# Electron microphysics at plasma-solid interfaces

F. X. Bronold, K. Rasek and H. Fehske  Institut für Physik, Universität
Greifswald, 17489 Greifswald, Germany

###### Abstract

The most fundamental response of a solid to a plasma and vice versa is
electric. An electric double layer forms with a solid-bound electron-rich
region–the wall charge–and a plasma-bound electron-depleted region–the plasma
sheath. But it is only the plasma sheath which has been studied extensively
ever since the beginning of plasma physics. The wall charge received much less
attention. Especially little is known about the in-operando electronic
structure of plasma-facing solids and how it affects the spatio-temporal
scales of the wall charge. The purpose of this perspective is to encourage
investigations of this terra incognito by techniques of modern surface
physics. Using our own theoretical explorations of the electron microphysics
at plasma-solid interfaces and a proposal for measuring the wall charge by
infrared reflectivity to couch the discussion, we hope to put together enough
convincing reasons for getting such efforts started. They would open up–at the
intersection of plasma and surface physics–a new arena for applied as well as
fundamental research.

##  I Introduction

Low-temperature plasmas, ionized gases with electron and ion temperatures of
at most a few tens of an electron volt, are technologically extremely
successful. They are used in devices for particle detection, lighting,
welding, sterilization, pollutants management, and ozone production, as well
as in various sorts of materials modifications and syntheses. In all these
applications it is either the chemistry inside the bulk plasma or the
structural and chemical aspects of plasma-solid interaction which are
commercially exploited. A number of roadmaps have been layed out suggesting
how plasma physics should evolve to make it economically even more valuable
than it already is Weltmann  _et al._ (2019); Adamovich  _et al._ (2017);
Charles (2014). In this perspective we will not compete with them. Instead we
will focus on another aspect of plasma-solid interaction: the electric
response of the solid to the plasma and vice versa leading to an electric
double layer at the interface. Particularly the solid-bound part of the double
layer has been largely overlooked. Including it into the physical
investigations, and considering it as part of the in-operando modification of
the electronic properties of a plasma-facing solid, new vistas for fundamental
as well as applied research at the intersection of plasma and solid state
physics may open up.

Double layers are ubiquitous in nature. They arise at interfaces due to charge
separation. Perhaps the most prominent double layer occurs at the basic
building block of a battery, the electrolyte-metal interface Sass (1983);
Schmickler (1996). But also the Schottky contact Tung (2014), the
semiconductor-metal interface omnipresent in solid state electronics, gives
rise to a double layer. In fact, there are many more interfaces and hence
double layers of technological relevance. Of particular current interest are,
for instance, interfaces of large energy gap tertiary oxides Klein (2016);
Mannhart and Schlom (2010). The buried electron gases forming there and being
the objects of study are the negative legs of electric double layers. In
plasma physical settings, double layers arise when two gaseous plasmas face
each other Raadu (1989); Charles (2007) or when a plasma faces a solid.

The double layer developing at the plasma-solid interface, with a solid-bound
electron-rich and a plasma-bound electron-depleted region, is the object of
this perspective. It is known since the very beginning of plasma physics
Langmuir and Mott-Smith (1924). In the simplest case, it arises because
electrons outrunning the ions are deposited more efficiently into or onto the
solid, depending on its electronic structure, than they are extracted from it
by the neutralization of the ions. The plasma-bound part of the double
layer–the plasma sheath–has been thoroughly investigated Hershkowitz (2005);
Robertson (2013), most notably, its matching with the bulk plasma Schwager and
Birdsall (1990); Brinkmann (2009); Franklin (2003); Riemann (2003) and its
structural changes due either to negative ions Palop  _et al._ (2007) or the
emission of electrons form the wall Taccogna  _et al._ (2004); Sheehan  _et
al._ (2013); Campanell and Umansky (2016). The structure of the sheath in
front of biased electrodes received recently also substantial attention
Baalrud  _et al._ (2020). But little is known about the solid-bound part–the
wall charge. No systematic investigations exist about its material dependence
or its depth profile perpendicular to the interface. The magnitude Kindel and
Arndt (1980); Pangal  _et al._ (1996); Slikboer  _et al._ (2019) of the wall
charge and also its lateral distribution Kawasaki  _et al._ (1994); Tschiersch
_et al._ (2014) can be measured by particularly designed setups, but the
electronic states hosting it are unknown.

From our point of view, the electron microphysics at plasma-solid interfaces,
comprising electron deposition and extraction to and from the solid, the
build-up of the double layer, and–most importantly–the in-operando electronic
structure of the plasma-facing solid, is an unexplored territory. Very often
the solid is only treated as an electron reservoir, characterized by a
geometric boundary and probabilities for electron extraction and deposition.
As long as the probabilities are simply adjustable parameters, the effect of
the wall on the properties of Hall thrusters Raitses  _et al._ (2005), radio-
frequency discharges Raizer  _et al._ (2019), dusty plasmas Mendis (2002);
Ishihara (2007), and dielectric barrier discharges Kogelschatz (2003);
Brandenburg (2017), to name only a few of the many technologically relevant
plasma applications, can be studied without going into the details of the
wall’s electron microphysics. It is then also sufficient to model the wall
charge as an idealized infinitely thin surface charge. However, if one wants
to have control over the surface parameters, or if one wants to understand how
the wall charge affects the functionality of the discharges, one has to
explore the electronic processes inside the plasma-embracing solid in detail.

The memory effect in dielectric barrier discharges Massines  _et al._ (2003);
Wagner  _et al._ (2005), for instance, that is, the phenomenon that the
electric breakdown preferentially occurs at locations where wall charges from
the previous one still reside on or inside the dielectric, depends on the type
of electronic states the charges are bound to. On the macroscopic scale, the
memory effect has been investigated intensively Nemschokmichal  _et al._
(2018); Peeters  _et al._ (2016). What is missing is a microscopic inquiry.
With the insights gained from it it would be perhaps possible to manipulate
the discharge by a judicious choice of the dielectric material. For the
control of catalytic surface reactions, often the technological purpose of
this type of discharges, a microscopic understanding of what is going on
electronically inside the dielectric would be also rather useful Neyts  _et
al._ (2015); Patel  _et al._ (2019). This statement even holds for ozonizers,
perhaps the oldest application of barrier discharges, where optimization
procedures for the dielectric stacks are still guided by empiricism.

Figure 1: Electrons and ions generated inside a plasma by impact ionization
hit a plasma-facing solid. Usually probabilities for electron sticking
($s_{e}$), reflection ($r_{e}$), secondary emission ($\gamma_{e}$), and ion
neutralization ($\alpha_{i}$) are used to characterize the effect the solid
has electronically on the plasma. What the holes and electrons injected into
the solid and making up the negative leg of an electric double layer, whose
positive one is the plasma sheath, are doing is ignored. We argue in this
perspective that the electron microphysics at the interface, comprising the
charge dynamics on both sides of it and the in-operando electronic structure
of the plasma-facing solid, should be the subject of physical inquiry.

Miniaturized semiconductor-based microdischarges are another type of
discharge, where electronic processes inside the solid play an increasingly
important role Kulsreshath  _et al._ (2012); Eden  _et al._ (2013); Tabib-Azar
and Pai (2017); Michaud  _et al._ (2018). For future progress, it seems to be
essential to treat the charge kinetics inside the plasma and the solid on an
equal footing Rasek  _et al._ (2020); Arumugam  _et al._ (2020); Bonitz  _et
al._ (2019); Sun  _et al._ (2019); Bronold and Fehske (2017a). At least it is
conceivable to realize by continuing miniaturization situations, where the
scales for charge transport and relaxation are comparable for both subsystems.
Electrons inside the solid may then be equally hot as the electrons in the
plasma, that is, on the order of a few electron volts, which should strongly
affect, for instance, the in-operando interface resistance and hence the
electronic characteristics of the discharge. Since these devices combine
aspects of gaseous and solid state electronics they may be of interest for
optoelectronic applications Tabib-Azar and Pai (2017). Assuming materials
issues, such as structural damage due to ion implantation Michaud  _et al._
(2018), which are a problem at the moment, can be solved, progress will still
depend on an holistic approach considering electronic processes inside the
solid and the plasma as equally important. Only then will it be possible to
tap the full technological potential of this type of microdischarges.

This perspective strongly bats for a systematic study of the electron
microphysics at the plasma-solid interface. It asks not only for calculating
the probabilities for electron deposition and extraction from microscopic
models for the surface but also for the selfconsistent description of the
chain of events shown in Fig. 1. A kinetic theory is thus the goal, which
tracks the charges created by impact ionization on the gaseous side of the
interface to the inside of the solid, where they annihilate to phonons or
photons. Essential for both theoretical tasks is a precise knowledge of the
in-operando electronic structure of the plasma-facing solid. It is thus
necessary to enlarge the experimental toolkit by techniques of modern surface
diagnostics Brillson (2010), which fortunately are no longer limited to
vacuum-solid interfaces but cover nowadays also liquid-solid Liu  _et al._
(2020); R. Hausbrand, M. Fingerle, T. Späth, and C. Guhl (2017); Schnippering
_et al._ (2011); Barritt  _et al._ (2010); Lange  _et al._ (1983) and solid-
solid interfaces Varsha and Nageswaran (2020); Zborowski  _et al._ (2018);
Peeters  _et al._ (2017); Tselev  _et al._ (2016); Copuroglu  _et al._ (2013);
Rubio-Zuazo and Castro (2008); Aarts  _et al._ (2007, 2005). We strongly call
for applying these techniques to plasma-solid interfaces. So far, this has
been done only occasionally Tselev  _et al._ (2018); Laha  _et al._ (2014);
Tai  _et al._ (2013); Guha  _et al._ (2007); Nitschke and Meichsner (1997). It
is the combined theoretical and experimental effort we hope to initiate by
this perspective which would provide an unprecedented view of the electronics
of the plasma-solid interface. In the long run, we are convinced, this will be
the basis for new concepts of solid-bound low-temperature gas discharges.

In the next section we use our own work to exemplify the two basic theoretical
aspects of the interface’s electron microphysics by, respectively, a
calculation of the probability with which an electron is absorbed by a metal
at energies low enough to be of relevance for plasma applications and a
semiclassical kinetic model for the double layer at a dielectric plasma-solid
interface. In addition, we present an experimental scheme for investigating
the wall charge by infrared reflection spectroscopy. Exploratory calculations
are rather promising and we hope experimentalists put it into work. Having
discussed results of our own work, we next plea for in-operando studies of the
electronic structure of plasma-facing solids by photoemission and electron
spectroscopy. The techniques cannot be applied directly to the interface. We
envisage therefore spinning wall and from-the-back arrangements. Albeit
expensive, and certainly requiring long term commitment, having them on board
would be especially beneficial because they provide the microscopic
information most relevant for the electronics of the plasma-solid interface:
chemical composition, presence or absence of surface states, and energy
barriers due to the profile of the electric potential. In the course of the
plea we also indicate materials which we consider to be of interest for
fundamental as well as applied research. In a concluding synopsis we finally
summarize our main points for quick reference.

##  II Work we are doing

The pragmatic approach towards the plasma wall is to consider it as an
electron reservoir. Leaving structural damage due to heavy particles aside,
the wall is then characterized by a fixed geometric boundary and probabilities
for electron sticking ($s_{e}$), reflection ($r_{e}$) and (secondary) emission
($\gamma_{e}$). The latter may arise due to electron impact or charge-
transferring encounters with heavy particles, such as ions or radicals, giving
rise to various Auger-type processes. As far as electrons are concerned,
impact energies are typically well below $1\,\mathrm{keV}$, making theoretical
approaches and techniques developed for the description of electron
spectroscopy and microscopy not applicable. In addition, it has been noticed
that in this energy range measured data are also sparse Tolias (2014a, b),
triggering hence efforts to calculate Marbach  _et al._ (2012); Pamperin  _et
al._ (2018); Bronold and Fehske (2015, 2017b) or to measure some of the
probabilities, usually called surface parameters, by beam Gonzalez  _et al._
(2017); Pierron  _et al._ (2017); Corbella  _et al._ (2016); Marcak  _et al._
(2015); Dunaevsky  _et al._ (2003) or plasma experiments Zobnin  _et al._
(2018); Demidov  _et al._ (2015); Daksha  _et al._ (2016, 2019).

Beam experiments for measuring $\gamma$-coefficients due to ion Marcak  _et
al._ (2015) or electron impact Gonzalez  _et al._ (2017); Pierron  _et al._
(2017); Dunaevsky  _et al._ (2003) are highly relevant but have the drawback
of not being made under plasma operation. Plasma experiments, on the other
hand, enable an in-operando determination of electron sticking Zobnin  _et
al._ (2018); Demidov  _et al._ (2015) and $\gamma$-coefficients Daksha  _et
al._ (2016, 2019). They depend however on simulations of the gas discharge
under the assumption that the coefficient to be determined is the only one
unknown. Calculating the coefficients from first principles is also rather
challenging. Even the simplest projectile, the electron, gives rise to
scattering cascades involving many different collision paths. Ions and
radicals, having internal degrees of freedom, lead to even more involved
collisions Rabalais (1994); Monreal (2014).

