A New Spin on Photoemission Spectroscopy Page: 62 of 259
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2-dimensional systems such as surface states or quasi-2D materials, this gives a complete
determination of the (spin-integrated) electronic states. This makes such 2-dimensional
systems ideal for ARPES experiments.
Full 3-dimensional systems are more complicated, in part because of the loss of kz
conservation through the surface in step 3. Unfortunately, this means that kz cannot be
directly extracted from the measurements of EK, 0, and ,3 alone, and proper determination
requires complete knowledge of the exact "final states" (the unoccupied bandstructure above
EF) the electrons are photoexcited to before exiting the sample. Often times, the final state
dispersions are approximated as parabolic free-electrons. This technique becomes more
appropriate at high kinetic energies (thus high photon energies) where effects of the crystal
potential are weaker. In the free-electron approximation, the states are assumed to be free
but in an extra potential, Vi, called the inner potential which can be thought of as the
potential step the electrons must pass through to exit the sample. Thus, going back the
first step we have
E f = 2me (kf + kf + k f ) - VTin. (2.9)
Solving for kz,f and using equations 2.3 and 2.4, the initial momentum along z is
kzj = 2m, (hi, - ID - EBI - In) - k i - Iy i- (2.10)
Experimentally, Vin can be determined by looking at normal emission (0 = 1 = 0 = kxyz)
and varying the photon energy, hw. in can then be solved for by fitting the periodicity to
the known Brillouin zone size.
The kz component causes other complications for ARPES experiments. As mentioned
earlier, the final-state electrons have short inelastic mean free paths as they travel through
the solid. This concept can also be described by a finite photoelectron lifetime. A way
to express this lifetime is through an imaginary part to the momentum, implying a finite
broadening to kf. Again, due to the surface discontinuity in the z-direction, only kz can
have an imaginary component, giving a broadening of
Ak = 1
Skz, f = (2.11)
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Jozwiak, Chris. A New Spin on Photoemission Spectroscopy, thesis or dissertation, December 1, 2008; United States. (https://digital.library.unt.edu/ark:/67531/metadc1014237/m1/62/: accessed April 23, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.