Accepted Manuscript Nano-holes as standard leak elements Vincenzo Ierardi, Ute Becker, Sarantis Pantazis, Giuseppe Firpo, Ugo Valbusa, Karl Jousten PII: S0263-2241(14)00397-2 DOI: http://dx.doi.org/10.1016/j.measurement.2014.09.017 Reference: MEASUR 2995 To appear in: Measurement Received Date: 2 May 2014 Revised Date: 2 July 2014 Accepted Date: 5 September 2014 Please cite this article as: V. Ierardi, U. Becker, S. Pantazis, G. Firpo, U. Valbusa, K. Jousten, Nano-holes as standard leak elements, Measurement (2014), doi: http://dx.doi.org/10.1016/j.measurement.2014.09.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Received Date: 2 May 2014Revised Date: 2 July 2014Accepted Date: 5 September 2014
Please cite this article as: V. Ierardi, U. Becker, S. Pantazis, G. Firpo, U. Valbusa, K. Jousten, Nano-holes as standardleak elements, Measurement (2014), doi: http://dx.doi.org/10.1016/j.measurement.2014.09.017
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customerswe are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, andreview of the resulting proof before it is published in its final form. Please note that during the production processerrors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
KEYWORDS: focused ion beam, nanotechnology, Si3N4, SEM, STEM, AFM, standard leak,
gas flow meter, leak rate measurement, vacuum metrology, DSMC.
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1. Introduction
Standard leaks are an important tool to calibrate leak detectors and to provide safety and functionality
of tested objects, e.g. pace makers. Most leak rate measurements are carried out with helium as test
gas. The container to be tested, however, often contains a different gas or vapour. It is not easy to
predict the leak rate for these kinds of gases when the helium leak rate had been measured, since the
conversion formula depends on the type of flow, which is often unknown [1]. In addition, when the
transitional flow regime between molecular and viscous flow is involved, there are no analytical
formulas available [2].
For this reason, it would be advantageous to have standard leak elements which are predictable for any
gas species, either from geometry or from a calibration for just one gas species. With such elements it
is possible to measure real leaks for any gas and check empirical conversion formulas.
In the molecular flow regime where the mean free path of molecules is greater than a characteristic
dimension of an object, conductances and flow rates can be easily predicted from geometrical
dimensions. Very small elements like tubes, channels or orifices of the order of 100 nm in lateral
dimension will both exhibit molecular flow through them up to relatively high upstream pressures
(100 kPa) and generate suitably large leak rates at the same time.
For this reason, such elements were produced by the focused ion beam (FIB) technique and tested for
their suitability as standard leak. In the following section we will report on the leak elements
fabrication and their geometrical characterization, while in Section 3 we will present the calculation of
their conductance by Direct Simulation Monte Carlo (DSMC). Section 4 explains the experimental
set-up of the conductance measurements by vacuum technique. The results are discussed in Section 5
before we conclude in Section 6.
2. Leak elements fabrication and their geometrical characterization
The nano-holes are drilled in low stress silicon nitride (Si3N4) membranes (Figure 1) by a focused ion
beam (FIB) [3][4]. The membranes are custom made on chips with size of 5 mm × 5 mm, window
size 100 µm x 100 µm, thickness 200 nm and are able to resist 100 kPa of pressure differential.
Considering the geometric characteristics (length and diameter) of the nano-holes used in this work,
the FIB machining overcomes the speed limitation of others techniques used to drill nano-holes, and in
addition does not need chemically assisted etching by gas injection, such as in the case of others
techniques used for this purpose [4][5][6].
To drill the nano-holes we have used a CrossBeam® workstation 1540XB model by Zeiss, which
combines a SEM and a FIB. The Ga+ ion beam has an energy range from 5 to 30 keV, current of 1 pA
to 20 nA, and a minimum spot size of 7 nm [7]. For this investigation a nano-hole has been drilled by
means of 30 keV Ga+ beam, using a current of 200 pA and a dwell time of 1s [8].
3
We have determined the nano-hole geometry by means of Scanning Electron Microscopy (SEM),
Scanning Transmission Electron Microscopy (STEM) and Atomic Force Microscopy (AFM). Soon
after the nano-hole fabrication we have collected its SEM and STEM images in situ. From these
images it is possible to measure the open area and determine the equivalent diameter of the nano-hole
with a relative uncertainty of 2.5 %. This value of uncertainty was achieved by calibrating SEM,
STEM and also the AFM with a certified calibration gratings supplied by the Italian National
Metrology Institute (INRIM).
