Paarl Africa Underground Laboratory

Robert Adam5,1, Claire Antel14, Munirat Bashir23, Driss Benchekroun18, Xavier
Bertou20, Markus Böttcher8, Andy Buffler7, Andrew Chen4, Rouven Essig22, Jules
Gascon12, Mohamed Gouighri19, Trevor Hass1, Gregory Hillhouse6, Abdeslam
Hoummada18, Anslyn John1, Pete Jones3, Youssef Khoulaki18, Luca Lavina13,
Lerothodi Leeuw2, Mantile Lekala9, Robert Lindsay2, Roy Maartens2, Yin-Zhe
Ma1, Fairouz Malek11,∗, Peane Maleka3, Jacques Marteau12, Rachid Mazini21,
Thebe Medupe8, Bruce Mellado Garcia4, Marcello Messina15, Lumkile Msebi2,
Chilufya Mwewa26, Zina Ndabeni3,7, Richard Newman1, George O’Neill16, Fabrice
Piquemal10, Lydia Roos13, Daniel Santos11, Silvia Scorza11, Fedor Simkovic24,
Ivan Stekl25, Yahya Tayalati17, Smarajit Triambak2, Zeblon Vilakazi4, Shaun
Wyngaardt1, JJ van Zyl1

1Stellenbosch University-South Africa;
2University of the Western Cape-South Africa;
3iThemba LABS-South Africa;
4School of Physics, University of the Witwatersrand Johannesburg-South Africa;
5Square Kilometre Array Observatory-South Africa;
6Botswana International University of Science and Technology-Botswana;
7University of Cape Town-South Africa;
8North West University Potchefstroom-South Africa;
9The University of South Africa;
10LP2I, CNRS-IN2P3, Université Bordeaux-France;
11LPSC, CNRS-IN2P3, Université Grenoble Alpes-France;
12IP2I, CNRS-IN2P3, Université Claude Bernard Lyon-France;
13LPNHE, CNRS-IN2P3, Sorbonne Université Paris-France;
14 Université de Genève-Switzerland;
15 LNGS, Gran-Sasso-Italy;
16European Spallation Source ERIC, Lund,-Sweden;
17Mohammed V university of Rabat-Morocco;
18Hassan II university of Casablanca-Morocco;
19Ibn Tofail University of Kenitra-Morocco;
20Centro Atómico Bariloche, CNEA/CONICET-Argentina;
21Institute of Physics, Academia Sinica, Taipei-Taiwan;
22Stony Brook University, USA;
23Ibrahim Badamasi Babangida University-Nigeria;
24Comenius University Bratislava-Slovakia;
25IEAP CTU Prague-Czechia;
26Brookhaven National Laboratory, USA.

*Contact<EMAIL_ADDRESS>
###### Abstract

Establishing a deep underground physics laboratory to study, amongst others,
double beta decay, geoneutrinos, reactor neutrinos and dark matter has been
discussed for more than a decade within the austral African physicists’
community. PAUL, the Paarl Africa Underground Laboratory, is an initiative
foreseeing an open international laboratory devoted to the development of
competitive science in the austral region. It has the advantage that the
location, the Huguenot tunnel, exists already and the geology and the
environment of the site is appropriate for an experimental facility. The paper
describes the PAUL initiative, presents the physics prospects and discusses
the capacity for building the future experimental facility.

##  1 Introduction

Neutrinos and the search for dark matter have been big drivers in the field of
experimental physics, sitting at the frontier between nuclear physics,
particle physics, astroparticle physics and cosmology. These studies are only
possible in underground laboratories (ULs) where the experiments are shielded
from cosmic rays by at least $\sim$ 1000 m of rock. These laboratories are
located in mines and tunnels, and most of the operating laboratories are to be
found in the northern hemisphere: Europe, USA, Russia, Canada and Japan.
Indeed, Europe is an example of such an integration with a network of
investigation programs, common laboratories and integrated systems for
education and investigation, allowing knowledge transfer to its institution
and the society.

Figure 1:  Map of the existing or planned underground laboratories. Green
dots: the operating facilities. Orange dot: under commissioning SUPL facility
in Australia . Red dots: ANDES at the Argentina-Chile border and INO in India.
Yellow star: the future Paarl Africa underground lab facility, in the Western
Cape Province of South Africa

Figure 1 shows the map of the existing or planned underground laboratories. At
present, there are 11 ULs in operation (green dots), one is under
commissioning (orange dot, SUPL in Ausralia), and two more (red dots) are
planned: ANDES at the Argentina-Chile border and INO in India. ANDES (Ague
Negra Deep Experiments Site)[1] aims to create a 4000 m2 general purpose deep
underground laboratory with 1750 m of overburden in the Agua Negra tunnel
between Argentina and Chile. INO (India-based Neutrino Observatory)[2] is a
planned laboratory in Pottipuram, India, with 1200 m of overburden to host an
iron calorimeter (ICAL) for neutrino studies. The yellow star is the foreseen
location of the future Paarl Africa underground lab facility, in the Western
Cape Province of South Africa, under the 1300 m Du Toitskloof mountain with
$\sim$ 800 m of rock overburden for the Huguenot tunnel itself. About 100
experiments are running or are under construction in the ULs currently in
operation and roughly 6000 researchers are involved.

From the experimental point of view, the events that may lead to the
identification of dark matter particles and rare electroweak decays have
extremely low counting rates and would be observed at energies where an
overwhelming background of radioactive decays and cosmic rays is present.
Given the high energy and penetration of the cosmic radiation, the detectors
need to be shielded by going underground. One can get a reduction of the
cosmic radiation effects by orders of magnitude by digging labs some
kilometers under the surface, or by building them under high geological
features - see Figure 2 showing the laboratory depth (in meter water
equivalent 111The meter water equivalent (m.w.e.)[3] is a standard measure of
cosmic ray attenuation in underground laboratories. A laboratory at a depth of
1000 m.w.e is shielded from cosmic rays equivalently to a lab 1000 m below the
surface of a body of water.) as a function of the cosmic-ray muon flux for
each continent. Circles represent the volume of science space.

