Increasing the performance of a superconducting spin valve using a Heusler alloy

  1. Andrey A. Kamashev1,2,
  2. Aidar A. Validov1ORCID Logo,
  3. Joachim Schumann2,
  4. Vladislav Kataev2ORCID Logo,
  5. Bernd Büchner2,3,
  6. Yakov V. Fominov4,5 and
  7. Ilgiz A. Garifullin1ORCID Logo

1Zavoisky Physical-Technical Institute, Russian Academy of Sciences, 420029 Kazan, Russia
2Leibniz Institute for Solid State and Materials Research IFW Dresden, D-01171 Dresden, Germany
3Institute for Solid State Physics, Technical University Dresden, D-01062 Dresden, Germany

4L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, 142432 Chernogolovka, Russia
5National Research University Higher School of Economics, 101000 Moscow, Russia

  1. Corresponding author email

Associate Editor: J. M. van Ruitenbeek
Beilstein J. Nanotechnol. 2018, 9, 1764–1769. doi:10.3762/bjnano.9.167
Received 31 Jan 2018, Accepted 25 May 2018, Published 12 Jun 2018


We have studied superconducting properties of spin-valve thin-layer heterostructures CoOx/F1/Cu/F2/Cu/Pb in which the ferromagnetic F1 layer was made of Permalloy while for the F2 layer we have taken a specially prepared film of the Heusler alloy Co2Cr1−xFexAl with a small degree of spin polarization of the conduction band. The heterostructures demonstrate a significant superconducting spin-valve effect, i.e., a complete switching on and off of the superconducting current flowing through the system by manipulating the mutual orientations of the magnetization of the F1 and F2 layers. The magnitude of the effect is doubled in comparison with the previously studied analogous multilayers with the F2 layer made of the strong ferromagnet Fe. Theoretical analysis shows that a drastic enhancement of the switching effect is due to a smaller exchange field in the heterostructure coming from the Heusler film as compared to Fe. This enables to approach an almost ideal theoretical magnitude of the switching in the Heusler-based multilayer with a F2 layer thickness of ca. 1 nm.

Keywords: ferromagnet; proximity effect; spin valve; superconductor


Historically, the first concept to manipulate the transition temperature Tc of a superconductor by sandwiching it between two ferromagnetic insulators was thought of by de Gennes [1]. Regarding the case of metallic ferromagnets, the physical principle of a superconducting spin valve (SSV) is based on the idea proposed by Oh et al. in 1997 [2] who calculated the pairing wave-function amplitude in a trilayer F1/F2/S (where F1 and F2 are ferromagnetic layers and S is a superconducting layer) and found out that the superconducting (SC) transition temperature Tc depends on the mutual orientation of the magnetizations M1 and M2 of the layers F1 and F2. Later, another construction based on three-layer thin films F1/S/F2 was proposed also theoretically [3,4]. According to the above theories, for the parallel (P) configuration of M1 and M2 the transition temperature [Graphic 1] should be always smaller than [Graphic 2] for the antiparallel (AP) orientation of the magnetic vectors. This is because in the former case the mean exchange field from the F-layers destructively acting on the Cooper pairs is larger. Thus, under favorable conditions the switching between AP and P configurations, which could be achieved by an appropriate application of a small external magnetic field, should yield a complete switching on and off of the superconducting current in such a construction.

A number of experimental studies have confirmed the predicted effect of the mutual orientation of magnetizations in the F1/S/F2 structure on Tc [5-9]. However, the major difficulty in a practical realization of an SSV, i.e., to obtain a difference between [Graphic 3] larger than the width δTc of the superconducting transition for a given configuration of M1 and M2, was not overcome in these works. One should note that the reported antiferromagnetically coupled [Fe/V]n superlattice [10] in which ΔTc could implicitly reach up to 200 mK cannot be considered as an SSV because this system can not be switched from the AP to P orientation of the magnetizations instantaneously.

In addition to that, the SSV effect becomes more complicated due to the following fact [11]: It is well known [12] that in the ferromagnetic layer the Cooper pair acquires a nonzero momentum due to the Zeeman splitting of electronic levels. Its wave function oscillates in space when moving away from the S/F interface. If the F layer is thin enough, the wave function is reflected from the surface opposite to the S/F interface. The interference of the incident and reflected functions arises. Depending on the thickness of the F layer, the interference at the S/F interface can be constructive or destructive. This should lead to an increase or decrease of the Tc of the S/F structure depending on the interference type.

