Magnetocaloric cycle with six stages: Possible application of graphene at low
temperature
M. S. Reis
Citation: Applied Physics Letters 107, 102401 (2015); doi: 10.1063/1.4930577
View online: http://dx.doi.org/10.1063/1.4930577
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APPLIED PHYSICS LETTERS 107, 102401 (2015)
Magnetocaloric cycle with six stages: Possible application of graphene
at low temperature
M. S. Reisa)
Instituto de Fısica, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/n,
24210-346 Niter
oi, RJ, Brazil
(Received 24 March 2015; accepted 28 August 2015; published online 9 September 2015)
The present work proposes a thermodynamic hexacycle based on the magnetocaloric oscillations of
graphene, which has either a positive or negative adiabatic temperature change depending on the
final value of the magnetic field change. For instance, for graphenes at 25 K, an applied field of
2.06 T/1.87 T promotes a temperature change of ca. 25 K/þ3 K. The hexacycle is based on the
Brayton cycle and instead of the usual four steps, it has six stages, taking advantage of the extra
cooling provided by the inverse adiabatic temperature change. This proposal opens doors for
C 2015 AIP Publishing LLC.
magnetic cooling applications at low temperatures. V
[http://dx.doi.org/10.1063/1.4930577]
The magnetocaloric effect (MCE) is an interesting ability of magnetic materials to either change their temperature
or exchange heat with a thermal reservoir under a magnetic
field change;1–3 and these two possible situations characterise the adiabatic and isothermal processes, respectively.
This effect is analogous to the thermo-mechanical vapour
compression-expansion approach and therefore the idea of a
magnetic cooling device based on the MCE is straightforward. Indeed, adiabatic demagnetization at low temperatures has been used since 1930s.4,5 Since these initial
studies, this still emerging technology6 has formed two
working communities: one for magnetic materials optimization and other for prototype and concept development.
There are some well-established magnetic cooling devices to enable working at low temperature, including refrigeration of infrared bolometers down to 0.1 K (Ref. 7) and
astronautics applications.8,9 There are, however, fewer
examples of working devices at room temperature3,6,10 and
they are still in the prototype stage, in spite of the prediction
of commercialisation of this technology by 2015.6 Amongst
the challenges are the difficulties to produce high magnetic
fields and large amounts of magnetic refrigerant material; to
overcome this problem, some groups are focusing on smaller
devices with much less magnetic material, which are
designed for specific applications such as the cooling of electronic devices.11 As a consequence of these challenges and
efforts, it is important to cite the organisation of the community on the Magnetic Cooling Working Party of the
International Institute of Refrigeration.12
When optimising magnetocaloric materials, their properties are fully optimised around the critical temperature and
golden materials have both, magnetic and structural transitions. For instance, some materials optimised to work at
room temperature are Gd5 ðSi2 Ge2 Þ; LaðFex Si1x Þ13 , Heusler
alloys, and all their families.13–15 However, there are also a
plenty of other compounds proposed to work at low temperatures, especially intermetallics,16 multiferroics,17,18 and molecular magnets.19–21 More recently, in spite of the absence
of magnetic transition, the magnetocaloric properties of diamagnetic materials (such as gold, lead, silver, and even graphene) have been deeply explored.22–31 In few words, those
efforts describe the oscillating MCE (OMCE), which arises
due to the crossing of the Landau levels through the Fermi
level of the system by changing the applied magnetic field.
This OMCE is governed by the same physical mechanism of
the de Haas-van Alphen32 and Shubnikov-de Haas effects33
and has been previously experimentally observed, mainly in
GaAs-like materials.34,35
Thus, the aim of the present effort is to take advantage
of the OMCE to propose a magnetocaloric hexacycle, based
on the ability of diamagnetic materials (especially graphene,29 which has a more pronounced OMCE in comparison with the other diamagnetic materials26,31) to tune the
effect as either normal or inverse, i.e., either increase or
decrease its temperature due to a magnetic field change. The
normal effect is used in the hexacycle in a similar fashion as
done in magnetocaloric Brayton cycles.3 However, due to
the ability of the material to also present an inverse effect by
changing the final value of the applied magnetic field, it is
taken as an advantage and an extra cooling is performed,
thus enhancing the cycle. Further details are given in this
text, and these results are indeed able to contribute to the
knowledge of this promising technology.
