Increasing stripe-type fluctuations in $A$Fe$_{2}$As$_{2}$ ($A$ = K, Rb, Cs) superconductors probed by $^{75}$As NMR spectroscopy

TL;DR

This study uses $^{75}$As NMR spectroscopy to show that increasing lattice size in $A$Fe$_{2}$As$_{2}$ superconductors enhances stripe-type spin fluctuations and electronic correlations, approaching a quantum critical point.

Contribution

It reveals how lattice expansion in $A$Fe$_{2}$As$_{2}$ compounds intensifies spin fluctuations and deviations from Fermi-liquid behavior, indicating proximity to a quantum critical point.

Findings

Enhanced stripe-type spin fluctuations with larger alkali ions.

Decreased Fermi-liquid behavior evidenced by relaxation rate exponents.

Approach to a quantum critical point with lattice expansion.

Abstract

We report $^{75}$As nuclear magnetic resonance measurements on single crystals of RbFe$_{2}$As$_{2}$ and CsFe$_{2}$As$_{2}$. Taking previously reported results for KFe$_{2}$As$_{2}$ into account, we find that the anisotropic electronic correlations evolve towards a magnetic instability in the $A$Fe$_{2}$As$_{2}$ series (with $A$ = K, Rb, Cs). Upon isovalent substitution with larger alkali ions, a drastic enhancement of the anisotropic nuclear spin-lattice relaxation rate and decreasing Knight shift reveal the formation of pronounced spin fluctuations with stripe-type modulation. Furthermore, a decreasing power-law exponent of the nuclear spin-lattice relaxation rate $(1/T_{1})_{H\parallel{ab}}$, probing the in-plane spin fluctuations, evidences an emergent deviation from Fermi-liquid behavior. All these findings clearly indicate that the expansion of the lattice in the $A$Fe$_{2}$As$_{2}$ series tunes the electronic correlations towards a quantum critical point at the transition to a yet unobserved, ordered phase.

Equations

$$R=\left\{\begin{array}[]{lll}0.5+0.5\left(\dfrac{A_{c}S_{c}}{A_{ab}S_{ab}}\right)^{2}&\text{no correlation}&\\ 0.5&(\pi,\pi)&\\ 0.5+\left(\dfrac{S_{ab}}{S_{c}}\right)^{2}&(0,\pi)\hskip 2.84526pt\text{or}\hskip 2.84526pt(\pi,0).&\end{array}\right.$$

Full text

The investigation of quantum critical points (QCPs) in iron pnictides provides a route for understanding the relation of their relatively high superconducting transition temperatures and the anomalous normal-state properties, such as enhanced quasiparticle masses and non-Fermi-liquid behavior BaFeAsP-QCP-London ; Shibauchi-QCP-review ; BaFeAsP-QCP-dHvA ; BaFeAsP-QCP-nFL-RT ; BaFeAsP-QCP-NMR ; BaFeAsP-FeCo-QCP-NMR ; QCP-JPCM-QSi ; QCP-Eilers ; QCP-Mizukami ; Wang-K122-pressure ; QCP-theory ; QCP-theory . Signatures of quantum criticality, driven by variation of the charge carrier concentration, have been reported for moderately Ni and Co doped BaFe2As2 Zhou2013 ; Ning2010 ; Chu2009 . Also, by variation of the internal pressure, a QCP, emerging from the interplay of electronic localization and itinerancy, was theoretically anticipated QCP-theory , and has been experimentally identified in BaFe2(As1-xPx)2 by various techniques BaFeAsP-QCP-London ; Shibauchi-QCP-review ; BaFeAsP-QCP-dHvA ; BaFeAsP-QCP-nFL-RT ; BaFeAsP-QCP-NMR ; BaFeAsP-QCP-NMR .

