It has been widely known in condensed matter and materials physics that the application of magnetic field to a superconductor will generally destroy the superconductivity as a usual scenario. There are two reasons responsible for this.
One is the Zeeman effect [1-2], where the alignment of the electron spins by the applied magnetic field can break apart the electron pairs for a spin-singlet (but not a spin-triplet) state. In this case, the electron spin pairs have opposite spins (such as the s-wave spins typical in type I superconductors). The applied magnetic field attempts to align the spins of both electrons along the field, thus breaking them apart if strong enough. In terms of energy, one electron gains energy while the other (as a pair) loses energy. If the energy difference is larger than the amount of energy holding the electrons together, then they fly apart and thus the superconductivity disappears. But this does not apply to the spin triplet superconductivity (p-wave superconductors) where the electron pairs already have their spins aligned (along the field) in a p-wave state.
The other is the orbital effect , which is a manifestation of the Lorentz force from the applied magnetic field since the electrons (as a pair) have opposite linear momenta, one electron rotating around the other in their orbitals. The Lorentz force on them acts in opposite directions and is perpendicular to the applied magnetic field, thus always pulling the pair apart. This does not matter with their spin pairing symmetries (s-wave, p-wave or d-wave). In type-II superconductors, the Meissner screening currents associated with the vortex penetration in the applied magnetic field can also increase the electron kinetic energy (and momentum). Once this energy becomes greater than the energy that unites the two electrons, the electron pairs break apart and thus superconductivity is suppressed. Therefore, the orbital effect could be even more important in type-II superconductors.
However, in certain complex compounds, especially in some low-dimensional materials, superconductivity can be enhanced [4 - 5] by the application of magnetic field. The enhancement of superconductivity by magnetic field is a counter-intuitive unusual phenomenon.
In order to understand this interesting phenomenon, different theoretical mechanisms have been proposed, while there are still debates and experimental evidence is needed. The first theory is the Jaccarino-Peter compensation effect , the second theory is the suppression effect of the spin fluctuations [7-9], and the third theory is the anti-proximity effect (in contrary to the proximity effect ) found in the nanowires recently . These theories will be briefly described in Section 3.
Experimentally, there are several effective techniques that can be used to the study of the superconductivity of the materials with magnetic field applications. They include electrical resistivity measurements, Nernst effect measurements, SQUID magnetic susceptibility measurements, and nuclear magnetic resonance (NMR) measurements, etc.
Among these experimental techniques, NMR is one of the most powerful ones and it is a versatile local probe capable of directly measuring the electron spin dynamics and distribution of internal magnetic field including their changes on the atomic scale. It has been widely used as a tool to investigate the charge and spin static and dynamic properties (including those of the nano particles). It is able to address a remarkably wide range of questions as well as testing the validity of existing and/or any proposed theories in condensed matter and materials physics.
The authors have extensive experience using the NMR and various other techniques for the study of the novel condensed matter materials. This chapter focuses on the NMR studies of the quasi-two dimensional field-induced superconductor –(BETS)2FeCl4. This is a chance to put some of the work together, with which it will help the science community for the understanding of the material as well as for the mechanism of superconductivity. It will also help materials scientists in search of new superconductors.
2. Field-induced superconductors
The discovery of field-induced superconductors is about a decade earlier than the discovery of the high-Tc superconductors (which have been found since 1986), while not many field-induced superconductors have been found. Both types of materials, field-induced superconductors and high-Tc superconductors, are highly valuable in science and engineering due to their important physics and wonderful potentials in technical applications.
Here are typical field-induced superconductors found so far, with chemical compositions as shown in the following
3. Theory for field-induced superconductors
In this section, we will briefly describe the theoretical aspects for the field-induced superconductors regarding their mechanisms of the field-induced superconductivity. We will mainly discuss the theory of Jaccarino-Peter effect, the theory of spin fluctuation effect, and the theory of anti-proximity effect.
