*5.1.2. Temperature dependence of the 77Se-NMR resonance frequency*

12 Superconductors – Materials, Properties and Applications

*5.1.1. 77Se-NMR spectrum* 

identical.

77Se nuclei.

**5.1. 77Se-NMR measurements in** λ**–(BETS)2FeCl4**

*H***0** 

**= 9 T ||** *a***'**

**Figure 7.** 77Se-NMR absorption spectrum at various temperatures with applied magnetic field

*H*0 = 9 T || *a'* in λ–(BETS)2FeCl4 [23].

The 77Se-NMR spectra of λ–(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

The sample used for the 77Se-NMR measurements was grown using a standard method [22] 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 μg with ~ 2.0 × 1015

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.

along the lattice *c*-axis (the needle direction) and it is also perpendicular to the applied field

φ

) whose rotation axis is

The sample and coil were rotated on a goniometer (rotation angle

The temperature (*T)* dependence of the 77Se-NMR resonance frequency (ν) 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 (*H*T) at the Fe3+ ions. This is shown in Fig. 8 (b), where the solid lines show the fit to the *Md*.

The resonance frequency ν 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 *H*0 = 9 T, a good fit to frequency ν(uncertainty ± 3 kHz) is obtained using

$$h\nu(T, H\_0) = a - bM\_d(T, H\_T)\_\prime \tag{2}$$

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 ν 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 *H*0, 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 π-electrons is (i.e., the π-d exchange field *H*πd) which is the central goal of our 77Se-NMR measurements.

According to the *H-T* phase diagram of λ–(BETS)2FeCl4 [Fig. 6 (c)], the magnitude of *H*πd = 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 ν from our experiments is shown in Fig. 9, which is plotted as a function of angle φ at several temperatures. The angle φ basically describes the alignment direction of the applied magnetic field *H*0 relative to the sample lattice.

Field-Induced Superconductors: NMR Studies of λ–(BETS)2FeCl4 15

*H***0** 

ν

*H***0**

≈+++ (3)

plotted at several temperatures

π

is from the

**Figure 9.** Angular dependence of the 77Se-NMR resonance frequency

*H***0** 

*-d exchange field* 

where *HIZ* is from the Zeeman contribution due the applied magnetic field, *hf HI*

From the theory of NMR [25], we can express the contributions to the Hamiltonian (*H*I) of

, *hf hf dip HH H H H I IZ I Id d* π

direct hyperfine coupling of the 77Se nucleus to the BETS π-electrons, while *hf HId* is from the indirect hyperfine coupling via the π-electrons to the 3*d* Fe3+ ion spins, and the last term

The π-d exchange field *H*πd comes from *hf HId* , and the term *dip Hd* produces dipolar field *H*dip. We calculated *H*dip from the summation of the near dipole, the bulk demagnetization and the Lorentz contributions. The cartoon of the π-d exchange interaction for the Jaccarino-Peter mechanism and the sample rotation direction in the magnetic field are shown in Fig.

for the rotation of *H*0 = 9 T about the *c* axis in λ–(BETS)2FeCl4 [23].

*H*

π

 *and Fe3+-d electrons)* 

*dip Hd* is from the dipolar coupling to the Fe3+ spins.

*5.1.4. Determination of the* 

the 77Se nuclear spins as

π

*(between the Se-*

10.

**Figure 8.** (a) 77Se-NMR frequency shift as a function of temperature, and (b) 77Se-NMR frequency shift vs the Fe3+ magnetization, with applied magnetic field *B*0 = 9 T || *a'* in λ–(BETS)2FeCl4 [23].

To understand the complexity of these sets of data, we need clarify the angle φ 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 π-electron orbital moment, *p*z, 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 *H*0.

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 *p*z is determined to have angle 76.4o from the *c* axis through our calculation according to the X-ray data of λ–(BETS)2FeCl4. All these are clearly drawn as that shown in the inset of Fig. 9.

Thus during a sample rotation the direction of *H*0 is always in the *xy*-plane, where *x*-axis is chosen to be in the *c-p*z plane, and the angle φis counted from the *x*-axis.

Therefore, an angle φ = 0° corresponds to *H*0 to be in the *c*-*pz* plane with the smallest angle between *H*0 and *pz* as φmin= 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 π-d exchange field *H*πd precisely.

**Figure 9.** Angular dependence of the 77Se-NMR resonance frequency ν plotted at several temperatures for the rotation of *H*0 = 9 T about the *c* axis in λ–(BETS)2FeCl4 [23].

#### *5.1.4. Determination of the* π*-d exchange field (between the Se*π *and Fe3+-d electrons)*

14 Superconductors – Materials, Properties and Applications

*H***0** 

**Figure 8.** (a) 77Se-NMR frequency shift as a function of temperature, and (b) 77Se-NMR frequency shift

**= 9 T ||** *a***'** *<sup>H</sup>***<sup>0</sup>**

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 π-electron orbital moment, *p*z, 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*

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 *p*z is determined to have angle 76.4o from the *c* axis through our calculation according to the X-ray data of λ–(BETS)2FeCl4. All these are clearly drawn as that shown in the inset of

Thus during a sample rotation the direction of *H*0 is always in the *xy*-plane, where *x*-axis is

Based on these data shown in Figs. 8 - 9, we can determine the magnitude of the π-d

is counted from the *x*-axis.