The magnetocaloric potentials described below focus on
graphene; however, the results are qualitatively similar to
other diamagnetic materials, e.g., gold. See Refs. 22, 24, 26,
30, and 31 for further details. Thus, the magnetic entropy per
graphene area is given by24,29
SðT; BÞ ¼ S0 ðTÞ þ Sosc ðT; BÞ;
where
S0 ðT Þ ¼ NkB
2p2 kB T
3F
(2)
is the field independent term24 and
Sosc ðT; BÞ ¼ 2NkB
a)
[email protected]
0003-6951/2015/107(10)/102401/4/$30.00
(1)
107, 102401-1
1
X
cosð jmpÞ
T ð jxÞ
jm
j¼1
(3)
C 2015 AIP Publishing LLC
V
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102401-2
M. S. Reis
Appl. Phys. Lett. 107, 102401 (2015)
depends on the magnetic field B through the dimensionless
quantity
m¼
N/0
:
B
(4)
pffiffiffiffiffiffiffi
The parameters on the above equations are F ¼ hvF Np,
which stands for the Fermi energy for a given Fermi velocity
vF (typically of 106 m/s)29 and the density of charge carries N
(typically of 1016 m2);29 /0 ¼ ph=e ¼ 2:06 1015 Tm2 is
the magnetic flux quantum
T ð yÞ ¼
yLðyÞ
sinh y
where LðyÞ ¼ coth y 1
y
(5)
and, finally,
x ¼ 2p2 m
kB T
:
F
(6)
From these results, the magnetic entropy change
DSðT; DB : 0 ! BÞ ¼ SðT; BÞ SðT; 0Þ
(7)
could be determined29
DSðT; BÞ ¼ 2NkB
1
X
cosð jmpÞ
T ð jxÞ:
jm
j¼1
(8)
Above we wrote DS(T, B) instead of DS(T, DB: 0 ! B) to
simplify the notation; however, it is important to stress that
the magnetic entropy change refers to a change on the
applied magnetic field, from zero to B. Analogous to the
above information, the adiabatic temperature change, based
on the condition S(T, 0) ¼ S(TB, B), could be obtained.24 It
should be noted that T is the initial temperature, i.e., the temperature of the system under zero magnetic field. The final
temperature TB after a magnetic field change from zero up to
B is given as follows:24
TB ¼ T 1
3F X
cosð jmpÞ
T ð jxB Þ;
2
p kB j¼1
jm
(9)
which must be solved numerically, due to the selfconsistence of the above equation, since xB ¼ x(TB) Thus,
from TB, the adiabatic temperature change DT ¼ TB T can
be easily evaluated.24
For the purpose of the present work, only the adiabatic
temperature change is important and therefore this quantity
is presented in Figure 1, as a function of m ¼ N/0 =B. Note
from Eq. (9) that, in spite of the self-consistent character, it
depends on cosðjmpÞ and therefore the oscillatory behaviour
of this quantity as a function of m is expected. This is the
main point of the present effort and we will take advantage
of this behaviour to propose the magnetocaloric hexacycle.
More precisely, note the magnetic field change from zero up
to a magnetic field of B ¼ 20.6/m [T] (for an odd value of m)
produces a positive adiabatic temperature change, i.e., the
graphene heats up. In a similar fashion, a magnetic field
change from zero up to a magnetic field of B ¼ 20.6/m [T]
(for an even value of m) produces a negative adiabatic temperature change, i.e., the graphene cools down. Thus,
FIG. 1. Adiabatic temperature change for graphenes at low temperature as a
function of m ¼ N/0 =B. For a magnetic field change from zero up to B1 (corresponding to m ¼ 11), the adiabatic temperature change is positive, while
changing the magnetic field from zero up to B2 (corresponding to m ¼ 10),
produces a negative temperature change. The oscillatory character of this
quantity as a function of the final value of the applied magnetic field is the
key rule to the enhanced magnetocaloric cycle proposed in the present work.
depending on the final value of the applied magnetic field,
we can tune either normal or inverse magnetocaloric effect.
Figure 1 runs m values from ca. 1 (corresponding to a
large magnetic field of ca. 20.6 T) up to 30 (corresponding to
an accessible value of magnetic field of ca. 0.69 T). Note the
oscillatory behaviour of this quantity and the extremely high
values of adiabatic temperature change. For instance, m ¼ 3
(corresponding to B ¼ 6.87 T) produces ca. þ35 K of temperature change. The inset of this figure focuses on the feasible
values of magnetic field reached by an assembly of permanent
magnets; more precisely, we highlight m ¼ 10 (corresponding
to B ¼ 2.06 T) and m ¼ 11 (corresponding to B ¼ 1.87 T), and
these values will be used further in the next paragraphs.
The focus of the present effort is to present a magnetocaloric hexacycle based on adiabatic processes (Brayton-like
cycle), and thus, the adiabatic temperature change as a function of temperature is also an important quantity to be seen,
and this is depicted in Figure 2. Note that as mentioned
before, for even values of m, we find an inverse magnetocaloric effect, in which the material (for our case graphene)
cools down under the corresponding magnetic field. In a similar fashion, for an odd value of m, we observe a normal temperature change, where the material heats up under the
corresponding magnetic field. It is quite important to stress
the large values of temperature change found for small values of magnetic field (those accessible with a simple assembly of permanent magnets).