Our calculations of electron surface parameters are based on semiempirical
effective Hamiltonians for the subset of electronic states which are most
relevant for the collision process under consideration. The matrix elements of
the Hamiltonians can be either determined experimentally or theoretically from
first principles. Initially, we applied the semiempirical approach to
secondary electron emission due to heavy particles Marbach  _et al._ (2012);
Pamperin  _et al._ (2018) and electron scattering off dielectric surfaces
Bronold and Fehske (2017b, 2015). But later on we also used it for the kinetic
modeling of the electric response of a dielectric plasma-solid interface
Bronold and Fehske (2017a); Rasek  _et al._ (2020). To identify the kind of
information needed about the interface’s electronic structure to make the
semiempirical approach work, we illustrate in the following the calculational
schemes with some new results. The link between our work and the work we hope
to initiate is the proposal for measuring the wall charge by infrared
reflectivity described at the end of this section. It calls for
experimentalists to implement it.

###  II.1 Electron absorption by metallic walls

An important surface parameter, controlling the charging of dust particles
immersed in a plasma Mendis (2002); Ishihara (2007), is the electron sticking
coefficient. Its complement, the electron emission yield due to electron
impact, plays an important role in Hall thrusters Raitses  _et al._ (2005) and
barrier discharges Kogelschatz (2003); Brandenburg (2017). Based on the
invariant embedding principle for electron scattering off surfaces Dashen
(1964); Vicanek (1999); Glazov and Pázsit (2007), we set up in our previous
work Bronold and Fehske (2015, 2017b) a scheme for calculating electron
sticking coefficients for dielectric surfaces at energies below the band gap.
In this subsection we apply it to a low energy electron scattering off a
metallic surface. The intention is to identify the parts of the electronic
structure of the metal which should be known as precisely as possible. If they
are affected by the plasma, they should be measured in-operando.

An electron hitting a metallic surface enters, after successful transmission
through the surface potential, the conduction band. As illustrated in Fig. 2,
elastic and inelastic scattering events inside the metal may push the electron
back to the interface, where it may traverse the surface potential in the
reverse direction to leave the solid again. The probability for coming back to
the plasma is thus the result of passing twice through the surface potential
and diffuse scattering in between. Notice, only electrons from the plasma with
an energy high enough to overcome the wall (sheath) potential $U_{w}$ have a
chance to perform such a cascade. Putting the origin of the energy axis to the
potential just outside the solid, we thus have to track only electrons
impacting with $E>0$. That the electron performing the cascade stems from the
plasma is irrelevant. The plasma affects the cascade only indirectly through
the depth $U_{0}$ of the surface potential, which is the sum of the Fermi
energy $E_{F}$ and the work function $\Phi$. Both may depend on the surface’s
chemical contamination and structural modification by the plasma. Hence,
$E_{F}$ and $\Phi$ should be measured in-operando. In case the metal is
biased, the situation is essentially the same except that, depending on the
polarity of the bias voltage $\pm V_{\rm bias}$, the wall potential $U_{w}$ is
reduced (positive bias) or increased (negative bias) by the amount $eV_{\rm
bias}$. The material parameters $E_{F}$ and $\Phi$, however, are unaffected.

Figure 2: Left: Definition of the variables used in (1) to calculate the
sticking probability $S(E,\xi)$ for an electron hitting a metallic solid with
energy $E$ and direction cosine $\xi=\cos\beta$. The probability for
transmission through the surface potential is given by $T$ while the
probability for scattering inside the solid from a state $(E,\eta)$ to a state
$(E^{\prime},\eta^{\prime})$, where $\eta=\cos\theta$ and
$\eta^{\prime}=|\cos\theta^{\prime}|$, is denoted by $B$. The metal is
modelled by an exponential potential step characterized by a width parameter
$a$ and a depth $U_{0}=E_{F}+\Phi$, where $E_{F}$ is the Fermi energy and
$\Phi$ is the work function. Also indicated is the electron energy
distribution function $f_{e}(E)$ and the wall potential $U_{w}$ an electron
from the plasma has to overcome to reach the solid. Biasing the metal by $\pm
V_{\rm bias}$ would lead to $U_{w}\mp eV_{\rm bias}$ while keeping $E_{F}$,
$\Phi$, and the origin of the energy axis fixed. Right: Electron emission
yield $\mathrm{EEY}=1-S(E,\xi)$ for $\xi=1$ and a Ag surface characterized by
$k_{B}T=0.03\,\mathrm{eV}$, $E_{F}=5.5\,\mathrm{eV}$, $\Phi=4.4\,\mathrm{eV}$,
$a=0.26\,$Å, and $n_{\rm imp}=n/1000$, where $n$ is the electron density
corresponding to $E_{F}$. Electron-electron, electron-impurity, and electron-
phonon scattering are taken into account. Using the electron-phonon coupling
function $\lambda(E)$ as an energy dependent fit parameter, the experimental
data Gonzalez  _et al._ (2017) can be nicely reproduced. The dashed vertical
line indicates the bulk plasmon energy, where nothing spectacular happens.

To describe the scattering cascades we use the variables and probabilities
defined in Fig. 2. The sticking probability $S(E,\xi)$, where $E$ is the
energy and $\xi$ the cosine of the angle $\beta$ with which the primary
electron hits the surface with respect to the inwardly directed surface normal
is then given by Bronold and Fehske (2015, 2017b)

$\displaystyle S$ $\displaystyle(E,\xi)=T(E,\xi)-T(E,\xi)$
$\displaystyle\times\int^{1}_{\eta_{\rm
min}(E)}\\!\\!\\!\\!\\!\\!\\!\\!\\!d\eta^{\prime}\int^{E}_{E_{\rm
min}(\eta^{\prime})}\\!\\!\\!\\!\\!\\!\\!\\!\\!dE^{\prime}\rho(E^{\prime})B(E\eta(\xi)|E^{\prime}\eta^{\prime})T(E^{\prime},\xi(\eta^{\prime}))~{},$
(1)

where $\xi=cos\beta$, $\xi^{\prime}=|cos\beta^{\prime}|$, and
$0<\beta\leq\frac{\pi}{2}<\beta^{\prime}\leq\pi$. The function $T(E,\xi)$ is
the probability for quantum-mechanical transmission through the surface
potential and $B(E\eta|E^{\prime}\eta^{\prime})$ is the probability for
scattering inside the solid from the state $(E,\eta)$ to the state
$(E^{\prime},\eta^{\prime})$, where $\eta=cos\theta$ and
$\eta^{\prime}=|cos\theta^{\prime}|$ are the direction cosines inside the
solid. The upper limit to the energy integral accounts for the fact that the
impacting electron cannot gain energy. The lower limits to the angle and
energy integrals, $\eta_{\rm min}(E)$ and $E_{\rm min}(\eta^{\prime})$, as
well as the functions $\eta(\xi)$ and $\xi(\eta)$, which allow to switch
between internal and external direction cosines, depend on the depth $U_{0}$
of the surface potential and the effective mass $m_{e}^{*}$ of the electron in
the conduction band of the metal. The functions arise from the conservation of
lateral momentum and total energy.

In our previous work on dielectric surfaces Bronold and Fehske (2015, 2017b)
the scattering cascades were driven by an optical phonon. Neglecting its
dispersion, it was possible to approximately reduce the calculation of
$B(E\eta|E^{\prime}\eta^{\prime})$ to the solution of an algebraic recursion
relation. For metals, this is no longer possible because the finite density of
electrons in the conduction band makes electron-electron collisions a main
actor in the scattering cascades. The continuous energy losses which they
gives rise to cannot be treated algebraically. Since we found it necessary to
also include electron-impurity and electron-phonon scattering, the theoretical
treatment of metallic surface is quite involved. For the purpose of this
perspective we describe the approach only as much as it is necessary to make
our points. The technical details will be given elsewhere.

The function $B(E\eta|E^{\prime}\eta^{\prime})$ can be obtained from the
invariant embedding principle for electron backscattering from surfaces Dashen
(1964); Vicanek (1999); Glazov and Pázsit (2007) and a normalization procedure
which takes into account that electron-electron scattering provides two
possibilities for ending up in the final state $(E^{\prime},\eta^{\prime})$,
whereas for electron-impurity and electron-phonon scattering there is only
one. The principle leads to an equation for the unnormalized backscattering
probability $Q(E\eta|E^{\prime}\eta^{\prime})$. Forward and backward
scattering events are encoded into different kernels,
$K^{+}(E\eta|E^{\prime}\eta^{\prime})$ and
$K^{-}(E\eta|E^{\prime}\eta^{\prime})$, respectively, which can be obtained
from the golden rule scattering rates $W^{+}(E\eta|E^{\prime}\eta^{\prime})$
and $W^{-}(E\eta|E^{\prime}\eta^{\prime})$ and thus from the matrix elements
for electron-electron, electron-impurity, and electron-phonon interaction.

To figure out which electronic states should be used in the matrix elements, a
look at the universal curve for the mean free path of an electron inside a
solid Seah and Dench (1979) is helpful. It suggests that an electron hitting a
solid with energies below $100\,\mathrm{eV}$, which is the range most
important for plasma applications, penetrates rather deeply into the solid.
The matrix elements, and also the transition rates, are thus the ones for bulk
electronic states. Approximations which have been worked out for them can be
also employed. In the high temperature limit, for instance, the rate for
electron-phonon scattering reads in the quasi-elastic approximation Harrison
(1979),

$\displaystyle
W^{\pm}(E\eta|E^{\prime}\eta^{\prime})=\frac{\lambda(E)}{2}\frac{k_{B}T}{m_{*}k_{F}}\delta(E-E^{\prime})~{},$
(2)

where we used atomic units (measuring energy in Rydbergs, length in Bohr
radii, and mass in electron masses) and introduced an energy dependent
coupling function $\lambda(E)$. In this approximation the rate does not depend
on the direction cosines and the coupling function $\lambda(E)$ is all what
remains from the electronic structure. The situation where an energy barrier
prevents a primary electron with energy $E>0$ from entering the solid, will be
discussed in subsection III.1. It occurs for surfaces with negative electron
affinity.

The embedding approach separates forward and backward scattering. It is thus
rather straightforward to take into account that forward scattering hardly
changes the direction cosines. Assuming hence
$K^{+}(E\eta|E^{\prime}\eta^{\prime})$ to be strongly peaked for
$\eta=\eta^{\prime}$, a saddle-point integration can be employed leading at
the end to an integral equation for $Q(E\eta|E^{\prime}\eta^{\prime})$, where
$\eta$ and $\eta^{\prime}$ are only parameters. Suppressing them, the equation
to be solved reads

$\displaystyle Q(E|E^{\prime})=K^{-}(E|E^{\prime})$
$\displaystyle+\int_{E^{\prime}}^{E}dE^{\prime\prime}K^{+}_{1}(E|E^{\prime\prime};E^{\prime})Q(E^{\prime\prime}|E^{\prime})$
$\displaystyle+\int_{E^{\prime}}^{E}dE^{\prime\prime}Q(E|E^{\prime\prime})K^{+}_{2}(E^{\prime\prime}|E^{\prime};E)$
$\displaystyle+\int_{E^{\prime}}^{E}dE^{\prime\prime}\int_{E^{\prime}}^{E^{\prime\prime}}dE^{\prime\prime\prime}Q(E|E^{\prime\prime})K^{-}(E^{\prime\prime}|E^{\prime\prime\prime})Q(E^{\prime\prime\prime}|E^{\prime})~{}.$
(3)

Two forward scattering kernels, $K_{1}^{+}$ and $K_{2}^{+}$, appear because
the saddle-point integrations leading to the second and the third term on the
rhs were done with respect to different energy variables. Notice also, one of
the three energy variables in $K_{1}^{+}$ and $K_{2}^{+}$ is always unaffected
by the integrations.

To solve (3) numerically we expand $Q(E|E^{\prime})$ in the number of
backscattering events, that is, in powers of the kernel $K^{-}$. In each
order, a linear Volterra-type integral equation has then to be solved.
Empirically, we found for a silver surface convergence achieved at
$17^{\mathrm{th}}$ order. Compared to dielectric surfaces, where we only had
to solve an algebraic recursion relation, metallic surfaces are thus indeed
computationally expensive. The details of the numerics do not matter at this
point and will be described elsewhere. Important for the perspective is, the
surface scattering problem is solved but its solution depends, via the
kernels, on matrix elements entering the transition rates. The sticking
coefficient depends in addition also on the surface potential, defining the
quantum mechanical transition probability $T$. These are the quantities which
have to be obtained either from first principle calculations or from surface
diagnostics.