From the experimental point of view, the results obtained for both theoretical designs of the SSV suggested that the scheme by Oh et al. [2] may be the most promising for the realization of the full SSV effect. Indeed, this approach turns out to be successful. Previously we have demonstrated a full switching between the normal and uperconducting states for the CoOx/Fe1/Cu/Fe2/In spin-valve structure [13]. Later on we replaced the superconducting In by Pb in order to improve superconducting parameters [14] and introduced an additional technical Cu interlayer (N2) in order to prevent degradation of the samples [15]. Thus, the final design of the SSV structures was set as AFM/F1/N1/F2/N2/S. In this construction the Cu interlayer (N1) decouples magnetizations of the Fe1 (F1) and Fe2 (F2) layers and the antiferromagnetic (AFM) CoOx layer biases the magnetization of the Fe1 layer by anisotropy fields. Despite substantial experimental efforts in optimizing the properties of the In- and Pb-based SSVs [16,17], in particular in reducing the width δTc, our theoretical analysis of the properties of such multilayers in the framework of the theory of [11] has shown that the experimentally achieved magnitude of ΔTc of the SSV effect of 20 mK and 40 mK for the two types of the S layer, respectively, was substantially smaller as expected on theoretical grounds. Recently the interest on SSVs increased considerably (see the review in [18] and the very recent publications [19-24]).

Here, we present experimental results that evidence a significant improvement of the magnitude of ΔTc in a Pb-based SSV by using the ferromagnetic Heusler alloy (HA) Co2Cr1−xFexAl as a material for the F2 layer. Prepared under well-defined conditions [25] the HA layer produces a substantially smaller exchange field acting on the superconducting Cooper pairs as compared to the Fe layer of the same thickness. This opens a possibility to grow heterostructures where the theoretically desired parameters for the maximum SSV effect could be practically realized yielding the doubling of the magnitude of the SSV effect up to the almost ideal theoretical value.


Technical particularities of the fabrication of the SSV heterostructures that have been studied in the present work have been reported in detail previously (see Supporting Information File 1). The new HA-based part of the multilayer F2/N2/S = HA/Cu/Pb has been investigated in detail with the focus on the S/F proximity effect very recently. It was shown [25] that the degree of the spin polarization of the conduction band of the HA film amounts to 30% for the films prepared at a particular substrate temperature of Tsub = 300 K during the growth of the HA layer and to 70% at Tsub = 600 K. In the AFM/F1/N1/F2/N2/S structure it would be advantageous to achieve a penetration depth of the Cooper pairs into F2 ferromagnetic layer as large as possible. This means that the spin polarization of the conduction band should be small. To fulfill this requirement we have prepared a set of samples CoOx/Py(5 nm)/Cu(4 nm)/Co2Cr1−xFexAl/Cu(1.5 nm)/Pb(50 nm) with the HA layer of different thickness grown at Tsub = 300 K. Representative superconducting transition curves are shown in Figure 1.


Figure 1: Superconducting transition curves for CoOx/Py(5)/Cu(4)/Co2Cr1−xFexAl/Cu(1.5)/Pb(50) multilayers with different thicknesses of the HA layer dHA for P (open circles) and AP (closed circles) mutual orientation of the magnetizations M1 and M2 of the Py and Co2Cr1−xFexAlx ferromagnetic layers, respectively: (a) dHA = 1 nm; (b) dHA = 3 nm; (c) dHA = 4 nm.

A clear shift of the curves upon switching the mutual orientation of the magnetizations M1 and M2 of the ferromagnetic layers between P and AP configurations characteristic of the SSV effect is clearly visible. The superconducting transition temperature was determined as a midpoint of the transition curve. The dependence of the magnitude ΔTc of the SSV effect on the thickness of the HA layer is presented in Figure 2.