To describe then the magnetocaloric hexacycle, we need
first to see an S–T diagram, i.e., an entropy-temperature
picture. This is depicted in Figure 3 for three cases: m ¼ 10
(even, and corresponding to 2.06 T of applied magnetic
field), m ¼ 11 (odd, and corresponding to 1.87 T), and B ¼ 0.
To optimise the understanding of this hexacycle, we also
propose a corresponding device, based on a rotatory wheel,
analogous to other magnetocaloric devices,3 with six sectors
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102401-3
M. S. Reis
FIG. 2. Adiabatic temperature change as a function of temperature for those
two cases explored in this work: even/odd value of m and the corresponding
negative/positive temperature change. This result is the key rule to the proposed magnetocaloric hexacycle. Note also the low values of applied magnetic field and corresponding quite high temperature change.
filled with magnetic material (in our case, diamagnetic graphene). It then rotates counter-clockwise and receives an
exchange fluid that flows along the circuit “Heat sourcerotatory wheel-heat sink.” See Figure 4 for a better understanding of this device.
Thus, below is the detailed description of the magnetocaloric hexacycle, step-by-step, based on Figures 3 and 4.
Step 1 ! 2: The graphene is placed under a magnetic
field corresponding to an odd value of m, for instance,
B1 ¼ 1.87 T, and then it heats up from T1 to T2.
Step 2 ! 3: At this stage, the amount of graphene keeps
under the same value of magnetic field (B1), however, the
corresponding sector receives a fluid (“green” temperature)
from the heat source. The graphene-fluid thermal
FIG. 3. S–T diagram for graphenes. It includes the normal and inverse magnetocaloric effect, based on odd and even values of m. The violet closed circuit represents the proposed hexacycle, based on those two values of applied
magnetic field.
Appl. Phys. Lett. 107, 102401 (2015)
FIG. 4. Proposed device based on the magnetocaloric hexacycle. The circle
represents a counter-clockwise rotatory wheel with six sectors, filled with regenerator material (in our example, graphene). Coloured lines represent
tubes which contain a fluid flow at different temperatures. Light and dark
grey sectors are those under B1 and B2 values of magnetic field, respectively.
equilibrium is reached (at T3), and then the sector drains the
fluid (“red” temperature ¼ T3) to the heat sink.
Step 3 ! 4: The graphene then leaves the magnetic field
and decreases its temperature from T3 down to T4. While
this occurs, the fluid is releasing heat at the heat sink,
remaining in thermal equilibrium with a thermal reservoir
in which the heat sink is embedded (“purple” temperature).
Step 4 ! 5: Here is the main advantage of the oscillating
magnetocaloric effect. In spite of to close the cycle, as done
in usual Brayton cycles and even in other magnetocaloric
cycles,3 the cycle now takes advantage of the inverse temperature change of the graphene to further cool down its
temperature. It is then placed under a magnetic field corresponding to an even value of m, for instance, B2 ¼ 2.06 T;
then it goes from T4 down to T5.
Step 5 ! 6: The regenerator material is then kept under the
same value of magnetic field (B2) and the sector receives
the fluid from the heat sink to then reach the thermal equilibrium at T6, i.e., “yellow” temperature (the coldest one to
the fluid).
Step 6 ! 1: Finally, to close the cycle, the graphene package leaves the magnetic field and then its temperature
increases from T6 up to T1, while the fluid (now at its lowest
temperature) goes through the heat source. After this, the
heat source releases the fluid at the “green” equilibrium
temperature.
In summary, this work focus on the proposal of the
magnetocaloric hexacycle, based on the oscillating magnetocaloric effect, predicted to occur in diamagnetic materials.22–31 This effect can be controlled as either inverse or
normal, i.e., a decrease or increase of temperature due to a
magnetic field change, by tuning the final value of magnetic
field after a magnetic field change. In the work presented
here, we chose graphene as a refrigerant material (its magnetocaloric properties have been described in detail in the previous publications22–25,27–29) and we found a huge inverse/
normal temperature change of 25 K/þ3 K for 2.06 T/1.87 T
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102401-4
M. S. Reis
of magnetic field change (easily obtained with a simple assembly of permanent magnets). This interesting property
opens doors to the proposed magnetocaloric hexacycle,
which is based on six stages: two of those based on the normal MCE and another two taking advantage of an extra cooling due to the inverse MCE. Thus, magnetic cooling at low
temperatures can be further improved and optimised by considering this enhanced cycle. These ideas can be further
developed and discussed by considering, for instance, the
conception of an improved cycle with both, normal and
inverse MCE; however, without those intermediate steps
where the magnetic field goes to zero. It is a challenge and
still an open question to be deeply discussed.
We acknowledge FAPERJ, CAPES, CNPq, and
PROPPI-UFF for financial support.
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