I'm trying to understand cyclotron resonance measurements of electron effective mass in intrinsic silicon. I need to understand the theory used to make the computations of effective mass in non-parabolic bands.(adsbygoogle = window.adsbygoogle || []).push({});

A basic introduction to the cyclotron technique is here, but only for parabolic bands:

http://large.stanford.edu/courses/2007/ap273/hellstrom1/

In non-parabolic bands, carriers with increasing energy levels have increasing effective (classical) masses M; but I'm having a difficult time isolating exactly which carriers at what energies affect the cyclotron measurement of mass, and how.

I am using this theory (K*p theory): https://arxiv.org/pdf/1701.07067.pdf

See p.6 "At low temperatures the mass is usually measured at the Fermi energy within the band. As a consequence, one often plots the mass as a function of the electron density which determines the Fermi energy."

Fermi Energy (or Fermi Level?) has two meanings that I know of; it's the energy of the topmost electrons in a densely filled band (cold electron gas). It's also the energy that marks the 50% probability of being occupied. In an intrinsic semiconductor (which is what I'm studying for silicon), Fermi level would be very near the middle of the bandgap; There are no carriers at that energy level in an E-k diagram ... therefore I'm not sure what the author means by saying that the mass is 'usually' measured at that level of energy.

However, in many cyclotron experiments, the discussion follows a different tack:

A.K.Walton, 1970, p.1415, journal of solid state physics, talking about Stradling and Ukhov's 1966 Cyclotron experiment:

Which isn't very helpful, because I don't know what kind of "average" he is talking about and he doesn't cite a source that I have access to. Now the observed value of m_{T}is the average over the Boltzmann energy distribution which applies to these experiments. It's easily shown that average E_{T}= k_{B}T ...

A slightly less baffling comment is made in "Cyclotron resonance measurements of the non-parabolicity of the conduction bands in silicon and germanium", 1976, J.C.Ousset et al.

Both comments seem to mean that cyclotrons measure the mass of the particular carrier having an E=kT from the conduction band's bottom, although I don't know what's special about that energy level. When I use generic density of states calculations, I come up with an estimate of kT/2 being the energy level that is occupied by the most electrons ; I naively expected that the mass of the most common carrier would dominate the results; but it clearly doesn't and I don't know why. The cyclotron resonance absorption at temperatures where kT > ħω is composed of unresolved contributions from a considerable number of Landau levels above the band edge. The maximum contribution to the resonance line comes from transitions between Landau states with quantum numbers n ≈ kT/ħω for kT/ħω >> 1. Thus the peak absorption under these conditions should correspond to a an energy ~kT away from the bottom of the band.

Ousset used a 337μ exciting wavelength, so that's a frequency of 889.6 GHz, Therefore: ħω=hf ≅ 3.68 m_eV

kT=3.68meV corresponds to a temperature T=42.7 Kelvin. The data measures mass from T=40K to 240K, but n~=1 at 40 Kelvin and only reaches n~=5 at 240 Kelvin. Ousset doesn't even seem to care that n isn't >> 1 in much of his data.

How does anyone derive the result that peak absorption corresponds to the mass of a carrier with average thermal energy kT? What's the connection between Landau levels and a Fermi-Dirac (or Boltzmann approximation) carrier distribution?

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# I Cyclotron resonance and Landau Levels

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