Figure 2: Dependence of [Graphic 4] on the thickness of layer F2, dF2 in the SSV heterostructures AFM/F1/N1/F2/N2/S. Triangles are the data points for CoOx/Py(5)/Cu(4)/Co2Cr1−xFexAlx/Cu(1.5)Pb(50) from the present work. For comparison previous results for CoOx/Fe1/Cu/Fe2/Cu/Pb multilayers [17] are plotted with squares in the main panel and in the insert in which, additionally, the data for the CoOx/Fe1/Cu/Fe2/In SSV from [16] are plotted with circles for comparison. Solid and dashed lines present the results of theoretical modeling.

The dependence ΔTc(dHA) reveals an oscillating behavior due to the interference of the Cooper pair wave functions reflected from both surfaces of the ferromagnetic F2 layer (of the order of 4 nm) proximate to the superconducting layer. This yields for certain thicknesses of the F2 layer an inverse SSV effect ΔTc < 0 [26]. The most remarkable result of the present study is the magnitude of the direct SSV effect, which reaches for dHA = 1 nm (about two monolayers of HA) the maximum value of 80 mK (triangles in Figure 2). This surpasses the result for the analogous heterostructure with Fe as the F2 layer [17] by a factor of 2 (see the data comparison in Figure 2). As we will discuss below, the achieved SSV effect in the Pb-based heterostructure with the HA layer approaches the maximum value predicted by theory. The scattering of ΔTc is mainly due to some uncertainty in the determination of the thickness of the HA layer, which indirectly affects the accuracy of the determination of ΔTc.


To set up the basis for discussion we fist summarize the parameters of the theory [11] describing the SSV effect in the above systems. As described in [17], in order to estimate these parameters characterizing the properties of the S layer we use our experimental data on the resistivity and on the dependence of Tc on the S-layer thickness at a large unchanged thickness of the F layer in the S/F bilayer. The residual resistivity ρS = ρ(Tc) can be determined from the residual resistivity ratio RRR = R(T = 300 K)/R(Tc= [ρ(300 K) + ρ(Tc)]/ρ(Tc). Since the room-temperature resistivity of the Pb layer is dominated by the phonon contribution ρph(300 K) = 21 μΩ·cm [27] we obtain the ρS values presented below in Table 1. Then with the aid of the Pippard relations [28] the following equality can be obtained [17]:


Here γe denotes the electronic specific heat coefficient, vF is the Fermi velocity of the conduction electrons, and l is the mean-free path of the conduction electrons. Using for Pb γe = 1.6 × 103 erg/K2·cm3 [27], from Equation 1 one can find the mean-free path lS, the diffusion coefficient of conduction electrons DS and the superconducting coherence length


The same procedure can be applied for the F layers taking into account the definition of the superconducting coherence length in the F layers [29],


where DF is the diffusion coefficient for the conduction electrons in the F layer and TcS is the superconducting transition temperature for an isolated S layer.

The theory contains also the material-specific parameter γ and the interface transparency parameter γb. The first one is defined as


the second one can be calculated from the critical thickness of the S layer, [Graphic 5], which is defined as the thickness below which there is no superconductivity in the S/F bilayer: [Graphic 6].

In the limiting case [Graphic 7], the thickness [Graphic 8] can be calculated explicitly as [29]


Here γE ≈ 1.78 is the Euler constant. Our data yield [Graphic 9]b = 1.95) for the Fe/In system, [Graphic 10]b = 2.7) for the Fe/Cu/Pb system, and [Graphic 11]b = 0.37) for the HA/Cu/Pb system. All obtained parameters are presented in Table 1. The larger value of γb for the Fe/Cu/Pb system compared to the Ha/Cu/Pb system makes, of course, sense. Indeed, the difference between the HA system and the Fe system is seen in a difference of γb, which helps to rationalize the use of a weaker F-layer.