= 0° corresponds to *H*0 to be in the *c*-*pz* plane with the smallest angle

φ

min= 13.6° as mentioned in Section 5.1.1.

φ

**= 9 T ||** *a***'**

first as there

vs the Fe3+ magnetization, with applied magnetic field *B*0 = 9 T || *a'* in λ–(BETS)2FeCl4 [23].

To understand the complexity of these sets of data, we need clarify the angle

(the needle direction) in the applied magnetic field *H*0.

chosen to be in the *c-p*z plane, and the angle

φ

φ

Fig. 9.

Therefore, an angle

between *H*0 and *pz* as

exchange field *H*πd precisely.

From the theory of NMR [25], we can express the contributions to the Hamiltonian (*H*I) of the 77Se nuclear spins as

$$H\_I = H\_{I\mathcal{Z}} + H\_{I\mathcal{x}}^{hf} + H\_{Id}^{hf} + H\_d^{dip} \, , \tag{3}$$

where *HIZ* is from the Zeeman contribution due the applied magnetic field, *hf HI*π is from the direct hyperfine coupling of the 77Se nucleus to the BETS π-electrons, while *hf HId* is from the indirect hyperfine coupling via the π-electrons to the 3*d* Fe3+ ion spins, and the last term *dip Hd* is from the dipolar coupling to the Fe3+ spins.

The π-d exchange field *H*πd comes from *hf HId* , and the term *dip Hd* produces dipolar field *H*dip. We calculated *H*dip from the summation of the near dipole, the bulk demagnetization and the Lorentz contributions. The cartoon of the π-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 νis

$$\begin{aligned} &\mathcal{V}(\boldsymbol{\phi}^{\prime}, \boldsymbol{H}\_{0}, \boldsymbol{T}) \\ &= \prescript{\boldsymbol{\nabla}}{}{\mathcal{Y}} \Big[ \prescript{\boldsymbol{H}}{}{\boldsymbol{H}}\_{0} + \prescript{\boldsymbol{H}}{}{\operatorname{div}}(\boldsymbol{H}\_{0}, \boldsymbol{T}) \Big] \Big[ \prescript{\boldsymbol{\nabla}}{}{\boldsymbol{1}} + \prescript{\boldsymbol{K}}{}{\operatorname{s}}(\boldsymbol{\phi}^{\prime}) \Big] \ + \prescript{\boldsymbol{\nabla}}{}{\boldsymbol{\gamma}} \prescript{\boldsymbol{K}}{}{\operatorname{s}}(\boldsymbol{\phi}^{\prime}) \boldsymbol{H}\_{\pi \mathsf{d}}(\boldsymbol{H}\_{0}, \boldsymbol{T}) \Big. \end{aligned} \tag{4}$$

Field-Induced Superconductors: NMR Studies of λ–(BETS)2FeCl4 17

2 = 0° as shown from Fig. 8, it gives

<sup>−</sup> Δ =× . The value of

 φ

Alternatively, for better accuracy we obtained the following expression for the π-d exchange

1 2 01 2

*dip dip*

φ φ

<sup>Δ</sup> <sup>−</sup> =− − <sup>Δ</sup> <sup>Δ</sup> (6)

φ

1 2 1 2

0 0 77 0

*T H H*

φ

<sup>−</sup> − =× T is calculated with *H*0 = 9.0006 T. With these values we

obtained *H*πd = (- 32.7 ± 1.5) T at temperature *T* = 5 K and applied field *H*0 = 9 T. This is very close to the expected value of −33 T obtained from the electrical resistivity measurement [27,

If the applied field is *H*0 = 33 T, by using our modified Brillouin function with the average of

This large value of negative π-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

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 λ–(BETS)2FeCl4, while most of our detailed research NMR work including

Our 77Se-NMR experiments revealed large value of negative π-d exchange field (*H*π<sup>d</sup> ≈ 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-

Future high field NMR experiments (*H*<sup>0</sup> ≥ 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

field-induced superconductivity in this quasi-2D superconductor λ–(BETS)2FeCl4.

*an an*

() () (,, ) ( ,) . (,) (,)

*H HT H K K*

 φφ

1 = 90°, and

ν φφ

γ

(5 , 90 , 0 ) *<sup>K</sup>* = 880 ± 26 kHz, and <sup>3</sup> (90 , 0 ) 4.42 10 *Kan*

the Fe3+ spins, we expected the value of the *H*πd (33 T, 5 K) = (- 34.3 ± 2.4) T.

both proton NMR and 77Se-NMR were reported in refs.[20, 23, 24].

φ

field *H*πd from Eqs. (4) – (5) to be,

Thus from the data *T*0 = 5 K,

<sup>3</sup> (90 ) (0 ) 3.63 10 *H H dip dip*

28] and the theoretical estimate [29].

superconductor λ–(BETS)2FeCl4.

novel field-induced superconductor.