Figure 2:  Effective depth (in meter water equivalent[3]) vs decreasing
cosmic-ray muon flux (cm2 s-1)[4]. The yellow star indicate the corresponding
numbers for the future Paarl Africa underground lab facility in the Western
Cape Province of South Africa, under the 1300 m Du Toitskloof mountain with
$\sim$ 800 m of rock overburden for the Huguenot tunnel.

The scientific work can expand to research projects in multiple fields
including biology, geosciences, chemistry, mining technology and underground
construction and architecture.

Discussions about an underground research facility in South Africa started in
2011. As one of the world’s largest producers of gold, South Africa has a
number of the world’s deepest gold mines, including the recently-closed
TauTona Gold Mine with a depth of 3900 m, and Mponeng deeper still. In 1965
the Nobel Prize laureate Frederich Reines along with South African Physicist,
Friedel Sellschop detected the first atmospheric neutrino events in the East
Rand mine near Johannesburg, South Africa (SA) [5]. Initial focus by the South
African nuclear physics community was on establishing an underground facility
in one of South Africa’s deep gold mines. The alternative is to develop such
an underground laboratory inside the Huguenot tunnel, which is conveniently
located between the towns of Paarl and Worcerster in the Western Cape Province
of South Africa, see Figure 1 and section 4.

##  2 Research Purpose

The underground laboratory will provide a dynamic environment for advances in
ultra-sensitive detectors and ultra-low radiation techniques and highly
trained graduates ready to lead innovation in both the global search for rare
events and cutting-edge technological development to benefit South Africa
industry. However, physics research is not the only field of interest.

The need for very low radioactive material for dark matter and neutrino
underground experiments gave birth to the study of new detectors able to
measure extremely low radiation levels. These very sensitive detectors, able
to detect levels of radiation a millionth of the natural radiation of the
human body, have to be located deep underground to be shielded from cosmic
radiation. The industry has shown interest in these techniques to select pure
materials with almost no radioactive content. Researchers involved in this
work can contribute to many needs in South Africa for accurate measurements,
such as the detection of the radioactive gas radon that has been identified as
a major radiation hazard in South African underground mines [6].

The research of endolithic bacteria and technologies for bio-leaching are a
subject of continuously increasing interest. New ideas in underground
biology[7, 8, 9] have also included initiatives related to astrobiology[10],
examining the impact of radiation (or the lack of it) to evolutionary
processes or formation of bio-aerosols.

The possibility to control the lighting conditions and other environmental
parameters makes underground laboratories ideal places to experiment with
hydroponic farming. They provide an ideal sandbox to experiment with lighting,
determining the ability to use underground conditions as a working environment
and even landscaping the tunnels as habitable environments, particularly with
the very stable temperatures that occur underground.

The underground laboratory can also provide new insight on environmental
topics. In glaciology, the study of ice samples from the Arctic, Antarctic
etc. allows mapping of the evolution of climatic parameters and contamination
both in space and over time for the last centuries. The measurement of 137Cs
and 241Am is the only way to get a precise dating of ice samples and has to be
done in underground laboratories[11].

Underground laboratories are excellent sites to investigate new technologies
for geothermal energy production or energy storage as well as the
characterization of the geology of the earth[12]. The Cape Supergroup, where
the lab would sit, has been identified as a region of interest for geothermal
research in particular.

##  3 Landmark Physics and Flagship astroparticle experiments

Figure 3:  Current status of the Spin-Independent WIMP-nucleon interaction,
one of the most common models used to look for WIMPs[13].

Underground laboratories provide the low-background radioactive environment
necessary for astroparticle physics to explore extremely rare phenomena. The
underground location naturally guarantees high suppression of muons and
cosmic-ray particles produced in the atmosphere and, consequently, of
cosmogenic by-products.

In the case of astroparticle physics, the leading fields are the direct search
for dark matter and the study of neutrino properties. Figure 3 shows the
current landscape for direct dark matter searches [13]. The figure is scoping
the WIMP (Weakly Interacting Massive Particle) model (one of the most
promising models in which dark matter is typically assumed to have been
produced thermally in the early universe) and parameters are constrained by
the measured relic abundance of dark matter today. When looking for very low
cross-sections, those detectors will soon start detecting neutrinos (from our
Sun, from the atmosphere, from supernovae, etc…). This so-called neutrino
floor, acting as background, will make the dark matter search more
challenging, but it will also open new doors to neutrino physics studies. As
can be seen from the figure, two regions of the parameter space still remain
to be explored, which require different experimental efforts.

At high WIMP masses, only detectors using noble liquids (Xenon and Argon) can
reach the required sensitivity. Besides the continuous reductions of the
radiogenic background, the next generation experiments aim to build larger
detectors to reach the neutrino floor, often surrounded by even more large
shielding systems. This implies larger volumes and, as a consequence, wide
footprints to host all subsystems. The timescale of next generation dark
matter experiments foresees data taking ranging from 2030 (DARWIN [14], seen
as the natural step after the currently running experiments XENONnT [15] and
LZ [16]) and beyond (Argo, which is the next step of DarkSide-20k [17], still
under construction). While the underground site for those experiments is not
yet defined, a novel underground site that does not surpass the existing ones
in terms of depth can hardly be a good choice for them.

At small dark matter masses, however, there are many new opportunities to
which a novel underground laboratory can contribute (for recent reviews with
references see e.g. [18, 19]). Dark matter with mass below the proton (sub-
GeV) typically lies in a dark sector, which does not interact directly with
any of the Standard Model forces. Instead, new particles (such as dark
photons, scalars, or pseudo-scalars) can mediate interactions between the dark
matter and the ordinary matter. Dark matter particles can scatter off
electrons, nuclei (via the Migdal effect), or collective modes in a detector
target, leaving small signals consisting of ionization or phonons.