Table 1: Parameters used for fitting of the theory to the experimental results [17].

parameter 1 2 3
Fe2/In Fe2/Cu/Pb HA/Cu/Pb
ρS, μΩ·cm 0.2 1.47 1.47
lS, nm 300 17 17
DS, cm2/s 1100 100 100
ξS, nm 170 41 41
ρF, μΩ·cm 10 10 130
lF, nm 10 10 6.41
DF, cm2/s 3.3 3.3 21.4
ξF, nm 7.5 7.5 14
ξh, nm 0.5 0.3 1.25
γ 0.45 0.78 0.03
γb 1.95 2.7 0.37

Figure 2 summarizes the experimental values of ΔTc(dF2) for SSV heterostructures with HA as the F2 layer obtained in the present work and our previous results on Fe-based SSVs [16,17]. Solid lines in Figure 2 are theoretical results using the parameters listed in Table 1. The general feature of the SSVs with F2 = Fe is that the measured points at small thicknesses dF2 lie much lower than the theoretically expected positive maximum of ΔTc (direct SSV effect). One should note that the difference in the theoretical maximum values of ΔTc for In- and Pb-based systems (inset in Figure 2) is caused by the different values of the superconducting transition temperature of the single S layer (Tc = 3.4 K for In and Tc = 7.18 K for Pb). Obviously in both cases, to reach the expected maximum it would be necessary to further decrease the thickness of the F2 layer. It should be emphasized that the thickness dF2 is one of the crucial parameters for the functionality of the spin valve. It determines the number of the Cooper pairs that experience the influence of the exchange fields of both F layers in the heterostructure. In general, to get the maximum magnitude of the spin-valve effect ΔTc, the thickness dF2 of the F2 layer proximate to the S layer should be of the order or smaller than the penetration depth of the Cooper pairs into the F2 layer, [Graphic 12]. Here h is the exchange splitting of the conduction band of a ferromagnet. The thinner the F2 layer is, the more Cooper pairs can reach the contact region between the two ferromagnetic layers where at certain thicknesses of the F1- and F2-layers the compensation effect of the exchange fields can take place in the AP case. For the previously studied Fe-based systems, h was of the order of 1 eV and ξh amounted to 0.6–0.8 nm [16,17]. According to theory [11], this means that the maximum of ΔTc should occur in the interval between 0.3 and 0.4 nm (inset of Figure 2). With the available experimental setup, it is practically impossible to grow a continuous iron film of such small thickness.

It is well known [30,31] that in dilute alloys, e.g., in PdNi alloys with 10% of Ni, ξh is of the order of 5 nm, which is an order of magnitude larger compared to pure ferromagnetic elements such as Fe, Ni, or Co. As our present experimental results demonstrate, the use of a Heusler alloy for the growth of the F2 layer is very beneficial. It greatly relaxes the stringent condition on the minimum thickness of the F2 layer. Indeed, according to the previous analysis of the Pb/Cu/Co2Cr1−xFexAl trilayers, the HA film grown at the substrate temperature of Tsub = 300 K can be classified as a weak ferromagnet with a relatively small exchange field hHA ≈ 0.2 eV [25]. As can be seen in Figure 2, this reduction of h shifts the peak of the theoretical values of ΔTc(dF2) for F2 = HA to larger thicknesses of the order of 1 nm, which can be easily reached experimentally. Under these conditions the measured maximum magnitude of ΔTc is two times larger compared to the best previous result on the Fe-based SSVs (Figure 2). In fact, it almost reaches the theoretically predicted value suggesting that further optimization of the properties of the F2 layer is unlikely to significantly increase the SSV effect. In this respect it would be very interesting to explore theoretically and experimentally the option of optimization of the F1 layer in the SSV AFM/F1/N1/F2/N2/S heterostructure.

Recently, Singh et al. [32] reported a huge SSV effect for a S/F1/N/F2 structure made of amorphous MoGe, Ni, Cu and CrO2 as S, F1, N and F2, respectively. This structure exhibited a ΔTc of ca. 1 K when changing the relative orientation of magnetizations of two F layers. The reason for such a surprisingly strong SSV effect remains unclear [33]. Gu et al. [34,35] reported ΔTc ≈ 400 mK for three-layered Ho/Nb/Ho films.

Finally, a discrepancy between the theoretical curves and experimental data at larger thicknesses dF2 in the regime of the inverse (negative) SSV effect found for all the above discussed systems (Figure 2) needs to be commented. In this respect, we note that the assumptions of theory [11] do not fully comply with the properties of our samples. While the assumption of F layers being weak ferromagnets (exchange energy much smaller than the Fermi energy) is satisfied for the Heusler alloy, iron is closer to the limit of strong ferromagnets (exchange energy starts to be comparable with the Fermi energy). Accurate theoretical description of ferromagnets with large exchange splitting requires taking into account different densities of states in different spin subbands and modified boundary conditions at SF interfaces [36,37]. At the same time, the major inconsistency between theory and experiment in our case is probably related to the assumption of the dirty limit (mean free path much smaller than the coherence length). In our samples, these assumptions are close to the border of applicability or even not satisfied (depending on the specific material). The ferromagnets turn out to be strong enough so that the condition [Graphic 13] is not satisfied at all. Therefore, we cannot expect theory [11] to describe quantitative details of our results. Still, we observe that the theory captures main qualitative features of the experiment.


