Physisorption of an electron in deep surface potentials off a dielectric surface

Abstract

We study phonon-mediated adsorption and desorption of an electron at
dielectric surfaces with deep polarization-induced surface potentials where
multi-phonon transitions are responsible for electron energy relaxation.
Focusing on multi-phonon processes due to the nonlinearity of the coupling
between the external electron and the acoustic bulk phonon triggering the
transitions between surface states, we calculate electron desorption times for
graphite, MgO, CaO, (\text{Al}_2\text{O}_3), and (\text{SiO}_2) and electron
sticking coefficients for (\text{Al}_2\text{O}_3), CaO, and (\text{SiO}_2). To
reveal the kinetic stages of electron physisorption, we moreover study the time
evolution of the image state occupancy and the energy-resolved desorption flux.
Depending on the potential depth and the surface temperature we identify two
generic scenarios: (i)adsorption via trapping in shallow image states followed
by relaxation to the lowest image state and desorption from that state via a
cascade through the second strongly bound image state in not too deep
potentials and (ii)adsorption via trapping in shallow image states but followed
by a relaxation bottleneck retarding the transition to the lowest image state
and desorption from that state via a one step process to the continuum in deep
potentials.

Full text

Page 1

arXiv:1102.0632v1 [cond-mat.other] 3 Feb 2011
Physisorption of an electron in deep surface potentials off a dielectric surface
R. L. Heinisch, F. X. Bronold, and H. Fehske
Institut f¨ ur Physik, Ernst-Moritz-Arndt-Universit¨ at Greifswald, 17489 Greifswald, Germany
(Dated: May 9, 2011)
We study phonon-mediated adsorption and desorption of an electron at dielectric surfaces with
deep polarization-induced surface potentials where multi-phonon transitions are responsible for elec-
tron energy relaxation. Focusing on multi-phonon processes due to the nonlinearity of the coupling
between the external electron and the acoustic bulk phonon triggering the transitions between sur-
face states, we calculate electron desorption times for graphite, MgO, CaO, Al2O3, and SiO2 and
electron sticking coefficients for Al2O3, CaO, and SiO2. To reveal the kinetic stages of electron
physisorption, we moreover study the time evolution of the image state occupancy and the energy-
resolved desorption flux. Depending on the potential depth and the surface temperature we identify
two generic scenarios: (i) adsorption via trapping in shallow image states followed by relaxation to
the lowest image state and desorption from that state via a cascade through the second strongly
bound image state in not too deep potentials and (ii) adsorption via trapping in shallow image
states but followed by a relaxation bottleneck retarding the transition to the lowest image state and
desorption from that state via a one step process to the continuum in deep potentials.
I. INTRODUCTION
Image states, arising from the polarization-induced in-
teraction between an electron and a surface, offer the
possibility for electron trapping at a surface.
their original prediction1for the surface of liquid and
solid He, they have been extensively studied for metal-
lic surfaces.2–6But image states also exist for dielectric
surfaces provided the electron affinity of the dielectric is
negative, that is, the vacuum level falls inside the gap
between the valence and the conduction band. Image
states are then the lowest unoccupied states and should
hence allow for temporary trapping of external electrons.
So far image states at a dielectric surface have been only
observed for graphite,7but they are expected for other
dielectrics with negative electron affinity as well, for in-
stance, boron nitride8and the alkaline earth oxides.9
Based on the idea of a two-dimensional electron sur-
face plasma,10–13electron trapping in image states has
been suspected for a long time to be responsible for the
build-up of surface charges at plasma walls. We have
recently proposed therefore to consider the charging of
a plasma wall as an electron physisorption process.14,15
Indeed, for plasma walls with negative electron affinity
image states should contribute to the very beginning of
the charging process, when the wall carries no charges
yet and the image states thus fall inside the energy gap
of the wall. Only with increasing surface charge image
states are expected to play a less important role because
the Coulomb barrier due to the electrons already resid-
ing on the wall shifts image states to an energy range
where they are destabilized by unoccupied bulk states.