Figure 2 shows the SEM and STEM images of the nano-hole under investigation. The equivalent
radius calculated from the open area of nano-hole is 92 nm. In Figure 2 it is also possible to see the
difference between STEM and SEM images of these structures: in the STEM image the edge of the
nano-hole shows a stronger contrast than in the SEM image, because the image is obtained detecting
the transmitted electrons. This stronger contrast allows a more precise measurement of the open area.
Subsequently, we have characterized the nano-hole by means of AFM collecting images of the nano-
hole in tapping mode using tips Olympus OMCL AC160TS with an apical radius of 7 nm. The AFM
analysis of the nano-hole shows the surface morphology of the region drilled, i.e. a portion of the
surface containing the nano-hole. In spite of some limitations of the AFM analysis, this technique
provides a very detailed 3D reconstruction of the inlet region of the nano-hole and of the surrounding
area. Figure 2 shows an example of the nano-hole inlet region morphology.
Due to the nature of the technique, AFM investigates only the inlet area, while the inner and exit part
is not accessible (Figure 4). Since the energy distribution in the ion beam of the FIB has a Gaussian
profile, the same shape is partially reproduced in the inlet region. For this reason, the nano-holes inlet
area is a funnel with a Gaussian profile. While, to study the inner part of the nano-holes we have used
an indirect approach, i.e. we cut and analyzed cross-sections of nano-holes, obtained with the same
FIB parameters, by means of the FIB and the SEM. Before cutting we have deposited a layer of
platinum onto the nano-hole surface in order to prevent damage or FIB artefacts. Figure 5 shows
examples of nano-hole cross-sections. Combining together the information obtained from these
techniques, it is possible to duplicate precisely the actual shape of the nano-holes tested without
destroying them. This procedure is fully described in [8].
In order to study with the AFM the outlet region of the nano-holes, we have followed an indirect
approach as what was used to study their inner part, i.e. we have drilled other nano-holes turning the
chip showed in Figure 1 upside down: In this way the outlet region is accessible and can be studied
with AFM. The AFM analysis of the outlet region of several nano-holes showed a small amount of
deposited material forming a pile up around the nano-holes with height ranging from 2 nm to 8 nm,
which very likely is material deposition caused by Ga+ ion beam. The accurate and precise geometric
characterization of the nano-hole has allowed the calculation of the conductance by means of the
Direct Simulation Monte Carlo (see Section 3).
4
After the fabrication and the characterization of the inlet shape and open area of the nano-hole, the
membrane was mounted on a perforated copper disk and clamped between two CF16 flanges in order
to test their performance in gas flow generation (see section 4). The copper disk is compatible and
could replace the CF16 sealing. The disk is shaped in order to host the membrane, which is fixed to
the disk by means of the Torr-Seal (trade name by Agilent Company) glue. Figure 6 shows the
assembling scheme of the nano-hole between the CF16 flanges.
Finally, after the test we have analyzed the nano-hole by means of SEM, STEM and AFM. The nano-
hole looks identical as before the test (images not shown).
3. Calculation method of conductance by DSMC
The DSMC method [9], well known for its ability to deal with non-linear rarefied flows, was used for
the simulation of flow through the nano-hole. This method provides solutions equivalent to the ones
obtained by the Boltzmann equation, while remaining relatively simple and efficient. The fundamental
idea of the method is the decomposition of the molecular behaviour within one time step in two
sequential steps: the free motion, performed deterministically, and the process of intermolecular
collisions, performed stochastically. The method has been well described elsewhere [9] [10]and
therefore we will only mention the models used here.
Similar works for short channels have been performed in the past, such as [11] and [12], but the
geometry were cylindrical channels with abrupt openings at the inlet/outlet ends. The detailed
geometrical analysis of the nano-hole allowed us to take into account the curved wall boundaries in the
radial direction.
A modified version of the dsmcFoam solver [13] of the OpenFOAM® computational package has
been employed for this purpose. The most important modifications concern the introduction of
molecules from a Maxwellian distribution function of a specified number density at the open
boundaries in some distance in front of and behind the channel, as well as the appropriate initialization
of particles in the domain.
A mesh consisting of tetrahedral elements has been produced using the SALOME meshing suite [14],
allowing a very good approximation of the measured geometry. The geometry was assumed to possess
rotational symmetry around the axis and therefore we considered only a 10° slice of the full 360°
domain. The cell size varied in three levels, with the smallest cells concentrated near the wall
boundaries, in order to capture the large gradients of the macroscopic quantities (pressure,
temperature, velocity) and provide sufficient accuracy. In all cases, the cell size was much smaller
than the mean free path, as it is required by the DSMC method. A region representing part of the
upstream container has been included in the computational domain for the correct imposition of the
corresponding conditions. Its dimensions were equal to 8 times the orifice minimum radius in the
radial and axial direction, similar to previous works [11]. The corresponding part downstream has
5
been omitted, since vacuum conditions prevail behind the outlet cross-section and therefore backflow
due to gas injection or intermolecular collisions is negligible. The resulting cell grid is shown in
Figure 7. The No-Time Counter (NTC) scheme and the Variable Hard Spheres model [9] were
employed for the calculation of the potential collision pairs. The diffuse scattering kernel has been
employed for wall-particle interactions.