In summary, we have experimentally demonstrated that using for the F2 layer in a CoOx/F1/Cu/F2/Cu/Pb heterostructure a specially prepared thin film of the Heusler alloy Co2Cr1−xFexAl with a small degree of the spin polarization of the conduction band significantly increases the magnitude of the superconducting spin valve effect ΔTc as compared to similar systems with the F2 layer made of the strong ferromagnet Fe. It follows from our theoretical analysis that the experimentally achieved value is close to the maximum predicted by theory. The use of the Heusler alloy did not increase this maximum value beyond the theoretical result but enables to reach experimentally the maximum possible value of ΔTc at a larger, technically realizable thickness of the F2 layer, in a full agreement with theory. It seems unlikely that further optimization of the material of the F2 layer would yield substantially larger values of ΔTc. An interesting alternative would be to optimize the parameters of the F1 layer, which is tempting to explore in the future.

Supporting Information

Supporting Information File 1: Fabrication of the SSV heterostructures.
Format: PDF Size: 104.9 KB Download


A. K. gratefully acknowledges the Leibniz - DAAD Fellowship. Ya. F. was partially supported by the RF Ministry of Education and Science (grant No. 14Y.26.31.0007), by the RFBR (grant No. 17-52-50080), and by the Basic research program of HSE.


