EPR Study of Superconductors

Abstract

Historical review on the studies of the electron paramagnetic resonance in superconductors performed in the period from 1970 to 1990 at the Kazan Physical Technical Institute of the Russian Academy of Sciences (group of Dr. E. G. Kharakhash’yan) in collaboration with Kazan State University (group of Prof. B. I. Kochelaev) and with the Institute for Physical Problems of Russian Academy of Sciences (group of Prof. N. E. Alekseevskii) is presented. We have observed for the first time the electron paramagnetic resonance of impurities in a type II superconductor; found indication for a long-range exchange interaction between magnetic impurities arising due to the superconducting correlations; observed the magnetic ordering of impurities in the superconducting state; and, finally, we found one of the first evidences for heterogeneity of the 1:2:3 high-$T_c$ superconductor which is its natural property.

Appendices

Appendix 1: EPR Line Shape in a Superconductor

In the previous Section we present the experimental evidence for the possibility to observe EPR of a localized moment in the superconducting state. Surprisingly the observed linewidth did not exceed its value in the normal state. This means that in order to understand the reason for unchanged linewidth upon transition from the normal to the mixed superconducting state it is necessary to calculate line shape in the Abrikosov [6, 7] vortex state taking into account the magnetic field distribution [76].¹ This mixed state is characterized by the presence of an ordered lattice of tubes of magnetic flux, or vortices, surrounded and maintained by superconducting currents. The magnetic field decreases from a maximum value at the center of an isolated vortex over a characteristic distance $\lambda$, the London penetration depth. The superconducting order parameter grows from zero at the center of the vortex over a distance of the order of $\xi (T)$, the coherence length, with $\xi (T) < \lambda (T)$. For applied fields far from both $H_{c1}$ and $H_{c2}$ (the lower and upper critical fields, respectively) the vortex density is not too great. Experiments by Redfield [5] in this region confirmed the ordered arrangement of the vortices, and verified that the lattice symmetry was triangular.

The distribution of the probabilities of encountering a given magnetic field in a triangular vortex lattice can be approximated by the analytic function [22]

$$\begin{aligned} f(x)= \left\{ \begin{array}{lcl} 0.837-0.500\ln {(-x)},&{}\quad 0.08<x<0 \\ 0.236-0.576\ln {x}, &{}\quad 0<x<0.92 \end{array} \right. \end{aligned}$$

(3)

Here $x = (H - H_s)/(Hv - H_c)$, $H_v$ and $H_c$ are the maximum and minimum fields in the lattice, and $H_s$ is the field at the saddle point of the unit cell of the vortex lattice. If the shape of the homogeneously broadened EPR line in metal is determined by the Lorentz line

$$\begin{aligned} I_{norm}(H)={{1}\over {\pi }}{{{\Delta H}+H}\over {(\Delta H)^2+H^2}} \end{aligned}$$

(4)

where $\Delta H$ is the half-width of the line at half-height and $H$ is the magnetic field relative to the center, then the EPR line shape with allowance for the broadening due to the vortex lattice is determined by the convolution

$$\begin{aligned} I_{sup}(H)=\int {I_{norm}(H_1-H)f\left( {{H_1-H_c}\over {H_v-H_c}}\right) dH_1} \end{aligned}$$

(5)

The observed behavior of the spectrum is qualitatively consistent with the absence of pronounced broadening in the superconducting state due to the vortex structure at certain values of parameters. The average internal field is smaller than in the normal state, that is, a part of the flux is excluded. Increasing the amplitude of the high-field wing of the resonance line is due to the discontinuity at the maximum field $H_v$.

Appendix 2: Electron Bottleneck and $T_c$-Suppression by Magnetic Impurity

For nonmagnetic metals, NMR relaxation rate always complies to the Korringa law [23], that is, the linear dependence of the spin-lattice relaxation rate on the temperature. In contrast to NMR in EPR in metals, conduction electrons cannot generally be considered to be in equilibrium and this leads to some complications in the theory. The best known of these is the “electron bottleneck” effect. If cross relaxation between the localized moments and conduction electrons is rapid, spin angular momentum transferred to the conduction electrons can be transferred back to the localized moments before it has time to decay to the lattice. The apparent rate of relaxation of the local moments is then determined by the conduction electron’s spin-lattice relaxation (see Fig. 12). The bottleneck affects the position as well as the width of the resonance line.