There is nowadays a plethora of experimental techniques that are trying to
gain sensitivity to such small signals. In this case, the challenge is to
develop detectors with very low energy thresholds and excellent control over
detector backgrounds, rather than to build large detectors that are highly
demanding in terms of occupied volumes in an underground laboratory. In fact,
the last few years has seen significant efforts in the direct-detection
community on building small-scale experiments that can probe low-mass dark
matter.

A range of target materials, including liquid xenon, liquid argon, superfluid
helium, silicon, germanium, gallium-arsenide, sapphire, diamonds, molecular
liquids and gases, quantum dots, etc. are being used, or have been proposed,
as target materials. For example, the DarkSide [17] collaboration plans to
have a ”Low Mass” version of their technology (DarkSide-LM), tuned to have a
radiogenic background optimised for low mass searches, using only one-ton of
liquid Argon. So far, there is no established schedule on when and where this
detector will be built. This could be a good opportunity for PAUL.

Another technology that has recently come to the forefront is the Skipper-CCD
[20], which has achieved single-electron sensitivity. Charge Coupled Devices
(CCDs) are known to be very useful in digital cameras and widely used in
research for it. Experiments like SENSEI [21] and DAMIC-M [22] have already
achieved several results using small detectors [23, 24, 25, 26], and are now
working to increase the target mass. Both collaborations are planning a next-
generation detector called Oscura [27, 28].

Cryogenic detectors, operating at temperature well below 1 Kelvin, have also
achieved a sensitivity at the single-electron level [29, 30]. At the same time
work is done to extend the particle-identification capabilities offered by
this technology to effectively control the new backgrounds that appear as new
energy domains are explored [31].

There are many other efforts and plans, and since the hunt for low-mass dark
matter is relatively young, there is space for new experiments and new
underground sites. The search for light dark matter particle is particularly
attractive for these new technologies as significant advances can be achieved
with kg-scale detector arrays.

One of the most interesting facts about having the possibility to perform an
experiment of direct dark matter detection in an underground laboratory
located in the Southern Hemisphere is to compare the eventual systematic
errors or modulation with respect to the same detector in the Northern
Hemisphere. Any systematic error or annual modulation correlated to a seasonal
variation will have an opposed phase, giving the opportunity to discriminate
them with respect to a dark matter signal. It also opens different regions of
parameter space when searching for daily modulations [32, 33, 34].

The other advantage to build an UL facility in South Africa is to combine the
direct detection with indirect dark matter detection from radio astronomy
surveys that South Africa is leading. As is well known, South Africa has been
leading the world-astronomy collaboration “Square Kilometre Array” (SKA) mid-
frequency arrays [35], and has already built 64-dishes precursor “MeerKAT”
telescopes in 2018 [36]. The MeerKAT telescopes are equipped with L-band
($856$-$1711$ MHz) and UHF-band ($544$-$1087$ MHz) receivers with average
channel width $16\,$kHz. The dark matter (WIMP) annihilation into standard
model particles (e.g. $b\bar{b}$, $\tau^{+}\tau^{-}$, $\mu^{+}\mu^{-}$) can
eventually cascade to $e^{+}e^{-}$ pairs, which can lose their energies by
inverse Compton-scattering and synchrotron radiation. This cascade process can
generate fluxes in X-rays and radio wavelength respectively, with detailed
variation determined by the dark matter mass and the astrophysical
environment. In Ref. [37], we have shown that with L-band receiver the radio
antenna can place constraints on the WIMP cross-section in the mass range of
$\sim 10$-$100$ GeV. In addition, the annihilation and decay of WIMP during
dark ages can leave imprints in the cosmic microwave background (CMB)
radiation and alter the cosmic reionization proces [38], which can be measured
by high-sensitivity Hydrogen-Epoch Reionization Array (HERA [36, 39]) and CMB
Stage-4 survey [40]. Both experiments have South Africa’s deep participation
and involvement. Therefore, the strong synergy between the astrophysical
(indirect) probes and Paarl Africa Underground Laboratory (direct probe) can
jointly measure and constrain dark matter effect, which may shed lights on new
physics.

##  4 The Paarl Africa Underground Laboratory and the experimental facility

The development of the Huguenot tunnel as an underground low-level radiation
facility holds a number of strategic advantages. Such a facility is located
approximately 25 km from Stellenbosch University and 40 km from the iThemba
LABS. It offers a unique possibility to build up a scientific complex suitable
for the detection of events under ultra-low radiation exposure, as required
for dark matter and neutrino studies, as well as for other areas of research.

Figure 4:  Map of the Western Coast of Cape Town and the location of the
Huguenot tunnel. Modified image extracted from Google Map.

The tunnel provides quick and easy access to the local research communities.
Research programs done at such a facility will also support postgraduate
training programs at Stellenbosch University, the University of the Western
Cape and the University of Cape Town. Furthermore, the research at the PAUL
will support national and international research activities in astronomy,
nuclear and particle physics, as well as many of the research topics already
discussed in section 2.

According to the geological survey, which was done during the excavation
period (1973 - 1984), the composition of the mountain range consists mainly of
quartzitic sandstone (also referred to as Table mountain sandstone), see
Figure 5. It is therefore expected that radon concentration levels within the
tunnel should be low, and should therefore not have a considerable
contribution to the background radiation signature. The height of the Du
Toitskloof mountain range is about 1300 m, providing $\sim$ 800 m of rock
overburden for the Huguenot tunnel itself [41].

Figure 5:  Geological mapping and analysis of the rock of the tunnel [41].

In 2013, the environmental radiation groups at Stellenbosch University and
iThemba LABS performed a preliminary radon measurement [42].The concentrations
measured at three sites confirm that the level of radon is well below any
degree of consideration, with a mean level of radon no more than 64.9 Bqm-3
(others showing mean levels as low as 45.4 Bqm-3). In addition to the gamma
ray spectra and radon monitoring, the background measurements were performed
alongside biomonitors placed at strategic locations within the tunnel in order
to measure the air pollution levels within the tunnel.