In the case of the heavily hole-doped $A$Fe2As2 ($A$ = K, Rb, Cs) series, the Fe $3d$ orbitals are nominally filled with $N=5.5$ electrons. The proximity to the Mott-insulating phase, expected for $N=5.0$, promotes phenomena that arise from localization and increased electronic correlations. Specifically, it was proposed that a QCP is approached with increasing chemical pressure, driven by an isovalent substitution of K*+* with Rb*+* or Cs*+* QCP-Eilers ; QCP-Mizukami ; Wang-K122-pressure . In KFe2As2, a pronounced mass enhancement due to strong correlation effects, as well as deviations from standard Fermi-liquid behavior were reported from several studies Terashima2010 ; K122-crossover ; Werner-NPhys-crossover ; Liu-Lifshitz-crossover ; Werner-NPhys-crossover ; WQYu-K122 ; Dong-Non-fermi ; Dong-Non-fermi . An increase of the negative pressure in RbFe2As2 and CsFe2As2 does not affect the Fermi surface topology, but leads to enhanced quasiparticle masses, presumably driven by an orbital-selective increase of correlations Hardy2016 ; K122-crossover ; Zhang-RbNonfermi ; Rb122-gamma180 ; QCP-Eilers ; Backes-KRbCs-Hund . In line with these findings, the application of positive pressures was reported to decrease the Sommerfeld coefficient $\gamma$ in KFe2As2 K122-press-gamma . In a recent dilatometry and local-density approximation (LDA) study of the $A$Fe2As2 series, Eilers et al. suggested that the enhancement of correlations originates from a pressure-dependent hybridization of the in-plane Fe 3$d_{xy}$ orbitals QCP-Eilers . The low-energy fluctuations close to the QCP seem to suppress $T_{c}$, in contrast to the usually propagated picture of superconductivity stimulated by critical fluctuations.

As for the yet unobserved, ordered phase beyond the QCP, a combination of components from spin, orbital and charge degrees of freedom was proposed in a recent work by Drechsler et al. Drechsler2017 . For a $d_{x^{2}-y^{2}}$ superconducting order parameter, related critical spin fluctuations arising from the Fe 3$d_{xz}$/$d_{yz}$ bands could eventually enhance the quasiparticle masses without strengthening the formation of superconductivity. In support of the multi-component order, a tendency for charge order was experimentally observed in nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR) studies of KFe2As2 and RbFe2As2 Wang-K122-pressure ; Civardi2016 .

In general, approaching a magnetic QCP is reflected by non-Fermi-liquid behavior and a drastic increase of spin fluctuations, yielding a power-law behavior of related physical quantities with exponents differing from those of a Fermi liquid RMP-QCP-NonFermi . Whereas the findings of Eilers et al. clearly indicate that a quantum critical point is approached with increasing FeAs-cell volume, other experimental reports seem much less conclusive. Different power-law dependencies between $\propto T^{1.5}$ and $\propto T^{2.0}$ were reported for the resistivity of the $A$Fe2As2 series K122-crossover ; Dong-Non-fermi ; Zhang-RbNonfermi ; Hong-CsNonfermi ; Khim2016 . From NMR measurements of all three $A$Fe2As2 compounds with $H\parallel{c}$, the same power law of $1/T_{1}\propto T^{0.75}$ was reported, i.e., giving no indication of approaching quantum criticality Wu-XHChen . In order to resolve this discrepancy between the different experimental findings, we performed a detailed NMR study of the anisotropic magnetic correlations.

In this paper, we present 75As NMR measurements on high-quality single crystals of RbFe2As2 and CsFe2As2 for both in-plane ($H\parallel{ab}$) and out-of-plane ($H\parallel{c}$) orientations of the magnetic field and temperatures between 1.6 and 300 K. Taking previously reported results for KFe2As2 into account, we find a continuous reduction of the Knight shift and a drastic increase of spin fluctuations along the $A$Fe2As2 series. Based on the anisotropy of $1/T_{1}$, we provide evidence for a stripe-type modulation of the spin fluctuations in all three $A$Fe2As2 compounds. Furthermore, a successively reduced power-law exponent of $(1/T_{1})_{H\parallel{ab}}$ $\propto T^{\eta}$ at low temperatures shows increasing deviation from the unity exponent of a Fermi liquid. Our results clearly indicate that the electronic correlations are evolving from proximity to a Fermi liquid towards a state with strong correlations in close vicinity to a QCP.

Single crystals of $A$Fe2As2 ($A$ = Rb, Cs) were grown by use of the self-flux technique Khim2016 . Their stoichiometry was confirmed by energy-dispersive x-ray spectroscopy. For a precise sample orientation with field, a single-axis goniometer with a resolution of $0.1^{\circ}$ was used. The 75As NMR central transition spectra and nuclear spin-lattice relaxation rate $1/T_{1}$ were measured at temperatures from 1.6 to 300 K at $\mu_{0}H=5.513$ T for RbFe2As2 and at $\mu_{0}H=5.900$ T for CsFe2As2. $T_{1}$ was obtained from fitting $M_{z}(t)=M_{0}\left[1-f(0.9e^{-6t/T_{1}}+0.1e^{-t/T_{1}})\right]$ to the recovery of the nuclear magnetization after inversion stretchT1 .

