Cooling achieved by rotating an anisotropic superconductor in a constant magnetic field : A new perspective

A new type of rotary coolers based on the temperature change (ΔTrot) of an anisotropic superconductor when rotated in a constant magnetic field is proposed. We show that at low temperature the Sommerfeld coefficient γ(B,Θ) of a single crystalline superconductor, such as MgB2 and NbS2, sensitively depends on the applied magnetic field (B) and the orientation of the crystal axis (Θ), which is related to the electronic entropy (SE) and temperature (T) via the expression: SE=γT. A simple rotation of the crystal from one axis to one another in a constant magnetic field results in a change in γ and hence SE: ΔSE=ΔγT. A temperature change −ΔTrot ∼ 0.94 K from a bath temperature of 2.5 K is achieved by simply rotating the single crystal MgB2 by 90° with respect to the c-axis direction in a fixed field of 2 T. ΔTrot can be tuned by adjusting the strength of B within a wide magnetic field range. Our study paves the way for development of new materials and cryogenic refrigerators that are potentially more energy-eff...

4][5][6] Since this technology uses the magnetizing/demagnetizing principle to lower temperature of a magnetocaloric material in a refrigeration cycle, 3 a large mechanical energy is practically required for moving the material in and out of the magnetic field source. 7][10][11][12] Due to the anisotropic property, a large difference in magnetic entropy can be expected when the material is magnetized along the easy and hard magnetization axis directions. 10n other words, the magnetic entropy change and hence the temperature change of a magnetocaloric material can be realized by simply rotating the material by 90 o from the hard magnetization axis to the easy magnetization axis in a fixed magnetic field. 10,11Keeping the magnetocaloric material within a constant magnetic field enables the conception of rotary magnetic refrigerators working at high frequency, thus leading to the cooling power enhancement and the increased compactness of the devices. 8,111][12] Continuing efforts are to search for materials with enhanced anisotropic magnetocaloric properties.
4][15] This technology relies primarily on the phenomenon that a normal metal -insulator -superconductor junction exhibits electronic cooling in the normal metal, when biased at a voltage just below the superconductor's gap.A refrigerator based on vanadium (V) with a critical temperature of ∼4 K has been reported to efficiently cool down electrons in an Al island from 1 K to 0.4 K. 15 However, this type of refrigerator usually suffers from a poor evaluation of highly energetic quasiparticles in the superconducting electrodes.
In this Letter, we propose a new type of solid-state cooling devices based on the concept of a rotary magnetic refrigerator and anisotropy of the Sommerfeld coefficient of a superconducting material.We show that cooling can be achieved by simply rotating an anisotropic superconductor by 90 o within a constant magnetic field.The potential importance of the proposed device for cryogenic refrigeration is discussed.
According to the thermodynamic theory, the total entropy (S T ) of a material is a sum of the lattice entropy (S L ), electronic entropy (S E ), and magnetic entropy (S M ), which can be expressed as follows: with S L being described by 8,9 S E being described by and S M being described by 16 where N is the number of spins, R the universal gas constant, T D the Debye temperature, γ the Sommerfeld coefficient, k B the Boltzmann constant, µ the chemical potential, ε the energy level, f (ε) the Fermi distribution function, and ρ d σ (ε) the density of states.Here we intentionally describe S M for itinerant electron systems, 16 since type-II superconductors such as MgB 2 and NbS 2 , as discussed in this paper, are considered like common metals.From equations (2), (3), and (4), it appears that S M depends on both magnetic field (B) and temperature (T ), while S L and S E depend on temperature only.
In most of the previous MCE studies 3,4,6 only the change in magnetic entropy (∆S M ) with respect to magnetic field was considered in contributing to the total entropy change (∆S T ).However, we recall that a volume change can affect the phonon system and hence the Debye temperature (T D ) by where η is the Gruneisen parameter, which is between 1 and 3 for many systems. 17It follows that application of an external magnetic field can cause the volume of the material to change significantly, especially around its structural/magnetic phase transition temperature, leading to the change in T D and hence ∆S L , which, in addition to the ∆S M , contributes to the ∆S T . 18,19In this regard, S L clearly depends not only on temperature, but also on the applied magnetic field.Through direct measurements of the MCE in NiCoMnIn alloys, Kihara et al. obtained an extremely large value of ∆S L (∼51 J/kg K), demonstrating the dominant contribution of ∆S L to the ∆S T in this type of material. 191][22] Since there is a large difference in γ between the antiferromagnetic state and the ferromagnetic one in metamagnetic materials, such as CoMnSi alloys, 18 application of a sufficiently high magnetic field can convert the AFM into the FM phase, resulting in a large change in γ and hence the large ∆S E via the relation: the electronic density of states) and hence the dominant contribution of ∆S E to the ∆S T in this material. 18It has also been noted that γ is different with respect to orientation of the material (γ depends on Θ -the angle between the applied magnetic field and the crystal axis), especially for single crystals with strong magnetocrystalline anisotropy. 21,22As a result, a rotation of the crystal by 90 o from the hard magnetization axis to the easy magnetization axis can lead to a change in γ and hence ∆S E .This follows that in addition to the rotating magnetic entropy change (∆S rot M ), this rotating electronic entropy change (∆S rot E ) may also contribute significantly to the total rotating entropy change (∆S rot T ) of the material, [8][9][10][11][12][13][14][15][16][17][18][19]21,22 but the prediction of which remains to be verified experimentally.