The later stages of charge collection most probably oc-
curs via surface resonances or empty volume states.16
Regardless of its importance for charge collection at
dielectric plasma walls, the electron kinetics in the im-
age states of a dielectric surface is an interesting phe-
nomenon in its own right. In addition, it is relevant in
other physical contexts as well. For instance, (i) in elec-
Since
tron emitters, such as cesium-doped silicon oxide films
with negative electron affinity, electron emission via im-
age states reduces the operational voltage considerably,17
(ii) in gallium arsenide based heterostructures surface
charging can be used for the contactless gating of field
devices,18and (iii) for the alkaline earth oxides, studied
in the field of heterogeneous catalysis,19–22the electronic
surface states provide the environment for catalytic reac-
tions. Some situations encompass electronic transitions
from bulk to surface states, as it is the case for electron
emitters, while for others, the electron does not penetrate
into the bulk and the electron kinetics takes only place in
surface states. Interesting questions in this case are the
probability for temporary trapping in these states, the
mechanism of electron energy relaxation at the surface,
and the time after which a trapped electron is released.
This is the concluding paper out of a series of three
on the phonon-mediated physisorption of an electron in
the image states of a dielectric surface. As in our pre-
vious work, Refs. 23 and 24 (thereafter referred to as I
and II), we investigate adsorption and desorption of an
electron at finite temperatures assuming an acoustic lon-
gitudinal bulk phonon to control energy relaxation at the
surface. For the dielectric material we are considering,
the level spacing of the lowest two bound states typically
exceeds the Debye energy, implying that multi-phonon
processes have to be taken into account. In I and II, we
have studied desorption and sticking using an expansion
of the energy dependent T matrix,25–27allowing to calcu-
late one- and two-phonon transition probabilities. This
approach is however limited to very few materials, for in-
stance, graphite and MgO. In the following we will adopt
a different strategy, calculating multi-phonon transition
probabilities due to the nonlinear electron-phonon cou-
pling non-perturbatively. This allows us to calculate the
desorption time and the sticking coefficient for the deeper
surface potentials of CaO, Al2O3and SiO2.
The remaining paper is structured as follows.
Sec. II we briefly recall the quantum-kinetic approach
In

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to physisorption. In Sec. III, we calculate the multi-
phonon state-to-state transition probabilities. In Sec. IV
we present our results for the desorption time and the
prompt and kinetic energy-resolved and energy-averaged
sticking coefficient. In this section we also discuss the
time evolution of the bound state occupancy and the
energy-resolved desorption flux. Section V is devoted to
the analytic treatment of a simplified two-state model,
used to identify two generic physisorption scenarios into
which we can classify the results of this paper as well as
our previous results, before we conclude in Sec. VI.
II.ELECTRON KINETICS
As in I23and II24we describe the time evolution of the
occupancy of the bound surface states with a quantum-
kinetic rate equation.28,29It captures all three character-
istic stages of physisorption:30,31initial trapping, subse-
quent relaxation and desorption.
The time dependence of the occupancies of the bound
states is given by28,29
d
dtnn(t) =
?
−
n′
[Wnn′nn′(t) − Wn′nnn(t)]
?
Tnn′nn′(t) +
k
Wknnn(t) +
?
k
τtWnkjk(t) (1)
=
?
n′
?
k
τtWnkjk(t) , (2)
where Wn′nis the probability per unit time for a transi-
tion from a bound state n to another bound state n′, Wkn
and Wnkare the probabilities per unit time for a transi-
tion from the bound state n to the continuum state k and
vice versa and τt= 2L/vzis the transit time through the
surface potential of width L, which, in the limit L → ∞,
can be absorbed into the transition probability. The ma-
trix Tnmis defined implicitly by the above equation. The
last term in Eqs. (1) and (2), respectively, gives the in-
crease in the bound state occupancy due to trapping of
an electron in bound surface states.
The probability for an approaching electron in the con-
tinuum state k to make a transition to any of the bound
states is given by the prompt energy-resolved sticking
coefficient,
sprompt
e,k
= τt
?
n
Wnk. (3)
Treating the incident electron flux as an externally
specified parameter, the solution of Eq. (1) describes
the subsequent relaxation and desorption. It is given by
nn(t) =
?