Let us now turn to results obtained for an electron hitting a silver surface
with energies less then $20\,\mathrm{eV}$, that is, with energies where the
separation into true secondaries and backscattered electrons is no longer
meaningful. Empirical formulae for the emission yield are thus not applicable.
For the data shown, we statically screened the electron-electron and electron-
impurity interaction. The coupling with phonons was considered in the quasi-
elastic high-temperature approximation as discussed above. To account for
image charge effects, we used moreover an exponential surface barrier ($z<0$,
see Fig. 2),

$\displaystyle V_{s}(z)=-\frac{U_{0}}{e^{-z/a}+1}$ (4)

with a width $a=0.26\,$Å and a depth $U_{0}=E_{F}+\Phi$, where
$E_{F}=5.5\,\mathrm{eV}$ and $\Phi=4.4\,\mathrm{eV}$. The step potential also
shown in Fig. 2 is recovered by setting $a=0$. But it turned out not to be
appropriate. Whereas $E_{F}$ and $\Phi$ are at least known for free-standing
Ag surfaces, the parameter $a$ is essentially unknown. We use it therefore as
a fit parameter. For $a=0.26\,$Å , we found best agreement with experimental
data Gonzalez  _et al._ (2017). The model (4) could be avoided by knowing
$V_{s}(z)$ either from first principle calculations or measurements. Since the
plasma may chemically and structurally affect the surface, it may also modify
$V_{s}(z)$. Calculations for $V_{s}(z)$ have thus to take the plasma into
account and the experimental work concerning this quantity has to be done in-
operando.

Figure 3: Energy and angle dependence of the sticking probability $S(E,\xi)$
for the Ag surface specified in the caption of Fig. 2. In contrast to what one
would expect, a metallic surface is not a perfect absorber for low energy
electrons, that is, for electrons with energies below a few tens eV. Diffuse
backscattering inside the metal makes the probability less than unity. It
approaches unity only for $\xi=1$ and $E\approx 5\,\mathrm{eV}$.

Figure 2 compares results for perpendicular incident with experimental data
Gonzalez  _et al._ (2017). Since the data are for the electron emission yield,
we plot $1-S(E,\xi)$ instead of $S(E,\xi)$. The agreement with the data is
rather good. To obtain it, we had to include electron-phonon scattering.
Electron-electron scattering alone was not enough. The structure of (3) gives
the reason. Being a Volterra-type integral equation, elastic backscattering,
encoded into the diagonal $K^{-}(E|E)$, acts as a seed for the solution
$Q(E|E^{\prime})$ with $E\geq E^{\prime}$. The elastic component of electron-
electron scattering, however, is too weak. It is proportional to
$k_{B}T/E_{F}$. To get the scattering cascade running, other scattering
processes are thus necessary. Electron-impurity scattering is also rather
weak, at least for reasonable impurity concentrations. In the absence of
Rutherford scattering on the nuclei (which may play a role), it is thus the
interplay of electron-phonon and electron-electron interaction which affects
most strongly the probability with which an electron scatters off a metal
surface at low energies. We used the electron-phonon coupling function
$\lambda(E)$ as an energy dependent fit parameter. For the results shown,
$\lambda(E)=0.1,0.3$, and $0.5$ for $0<E<5\,\mathrm{eV}$, $5\leq
E<8\,\mathrm{eV}$, and $E\geq 8\,\mathrm{eV}$ consistent with values expected
from studies of electron heating in solids Lin  _et al._ (2008). To avoid the
empirical strategy, $\lambda(E)$ could be calculated from a model for
electron-phonon interaction. Since the experimental data show no feature at
the bulk plasmon energy, static screening of the Coulomb interaction seems to
be in order. Although at room temperature electron-phonon dominates electron-
impurity scattering, we kept the latter for completeness assuming an impurity
concentration of $n_{\mathrm{imp}}=n/1000$, where $n$ is the electron density
in the conduction band of Ag which can be obtained from $E_{F}$. But it
actually has no effect on the results.

For the parameters reproducing the experimental data Gonzalez  _et al._ (2017)
we also calculated the angle dependence of $S(E,\xi)$ for the whole range
$0<\xi\leq 1$. It is shown in Fig. 3. The result is rather course grained but
a finer grid would have increased the numerical costs dramatically. For almost
all angles and energies $S(E,\xi)$ is less than unity. Only for $\xi=1$ and
$E\approx 5\,\mathrm{eV}$ is $S$ close to unity. Even a metallic Ag surface is
thus no perfect absorber for electrons. In preliminary calculations we
obtained similar good results for Cu and Au indicating that the semiempirical
model captures the essential physics involved in the backscattering of low
energy electrons from metallic surfaces. Combined with first-principle
calculations or measurements of the surface potential $V_{s}(z)$ and the
electron-phonon coupling function $\lambda(E)$ we expect the model to yield
reliable electron sticking coefficients also for other metals. Since the
original version of the model has already shown its usefulness for dielectrics
Bronold and Fehske (2015, 2017b), the lack of experimental data for $S(E,\xi)$
could indeed be compensated by calculations of the type described in this
subsection.

###  II.2 Kinetics of the electric double layer

From a plasma physics point of view, it is tempting to ignore the solid-state
physics behind the surface parameters. For instance, the scattering cascades
discussed in the previous subsection determining $S(E,\xi)$ are not part of
traditional plasma modeling. The same holds for the microscopic processes
determining secondary electron emission due to impacting heavy particles. In
both cases, the physics is outsourced to specialists. The prospect, however,
to calculate parameters which eventually are only buried in a plasma
simulation, acquiring thereby a merely supporting character, is not too
motivating. More rewarding, and hence motivating, for condensed matter
theorists is to overcome the parameters and to attempt a holistic modeling of
the plasma-solid interface. It is then at least conceivable to discover so far
overlooked scenarios arising from the interplay of processes taking place,
respectively, in the solid and the plasma. Getting away from supporting-type
calculations, we now consider the electronics inside plasma-facing solids as
an integral part of the physical inquiry, to be studied at the same footing as
the charge dynamics of the plasma it is bounding.

For this type of modeling electron sticking and secondary emission
coefficients (due to electron impact) are obsolete. All what is needed is the
quantum-mechanical probability $T$ for an electron crossing the surface
potential $V_{s}(z)$. We expect the holistic approach to be particularly
rewarding for semiconductor-based miniaturized microdischarges Eden  _et al._
(2013) which are closest to the interest of solid-state physicists. In these
systems transit times across the gas volume may be soon on the same order as
the transport times inside the solid. Between two subsequent electron
encounters the solid may thus stay in an excited state. Electron reflection
and sticking should hence depend on this state.

In passing let us be a bit more basic by pointing out a fundamental difference
between quantities characterizing collisions in the volume of the plasma
(cross sections) and quantities describing collisions with the solid embracing
the plasma (surface parameters). The former can always be calculated without
considering the plasma environment. They are independent of it. Surface
parameters, in contrast, may depend on it, because the plasma may in principle
modify the surface chemically, structurally, and electrically. Rigorously
speaking, the calculation of surface parameters has thus to take the plasma
into account. The feedback of the plasma onto the surface parameters may be
small in some instances. But from a fundamental point of view it is always
there. For the calculation of the sticking coefficient, for instance, the
modifications of the surface’s electronic structure due to the plasma has to
be considered. It is this modification which is the main theme of this
perspective.

Figure 4: Left: Interface model for an electric double layer with negative
space charge inside the solid and positive charge in front of it. Shown are
the edges of the conduction band ($U_{*}$) and valence band
($U_{\mathrm{vb}}$), the edges for the motion of valence band holes $(U_{h}$),
and the potential energies for electrons ($U_{e}$) and ions ($U_{i}$) on the
plasma side. Hole injection due to the neutralization of ions, electron/hole-
phonon scattering, and electron-hole recombination via traps in the energy gap
are also illustrated together with reflection and transmission (where it
applies) at the potential profile and the interface. The wall (sheath)
potential on the plasma side is $U_{p}$ while $U_{w}-U_{p}$ is the potential
drop due to the plasma source at $z=z_{w}$. Right: Potential energy and net
charge density profiles (upper panel) and distribution function for the
surplus electrons inside the solid at $z=0.13\,\lambda_{D}^{w}$ (lower panel)
as obtained from the numerical solution of (5) and (6) with the matching and
boundary conditions specified in the text. On the solid (plasma) side, the
potential is given in units of the band bending (sheath potential),
$U_{1}=-0.1~{}\mathrm{eV}$ and $U_{p}=4.7~{}\mathrm{eV}$, respectively, and
the densities are given in units of $n_{0}^{w}=-10^{13}~{}\mathrm{cm}^{-3}$
and $n_{0}^{w}=5\cdot 10^{13}~{}\mathrm{cm}^{-3}$. Distances from the
interface are measured in units of $\lambda_{D}^{w}=0.29~{}\mu\mathrm{m}$ and
$\lambda_{D}^{p}=3.7~{}\mu\mathrm{m}$. The model parameters are given in Table
1.

The set of equations for the holistic modeling of the charge dynamics on both
sides of the plasma-solid interface consists of Boltzmann equations for the
distribution functions of the involved charge carriers, the Poisson equation
for the electric field, and matching conditions at the interface. For the
dielectric plasma-solid interface shown in Fig. 4, the equations read (again
in atomic units measuring energy in Rydbergs, length in Bohr radii, and mass
in electron masses) Bronold and Fehske (2017a); Rasek  _et al._ (2020)

$\displaystyle\big{[}\pm v_{s}(z,E,K)\frac{\partial}{\partial
z}+\gamma_{s}(z,E,K)\big{]}F^{\gtrless}_{s}(z,E,K)$
$\displaystyle=\Phi_{s}^{\gtrless}(z)~{},$ (5)
$\displaystyle\frac{d}{dz}\varepsilon(z)\frac{d}{dz}U_{c}(z)$
$\displaystyle=8\pi n(z)~{},$ (6)

where we introduced distribution functions $F_{s}^{\gtrless}$ for left- ($<$)
and right ($>$) moving particles having, respectively, negative and positive
velocity components in $z-$direction, and split the collision integral in an
out-scattering and an in-scattering term, $\gamma_{s}$ and
$\Phi^{\gtrless}_{s}$, respectively. The index $s=*,h,e,i$, denotes,
respectively, conduction band electrons, valence band holes, electrons, and
ions. Independent variables are the spatial coordinate $z$ perpendicular to
the interface, the total energy $E$, and the kinetic energy $K$ in the lateral
dimensions. The function

$\displaystyle v_{s}(z,E,K)=2\sqrt{m_{s}^{-1}[E-U_{s}(z)-K]}$ (7)

is the magnitude of the velocity perpendicular to the interface with
$U_{i}=U_{c},U_{e}=-U_{c}$, $U_{*}=-U_{c}-\chi$, and $U_{h}=U_{c}+E_{g}+\chi$
the potential energies for ions, electrons, conduction band electrons, and
valence band holes. The source $n(z)$ of the Poisson equation is the charge
distribution of the double layer consisting of a net negative (positive)
charge inside the solid (plasma) to be selfconsistently obtained from the
distribution functions.

To complete the set of equations we need conditions for matching the half-
space solutions for the solid ($z<0$) and the plasma ($z>0)$ at the interface
($z=0$). For the electric potential energy $U_{c}$ the standard continuity
conditions of electrostatics apply, while the distributions functions obey

$\displaystyle F_{e,*}^{>,<}(0,E,K)$
$\displaystyle=[1-T(E,K)]F_{e,*}^{<,>}(0,E,K)$
$\displaystyle+T(E,K)F_{*,e}^{>,<}(0,E,K)~{},$ (8) $\displaystyle
F_{h}^{<}(0,E,K)$ $\displaystyle=F_{h}^{>}(0,E,K)+S_{h}^{<}(E,K)~{},$ (9)
$\displaystyle F_{i}^{>}(0,E,K)$ $\displaystyle=0~{},$ (10)

where $T(E,K)$ is the quantum-mechanical electron transmission coefficient for
the surface potential $V_{s}(z)$ and $S^{<}_{h}(E,K)$ is a function describing
the injection of a valance band hole due to neutralization of an ion. For
simplicity it is assumed that an ion hitting the surface is resonantly
neutralized with unit probability but Auger neutralization could be also
included. The source function $S^{<}_{h}(E,K)$ requires a model for hole
injection and a normalization to ensure the equality of electron and ion
fluxes at the interface, $j_{e}(0)=j_{i}(0)$, which has to be also satisfies.
Augmented by boundary conditions, ensuring quasi-neutrality and the absence of
electric fields in the bulk regions of the solid and the plasma, respectively,
as well as plasma generation on the plasma side of the interface, the
equations encode the chain of events shown in Fig. 1. Since the scattering
inside the solid may bring electrons back to the interface, where they may be
transmitted to the plasma, the electron emission yield and its complement, the
sticking coefficient, could be also obtained from the present scheme.

So far we considered a floating dielectric solid in contact with a plasma,
keeping the model as simple as possible without affecting its main mechanisms.
Details can be found in the literature Rasek  _et al._ (2020). For the purpose
of this perspective it suffices to list its main features. The plasma is
generated by a source selfconsistently attached deep inside the plasma by a
standard construction Schwager and Birdsall (1990). The interface is perfectly
absorbing from the plasma and impenetrable from the solid side. Due to
numerical constraints, the energy domain is truncated, requiring to enclose
electron and hole injection into effective (Gaussian) source functions,
centered with a width $\Gamma^{\mathrm{in}}$ around
$E-U_{*,h}=I_{*,h}^{\mathrm{in}}$, below the actual injection points, which
are too far above the band edges to be numerically accessible at the moment.
The physics of the model is however unaffected by this construction.
Collisions, finally, are only included on the solid side, where electrons and
holes relax due to scattering on an optical phonon and recombine
nonradiatively via traps in the energy gap of the dielectric. Again due to
numerical constraints, at the moment we have to take an artificially high trap
density. The plasma in front of the solid is collisionless. We also included a
finite background doping by acceptors. Needless to say, the truncation of the
energy domain as well as the high trap density can be avoided by investing
into computing power.

