# Electron drag force in EUV induced pulsed hydrogen plasmas

M. Chaudhuri∗, A. Yakunin, M. van de Kerkhof and R. Snijdewind  ASML The
Netherlands B.V., P.O.Box 324, 5500AH Veldhoven, The Netherlands

###### Abstract

Extreme ultraviolet (EUV) induced pulsed plasma is unique due to its transient
characteristics: the plasma switches between non-thermal state (when EUV power
is ON at the beginning of the pulse) and thermal state (end of the pulse at
$\sim$ 20 $\mu$s). It is shown that although electron drag force acting on nm
size particles in hydrogen plasma is negligible compared to the ion drag force
at the beginning of the pulse, however it can be dominant at the end of the
pulse and can play important role in particle transport leading to defectivity
issues for semiconductor chip production technologies.

The impact of semiconductor integrated circuits (IC)/chips on modern lifestyle
is hard to overstate. From computers to communication, education to
entertainments, the growth of electronics technology accompanied by the
advancement of semiconductor chip manufacturing technology has been
phenomenal. The fascinating evolution from single transistors to current
memory chips and microprocessors with billion-transistors is a remarkable
story. The fabrication of such ICs involves a great variety of physical and
chemical processes performed on a silicon substrate. Fundamental of all of
these processes is optical projection lithography which is basically a
photographic process where an image of the mask (pattern) is exposed and
projected onto the wafer which is covered by a photoresist (light-sensitive
polymer). After development, the 3D images of the pattern can be obtained on
the substrate Mack_litho . The lenses/optical elements used in the projection
optics are almost perfect so that the resulting image is almost aberration
free and mainly diffraction-limited. For such images, the resolution is
determined by the wavelength of the imaging light ($\lambda$) and the
numerical aperture (NA) of the projection lens according to the Rayleigh
resolution criterion: $\propto\lambda/$NA. To obtain best possible resolution,
the EUV lithography has been introduced in recent times which uses highly
energetic EUV photons (energy $\sim$ 92 eV) with much shorter illumination
wavelength (13.5 nm). One of the unavoidable side effect of this development
is the generation of EUV photon induced plasma due to the interaction of such
highly energetic photons with the low pressure (1-10 Pa) background hydrogen
gas BanineJAP2006 ; van_der_Horst_2014 ; Roderik_2018 ; Kerkhof_2019 ;
Becker_2019 ; Kerkhof_2020 ; Kerkhof_2021 . It is to be noted that the EUV
photon induced plasmas differ from traditional low-temperature laboratory
plasmas (DC discharge, capacitively coupled radio frequency plasmas,
inductively coupled plasma, dielectric barrier discharges, etc), in a sense
that there exists no continuous external power supply to sustain the
discharge. Although the photon induced plasmas is an active subject of
investigation for astrophysical plasmas for some time Goertz1989 ;
Rosenberg1995 ; Fortov1998 ; PhysRevLett.84.6034 ; 2001_Weigartner , the
experimental research has gained momentum recently due to industrial
applications in the form of EUV lithography.

The spatial and temporal evolution of the EUV plasmas has been investigated
experimentally and with particle-in-cell (PIC) model van_der_Horst_2016 ;
Astakhov_2016 ; Beckers_2016 ; van_de_ven_2018 . When the plasma decay time is
much shorter than the EUV pulse repetition time (20 $\mu$s) then the plasma
evolution occurs in different stages: at first the plasma creation happens due
to direct photoionization of the hydrogen gas (ionization energy of 13.6 eV)
during the passage of highly energetic EUV photons. In this phase, the plasma
contains highly energetic electrons with excess photon energy and non-
Maxwellian energy distribution. A part of these electrons then move towards
the wall leaving behind a positive space charge region which confines the
remaining electrons and allow positive ions to accelerate towards walls. The
confined electrons lose their energy within few tens of nanoseconds due to
electron impact ionization and increase the plasma density. After this stage,
plasma starts to expand and the local electron density decreases rapidly.
During the last stage of the process, the electrons are supposed to reach
their equilibrium temperature with plasma density continues to decrease due to
ambipolar diffusion and recombination processes at wall. The energy is
supplied within few tens of nanoseconds by the EUV pulse while it takes $>$20
$\mu$s for the plasma to completely extinguish which makes the highly
transient plasmas between nonthermal ($T_{e}>>T_{i}$) and thermal states
($T_{e}\sim T_{i}$). This effect leads to a buildup process over multiple
pulses. Here, $T_{e(i)}$ are the electron (ion) temperatures. Such transient
plasmas generates an electric field ($E$) when it comes in contact with the
surface and is also responsible for charging processes of the nano particles
which are present in the system as part of the contamination due to lack of
sufficient cleanliness of the surfaces or due to other mechanisms such as
blistering or spitting. In presence of such transient electric fields, charge
dependent volume forces such as electrostatic force ($F_{\rm el}$), ion drag
($F_{\rm id}$) and electron drag ($F_{\rm ed}$) forces play important roles
for nano particle dynamics. The goal of this work is to show that the electron
drag force which was always neglected earlier in EUV photon induced plasmas
can play important roles for nano particle transport towards critical zones
under typical operational conditions in EUV lithography machines.