There were approximately 30 particles per cell and the sampling took place between 106-109 time
steps. The time step was 10-12
seconds, significantly less than the mean time between collisions. Using
these settings, an excellent agreement (maximum deviation 0.23%) with molecular flow fitting
equations for straight channels of the same length [15] has been confirmed. Parameterization studies
have also been carried out for the numerical parameters and showed that a numerical accuracy of
about 2% may be expected.
4. Measurement procedure of conductance by a primary gas flowmeter
The experimental set-up of our measurement system for the conductances of the leak elements is
shown in Figure 8. The measurement is carried out by a direct comparison of the unknown flow rate
qnh from the nano-hole with the known flow rate from the primary gas flowmeter qfm. The signal of a
Quadrupole Mass Spectrometer (QMS) serves to compare the two flow rates. For this purpose, the
nano-hole and the flowmeter are mounted at equivalent places on the vacuum system with respect to
the QMS. The two exits are at a distance of 1 m from the ion source of the QMS with a diffuser (not
shown in the figure) in between. The length of the tube which is about 1 m from the gas source to the
QMS, the diffuser and the position of the QMS, ensure that equal flows from both sources (the
flowmeter and the nano-hole) generate the same signal on the QMS.
The flowmeter is a primary measurement device and was described in [16]. It generates known gas
flow rates from 10-7 Pa L s-1 to 10-2
Pa L s-1 at temperatures near room temperature. The flowmeter was
modernized since the publication [16], but the measurement method remained the same. A very
similar flowmeter in our laboratory was described in [17] with an extended measurement uncertainty
discussion of the flow rate.
The QMS is mounted on a 6-way DN63CF cross. The cross is pumped by a 180 L/s (for helium)
turbomolecular pump with Holweck stage backed by a membrane pump.
To establish the upstream pressure of the nano-hole, the upstream part is connected to a volume Vup of
about 1 liter in which the desired gas species with pressure pup can be injected. The nano-hole can be
by-passed for quick venting for dismantling and pumping after mounting.
In the upper range of the generated flow (> 10-5
Pa L s-1
) the flowmeter is used in the constant pressure
mode [16], [17]. The gas flow exits via a leakage, which is a fine control leak valve, with a
conductance of about 10-6
L s-1
. The pressure decrease can be compensated by changing the volume by
squeezing a bellows displacer volume. The pressure is measured by a differential capacitance
6
diaphragm gauge with respect to a constant pressure reference volume. The volume change ∆V by the
displacer to keep the pressure constant within a measured time interval gives the conductance of the
leakage at the prevailing pressure pfm in the flowmeter:
t
VC
∆
∆=fm ( 1 )
The flow rate is obtained by multiplying Cfm with the pressure pfm.
In the lower range of the generated flow (≤ 10-5 Pa L s-1) the flowmeter is used in the constant
conductance mode. Here, the flow through the leakage is molecular and the conductance independent
of pressure which is below 80 Pa. The conductance is measured near this pressure and then the
pressure is reduced to give the desired flow rate.
In both measurement modes, the molar gas flow rate of the flowmeter qν,fm is determined by
fm
fmfmfm,
RT
Cpq =ν (2 )
where R denotes the molar gas constant and Tfm the temperature in the flowmeter.
The flow rate through the nano-hole is determined by
0fm
0nh,fmnh,
II
IIqq
−
−= νν ( 3 )
Where: Inh is the signal of the QMS for the relevant gas species, when this is exposed to the unknown
flow from the nano-hole, Ifm, when it is exposed to the known flow from the flow meter of the same
gas species, and I0 the offset at residual pressure. The quotient term, Z, describes the ratio of the ion
current in the mass spectrometer exposed to either the nano-hole or the flowmeter and is kept close to
1 during the measurements to avoid inaccuracies due to non-linearity of the QMS signal.