Numerically the model can be solved by rewriting the kinetic equations (5) as
integrals and applying an iterative approach initially developed for solid-
solid interfaces Grinberg and Luryi (1992); A. R. St. Denis and D. L. Pulfrey
(1998); Konistis and Hu (2002). Representative results are plotted on the rhs
of Fig. 4 for the model parameters given in Table 1. The absolute numbers,
which depend on the truncations, are not of main concern at this point. More
important is that a working scheme has been setup which extends the kinetic
modeling into the solid. Let us first have a look at the potential and charge
density profiles. Due to the difference in the Debye screening lengths, the
charge neutrality of the double layer is not obvious from the plot but of
course satisfied. The kink in the potential profile at $z=0$ is due to the
difference in the dielectric functions of the solid and the plasma, to be
taken as $\varepsilon=11.8$ and $\varepsilon=1$, respectively. Also seen can
be the band bending $U_{1}$ induced by the surplus electrons inside the plasma
and the sheath potential $U_{p}$ on the plasma side. The model determines both
selfconsistently, together with the strength of the plasma source, which is
also no more a free parameter but fixed by the scattering and recombination
processes inside the solid Rasek  _et al._ (2020).

The distribution function for the solid’s surplus electrons originating from
the plasma is plotted in the lower panel of the rhs of Fig. 4 for a spatial
position immediately after the interface. Left- and right-moving
distributions, $F_{*}^{<}$ and $F_{*}^{>}$, are distinguished by attaching an
artificial sign to the variable $K_{z}=E-U_{*}-K$. Three features can be
identified: (i) The peak at $E-U_{*}=0.5\,\mathrm{eV}$ due to electron
injection, (ii) the replicas of this peak due to electron-phonon scattering,
and (iii) the step at $K_{z}=0$, separating left- ($K_{z}<0$) from right-
moving $(K_{z}>0$) electrons. Since the latter can only arise due to
backscattering, which is less likely then forward scattering, the distribution
function for right-moving electrons is suppressed compared to the distribution
function for left-moving ones. Also seen is the overall decay of the
distribution function in the variable $E-U_{*}$, signaling that the surplus
electrons pile up at $E=U_{*}$, the bottom of the conduction band, and vanish
high above it.

Table 1: Material parameters used to obtain the results shown on the rhs of
Fig. 4. Conduction band electrons and valence band holes scatter on an optical
phonon with energy $\hbar\omega_{0}$ and recombine nonradiatively via traps at
energy $E_{t}$ (measured with respect to the valence band edge), having a
density $N_{t}$ and a capture cross section $\sigma_{t}$. The dielectric has
an energy gap $E_{g}$, an intrinsic density $n_{\mathrm{int}}$, and is doped
with an acceptor density $n_{A}$. As discussed in the text, we need source
functions $S_{*,h}^{<}$ for injecting electrons and holes. They are defined in
Rasek  _et al._ (2020) and require the parameters $\Gamma^{\mathrm{in}}$ and
$I_{*,h}^{\mathrm{in}}$. The remaining parameters are the thermal energies and
masses of the charge carriers.
$\hbar\omega_{0}[\text{meV}]$ | 0.1
---|---
$E_{t}[\text{eV}]$ | 0.35
$N_{t}[\text{cm}^{-3}]$ | $10^{20}$
$\sigma_{t}[\text{cm}^{2}]$ | $10^{-15}$
$E_{g}[\text{eV}]$ | 1
$n_{\mathrm{int}}[\text{cm}^{-3}]$ | $5\cdot 10^{10}$
$n_{\mathrm{A}}[\text{cm}^{-3}]$ | $10^{14}$
$\Gamma^{\mathrm{in}}[\text{eV}]$ | 0.1
$I_{*,h}^{\mathrm{in}}[\text{eV}]$ | 0.5
$k_{B}T_{*,h,i}[\text{eV}]$ | 0.025
$k_{B}T_{e}[\text{eV}]$ | 2
$m_{*,h,e}[m_{e}]$ | 1
$m_{i}[m_{e}]$ | 1836

The overall picture encoded in Fig. 1 is thus nicely emerging from the kinetic
theory spelled out in this section: Electron and ions created by a plasma
source recombine inside the (dielectric) solid as conduction band electrons
and valence band holes, after energy relaxation and transfer through the
surface potential $V_{s}(z)$. The strength of the source is fixed by the
electron microphysics inside the wall, which is thus of equal importance as
the processes creating electrons and ions inside the plasma. Obviously, to set
up this type of modeling, the electronic structure of the interface has to be
known. It affects the electron transmission coefficient $T(E,K)$ as well as
the scattering channels which have to be taken into account. For instance, in
case surface states are present, surplus electrons may not only scatter into
bulk states of the conduction band but also into surface states. Energy
barriers, that is, the electron affinity or the work function, are also
affected by surface states because they are usually charged giving rise to
band bendings and surface dipoles. Hence, a quantitative description of the
double layer at a plasma-solid interface, in particular, of its solid-based
part, requires input data which can be only obtained by making the electronic
structure of plasma-facing solids the object of experimental and theoretical
inquiry.

###  II.3 Infrared diagnostics for the wall charge

Based on assumptions about the electronic structure of a dielectric plasma-
solid interface we presented in the previous section a kinetic theory which
determines the distribution functions and the depth profiles of surplus
electrons and holes making up the wall charge. To test and guide theoretical
approaches of this kind, it is also necessary to access the wall charge by
experimental techniques. So far, methods exist for measuring the accumulated
charge per unit area by electric probes Kindel and Arndt (1980),
optomechanical sensors Pangal  _et al._ (1996), and the optoelectric Pockels
effect Slikboer  _et al._ (2019). The latter allows also to extract the
lateral charge distribution Kawasaki  _et al._ (1994); Tschiersch  _et al._
(2014). No attempts have been made, however, to measure charge profiles
perpendicular to the interface or to determine the electronic states hosting
the charge. Thermostimulated current Li  _et al._ (2008) and luminescence
Ambrico  _et al._ (2014, 2009) techniques have been used to estimate the
binding energy of an electron trapped in or onto a plasma-facing dielectric,
but the electronic states involved could not be determined by them. Hence,
there is a need to improve the charge diagnostics. In particular, it is
necessary to make it microscopic enough to explore charge distributions inside
the solid. Combined with theoretical modeling, the character of the electron
states the wall charge is bound to (bulk vs. surface states) could then be
also uncovered.

In this subsection we discuss the possibility of using infrared reflection
spectroscopy as a diagnostics for the charge inside a plasma-facing dielectric
material. In contrast to our previous proposals for using optical Rasek  _et
al._ (2018) or electron Thiessen  _et al._ (2019) spectroscopy to determine
the charge of a planar plasma-facing solid, the new proposal does not rely on
a from-the-back geometry and does also not require a layered structure.
Instead, we now suggest to use the dielectric wall as an internal reflection
element. The charge is thus determined by passing p-polarized infrared light
directly through the wall. Since we base our analysis in addition also on
surface response functions Kempa and Gerhardts (1985) we are now able to treat
inhomogeneous charge distributions. The homogeneous Drude model we used
previously for the charge residing in the wall can thus be avoided and with it
the artificial confinement of the charge by layering the solid structure in
contact with the plasma.

The idea of the proposal is shown on the lhs of Fig. 5. Infrared light passes
through a dielectric parallelepiped which serves at the same time as the wall
of the plasma. Provided the optical loss inside the dielectric is small, the
wall can be macroscopically thick and hence mechanically stable. The angle of
the entrance surface is large enough to ensure total reflection at the plasma-
solid interface. On the opposite interface the parallelepiped should be
polished to ensure perfect reflectivity. Since total reflection at the plasma-
solid interface depends on the dielectric function in the vicinity of the
interface, and hence on the charge distribution around it, the intensity of
the transmitted light should be sensitive to the wall charge. Indeed,
exploratory calculations show the feasibility of the scheme. We will now
present it in some detail hoping to motivate experimentalists to implement the
approach.

Figure 5: Left: Principle of the experimental setup for measuring the charge
accumulated inside a plasma-facing dielectric by infrared spectroscopy. The
transmitted light through the dielectric serving as an internal reflection
element is detected in the vicinity of the wave number where internal
reflection sets in, that is, where the real part of the dielectric function
becomes larger than unity. For sapphire $\lambda^{-1}\approx
1040\mathrm{cm}^{-1}$ (see dashed line in the plot for
$\varepsilon(\lambda^{-1})$). Right: Theoretically predicted change of
transmissivity $\Delta T=T-T_{0}$, where $T$ and $T_{0}$ are, respectively,
the transmissivities for plasma-on and plasma-off, as a function of $n_{s}$
for a reflection element made out of sapphire. Fitting experimental data for
$\Delta T$ to theoretical curves, arising as discussed in the main text from
models for the space charge layer inside the solid, will enable one to
determine $n_{s}=\int dzn(z)$ and maybe even the charge density profile $n(z)$
itself.

In the incoherent limit, where the thickness of the optical element is much
larger than the wavelength of the light, the transmissivity $T$ of the
element, that is, the ratio of the transmitted $(I_{T})$ to the incoming
$(I_{0})$ light intensity is given by Milosevic (2012)

$\displaystyle
T=\frac{I_{T}}{I_{0}}=\frac{R_{s}^{N}(1-R_{0})^{2}}{1-R_{s}^{2N}R_{0}^{2}}$
(11)

with $R_{0}$ the reflectivity at the entrance and exit interface and $R_{s}$
the reflectivity at the plasma-solid interface. Whereas $R_{0}$ is simply
given by the Fresnel formula for perpendicular incident at a dielectric-vacuum
interface, $R_{s}$ is a non-Fresnel reflectivity for angle of incident
$\theta$. It takes the charge inhomogeneities at the dielectric-plasma
interface into account. Compositional or structural inhomogeneities could be
also included. But we focus in this subsection only on the charge.

In order to calculate $R_{s}$ we employ an approach based on surface response
functions Kempa and Gerhardts (1985). Indexing the materials as shown in Fig.
5 and denoting Fresnel reflection coefficients between media $i$ and $j$ by
$\bar{r}_{ij}$, the reflectivity at the entrance and exit interfaces reads
$R_{0}=|\bar{r}_{12}|^{2}$, while $R_{s}=|r_{23}|^{2}$ with (suppressing the
$\omega=2\pi c/\lambda$ dependence where it applies) Kempa and Gerhardts
(1985)

$\displaystyle r_{23}=\bar{r}_{23}(1+C_{23})$ (12)

and

$\displaystyle
C_{23}=2ip_{2}\frac{p_{3}^{2}\bar{\varepsilon}_{2}d_{\parallel}-k^{2}\bar{\varepsilon}_{3}d_{\perp}}{\bar{\varepsilon}_{2}p_{3}^{2}-\bar{\varepsilon}_{3}k^{2}}~{},$
(13)

where $\bar{\varepsilon}_{i}$ and $p_{i}$ denote the (homogeneous) bulk
dielectric function and the perpendicular component of the wavevector of the
light in medium $i$, $k$ is the conserved parallel component of the
wavevector, and

$\displaystyle d_{\perp}$ $\displaystyle=\frac{\int
dz\big{\\{}\varepsilon_{zz}^{-1}(z)-\big{[}\bar{\varepsilon}_{2}^{-1}\theta(-z)+\bar{\varepsilon}_{3}^{-1}\theta(z)\big{]}\big{\\}}}{\bar{\varepsilon}_{2}^{-1}-\bar{\varepsilon}_{3}^{-1}}$
(14) $\displaystyle d_{\parallel}$ $\displaystyle=\frac{\int
dz\big{\\{}\varepsilon_{xx}(z)-\big{[}\bar{\varepsilon}_{2}\theta(-z)+\bar{\varepsilon}_{3}\theta(z)\big{]}\big{\\}}}{\bar{\varepsilon}_{2}-\bar{\varepsilon}_{3}}$
(15)

are the surface response functions. They depend on integrals,

$\displaystyle\varepsilon_{xx}(z)$ $\displaystyle=\int
dz^{\prime}\varepsilon_{xx}(z,z^{\prime})~{},$ (16)
$\displaystyle\varepsilon^{-1}_{zz}(z)$ $\displaystyle=\int
dz^{\prime}\varepsilon^{-1}_{zz}(z,z^{\prime})~{},$ (17)

over the nonlocal dielectric function (and its inverse) containing the charge
inhomogeneity via a Drude term. The $z$-integrals run over the dielectric
($z<0$) as well as the plasma halfspace ($z>0$).