There is a real opportunity for a dedicated underground research facility to
be established inside the Huguenot tunnel. The ideal would be to build the
laboratory at the north of the service tunnel [43] (to be built itself to the
north of the current one, driving from Paarl to Rawsonville). The laboratory
will consist of a surface facility, located near the road freeway Huguenot
tunnel, and potentially extensive underground spaces and various connecting
tunnels. The experimental hall will be covered by about 800 m of rock, under
the Du Toitskloof Mountains, protecting the experiments from cosmic rays. It
will be easily accessible, with the ability to drive to it rather than using
mine-shaft elevators, similarly to the LSM [44] or LNGS [45] facilities.

For a small or medium size lab, the LSM concept [44], where one has a
deceleration line which stops on the side, is an appropriate design. The fact
that one access tunnel to the lab is large allows one to stop a large truck in
the proper space and then discharge the truck contents with a crane to move it
inside the cavern. See Figure 6 for a minimal design of a possible 600 m2
laboratory (40x16x16 m3), based on the medium cavern design of ANDES [1].

Figure 6:  Mock-up of a possible 600 m2 laboratory (40x16x16 m3) in the
Huguenot tunnel. Courtesy: Joaquín Venturino (CNEA), April 2023.

The future underground laboratory is currently being designed; It directly
involves the company operating the Huguenot tunnel since earthworks and
infrastructure construction are planned over the next five to ten years.

Apart from the operating surface and the digging work (in the tunnel) which
involves large-scale building work, minimum services are necessary for the
viability of the scientific installation. They are for example:

  * •

Ventilation equipment capable of renewing the total volume of air in the
laboratory every hour and of filtering the radon present in the air;

  * •

Air conditioning to maintain the laboratory at 21° Celsius. It is also
necessary to control the relative humidity to guarantee the operation of
equipment and the hygrothermal comfort of people;

  * •

Power supply equipment and generator;

  * •

Storage tanks and treatment plant for water and other effluents;

  * •

Provision of basic services to the experiments: Compressed air, water, data
storage and processing, communications;

  * •

The computer center must have a fiber optic connection;

  * •

The medical care kit in the event of an accident;

  * •

Fire fighting systems, surveillance and security systems.

All these services require collaboration with industry, locally and
regionally. Some state-of-the-art equipment are very typical of the discipline
and can, sometimes, only be made with the help of international technological
companies.

##  5 Conclusion

PAUL is foreseen as an open international laboratory, a unique opportunity for
Africa devoted to the development of a competitive science in the region. It
has the advantage that the location, the Huguenot tunnel, exists already and
the geology and the environment of the site is appropriate for an experimental
facility.

PAUL will host international collaborations in experimental areas of great
impact in physics, cosmology, geology, biology, material sciences, climate
science etc. The activities of the laboratory will be accompanied by
supporting centers and universities.

PAUL has a strong synergy with the radio astronomy probe of dark matter by
South Africa’s MeerKAT, HERA and SKA mid-array, through the WIMP annihilation
and decay effects which eventually cascade down into synchrotron radiation.
The strong synergy between “ground” (PAUL) and “sky” (SKA) can be combined to
explore much larger parameter space with high-sensitivity, which may reveal
some new physics beyond Standard Model.

The proposal will offer the chance to integrate with other facilities,
laboratories and experiments and increase the potential of the region by
increasing its academic activities, the formation of new human resources and
the development of new basic research and technologies.

The initiative will be managed by a steering committee composed by
representatives from South African universities and the South African Research
Council, alongside experts from international facilities. The future
laboratory will have a structure similar to an international facility such as
Laboratoire Souterrain de Modane in France[44] or Laboratori Nazionali Del
Gran Sasso in Italy[45].

This is a unique opportunity to build an excellent deep underground
laboratory, the only one in Africa, with a strong impact on regional
integration.

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# Interaction of a graphene sheet with a ferromagnetic metal plate

Anh D. Phan1, N. A. Viet2, Nikolai A. Poklonski3, Lilia M. Woods1, Chi H. Le4
1Department of Physics, University of South Florida, Tampa, Florida 33620, USA
2Institute of Physics, 10 Daotan, Badinh, Hanoi, Vietnam  3Physics Department,
Belarusian State University, Minsk 220030, Belarus  4School of Engineering,
University of Greenwich, Medway, United Kingdom

###### Abstract

Nanoscale surface forces such as Casimir and the van der Waals forces can have
a significant influence on fabrication, handling and assembly processes as
well as the performance of micro and nano devices. In this paper, the
investigation and the calculation of the Casimir force between a graphene
sheet and a ferromagnetic metal substrate in a vacuum are presented. The
reflection coefficients of graphene are graphene-conductivity dependent, and
the conductivity of graphene is described by the Kubo formalism. There is an
effect of magnetic properties of the metal on the Casimir interaction. The
magnetic effect plays a significant role at low temperatures or high value of
chemical potential. The numerical results also demonstrate that the thickness
of a metal slab has a minor influence on the Casimir force. The investigation
and findings about the Casimir force in this study would lead to useful
information and effective solutions for design and manufacturing of micro and
nano devices, especially in the areas of micro and nano machining,
fabrication, manipulation, assembly and metrology.

###### pacs:

78.20.Ls, 31.30.jh, 78.67.Wj, 12.20.Fv

##  I Introduction

Micro and nano technologies are potential economic engines and are considered
as enabling technologies with exceptional economic capabilities. There have
been increasingly more products in which the integrated micro and nano-
electromechanical systems (MEMS/NEMS) are crucial in today’s market because
they allow for improved functionality, lower costs and higher quality. With
the decrease in the size of MEMS and NEMS, additional nanoscale surface
forces, such as the Casimir force and the van der Waals force, should be
considered 1 ; 2 , especially in the areas of micro and nano machining,
fabrication, manipulation, assembly and metrology. Therefore, a fundamental
understanding of the Casimir force has recently been under intense discussion
in the research and technological development (RTD) communities.