  1. De Gennes, P. G. Phys. Lett. 1966, 23, 10. doi:10.1016/0031-9163(66)90229-0
    Return to citation in text: [1]
  2. Oh, S.; Youm, D.; Beasley, M. R. Appl. Phys. Lett. 1997, 71, 2376–2378. doi:10.1063/1.120032
    Return to citation in text: [1] [2]
  3. Tagirov, L. R. Phys. Rev. Lett. 1999, 83, 2058–2061. doi:10.1103/PhysRevLett.83.2058
    Return to citation in text: [1]
  4. Buzdin, A. I.; Vedyayev, A. V.; Ryzhanova, N. V. Europhys. Lett. 1999, 48, 686. doi:10.1209/epl/i1999-00539-0
    Return to citation in text: [1]
  5. Gu, J. Y.; You, C.-Y.; Jiang, J. S.; Pearson, J.; Bazaliy, Ya. B.; Bader, S. D. Phys. Rev. Lett. 2002, 89, 267001. doi:10.1103/PhysRevLett.89.267001
    Return to citation in text: [1]
  6. Potenza, A.; Marrows, C. H. Phys. Rev. B 2005, 71, 180503. doi:10.1103/PhysRevB.71.180503
    Return to citation in text: [1]
  7. Moraru, I. C.; Pratt, W. P., Jr.; Birge, N. O. Phys. Rev. Lett. 2006, 96, 037004. doi:10.1103/PhysRevLett.96.037004
    Return to citation in text: [1]
  8. Moraru, I. C.; Pratt, W. P., Jr.; Birge, N. O. Phys. Rev. B 2006, 74, 220507. doi:10.1103/PhysRevB.74.220507
    Return to citation in text: [1]
  9. Miao, G.-X.; Ramos, A. V.; Moodera, J. S. Phys. Rev. Lett. 2008, 101, 137001. doi:10.1103/PhysRevLett.101.137001
    Return to citation in text: [1]
  10. Westerholt, K.; Sprungmann, D.; Zabel, H.; Brucas, R.; Hjörvarsson, B.; Tikhonov, D. A.; Garifullin, I. A. Phys. Rev. Lett. 2005, 95, 097003. doi:10.1103/PhysRevLett.95.097003
    Return to citation in text: [1]
  11. Fominov, Ya. V.; Golubov, A. A.; Karminskaya, T. Yu.; Kupriyanov, M. Yu.; Deminov, R. G.; Tagirov, L. R. JETP Lett. 2010, 91, 308–313. doi:10.1134/S002136401006010X
    Return to citation in text: [1] [2] [3] [4] [5] [6]
  12. Bergeret, F. S.; Volkov, A. F.; Efetov, K. B. Rev. Mod. Phys. 2005, 77, 1321–1373. doi:10.1103/RevModPhys.77.1321
    Return to citation in text: [1]
  13. Leksin, P. V.; Garif’yanov, N. N.; Garifullin, I. A.; Schumann, J.; Vinzelberg, H.; Kataev, V.; Klingeler, R.; Schmidt, O. G.; Büchner, B. Appl. Phys. Lett. 2010, 97, 102505. doi:10.1063/1.3486687
    Return to citation in text: [1]
  14. Leksin, P. V.; Garif’yanov, N. N.; Garifullin, I. A.; Fominov, Y. V.; Schumann, J.; Krupskaya, Y.; Kataev, V.; Schmidt, O. G.; Büchner, B. Phys. Rev. Lett. 2012, 109, 057005. doi:10.1103/PhysRevLett.109.057005
    Return to citation in text: [1]
  15. Leksin, P. V.; Kamashev, A. A.; Garif’yanov, N. N.; Garifullin, I. A.; Fominov, Ya. V.; Schumann, J.; Hess, C.; Kataev, V.; Büchner, B. JETP Lett. 2013, 97, 478–482. doi:10.1134/S0021364013080109
    Return to citation in text: [1]
  16. Leksin, P. V.; Garif’yanov, N. N.; Garifullin, I. A.; Schumann, J.; Kataev, V.; Schmidt, O. G.; Büchner, B. Phys. Rev. B 2012, 85, 024502. doi:10.1103/PhysRevB.85.024502
    Return to citation in text: [1] [2] [3] [4]
  17. Leksin, P. V.; Garif’yanov, N. N.; Kamashev, A. A.; Fominov, Ya. V.; Schumann, J.; Hess, C.; Kataev, V.; Büchner, B.; Garifullin, I. A. Phys. Rev. B 2015, 91, 214508. doi:10.1103/PhysRevB.91.214508
    Return to citation in text: [1] [2] [3] [4] [5] [6] [7] [8]
  18. Linder, J.; Robinson, J. W. A. Nat. Phys. 2015, 11, 307–315. doi:10.1038/nphys3242
    Return to citation in text: [1]
  19. Flokstra, M. G.; Cunningham, T. C.; Kim, J.; Satchell, N.; Burnell, G.; Curran, P. J.; Bending, S. J.; Kinane, C. J.; Cooper, J. F. K.; Langridge, S.; Isidori, A.; Pugach, N.; Eschrig, M.; Lee, S. L. Phys. Rev. B 2015, 91, 060501. doi:10.1103/PhysRevB.91.060501
    Return to citation in text: [1]
  20. Alidoust, M.; Halterman, K.; Valls, O. T. Phys. Rev. B 2015, 92, 014508. doi:10.1103/PhysRevB.92.014508
    Return to citation in text: [1]
  21. Mironov, S.; Buzdin, A. Phys. Rev. B 2015, 92, 184506. doi:10.1103/PhysRevB.92.184506