The central object is the dielectric function. In tensor notation with respect
to the spatial coordinates,

$\displaystyle\underline{\varepsilon}(z,z^{\prime})=\underline{1}\delta(z-z^{\prime})\big{[}\bar{\varepsilon}_{2}\theta(-z)+\bar{\varepsilon}_{3}\theta(z)\big{]}+\frac{4\pi
i}{\omega}\underline{\sigma}(z,z^{\prime})~{},$ (18)

where $\underline{\sigma}(z,z^{\prime})$ is the conductivity tensor. It can be
obtained from the kinetic theory described in the previous subsection by
including an additional force term due to the electric field of the infrared
light send through the wall and linearizing the new set of Boltzmann equations
around the solution of the electric double layer. From the electric current
produced by this procedure $\underline{\sigma}(z,z^{\prime})$ can be
identified. The dielectric function follows straight and its inverse can be
obtained numerically.

We did not yet implement the full scheme. To obtain first results we adopted a
local approximation. Under the assumption that the charge inhomogeneities are
due entirely to conduction band electrons and ions on the solid and plasma
side of the interface, respectively, we write

$\displaystyle\underline{\varepsilon}(z,z^{\prime})=\underline{1}\delta(z-z^{\prime})\big{[}\varepsilon_{2}\theta(-z)+\varepsilon_{3}\theta(z)\big{]}$
(19)

with

$\displaystyle\varepsilon_{i}=\bar{\varepsilon}_{i}-\frac{4\pi
e^{2}}{m_{i}}n_{i}(z)~{},$ (20)

where $m_{2}=m_{*}$, $m_{3}=m_{+}$, $n_{2}(z)=n_{*}(z)$, and
$n_{3}(z)=n_{+}(z)$. Going through the formulae for the surface response
functions (14) and (15), one realizes that for $m_{+}\gg m_{*}$ the integrals
over the plasma side of the interface can be neglected in leading order. The
functions depend then only on $n_{*}(z)$. Using, for purpose of demonstration,
a rough model $n_{*}(z)\sim e^{z/a}$, the integrals can be work out easily.
Normalizing finally $n_{*}(z)$ over the width of the reflection element to a
total surface density $n_{s}$, we get at the end the transmissivity (11) as a
function of $n_{s}$.

Results obtained for this simple model are shown on the rhs of Fig. 5. The
wavenumbers of interest are the ones around the threshold for total reflection
at the plasma-solid interface, defined by $\bar{\varepsilon}(\lambda^{-1})=1$.
Taking sapphire as an example, $\lambda^{-1}\approx 1040\,\mathrm{cm}^{-1}$,
as can be seen from the dielectric function on the left. To determine $n_{s}$
experimentally, it is best to focus on the change of transmissivity when the
plasma is turned on. For plasma off, $n_{s}=0$ and–since we neglect other
possibilities of inhomogeneities–$R_{s}=\bar{R}_{s}=|\bar{r}_{23}|^{2}$, while
for plasma on $R_{s}=\bar{R}_{s}|1+C_{23}|^{2}$. Inserted into (11) two
transmissivities, $T$ and $T_{0}$, result whose difference $\Delta T=T-T_{0}$
is shown on the right. A clear signature can be observed for values of $n_{s}$
typical for dielectric barrier discharges. The magnitude of $\Delta T$ is in
this example rather small, but a photodetector with a high enough sensitivity
should be able to measure it. In other applications of reflection
spectroscopy, sensitivities up to $10^{-3}$ have been achieved decades ago
Ling  _et al._ (1993); Nitschke and Meichsner (1997). We expect modern
instrumentation to be actually better and are thus convinced that the proposed
method can be realized. It is also conceivable to place the optical element
between two highly reflecting mirrors and to measure the absorbance $A=-\log
T$ of the element by cavity ring down spectroscopy O’Keefe and Deacon (1988)
which is known to be an extremely sensitive technique. The mirrors can be also
integrated into the optical element itself Peeters  _et al._ (2017); Aarts
_et al._ (2007, 2005) by coating the entrance and exit interfaces
appropriately. In the next section we will say more about this particular
technique because it can be perhaps also applied to investigate in-operando
the infrared active parts of the electronics of plasma-facing solids.

Figure 6: Illustration of three possibilities for the band structure of a
dielectric surface. Which one may be realized depends on the plasma exposure.
(a) Positive electron affinity without surface states. The model we currently
employ. (b) Positive electron affinity with surface donors ($D^{+}$) or
acceptors ($A^{-}$) leading to band bending. (c) Negative electron affinity
leading to image states in front of the surface. In situations (b) and (c),
the impinging electron may also scatter into the respective surface states. As
in Fig. 2, the origin of the energy axis coincides with the potential just
out-side the surface.

The advantage of the charge diagnostics described in this subsection is that
it requires only the material to be transparent for infrared light. Many
dielectrics of interest for plasma applications obey this criterion. With this
type of spectroscopy, it is also possible to monitor in-operando chemical and
structural modifications of the plasma-solid interface, which in general lead
also to changes in the dielectric function around the interface and hence to
non-Fresnel reflectivities from which information about the modifications is
gained. For that purpose the standard modus of operating reflection
spectroscopy is employed: looking for changes in transmissivity above the
threshold for total reflection at the plasma-solid interface. The charge
diagnostics, on the other hand, focuses on changes close to the threshold.

##  III Work we hope to initiate

In the previous section we continued our ongoing research program on electron
kinetics at plasma-solid interfaces, comprising the calculation of surface
parameters, the modeling of the electric double layer, and a proposal to
measure the solid-bound part of the double layer by infrared reflectivity. The
models on which our work is based employ parameters and functions associated
with the electronic structure of the plasma-facing solid. Particularly
important are energy barriers, that is, for metals the work function and for
dielectrics the electron affinity. For the calculation of the electron
sticking coefficient, the shape of the surface potential and the electron-
phonon coupling function turned out to be also essential, while the modeling
of the electric double layer required, among others, information about
recombination cross sections and trap densities. All these quantities are
related to the electronic structure of the plasma-facing solid. To make the
models predictive, it is thus necessary to know as much as possible about it,
either from experiment or ab-initio theory.

First-principle calculations of the electronic structure are only practical
for properties which are not affected by the presence of the plasma. The
electron-phonon coupling function used in the calculation of the electron
sticking coefficient, for instance, is such a quantity. It depends only on the
bulk electronic structure which is shielded from the influence of the plasma.
Energy barriers, in contrast, are not. They depend on what the plasma
initiates chemically and structurally on the surface. A calculation of the
barriers requires therefore to treat the interaction between a plasma and a
solid in all its electrical, structural, and chemical manifestations. A
hopeless endeavor–at least if it is unguided by experimental investigations of
the electronics, structure, and chemistry of the interfaces.

More promising is to determine the information about the electronic structure
of the plasma-solid interface experimentally by photon and electron
spectroscopy Brillson (2010). Energy barriers, the energetic position of
surface states, and the surface potential, to name only the most important
electronic interface parameters, can be for instance measured by photoemission
spectroscopy. Up to now, this type of spectroscopy has neither been performed
ex-sito (plasma off) nor in-operando (plasma on) on a plasma-facing solid.
Both modii are challenging because the sample to be investigated is outside
the vacuum equipment required for the electron beams part of the interface
diagnostics. Primarily because in-operando techniques can also provide
information about the depth profile of the wall charge, we argue in this
subsection for setting up in-operando experiments. Although more complicated
then ex-sito experiments, they would yield insights about the electronics of
the plasma-solid interface which we never had before.

###  III.1 Motivation for in-operando diagnostics

Besides being able to explore the profile of the wall charge there is also a
fundamental reason favoring in-operando over ex-sito characterizations of the
plasma-solid interface.

The electronic structure of a surface is a thermodynamic property, arising
from the minimization of its free energy Bechstedt (2003). As a result, the
positions and bondings of the atoms in the first few crystallographic planes
of the surface usually differ from the ones appearing in the simply truncated
corresponding solid. It is this reconstruction of the surface which determines
the electronic structure and hence also the energy barriers electrons have to
overcome if they want to enter or leave the solid. The plasma exposure affects
of course the positions of the atoms on the surface, due to particle and
energy influx. The minimization of the free energy the surface has to perform
is thus constrained by the plasma. It could thus well be that the in-operando
electronic structure of a plasma-facing solid strongly deviates from its ex-
sito counterpart. Characterizing it ex-sito may thus not be sufficient.

To motivate in-operando experiments further let us have a look at the
electronics which may take place at plasma-facing dielectrics. Due to their
relevance for solid-bound microdischarges Kulsreshath  _et al._ (2012); Eden
_et al._ (2013); Tabib-Azar and Pai (2017); Michaud  _et al._ (2018) we focus
on this class of materials.

The electronic structure of a dielectric surface is strongly affected by its
termination. Particularly the presence of surface states depends on it, which
in turn affects the distribution of intrinsic and extrinsic charges across the
surface. The former may lead to a surface dipole, and hence to a modification
of the energy barrier an electron has to overcome by leaving or entering the
solid, while the latter concerns the solid-bound part of the double layer.
Surface states may also open up additional channels for electron capture from
the plasma. As shown in Fig. 6, an electron impinging on an interface with
surface states may not only scatter into bulk states (as assumed in subsection
II.1) but also into acceptor-like ($A^{-}$) or donor-like ($D^{+}$) states at
the interface. Besides a wide space charge layer, the negative leg of the
double layer may thus also consist of a strongly localized part. A
particularly interesting situation arises for dielectrics with negative
electron affinity, where electrons may be trapped in front of the surface by
polarization-induced image states Heinisch  _et al._ (2011, 2012), not unlike
to what happens to electrons on top of a liquid helium film Cole (1974).
Materials with this property are diamond Rutter and Robertson (1998); Cui  _et
al._ (1998); Yamaguchi  _et al._ (2009), boron nitride Loh  _et al._ (1999),
and MgO Rohlfing  _et al._ (2003); McKenna and Shluger (2008). Since the
plasma affects the termination of a plasma-facing dielectric chemically as
well as structurally, its electronic structure, especially the important class
of surface states, will depend on the plasma. It thus has to be studied in-
operando.

Figure 7: Example for the dependence of the energetic position of a surface
state on the arrangement of the atoms in the first few atomic layers, that is,
on the reconstruction of the surface. Shown is the band structure for the
$(10\bar{1}0)$ surface of ZnO together with its Brillouin zone. The position
of the surface state $S_{1}$ near the top of the valence band before (after)
reconstruction is indicated by the dot-dashed (solid) line. The reconstructed
surface has further surface states $S_{i}$ (solid lines) as well as surface
resonances (dashed lines). Anticipating now the $(10\bar{1}0)$ ZnO surface is
exposed to a plasma. Due to the permanent influx of energy and particles, the
geometric ordering of the surface atoms is not clear. Hence, it is uncertain
where, if present at all, in the in-operando band structure the surface state
$S_{1}$ will be sitting. Adapted with permission from Wang and Duke, Surf.
Sci. 192, 309 (1987) Wang and Duke (1987).

To make this point more explicit let us discuss ZnO as an example Wang and
Duke (1987); McNeill  _et al._ (2016). From band structure calculations it is
known that the electronic structures of reconstructed and non-reconstructed
ZnO surfaces are different Wang and Duke (1987). In particular, the energetic
position of the surface states depends on the organization of the atoms in the
first few atomic layers. For the $(10\bar{1}0)$ surface this can be seen in
Fig. 7. Since the energy and particle flux from the plasma disturbs the atoms
in the top layers, the position of the surface state $S_{1}$ may change in the
course of plasma exposure. The state may be even absent and the in-operando
electronic structure completely different from the one of the free-standing
reconstructed $(10\bar{1}0)$ ZnO surface. Only in-operando diagnostics could
tell if this is indeed the case. Chemical modifications of the electronic
structure of a plasma-facing ZnO surface are also conceivable. Adlayers of
H-atoms and OH-groups, for instance, affect the band bending at a ZnO surface
McNeill  _et al._ (2016). Traces of these substances inside the plasma will
thus most probably influence the electron kinetics across a plasma-ZnO
interface –via the surface states the band bending must be associated with,
which in turn also affect energy barriers and capture cross sections. In-
operando techniques could provide information about the chemically modified
electron kinetics at the plasma-ZnO interface even if the adlayers are only
present while the plasma is on.

Having argued up to this point that the modeling of the electron microphysics
at the plasma-solid interface depends on the in-operando electronic structure
of the plasma-facing solid, and hence it should be investigated
experimentally, the question remains, are the experiments good for anything
more than only providing input parameter for the modeling. In our view, they
are because they may help to establish a new research arena at the
intersection of plasma and surface physics.

In this arena it would be possible, for instance, to work towards designing
the electric properties of plasma-solid interfaces by a judicious choice of
the solid and the feedstock gas. The interface resistance could perhaps be
engineered as well as the shape of the double layer. Material science provides
an almost inexhaustible reservoir of materials with surfaces having rather
sensitive electronic structures. Subjecting them to various low-temperature
plasmas could be part of a systematic search for discharges with new operation
modii or functionalities.

