Oscillating magnetocaloric effect - UFF

APPLIED PHYSICS LETTERS 99, 052511 (2011)

Oscillating magnetocaloric effect M. S. Reisa) Instituto de Fı´sica, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/n, 24210-346, Nitero´i-RJ, Brazil

(Received 16 May 2011; accepted 1 July 2011; published online 5 August 2011) This Letter presents the oscillatory behavior found in the magnetic entropy change of diamagnetic materials. We show that this quantity depends on the oscillating term and, as a consequence, the magnetocaloric potential can be tuned as either inverse or normal, depending on the value of the magnetic field change. A quite small change (103 T) of the magnetic field change is able to invert the magnetic entropy change. These results open doors for applications at quite low temperatures and can be further developed to be incorporated into adiabatic demagnetization C 2011 American Institute of Physics. refrigerators, as well as sensible magnetic field sensors. V [doi:10.1063/1.3615296]

The magnetocaloric effect (MCE) is intrinsic to magnetic materials and is induced via coupling of the magnetic sublattice with the magnetic field, which alters the magnetic part of the total entropy due to a corresponding change of the magnetic field. The MCE can be estimated via the magnetic entropy change DSðT; DBÞ, and is a function of both temperature T and the magnetic field change DB. In addition, the MCE has a significant technological importance, since magnetic materials with large MCE values could be employed in various thermal devices.1 However, the magnetocaloric potential of materials is not limited to room temperature applications; and the best example is the adiabatic demagnetization refrigerator, which can reach mK scale.2 In spite of those direct applications, the inverse magnetocaloric effect (negative magnetic entropy change), is opening doors for devices, since the reverse operation can be done (cooling instead of heating, when submitted to a magnetic field change under adiabatic conditions). In this sense, a lot of efforts have been made by the community.3–9 On the other hand, oscillations are usually found in thermodynamic quantities due to the de Haas-van Alphen effect, as a consequence of the crossing of the Landau levels through the Fermi energy.10 It was observed in superconductors,11,12 heavy fermions,13 and even simple diamagnetic metals, such as Au.14 Thus, the aim of the present Letter is to connect the oscillations found in thermodynamic quantities of diamagnetic materials with potential applications due to its inverse and normal magnetocaloric effect at low temperatures. From now on, let us survey some results of an electron gas, as well as this model with an external perturbation, such as magnetic field. To achieve reasonable results, some approximations are needed, and for the present survey, we will consider eF kB T and lB B > kB T. The magnetic entropy SðT; BÞ depends on the grand canonical partition function ZðT; BÞ through SðT; BÞ ¼ kB

@ ½T ln ZðT; BÞ; @T

a)

(1)

Author to whom correspondence should be addressed. Electronic mail: [email protected]

0003-6951/2011/99(5)/052511/3/$30.00

where lnZ ¼

ð1

de gðeÞlnð1 þ zebe Þ;

(2)

0

z ¼ elb is the fugacity and g(e) the density of states, given by10 þ1 X 3 3=2 1=2 ð1Þl lp p 1=2 e gðeÞ ¼ NeF cos e þa 2 a 4 l1=2 l¼1 ¼ g0 ðeÞ þ gB ðeÞ: (3) Above, l is the chemical potential, b ¼ 1/kBT, and a ¼ lBB, where B is the applied magnetic field. Note the density of states has two contributions: gB(e), explicitly dependent on the magnetic field B; and g0(e), that does not depend on B, and is the same of that for a non-perturbed electron gas. Thus, from Eqs. (2) and (3) it is possible to write ln Z ¼ ln Z 0 þ ln Z B :

(4)

The first term of the above equation is well known and does not need to be described here, since it rules the non-perturbed electron gas. On the contrary, the second term needs an additional attention. After integration by parts (two times), we achieve 1 1 b be lnZ B ¼ GðeÞlnð1 þ ze Þ þ GðeÞ 1 be e þ 1 z 0 0 ð1 @ 1 ; (5) b de GðeÞ @e z1 ebe þ 1 0 where 3=2 X þ1 3 a ð1Þl lp p GðeÞ ¼ N e ; sin 2p eF a 4 l3=2 l¼1

(6)

and 3=2 X þ1 3 a ð1Þl lp p GðeÞ ¼ 2 Na e : cos 2p eF a 4 l5=2 l¼1

99, 052511-1

(7)

C 2011 American Institute of Physics V

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052511-2

M.S. Reis

Appl. Phys. Lett. 99, 052511 (2011)

DS 3 a 3=2 p p ¼ cos eF T ðxÞ: NkB 2 eF a 4

Remembering, we are considering eF kB T, and then: l eF. This condition implies to z 1 and Eq. (5) resumes as o ln Z B ¼ ln Z no B þ ln Z B ;

(8)

ln Z no B ¼ Gð0ÞeF b Gð0Þb

(9)

where

has a non-oscillatory character and ln

Z oB

3=2 X þ1 3 a ð1Þl lp p eF ½sinhðxl Þ1 ¼ N cos 3=2 2 eF a 4 l l¼1

Note only T ðxÞ contains information on the temperature, and the amplitude of the magnetic entropy change can be maximized and minimized by the cosine term. Thus, we can consider the condition p p cos eF ¼ 61; (17) a 4 and then, eF 1 ¼ n: a 4