κ
e−λκt
?t
−∞
dt′eλκt′e(κ)
n
?
kl
˜ e(κ)
l
τtWlkjk(t′) ,
(4)
where e(κ)
the eigenvalue −λκof the matrix T.
n
and ˜ e(κ)
n
are the right and left eigenvectors to
If the modulus of one eigenvalue, λ0, is considerably
smaller than the moduli of the other eigenvalues, λκ,
a unique desorption time and a unique sticking coeffi-
cient can be identified.29In this case λ0governs the long
time behavior of the equilibrium occupation of the bound
states, neq
with the desorption time, λ−1
bound state occupancy nn(t) splits into a slowly varying
part n0
n(t) given by the κ = 0 summand in Eq. (4) and a
quickly varying part nf
n(t) given by the sum over κ ?= 0
in Eq. (4).
The ”adsorbate“, i.e. the fraction of the trapped elec-
tron remaining in the surface states for times on the or-
der of the desorption time, is given by the slowly varying
part only, n0(t) =?
d
dtn0(t) =
k
q∼ e−Eq/kBTs, and its inverse can be identified
0
= τe. In this case the
nn0
n(t). Differentiating n0(t) with
respect to the time,
?
skinetic
e,k
jk(t) − λ0n0(t) ,(5)
we can, following Brenig31, identify the kinetic energy-
resolved sticking coefficient
skinetic
e,k
= τt
?
n,n′
e(0)
n′ ˜ e(0)
nWnk, (6)
giving the probability for both, initial trapping and sub-
sequent relaxation.
If the incident unit electron flux corresponds to an
electron with Boltzmann distributed kinetic energies, the
prompt or kinetic energy-averaged sticking coefficient is
given by
s...
e=
?
ks...
?
e,kke−βeEk
kke−βeEk
, (7)
where β−1
The desorption flux, that is, the flux due to an electron
that is not instantly reflected at the boundary but sticks
to the surface for a finite time can also be calculated from
the occupancy of the bound surface states. From Eq. (1),
we infer that the losses of the bound state occupancy
increase the continuum state occupancy by
e
= kBTeis the mean electron energy.
dnk
dt
=
?
n
Wknnn(t) .(8)
As the electron remains in the surface potential for the
time it needs to travel trough the surface potential the
occupancy of the continuum state k is given by nk =
τt ˙ nk. To obtain the energy-resolved desorption flux we
multiply the occupancy of the continuum state k with
the flux jbox
k
, associated with the box-normalized state
|φk?.23Thus, the energy-resolved desorption flux is given
by
jk(t) = τtjbox
k
?
n
Wknnn(t) , (9)
which is well defined in the limit L → ∞.

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III. TRANSITION PROBABILITIES
The kinetic equations presented in the last section rely
on the knowledge of the transition probabilities. They
have to be calculated from a microscopic model for the
electron-surface interaction.
For a dielectric surface, the transitions are driven by
phonons, whose maximum energy is, within the Debye
model, the Debye energy ?ωD. Measuring energies in
units of the Debye energy, important dimensionless pa-
rameters characterizing the potential depth are
ǫn=
En
?ωD
and∆nn′ =En− En′
?ωD
,(10)
where En< 0 is the energy of the nthbound state.
In I, we introduced the following classification for the
potential depth. If −n + 1 > ∆12> −n, we call the po-
tential n-phonon deep. For the calculations in I and II,
we considered only one- or two-phonon deep potentials,
for which one- and two-phonon transition probabilities
are sufficient. Dielectrics with two-phonon deep poten-
tials, such as graphite or MgO, are however an exception.
Many dielectrics, for instance, Al2O3, CaO, GaAs, or
SiO2have more than two-phonon deep potentials. Hence,
the more relevant situation is physisorption in deep sur-
face potentials for which multi-phonon transition proba-
bilities are required.
