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28 2. Theoretical background
III. The inelastic mean free path A critical parameter for the probing depth in any photoemission
experiment is the inelastic mean-free path (IMFP) λ of hot electrons in a solid.
Calculated curves of λ versus E kin are depicted in Fig. 2.18. The similarity in the slope for
many elemental solids justifies the use of the term “universal” curve. We recognize that with
energies of 10 keV IMFPs of up to 200 Å can be reached.
Figure 2.18.: Universal curve of photoelectron
emission in solids. The inelastic
mean free path (IMFP) λ is the key parameter
to describe inelastic scattering
of photoelectrons in solids.
Extending the escape depth of the
photoelectrons up to several tens of
nanometers (with λ ∝ Ekin 0.78 ), 85 the
HAXPES regime is perfectly suited to
investigate the electronic properties of
buried interfaces or bulk properties.
This recent calculation of IMFPs is
from Tanuma et al. (2011). 84
It is evident that, in reality, the IMFP curves can be very different depending on the physical
properties of an individual material. Density, dielectric function and band gap of the solid as
well as the elastic scattering can also alter the trajectory of a photoexcited electron. If these
effects are included into λ, one obtains the effective attenuation length (EAL). One can then
express a mean escape depth down to which unscattered photoelectrons can be expected as
λ ∗ = λ EAL (E kin )· cosα, (2.26)
where λ EAL denotes the experimentally determined effective attenuation length, 80 and α the
off-normal exit angle of the photoelectron.
As evident from eq. (2.26), the bulk-vs-surface sensitivity of the photoemission measurement
can be controlled the kinetic energy or by varying the photoelectron’s off-normal exit angle
α. The behavior is determined by cosine, which gives e. g. a halving of the mean escape
depth at λ ∗ (60 ◦ )=1/2λ ∗ (0 ◦ ). This phenomenon is extremely useful, since simply by tilting
the sample one can obtain depth-resolved information, 86 which we apply for chemical depth
profiles of EuO/Si in Ch. 5. Most intuitive for depth-selective measurements, however, is
the information depth ID, from which 95% of the detected photoelectrons originate. This
definition coincides, in practice, with the detection limit for chemical species in buried layers
(see Ch. 5.1), and equals approximately ID 95% ≈ 3λ ∗ .
Concluding, hard X-ray excitation offers the advantage of a largely tunable probing depth up
to tens of nanometers, thus allowing to investigate buried layers and functional interfaces.
Minor drawbacks are reduced photoemission cross-sections and Debye-Waller factors. Those,
however, can be compensated by nowadays highly brilliant hard X-ray radiation sources, as
introduced in Ch. 3.4.3.