3.1. Theory of Jaccarino-Peter effect
This theory was proposed by Jaccarino, V. and Peter, M. in 1962 . It means that if there is an existence of localized magnetic moments at a state and conduction electrons as well at the same state in a material, then it could lead to a negative exchange interaction J between the conduction electrons and the magnetic moments when an external magnetic field (H) is applied. This negative exchange interaction J is formed due to the easy alignment of the localized magnetic moments (along the external magnetic field direction) while the external magnetic field H is applied. Thus the spins of the conduction electrons will experience an internal magnetic field (HJ), HJ = J<S>/gB, created by the magnetic moments [proportional to the average spins (<S>) of the moments]. Here the internal magnetic field HJ is called the exchange field and the direction of the exchange field HJ is opposite to the externally applied magnetic field H. This picture is sketched as that shown in Fig. 1.
In some cases, this HJ could be very strong. Therefore, if the exchange field HJ is strong enough to cancel the externally applied magnetic field H, i.e., HJ = H, then the resultant field in total that the conduction electron spins experience becomes zero (also a complete suppression of the Zeeman effect), and thus the superconductivity is induced in this case.
Certainly, superconductivity could also be possible in this case (as a stable phase), even if without the external field H (i.e. H = 0). This is because the magnetic moments point to random directions (without H) and cancel each other, i.e., HJ = 0, and thus the conduction electron spins also feel zero in total field.
3.2. Theory of spin fluctuation effect
This theory was mainly reported by Maekawa, S. and Tachiki, M.  in 1970s, with the discovery of field-induced superconductors EuxSn1-xMo6S8-ySey. These types of materials have rare-earth 4f-ions and paired conduction electrons from the 4d-Mo-ions. The rare-earth 4f-ions have large fluctuating magnetic moments, while the conduction electrons from the 4d-Mo-ions have strong electron–electron interactions and they form Cooper pairs.
Without externally applied magnetic field (H = 0), the fluctuating magnetic field from the rare-earth 4f-ion moments at the 4d-Mo conduction electrons are so strong that it weakens the BCS coupling of the Mo-electrons. Thus there is no superconductivity without externally applied magnetic field.
However, when external magnetic field is applied (H ≠ 0) it suppresses the spin fluctuation, causing an increase of the BCS coupling among the conduction electrons. Thus, superconductivity appears in the presence of an applied magnetic field. This scenario can be seen from a typical H-T phase diagram shown in Fig. 2.
3.3. Theory of anti-proximity effect
3.3.1. Superconductivity in nanowires enhanced by applied magnetic field
Unlike the bulk superconductors, a nanoscale system can have externally applied magnetic field H to penetrate it with essentially no attenuation at all throughout the whole sample.
But it has been observed that the application of a small magnetic field H can decrease the resistance in even simple narrow superconducting wires (i.e., negative magnetoresistance) [12, 13], while larger applied magnetic field H can increase the critical current (IC) significantly . These indicate an enhancement of superconductivity in nanowires by the application of magnetic field. But to understand the enhancement of superconductivity by magnetic field in nanoscale systems is very challenging currently in the science community.
3.3.2. Proximity effect
On the other hand, when a superconducting nanowire is connected to two normal metal electrodes, generally a fraction of the wire is expected to be resistive, especially when the wire diameter is smaller than the superconducting coherence length. This is called the proximity effect .
Similarly, when a superconducting nanowire is connected to two bulk superconducting (BS) electrodes, the combined sandwiched system is expected to be superconducting (below the TC of the superconducting nanowire and the BS electrodes), and the superconductivity of the nanowire is then expected to be more supportive and more robust through its coupling with the superconducting reservoirs. This is also actually what is theoretically expected .
3.3.3. Anti-proximity effect
Contrary to the proximity effect, it has been found in 2005 [11, 16] that, in a system consisting of 2 long, 40 nm diameter Zinc nanowires sandwiched between two BS electrodes (Sn or In), superconductivity of Zinc nanowires is completely suppressed (or partially suppressed) by the BS electrodes when the BS electrodes are in the superconducting state under zero applied magnetic field. However, when the BS electrodes are driven normal by an applied magnetic field (H), the Zinc nanowires re-enter their superconducting state at ~ 0.8 K, unexpectedly. This is called ‘‘anti-proximity effect’’.