Figure 1: A simple illustration of EUV plasma generation using pulsed power
mode. Each pulse is 20$\mu$s duration within which the EUV power is ON for 100
ns. During this initial phase of power ON mode, the plasma is non-thermal
($T_{e}\gg T_{i}$). After this initial phase, the electron temperature
decreases rapidly and at the end of the pulse, the plasma becomes thermal
($T_{e}\sim T_{i}$). For each pulse, the plasma exhibits the transient
characteristics by switching between non-thermal and thermal states.

The electrostatic force $F_{el}=-ZeE$ on the charged particle is generated as
direct impact of electric field where $Z$ is the charge number for each
individual particle. The indirect effect of electric field is to produce
$F_{\rm id}$ and $F_{\rm ed}$ on the grains which are nothing but the momentum
transfer rates from drifting ions and electrons to the grains within the
electric field. Depending on the plasma regime, different forces respond
differently and the competition amongst them are responsible for the
equilibrium states and transport properties of the grains Fortov2005PR . Often
the ion drag force dominates over the electron drag force because of large
ion-to-electron mass ratio and plays very important role to understand
particle dynamics in laboratory and microgravity experiments
PhysRevE.70.056405.khrapak . However, a complete self-consistent model for ion
drag force is yet to be constructed which can describe all cases of interests.
Below, we first briefly describe present level of theoretical understanding of
the ion drag force and then we justify to use suitable analytical expressions
for estimations depending on our experimental parameter regime. There are
mainly two approaches to calculate ion drag force: binary collision (BC)
approach and linear response (LR) formalism Fortov2005PR . Each of them has
pros and cons, but they are complementary to each other khrapak.pop.2005 . In
the BC approach, the velocity dependent momentum transfer cross-section
$\sigma(v)$ is obtained after analyzing ballistic ion trajectories in the
isotropic attractive Debye-Hückel (Yukawa) potential of the particle. Then the
ion drag force can be calculated by integrating $\sigma(v)$ with suitable ion
velocity distribution function Khrapak_PhysRevE.66.046414 . The scattering
parameter ($\beta$) is an important quantity which determines the strength of
ion-particle coupling for momentum transfer process and is defined as,
$\beta=Ze^{2}/(m_{i}v^{2}\lambda)$, where $\lambda$ and $m_{i}$ denote the
effective screening length and ion mass. The advantage for BC approach is that
it can be applied for any strength of ion-particle coupling but it completely
neglects collisional effects (for example ion-neutral charge exchange
collisions) and ion flow effects which creates potential anisotropy. In the LR
approach, the ion drag force is calculated by solving Poisson equation coupled
to the kinetic (or hydrodynamic) equations for the ions and electrons
PhysRevLett.92.205007.ivlev ; PhysRevE.71.016405.ivlev ; chaudhuri.pop.2007 ;
chaudhuri.pop.2008 ; khrapak.IEEE.2009 . The advantage for the LR approach is
that the collisional effects and the ion flow induced potential anisotropy are
taken into account self-consistently for ion drag force calculation, but it is
applicable only for weak ion-particle coupling $\beta<1$. However, the
dominance of the ion drag force over the electron drag force may change when
the electrons drift much faster than the ions because of their much higher
mobility. It was shown in an earlier work by Khrapak and Morfill
Khrapak_PhysRevE.69.066411  that in the collisionless regime the electron drag
force calculated using BC approach can indeed dominate over the electric and
ion drag forces provided the electron-to-ion temperature ratio is not too
high. In this case, using the Coulomb scattering theory for point-like
particles with equal magnitude of Coulomb logarithm for electron-particle and
ion-particle interactions, the ratio of ion drag and electron drag forces for
subthermal drifts can be written as Khrapak_PhysRevE.69.066411 ,

$\frac{F_{id}}{F_{ed}}\sim\gamma\mu^{1/2}\tau^{3/2}$ (1)

Figure 2: An illustration of the charge induced volume forces acting on the
negatively charged stationary particle in the presence of an external electric
field (E): electric force ($F_{el}$), ion drag force ($F_{id}$) and electron
drag force ($F_{ed}$)

.

Here, $\gamma=u_{i}/u_{e}$ is the ratio of ion-to-electron drift velocities,
$\mu=m_{i}/m_{e}$ is the ion-to-electron mass ratio and $\tau=T_{e}/T_{i}$ is
the electron-to-ion temperature ratio. For the particle movements in a
stationary hydrogen plasma, $u_{i}\sim u_{e}$. Then at the beginning of the
pulse for non thermal plasma condition, $F_{id}>>F_{ed}$. But then ion drag
force decreases with time and $F_{id}\sim F_{ed}$ when $\tau\sim 12$. However,
in case of weakly ionized plasmas in an external large scale electric field
with mobility limited electron and ion drifts, the electrons drift much faster
than ions, $u_{e}\gg u_{i}$. Such a situation was considered earlier in the
regime of collisionless ion and electron trajectories in the vicinity of the
grain (BC approach) and the ratio of ion drag and electron drag force was
estimated as Khrapak_PhysRevE.69.066411 ,

$\frac{F_{id}}{F_{ed}}\sim\frac{\omega[1+(z\tau/2)+0.5(z\tau)^{2}\ln(2\xi/z\tau)]}{(1+z/2)e^{-z}+0.5z^{2}\ln(2\xi/z)}$
(2)

Here, $\omega=\sigma_{en}/\sigma_{in}$ is the ratio of electron-neutral and
ion-neutral collision cross-section. The particle size is normalized by the
effective debye length, $\xi=\lambda_{\rm D}/a$.

Later, Chaudhuri et. al. extended the work of electron drag force to the
collisional regime using LR formalism with hydrodynamic approach
chaudhuri.pop.2007.054503 . In case of weakly collisional
($\ell_{i}>\lambda_{\rm D}$) and highly collisional plasma
($\ell_{i},\ell_{e}<\lambda_{D}$), ion and electron drag forces (LR)
experienced by a nonabsorbing particle are directed along the drift velocities
of ions and electrons respectively i.e. they act in opposite directions. The
ratio of absolute magnitude of ion-to-electron drag forces acting on a
nonabsorbing particle is $|F_{id}/F_{ed}|\sim\tau^{2}$ which implies that for
thermal plasmas, they exactly cancel each other chaudhuri.pop.2007.054503 .
But for nonthermal plasma, the ion drag force (directed opposite to the
electric force) always dominates. However, for absorbing particle in the
highly collisional regime, it is possible to obtain negative ion drag force as
ion absorption reduces the magnitude of the ion drag force. On the other side,
the electron absorption increases the magnitude of the electron drag force.
The absolute ratio of ion-to-electron drag force for an absorbing particle is
$|F_{id}/F_{ed}|\sim\tau$ chaudhuri.pop.2007.054503 .

Figure 3: The variations of ratio of ion drag force and electron drag force
with particle size (diameter) in thermal plasma for three different cases:
Eqn-1 (collisionless, subthermal drifts for ions and electrons), Eqn-2
(collisionless, mobility limited drifts with $u_{e}\gg u_{i}$) and this work
(collisionless electrons, weakly collisional ions with suprathermal drifts and
weak ion-grain coupling). The relevant parameter values have been described in
the text.

To understand the importance of electron drag force for particle transport in
transient EUV induced pulsed plasmas, we consider a single pulse duration
where the plasma switches between nonthermal (initial stage of the pulse when
EUV power is ON with $\tau>>1$) to thermal states (end of the pulse when
$\tau\sim 1$) where $\tau=T_{e}/T_{i}$ is the electron-to-ion temperature
ratio. Typically, the EUV lithography machines operate with following
conditions Kerkhof_2021 : background hydrogen gas pressure at $p\sim 5$ pa,
plasma (electron) density with $n_{e}\sim 4e8/cm^{3}$, E $\sim$ 5 V/cm,
$T_{e}\sim 10eV$ and $T_{i}\sim 0.025eV$ (room temperature). For the thermal
state, $T_{e}\sim T_{i}\sim 0.025$ eV. The charge of a particle is an
important parameter which plays an important role on particle dynamics. The
orbital motion limited (OML) theory is commonly used to estimate particle
charge which assumes collisionless, ballistic ion and electron trajectories in
the vicinity of an isolated particle without the presence of any kind of
interaction potential barrier Allen_1992 ; kennedy_allen_2003 . In this case,
the electron flux can be written as,
$I_{e}\approx\sqrt{8\pi}a^{2}n_{e}v_{Te}\exp(-z)$ where $n_{e}$ is the
electron density, $v_{Te}=\sqrt{T_{e}/m_{e}}$ is the electron thermal
velocity. The ion flux can be written as,
$I_{i}\approx\sqrt{8\pi}a^{2}n_{i}v_{\rm Ti}(1+z\tau)$ where $a$ is the
particle radius, $n_{i}$ is the ion density, $v_{\rm Ti}=\sqrt{T_{i}/m_{i}}$
is the ion thermal velocity and $z=Ze^{2}/aT_{e}$ is the normalized particle
charge. The estimated charge for a 100 nm diameter particle in nonthermal
plasma condition mentioned above is $Z\sim 325$. However, when the experiments
are performed at 5 pa background gas pressure then the collisionality index
(normalized ion mean free path w.r.t effective debye length),
$\eta=\lambda_{\rm D}/\ell_{i}\sim 0.07$ which represents weakly collisional
regime with effective debye length $\lambda_{\rm D}\sim 61\mu$m. In this
regime, the OML expression for the ion flux should not be applied for particle
charge determination but rather the collision enhanced ion model should be
used, $I_{i}\approx\sqrt{8\pi}a^{2}n_{i}v_{\rm
Ti}(1+z\tau+0.1z^{2}\tau^{2}\eta)$ and the estimated charge for 100 nm
diameter particle is $Z\sim 190$. The result shows that the charge values in
weakly collisional regime are significantly less than that calculated in
collisionless regime using OML theory and consistent with previous
experimental results PhysRevLett.93.085001.Ratynskaia ;
PhysRevE.72.016406.Khrapak ; Khrapak_2012 ; Antonova_2019 . To estimate the
magnitude of electron and ion drag forces, we consider collisionless electron
drag force Khrapak_PhysRevE.69.066411 ,

$F_{ed}=\frac{2}{3}\sqrt{\frac{2}{\pi}}\left(\frac{T_{i}}{e}\right)^{2}\xi^{-2}\tau^{2}(1+\tau)^{-1}M_{e}\Phi(z,\xi)$
(3)

Here, $M_{e}=u_{e}/v_{Te}$ is the thermal mach number for electrons. The
parameter $\Phi(z,\xi)$ accounts for the electrostatic interaction between
electrons and charged particles which can be estimated as contribution from
direct collisions $\approx(1+z/2)e^{-z}$ and from scattering
$\approx(z^{2}/2)\ln(2\xi/z)$. However, for the ion drag force, the expression
for weakly collisional regime as was obtained by LR formalism has been taken
into account Fortov2005PR ,

${F_{id}}=\sqrt{\frac{2}{\pi}}\left(\frac{T_{i}}{e}\right)^{2}ln\left(\frac{4M_{i}}{\eta\beta}\right)\frac{\beta^{2}}{M_{i}}$
(4)

Figure 4: The variations of electric force, ion drag force and electron drag
force with particle size (diameter) in (a) non-thermal plasma and (b) thermal
plasma. In the non-thermal plasma regime, the ion drag force always dominates
over the electron drag force over the entire range of particle size under
consideration. But in this regime, the electric force is the most dominant
one. On the other side in thermal plasma, the electron drag force always
dominates over the ion drag force over the entire range of particle size.
However, in this regime, the electric force dominates over electron drag force
for small particles, but there is a cross-over at $\sim$ 100 nm particle size
after which electron drag force dominates over electric force. The plasma
parameters used for these plots are discussed in the text. Figure 5: (a) The
variation of the cross-over particle size (when $F_{el}=F_{ed}$ as represented
by the solid line) with pressure which can be represented by the best fit
quadratic equation: $d_{c}=0.27\times p^{2}+21\times p-17$. Here $d_{c}$ is
the cross-over particle size and $p$ is pressure. (b) The variation of the
cross-over particle size (when $F_{el}=F_{ed}$ as represented by the solid
line) with electron-to-ion temperature ratio ($\tau$) which can be represented
by the best fit quadratic equation: $d_{c}=20\times\tau^{2}+65\times\tau+20$.
The data point with blue circle in both figures represents the baseline
configuration for thermal plasma ($\tau=1$) at 5 pa gas pressure.

Here, $M_{i}=u_{i}/v_{i}$ is the thermal mach number for ions and
$\beta=z\tau/\xi$ is the scattering parameter. To highlight the new results of
our work, the ion drag-to-electron drag force ratio variation over particle
size is shown in Fig. 3. It is found that electron drag force is much stronger
than ion drag force in the thermal state of the EUV photon induced plasma
under operational conditions.

The variation of the electric, ion drag and electron drag forces with particle
size is shown in Fig. 4. At the beginning of the pulse when the plasma is non-
thermal ($\tau\sim 400$) with background gas pressure at 5 pa, the electric
and ion drag forces dominates over the electron drag force by 1-2 orders of
magnitude. A cross-over diameter $\sim 7\mu$m is observed between electric and
ion drag forces below which electric force dominates over ion drag force and
above which opposite occurs. However, at the end of the pulse with thermal
plasma condition ($\tau\sim 1$) and same background pressure at 5 pa, the
electron drag force is always higher than the ion drag force by almost 3
orders of magnitude. Furthermore, the important fact is that in this weakly
collisional regime, a cross-over particle size $d_{c}\sim 100$ nm is obtained.
For smaller particle size ($d<d_{c}$) the electric force dominates over
electron drag force and for bigger particle size ($d>d_{c}$) the electron drag
force dominates over electric force. This is because $F_{el}\sim a$ and the
electron drag force $F_{ed}\sim Z^{2}\sim a^{2}$. Fig. 5 shows such cross-over
particle size ($d_{c}$) variation with pressure and electron-to-ion
temperature ratio respectively. Each marker in Fig. 5a represents the cross-
over particle size between electric force and electron drag force in the
thermal plasma at the end of the pulse for a particular pressure. One such
cross-over particle size (100 nm) is shown in Fig. 4b for 5 pa gas pressure.
Similar cross-over size estimations are made for different pressures until 50
pa and a best fit was made as solid line as mentioned in the caption. Similar
estimations have been made for varying electron-to-ion temperature ratio at a
particular pressure (5 pa). From the above plots, it is possible to classify
the particle sizes for which electron drag force can dominate over electric
force under typical operational conditions.

In conclusion, it is found that particle transport within each EUV pulse
strongly depends on the plasma regime. At the beginning of the pulse when the
plasma is non-thermal, the electric and ion drag forces dominates over the
electron drag force. However, at the end of the pulse when the plasma becomes
thermal, the electron drag force dominates over the ion drag force for all
size of particles. The electron drag force becomes most dominant (stronger
than electric force) for bigger size particles. It is to be noted that this
work is based on single pulse analysis. In case of multiple pulse scenario,
residual charge may play an important role and its influence on the steady
state charge as well as drag forces estimations should be performed self-
consistently. It is also important to perform dedicated experiments for
successful realization of the predictions made in this work for EUV plasma and
this has been kept as future work.

$*$ Corresponding author<EMAIL_ADDRESS>
The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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