By measuring the central ($I_{z}=-1/2\rightarrow+1/2$ ) and satellite ($-3/2\rightarrow-1/2$, $+1/2\rightarrow+3/2$) transition frequencies for $H\parallel{ab}$ and $H\parallel{c}$, we determined the principal axis of the local electric field gradient (EFG) tensor $V_{ZZ}$ to be parallel to the $c$ axis, and the quadrupole frequency $\nu_{Q}\propto V_{ZZ}$ at 5 K as 14.4 MHz for RbFe2As2 and 13.7 MHz for CsFe2As2, larger than $\nu_{Q}=$ 12.2 MHz, reported for KFe2As2 Wang-K122-pressure . Between 5 to 300 K, $\nu_{Q}$ shows, in good accordance with the thermal expansion of the lattice, a weak variation ($\lesssim 1\%$) for both RbFe2As2 and CsFe2As2, thus proving the absence of a structural transition K122-crossover ; Meingast2012 . The 75As central-line spectra of RbFe2As2 and CsFe2As2 for $H\parallel{ab}$ and $H\parallel{c}$ are shown in Fig. 1. The full width at half maximum (FWHM) is, except for RbFe2As2 with $H\parallel{c}$, $\approx 30$ kHz and only weakly temperature dependent Misorientation-FWHM-broadening . This is even considerably lower than the FWHM of $50-80$ kHz reported for KFe2As2 Hirano-jpsj ; Wang-K122-pressure , proving the very high quality of the single crystals used for our NMR study TdepIntRb122 .

The Knight shift $K$ probes the uniform local susceptibility. It is defined by $f_{res}=(1+K)f_{0}$, where $f_{0}={\gamma_{n}}\mu_{0}H/2\pi$ is the Larmor frequency of a bare nucleus with gyromagnetic ratio ${\gamma_{N}}$ in a magnetic field $H$, and $f_{res}$ is the observed resonance frequency, which we extract as the first moment from the recorded spectra. For $H\parallel{ab}$ ($\perp{V_{ZZ}}$), a second-order quadrupole shift $3\nu_{Q}^{2}/16f_{0}$ contributes to $f_{res}$, and was subtracted from the data using the quadrupole frequencies specified above. As shown in Fig. 2 (a) and (b), we find a similar anisotropy for all three $A$Fe2As2 compounds. In general, the in-plane Knight shift $K_{ab}$ is slightly larger than the out-of-plane shift $K_{c}$, and both continuously decrease from KFe2As2 to CsFe2As2. This trend indicates a reduction of the uniform local spin susceptibility, in compatibility with an increase of antiferromagnetic correlations. Also, $K_{ab}$ shows a stronger temperature dependence than $K_{c}$, yielding a broad maximum that shifts to lower temperatures and becomes increasingly pronounced.

In line with the Knight shift data, measurements of the macroscopic susceptibility of KFe2As2 revealed a maximum around 100 K K122-crossover , and are considered as evidence for a coherence-incoherence crossover mechanism K122-crossover ; Liu-Lifshitz-crossover ; Wu-XHChen . It is suggested that localized spins dominate the susceptibility in the incoherent state at high temperatures, whereas the low-temperature coherent state is a metal with enhanced Pauli susceptibility. By defining $T^{*}$ as the maximum of the temperature-dependent Knight shift for $H\parallel{ab}$, we obtain $T^{*}=$133$\pm$3 K, 123$\pm$2 K, and 111$\pm$2 K for KFe2As2, RbFe2As2, and CsFe2As2, respectively.

