydrodynamic and drift diffusion method 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. The DD scheme is the most classical approach and usually solves the Poisson equation and the continuity equations for carriers considering the drift and diffusion components. In this approach, the charge transit time is assumed to be very large in comparison to the energy relaxation time. On the other hand, the HD method solves the DD scheme with the energy balance equations obtained from the moments of BTE. Thus, one may capture and calculate physical details such as carrier heating and the velocity overshoot effect. 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.
Comparison of semiclassical models
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. As the channel length becomes smaller, the velocity is no longer saturated in the high field region, but it overshoots the predicted saturation velocity. The cause of this phenomenon is that the carrier transit time becomes comparable to the energy relaxation time, and therefore the mobile carriers do not have enough time to reach equilibrium with the applied electric field by scattering in the short channel devices. The summary of simulation results (Illinois Tool: MOCA) with DD and HD model is shown in figure beside. In the figure (a), the case when the field is not high enough to cause the velocity overshoot effect in the whole channel region is shown. Note that at such limit, the data from the DD model fit well to the MC model in the non-overshoot region, but the HD model overestimate the velocity in that region. 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. For high field conditions as shown in the figure (b) the velocity overshoot effect almost all over the channel and the HD results and the MC results are very close in the channel region.
Monte Carlo for semiconductor transport
Band structure Band structure describes the relationship between energy(E) and wave vector(k). 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. Silicon band structure and its Brillouin zone are shown in figure below, but there is no analytical expression which satisfies entire Brillouin zone. By using some approximation, there are two analytical models for band structure, namely the parabolic and the non-parabolic modes.
Parabolic band structure For the concept of band structure, parabolic energy bands are generally assumed for simplicity. Electrons reside, at least when close to equilibrium, close to the minima of the E(k) relation. Then the E(k) relation can be extended in a Taylor series as
Because the first derivative vanishes at the band minimum, so the gradient of E(k) is zero at k = 0. Thus,
which yields the definition of the effective mass tensor
This expression is true for semiconductor which has isotropic effective mass, for instance GaAs. 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 describe longitudinal and transverse effective mass, respectively.
Non-parabolic band structure For higher applied fields, carriers reside above the minimum and the dispersion relation, E(k), does not satisfy the simple parabolic expression described above. This non-parabolicity is generally described by
where is a coefficient of non-parabolicity given by
where is the electron mass in vacuum, and is the energy gap.
Full band structure For many applications, non-parabolic band structure provides reasonable approximation. However, in case of very high field transport, which requires the better physical model of the full band structure. For full band approach, numerically generated table of E(k) is used. Full band approach for Monte Carlo simulation was first used by Karl Hess at the University of Illinois at Urbana-Champaign. This approach is based on empirical pseudopotential method suggested by Cohen and Bergstresser . Full band approach is computationally expensive, however, following the advancement of the computational power, it can be used as a more general approach.
Types of Monte Carlo simulation
One-particle Monte Carlo For this type of simulation, one carrier is injected and the motion is tracked in the domain, until it exits through contact. Another carrier is then injected and the process repeated to simulate an ensemble of trajectories. This approach is mostly useful to study bulk properties, like the steady state drift velocity as a function of field.
Ensemble Monte Carlo Instead of single carrier, a large ensemble of carriers is simulated at the same time. This procedure is obviously a good candidate for super-computation, since one may apply parallelization and vectorization. Also, it is now possible to perform ensemble averages directly. This approach is suitable for transient simulations.
Self-consistent ensemble Monte Carlo This method couples the ensemble Monte Carlo procedure to Poisson's equation, and is the most suitable for device simulation. Typically, Poisson's equation is solved at fixed intervals to update the internal field, to reflect the internal redistribution of charge, due to the movement of carriers.
Random flight selection The probability that the electron will suffer its next collision during dt around t is given by
where P[k(t)]dt is the probability that an electron in the state k suffers a collision during the time dt. Because of the complexity of the integral at the exponent, it is impractical to generate stochastic free flights with the distribution of the equation above. In order to overcome this difficulty, people use a fictitious “self-scattering” scheme. By doing this, the total scattering rate, including this self-scattering, is constant and equal to, say, . By random selection, if self-scattering is selected, k′ after the collision is the same as k and the carrier continues its flight without perturbation. Introducing a constant , the above equation reduces to
Random numbers r can be used very simply to generate stochastic free flights, which duration will then be given by . The computer time used for self-scattering is more than compensated for by the simplification of the calculation of the free-flight duration. To enhance the speed of free flight time calculation, several schemes such as “Constant Technique”, and “Piecewise Technique” are used to minimize the self-scattering events.
Scattering mechanisms
General background in solid-state physics 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. It is thus of great importance for a semiconductor device simulation to capture the physics of such mechanisms. The duration of the free flights is determined from the scattering rates. At the end of each flight, the appropriate scattering mechanism must be chosen in order to determine the final energy of the scattered carrier, or equivalently, its new momentum and scattering angle. In this sense, one will distinguish two broad types of scattering mechanisms which naturally derive form the classic kinetic theory of collision between two bodies:
Elastic scattering, where the energy of the particle is conserved after being scattered. Elastic scattering will hence only change the direction of the particle's momentum. Impurity scattering and surface scattering are, with a fair approximation, two good examples of elastic scattering processes.
Inelastic scattering, where energy is transferred between the scattered particle and the scattering center. 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, when running semiconductor Monte Carlo simulations, one has to deal mainly with the following types of scattering events: