Magnetic field dependence of spin-lattice relaxation in the $s_{\pm }$ state of Ba${}_{0.67}$K${}_{0.33}$Fe${}_2$As${}_2$

The spatially averaged density of states $\langle$$N\left(0\right)$$\rangle$ of an unconventional $d$-wave superconductor is magnetic field dependent, proportional to $H^{1/2}$, owing to the Doppler shift of quasiparticle excitations in a background of vortex supercurrents [G. E. Volovik, J. Phys. C 21, L221 (1988); JETP Lett. 58, 469 (1993)]. This phenomenon, called the Volovik effect, has been predicted to exist for a sign-changing $s_{\pm }$ state [Y. Bang, Phys. Rev. Lett. 104, 217001 (2010)], although it is absent in a single-band $s$-wave superconductor. Consequently, we expect there to be Doppler contributions to the NMR spin-lattice relaxation rate, $1/T_1\propto \langle N{\left(0\right)}^2\rangle$, for an $s_{\pm }$ state which will depend on the magnetic field. We have measured the ${}^{75}$As $1/T_1$ in a high-quality single crystal of Ba${}_{0.67}$K${}_{0.33}$Fe${}_2$As${}_2$ over a wide range of field up to 28 T. Our spatially resolved measurements show that indeed there are Doppler contributions to $1/T_1$ which increase closer to the vortex core, with a spatial average proportional to $H^2$, inconsistent with recent theory [Y. Bang, Phys. Rev. B 85, 104524 (2012)].

Figure 1

${}^{75}$As NMR spectra of Ba${}_{0.67}$K${}_{0.33}$Fe${}_2$As${}_2$ measured by a frequency-sweep technique in 13 T with $H\left|\right|$$c$-axis of the crystals. Below $T_c=32$ K (blue trace), the spectra shift noticeably to a lower frequency with decreasing temperature. The linewidths of the spectra decrease in the superconducting state, as reported in some other compounds (Refs. 15 and 16) where this was attributed to reduction in the local-field distribution from impurities.

Figure 2

The total Knight shift $K\left(T\right)$ is shown for $H=13$ T where the temperature dependence is associated with the spin part, $K_s$, that decreases below $T_c$, consistent with spin-singlet superconductivity. The black arrow indicates $T_c=32$ K. The data can be fit phenomenologically assuming that the low-temperature spin shift is proportional to an average density of states, given by Eq. (1), represented by the solid curve, with an orbital shift of $K_{\mathrm{orb}}=0.21\%$.

Figure 3

The spin-lattice relaxation rate $1/T_1$ of Ba${}_{0.67}$K${}_{0.33}$Fe${}_2$As${}_2$ in magnetic fields, $H=6.4,10.8,14,16.5$, and 27 T. The rates were measured at the peak position in the spectrum. The data is consistent with our two-gap model (solid curves) provided the magnetic field dependence at low temperature is $\propto H^2$. The black arrow indicates $T_c$.

Figure 4

(a)–(e) $1{/}^{75}{T}_{1}T$ with the best-fit curve in each magnetic field. The unit of the $a\left(H\right),b_0,$ and $c_0$ is (s${}^{-1}$ K${}^{-1}$), and the gaps are in meV. (f) $1{/}^{75}{T}_{1}T$ in BaK122 at 4 K is shown for comparison with the Zeeman contributions to $1{/}^{17}{T}_{1}T$ in Y123 and Bi2212 at 5 K. Assuming that the electronic $g$ factor is the same for BaK122 as for the cuprates, we argue that the Zeeman contribution to the field dependence of the average rate we have measured in BaK122 is significantly smaller than from Doppler contributions. In the case of the cuprates, the Zeeman contributions to the spin-lattice relaxation rate were isolated using frequency-resolved measurements performed at the saddle point of the local-field distribution (peak of the spectrum), where Doppler contributions cancel based on symmetry of the supercurrents from near-neighbor vortices.

Figure 5

Spin-lattice relaxation rate across spectrum in the (a) normal state and (b)–(d) superconducting states in 16.5 T, with $H\left|\right|c$-axis. In the normal state, at 40 K, there is no significant frequency dependence in $1/T_1$. However, the rate becomes dependent on frequency as the sample is cooled deep into the superconducting state.

Figure 6

Spin-lattice relaxation rate across spectrum at 4 K in (a) 24 T and (b) 28 T with $H\left|\right|c$-axis. The inhomogeneous frequency dependence of the rate is observed in both magnetic fields similar to $H=16.5$ T; see Fig. 5d. Additionally, the increase in the high-frequency part of the spectra can be understood as an asymmetry from the vortex lattice.