This is also a counterintuitive unusual phenomenon, never reported before 2005.
The schematic of the electrical transport measurement system exhibiting the anti-proximity effect with the Zinc nanowires sandwiched between two BS electrodes is shown in Fig. 3.
3.3.4. Theory of anti-proximity effect
There are several theoretical models that could be used for the theoretical explanation of the magnetic field-induced or -enhanced superconductivity in nanowires, while some of which were proposed long before the anti-proximity effect was reported. Thus they are not generally accepted.
a. Phase fluctuation model
This model proposes that there is an interplay between the superconducting phase fluctuations and dissipative quasiparticle channels .
When the bulk electrodes are superconducting, there is a supercurrent flowing between the nanowire and BS electrodes, and the contact resistances (R) vanish (R = 0). Thus the circuit frequency becomes low, and the quantum wire is shunted by the capacitor (C) if the energy (frequency f) is less than the bulk superconducting gap energy of the electrodes. In this case, the quantum fluctuations of the superconducting phase drive the superfluid density () of the zinc nanowire to zero (even when temperature T = 0). As soon as the superfluid density vanishes, the Cooper pairs dissociate and the nanowire becomes normal (with a normal resistor with resistance RN). Thus this explains the resistive behavior at zero applied magnetic field (H= 0).
When the bulk electrodes are driven normal by the applied magnetic field (H 30 mT), the contact resistances R ≠ 0. Similarly, if the electrodes are normal but the nanowire is superconducting (or vice versa), there will be a resistance due to charge conversion processes . This results in a high circuit frequency f = 1/(2RC) ~ 1 GHz, with a behavior like a pure resistor (i.e., impedance XC =1/2fC 0). If this shunting resistance is less the quantum of resistance (h/4e2 ~ 6.4 k), then it will damp the superconducting phase fluctuations, and thus stabilizing the superconductivity. On the other hand, in a dissipation environment [Fig. 4 (c)], the superfluid density () of the zinc nanowire cannot screen the interaction between bulk vortices completely. As a result, the superconducting phase becomes stable for sufficiently small shunt resistance .
In order words, when the magnetic field is applied (H ≠ 0) to the bulk electrodes, the dissipations between the two ends of the electrodes will be enhanced and meanwhile the superconducting phase fluctuations are damped. This leads to the stabilization of the superconductivity of the nanowires between the bulk electrodes.
b. Interference model
The interference model proposes that there is an interference between junctions of two superconducting grains, with random Josephson couplings J and J' associated with disorder, as sketched shown in Fig. 5. It produces a configuration-averaged critical current <IC> as 
This is a periodic function of [cos2(2/0)] with a period of half flux quantum (0/2 = hc/4e), where is the magnetic flux through each hole due to the existence of disorder in the sample (note, the sample has an array of holes through each of which there has a flux ).
Thus when is small, <IC> increases as increases, and this corresponds to a negative magnetoresistance, i.e., when applied magnetic field H increases, the electrical resistivity of the nanowires drops down. Thereby the superconductivity is enhanced.
c. Charge imbalance length model
This model proposes that there is a charge-imbalance length (or relaxation time) associated with the normal metal - superconductor boundaries of phase-slip centers . Applying magnetic field reduces the charge-imbalance length (or relaxation time), resulting in a negative magnetoresistance at high currents and near Tc. Thus the superconductivity in the nanowires is enhanced.
d. Impurity model
The impurity model deals with the superconductivity for nanoscale systems that have impurity magnetic moments with localized spins as magnetic superconductors , in which there is a strong Zeeman effect. According to this model, superconductivity is enhanced with the quenching of pair-breaking magnetic spin fluctuations by the applied magnetic field.