(10) has an oscillating behavior. In addition, xl ¼ lx and x¼

p2 kB T ¼ p2 : ab lB B

(11)

Summarizing, the grand canonical partition function (Eq. (4)) has two contributions: one that depends on B and other that does not. The first one has also two contributions: one that depends on B and T as a power law (Eq. (9)), and other that oscillates depending on the magnetic field (Eq. (10)). From the results above, it is possible to evaluate the magnetocaloric properties of this system. From Eq. (4): SðT; BÞ ¼ SðT; 0Þ þ SB ðT; BÞ and then, the magnetic entropy change, defined as DSðT; DBÞ ¼ SðT; BÞ SðT; 0Þ, is simply DSðT; DBÞ ¼ SB ðT; BÞ, since SðT; 0Þ ¼ SðT; 0Þ. It is a quite interesting result, since the magnetic entropy change only depends on the term explicitly dependent on B, i.e., SB ðT; BÞ ¼ kB

@ ½T ln Z B ðT; BÞ: @T

(12)

o However, from Eq. (8): SB ðT; BÞ ¼ Sno B ðT; BÞ þ SB ðT; BÞ, i.e., there are two contributions: one without oscillations and other oscillating. However, we obtained: Sno B ðT; BÞ ¼ 0, and the magnetic entropy change only depends on the oscillating term. Thus,

(16)

(18)

Note an interesting result: n even (and zero), implies in a negative (and minimized) DS; n odd, implies in a positive (and maximized) DS; and, finally, a semi-half integer makes zero the magnetic entropy change (and then the magnetocaloric potential of the material). We can re-write Eq. (16) as 3=2 DS 3 1 ¼ cosðnpÞT ðxÞ; NkB 2 n þ 1=4

(19)

where 1 x¼p y nþ ; 4 2

(20)

and y ¼ kBT/eF ( 1, as imposed before). Considering Au (eF ¼ 5.51 eV), n ¼ 0 implies in B ¼ 380 kT (absolutely out of the laboratory range) and, analogously, n ¼ 104 implies in B ¼ 10 T. Following this example, note that Figure 1 presents the magnetic entropy change as a function of y ¼ kBT/eF. The scale of y is of the order of 105 and then, considering Au, the temperature is of the order of 1 K. Thus, for Au, this effect is comfortable visible for 10 T and 1 K. In addition, as mentioned above, the magnetic entropy change can change its sign depending on the value of n, and this behavior can also be seen in Figure 1.

DSðT; DBÞ ¼ SoB ðT; BÞ 3=2 X þ1 3 a ð1Þl lp p ¼ NkB eF T ðxl Þ; cos 2 eF a 4 l3=2 l¼1 (13) where T ðxl Þ ¼ ½sinhðxl Þ1 xl Lðxl Þ;

(14)

and L(xl) is the Langevin function Lðxl Þ ¼ cothðxl Þ

1 : xl

(15)

As usually considered for thermodynamic quantities derived in this limit of low temperatures,10 we will consider only l ¼ 1 term. Of course, it is an approximation; however, after this consideration, we can understand the physics behind this model. Then,

FIG. 1. Magnetic entropy change as a function of y (proportional to the temperature). See Eq. (19).


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M.S. Reis

Appl. Phys. Lett. 99, 052511 (2011)

In conclusion, the magnetocaloric potential of the system discussed here is indeed smaller compared to the standard ferromagnetic materials.1 However, those oscillations can be useful if improved in terms of the magnetocaloric potential, especially for magnetic cooling at quite low temperatures, since the sign of the magnetic entropy change can be ruled by the final value of the applied magnetic field. This effect can indeed be used, if improved, in adiabatic demagnetization refrigerators. In addition, the contrary proposal is also true. Since this thermal system is sensible to 103 T of change of magnetic field change (within 10 T), it is possible to works as a highly sensible magnetic field sensor (with huge magnetic field on the background).

FIG. 2. (Color online) Magnetic entropy change as a function of the n (a function of the inverse magnetic field B). See Eq. (19).

We are in debt with A. Magnus Carvalho, P. J. von Ranke, N. A. Oliveira and A. M. Gomes for their fruitful comments. This work is supported by Brazilian agencies: FAPERJ, CNPq, CAPES, and PROPPi-UFF. 1

The magnetocaloric effect is due to a magnetic field change DB: 0 ! B(n). The final value of the magnetic field B(n) can tune the sign of DS; and the change of the magnetic field change that is able to do this inversion is (for n 1) jDðDBÞj ¼ jDBðn þ 1Þ DBðnÞj

eF 1 : lB n2

(21)

Again for Au, note that around 10 T of magnetic field change, an increasing of the final value of magnetic field in 103 T is enough to invert the magnetic entropy change. In other words, this effect is sensible to a magnetic field 10 000 times smaller then the applied one. It opens doors for applications, as discussed below. Finally, the magnetic entropy change as a function of magnetic field (for a constant temperature, say y ¼ 105; that for Au is around 1 K) is depicted in Figure 2. Note the oscillatory behavior of this quantity: maximized for n even, zero for n half-integer, and minimized for n odd. The modulation presented is given by Eq. (19) without the cosine term.

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