Chemistry Reference
In-Depth Information
10.2.5 Theory Describing Inelastic Electron Tunneling Action
Spectroscopy
Following Refs. [
8
,
9
] and references herein, I describe a theory to quantify the
relation between the reaction yield and vibrational excitation. The purpose of the
following formulation is to reproduce the voltage dependence of the reaction rate
(yield), namely STM-AS, where the vibrational excitation drives the reaction or
motion of adsorbates via the indirect process (anharmonic coupling between the
high frequency mode and reaction coordinate modes). The outcome is applicable
to the systems which have an essentially same mechanism. As already mentioned,
the reaction yield Y(V) is given by.
Y
ð
V
Þ¼
R
ð
V
Þ
I
tot
ð
V
Þ
e
ð
10
:
1
Þ
where R(V) is the reaction rate and I
tot
(V) is the total tunneling current and e is the
elementary electric charge. The definition of power law of R(V) gives
R
ð
V
Þ¼
k
ð
I
in
ð
V
ÞÞ
n
ð
10
:
2
Þ
with k being the rate constant, I
in
being the inelastic tunneling current, and n being
the reaction order, namely the number of tunneling electron required to induce
reactions or motions of an adsorbate. Here I
tot
= I
el
? I
in
(I
el
is the elastic
tunneling current) and the Iin vanishes if the energy of the tunneling electron, eV,
is lower than a specific vibrational energy hX whereas it increases linearly above
hX. Here I consider only the case of n = 1 (one-electron process) and assume the
indirect process where the reaction coordinate mode is indirectly excited via the
anharmonic coupling of the high frequency mode. In the framework of IET
process in which the tunneling electron induces the molecular vibration via
resonance
scattering,
I
in
is
proportionate
to
the
vibrational
generation
rate
I
in
= k'C
iet
, Thus.
R
ð
V
Þ¼
KC
iet
ð
V
Þ
ð
10
:
3
Þ
where K is the prefactor that is determined by the elementary process like elec-
tron-hole damping rate and anharmonic coupling between the high frequency and
reaction coordinates. Given a vibration mode characterized by a certain vibrational
density of state q
ph
(x) and define the vibrational generation rate in terms of a
spectral representation, C
iet
can be written by.
C
iet
ð
V
Þ¼
Z
1
0
dxq
ph
ð
X
Þ
C
in
ð
x
;
V
Þ
ð
10
:
4
Þ
where C
in
(X,V) is the spectral generation rate corresponding to an excitation of
energy hX. The spectral generation rate is given by.
