A particle kinetic energy, as everyone knows, is given by one-half the product of its mass by the square of its velocity. This is also true for an electron in a crystal. However, the presence of the ions, which constitute the crystal, complicates tremendously the motion of the electrons, since they are subject to the pushes and pulls of each ion. Solid state physics has taught us that we can "ignore" the presence of the ions (forgetting for now their thermal agitation) provided we change the relation between the kinetic energy and the velocity of the electrons. In particular, for electrons having a kinetic energy of the same magnitude of their thermal agitation, the energy is still given by "one-half the product of its mass by the square of its velocity", but now the "mass" is different, as required by the action of the ions. This new mass is called "effective mass". As the electron energy increases, the nice "parabolic relation" connecting the energy to the square of the velocity is lost and full quantum mechanical calculation of this energy-velocity relationship must be performed. This relationship is called the "band structure" of the crystal. Instead of velocity, physicists use "momentum" (the product of velocity times mass, provided a mass can be defined!) or a quantity called "wavevector" (denoted by k and denoting the inverse of the wavelength associated with a particle of a particular momentum). In the figure below one can see the energy(E)-wavevector(k) relation for Si. The wavevector runs along some particular symmetry directions in the unit cell of the Si (cubic) lattice, directions labeled by capital Greek and Roman letters. The 'bottom' of the conduction band, where "thermal", "cold" electrons reside, is close to the symmetry point labeled X. The zero of the energy - an arbitrary quantity - has been made to coincide with the top of the valence band, at the symmetry point labeled Gamma. This band-structure has been computed, to use some technical jargon, using nonlocal empirical pseudopotentials with spin-orbit interaction.

Additional numerical details about the calculation, tabulation, interpolation and use of the empirical-pseudopotential band-structure are given elsewhere in these pages.

The band-structure enters not only the "kinematics" (i.e., how carriers move freely under the action of the electric field, but without scattering), but also the "dynamics", i.e., how often carriers collide (see details on collision processes from the the Computational Electronics Group, University of Illinois, Urbana-Champaign), what is their new energy and velocity after a collision. This has to do with "density of states": It is difficult to parallel-park a car, since one has to place the car in a particular "state": Zero velocity and in a well defined position. Using the jargon of physicists, this is a "low density of states" situation. It is easier to drive the car out, since there is more tolerance on both the velocity and the position which the car can have while exiting the parking place. This is a "high density of states" situation. Thus, the "parking rate" is low when the density of final states is small, i.e., when parking (it takes longer to hit the perfect spot rather than the curb), yet high when driving out, i.e., when the density of final states is large. Thus one sees how the density of final states controls the frequency at which electrons undergo collisions of various types, and so, ultimately, how fast they move.

The figure below shows the "density of states" in Si (top) and the frequency of the collisions with the thermal vibrations of the Si ions (the electron/hole-phonon scattering rate, bottom): We see how they track each other. In the figure energies are negative, by convention, for holes (carriers in the valence bands) and positive (and shifted by the band-gap, 1.17 eV, from the zero of the energy scale) for electrons. At the temperature of liquid nitrogen (77 K) scattering with the lattice vibrations is less frequent than at room temperature (300 K), because the Si ions are subject to a smaller amount of thermal agitation and perturb less the motion of the carriers. At lower temperatures, electrons and holes move faster, because they scatter less frequently. Additional details about the physics and numerics concerning the calculation of the scattering rates of electrons in Si employed in DAMOCLES are available elsewhere in these pages and an overview of the main scattering mechanisms in semiconductors is available from the Computational Electronics Group at the University of Illinois at Urbana-Champaign).

Most Monte Carlo programs used in the past employed the "effective mass" approximation, since practical considerations (small electronic energies in large devices) did not require anything fancier. Scattering rates were also computed using the "parabolic band approximation". But the higher energies reached by carriers in small devices now require the use of the "correct" energy-velocity relationship, bypassing the simple square-law (i.e., parabolic) approximation. First to find the courage to use the full band-structure in MC simulation was, in the early 1980s, the group headed by Karl Hess at the Beckman Institute, University of Illinois at Urbana-Champaign. DAMOCLES was the first example of a MC program employing the full-band structure to treat also the carrier dynamics (i.e., the scattering rates) and to make use of these advances in the context of "real" devices, with electric field and dopant densities varying in space.