The Casimir force can cause the small elements in a device to stick together.
At small scales, the nanoscale surface forces may overcome elastic restoring
actions in the device and lead to the plates’ sticking during the fabrication
process 1 . It has been recently shown that Casimir forces may hamper the
functioning of MEMS and NEMS devices by providing a pull-in instability 36 .
The Casimir force between two objects can induce adhesion and stiction leading
to failures in devices 2 ; 3 ; 32 ; 36 . Particularly, when the size of a
material is under the threshold length, the influence of the loading
interaction on design and manufacturing of micro and nanodevices becomes much
more prominent.

The Casimir interaction originates from the quantum electromagnetic
fluctuations between two objects 2 ; 3 ; 40 . It provides a fundamental
understanding about nanoscience and has a significant influence on the high
performance of devices. Over six decades, the Lifshitz theory has been
employed to investigate this force in metal-metal 4 ; 5 , semiconductor-
semiconductor 6 , metal-superconductor 7 ; 28 , metamaterials-metamaterials 8
; 9  and graphene-graphene systems 10 ; 11 ; 12 ; 34 . Results of some
configurations have been experimentally examined and have good agreement with
theoretical calculations 3 ; 13 .

The interaction between a non-magnetic metal with a magneto-dielectric
material has also recently been discussed 29 ; 14 ; 15 . These studies
illustrate that the magnetic properties can significantly influence in the
Casimir pressure. Theoretical and experimental studies also show that it is
possible to obtain the repulsive Casimir force in such systems 2 ; 15 ; 16 ,
and the results could provide possible solutions to handle the stiction and
adhesion problems found in MEMS/NEMS devices. In terms of theory, the
calculations depend substantially on the model that describes the dielectric
function of metals.

Graphene, an atomically thin layered material with novel properties, has
gained a great deal of attention since its discovery 30 . Researchers from a
wide range of scientific fields have spent considerable effort in exploring
its nature and its potential for practical applications 17 ; 18 ; 19 ; 20 .
Recently obtained results demonstrate that graphene can be a promising
candidate for next-generation electronic devices 31 . The functions of
graphene have been applied to diverse areas from biology to material science.

Figure 1: (Color online) Schematics of a two-dimensional layer (graphene) and
a substrate (FM metal) with thickness $D$.

In this paper, we calculate the Casimir force between a graphene layer and a
ferromagnetic (FM) substrate shown in Fig.1. The FM material is taken to be Fe
described via a dielectric and magnetic response properties. We utilize the
Lifshitz theory to examine the role of the separation, temperature, and
thickness of the substrate. We also investigate the influence of the magnetic
response of the FM and show that in most cases its contribution is relatively
small compared to the one from the dielectric response.

This paper is organized as follows: Sec. II presents the detailed expressions
and calculations of the thermal Casimir interactions. The results and
discussions are mentioned in Sec. III. Finally, Sec. IV presents the
conclusions.

##  II CASIMIR INTERACTION BETWEEN GRAPHENE AND A METAL SUBSTRATE

Here we calculate the Casimir force between the graphene sheet and a substrate
in vacuum at a separation $a$. The force per unit area at temperature $T$ is
given as 3 ; 21 ; 34 ; 24

$\displaystyle
F(a,T)=-\frac{k_{B}T}{\pi}\sum_{l=0}^{\infty}\left(1-\frac{1}{2}\delta_{l0}\right)\int_{0}^{\infty}q_{l}k_{\perp}dk_{\perp}$
$\displaystyle\times\left(\frac{r_{TE}^{(1)}r_{TE}^{(2)}}{e^{2q_{l}a}-r_{TE}^{(1)}r_{TE}^{(2)}}+\dfrac{r_{TM}^{(1)}r_{TM}^{(2)}}{e^{2q_{l}a}-r_{TM}^{(1)}r_{TM}^{(2)}}\right),$
(1)

where $k_{B}$ is the Plank constant, $r_{TM}^{(1,2)}$ and $r_{TE}^{(1,2)}$ are
the reflection coefficients corresponding to the transverse magnetic (TM) and
transverse electric (TE) field modes.

The reflection coefficients for the substrate are given as 2 ; 14 ; 15

$\displaystyle r_{TE}^{(1)}\equiv
r_{TE}^{(1)}(i\xi_{l},k_{\perp})=\frac{\mu_{1}(i\xi_{l})q_{l}-k_{1}}{\mu_{1}(i\xi_{l})q_{l}+k_{1}},$
$\displaystyle r_{TM}^{(1)}\equiv
r_{TM}^{(1)}(i\xi_{l},k_{\perp})=\frac{\varepsilon_{1}(i\xi_{l})q_{l}-k_{1}}{\varepsilon_{1}(i\xi_{l})q_{l}+k_{1}},$
(2)

where

$\displaystyle q_{l}\equiv
q_{l}(i\xi_{l},k_{\perp})=\sqrt{k_{\perp}^{2}+\frac{\xi_{l}^{2}}{c^{2}}},$
$\displaystyle k_{1}\equiv
k_{1}(i\xi_{l},k_{\perp})=\sqrt{k_{\perp}^{2}+\mu_{1}(i\xi_{l})\varepsilon_{1}(i\xi_{l})\frac{\xi_{l}^{2}}{c^{2}}}.$
(3)

It is important to note that $k_{\perp}$ is the wave vector component
perpendicular to the plate, $c$ is the speed of light, $\xi_{l}=2\pi
k_{B}Tl/\hbar$ is the Matsubara frequencies. The response properties of the
metal are characterized by the dielectric function $\varepsilon_{1}(i\xi_{l})$
and the permeability function $\mu_{1}(i\xi_{l})$ which are frequency
dependent along the imaginary axis ($\omega_{l}=i\xi_{l}$). We use the Drude
model $\varepsilon_{1D}$ and the plasma model $\varepsilon_{1P}$ to describe
the dielectric function of metals 3 ; 14 ; 29