Turning to the dynamic correlations, $1/T_{1}T$ probes the fluctuating hyperfine fields at the nuclear site, $1/T_{1}T\propto\sum_{q,\alpha,\beta}F_{\alpha\beta}(q)\chi_{\alpha\beta}^{\prime\prime}(q,f_{res})/f_{res}$, with $\alpha,\beta=\left\{x,y,z\right\}$. Here, $F_{\alpha\beta}$ denotes the hyperfine form factors and $\chi_{\alpha\beta}^{\prime\prime}$ is the imaginary part of the dynamical electronic susceptibility. The low-temperature part of $(1/T_{1}T)$ increases drastically along the $A$Fe2As2 series, as shown in Fig. 2(c) and (d). This reveals a significant increase of both in-plane and out-of-plane hyperfine-field fluctuations and, therefore, gives strong indications of an increasing dynamical susceptibility upon approaching a magnetic instability. Note that for all $A$Fe2As2 compounds, $(1/T_{1}T)_{H\parallel{c}}$ is generally smaller than $(1/T_{1}T)_{H\parallel{ab}}$.

As was reported for BaFe2As2 and related compounds with K, Co, or Cu doping, the anisotropy ratio of the nuclear spin-lattice relaxation rate, $R=(1/T_{1})_{H\parallel{ab}}/(1/T_{1})_{H\parallel{c}}$, allows to conclude on the $q$ modulation of electronic spin fluctuations Hirano-jpsj ; Grafe2014 . As detailed in Refs. Kitagawa2010, ; Hirano-jpsj, , assuming that the hyperfine field at the As site is given as the sum of the fields from the nearest-neighbor Fe spins with tetragonal symmetry (in-plane isotropy of diagonal hyperfine coupling terms, i.e., $A_{ab}=A_{a}=A_{b}$), the anisotropy ratio $R$ is given as

[TABLE]

Here, $S_{ab}$ and $S_{c}$ are the in-plane and out-of-plane components of the fluctuating Fe spins. For our analysis of $R$, we consistently use the ratio $A_{c}/A_{ab}=0.5$, reported for KFe2As2 Hirano-jpsj . Since all three members of the $A$Fe2As2 series are isovalent and the structural parameters vary only slightly QCP-Eilers , the hyperfine coupling and chemical shift can be taken as invariant. By further assuming $S_{ab}=S_{c}$, $R\approx 0.6$ indicates an absence of correlations, $R=0.5$ corresponds to checkerboard-type fluctuations with $q=(\pi,\pi)$, and $R=1.5$ results from a fully developed commensurate stripe-type [$q=(0,\pi)$ or $q=(\pi,0)$] modulation.

Lee et al. reported two-dimensional incommensurate spin fluctuations in KFe2As2 at $\left(\pi(1\pm 2\delta),0\right)$, with $\delta=0.16$, determined by neutron scattering Lee2011 . For all three $A$Fe2As2 compounds, we find, within experimental error, $R\approx 1.2$ at 300 K, see Fig. 3. The same room temperature value of the anisotropy ratio was reported for Cu- and Co-doped BaFe2As2 Grafe2014 . Therefore, we conclude that stripe-type fluctuations, in full compatibility with the aforementioned neutron-scattering results are already well developed at room temperature. Further, the presence of these fluctuations implies that the proximite ordered state is not a purely magnetic Mott phase, as this would yield a checkerboard-type modulation. As a general trend, $R$ increases towards low temperatures. For KFe2As2 and RbFe2As2, no change of $R$ is found at $T^{*}$, even though the temperature-dependent exponent of $(1/T_{1})$ for both applied field directions changes at $T^{*}$, as will be discussed below. For CsFe2As2, the $1/T_{1}T$ data in Figs. 2 (c) and (d) show a hump-like structure at temperatures below about 100 K, which also seems to be reflected by a weak variation of $R$ at intermediate temperatures. Towards lowest temperatures, $R$ increases to $\approx$ 2 for RbFe2As2, and to $\approx$ 3 for CsFe2As2.

As was shown for the case of Cu- and Co-doped BaFe2As2, in the vicinity to a quantum critical point, possibly coupled to a structural instability, $R$ tends to increase to much larger values than 1.5 Grafe2014 . In the present case of the $A$Fe2As2 compounds, we have no experimental indication of a nearby structural transition. Therefore, an increase of $R$ beyond 1.5 can only result from an increase of $S_{ab}/S_{c}$, proving the increasingly anisotropic character of correlations close to the QCP. Moreover, these findings are compatible with an electronic localization in specific Fe $3d$ orbitals. LDA + dynamical mean field theory (DMFT) calculations have predicted an increasing localization in Fe $3d_{z^{2}}$ and $3d_{xy}$ orbitals in the $A$Fe2As2 series Backes-KRbCs-Hund . In line with these results, the study of Eilers et al. attributes a strong enhancement of effective quasiparticle masses to the hybridization of the $3d_{xy}$ orbitals QCP-Eilers .