Fig. 12

The relaxation paths involved in the EPR bottleneck. The rates $\delta _{ie}$ and $\delta _{ei}$ result from the exchange interaction; these rates denote transfer of magnetization from and to the localized moments and conduction electron subsystems

There are two regimes for EPR. First, isothermal regime ($\delta _{ei}<<\delta _{eL}$). The thermal broadening of the EPR linewidth is determined by the Korringa law

$$\begin{aligned} \delta _{ie}^{Kor}={{\pi }\over {\hbar }}[J_{sf} \rho (E_f)]^2 k_B T, \end{aligned}$$

(6)

and the $g$-shift

$$\begin{aligned} \Delta g_{max}=J_{sf}\rho (E_f), \end{aligned}$$

(7)

where $\hbar$ is the Plank constant and $\rho (E_f)$ is the density of states of the conduction electrons with given spin orientation on the Fermi surface, and $J_{sf}$ is the $sf$-exchange integral between localized moments and conduction electrons.

Experimentally one observes in addition a residual linewidth $a$

$$\begin{aligned} \Delta H=a+bT,\ \ \ b={{\hbar \delta _{ie}^{Kor}}\over {g\mu _B T}}. \end{aligned}$$

(8)

Second, bottleneck regime ($\delta _{ei}\ge \delta _{eL}$).

Very often for high concentration of impurity, $\delta _{ei}$ may be of the same order of magnitude as $\delta _{eL}$. Then the subsystem of the conduction electrons is not longer in thermal equilibrium with lattice. The effective relaxation rate $\delta _{ie}$ is reduced. This was first calculated by Hasegawa [15].

Assuming $g_i \simeq g_e$, the theory yields

$$\begin{aligned}&\displaystyle \delta _{ie}={{\delta _{eL}}\over {\delta _{ei}+\delta _{eL}}}\delta _{ie}^{Kor},\ \ b={{\hbar }\over {g\mu _B}}\left( {{\delta _{eL}\over {\delta _{ei}+\delta _el}}}\right) {{\delta _{ie}^{Kor}\over {T}}}.&\end{aligned}$$

(9)

$$\begin{aligned}&\displaystyle \Delta g=\left( {{\delta eL}\over {\delta _{ei}+\delta _{eL}}}\right) \Delta g_{max}.&\end{aligned}$$

(10)

Thus $\delta _{ie}^{Kor}$ corresponds to the “Korringa rate” and $\delta _{ei}$ to the “Overhauser rate” which can be also obtained from “detailed balance” to susceptibility of the two subsystems $\chi _i/\chi _e=\delta _{ei}/\delta _{ie}$. Thus

$$\begin{aligned} \delta _{ei}={{2\pi }\over {\hbar }}\rho (E_f)J_{sf}^2 S(S+1)c. \end{aligned}$$

(11)

From another side, $\delta _{ei}$ is the controlling parameter for pair breaking in the superconducting state. The theory by Abrikosov-Gor’kov [24, 25] well describes the $T_c$-suppression by magnetic impurity.

$$\begin{aligned} \ln (T_c/T_{c0}=\psi \left( {{1}\over {2}}\right) -\psi \left( {{1}\over {2}}+0.14{{\alpha }\over {\alpha _{cr}}} {{T_{c0}}\over {T_c}}\right) . \end{aligned}$$

(12)

The pair-breaking parameter $\alpha$ and its critical value $\alpha _{cr}$, where superconductivity is completely destroyed, are given by the theory in the first Born approximation

$$\begin{aligned} 4\hbar \alpha _{cr}&= k_BT_{c0}/\gamma ;\end{aligned}$$

(13)

$$\begin{aligned} 4\hbar \alpha _{cr}&= c \rho (E_f)J_{sf}^2 S(S+1). \end{aligned}$$

(14)

This equation yields

$$\begin{aligned} \left| {{\mathrm{d}T_c}\over {\mathrm{d}c}}\right| ={{3\hbar \pi }\over {16k_B}}{{\mathrm{d}\delta _{ei}}\over {\mathrm{d}c}}. \end{aligned}$$

(15)

This means that

$$\begin{aligned} \left| {{\mathrm{d}T_c}\over {\mathrm{d}c}}\right| =-{{\pi ^2}\over {8k_B}} c\rho (E_f)J_{sf}^2S(S+1). \end{aligned}$$

(16)