# Semiconductor physics, effective mass calculations, band edge. vs. DOS vs cyclotron resonance

I am trying to understand effective mass for electron carriers in the conduction band as found in experiments. I am using the equations in a simulator. The simulator models power mosfets from -85C to 150C. I would like to understand the models and measurements better.

In particular, I'm trying to get a very accurate estimate of the effective mass of electron carriers over temperature. I want to calculate the density of states mass (AKA carrier concentration mass) that is most accurate for a given intrinsic silicon temperature.

I have researched carefully. I discovered that the most precise data on electron mass comes from fitting cyclotron resonance experiments. That's why I want to use data from studies like this one which is based on J.C. Ousset et al's cyclotron data: Research gate, cyclotron resonance graphs from study

I am able to curve fit these graphs and produce simple equations that reproduce the graphs very accurately. I mean for the effective mass at the "band edge", and also for the "curvature" graph. See figures/graphs 1 and 3. These masses are interchangeable. The "band edge" is the same as "curvature" with E set to 0, while T is varied from 50 to 500 Kelvin.

I made a combined equation that can reproduce both graphs 1 and 3. There are two numerical fits to my equation. One fit is for transverse mass, the other for longitudinal mass. I obtained a very precise band-gap equation from other websites and am not using graph 2 of the research gate hosted article.

I want to produce graph 4 of the research Gate article from my curve fits of the other graphs, and the precise band-gap equation. I am having trouble.

I tried using an equation, given in undergraduate classes, that converts longitudinal and transverse masses into an isotropic equivalent mass.

$m_{iso}=( 6^2\cdot m_L\cdot m_t^2)^{ 1 \over 3 }$

This equation already takes into account 6 redundancies in silicon crystal structure. However, the results of the calculation are different than the DOS mass (AKA Carrier Concentration mass) that is plotted in the fourth graph of the Research Gate hosted article.

The fourth graph contains Barber's isotropic masses in the range of ~= 1.057 to 1.18 (from 0 to 300 Kelvin); Barber's data at temperature significantly above room temp might be non-cyclotron data (I can't check.)

However, when I plug the "band edge" mass values into the conversion equation, I get a mass that is ~= 1.062 to 1.07 (from 0 to 300 Kelvin) See my graph, below. The red line is my calculation, the grey is Barber's. My results don't aren't anywhere close to what the research gate author's got or Barber's results.

So, I assume "band edge" (A.K.A "curvature") masses measured by cyclotron resonance are different from "density of state" (AKA "carrier concentration") masses talked about in undergraduate courses. There must be a conversion between them. In the Research Gate hosted plots, the longitudinal and transverse mass in graphs 1 and 3 obviously have different values than the longitudinal and transverse masses in graph 4. Some kind of conversion was done.

I came across the following article: ARXIV article, review of relativity model for effective mass The study bolsters my suspicion that cyclotron data uses a slightly different definition of mass than undergraduate courses in semiconductor materials science. See section III of the article.

Is anyone familiar with these equations? Could you give me an idea of how to go about converting cyclotron "curvature" masses into "Carrier concentration" masses"? { EDIT: See comments after this post, and the physics forums links for more info. }

I assumed the most probable electron energy for a given temperature dominates the "curvature" mass that I want to convert to a "carrier concentration" mass.

I attempted to use Fermi-Dirac statistics and a generic density of states equation (parabolic approximation) to figure out what energy I need to plug into the curvature mass equation. I was able to figure out that the highest density of electrons is about kT/2 electron volts away from the band-gap edge. ( Boltzman k~=8.6e-5 eV/kelvin)

I can compute the "band edge"/"curvature" masses for that energy (green and yellow plots, below), but I don't get agreement with Barber.

Edit:Aug 10,2017 I have figured out that if I take the semi-relativistic dispersion equation from ARXIV ( eqn. (16) ), and take the second derivative of carrier energy (epsilon) with respect to quasi-momentum; I get a conversion formula: $M^{*} = m^{*} \cdot \left( { { 2 \epsilon } \over { E_g(T) } } \right)^2$

Where the capital $M$ is a scalar value which hopefully corresponds to $F=M \cdot a$ type mass. I converted the red plotted values into orange as a test of the conversion at epsilon=kT/2. The answer's correction is about half of what's needed to fit Barber's data.

• Richard Feynman discusses more and more accurate models of electron masses in his 3-volume "Feynman Lectures". Quantum is the 3rd volume, but even in volume one he presents models with resonances. – analogsystemsrf Jun 5 '17 at 4:13
• I didn't see anything helpful in volume I. Caltech has all volumes at http//www.feynmanlectures.caltech.edu. Which chapter do you think would help? In book 3, chapter 13 discusses effective mass for a wave packet; but that doesn't take into account relativistic limits on the speed of an electron in a crystal. It's just the same as classic undergraduate effective masses. There's non discussion of "non-parabolicity" of real E-k curves, that I can see or of cyclotrons. But the E-k curves for silicon come from cyclotrons... so there has to be a way to generate them. – Andrew of Scappoose Jun 5 '17 at 5:21
• Might I suggest migrating this to the physics stack exchange? While it's not off topic here, you may get a better response there. Let me know, or flag the question. – W5VO Aug 10 '17 at 11:58
• I don't have the physics background to do the tensor math that is expected in a physics forum; that's one of the reasons I asked the question here. The other is that, I believe, undergraduate engineers will look for the answer to this question here where it's easy to find. Physicists tend to digress or vaguely answer questions; eg: I asked on a forum; physicsforums.com/threads/… If the thread is moved, is there a way to keep a place holder so links don't get broken and people can find it in the future? – Andrew of Scappoose Aug 10 '17 at 20:13
• When I read Ousset's original article, he makes the comment that cyclotron's peak absorption is at around kT ; if I plug that into the equation instead of kT/2, my data matches Barber's even better than the research gate article. I'm seeing if I can understand Ousset's comments, here: physicsforums.com/threads/… – Andrew of Scappoose Aug 21 '17 at 21:06