These are major theoretical models for the explanation of the anti-proximity effect in nanoscale systems. Their validity needs more experimental evidence.
4. Field-induced superconductor –(BETS)2FeCl4
The field-induced superconductor –(BETS)2FeCl4 is a quasi-two dimensional (2D) triclinic salt (space group P ) incorporating large magnetic 3d-Fe3+ ions (spin Sd = 5/2) with the BETS-molecules inside which have highly correlated conduction electrons (-electrons, spin S = 1/2) from the Se-ions, where BETS is bis(ethylenedithio)tetraselenafulvalene (C10S4Se4H8). It was first synthesized in 1993 by Kobayashi et al. [4, 8, 9, 19].
–(BETS)2FeCl4 is one of the most attractive materials in the last two decades for the observation of interplay of superconductivity and magnetism and for the synthesis of magnetic conductors and superconductors.
We expect it to show strong competition between the antiferromagnetic (AF) order of the Fe3+ magnetic moments and the superconductivity of the material, where the properties of the conduction electrons are significantly tunable by the external magnetic field, together with the internal magnetic field generated by the local magnetic moments from the Fe3+ ions as well. Thus it has been of considerable interest in condensed matter and materials physics.
This interplay originates from the role of the magnetic 3d-Fe3+ ions moments including the effect of their strong interaction with the conduction -electrons. Because of this interplay, –(BETS)2FeCl4 has an unusual phase diagram [Fig. 5 (c)], including an antiferromagnetic insulating (AFI) phase, a paramagnetic metallic (PM) phase, and a field-induced superconducting (FISC) phase [4, 8].
The crystal structure of –(BETS)2FeCl4 in a unit cell is shown in Fig. 6 (a) . In each unit cell, there are four BETS molecules and two Fe3+ ions. The BETS molecules are stacked along the a and c axes to form a quasi-stacking fourfold structure.
Noticeably, the conducting layers comprised of BETS are sandwiched along the b axis by the insulating layers of FeCl4− anions. The least conducting axis is b, the conducting plane is ac, and the easy axis of the antiferromagnetic spin structure is ~30° away from the c axis (parallel to the needle axis of the crystal) [21, 22].
At the room temperature (298 K), the lattice constants are: a = 16.164(3), b = 18.538(3), c = 6.592(4) Angstrom (Ao), α = 98.40(1)o, = 96.69(1)o, and = 112.52(1)o. The shortest distance between Fe3+ ions is 10.1 Ao within a unit cell, which is along the a-direction, and the nearest distance of Fe3+ ions between neighboring unit cells is 8.8 Ao .
5. NMR studies of –(BETS)2FeCl4
In order to study the mechanism of the superconductivity in –(BETS)2FeCl4 and to test the validity of the Jaccarino-Peter effect, as well as to understand the multi-phase properties of the material as show in the unusual phase diagram [Fig. 6 (c)], we successfully conducted a series of nuclear magnetic resonance (NMR) experiments.
5.1. 77Se-NMR measurements in –(BETS)2FeCl4
5.1.1. 77Se-NMR spectrum
H0 = 9 T || a’ in –(BETS)2FeCl4 .
The 77Se-NMR spectra of –(BETS)2FeCl4 at various temperatures are shown in Fig. 7. The spectrum has a dominant single-peak feature which is reasonable as a spin I = 1/2 nucleus for the 77Se, while it broadens inhomogeneously and significantly upon cooling (the linewidth increases from 90 kHz to 200 kHz as temperature is lowered from 30 K to 5 K). What the77Se-NMR spectrum measures is the local field distribution in total at the Se sites. Apparently, these spectrum data indicate that all the Se sites in the unit cell are essentially identical.
The sample used for the 77Se-NMR measurements was grown using a standard method  without 77Se enrichment (the natural abundance of 77Se is 7.5%). The sample dimension is a* b* c = 0.09 mm 0.04 mm 0.80 mm corresponding to a mass of ~ 7 g with ~ 2.0 1015 77Se nuclei.