*Department of Physics, University of West Florida, USA* 

**Author details** 

Guoqing Wu

**6. Summary** 

Δν *d*

π

where φ*'* is the angle between *H*0 and the *pz* directions, and *Kc* and *Ks*(φ*'*) are, respectively, the chemical shift and the Knight shift of the BETS Se π-electrons.

**Figure 10.** (a) Cartoon of the interactions for the Jaccarino-Peter mechanism. (b) Sketch of the sample rotation direction used for the 77Se-NMR measurements [23].

Here,

$$\left(K\_s(\phi')\right) \ = K\_{\rm iso} + K\_{\rm an}(\phi') = K\_{\rm iso} + K\_{\rm an} \left[3\cos^2\phi\cos^2\phi\_{\rm min} - 1\right]\_{\prime} \tag{5}$$

where *K*iso and *K*an (φ*'*) are the isotropic and axial (anisotropic) parts of the Knight shift, respectively. *K*iso(ax) is a constant determined by the isotropic (axial) hyperfine field produced by the 4*p*π spin polarization of the BETS Se π-electrons [26].

The dashed lines in Fig. 8 are the fit to Eqs. (4) – (5). The gyromagnetic ratio of Se nucleus is <sup>77</sup>γ = 8.131 MHz/T. The value of *K*ax= 15.3 × 10-4 can be obtained precisely from the BETS molecule magnetic susceptibility and the π-electron spin polarization configuration [20, 26]. From the fit, now we can obtain the value of the π-d exchange field *H*πd.

Alternatively, for better accuracy we obtained the following expression for the π-d exchange field *H*πd from Eqs. (4) – (5) to be,

$$H\_{\pi d}(H\_0, T\_0) = \frac{\Delta \nu(T\_0, \phi\_1, \phi\_2)}{\tau^{\tau}} - \frac{H\_{\text{dip}}(\phi\_1) - H\_{\text{dip}}(\phi\_2)}{\Delta K\_{\text{ an}}(\phi\_1, \phi\_2)} - H\_0. \tag{6}$$

Thus from the data *T*0 = 5 K, φ1 = 90°, and φ2 = 0° as shown from Fig. 8, it gives Δν (5 , 90 , 0 ) *<sup>K</sup>* = 880 ± 26 kHz, and <sup>3</sup> (90 , 0 ) 4.42 10 *Kan* <sup>−</sup> Δ =× . The value of <sup>3</sup> (90 ) (0 ) 3.63 10 *H H dip dip* <sup>−</sup> − =× T is calculated with *H*0 = 9.0006 T. With these values we obtained *H*πd = (- 32.7 ± 1.5) T at temperature *T* = 5 K and applied field *H*0 = 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 [29].

If the applied field is *H*0 = 33 T, by using our modified Brillouin function with the average of the Fe3+ spins, we expected the value of the *H*πd (33 T, 5 K) = (- 34.3 ± 2.4) T.

This large value of negative π-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 λ–(BETS)2FeCl4.

## **6. Summary**

16 Superconductors – Materials, Properties and Applications

0

*H T*

( ', , )

= + γ

ν φ

*H* π**d**

*H***0**

where φ

Here,

where *K*iso and *K*an (

From Eq. (3) the corresponding 77Se NMR resonance frequency

chemical shift and the Knight shift of the BETS Se π-electrons.

77 77

*'* is the angle between *H*0 and the *pz* directions, and *Kc* and *Ks*(

**Figure 10.** (a) Cartoon of the interactions for the Jaccarino-Peter mechanism. (b) Sketch of the sample

min ( ') ( ') 3cos cos 1 , *K KK KK <sup>s</sup> iso an iso ax*

respectively. *K*iso(ax) is a constant determined by the isotropic (axial) hyperfine field produced

The dashed lines in Fig. 8 are the fit to Eqs. (4) – (5). The gyromagnetic ratio of Se nucleus is <sup>77</sup>γ = 8.131 MHz/T. The value of *K*ax= 15.3 × 10-4 can be obtained precisely from the BETS molecule magnetic susceptibility and the π-electron spin polarization configuration [20, 26].

 φ=+ =+

From the fit, now we can obtain the value of the π-d exchange field *H*πd.

2 2

 φ

<sup>−</sup> (5)

*H***0**

φ

*'*) are the isotropic and axial (anisotropic) parts of the Knight shift,

rotation direction used for the 77Se-NMR measurements [23].

by the 4*p*π spin polarization of the BETS Se π-electrons [26].

φ

φ

++ +

0 0 0

*H H HT K K K H HT*

φ

( , ) 1 ( ') ( ') ( , ), *dip c s s d*

νis

 γφ

π

φ

(4)

*'*) are, respectively, the

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 λ–(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 π-d exchange field (*H*π<sup>d</sup> ≈ 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 magneticfield-induced superconductivity in this quasi-2D superconductor λ–(BETS)2FeCl4.

Future high field NMR experiments (*H*<sup>0</sup> ≥ 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.

## **Author details**

Guoqing Wu *Department of Physics, University of West Florida, USA*  W. Gilbert Clark *Department of Physics and Astronomy, University of California, Los Angeles, USA* 
