Can you guess what this first giff shows?
(ps. this post contained a number of errors originally. i edited it, hopefully fixing them, on Feb 26.)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUMrSqFV0i0MsGvZrlLpfdF3RbNd7-KlhPAUBAhjhhhnklfWdtKRVs-FkjAUTKaIDh9Qcg4S7HsgyEBJgAjr3OtuAqUzgkz7fGVH6jNnO5_zqY1oFhiF2DgbD4rceHh8VmPphoSq9IbUI/s320/PlaneWave.gif)
A lot of our discussion of electron movement in semiconductors is based on a belief in electron states that are somewhat localized (maybe within 20 lattice sites or therabouts), which maintain their integrity for some bit of time, and can move in a crystal. This belief is based in wave-packet theory. Here is an outline of how one might make some single-electron wave-packet states in free space or in a crystal:
Free space:
In free space we would use the spatial energy eigenstates, $e^{ikx}$ to make the wave packet. (One always has to use the energy eigenstates. In a crystal they are more complex.) In free space they are all of the form $e^{ikx}$ and have energies of $E(k) = \hbar^2 k^2/(2m)$.
A typical wave-packet state is a mixed state made of lots of these energy eigenstates, each with its appropriate time dependence. For example.
$\Psi(x,t) = \int g(k) e^{-ikx} e^{-i \omega(k) t} dk$
where
$g(k) = \sqrt{30/\pi} e^{-30^2 (k-\pi/20)^2}$.
The E vs k relationship:
$E(k) = \hbar^2 k^2/(2m)$,
leads to something like:
$\omega (k) = 1972 (3 \times 10^{18}) k^2/(1.022 \times 10^6)$
where k is in inverse angstroms (and c in there is in angstroms/second.)
For an electron in a crystal the E vs k relationship is different:
$\omega (k) = (E_o - (B/2) cos (kb))/\hbar$
$= (-10/6.6-(2/6.6) cos(k)) \times 10^{16}$
for a crystal with b = 1 Angstrom, B/2 = 2 eV and with $\hbar$ in eV-sec.
(With k in inverse angstroms.)
(Hmmm. It might be better to program with hbar in eV-picoseconds and then have t in picoseconds in the equation for Psi. (Smaller exponents))
Also, the energy eigenstates would be different for an electron in a crystal. (They would be the Bloch states created by a sum over all the lattice sites that we talk about a lot.) I think that at low k, near the bottom of the conduction band, these long wave-length states are pretty similar to free electron states. A bit surprising, but i think it is true.
Anyway, I think what either of those two possibilities will give you is a one-electron wave-packet state that starts out about 10 or 20 lattice spacings (Angstroms) wide and moves to the right as a function of time. The reason I think it moves to the right is that the k values that are used center around k=pi/20, which is positive. I think the packet profile, g(k), is pretty narrow in k; that was my intention.
One could start by just integrating (over k) at t=0 to investigate the nature of $\Psi (x,0)$. The integral is from k = -infinity to infinity.
And then try a small finite t and integrate (k) again to see the difference.
Here is an intriguing result from using a pretty narrow (in k) gaussian "packet" of crystal states centered at k=pi/10.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhS2yhXVynptr67Iy7BeZpYqvvGVs6fCMfJeUCYjRDsliTwlgIwaaDJjGmfl548Mrn7AED6H9mdjlmPWRk1ewteGW6KyNCEX1w_NtVOyHBPhgrPcxlsucLnE82cLpAv_85E6e0Uhwi7d30/s320/CrystalWavepacketHP2.gif)
Where do the coefficients in g(k) and w(k) come from? What are the units of g(k) and what is w(k) telling us here? The units for w(k) is Hz^2?
ReplyDeletew(k) is E/hbar. So that is that frequency (sec-1) that goes in the time dependence of the energy eigenstate $e^{ikx}$. Does that make sense?
DeleteIn the equations above it has been worked out so that with k in inverse angstroms, w(k) will be in sec-1. (A lot of careful (hopefully) steps were left out. One has to be careful especially with the management and control of the units when putting in numbers. There are really two calculations in parallel. One for the number; the other keeping track of units.
Deleteg(k) is a new thing, although you may have seen it before if you did a spreading free-electron wave packet. It determines the distribution of momentum eigenstates that are "in" the wave packet.
Delete