We already listed materials with negative electron affinity. Especially
diamond, whose electron affinity can be tuned chemically from positive to
negative Maier  _et al._ (2001); Rietwyk  _et al._ (2013); Beattie  _et al._
(2019), is an interesting candidate for establishing new types of low-
temperature gas discharges. The tuning of the electron affinity can very well
be performed by the plasma itself. It is thus conceivable to come up with a
plasma-diamond interface with tailor-made electron affinity and hence electron
microphysics. Using diamond layers in a dielectric barrier discharge and
tuning the electron affinity from positive to negative by changing the
chemical composition of the feedstock gas, while simultaneously measuring the
electron affinity and some key plasma parameters, could be a research project
in this new arena.

Another project could involve the photocatalyst $\mathrm{TiO}_{2}$. The
electronic structure of its surface can be controlled by oxygen and UV light
T.C. Rödel, F. Fortuna, F. Bertran, M. Gabay, M. J. Rozenberg, A. F.
Santander-Syro and P. Le Févre (2015); Yin  _et al._ (2018). Using
$\mathrm{TiO}_{2}$ in a barrier discharge, whose feedstock gas contains traces
of oxygen, and monitoring its electronic structure together with the plasma
may thus be also an interesting study. Many more projects are conceivable and
could be performed once the tools of in-operando interface diagnostics are in
place.

###  III.2 Implementation of in-operando diagnostics

Experimental probes most suited for investigating the geometry, chemical
composition, and electronic structure of free-standing surfaces are electron
and photon spectroscopy Brillson (2010). Applying them in-operando also to
plasma-facing solids would yield a host of data we so far have no access to.
Unfortunately, the presence of the plasma prevents the techniques to be
applied directly to the interface of interest. The electric field in the
sheath disturbs incoming and outgoing electron beams making electron and
photoemission spectroscopy (which involves an outgoing electron) from the
front impossible. The standard setups do not work. Thus, one has to come up
with alternatives. Two are shown in Fig. 8: The spinning wall Guha  _et al._
(2007) and the from-the-back geometry.

Figure 8: Two possibilities for performing in-operando photoemission and
electron energy loss spectroscopy of a plasma-facing solid. (a) Spinning wall
setup Guha  _et al._ (2007) and (b) from-the-back geometry. In photoemission
spectroscopy a photon ($\gamma$) of fixed energy excites the electronic system
of the target. The information about its electronic structure is then encoded
into photoelectrons as well as secondary electrons. Electron energy loss
spectroscopy, on the other hand, is an electron reflection technique. It
utilizes the coupling of the incoming electron $(e)$ to the dipole fields of
the charge fluctuations inside the target. Detecting the nearly specularly
reflected electron beam enables then to read out information about the charge
distributions. The thick red line indicates the interface of interest.

Let us first discuss the spinning wall technique. It is shown on the lhs of
Fig. 8 and has proven its feasibility for in-operando Auger electron
spectroscopy Guha  _et al._ (2007). We expect it therefore to be also suitable
for other types of electron and photon spectroscopy. In particular,
photoemission spectroscopy Brillson (2010) for chemical, structural, and
electronic analysis could be performed in such a setup. The trick of the wall
is to alternately expose the surface to the plasma environment and the vacuum
necessary for its diagnostics. The simplest way to do this is to use a
cylinder made out of (or covered with) the material to be studied, place it
properly sealed inside the wall of the discharge vessel, and rotate it with a
constant velocity. Since the diameter of the cylinder is a few $\mathrm{cm}$,
on the atomic scale the surface remains flat. The photon beam hitting in a
photoemission study the circumference of the cylinder on the vacuum side of
the device, and hence in the standard manner, probes thus an atomically flat
surface. It may thus even be possible to investigate it by photoemission
electron microscopy Ade  _et al._ (1998).

A drawback of the spinning wall is that it is only in-operando for plasma-
induced processes persisting at least for the time a full rotation takes. The
modification of the surface due to the plasma, be it chemical, structural or
electrical, should also not be undone by passing through the seals. For the
rotation velocity used in the Auger electron diagnostics of the plasma wall
Guha  _et al._ (2007), processes decaying slower than a few milliseconds could
de-facto be observed in-operando. Chemical desorption takes place on a much
longer time scale. Hence, chemical modifications due to adlayers and the
changes in the electronic structure they give rise to (energy barriers, band
bending etc.) stay intact during the rotation and should thus be observable by
this technique. Provided surplus electrons making up the wall charge stay long
enough on the surface, the spinning wall can be also used to measure their
total amount per unit area by electron energy loss spectroscopy to be
discussed at the end of this section. Indeed, electron residence times on
dielectric surfaces can be very long. On a bismuth silicon oxide (BSO)
Tschiersch  _et al._ (2014) or a sapphire Ambrico  _et al._ (2009, 2014)
surface, for instance, some electrons appear to be trapped for at least
minutes, long enough to be even measurable by ex-sito setups, that is, when
the plasma is off. The depth profile of the wall charge, however, a quantity
we are particularly interested in, cannot be determined by a spinning wall
setup since the restoring force of the sheath, which affects the profile, is
absent when the surface is on the vacuum side for diagnostics. It may be
however explored in a from-the-back geometry to which we now turn.

Such a setup is shown on the rhs of Fig. 8. It is based on a layered
structure, which is thin enough to allow information about the plasma-solid
interface to be read out from the interface opposite to it and at the same
time thick enough to guarantee mechanical stability. Experimentally, one now
faces the problem of investigating a buried interface. The progress made in
this field, especially with respect to buried liquid-solid Liu  _et al._
(2020); R. Hausbrand, M. Fingerle, T. Späth, and C. Guhl (2017); Barritt  _et
al._ (2010); Lange  _et al._ (1983) and solid-solid Varsha and Nageswaran
(2020); Zborowski  _et al._ (2018); Tselev  _et al._ (2016); Copuroglu  _et
al._ (2013); Rubio-Zuazo and Castro (2008) interfaces, where information
depths up to $70~{}\mathrm{nm}$ have been realized Zborowski  _et al._ (2018),
makes us rather optimistic that the from-the-back setup may actually work for
the plasma-solid interface.

The challenge is to have an information depth large enough to allow structures
to be build which are also mechanically stable. Since in photoemission
spectroscopy the information is carried by electrons, the thickness of the
stack cannot exceed the inelastic mean free path for an electron. From the
universal curve Seah and Dench (1979) it then follows that, if at all, the
method may work for electrons with rather low or rather high energy. For them
the mean free paths are longest, on the order of $10-100$ monolayers. In
practice, the method is thus limited to sub-100 nm thick structures and
electron energies of a few eV or a few keV. The mechanical stability of
sub-100 nm thick structures is obviously a critical issue. Fortunately, there
are materials such as $\mathrm{SiO}_{2}$, $\mathrm{Al}_{2}\mathrm{O}_{3}$, and
$\mathrm{Si}_{3}\mathrm{N}_{4}$ which are hard and robust enough to make such
a setup conceivable. In particular, $\mathrm{Si}_{3}\mathrm{N}_{4}$ has proven
its usefulness as a sub-100 nm window in from-the-back microscopy at vacuum-
liquid Tselev  _et al._ (2016) as well as vacuum-plasma interfaces Tai  _et
al._ (2013); Tselev  _et al._ (2018). It could thus be coated with the
material of interest and inserted into the wall as shown on the rhs of Fig. 8.

A number of technical details beyond mechanical stability have to be of course
also clarified before experiments of this sort can be put into place. Not only
the vacuum side of the setup, where the diagnostics takes place, has to be
designed carefully. The plasma chamber with its recess for the measuring
window needs also attention. In case the setup is utilized to study a floating
plasma-solid interface, the recess has to be electrically isolated from the
rest of the wall. The measuring window in turn has to be optimized by model
calculations for different stacks of materials having various thicknesses.
Ideally, the plasma-facing layer is thick enough for the wall charge to
develop its full depth profile. Initially, however, the required thickness is
unknown. It depends on the electronic structure of the plasma-facing solid the
experiment is supposed to reveal. Based on assumptions about the electronic
structure, model calculations can however estimate the depth. In an iterative
process, involving calculations and measurements, the optimal configuration
can hence be found.

With an optimized configuration for the measuring window the plasma-solid
interface could be analyzed in-operando in the same manner as free-standing
surfaces in the from-the-top geometry Brillson (2010). For instance, using
hard X-rays in the keV energy range, the chemical composition could be
analyzed. Depth-profiling of selected lines could provide information about
the band bending and hence about the surface potential and energy barriers.
The filling of the electronic states at the interface could be studied by
direct and inverse photoemission spectroscopy in the eV energy range, using
ultraviolet light. It would thus be possible to determine for the first time
the states hosting the wall charge.

Since we recently proposed from-the-back electron energy loss spectroscopy as
a diagnostics for the wall charge Thiessen  _et al._ (2019), we also say a few
words about this work. The setup is identical to the one shown on the rhs of
Fig. 8. Instead of a photon the inquiring particle is an electron which
however does not enter the solid structure. The electron mean free path is
hence not the critical length scale for reading out information about the
plasma-solid interface on the opposite side. Instead it is the range of the
dipole fields of the charge fluctuations inside the solid which has to be
comparable to the thickness of the stack for the from-the-back geometry to
work. To avoid surface response functions, we assumed the wall charge to be
homogeneously confined within the plasma-facing film by an electronegative
substrate layer. The strength of the signal we found for a stack which we
considered to be still mechanically stable, was however rather weak. Only by
pre-doping the plasma-facing film with electrons the signal passed a plausible
detection limit.

Whereas from-the-back electron energy loss spectroscopy does not look too
promising, it may be feasible to do it with a spinning wall. There, while on
the vacuum side, the interface is subjected to the electron beam directly.
From-the-front, however, space charge layers have been successfully
investigated by electron energy loss spectroscopy Mahboob  _et al._ (2004);
Dou  _et al._ (1998); Lüth (1988). We expect therefore the spinning wall to
enable electron energy loss spectroscopy of the wall charge. To determine the
total magnitude of the wall charge per unit area, and possibly also the depth
profile of the wall charge, a theoretical analysis of the measured signal is
however necessary. Unlike to what we did in our exploratory work Thiessen  _et
al._ (2019), the theoretical analysis has to take the charge inhomogeneity
perpendicular to the interface into account. For that purpose, it is necessary
to generalize the calculation of the cross section for electron energy loss
Ibach and Mills (1982) to non-Fresnel interfaces by combining it with surface
response functions Kempa and Gerhardts (1985).

The in-operando diagnostics we focused on so far are of the type photon in and
electron out (photon spectroscopy) or of the type electron in and electron out
(electron energy loss spectroscopy). A technique employing only photons is
infrared spectroscopy. In subsection II.3 we proposed to use it as a
diagnostics for the wall charge. Especially in combination with the cavity
ring down methodology Peeters  _et al._ (2017); Aarts  _et al._ (2007, 2005);
O’Keefe and Deacon (1988), we expect it to be rather sensitive. The ring down
approach may be however also useful for studying in-operando the infrared
active parts of the electronics of plasma-facing dielectrics. The setup for
this purpose is identical to the one used for the in-growth investigation of
dangling bonds in a hydrogenated amorphous silicon (a-Si:H) film Peeters  _et
al._ (2017); Aarts  _et al._ (2007). It is schematically shown in Fig. 9,
together with the labeling for the application we have in mind. The a-Si:H
film is deposited on the total internal reflection (TIR) side of a prism which
acts also as an optical cavity because of highly reflective coated entrance
and exit interfaces. Due to the coupling of the dangling bonds to the
evanescent electric field leaking from the prism into the film, light
intensity is lost from the cavity. Exciting the prism by an optical pulse and
tracking in time the optical losses of the prism provides thus information
about the density Aarts  _et al._ (2007) and kinetics Peeters  _et al._ (2017)
of the bonds while the film keeps growing.

Figure 9: Illustration of the setup used for studying dangling bonds in a-Si:H
films during growth Peeters  _et al._ (2017); Aarts  _et al._ (2007). The
prism acts as a total internal reflection (TIR) element and as an optical
cavity which is fed by an infrared pulse. Due to the evanescent field leaking
from the cavity into the film additional optical losses occur. From the change
of the pulse’s ring down time information about the dangling bonds can be
obtained. With a similar setup the infrared active parts of the electronics of
a plasma-facing dielectric can be perhaps also investigated. The dielectric
could be deposited as a film on top of the TIR side of the prism or used as
the material from which the prism itself is made from. In the latter case, it
is the absorbance of the propagating wave inside the prism which carries
information about the in-operando modifications of its electronics. The dashed
line indicates the region of the wall charge which we expect to be also
measurable by such a setup.

As suggested by the labeling in Fig. 9 a similar setup can be perhaps used to
investigate the infrared active parts of the electronics of a dielectric in
contact with a plasma. Two operation modii are conceivable. Either one
deposits the dielectric on top of the TIR side of the prism and utilizes the
evanescent component of the electric field leaking out of the cavity, as it is
done in the investigation of the a-Si:H film. This would be the canonical way
to study a dielectric plasma-solid interface by evanescent wave cavity ring
down spectroscopy Schnippering  _et al._ (2011); Aarts  _et al._ (2005). It
has however the drawback to be limited to thin films. An alternative would be
to use the dielectric of interest itself as a prism material. Since most
dielectrics are transparent in the infrared this should be possible. It would
then be the plasma-induced change of the absorbance of the propagating wave
inside the prism, caused, for instance, by surplus carriers in the space
charge layer (wall charge) or by modifications of the subgap defect or surface
states, which affects the ring down time of the cavity. Using–by
construction–the plasma-facing dielectric as an optical element in the
infrared should thus provide access to the infrared active parts of its
electronics. The measurements can be done while the plasma is on, an in-
operando investigation should hence be possible.

