Association

54 3. Experimental details
Fig. 3.19a,
I rl (c,d,λ)=e −
I sub (c,d,λ)=e −
∫
d d+c
λcap cosα
e −
d
∫ ∞
d+c
λcap cosα
e −
d+c
x
λ rl cosα
dz, and (3.13)
x
λ sub cosα
dz. (3.14)
In the next step, we express the fraction of spectral contribution of the reaction layer f rl with
respect to the bulk intensity. Here, the measured intensities I have to be normalized by λ and
the density n of contributing atoms in the particular layer,
I rl (c,d,λ) calc meas
= f
I sub (c,d,λ) rl = I rl
· λsubnsub
. (3.15)
I sub λ rl n rl
This equation (3.15) contains the thickness of the buried reaction layer c. Because it is not
analytically solvable, a Taylor expansion of c to the 2nd order is applied which lets c be
expressed as
λ
λ rl λ cap ∗λ sub cosα
e rl cosα
λ sub cosα √ λ rl λ cap ∗
c =
+
·
λ sub λ cap ∗ − 2λ sub λ rl − 2λ rl λ cap ∗ λ sub λ cap ∗ − 2λ sub λ rl − 2λ rl λ cap ∗
d
d
λ
√2f rl λ cap ∗λ sub e rl cosα λ
− 4f rl λ rl λ sub e rl cosα λ
− 4f rl λ rl λ cap ∗e rl cosα λ
− λ rl λ cap ∗e sub cosα
( )
d
d 2
.
λ
−e sub cosα λ
e rl cosα
d
d
d
(3.16)
In order to determine the thickness of possible reaction layers of EuO on a silicon surface
(SiO x or EuSi 2 ), this expression is applied to Si 1s or Si 2p HAXPES spectra of EuO/Si heterostructures
investigated in this work.
A consistent chemical characterization of the reaction layer at the functional EuO/Si interface
can be achieved by the additional evaluation of HAXPES spectra from the magnetic oxide
layer in direct contact with the Si substrate. Here, the photoelectrons from the bottom layer
in the EuO oxide (ox) layer carry information about the electronic structure of a possible
reaction layer (rl). Hence, the HAXPES intensities are modeled according to Fig. 3.19b as
I rl (a,b,c,λ)=e −
I ox (a,b,λ)=e −
∫
a+b a+b+c
λ cap ∗ cosα
e − x
a+b
∫
a a+b
λcap cosα
e − x
a
λ rl cosα
dz, and (3.17)
λox cosα
dz. (3.18)
The measured photoelectron spectra have to be normalized (similar to eq. (3.15)), and this
fraction is compared with the modeled fraction, so that I rl(a,b,c,λ) calc meas
= f
I ox (a,b,λ) rl = I rl
I · λoxn ox
ox λ rl n
can be
rl
solved analytically to extract the thickness c of the reaction layer as
⎛
⎞
c = −λ rl ln⎜⎝ f λ ox
rl e − aλox+aλcap+bλcap
λcapλox cosα
+ e − (a+b)(λ cap ∗ +λ rl)
λ rl λ cap ∗ cosα
λ
− f ox
λ rl e − a(λox+λcap)
λcapλox cosα
rl λ rl
⎟⎠ · cosα
(3.19)
+ aλ rl + bλ rl + aλ cap ∗ + bλ cap ∗
.
−λ cap ∗