Supplemental Information A: Apparatus Function of a Time-of-Flight
Detector
In a typical TOF spectrometer for time and angle resolved TPPE, a detector is
positioned at a distance R from the sample and the direction to the center of the detector
forms an angle 0 with the surface normal (Fig. A1). The goal of this Supplemental
Information is to relate the apparatus function of the detector, W(t), to R and 0. The
apparatus function of the detector gives the detector response in a (rather unphysical)1
case when the electronic states at the sample surface are “instantaneously” ionized at time
t = 0 forming an infinitely thin “expanding sphere” of electrons which reaches the
detector at some time later.
Dividing the whole surface area of the detector by small elements Si, the
response of the detector as a function of time can be written as
W t , 0    wi t  ,
(A1)
i
where wi(t) is the response of the i-th bin. For sufficiently small bins, wi(t) is
wi t    d N i  t  t i  ,
(A2)
where d is the detector sensitivity, Ni is the number of electrons arriving to Si, and (t
– ti) is a delta function peaked at the time of electron arrival to the i-th element given by
t i  me R p i ,
(A3)
where pi is the momentum of electrons detected by Si, and the length of the flight path
to any bin is taken to be the same. (Indeed, for our A  B = 17.8  7.1 mm detector and R
= 135 mm flight path the difference in the flight time to different parts of the detector is
only 0.25%, which is below the experimental resolution.)
The strategy is as follows: Ni and ti will be related to the angles of the
photoemission (in the spherical coordinates), after which Eqs. (A3) and (A4) can be
substituted into Eq. (A1) and the sum in Eq. (A1) can be replaced by an integral over the
detector. It is useful to introduce the following dimensionless quantities:
and
  K K0 ,
(A4.1)
  t t0 ,
(A4.2)
  m * me ,
(A4.3)
K  p 2 2me ,
(A4.4)
where
is the photoemitted electron kinetic energy, which can be a function of , K0 ≡ K( = 0) is
the kinetic energy of an electron photoemitted along the surface normal and
t0 
me R 2
2E 0
(A4.5)
is the time it takes for an electron to reach the detector along the surface normal. In order
to derive an explicit form of W(t), two cases must be considered separately:
Photoemission from delocalized interfacial states
We will restrict the discussion to interfacial states which have a good quantum
number p|| on the time scales relevant to photoionization and assume cylindrical
symmetry of the 2D band structure. As it was discussed before, p|| must be conserved
leading to the following set of conditions:
Ki 
Kf 
p||2
2m *
,
p||2  p 2
2me
(A5.1)
,
(A5.2)
E  K f  K i    Ebind ,
(A6)
where p is the normal, i.e. z-component of the photoemitted electron momentum, Ki and
Kf are the initial and the final electronic kinetic energies, and E is the electron kinetic
energy change in photoionization, which is selected by the photon energy, ħ. The
obvious relationships between the components of the momentum are
and
p 2  p||2  p2 ,
(A7)
tan   p|| p  .
(A8)
After trivial algebra the components of the momentum and the flight time can be
expressed through the photoemission angle :
  sin 2 
,

(A9.1)
p||2 
2m * E
sin 2  ,
2
  sin 
(A9.2)
p 2 
2m * E
cos 2  .
2
  sin 
(A9.3)
2 
The number of electrons photoemitted in a solid angle limited by and  is
N  I e n pr p pr p ,
(A10)
where I is the light intensity, e is the photoionization probability which can be obtained
from Eq. (13) upon proper normalization of the photon flux, n pr p is the electron number
density in the 2D momentum space, pr and p refer to the radial and tangential
components of p|| and
p r 
p||