Again, numerous technical details have to be clarified before experiments of
this sort can start. For instance, the spectral range of the light which can
be coupled into the cavity depends on its eigenmodes and thus on the geometry
of the prism which hence has to be constructed carefully. The light pulses
have to be moreover short enough to enable a detection of the ring down time.
In addition, the relative weight of the absorbance in the bulk and the
interface regions of the prism has to be quantified. Only the latter provides
information about the electronic structure of the plasma-solid interface. To
what extend this can be done within the framework of generalized
reflectivities introduced in subsection II.3 is an open issue and should be
part of theoretical studies guiding the planning of the experiments.

##  IV Synopsis

Traditionally, in plasma physics, the electron microphysics at plasma-solid
interfaces is associated with probabilities for electron deposition and
extraction. The in-operando electronic structure of the interface, which also
includes its charging due to the plasma, has so far not been the subject of
systematic investigations. Yet, it is intrinsically coupled to the plasma due
to particle and energy influx. Knowing the interface’s in-operando electronic
structure–in contrast to the electronic structure of the solid without plasma
exposure–seems to us essential for taping the full technological potential of
bounded low-temperature gas discharges. Particularly the electric properties
of miniaturized solid-based dielectric barrier discharges call for an in-
operando investigation, since the solid becomes an integral part of the plasma
device.

Couching the perspective by our own efforts concerning the calculation of
electron emission yields, the selfconsistent description of electric double
layers, and the infrared diagnostics of the wall charge, we bat for an
investigation of the plasma-solid interface’s electronic structure by in-
operando techniques of surface physics. As exemplified by a discussion of
dielectric materials, important parameters of the interface’s electronic
structure are likely to change in the course of plasma exposure. In
particular, information about energy barriers, band bendings, and the presence
or absence of surface states has to be obtained in-operando, that is, in
experimental settings, where the plasma is on. The most powerful techniques
for this purpose are infrared reflection, photoemission, and electron energy
loss spectroscopy. However, due to the plasma, they cannot be applied directly
to the interface of interest. Alternative setups need to be developed. A
possibility for infrared spectroscopy is a setup which uses the solid as an
internal reflection element. Photoemission and electron energy loss
spectroscopy can be applied in a spinning wall or a from-the-back geometry.

The experiments are challenging but within reach of modern instrumentation.
They would provide a wealth of information from which essentially all of
present day plasma technology would benefit. Most importantly, however, it
would guide the development of a selfconsistent theory of the interface’s
electronics, including the build-up of the wall charge. Having such a theory
at hand, it would be possible to search for ways to manipulate the fate of
electrons crossing the interface in either way. Since the operation modii and
the surface chemistry of solid-bound gas discharges depend on it, it is thus
conceivable that the efforts we hope to initiate by this perspective will in
the long run culminate in gas discharges with new functionalities.

## Acknowledgments

It is a pleasure to thank Profs. M. Bauer, M. Bonitz, and K. Rossnagel from
the University Kiel, Prof. J. Meichsner from the University Greifswald, and
Dr. J. P. van Helden from the INP Greifswald for valuable discussions. Support
from the Deutsche Forschungsgemeinschaft through Project No. BR-1994/3-1 is
also greatly acknowledged.

## Availability statement

Data supporting the findings of this study are available from the
corresponding author upon reasonable request.

