Protocol for drain current measurements
1
Basics: the naive model of drain current
A beam of intensity I0 (number of photons per unit area) and cross section
S impinges with a grazing angle Φ on a sample S. Neglecting all the effects of
reflection/refraction/multiple interference, and being Λ the electron free collision
path, the excited volume contributing to the drain current can be written as
S
(0.1)
,
V
sin 
which gives, within a linear approximation (i.e.  dI  I dz ), for the drain
current
S
C  I0V C ,
(0.2)
YD  I0
sin 
where  is the optical absorption coefficient and C is the function
converting the absorbed photon into the electrons contributing to drain current.
In the case that the reflectivity R of the sample is not negligible, the drain
current is given by
(0.3)
YD  I0 1  R V C .
The reflectivity of the sample depends on the degree of linear polarization
S12  S2 2
of the incoming light, and is given by
P
S0
 1 P 
 1 P 
(0.4)
R0  R0S 
 R0P 

,
 2 
 2 
where the reflectivities S and P are related to the real and imaginary part
of the refraction index n  n  ik by
R0S 
2
sin   n  cos2 
2
(0.5)
2
sin   n  cos2 
and
R0P 
2
2
n  cos2   n sin 
2
.
(0.6)
2
2
2


n  cos   n sin 
Being that the extinction coefficient k is related to the absorption
coefficient by
4 k
,
(0.7)


and the refraction index (real part) n is related to the extinction coefficient
k by the KK relations, the determination of the absorption coefficient by a drain
current measurement becomes more complicated. As can be seen from Figure
1.1, at the carbon K edge, at around 284 eV, the reflectivity of an ideal Ag mirror
(with negligible roughness) is below 1% and can be neglected for angles greater
12 degrees. For angles below 12 degrees, the measurement of reflectivity of the
sample should be done.
Figure 1.1: Calculated reflectivity of an ideal Ag mirror at photon energy = 284 eV
as a function of the grazing angle.
Let us suppose the case of an experiment whose goal is the measure of
the optical absorption coefficient αa of an overlayer, with electron free collision
path Λ, deposited onto a semi-infinite substrate of optical absorption coefficient
αs , with electron free collision path Λs .
Note: the substrate absorption coefficient is supposed to be flat, i. e.
 S ( )  cos t . Otherwise or a numerical correction (f.i. by Henke table,
tabulated cross section) or a third calibration measurement is needed to
correct for the shape of the substrate absorption coefficient.
The measurement proceeds in two steps.
First step: measurement of substrate drain current, YDO, before depositing the
overlayer
(the
incoming
flux
is
simultaneously
monitored
for
normalization/comparison purposes by measuring the drain current, WDO, of a
tungsten mesh. See also below).
In summary the result of the measurement can be written
S
(0.8)
 S S   CS ; I0  , t   I0 ( )f  t 
sin 
with a mesh drain current WDO ( , t ) , where t is the instant at which the
measurement is taken.
YDO  , t   I0 , t 1  R 0 
Let us introduce also the quantity YDO  , t  
YDO  , t 
WDO  , t 
of use in the data
reduction (note that YDO ( , t ) being the ratio of two currents, is an a-dimensional
quantity).
Moreover by knowing  S at one energy point (e.g. from Henke tables) and
measuring independently (e.g. by a calibrated photodiode I 0 ( , t ) ) the values of
S
S
 S S ( )CS can be
 SCS in the YDO (, t )  I0 ( , t )
the quantity
sin 
sin 
determined.
Second step: it consists in the measurement of the drain current YDA, in presence
of the overlayer of thickness d (the incoming flux is simultaneously monitored for
normalization/comparison purposes by measuring the drain current, WDA, of a
tungsten mesh).
Two cases must be considered
a) overlayer thickness d  
The drain current is given, neglecting reflectance, by
S
S
(  s  d ) sCs ,
(0.9)
YD  , t '   I0  , t ' 
d aCa  I0  , t ' 
sin 
sin 
with a mesh drain current WD ,t  , where the optical absorption in the
overlayer can be neglected disregarding the absorption of I0 in the adsorbate
layer.
Dividing by the drain current measured in the step 1 (absence of
overlayer) and utilizing the mesh currents, WDO  , t  and WD  , t ' , for relative
normalization one obtains
I0 ( , t ' ) S
I0 ( , t ' ) S