 
p|| 
  sin 2   tan 
(A11.1)
p  p|| 
and
(A11.2)
bounds the allowed values of p|| (see Fig. A2).
Note, that these N electrons arrive at the element S during the time interval
t   t 0
sin  cos
   sin 2  
 .
(A12)
The response of the element as a function of time is approximately proportional to
N
ht  t 2 ht 2  t   N t  ,
t 0
t
where h(t) is the Heaviside step function. This validates the use of the delta-function in
Eq. (A2).
Collecting Eqs. (A2) and (A9)-(A11) into Eq. (A1):
W t , 0   2m * EI d  e   n pr p  p||,i 
i
sin  i cos i  i  i
  sin
2
i 
2
 t  t i  .
(A13)
In order to convert the sum into an integral over the detector, the delta function should
also be converted to the angular variables. Since the time of arrival does not depend on
, this is simply
 t  t i  


   sin 2 
1
    i   
    i 
dt d
t 0 sin  cos
(A14)
where, according to Eq. (A9.1),
 i  arcsin  1   i2 
(A15)
and Eq. (A13) becomes
2m * EI d  e 
W t , 0  
t0
2m * EI d  e

t 0 3
3
2

over the
detector at  0
n pr p  p||  
  sin  
3
2
2
 n  p     arcsin
pr p


   arcsin  1   2  dd
||

 1   2  dd
(A16)
over the
detector at  0
pr p
The electron number density, n
, is
n pr p 
2 Ne
.
p r p
(A17)
Since cylindrical symmetry was assumed, one can introduce
and
n p|| 
dN e 2
  n pr p p|| d  2p|| n pr p
dp||
0
(A18.1)
n Ei 
dN e dN e dp||

 2m * n pr p .
dEi
dp|| dEi
(A18.2)
The density of states (per unit energy interval) is constant in 2D 2 and, assuming that the
population of the band does not change drastically within the detector acceptance range
and denoting
  ,  , 0  
    arcsin
over the
detector at  0

 1   2  dd
(A19.1)
or
  ,  , 0  

over the
detector at  0

   arcsin 

  1 
dd
 
(A19.2)
which will be dealt with in Supplemental Information B, we, finally, have
W t , 0   I d  e nEi
E
  ,  , 0 
t0 3
(A20)
E
  ,  , 0  .
2t 0
(A21)
or, in the energy variables,
W  , 0   I d  e n Ei
Figure A3 shows the integrated detector sensitivity

P 0    W  , 0 d
(A22)
0
for the detector used in the experiment as a function of the detector angle. An additional
factor of cos4 appears when the decrease of the normal component of light on the
surface is taken into account for both the pump and probe pulses. The detector sensitivity
is a weak function of angle which allows direct comparison of peak amplitudes from
spectra taken at different angles of observation.
Photoemission from localized interfacial states
The kinetic energy of electrons photoemitted from a localized state does not
depend on the photoemission angle. Thus, the value of the total momentum given by Eq.
(A7) is independent of the angle:
p 2  2me K 0 ,
(A23)
K 0    Ebind
(A24)
where
is the photoemitted electron kinetic energy. Electrons arrive at different parts of the
detector simultaneously, which makes the delta function argument in Eq. (A2)
independent of the summation index and the detector response is instantaneous: W(t, 0)
~ (t – t0). The number of electrons that arrived to the detector is given by analogy to Eq.
(A10):
N  I e nloc  loc  p||  p x p y ,
2
(A25)
where nloc is the number density of the localized electrons. In order to take the integral
over the detector we note that only those electrons contribute to the signal for which
x
py
px
t 0 , and y 
t0
me
me
(A26)
belong to the detector projection on the (x, y) plane. Replacing the value of the wave
function amplitude by its average value for a small detector we have
N  I e nloc  loc  p|| 
2
 me 
 
 t0 
2
 dxdy ,
detector projection
on the  x , y  plane
(A27)
where p|| is the average detected value of the parallel momentum. The integral is
proportional to the area of the detector projection on the (x, y) plane, S·cos0 and the
signal is simply
W t , 0  
2
I e d Sme2
nloc  loc  p||  cos 0 t  t 0  ,
2
t0
(A28)
The detector sensitivity is thus
P 0   nloc cos 0
(A29)
It is again, a weak function of the observation angle.
Supplemental Information B: Detector Shape Factor (, , 0)
The detector shape function (, , 0) shows the angular length of the arc which
collects signal at given time:
  ,  , 0  
    asin