To calculate multi-phonon transition probabilities for
the one-dimensional microscopic model used in I and II,
we briefly recall its main features. In short, for a di-
electric surface, the main source of the attractive static
electron-surface potential is the coupling of the electron
to a dipole-active surface phonon.32Far from the surface
the surface potential arising from this coupling merges
with the classical image potential and thus ∼ 1/z. Close
to the surface, however, the surface potential is strongly
modified by the recoil energy resulting from the momen-
tum transfer parallel to the surface when the electron
absorbs or emits a surface phonon. Taking this effect
into account leads to a recoil-corrected image potential
∼ 1/(z + zc) with zca cut-off parameter defined in I.
Transitions between the eigenstates of the recoil-
corrected image potential are due to dynamic perturba-
tions of the surface potential. The surface potential is
very steep near the surface. A particularly strong per-
turbation arises therefore from the longitudinal-acoustic
bulk phonon perpendicular to the surface which causes
the surface plane to oscillate.
The Hamiltonian from which we calculate the transi-
tion probabilities was introduced in I where all quanti-
ties entering the Hamiltonian are explicitly defined. It is
given by
H = Hstatic
e
+ Hph+ Hdyn
e−ph, (11)
where
Hstatic
e
=
?
q
Eqc†
qcq
(12)
describes the electron in the recoil-corrected image po-
tential,
Hph=
?
Q
?ωQb†
QbQ
(13)
describes the free dynamics of the bulk longitudinal
acoustic phonon responsible for transitions between sur-
face states, and
Hdyn
e−ph=
?
q,q′
?q′|Vp(u,z)|q?c†
q′cq
(14)
denotes the dynamic coupling of the electron to the bulk
phonon.
The perturbation Vp(u,z) can be identified as the dif-
ference between the displaced surface potential and the
static surface potential. It reads, after the transforma-
tion z → z − zc,
Vp(u,z) = −e2Λ0
z + u+e2Λ0
z
, (15)
where Λ0= (ǫs− 1)/4(ǫs+ 1) with ǫsthe static dielec-
tric constant. In general, multi-phonon processes can
arise both from the nonlinearity of the electron-phonon
coupling Hdyn
e−phas well as from the successive actions of
Hdyn
e−phencoded in the T matrix equation,
T = Hdyn
e−ph+ Hdyn
e−phG0T ,(16)
where G0is given by
G0=?E − Hstatic
e
− Hph+ iǫ?−1. (17)
The transition probability per unit time from an elec-
tronic state q to an electronic state q′encompassing both
types of processes is given by25
Wq′q=2π
?
?
s,s′
e−βsEs
s′′e−βsEs′′|?s′,q′|T|s,q?|2
?
× δ(Es− Es′ + Eq− Eq′) ,(18)
where βs = (kBTs)−1, with Ts the surface temperature
and |s? and |s′? the initial and final phonon states. We
are only interested in the transitions between electronic
states. It is thus natural to average in Eq. (18) over
all phonon states. The delta function guarantees energy
conservation.
In our previous work, we have used an expansion of
the T matrix to calculate multi-phonon transition rates.
In principle this ensures that both linear and nonlinear
terms in the interaction as well as successive actions of
the perturbation are taken into account up to a certain
order of the phonon process. However even for a two-
phonon deep potential, taking all two-phonon processes
into account is nearly impossible. The calculation be-
comes feasible if two-phonon processes are only taken

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4
into account for transitions not already enabled by a one-
phonon process. This amounts to computing only the
lowest required phonon order for a given transition, ne-
glecting higher order corrections to it. For higher order
phonon processes even this simplified strategy becomes
unfeasible. A different approach is thus needed.