The conductance of the nano-hole is then given by
up
nh,
nhp
RTqC
nhν= , ( 4 )
where Tnh is the temperature of the gas travelling through the nano-hole. By inserting Eq. (2) and (3) in
(4), the relative standard uncertainty of Cnh can be determined, following the indication of the GUM
[18], by
2
up
up
2
nh
nh
22
fm
fm
2
fm
fm
2
fm
fm
nh
nh)()()()()()()(
+
+
+
+
+
=
p
pu
T
Tu
Z
Zu
T
Tu
C
Cu
p
pu
C
Cu , ( 5 )
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As will be seen in Tables 1 and 2, this relative expanded uncertainty (two times the standard
uncertainty) varies between about 1.6 % and 4.1 %.
5. Results and discussion
Helium, argon and nitrogen were used as test gases. The results for the measured conductance in terms
of the upstream pressure for nano-hole are shown in Table 1 and Figure 9 for helium and Table 2 and
Figure 10 for argon. The uncertainties were calculated according to Eq. (5). The molecular regime,
where the conductance should be independent of pressure, extends up to about 10 kPa for helium and
3 kPa for argon. This is reasonable as the mean free path of helium is a factor of 3.2 larger than the
one for argon at the same pressure. The corresponding mean values in the molecular range were
(5.25±0.17)⋅10-9
L/s for helium and (1.70±0.05)⋅10-9
L/s for argon. No Knudsen minimum can be
observed in the transitional regime between molecular and viscous flow within the uncertainties,
which is typical for orifices. The measurements for nitrogen were not as accurate due to the
interference with CO on the QMS and showed a mean value of (2.34±0.44)⋅10-9 L/s.
The ratio CHe/CAr for L2 was (3.09±0.13), while the ratio of the square root of the molecular masses is
16.3440 = being in very good agreement with the measurement. The ratio CHe/CN2 was
(2.24±0.42), while the expected one is 65.2428 = .
The calculated conductances from the known geometry are shown in Tables 1 and 2. In the molecular
regime the calculated values are (5.16±0.75)⋅10-9
L/s for helium and (1.63±0.24)⋅10-9
L/s for argon.
Since the geometrical effects on the uncertainty estimate of the simulation would have required an
extensive Monte Carlo investigation by changing the geometrical input parameters for the simulation
and repeating the simulation many times, we simplified the uncertainty estimate in the following
manner: The conductance of the nano-hole was expressed as conductance CA of a zero thickness
orifice multiplied with transmission probability PC. The uncertainty of CA was calculated from the
uncertainty of the diameter, while the uncertainty of PC was calculated from an analytical
approximation for short tubes (Eq. (4.158) in [15]). It was verified that the analytical approximation
reflected the simulation result within about 25% so that it could be used instead. Additionally, a
numerical uncertainty of 2% for PC was added quadratically according to GUM [18]. The uncertainty
for the diameter was estimated to 2.5% as evaluated from the STEM images. The uncertainty of the
length of the tube (equal to the thickness of the silicon nitride membrane) was given by the
manufacturer to be about 10 %. The total uncertainty for the conductance of the nano-hole is 7.3% or
the expanded uncertainty 15% (Tables 1 and 2). This means that measurements and calculations of the
predicted conductances of the simulation were in full agreement within the uncertainties.
We should note here that there are other influences that cannot be taken into account quantitatively
without a large effort: The cross-section of the nano-hole was assumed to be circular in the simulation
8
(see Fig.2 for measured shape). The outlet section was approximated by measurements from other
nano-holes which had to be destroyed for this purpose. For both simplifications we expect that their
influence is stronger in the transitional flow regime than in the molecular regime. In addition, the
diameter change in the simulation would shift the onset of the transitional regime. An accommodation
factor equal to 1 was used, but specular reflections may happen resulting in a lower accommodation
coefficient than 1.
6. Conclusion
Our results shown that the flow through a nano-hole with diameter of about 200 nm is molecular for
helium up to 10 kPa and 3 kPa for heavy gases. This means that by a calibrated conductance value for
a single gas, the conductance for any other gas can be predicted, provided that its molecular mass is
known. By knowing the upstream pressure, the flow rate through such an element can be calculated.
Even the flow rate for gas mixtures can be predicted, since the flow through the nano-hole is
molecular and the gas molecules do not interact in the orifice.
By a complete geometric characterization (STEM, SEM and AFM) of the nano-holes it was possible
to predict the conductances by the DSMC method within about 15 % in agreement with
measurements.
Acknowledgements
Support through the EMRP IND12 project is gratefully acknowledged. The EMRP is jointly funded by
the EMRP participating countries within EURAMET and the European Union. OpenFOAM is a
registered trademark of OpenCFD Limited. The authors thank Joachim Buthig and Mercede Bergoglio
for their contribution to this work.
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Tables
Table 1 Measured and calculated conductances and their expanded uncertainties (two times standard
uncertainties, k=2) for L2 and helium in dependence of upstream pressure