4][25] However, there is a large difference in γ for B cri 1 < B < B cri 2 (see FIG. 1b).This indicates that a large ∆S rot E can be realized by simply rotating the crystal by 90 o from the perpendicular (B ⊥ c) to parallel (B // c) direction of the c-axis.The corresponding temperature change (∆T rot ) due to this rotation can be estimated as where C p is the specific heat of the material.From the magnetic field dependence of ∆γ in FIG.1b and equation ( 6), and considering C P to be magnetic field independent in the considered low temperature range, the magnetic field dependences of ∆S rot E and ∆T rot can be simulated, the results of which are shown in FIG. 2. It can be observed that both ∆S rot E and ∆T rot reach the maximal values at a certain magnetic field under which the largest difference in γ is obtained.Interestingly, ∆T rot is tunable by adjusting the strength of B, which is beneficial for the design of active refrigerators.Taken ∆γ ∼ 1.5 mJ/mol K 2 and C P ∼ 10 mJ/mol K for MgB 2 for B = 2 T and T = 2.5 K from Refs.23 and 24, we have estimated ∆S rot E ∼ 3.75 mJ/mol K and −∆T rot ∼ 0.94 K.4][25] This is of particular interest in exploiting the proposed technique for ultralow temperature cooling applications.Nevertheless, we must note that in a degenerate electron system like MgB 2 , 23 spin is an index indicating electronic states rather than an independent local moment.As a result, in this system the magnetic entropy associated with spins of itinerant electrons near Fermi level cannot be simply separated from the total electronic entropy.Since the density of these spins is relatively small, their response to an applied external magnetic field may be small, possibly contributing a tiny cooling effect.Further experimental studies are needed to verify this.
Based on the above findings, we propose in FIG. 3 two possible ways for the design of new rotary refrigerators and their implementations.The first design is similar to that proposed previously for a rotary magnetic refrigerator, 19 where a refrigerator constituted of the superconducting single crystals blocks, instead of using the magnetocaloric single crystals blocks, can be designed (see FIG. 3a).By rotating the crystal axis by 90 o from the perpendicular to parallel direction of the c-axis, the ∆S rot E and hence ∆T rot can be realized.This method is potentially useful for the liquefaction of helium.The second design is based on the cooling principle of a normal metal/superconductor junction, but, instead of using an external voltage to bias the junction near the superconducting gap, a layered film material composed of a normal metal and an anisotropic superconductor can be simply rotated by 90 o , allowing the normal metal to be cooled down to a lower temperature due to the cooled superconducting layer (see FIG. 3b).Alternatively, the film is fixed while the magnetic field is rotated by 90 o .It has been reported that the sensitivity of ultralow temperature detectors, such as superconducting quantum interference proximity transistor sensors, is significantly improved when the devices are maintained to operate in the sub-50-mK temperature regime. 26Since the presently proposed technique allows cooling down within the sub-50-mK temperature region by selecting suitable superconducting materials, it potentially opens up new opportunities for applications of a cooling device for advanced microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS).
In summary, we have proposed a new approach for the development of rotary coolers using anisotropic superconductors and their various temperature responses with respect to rotating the material axis by 90 o in a constant magnetic field.The proposed method may find its usefulness in cryogenic refrigeration technology.Future research should focus on seeking superconducting materials that exhibit large rotating electronic entropy changes in low magnetic field range (< 2 T), in order to exploit this concept fully.

FIG. 1 .FIG. 2 .
FIG. 1.(a) Simulated magnetic field dependence of the Sommerfeld coefficient (γ) of an anisotropic superconductor at a given temperature (T ) for the magnetic field parallel (B // c) and perpendicular (B ⊥ c) to the c-axis; (b) Simulated magnetic field dependence of ∆γ as the difference in γ between B // c and B ⊥ c.

125022- 5 MFIG. 3 .
FIG.3.Proposed cooling applications using superconducting materials as refrigerants and the principle of rotating them in the presence of a constant magnetic field.(a) The first design can be used for the liquefaction of helium and (b) the second design can be used for cooling down a normal metal.

125022- 6 M
.-H.Phan and D. Mandrus AIP Advances 6, 125022 (2016) Barcza et al.reported ∆γ ∼ 4.7 mJ/mol K 2 and ∆S E ∼ 10 J/kg K for the CoMnSi alloy, demonstrating a sizable effect of the field-induced variation in γ (which is a measure of we have simulated in FIG.1a a general dependence of γ on B at a given T for B // c and B ⊥ c.It can be seen in this figure that γ is almost equal for B ≤ B cri