    Return to citation in text: [1]
  22. Flokstra, M. G.; Satchell, N.; Kim, J.; Burnell, G.; Curran, P. J.; Bending, S. J.; Cooper, J. F. K.; Kinane, C. J.; Langridge, S.; Isidori, A.; Pugach, N.; Eschrig, M.; Luetkens, H.; Suter, A.; Prokscha, T.; Lee, S. L. Nat. Phys. 2016, 12, 57–61. doi:10.1038/nphys3486
    Return to citation in text: [1]
  23. Halterman, K.; Alidoust, M. Phys. Rev. B 2016, 94, 064503. doi:10.1103/PhysRevB.94.064503
    Return to citation in text: [1]
  24. Srivastava, A.; Olde Olthof, L. A. B.; Di Bernardo, A.; Komori, S.; Amado, M.; Palomares-Garcia, C.; Alidoust, M.; Halterman, K.; Blamire, M. G.; Robinson, J. W. A. Phys. Rev. Appl. 2017, 8, 044008. doi:10.1103/PhysRevApplied.8.044008
    Return to citation in text: [1]
  25. Kamashev, A. A.; Leksin, P. V.; Schumann, J.; Kataev, V.; Thomas, J.; Gemming, T.; Büchner, B.; Garifullin, I. A. Phys. Rev. B 2017, 96, 024512. doi:10.1103/PhysRevB.96.024512
    Return to citation in text: [1] [2] [3]
  26. Leksin, P. V.; Garif’yanov, N. N.; Garifullin, I. A.; Schumann, J.; Kataev, V.; Schmidt, O. G.; Büchner, B. Phys. Rev. Lett. 2011, 106, 067005. doi:10.1103/PhysRevLett.106.067005
    Return to citation in text: [1]
  27. Kittel, C. Introduction to Solid State Physics; John Wiley & Sons: New York, NY, U.S.A., 1976.
    Return to citation in text: [1] [2]
  28. Pippard, A. B. Rep. Prog. Phys. 1960, 23, 176–266. doi:10.1088/0034-4885/23/1/304
    Return to citation in text: [1]
  29. Fominov, Ya. V.; Chtchelkatchev, N. M.; Golubov, A. A. Phys. Rev. B 2002, 66, 014507. doi:10.1103/PhysRevB.66.014507
    Return to citation in text: [1] [2]
  30. Kontos, T.; Aprili, M.; Lesueur, J.; Grison, X. Phys. Rev. Lett. 2001, 86, 304–307. doi:10.1103/PhysRevLett.86.304
    Return to citation in text: [1]
  31. Kontos, T.; Aprili, M.; Lesueur, J.; Genêt, F.; Stephanidis, B.; Boursier, R. Phys. Rev. Lett. 2002, 89, 137007. doi:10.1103/PhysRevLett.89.137007
    Return to citation in text: [1]
  32. Singh, A.; Voltan, S.; Lahabi, K.; Aarts, J. Phys. Rev. X 2015, 5, 021019. doi:10.1103/PhysRevX.5.021019
    Return to citation in text: [1]
  33. Ouassou, J. A.; Pal, A.; Blamire, M.; Eschrig, M.; Linder, J. Sci. Rep. 2017, 7, 1932. doi:10.1038/s41598-017-01330-1
    Return to citation in text: [1]
  34. Gu, Y.; Robinson, J. W. A.; Bianchetti, M.; Stelmashenko, N. A.; Astill, D.; Grosche, F. M.; MacManus-Driscoll, J. L.; Blamire, M. G. APL Mater. 2014, 2, 046103. doi:10.1063/1.4870141
    Return to citation in text: [1]
  35. Gu, Y.; Halász, G. B.; Robinson, J. W. A.; Blamire, M. G. Phys. Rev. Lett. 2015, 115, 067201. doi:10.1103/PhysRevLett.115.067201
    Return to citation in text: [1]
  36. Cottet, A.; Huertas-Hernando, D.; Belzig, W.; Nazarov, Y. V. Phys. Rev. B 2009, 80, 184511. doi:10.1103/PhysRevB.80.184511
    Return to citation in text: [1]
  37. Eschrig, M.; Cottet, A.; Belzig, W.; Linder, J. New J. Phys. 2015, 17, 083037. doi:10.1088/1367-2630/17/8/083037
    Return to citation in text: [1]

Interesting articles

Daniel Lenk , Vladimir I. Zdravkov , Jan-Michael Kehrle , Günter Obermeier , Aladin Ullrich , Roman Morari , Hans-Albrecht Krug von Nidda , Claus Müller , Mikhail Yu. Kupriyanov , Anatolie S. Sidorenko , Siegfried Horn , Rafael G. Deminov , Lenar R. Tagirov and Reinhard Tidecks

Stefan Kolenda , Peter Machon , Detlef Beckmann and Wolfgang Belzig

Chamanei Perera , Kristy Vernon , Elliot Cheng , Juna Sathian , Esa Jaatinen and Timothy Davis

© 2018 Kamashev et al.; licensee Beilstein-Institut.
This is an Open Access article under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The license is subject to the Beilstein Journal of Nanotechnology terms and conditions: (

Back to Article List