Due to the small number of spins, a small microcoil with a filling factor ~ 0.4 was used. For most acquisitions, 104–105 averages were used on a time scale of ~ 5 min for 104 averages. The sample and coil were rotated on a goniometer (rotation angle ) whose rotation axis is along the lattice c-axis (the needle direction) and it is also perpendicular to the applied field H0 = 9 T. According to the crystal structure of –(BETS)2FeCl4, our calculation indicates that the direction of the Se-electron pz orbital is 76.4° from the c-axis. Thus the minimum angle between pz and H0 during the rotation of the goniometer is min=13.6°.
5.1.2. Temperature dependence of the 77Se-NMR resonance frequency
The temperature (T) dependence of the 77Se-NMR resonance frequency () from the above experiment is shown in Fig. 8 (a). In order to understand the origin of this resonance frequency, we also plotted it as a function of the 3d-Fe3+ ion magnetization (Md), which is a Brillouin function of temperature T and the total magnetic field (HT) at the Fe3+ ions. This is shown in Fig. 8 (b), where the solid lines show the fit to the Md.
The resonance frequency is counted from the center of the 77Se-NMR spectrum peak (maximum). What it measures is the average of the local field in magnitude in total, including the direct hyperfine field from the conduction electrons and the indirect hyperfine field that coupled to the Fe3+ ions at the Lamar frequency of the 77Se nuclei (see details in Section 5.1.4).
Figure 8 indicates that in the PM state above ~ 7 K at the applied field H0 = 9 T, a good fit to frequency (uncertainty ± 3 kHz) is obtained using
where the fit parameters a = 73.221 MHz and b = 3.0158 [(mol.Fe/emu).MHz].
This result is a strong indication that the temperature T dependence of the77Se-NMR resonance frequency is dominated by the hyperfine field from the Fe3+ ion magnetization Md.
It is important to notice that the sign of the contribution from Md is negative in Eq. (2). Thus, this also indicates that the hyperfine field from the Fe3+ ion magnetization is negative, i.e., opposite to applied magnetic field H0, as needed for the Jaccarino-Peter compensation mechanism.
Now, to verify to validity of the Jaccarino-Peter mechanism, we need to find the field from the 3d Fe3+ ions at the Se -electrons is (i.e., the -d exchange field Hd) which is the central goal of our 77Se-NMR measurements.
According to the H-T phase diagram of –(BETS)2FeCl4 [Fig. 6 (c)], the magnitude of Hd = 33 T (tesla) at temperature T = 5 K.
5.1.3. Angular dependence of the 77Se-NMR resonance frequency
The angular dependence of the 77Se-NMR resonance frequency from our experiments is shown in Fig. 9, which is plotted as a function of angle at several temperatures. The angle basically describes the alignment direction of the applied magnetic field H0 relative to the sample lattice.
To understand the complexity of these sets of data, we need clarify the angle first as there are many other directions involved here as well. First, the crystal lattice has its a, b and c axes which have their own fixed directions. Second, the z component of the BETS molecule -electron orbital moment, pz, also has a fixed direction, which is perpendicular to the BETS molecule Se-C-S loop plane. Third, there is a direction of sample rotation which is along c (the needle direction) in the applied magnetic field H0.
To distinguish each of these directions, we used the Cartesian xyz reference system and choose the reference z axis to be parallel to the lattice c axis, then the direction of pz is determined to have angle 76.4o from the c axis through our calculation according to the X-ray data of –(BETS)2FeCl4. All these are clearly drawn as that shown in the inset of Fig. 9.
Thus during a sample rotation the direction of H0 is always in the xy-plane, where x-axis is chosen to be in the c-pz plane, and the angle is counted from the x-axis.
Therefore, an angle = 0° corresponds to H0 to be in the c-pz plane with the smallest angle between H0 and pz as min= 13.6° as mentioned in Section 5.1.1.
Based on these data shown in Figs. 8 - 9, we can determine the magnitude of the -d exchange field Hd precisely.