From I and II we qualitatively know the relevance of
the different types of multi-phonon processes for partic-
ular electronic transitions. For continuum-bound state
transitions, for instance, one-phonon processes are suffi-
cient at low electron energies. We will therefore compute
the transition probability between bound and continuum
states in the one-phonon approximation. For transitions
between bound states, we found that multi-phonon pro-
cesses due to the nonlinearity of the electron-phonon cou-
pling tend to be more important than the multi-phonon
processes due to the iteration of the T matrix, unlike to
what we found for bound state - continuum transitions
(see I) or to whatˇSiber and Gumhalter33–35found in
the context of atom-surface scattering. Indeed, multi-
phonon processes from the iteration of the T matrix give
a minor contribution, unless resonances arising from the
T matrix become relevant. This happens whenever the
energy difference between two bound states is a multi-
ple of the Debye energy. Resonances smoothen then the
abrupt steps in the transition probability at the depth
thresholds. Since the electronic matrix element between
the first and the second bound state is the largest one,
this effect is most pronounced for |∆12| = n.
In view of the above discussion we expect an approxi-
mation which takes only the nonlinearity of the electron-
phonon interaction non-perturbatively into account to
give an acceptable first estimate for the multi-phonon
transition rates. We denote this approximation the non-
linear multi-phonon approximation.In particular, it
should be sufficient for the identification of the generic
behavior of multi-phonon-mediated adsorption and des-
orption.
Calculating multi-phonon processes due to nonlinear
terms in the interaction potential36amounts to a dis-
torted wave Born approximation with the full interaction
potential. Thus, the transition probability per unit time
is given by
Wq′q=2π
?
?
s,s′
e−βsEs
s′′e−βsEs′′
?
????q,s|Hdyn
e−ph|s′,q′?
???
2
× δ(Es+ Eq− Eq′ − Es′) . (19)
To evaluate the multi-phonon transition probability,
we use Hdyn
e−phin the form of Eq. (15). The transition
matrix element in Eq. (19) is then given by
?q,s|Hdyn.
e−ph|q′,s′?
?∞
= ?s|
zc
dzφ∗
q(z)[v(z + u) − v(z)]φq′(z)|s′? , (20)
where v(z) = −(e2Λ0)/z.
variables x = z/aB, the Fourier transform of the static
potential
Introducing dimensionless
v(p) =
?∞
xc
dxeipxv(x) ,(21)
and the state-to-state matrix element
fqq′(p) =
?∞
xc
dxφ∗
q(x)e−ipxφq′(x) ,(22)
the transition probability can be rewritten as
Wq′q=2π
?
?
s,s′
e−βsEs
s′′e−βsEs′′
?
?
?∞
−∞
dp
2π
?∞
−∞
d˜ p
2πv(p)v∗(˜ p)fqq′(p)f∗
qq′(˜ p)?s|
?
e−i
p
aBu− 1
?
|s′?
× ?s′|ei
˜ p
aBu− 1
?
|s?δ(Es+ Eq− Es′ − Eq′) .(23)
Using the identity δ(x) = 1/(2π)?∞
Wq′q=
?2
−∞
−∞dt eixtand employing ?s|eiEst/?= ?s|eiHpht/?, the above expression becomes
?∞
se−βsEs?s|...|s?/?
??
?
1
?∞
dp
2π
?∞
−∞
d˜ p
2πv(p)v∗(˜ p)fqq′(p)f∗
qq′(˜ p)
−∞
dt ei(Eq−Eq′)t/???
?
e−i
p
aBu(0)− 1
??
ei
˜ p
aBu(t)− 1
?
??(24)
with ??...?? =?
s′′e−βsEs′′the average over phonon states. This average can be evaluated for
q ?= q′employing Glauber’s theorem37which yields
e−i
p
aBu(0)− 1
??
ei
˜ p
aBu(t)− 1
?
?? = e
−
1
2a2
B
p2??u(0)2??e
−
1
2a2
B
˜ p2??u(t)2??e
1
a2
B
p˜ p??u(0)u(t)??
(25)
with
??u(0)u(t)?? =
?
Q
?
2µNsωQ
?(1 + nB(?ωQ))e−iωQt+ nB(?ωQ)eiωQt?
(26)
the correlation function of the displacement field
u =
?
Q
?
?
2µωQNs
?
bQ+ b†
−Q
?
,(27)
where µ is the mass of the unit cell of the lattice and Ns

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5
is the number of unit cells.