$\displaystyle\varepsilon_{1D}(i\xi)=1+\frac{\omega_{p}^{2}}{\xi(\xi+\gamma_{p})},$
$\displaystyle\varepsilon_{1P}(i\xi)=1+\frac{\omega_{p}^{2}}{\xi^{2}},$ (4)

where $\omega_{p}$ is the plasma frequency and the damping parameter is
$\gamma_{p}$. At room temperature, the $l=0$ term of the Casimir-Lifshitz
force in the metallic system is dominant. This reason allows us to consider
the contribution of the static magnetic permeability $\mu(0)\gg 1$ on the
Casimir force. For higher orders of $l$, one should put $\mu(\xi_{l})=1$ 29 .
Both models lead to $r_{TM}^{(1)}(0)=1$. However, we can obtain two different
expressions of $r_{TE}^{(1)}(0)$ when applying these two models 29

$\displaystyle r_{TE,D}^{(1)}(0,k_{\perp})=\frac{\mu(0)-1}{\mu(0)+1},$
$\displaystyle
r_{TE,P}^{(1)}(0,k_{\perp})=\frac{\mu(0)ck_{\perp}-\sqrt{c^{2}k_{\perp}^{2}+\mu(0)\omega_{p}^{2}}}{\mu(0)ck_{\perp}+\sqrt{c^{2}k_{\perp}^{2}+\mu(0)\omega_{p}^{2}}},$
(5)

here $r_{TE,D}^{(1)}(0,k_{\perp})$ and $r_{TE,P}^{(1)}(0,k_{\perp})$ are the
TE reflection coefficients corresponding to the Drude and plasma model,
respectively.

For graphene, the reflection coefficients are found to be 11 ; 16

$\displaystyle r_{TE}^{(2)}\equiv
r_{TE}^{(2)}(i\xi_{l},k_{\perp})=-\frac{2\pi\xi_{l}\sigma/q_{l}c^{2}}{1+2\pi\xi_{l}\sigma/q_{l}c^{2}},$
$\displaystyle r_{TM}^{(2)}\equiv
r_{TM}^{(2)}(i\xi_{l},k_{\perp})=\frac{2\pi\sigma q_{l}/\xi_{l}}{1+2\pi\sigma
q_{l}/\xi_{l}},$ (6)

where $\sigma\equiv\sigma(i\xi)$ is the 2D conductivity described via the the
Kubo formalism 11

$\displaystyle\sigma(i\xi)=\frac{2e^{2}k_{B}T\ln(2)}{\pi\hbar^{2}\xi}+\frac{e^{2}\xi}{8\pi
k_{B}T}\int_{0}^{\infty}\frac{\tanh(x)dx}{x^{2}+(\frac{\hbar\xi}{4k_{B}T})^{2}}.$
(7)

The first term in Eq.(7) corresponds to the intraband contribution, while the
second term corresponds to the interband contribution. At low temperatures,
the graphene conductivity $\sigma(i\xi_{l})$ approaches to the universal value
$\sigma_{0}=e^{2}/4\hbar$.

Figure 2: (Color online) The normalized graphene conductivity
$\sigma(i\xi)/\sigma_{0}$ vs frequency at given temperatures.

Using Eq.(6) and Eq.(7) the TM reflection coefficient of graphene is given by

$\displaystyle
r_{TM}^{(2)}=\frac{\frac{2e^{2}k_{B}T}{\pi\hbar^{2}}+\frac{e^{2}\xi_{l}^{3}}{8\pi
k_{B}T}\int_{0}^{\infty}\frac{\tanh
xdx}{x^{2}+(\hbar\xi_{l}/4k_{B}T)^{2}}}{\frac{\xi_{l}^{2}}{2\pi
q_{l}}+\frac{2e^{2}k_{B}T}{\pi\hbar^{2}}+\frac{e^{2}\xi_{l}^{3}}{8\pi
k_{B}T}\int_{0}^{\infty}\frac{\tanh xdx}{x^{2}+(\hbar\xi_{l}/4k_{B}T)^{2}}}.$
(8)

The equation suggests $r_{TM}^{(2)}(0,k_{\perp})=1$. In the same way, it can
be found the expression of $r_{TE}^{(2)}(0,k_{\perp})$

$\displaystyle
r_{TE}^{(2)}(0,k_{\perp})=-\frac{(2\pi/c)8\ln(2)\sigma_{0}}{\pi\lambda_{T}k_{\perp}+(2\pi/c)8\ln(2)\sigma_{0}},$
(9)

where $\lambda_{T}=\hbar c/k_{B}T$ is the thermal wavelength in most
materials. The explicit expression of $r_{TE}^{(2)}(0,k_{\perp})$ shows that
it is susceptible to temperature. $r_{TE}^{(2)}(0,k_{\perp})$ is nonzero only
because of the intraband component of the graphene conductivity.

The first Matsubara frequencies $\xi_{1}$ are approximately 0.027, 0.162, and
0.325 $eV$ corresponding to the temperature 50, 300 and 600 $K$, respectively.
As seen in Fig. 2, for $l\geq 2$, it is possible to substitute $\sigma_{0}$
for $\sigma(i\xi_{l})$ in Eq.(6) to calculate the reflection coefficients and
the higher order terms in the Casimir-Lifshitz force formula.