(  s  d ) sCs

d
C
a a
YD ( , t ' ) WD ( , t ' ) sin 
WD (, t ' ) sin 


I0 ( , t )
S
YDO ( , t ' )
 S S ( )CS
WDO ( , t ) sin 
I0 ( , t ' )
I0 ( , t ' )

(   d ) sCs

d
C
WD ( , t ' ) a a WD ( , t ' ) s

I0 ( , t )
 S S ( )CS
WDO ( , t )
(0.10)
I 0 ( , t )  I 0 ( ) f (t )
, where I 0 ( ) is the photon
I 0 ( , t ' )  I 0 ( ) f (t ' )
distribution due to source plus transport optics and f(t) is the function taking into
account the time dependence of such distribution (e.g. due to time decay,
fluctuations etc.), the above expression becomes
I0 ( )f (t ' )
I0 ( )f (t ' )

d
C
(   d ) sCs

a a
I0 ( )f (t ')CW s
YD ( , t ' ) I0 ( )f (t ' )CW


I0 ( )f (t )
YDO (, t ' )
 S S ( )CS
I0 ( )f (t )CW
Introducing the relations
1
1
(  s  d ) sCs
d aCa 
d a ( )Ca  (  s  d ) s ( )Cs
CW
CW



1
 S S ( )CS
 S S ( )CS
CW
 a
(0.11)
 d
 d
dCa
d
 S
 S
 a
 S S ( )
S
 S S ( )CS
S
d
This expression shows that, through the known coefficient
, the
 S S ( )
ratio of the two measured drain currents is proportional to the absorption
coefficient of the overlayer plus a constant term, also in principle known.
In the case that reflectivity of the substrate is not negligible, the reflectivity
of the sample changes with the addition of adsorbate. Then, the ratio between
the drain current in step 2 and 1 has to take into account the reflectivity
measured only on the substrate R0 and on substrate + molecule R (is shown in
Fig. 3 the difference between them).
dCa 1  R 
 S  d 1  R 
YD (, t ' )
(0.12)



a
YDO (, t ' )
 S S ( )CS 1  R0 
 S 1  R0 
The reflectivity of the substrate + adsorbate depends on the substrate and
overlayer optical constants. The reflectivity of the system can be calculated by
 1 P 
 1 P 
the equation R  R S 
 RP 

 , where the reflectivity S and P are given
 2 
 2 
by the transmittance and reflectance Fresnel coefficients related to the interface
vacuum-adsorbate and adsorbate-substrate.
 R    r
S ,P
s,p
Va
   

  
2
s,p
s,p
s ,p
 2  2

r aS  tVa  r Va  exp  id
n a  sin2  
 
 .

s,p
s,p
2
2



1  r Va  r aS  exp  id
n a  sin2  



(0.13)
The variation of calculated reflectivity of the sample as the thickness is
shown in Figure 1.2 and Figure 1.3 in the case of an Ag sample + a C22H14 layer
with increasing thickness.
Figure 1.2: Calculated reflectivity of on an ideal Ag mirror + a C22H14 layer with
different thickness as a function of the grazing angle at photon energy = 284 eV.
The calculation was done at the carbon K edge at 284 eV and taking as
parameter P=0. The figures show that at grazing angles, where the substrate
reflectivity is not negligible, the variation of reflectivity changes significantly (also
with interference effects).
Figure 1.3: The same as Fig. 2 but for low grazing incidence angles.
b) overlayer thickness d  
At these coverages, the contribution of the substrate can be considered
S  d
vanishes and the drain current ratio is
negligible, the constant term
S
proportional to the overlayer absorption coefficient
I0 ( )f (t ' )
1
 aCa
 aCa
'
'
 aCa
I0 ( )f (t )CW
YD ( , t )
CW
. (0.14)



'
I0 ( )f (t )
1

YDO ( , t )

(

)
C
S
S
S
S S ( )CS
  ( )CS
CW
I0 ( )f (t )CW S S
Taking into account reflectivity, this ratio becomes
 aCa 1  R 
YD ( , t ' )
.
(0.15)

'
YDO ( , t )  S S ( )CS 1  R0 
Experimental: error evaluation
Let’s evaluate the error present in the determination of the ratio between
the drain current from the sample ID and the current from the monitor (mesh) WD.
The error of this measure is given by the usual error propagation, where  I2D and
 W2 are shown in Figure 1.4 and Figure 1.5:
D
2
 f  2  f  2
 f  
  I D  
  WD .
I
W