5.1.4. Determination of the -d exchange field (between the Se- and Fe3+-d electrons)
From the theory of NMR , we can express the contributions to the Hamiltonian (HI) of the 77Se nuclear spins as
where HIZ is from the Zeeman contribution due the applied magnetic field, is from the direct hyperfine coupling of the 77Se nucleus to the BETS -electrons, while is from the indirect hyperfine coupling via the -electrons to the 3d Fe3+ ion spins, and the last term is from the dipolar coupling to the Fe3+ spins.
The -d exchange field Hd comes from, and the term produces dipolar field Hdip. We calculated Hdip from the summation of the near dipole, the bulk demagnetization and the Lorentz contributions. The cartoon of the -d exchange interaction for the Jaccarino-Peter mechanism and the sample rotation direction in the magnetic field are shown in Fig. 10.
From Eq. (3) the corresponding 77Se NMR resonance frequency is
where ’ is the angle between H0 and the pz directions, and Kc and Ks(’) are, respectively, the chemical shift and the Knight shift of the BETS Se -electrons.
where Kiso and Kan (’) are the isotropic and axial (anisotropic) parts of the Knight shift, respectively. Kiso(ax) is a constant determined by the isotropic (axial) hyperfine field produced by the 4p spin polarization of the BETS Se -electrons .
The dashed lines in Fig. 8 are the fit to Eqs. (4) – (5). The gyromagnetic ratio of Se nucleus is 77 = 8.131 MHz/T. The value of Kax= 15.3 10-4 can be obtained precisely from the BETS molecule magnetic susceptibility and the -electron spin polarization configuration [20, 26]. From the fit, now we can obtain the value of the -d exchange field Hd.
Alternatively, for better accuracy we obtained the following expression for the -d exchange field Hd from Eqs. (4) – (5) to be,
Thus from the data T0 = 5 K, 1 = 90°, and 2 = 0° as shown from Fig. 8, it gives = 880 ± 26 kHz, and. The valueof T is calculated with H0 = 9.0006 T. With these values we obtained Hd = (- 32.7 ± 1.5) T at temperature T = 5 K and applied field H0 = 9 T. This is very close to the expected value of −33 T obtained from the electrical resistivity measurement [27, 28] and the theoretical estimate .
If the applied field is H0 = 33 T, by using our modified Brillouin function with the average of the Fe3+ spins, we expected the value of the Hd (33 T, 5 K) = (- 34.3 ± 2.4) T.
This large value of negative -d exchange field felt by the Se conduction electrons obtained from our NMR measurements verifies the effectiveness of the Jaccarino-Peter compensation mechanism responsible for the magnetic-field-induced superconductivity in the quasi-2D superconductor –(BETS)2FeCl4.
We have presented briefly the information about the field-induced-superconductors including the theories explaining the mechanisms for the field-induced superconductivity. We also summarized our 77Se-NMR studies in a single crystal of the field-induced superconductor –(BETS)2FeCl4, while most of our detailed research NMR work including both proton NMR and 77Se-NMR were reported in refs.[20, 23, 24].
Our 77Se-NMR experiments revealed large value of negative -d exchange field (Hd 33 T at 5 K) from the negative exchange interaction between the large 3d-Fe3+ ions spins and BETS conduction electron spins existing in the material. This result directly verified the effectiveness of the Jaccarino-Peter compensation mechanism responsible for the magnetic-field-induced superconductivity in this quasi-2D superconductor –(BETS)2FeCl4.
Future high field NMR experiments (H0 30 T) would be of interest, and NMR measurements with the alignment of applied magnetic field along the c-axis and sample rotation in the ac-plane (conducting plane) would further improve our understanding of this novel field-induced superconductor.
We are grateful to Prof. S. E. Brown at UCLA for helpful discussions and help with the NMR experiments. We also thank other coauthors J. S. Brooks, A. Kobayashi, and H. Kobayashi for our detailed 77Se-NMR work published in ref. .