##  III RESULTS AND DISCUSSIONS

In this section, we consider the Casimir interactions between graphene and a
Fe metal plate. The parameters of Fe are $\omega_{p}=4.09$ $eV$,
$\gamma_{p}=0.018$ $eV$, and $\mu(0)=10^{4}$ 4 . As discussed in previous part
$r_{TM}^{(1)}(0,k_{\perp})=r_{TM}^{(2)}(0,k_{\perp})=1$, the TM contribution
to the $l=0$ term of the Casimir force is given

$\displaystyle
F^{(0)}_{TM}(a,T)=-\frac{k_{B}T}{2\pi}\int_{0}^{\infty}\frac{k_{\perp}^{2}dk_{\perp}}{e^{2k_{\perp}a}-1}=-\frac{k_{B}T\zeta(3)}{8\pi
a^{3}}.$ (10)

The expression above is identical to the term with $l=0$ in the Casimir force
between two metals. The sum of the $l\geq 1$ is much smaller than that of the
metallic system due to the presence of the low graphene conductivity 11 . As a
result, the $l=0$ term is the dominating term. Now, to calculate the
contribution of the TE mode $F^{(0)}_{TE}(a,T)$ with $l=0$, it is necessary to
choose a good model for Fe. Both the plasma 14  and Drude model 22 , however,
has been widely used to compute the Casimir interaction and fit with
experimental data. Determining an accuracy of two models as compared to
measurement data has been a controversial issue. In the following work, we
utilize the plasma model to calculate the interactions. The expression of
$F^{(0)}_{TE}(a,T)$ is expressed by

$\displaystyle
F^{(0)}_{TE}(a,T)=-\frac{k_{B}T}{2\pi}\int_{0}^{\infty}\frac{k_{\perp}^{2}dk_{\perp}}{\frac{e^{2k_{\perp}a}}{r_{TE}^{(1)}(0,k_{\perp})r_{TE}^{(2)}(0,k_{\perp})}-1}.$
(11)

Note that $F^{(0)}_{TE}(a,T)$ is a unique term affected by the magnetic
property of the ferromagnetic substrate. In Fig.3, the ratio
$F^{(0)}_{TE}/F^{(0)}_{TM}$ is less than 0.75 $\%$ for distances $a$ in the
range from 0 to 1 $\mu m$ at given temperatures. It indicates that the effect
of $\mu(0)$ on the Casimir force can be neglectable. For this reason, Fe can
be treated as a regular non-magnetic metal $\mu(i\xi)=1$.

For a finite-thickness iron slab $D$, the reflection coefficients of the metal
material are modified as follows 15 ; 23

$\displaystyle
R_{TE,TM}(i\xi_{l},k_{\perp})=r_{TE,TM}^{(1)}\frac{1-e^{-2k_{1}D}}{1-(r_{TE,TM}^{(1)})^{2}e^{-2k_{1}D}}.$
(12)

Figure 3: (Color online) The ratios of $F^{(0)}_{TE}(a,T)/F^{(0)}_{TM}(a,T)$
and $F^{(2)}(a,T)/F^{(0)}_{TM}(a,T)$ at 50, 300, and 600 $K$ as a function of
separation distance $a$.

To study the influence of thickness on the Casimir force, we rewrite Eq.(1) in
the following way

$\displaystyle F(a,T)=F^{(0)}_{TM}(a,T)+F^{(0)}_{TE}(a,T)+F^{(1)}(a,T),$ (13)

here $F^{(1)}(a,T)$ is the sum of all $l\geq 1$ terms and is thickness-
dependent. Because $r_{TM}^{(1)}(0,k_{\perp})=1$, so
$R_{TM}^{(1)}(0,k_{\perp})=1$. From this, the expression of
$F^{(0)}_{TM}(a,T)$ is in Eq.(10) and is independent of $D$. The component
force $F^{(0)}_{TE}(a,T)$ is weaker than that for the thick plate and one can
consider $F^{(0)}_{TE}(a,T)\approx 0$. Therefore, only $F^{(1)}(a,T)$ is
sensitive to a variation of subtrate thickness.

Figure 4: (Color online) The Casimir pressures corresponding to the
approximate and full expression at different temperatures. Here
$F_{0}(a)=-\pi^{2}\hbar c/240a^{4}$.

Figure 3 shows that the thickness of the Fe slab does not influence much the
Casimir forces. At the same temperature, the separation between the two curves
corresponding to the finite plate and the semi-infinite plate is small. In
bulk materials, a reduction of the thickness gives rise to a small decrease of
the Casimir force. The thermal wavelength of graphene is
$\lambda_{T}/200\approx 38$ $nm$ at room temperature. At distances beyond the
thermal wavelength of a system, the contribution of the sum of $l\geq 1$ terms
can be ignorable. In the investigated range, the influence of $F^{(1)}(a,T)$
is not significant. This finding suggest that since graphene is a two
dimensional monolayer material, it significantly interacts only with the
nearest interface layers of the bottom substrate. Graphene’s effect on layers
separated by a sufficient number of layers is negligable. The Fe slab with
$D=20$ nm has more layers than that which are meaningful. Another interesting
special feature is that at large distance limit ($a\geq 0.8\mu m$) the
contribution of $F^{(1)}(a,T)$ to the total Casimir force is minor. It can be
explained that The $l\geq 2$ terms provide weak force components due to the
presence of the small universal conductivity in the reflection coefficients.
It is important to note that a universal graphene conductivity is a main
reason for less than 2 $\%$ difference as one compares the dispersion force
between two graphene sheets with $F_{0}(a)$ 11 . It demonstrates that the
approximate formula for the Casimir interaction equivalent to the $l=0$ term
of the Casimir-Lifshitz expression in this regime is
$F_{app}(a,T)=-k_{B}T\zeta(3)/8\pi a^{3}$. In Fig.4, we plot the normalized
Casimir force with the full expression (see Eq.(1)) and the approximate
expression. The $l=0$ term is dominant and can replace the full expression of
the Casimir interaction when $a\geq 0.8$ $\mu m$ for 300 $K$.

The critical $T_{c}$ of Fe is 1043 $K$. At $T>T_{c}$, the magnetic properties
of the ferromagnetic material nearly vanish. Spins in the Fe slab are
rearranged so that $\mu(i\xi)=1$ at all frequencies. Nevertheless, the
dielectric function is not affected by the directions of spins. Therefore, the
plasma frequency and damping parameter are unchanged by temperature increases
at $T\geq T_{c}$. If we consider the Casimir interaction between Fe and
another metal, the phase transition leads to a significant change in the
mutual interaction 14 ; 29 . The fact that graphene is quite transparent with
the presence of the magnetic properties, the dispersion force may not be
modified.