D 
 D

Figure 1.4: Noise in drain current measurements.
2
(0.16)
Figure 1.5: Noise in mesh current measurements.
Usually the signal of the monitor is lower then that of the sample and we
can assume
(0.17)
I D  WD  
and the ratio is given by
I
W 
.
(0.18)
f  D  D
WD
WD
Then, the error in the ratio between drain and monitor is given by
2
2
 1  2 1  WD    2
f  
  ID  
  WD 
4  WD 
 WD 
.
(0.19)
2
  2
1
1
 I2D  1 
  WD
WD
4  WD 
If the same instrument (or two equal instruments) is used for the measure
of ID and of WD, then we can assume that the errors in the two measures are the
same:

f 
 inst
WD
2
1
 
1  1 
 ,
4  WD 
(0.20)
being σinst the error of the instrument. Equations (0.16) and (0.20) show
that the error is minimum when the difference in magnitude of the two signals is
  0 , and it is

5
 f  inst
(0.21)
WD 4
2
Data analysis: smoothing with splines and energy shift
The data reduction shows little energy shifts due or to the grating position
or to source movements. Then, the spectra analysis requires before a smoothing
and then, often, a manual shift that permits to overlap them correctly. Energy
shifts generally can be related to the uncertainty in the reading of the position of
the grating axis. In the case of the carbon K edge, at 284 eV, for example, the
encoder resolution corresponds to about 6 meV. Another possible cause of
energy shift between spectra could be the movement of the light source The
formula that relates the variation ΔS to the energy shift ΔE is
E 2 cos  S
.
(0.22)
E 
2 c N qP 1  EXP
The algorithm for the smoothing with splines makes the minimization of the
quantity
xn
  g   x  
2
dx ,
(0.23)
x0
among all the functions g  x  for which
 g  xi   y i 
(0.24)

 S,

i
i 0 

is the value of the smoothing spline curve at xi , y i is the
n
where g  xi 
2
measured value at xi ,  i is the standard deviation, and S is the smoothing
factor. The number of point n for the smoothed signal can be chosen arbitrary.
The σ’s are essentially the instrumental errors of the ammeters. The best
estimates we have for them are:
  1013 A
for Keithley 1 for drain current
  10 14 A
for Keithley 2 for mesh current
The smoothing factor is S  1.0 . One starts interpolating the data using
these values producing a curve with about 200 points. Then the smoothing can
be refined changing S and/or the number of points for the output array. The
results of this smoothing procedure on the signal from the mesh is shown in
Figure 2.1 and in Figure 2.2..
Figure 2.1: smoothing of the mesh current data for scan #41.
Figure 2.2: zoom of the Figure 2.1.
In order to take into account the possible photon energy shift between the
two scans, the signals from the meshes, smoothed with the described procedure,
have been compared.
Figure 2.3: comparison of the mesh signals.
The minima of the curves, at energies around 285.3 eV, have been fitted
with two Gaussians (Figure 2.4): the difference of their positions give the energy
shift of the photon, since we can consider that the mesh is passivated and is
stable in time.
Figure 2.4: gaussian fit of the minima of the mesh signals.
The values for the centers of the two Gaussians are:
Emin  285.381 eV
.
Emin  285.355 eV
The difference is ΔE = 0.026 eV. An estimation of the same order of
magnitude is obtained starting the analysis from the signal of the BPM.
Figure 2.5: variation of the S value, as consequence of a movement of the beam
at the entrance of the beamline, between the two scans.
In the interval about 285 eV (from 284 and 286 eV), the mean values of S
are 0.0713 and 0.0270. Therefore ΔS = 0.0443. and then results ΔE = -0.0357,
close to the value obtained after the spline procedure.
3
Angular dependence of the intensity of an emitting dipole
In the reference frame of the laboratory the electric field of the radiation
has the following form:
 0 


i   t  k r  

(0.25)
E L  EH e
 1 .
i

e 


In the reference system of the sample, the components of the electric field
can be obtained using rotations around X axis (+ψC) and around the Y axis (-θM)
(in this order):
0
0  0 
 cos M 0 sin M  1