 1   2  dd
over the
detector at  0
 max
(B1)
 2  d 2 max   min 
 min
where 0 < max, min < /2 are the angles which bound the detector projection in the upper
half plane (Fig. B1). Indeed, for a given direction, the electron either hits the detector at
the instance , and then the integral over d gives unity, or misses the detector and does
not contribute to the integral. For a given detector size A  B, five cases can be
distinguished, depending on the detector position which are shown in Fig. B1 a-e. The
detector projection on the (x, y) plane is limited by
x0 
B
B
cos 0  x  x0  cos 0
2
2
 A 2  y  A 2.
and
(B2.1)
(B2.2)
where
x0  R sin  0
(B3)
The relationship between the kinetic energy and the radius of projection r is


r  R sin   R  1   2  R 
 1

(B4)
Introducing
C   sin  0 
B
cos 0
2R
(B5)
and
 1

(B6)
 t , 0   2 arccosC  
(B7.a)
 A 

 2 R 
(B7.b)
   1   2   
One can write for the cases a through e of Fig. B1
a)  min  0, cos  max  C   :
b)  min  0, sin  max 
A
:
2 R
 t , 0   2 arcsin 
c) cos  min  C  , sin  max 
A
:
2 R
 A 
  2 arccosC   
 2 R 
(B7.c)
 t , 0   2 arccosC    2 arccosC  
(B7.d)
 t , 0   2 arcsin 
d) cos  min  C  , cos  max  C   :
e) A proper combination of two cases a) – d) for the left and right parts.
The apparatus function for a few different angles is shown in Fig. B2. Although it
can have a significant contribution to the apparent linewidths for other systems, for the
present work it is negligible. More importantly, the integral of the apparatus function
allows correct normalization of the peak amplitudes when comparing data taken at
different observation angles. The integral is plotted in Fig. A3 and exhibits a weak
dependence on the angle of observation.
References:
1
Electrons are treated classically here.
2
C. Kittel, Introduction to Solid State Physics, Seventh Edition ed. (John Wiley
and Sons, New York, 1996).
Figure Captions:
Fig. A1. Schematic of a typical TOF spectrometer. Photoemitted electrons with
the sam kinetic energy expand forming a hemisphere. A sample is shown at the
hemisphere origin. The detector of size A  B (shaded) is positioned at the distance R
from the sample. Also, a small element of the detector, Si, is shown and its projection
onto the (x, y) plane, Pi. The former is limited by and .
Fig. A2. Projections of the momentum on the plane of the surface.
Fig. A3. The integral of the apparatus functions.
Fig. B1. Projections of the detector on the (x, y) plane (shaded rectangles) and of
the line on which the electrons strike the detector at a time  (circle).
Four cases are
possible (a)-(d); case (e) can be reduced to a combination of the first four.
Fig. B2. Detector apparatus functions for (left to right) 0°, 4°, 8°, 12°, 16°, and
20° detector angle for a unit effective mass.
Fig. C1. Time delays, observation angles, and approximate values of p||; at each
point of the grid experimental spectra were recorded. Large negative delays (not shown)
were used in the background subtraction. Typical spectra and fits for four dispersions
taken at different delay times (a) through (d) are presented in Fig. 2. Localized and
delocalized peak amplitudes obtained in different global fits are presented in Fig. C2.
Shading shows the area where peaks in individual spectra are distinguishable.
Fig. C2. Amplitudes of the localized (right column) and delocalized (left column)
states as a function of p|| at four different delay times; panels (a) through (c), matching
Figs. 2 and C1. Different symbols show results of different global fits. They are slightly
offset horizontally to avoid congestion. Amplitudes of the localized and of the
delocalized states are strongly anti-correlated at low values of p||. Typical error bars are
shown (95% confidence interval). The errors are obtained in a Monte-Carlo simulation
of the data sets.