A notable point is that the formula in Eq.(7) was used for pure graphene with
the chemical potential $\mu_{C}=0$. If $\mu_{C}\neq 0$, the graphene
conductivity has to be rewritten in the form of 25

$\displaystyle\sigma(\mu_{C},i\xi)=\frac{e^{2}k_{B}T\ln(2)}{\pi\hbar^{2}\xi}+\frac{e^{2}k_{B}T\ln(1+\cosh(\mu_{C}/k_{B}T))}{\pi\hbar^{2}\xi}$
$\displaystyle+\frac{e^{2}\xi}{\pi}\int_{0}^{\infty}\frac{\sinh(E/k_{B}T)}{\cosh(E/k_{B}T)+\cosh(\mu_{C}/k_{B}T)}\frac{dE}{(\hbar\xi)^{2}+4E^{2}}.$
(14)

The first two terms are known as the intraband conductivity, another is the
interband component. Eq.(14) shows that the intraband contribution of
$\sigma(\mu_{C},i\xi)$ still induces a non-zero value of
$r_{TE}^{(2)}(0,k_{\perp})$. Obviously, the graphene conductivity is strongly
susceptible to the chemical potential. The chemical potential of a graphene
sheet can be controlled by using an applied electric field $E_{d}$ 26 ; 27

$\displaystyle\frac{\pi\varepsilon_{0}\hbar^{2}v_{F}^{2}}{e}E_{d}=\int_{0}^{\infty}E\left(f(E)-f(E+2\mu_{C})\right)dE,$
(15)

here $f(E)$ is the Fermi distribution function, $v_{F}=c/300$ is the Fermi
velocity.

Other ways to modulate $\mu_{C}$ are an applied magnetic field 27  and
chemical doping. Using a magnetic field also varys the expression of graphene
conductivity and creates Landau energy levels. In the limit of extremely low
chemical potential, $r_{TE}^{(2)}(0,k_{\perp})$ is represented as Eq.(9). For
$\mu_{C}\gg k_{B}T$, the expression of the TE reflection coefficient with
$l=0$ is written by

$\displaystyle
r_{TE}^{(2)}(0,k_{\perp})\approx-\frac{4\sigma_{0}\mu_{C}/(\pi\hbar
c)}{k_{\perp}+4\sigma_{0}\mu_{C}/(\pi\hbar c)}.$ (16)

Figure 5: (Color online) The Casimir forces at the case with and without
taking into account the magnetic properties at $\mu_{C}=1$ $eV$.

In Fig.5, the Casimir interactions are calculated at low temperatures ($\leq
50$ $K$) and $\mu_{C}=1$ $eV$ with and without the magnetic properties. Unlike
the case of a pristine graphene sheet ($\mu_{C}=0$ $eV$), the influence of
permeability of Fe on the dispersion force is noticeable at 10 $K$. The
properties, however, have much less effect on the Casimir force at
temperatures greater than 50 $K$. The majority of $F(a,T)$ comes from the
interband conductivity of graphene. At 10 $K$, the contribution of the
intraband on the Casimir force can be considerable to that of the interband
conductivity. However, as temperature increases the interband contribution
increases substantially, and causing a decrease in the contribution from
intraband. For this reason, the magnetic properties nearly disappear at $T\geq
50$ $K$.

To investigate further the Casimir force dependence on chemical potential, we
consider the fluctuation interactions at $a=100$ $nm$ and various temperatures
versus $\mu_{C}$ shown in Fig.6. $F(a,T)/F_{0}$ changes from 0.018 to 0.034 at
$T=10$ $K$ and from 0.045 to 0.064 at $T=300$ $K$ when $\mu_{C}$ varys in the
regime below $1.2$ $eV$. In addition, at higher temperatures, the curves of
$F(a,T)/F_{0}$ is of the linear form.

Figure 6: (Color online) The Casimir pressures as a function of chemical
potential.

It is remarkable that Fe is not a superconducting material, so iron has no
temperature phase transition as temperature decreases to 10 $K$.
Alternatively, the universal value of conductivity $\sigma_{0}$ also
demonstrates that even if temperature is 0 $K$, graphene still does not change
phase and properties.

##  IV Conclusions

When design and manufacturing of micro and nano devices and components, it is
important and necessary to understand the loading and effects of related
forces on the systems and its components, especially the nanoscale surface
forces such as Casimir and the van der Waals. In this study, a comprehensive
investigation and discussion of the Casimir interaction between graphene and
FM materials is presented. We discussed about the Casimir force which is a
function of the graphene conductivity and the permeability constant of $Fe$.
It was shown that the conductivity of graphene heavily depends on the chemical
potential which again has a large impact on the dispersion force and causes a
difference between non-magnetic and magnetic calculations at the ultra-low
temperature. In other cases, there is no effect of the magnetic properties on
the Casimir force. One can apply a bias electric field to a graphene sheet to
tailor the chemical potential of graphene in order to control the magnitude of
the Casimir force. According to our numerical results, the force between a
graphene layer and the semi-infinite metal is smaller than the interaction
between two metals in 29  because the conductivity of metals is higher than
that of graphene. The numerical results also demonstrated that the thickness
of a metal slab had a minor influence on the Casimir force. The investigation
and above findings about the Casimir force in this study would lead to useful
information and effective solutions for design and manufacturing of micro and
nano devices where graphene and FM metal materials are utilised, especially in
the areas of micro and nano machining, fabrication, manipulation, assembly and
metrology.

###### Acknowledgements.

We give thanks to Dr. David Drosdoff for discussions. This research was
supported by the Nafosted Grant No. 103.06-2011.51. Lilia M. Woods
acknowledges the Department of Energy under contract DE-FG02-06ER46297.
Nikolai A. Poklonski acknowledges the ﬁnancial support from Belarusian
Republican Foundation for Fundamental Research Grant No. F11V-001.

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