i  t  k  r  



ES EH e
1
0  0 cos C sin C  1  
 0
  sin
 i 
0 cos M 
M

 0  sin C cos C   e 
, (0.26)
 sin M  e i cos C  sin C 
 

i  t  k  r 


 EH e
cos C   e i sin C


 cos M  e i cos C  sin C 


 
where  M and  C are both positive quantities. Note that the term k  r is
invariant under rotations since both vectors rotate in the same way.
Let us suppose we want to measure a physical quantity of the sample

represented by a vector p , which can be, for example, the direction of a bonding.
What we really measure is some signal proportional to
  2
f Ep i
.
(0.27)
I
2
E
 2
    * 
Therefore we must evaluate E  p  E  p E  p . p can be expressed, in the







reference system of the sample, in polar coordinates as:
 sin cos  



p  p  sin sin  
 cos 




 
i   t  k r 
a  sin M sin cos   cos M cos   b sin sin   , (0.28)
E  p  pEH e
where
a   e i cos C  sin C
.
(0.29)
b  cos C   e i sin C
It results

  *
 i   t  k r 
a *  sin M sin cos   cos M cos   b * sin sin   (0.30)
E  p  pEH e
The product of (0.28) and (0.30) gives



 2
2
E  p  p 2EH2 aa *  sin M sin cos   cos M cos   bb * sin2  sin2  

Let’s calculate separately the terms aa* , bb* , and ab* (note that ba*


*
* *
aa *   2 cos2  C  sin2  C   cos C sin C e i  e  i 
  cos  C  sin  C  2  cos C sin C cos 
2
2
(0.32)
2


bb *  cos2  C   2 sin2  C   cos C sin C e i  e  i 
(0.33)
 cos2  C   2 sin2  C  2  cos C sin C cos 



 1   e
ab *  cos C sin C  2  1   e i cos2  C   e  i sin2  C
ba *  cos C sin C
2
 i
(0.31)

  ab  ).
  sin M sin cos   cos M cos  sin sin  ab  ba
*
(0.34)
cos2  C   e i sin2  C
(0.35)
Combining (0.34) and (0.35) we obtain
 ab*  ba*   2 cos C sin C  2  1   cos   cos2  C  sin2  C  .


(0.36)
Finally we have:
 2
E  p  p 2EH2 

  sin sin cos  cos
  sin   2 cos sin cos   sin  sin  
sin    1   cos   cos   sin    
  2 cos2  C  sin2  C  2 cos C sin C cos 

 cos2  C
2 cos C
2
M
2
2
C
C
2
2
(0.37)
C
2
C
cos  
2
M
2
C
C
  sin M sin cos   cos M cos  sin sin  
Let us examine some limit cases:
Pure circular polarization
In this case   1 and   2 . The expression (0.37) reduces to:
 2
2
E  p  p 2EH2  sin M sin cos   cos M cos   sin2  sin2  ,


(0.38)
i.e. the “signal” is independent from the position of the chamber.
Pure linear polarization
For linear polarization is   0 . The (0.37) becomes
 2
2
E  p  p 2EH2 sin2  C  sin M sin cos   cos M cos   cos2  C sin2  sin2  

2cos C sin C  sin M sin cos   cos M cos  sin sin  
 2
2
E  p  p 2EH2 sin C  sin M sin cos   cos M cos   cos C sin sin   (0.39)
Horizontal chamber
It is  C  0 . Then

 2
2
E  p  p 2EH2  2  sin M sin cos   cos M cos   sin2  sin2  
2 sin sin  cos   sin M sin cos   cos M cos 
(0.40)
Vertical chamber
It is  C 

2
. Then
 2
E  p  p 2EH2
 sin
sin cos   cos M cos    2 sin2  sin2  
2
M
2 sin sin  cos   sin M sin cos   cos M cos 
(0.41)
 2
In the following figure is shown the aspect of E  p as function of  C for
four different values of ε.
Figure 3.1: calculated dipole intensity as a function of  c for   90 .
If during the measurements I use only the central part of the beam, on the
sample arrives an equal amount of light where  

2
  and light with

  , indicating with  the changing in the phase induced by the optics of
2
the beamline. Therefore the signal is composed by
I I   I  
 
  
  
2
Since
2


cos       sin 
2

,
  
cos       sin 
 2

it turns out that
I I

  
2
I

  
2
2I


(0.42)
2
Figure 3.2: calculated intensity as a function Figure 3.3: calculated intensity as a function
of  c for   110 .
of  c for   70
Figure 3.4: Sum of the intensities calculated for the two cases   70 and
  110 .