1-D, free particle 2-D, free particle 3-D, free particle
i~ddtΨ=−~
2
2 m∆Ψ i~dΨ
dt =−~^2
2 m∆Ψ i~dΨ
dt =−~^2
2 m∆Ψ
Eigenfunct. sol. Akei(kx−wt) Akei(k·x−wt) Akei(k·x−wt)
Disp. relation E=~ω=~(^2) k 2
2 m E=~ω=
~^2 k^2
2 m E=~ω=
~^2 k^2
2 m
DOSD(k) D(k) =^2 π D(k) =kπ D(k) =k
2
π^2
D(E) =D(k)dEdk D(E) =π^1 ~
√
2 m
E =
n
2 √EFE D(E) =
m
π~^2 =
n
EF D(E) =
(2m)^32
2 π^2 ~^3
√
E=^3 n
2 E(^32)
F
√
E
Fermi energy EF=π(^2) ~ (^2) n 2
8 m EF=
π~^2 n
m EF=
~^2
2 m(3π
(^2) n)^23
Chem. pot. ... ... ...
Table 1: Calculated thermodynamic quantities for free electrons
7.1.4 ARPES
The ARPES (angle resolved photo emission spectroscopy) is an experimental method to get the
dispersion relationship, which is needed to start the calculations. Here a photon (x-rays (XPS) or
ultraviolet light (UPS)) comes in, hits the sample and shoots out an electron. An electron analyser
finds out the energy of the electron. This is done just for a specific angle (angle resolved). The k-vector
of the incoming photon and the k-vector of the outgoing electron are known, also the energy of the
electron. From the variation of the energy of the incoming photon the dispersion relationship can be
found. The idea is that you have a density of states and the incoming photons shift the whole thing
up to free electrons. In the following figures 22 and 23 the principle of the arpes is shown.
Figure 22: ARPES(Angle resolved photo emission spectroscopy)