An increase of anisotropic dynamic correlations leads to an anisotropic power-law behavior with $1/T_{1}\propto T^{\eta}$ close to quantum criticality. Considering the given stripe-type modulation, in-plane spin fluctuations $S_{ab}$ contribute to $(1/T_{1})_{H\parallel{ab}}$ via off-diagonal hyperfine coupling terms, but not to $(1/T_{1})_{H\parallel{c}}$ Kitagawa2010 ; Smerald2011 ; Dioguardi2016 . In consequence, mostly the power-law exponent $\eta_{H\parallel{ab}}$ should evolve upon approaching a QCP driven by in-plane correlations.

Along this line, we now discuss the emergent non-Fermi-liquid character of the dynamic magnetic correlations by comparing $1/T_{1}$ for the $A$Fe2As2 compounds for both field orientations. For better comparability, Fig. 4 shows $1/T_{1}$ as a function of the reduced temperature $T/T^{*}$ Power-T* . As shown in Fig. 4(b), all $(1/T_{1})_{H\parallel{c}}$ curves follow almost the same power-law relation with $\eta_{H\parallel{c}}$ between 0.80 and 0.70 below $T^{*}$. For in-plane fields, however, $\eta_{H\parallel{ab}}$ successively decreases from 0.71 to 0.42 below $\approx 0.2~{}T/T^{*}$.

In Ba0.3K0.7Fe2As2, discussed in Ref. WQYu-K122, , $\eta$ is still close to unity and isotropic, as expected for a Fermi liquid. For KFe2As2, the reported values of $\eta$ range from 0.75 to 0.8, indicating enhanced spin fluctuations in contrast to a standard Fermi-liquid behavior WQYu-K122 ; K-power0.8 ; Wu-XHChen . In the present work, we find that $\eta$ becomes increasingly anisotropic for RbFe2As2 and CsFe2As2. Moreover, the decrease of $\eta_{H\parallel{ab}}$ evidences that the progressive deviation from Fermi-liquid behavior is mainly reflected by the evolution of the in-plane spin fluctuations $S_{ab}$. Our findings for $\eta_{H\parallel{c}}$ are, therefore, in general agreement with the results reported by Wu et al., but the strong evolution of $\eta_{H\parallel{ab}}$ is in sharp contrast to their statement on universality below $T^{*}$ Wu-XHChen .

In summary, we report a detailed investigation of the anisotropic magnetic correlations in the $A$Fe2As2 ($A$ = K, Rb, Cs) series by means of 75As NMR spectroscopy. The Knight-shift data indicate a successive reduction and increasing temperature dependence of the uniform spin susceptibility. The anisotropy of $1/T_{1}$ allows to conclude on a stripe-type modulation of spin fluctuations up to room temperature, implying that the proximite ordered state is not a purely magnetic Mott phase. Further, we find an increasing low-temperature anisotropy of spin fluctuations with increase of the alkali-ion radius. The power-law behavior of $(1/T_{1})_{H\parallel{ab}}$ gives evidence that the dynamic correlations evolve towards an increasing deviation from Fermi-liquid behavior, reflected mostly by the in-plane spin fluctuations. All these findings clearly indicate that the electronic correlations in the $A$Fe2As2 series evolve towards a quantum critical point at the transition to a yet unobserved, ordered phase. The associated unusual phenomenology, i.e. a decrease of $T_{c}$ in proximity of the QCP as well as indications for multicomponent order, call for further investigations, in particular, finding an experimental avenue to stabilize the ordered phase.

We thank S.-L. Drechsler, B. Büchner, and S. Aswartham for valuable discussions. Z.T.Z. was financially supported by the National Nature Science Foundation of China (Grant No. 11304321) and by the International Postdoctoral Exchange Fellowship Program 2013 (Grant No. 20130025). Further, support by the HLD at HZDR, a member of the European Magnetic Field Laboratory, and by the Deutsche Forschungsgemeinschaft (DFG) through the Priority Programme SFB 1458 and the Research Training Group GRK 1621 is gratefully acknowledged.