## References

  * Weltmann  _et al._ (2019) K.-D. Weltmann, J. F. Kolb, M. Holub, D. Uhrlandt, M. S̆imek, K. Ostrikov, S. Hamaguchi, U. Cvelbar, M. C̆ernák, B. Locke, A. Fridman, P. Favia, and K. Becker, Plasma Process Polym. 16, e1800118 (2019).
  * Adamovich  _et al._ (2017) I. Adamovich, S. D. Baalrud, A. Bogaerts, P. J. Bruggeman, M. Capelli, V. Colombo, U. Czarnetzki, U. Ebert, J. G. Eden, P. Favia, D. B. Graves, S. Hamaguchi, G. Hieftje, M. Hori, I. D. Kaganovich, U. Kortshagen, M. J. Kushner, N. J. Mason, S. Mazouffre, S. M. Thagard, H.-R. Metelmann, A. Mizuno, E. Moreau, A. B. Murphy, B. Niemira, G. S. Oehrlein, Z. L. Petrovic, L. C. Pitchford, Y.-K. Pu, S. Rauf, O. Sakai, S. Samukawa, S. Starikovskaia, J. Tenneyson, K. Terashima, M. M. Turner, M. C. M. van de Sanden, and A. Vardelle, J. Phys. D: Appl. Phys. 50, 323001 (2017).
  * Charles (2014) C. Charles, Frontiers in Physics 2, 00039 (2014).
  * Sass (1983) J. K. Sass, Vacuum 33, 741 (1983).
  * Schmickler (1996) W. Schmickler, Chem. Rev. 96, 3177 (1996).
  * Tung (2014) R. T. Tung, Appl. Phys. Rev. 1, 011304 (2014).
  * Klein (2016) A. Klein, J. Am. Ceram. Soc. 99, 369 (2016).
  * Mannhart and Schlom (2010) J. Mannhart and D. G. Schlom, Science 327, 1607 (2010).
  * Raadu (1989) M. A. Raadu, Phys. Reports 178, 25 (1989).
  * Charles (2007) C. Charles, Plasma Sources Sci. Technol. 16, R1 (2007).
  * Langmuir and Mott-Smith (1924) I. Langmuir and H. Mott-Smith, Gen. Electr. Rev. 27, 449 (1924).
  * Hershkowitz (2005) N. Hershkowitz, Phys. Plasma 12, 055502 (2005).
  * Robertson (2013) S. Robertson, Plasma Phys. Control. Fusion 55, 093001 (2013).
  * Schwager and Birdsall (1990) L. A. Schwager and C. K. Birdsall, Phys. Fluids B 2, 1057 (1990).
  * Brinkmann (2009) R. P. Brinkmann, J. Phys. D: Appl. Phys. 42, 194009 (2009).
  * Franklin (2003) R. N. Franklin, J. Phys. D: Appl. Phys. 36, R309 (2003).
  * Riemann (2003) K.-U. Riemann, J. Phys. D: Appl. Phys. 36, 2811 (2003).
  * Palop  _et al._ (2007) J. I. F. Palop, J. Ballesteros, M. A. Hernández, and R. M. Crespo, Plasma Sources Sci. Technol. 16, S76 (2007).
  * Taccogna  _et al._ (2004) F. Taccogna, S. Longo, and M. Capitelli, Phys. Plasma 11, 1220 (2004).
  * Sheehan  _et al._ (2013) J. P. Sheehan, N. Hershkowitz, I. D. Kaganovich, H. Wang, Y. Raitses, E. V. Barnat, B. R. Weatherford, and D. Sydorenko, Phys. Rev. Lett. 111, 075002 (2013).
  * Campanell and Umansky (2016) M. D. Campanell and M. V. Umansky, Phys. Rev. Lett. 116, 085003 (2016).
  * Baalrud  _et al._ (2020) S. D. Baalrud, B. Scheiner, B. T. Yee, M. M. Hopkins, and E. Barnat, Plasma Sources Sci. Technol. 29, 053001 (2020).
  * Kindel and Arndt (1980) E. Kindel and R. Arndt, Beitr. Plasmaphysik 20, 119 (1980).
  * Pangal  _et al._ (1996) K. Pangal, S. L. Firebaugh, and J. C. Sturm, Appl. Phys. Lett. 69, 1471 (1996).
  * Slikboer  _et al._ (2019) E. Slikboer, P. Viegas, Z. Bonaventura, E. Garcia-Caurel, A. Sobota, A. Bourdon, and O. Guaitella, Plasma Sources Sci. Technol. 28, 095016 (2019).
  * Kawasaki  _et al._ (1994) T. Kawasaki, T. Terashima, Y. Zhu, T. Takada, and T. Maeno, J. Phys. D: Appl. Phys. 27, 1646 (1994).
  * Tschiersch  _et al._ (2014) R. Tschiersch, M. Bogaczyk, and H.-E. Wagner, J. Phys. D: Appl. Phys. 47, 365204 (2014).
  * Raitses  _et al._ (2005) Y. Raitses, D. Staack, M. Keidar, and N. J. Fisch, Phys. Plasma 12, 057104 (2005).
  * Raizer  _et al._ (2019) Y. P. Raizer, M. N. Shneider, and N. A. Yatsenko,  _Radio-frequency capacitive discharges_ (CRC Press, Boca Raton, 2019).
  * Mendis (2002) D. A. Mendis, Plasma Sources Sci. Technol. 11, A219 (2002).
  * Ishihara (2007) O. Ishihara, J. Phys. D: Appl. Phys. 40, R121 (2007).
  * Kogelschatz (2003) U. Kogelschatz, Plasma Chemistry and Plasma Processing 23, 1 (2003).
  * Brandenburg (2017) R. Brandenburg, Plasma Sources Sci. Technol. 26, 053001 (2017).
  * Massines  _et al._ (2003) F. Massines, P. Segur, N. Gherardi, C. Khamphan, and A. Ricard, Surface and Coatings Technology 174-175, 8 (2003).
  * Wagner  _et al._ (2005) H.-E. Wagner, Y. V. Yurgelenas, and R. Brandenburg, Plasma Phys. Control. Fusion 47, B641 (2005).
  * Nemschokmichal  _et al._ (2018) S. Nemschokmichal, R. Tschiersch, H. Höft, R. Wild, M. Bogaczyk, M. M. Becker, D. Loffhagen, L. Stollenwerk, M. Kettlitz, R. Brandenburg, and J. Meichsner, Eur. Phys. J. D 72, 89 (2018).
  * Peeters  _et al._ (2016) F. J. J. Peeters, R. F. Rumphorst, and M. C. M. van de Sanden, Plasma Sources Sci. Technol. 25, 03LT03 (2016).
  * Neyts  _et al._ (2015) E. C. Neyts, K. Ostrikov, M. K. Sunkara, and A. Bogaerts, Chem. Rev. 115, 13408 (2015).
  * Patel  _et al._ (2019) H. Patel, R. K. Sharma, V. Kyriakou, A. Pandiyan, S. Welzel, M. C. M. van de Sanden, and M. N. Tsampas, ACS Energy Lett. 4, 2091 (2019).
  * Kulsreshath  _et al._ (2012) M. K. Kulsreshath, L. Schwaederle, L. J. Overzet, P. Lefaucheux, J. Ladroue, T. Tillocher, O. Aubry, M. Woytasik, G. Schelcher, and R. Dussart, J. Phys. D: Appl. Phys. 45, 285202 (2012).
  * Eden  _et al._ (2013) J. G. Eden, S.-J. Park, J. H. Cho, M. H. Kim, T. J. Houlahan, B. Li, E. S. Kim, T. L. Kim, S. K. Lee, K. S. Kim, J. K. Yoon, S. H. Sung, P. Sun, C. M. Herring, and C. J. Wagner, IEEE Trans. Plasma Sci. 41, 661 (2013).
  * Tabib-Azar and Pai (2017) M. Tabib-Azar and P. Pai, Micromachines 8, 117 (2017).
  * Michaud  _et al._ (2018) R. Michaud, V. Felix, A. Stolz, O. Aubry, P. Lefaucheux, S. Dzikowski, V. S. von der Gathen, L. J. Overzet, and R. Dussart, Plasma Sources Sci. Technol. 27, 025005 (2018).
  * Rasek  _et al._ (2020) K. Rasek, F. X. Bronold, and H. Fehske, Phys. Rev. E 102, 023206 (2020).
  * Arumugam  _et al._ (2020) S. Arumugam, M. Perumal, K. P. Anjana, S. V. M. Satyanarayna, and S. K. Sinha, Phys. Plasma 27, 023512 (2020).
  * Bonitz  _et al._ (2019) M. Bonitz, A. Filinov, J.-W. Abraham, K. Balzer, H. Kählert, E. Pehlke, F. X. Bronold, M. Pamperin, M. Becker, D. Loffhagen, and H. Fehske, Front. Chem. Sci. Eng. 13, 201 (2019).
  * Sun  _et al._ (2019) G.-Y. Sun, Y. Li, S. Zhang, B.-P. Song, H.-B. Mu, B.-H. Guo, A.-B. Sun, and G.-J. Zhang, Plasma Sources Sci. Technol. 28, 055001 (2019).
  * Bronold and Fehske (2017a) F. X. Bronold and H. Fehske, J. Phys. D: Appl. Phys. 50, 294003 (2017a).
  * Brillson (2010) L. J. Brillson,  _Surfaces and interfaces of electronic materials_ (Wiley-VCH Verlag, Weinheim, 2010).
  * Liu  _et al._ (2020) Y. Liu, Q. Sun, J. Liu, M. N. Banis, Y. Zhao, B. Wang, K. Adair, Y. Hu, Q. Xiao, C. Zhang, L. Zhang, S. Lu, H. Huang, X. Song, and X. Sun, ACS Appl. Mater. Interfaces 12, 2293 (2020).
  * R. Hausbrand, M. Fingerle, T. Späth, and C. Guhl (2017) R. Hausbrand, M. Fingerle, T. Späth, and C. Guhl, Thin Solid Films 643, 43 (2017).
  * Schnippering  _et al._ (2011) M. Schnippering, S. R. T. Neil, S. R. Mackenzie, and P. R. Unwin, Chem. Soc. Rev. 40, 207 (2011).
  * Barritt  _et al._ (2010) E. E. Barritt, C. I. Smith, D. S. Martin, K. Gentz, K. Wandelt, and P. Weightman, Europhys. Lett. 92, 57005 (2010).
  * Lange  _et al._ (1983) P. Lange, V. Glaw, H. Neff, E. Piltz, and J. K. Sass, Vacuum 33, 763 (1983).
  * Varsha and Nageswaran (2020) M. V. Varsha and G. Nageswaran, Frontiers in Chemistry 8, 23 (2020).
  * Zborowski  _et al._ (2018) C. Zborowski, O. Renault, A. Torres, C. Guedj, Y. Yamashita, S. Ueda, and G. Grenet, J. Appl. Phys. 124, 085115 (2018).
  * Peeters  _et al._ (2017) F. J. J. Peeters, J. Zheng, I. M. P. Aarts, A. C. R. Pipino, W. M. M. Kessels, and M. C. M. van de Sanden, J. Vac. Sci. Technol. A 35, 05C307 (2017).
  * Tselev  _et al._ (2016) A. Tselev, J. Velmurugan, A. V. Ievlev, and S. V. Kalinin, ACS Nano 10, 3562 (2016).
  * Copuroglu  _et al._ (2013) M. Copuroglu, H. Sezen, R. L. Opila, and S. Suzer, ACS Appl. Mater. Interfaces 5, 5875 (2013).
  * Rubio-Zuazo and Castro (2008) J. Rubio-Zuazo and G. R. Castro, Surf. Interface Anal. 40, 1438 (2008).
  * Aarts  _et al._ (2007) I. M. P. Aarts, A. C. R. Pipino, M. C. M. van de Sanden, and W. M. M. Kessels, Appl. Phys. Lett. 90, 161918 (2007).
  * Aarts  _et al._ (2005) I. M. P. Aarts, A. C. R. Pipino, J. P. M. Hoefnagels, W. M. M. Kessels, and M. C. M. van de Sanden, Phys. Rev. Lett. 95, 166104 (2005).
  * Tselev  _et al._ (2018) A. Tselev, J. Fagan, and A. Kolmakov, Appl. Phys. Lett. 113, 263101 (2018).
  * Laha  _et al._ (2014) R. Laha, A. Manivannan, and S. Kasiviswanathan, Rev. Sci. Instrum. 85, 035001 (2014).
  * Tai  _et al._ (2013) K. Tai, T. J. Houlahan, J. G. Eden, and S. J. Dillon, Sci. Reports 3, 1325 (2013).
  * Guha  _et al._ (2007) J. Guha, Y.-K. Pu, and V. M. Donnelly, J. Vac. Sci. Technol. A 25, 347 (2007).
  * Nitschke and Meichsner (1997) M. Nitschke and J. Meichsner, J. Appl. Polymer Science 65, 381 (1997).
  * Tolias (2014a) P. Tolias, Plasma Phys. Control. Fusion 56, 123002 (2014a).
  * Tolias (2014b) P. Tolias, Plasma Phys. Control. Fusion 56, 045003 (2014b).
  * Marbach  _et al._ (2012) J. Marbach, F. X. Bronold, and H. Fehske, Eur. Phys. J. D 66, 106 (2012).
  * Pamperin  _et al._ (2018) M. Pamperin, F. X. Bronold, and H. Fehske, Plasma Sources Sci. Technol. 27, 084003 (2018).
  * Bronold and Fehske (2015) F. X. Bronold and H. Fehske, Phys. Rev. Lett. 115, 225001 (2015).
  * Bronold and Fehske (2017b) F. X. Bronold and H. Fehske, Plasma Phys. Control. Fusion 59, 014011 (2017b).
  * Gonzalez  _et al._ (2017) L. A. Gonzalez, M. Angelucci, R. Larciprete, and R. Cimino, AIP Advances 7, 115203 (2017).
  * Pierron  _et al._ (2017) J. Pierron, C. Inguimbert, M. Belhaj, T. Gineste, J. Puech, and M. Raine, J. Appl. Phys. 121, 215107 (2017).
  * Corbella  _et al._ (2016) C. Corbella, A. Marcak, T. de los Arcos, and A. von Keudell, J. Phys. D: Appl. Phys. 49, 16LT01 (2016).
  * Marcak  _et al._ (2015) A. Marcak, C. Corbella, T. de los Arcos, and A. von Keudell, Rev. Sci. Instrum. 86, 106102 (2015).
  * Dunaevsky  _et al._ (2003) A. Dunaevsky, Y. Raitses, and N. J. Fisch, Phys. Plasma 10, 2574 (2003).
  * Zobnin  _et al._ (2018) V. A. Zobnin, A. D. Usachev, O. F. Petrov, V. E. Fortov, M. H. Thoma, and M. A. Fink, Phys. Plasma 25, 033702 (2018).
  * Demidov  _et al._ (2015) V. I. Demidov, S. F. Adams, I. D. Kaganovich, M. E. Koepke, and I. P. Kurlyandskaya, Phys. Plasma 22, 104501 (2015).
  * Daksha  _et al._ (2016) M. Daksha, B. Berger, E. Schuengel, I. Korolov, A. Derzsi, M. Koepke, Z. Donkó, and J. Schulze, J. Phys. D: Appl. Phys. 49, 234001 (2016).
  * Daksha  _et al._ (2019) M. Daksha, A. Derzsi, Z. Mujahid, D. Schulenberg, B. Berger, Z. Donkó, and J. Schulze, Plasma Sources Sci. Technol. 28, 034002 (2019).
  * Rabalais (1994) J. W. Rabalais, ed.,  _Low energy ion-surface interaction_ (Wiley and Sons, New York, 1994).
  * Monreal (2014) R. C. Monreal, Progr. Surface Science 89, 80 (2014).
  * Dashen (1964) R. Dashen, Phys. Rev. 134, A1025 (1964).
  * Vicanek (1999) M. Vicanek, Surf. Sci. 440, 1 (1999).
  * Glazov and Pázsit (2007) L. G. Glazov and I. Pázsit, Nucl. Instr. and Meth. B 256, 638 (2007).
  * Seah and Dench (1979) M. P. Seah and W. A. Dench, Surf. Interf. Anal. 1, 2 (1979).
  * Harrison (1979) W. A. Harrison,  _Solid state theory_ (Dover Publications Inc., New York, 1979).
  * Lin  _et al._ (2008) Z. Lin, L. V. Zhigilei, and V. Celli, Phys. Rev. B 77, 075133 (2008).
  * Grinberg and Luryi (1992) A. A. Grinberg and S. Luryi, Solid-St. Electron. 35, 1299 (1992).
  * A. R. St. Denis and D. L. Pulfrey (1998) A. R. St. Denis and D. L. Pulfrey, J. Appl. Phys. 84, 4959 (1998).
  * Konistis and Hu (2002) K. Konistis and Q. Hu, J. Appl. Phys. 91, 5400 (2002).
  * Li  _et al._ (2008) M. Li, C. Li, H. Zhan, and J. Xu, Appl. Phys. Lett. 92, 031503 (2008).
  * Ambrico  _et al._ (2014) P. F. Ambrico, M. Ambrico, L. Schiavulli, and S. D. Benedictis, J. Phys. D: Appl. Phys. 47, 305201 (2014).
  * Ambrico  _et al._ (2009) P. F. Ambrico, M. Ambrico, L. Schiavulli, T. Ligonzo, and V. Augelli, Appl. Phys. Lett. 94, 051501 (2009).
  * Rasek  _et al._ (2018) K. Rasek, F. X. Bronold, M. Bauer, and H. Fehske, Europhys. Lett. 124, 25001 (2018).
  * Thiessen  _et al._ (2019) E. Thiessen, F. X. Bronold, and H. Fehske, Plasma Sources Sci. Technol. 28, 095024 (2019).
  * Kempa and Gerhardts (1985) K. Kempa and R. R. Gerhardts, Surf. Sci. 150, 157 (1985).
  * Milosevic (2012) M. Milosevic,  _Internal reflection and ATR spectroscopy_ (Wiley, New York, 2012).
  * Ling  _et al._ (1993) L. Ling, S. Kuwabara, T. Abe, and F. Shimura, J. Appl. Phys. 73, 3018 (1993).
  * O’Keefe and Deacon (1988) A. O’Keefe and D. A. G. Deacon, Rev. Sci. Instrum. 59, 2544 (1988).
  * Bechstedt (2003) F. Bechstedt,  _Principles of surface physics_ (Springer Verlag, Berlin, 2003).
  * Heinisch  _et al._ (2011) R. L. Heinisch, F. X. Bronold, and H. Fehske, Phys. Rev. B 83, 195407 (2011).
  * Heinisch  _et al._ (2012) R. L. Heinisch, F. X. Bronold, and H. Fehske, Phys. Rev. B 85, 075323 (2012).
  * Cole (1974) M. W. Cole, Rev. Mod. Phys. 46, 451 (1974).
  * Rutter and Robertson (1998) M. J. Rutter and J. Robertson, Phys. Rev. B 57, 9241 (1998).
  * Cui  _et al._ (1998) J. B. Cui, J. Ristein, and L. Ley, Phys. Rev. Lett. 81, 429 (1998).
  * Yamaguchi  _et al._ (2009) H. Yamaguchi, T. Masuzawa, S. Nozue, Y. Kudo, I. Saito, J. Koe, M. Kudo, T. Yamada, Y. Takakuwa, and K. Okano, Phys. Rev. B 80, 165321 (2009).
  * Loh  _et al._ (1999) K. P. Loh, I. Sakaguchi, M. N. Gamo, S. Tagawa, T. Sugino, and T. Ando, Appl. Phys. Lett. 74, 28 (1999).
  * Rohlfing  _et al._ (2003) M. Rohlfing, N.-P. Wang, P. Krueger, and J. Pollmann, Phys. Rev. Lett. 91, 256802 (2003).
  * McKenna and Shluger (2008) K. P. McKenna and A. L. Shluger, Nature Materials 7, 859 (2008).
  * Wang and Duke (1987) Y. R. Wang and C. B. Duke, Surf. Sci. 192, 309 (1987).
  * McNeill  _et al._ (2016) A. R. McNeill, A. R. Hyndman, R. J. Reeves, and A. J. Downard, ACS Appl. Mater. Interf. 8, 31392 (2016).
  * Maier  _et al._ (2001) F. Maier, J. Ristein, and L. Ley, Phys. Rev. B 64, 165411 (2001).
  * Rietwyk  _et al._ (2013) K. J. Rietwyk, S. L. Wong, L. Cao, K. M. O’Donnell, L. Ley, and A. T. S. Wee, Appl. Phys. Lett. 102, 091604 (2013).
  * Beattie  _et al._ (2019) J. M. A. Beattie, J. P. Gross, M. J. Rayson, and P. R. Briddon, Diamond and Related Materials 94, 137 (2019).
  * T.C. Rödel, F. Fortuna, F. Bertran, M. Gabay, M. J. Rozenberg, A. F. Santander-Syro and P. Le Févre (2015) T.C. Rödel, F. Fortuna, F. Bertran, M. Gabay, M. J. Rozenberg, A. F. Santander-Syro and P. Le Févre, Phys. Rev. B 92, 041106 (2015).
  * Yin  _et al._ (2018) W.-J. Yin, B. Wen, C. Zhou, A. Selloni, and L.-M. Liu, Surf. Sci. Rep. 73, 58 (2018).
  * Ade  _et al._ (1998) H. Ade, W. Yang, S. L. English, J. Hartman, R. F. Davies, R. J. Nemanich, V. N. Litvinenko, I. V. Pinayev, Y. Wu, and J. M. J. Madey, Surf. Rev. Lett. 5, 1257 (1998).
  * Mahboob  _et al._ (2004) I. Mahboob, T. D. Veal, C. F. McConville, H. Lu, and W. J. Schaff, Phys. Rev. Lett. 92, 036804 (2004).
  * Dou  _et al._ (1998) Y. Dou, R. G. Egdell, T. Walker, D. S. L. Law, and G. Beamson, Surf. Sci. 398, 241 (1998).
  * Lüth (1988) H. Lüth, Vacuum 38, 223 (1988).
  * Ibach and Mills (1982) H. Ibach and D. L. Mills,  _Electron energy loss spectroscopy and surface vibrations_ (Academic Press, New York, 1982).