| General | Compact Modeling |
Semiconductor Physics
Question 007-07: “What methodology can be used for quantum approach to transport in nano devices which utilizes the minimum computation time?”
Answer 007-07: The specific method chosen depends on the problem being addressed. Among the many methods for quantum transport, one method has been broadly accepted; the so-called non-equilibrium Green's function (NEGF) method. This method has a firm basis in theory and has been successfully applied to problems from quantum transport in molecules, carbon nanotubes, semiconductor nanowires, nanoscale MOSFETs, spintronic devices, and more. For an introduction to the approach, visit www.nanoHUB.org, search "Datta" and look for the series of four lectures, "Concepts in Quantum Transport."
For simple nanodevices such as carbon nanotubes, the NEGF approach is often quite efficient, but for nanoscale MOSFETs, for example, the computational burden can be very large. (The NEGF approach is equivalent to solving the Boltzmann Transport Equation with one additional dimension.) For nanoscale MOSFETs, the so-called "density-gradient" or "effective potential" approach is often used. This method can be implemented by an added term to the drift-diffusion equation, and it can be used to "include" quantum transport" in Monte Carlo simulation. It is much more efficient than NEGF simulation, but needs to be carefully used and benchmarked against more rigorous methods such as NEGF.
Question 013-07: Is there a multi-particle Monte-Carlo simulator available in the public domain to simulate noise in semiconductor devices?
Answer 013-07: There are many multi-particle Monte Carlo simulators that can treat noise. However, to the very best of my knowledge, no public domain Monte Carlo device simulator with noise analysis capabilities is currently available. In general, though, any MC simulator can treat noise. But, one must be careful to distinguish numerical noise, which varies as 1/sq-rt(N), from real noise. The complexity of the problem in terms of the microscopic model and the range of validity for the concept of noise to be applied only allows for the set up of Monte Carlo simulators specific to the problem at hand.
The number of research groups with an internationally recognized reputation on the subject is very few. Concerning Monte Carlo simulation of noise in devices, several papers have appeared in the literature: one of the most active research groups is the Electronics Group at the University of Salamanca, in Spain (http://www.usal.es/~gelec/ingles/webgrupo.htm).
Question 018-08: What is the best way to derive an expression for the average thermal velocity of an electron assuming Fermi-Dirac Statistics? Are there any literatures with the derivation?
Answer 018-08: On thermal velocity: First of all, it is important to recognize that there are different ways to define a "thermal" velocity and that it is important to understand which thermal velocity is appropriate for the problem being addressed. For example, for a non-degenerate semiconductor in equilibrium (in three dimensions) with parabolic energy bands, the average kinetic energy of electrons in the conduction band is (3/2)(kB)(T), where kB is the Boltzmann’s constant and T is the temperature in Kelvin, so that:
(1/2)(m*)
Accordingly, the "rms average thermal velocity" is:
v_thermal (rms) = sqrt(
On the other hand, we could ask another question: What is the average velocity of electrons directed in the +x direction for a non-degenerate semiconductor in equilibrium? (Note that the overall average velocity is zero, because we are in equilibrium, but we can compute the average velocity of those in the +x direction, which is equal and opposite to the average velocity of those in the -x direction). We call this "thermal velocity" the "unidirectional thermal velocity, v_T."
A derivation shows: v_T = sqrt[(2)(kB)(T)/(pi)(m*)].
This velocity appears in thermionic emission problems. In fact, for thermionic emission, people frequently talk about the "Richardson thermal velocity, v_R” and v_R = v_T/2.
In general, these results depend on whether the electrons are free to move in 1, 2, or 3 dimensions, and if the semiconductor is degenerate, they involve Fermi-Dirac integrals (different ones depending on the dimensionality).
The only discussion and derivation of these results that I am aware of is in Lecture 3 of Professor Mark Lundstrom's course "Electronic Transport in Semiconductors," which is available online at: http://cobweb.ecn.purdue.edu/~ee656/lectures.html.
Question 029-08: We know that the group quantum states with the same principal quantum number are at the same energy level (quantum configuration degeneracy). But, while explaining the formation of band gap, invariably all text books on semiconductors start with two different energy levels for 3s and 3p states of isolated (infinite inter-atomic distance) silicon atoms. Why they are at two different energy levels even though they have the same principal quantum number (n = 3)?
Answer 029-08: There is an excellent discussion of this question in Vol. III, Chapter 19 of the famous Feynman Lectures. For a hydrogen atom, the energies are completely determined by the principle quantum number. All n = 3 levels, e.g. 3s and 3p, have the same energy. For multi-electron atoms, however, the situation is different. Recall that s-states have a higher probability of being located near the nucleus, where they feel the attractive charge, which lowers their energy. On the other hand, p-states have a higher probability to be located further away from the nucleus. The 3s electrons screen some of the charge on the nucleus so that the 3p-states feel less of an attractive force and have higher energy.