As in I and II we use for calculational convenience a
bulk Debye model for the longitudinal acoustic phonon,
although it is less justified for the high energy part of the
spectrum which also enters our calculation. Sums over
phonon momenta are thus replaced by
?
Q
··· =3Ns
ω3
D
?ωD
0
dω ω2... . (28)
In terms of the dimensionless variables
x =
ω
ωD
, δ =?ωD
kBTs
, and τ = ωDt ,(29)
the phonon correlation function becomes
??u(0)u(τ)?? =
3?
2µωD
?1
0
dxx
?
e−ixτ
1 − e−δx+
eixτ
eδx− 1
?
(30)
.
Hence, for the transition probability per unit time we
obtain
Wq′q=
e4Λ2
?2ωDa2
0
B
?∞
−∞
dp
2π
?∞
−∞
d˜ p
2πv(p)v(˜ p)fqq′(p)f∗
qq′(˜ p)e−1
2γp2q(0)e−1
2γ˜ p2q(0)
?∞
−∞
dτei∆qq′τ+γp˜ pq(τ),(31)
where
q(τ) =
?1
0
dxx
?
e−ixτ
1 − e−δx+
eixτ
eδx− 1
?
andγ =
3?
2µa2
BωD
.(32)
The transition probability (31) contains two Debye-
Waller factors, exp(−γp2q(0)/2) and exp(−γ˜ p2q(0)/2),
governing the reduction of the transition probability as
a function of the surface temperature. It also contains
phonon processes of all orders as can be most easily seen
from the Taylor expansion
eγp˜ pq(τ)= 1 + γp˜ pq(τ) +1
2[γp˜ pq(τ)]2+ ... .(33)
Clearly, the second term on the right hand side repre-
sents the one-phonon and the third term the two-phonon
process.From I we know that two-phonon processes
are much weaker than one-phonon processes.
pect therefore lower order phonon processes to dominate
their higher order corrections, so that the expansion (33)
converges quickly.
As higher order phonon processes are small compared
to lower order processes, we take, for a given ∆qq′ only
the leading term of exp(γp˜ pq(τ)) into account, that is,
the lowest order phonon process that enables a transition
between the states q and q′. The Fourier transformation
of powers of q(τ), however, required when (33) is used
in (31), cannot be evaluated in closed from, making it
necessary to construct an approximation for q(τ).
To derive an approximation for q(τ) subject to the con-
straint
We ex-
?∞
−∞
dτei∆qq′τqn(τ) = 0 for |∆qq′| > n , (34)
which states that an n-phonon process yields a non-
vanishing transition probability only for −n < ∆ < n
and vanishes otherwise, we split q(τ) = qs(τ)+qi(τ) into
a contribution arising from spontaneous phonon emission
qs(τ) and a contribution from induced phonon emission
or absorption qi(τ). They are respectively given by
qs(τ) =
?1
0
dx xe−ixτand qi(τ) = 2
?1
0
dx xcos(xτ)
eδx− 1.
(35)
The former can be evaluated giving
qs(τ) =cosτ − 1
τ2
+ iτ cosτ − sinτ
τ2
+sinτ
τ
. (36)
For qi(τ) we need to find an approximation. For that
purpose we look at the Fourier transform of qi
?∞
−∞
dτei∆τqi(τ) =
?2π
|∆|
eδ|∆|−1for − 1 < ∆ < 1
0else
.
(37)
Expanding the Fourier transform in terms of |∆|,
2π
|∆|
eδ|∆|− 1≈ 2π
?1
δ−1
2|∆| +1
12δ|∆|2+ O(δ2)
?
,
(38)
yields a high-temperature approximation which con-
verges quickly for the temperatures we are interested in
and guarantees at the same time that the one-phonon
contribution can only bridge energy differences up to
|∆qq′| = 1. Applying the inverse transformation gives
qi(τ) =
?2
+
δ− 1 +δ
?
6
?sin(τ)
?cos(τ)
τ
+
1
τ2
−1 +δ
3τ2
−δ
3
sin(τ)
τ3
+ O(δ2) , (39)