Abstract
A quantum critical point is a point in a system’s phase diagram at which an order is completely suppressed at absolute zero temperature (T). The presence of a quantum critical point manifests itself in the finiteT physical properties, and often gives rise to new states of matter. Superconductivity in the cuprates and in heavy fermion materials is believed by many to be mediated by fluctuations associated with a quantum critical point. In the recently discovered iron–pnictide superconductors, we report transport and NMR measurements on BaFe_{2−x}Ni_{x}As_{2} (0≤x≤0.17). We find two critical points at x_{c1}=0.10 and x_{c2}=0.14. The electrical resistivity follows ρ=ρ_{0}+AT^{n}, with n=1 around x_{c1} and another minimal n=1.1 at x_{c2}. By NMR measurements, we identity x_{c1} to be a magnetic quantum critical point and suggest that x_{c2} is a new type of quantum critical point associated with a nematic structural phase transition. Our results suggest that the superconductivity in carrierdoped pnictides is closely linked to the quantum criticality.
Introduction
In cuprate high temperature superconductors^{1} and heavy fermion^{2} compounds, the superconductivity is accompanied by the normalstate properties deviated from a Landau–Fermi liquid. Such deviation has been ascribed to the quantum critical fluctuations associated with a quantum critical point (QCP)^{3,4}, whose relationship to the occurrence of superconductivity has been one of the central issues in condensedmatter physics in the last decades. In iron–pnictide hightemperature superconductors^{5,6}, searching for magnetic fluctuations has also become an important subject^{7}. Quantum critical fluctuations of order parameters take place not only in spatial domain, but also in imaginary time domain^{8,9}. The correlation time τ_{0} and correlation length ξ are scaled to each other, through a dynamical exponent z, τ_{0}ξ^{z}. Several physical quantities, such as the electrical resistivity and spinlattice relaxation rate (1/T_{1}), can be used to probe the quantum critical phenomena. For the quasiparticles scattering dominated by the quantum critical fluctuations, the resistivity scales as ρT^{n}. For a twodimensional (2D) antiferromagnetic spindensity wave (SDW) QCP, the exponent n=1 is often observed^{1,2,10}. On the other hand, for a 2D order with q=(0,0) where z=3, n=4/3 at the QCP^{11,12}. Around a QCP, 1/T_{1} also shows a characteristic Tscaling^{11}.
BaFe_{2−x}Ni_{x}As_{2} is an electrondoped system^{13} where every Ni donates two electrons in contrast to Co doping that contributes only one electron^{14}. Therefore, Ni doping suffers less from disorder which is usually harmful for a QCP to exist. In this work, we find two critical points at x_{c1}=0.10 and x_{c2}=0.14, respectively. By NMR measurements, we identify x_{c1} to be a magnetic QCP and suggest that x_{c2} is a QCP associated with the tetragonaltoorthorhombic structural phase transition. The highest T_{c} is found around x_{c1}, which suggests that the superconductivity in the carrierdoped BaFe_{2}As_{2} is more closely tied to the magnetic QCP, while the unusual quantum criticality associated with the structural transition deserves further investigation.
Results
Electrical resistivity measurements
Figure 1 shows the inplane electrical resistivity data in BaFe_{2−x}Ni_{x}As_{2} for various x, which are fitted by the equation ρ_{ab}=ρ_{0}+AT^{n} (for data over the whole temperature range, see Supplementary Fig. S1 and Supplementary Note 1). For a conventional metal described by the Fermi liquid theory, the exponent n=2 is expected. However, we find n<1.5 for 0.09≤x≤0.14. This is a notable feature of nonFermi liquid behaviour. Most remarkably, a Tlinear behaviour (n=1) is observed for x_{c1}=0.10 and persists up to T=100 K. Another minimal n=1.1 is found for x_{c2}=0.14, which is in good agreement with a previous transport measurement^{15}. An equally interesting feature is that both the residual resistivity ρ_{0} and the coefficient A show a maximum at x_{c1} and x_{c2} as seen in Fig. 2a. The evolution of the exponent n with Ni content is shown in Fig. 2b.
NMR measurements
We use NMR to investigate the nature of x_{c1} and x_{c2}. We identify x_{c1} to be a SDW QCP and suggest that x_{c2} is a tetragonaltoorthorhombic structural phase transition QCP. Figure 3a,b display the frequencyswept spectra of ^{75}As NMR for x=0.05 and 0.07, respectively. The very narrow central transition peak above the antiferromagnetic transition temperature T_{N} testifies a good quality of the samples. When a magnetic order sets in, the spectra will split into two pairs as labelled in Fig. 3a, due to the development of an internal magnetic field H_{int}. For x=0.05, we observed two split broad peaks below T_{N}=74 K. This is due to a distribution of H_{int}, which results in a broadening of each pair of the peaks. Upon further doping, H_{int} is reduced and its distribution becomes larger, so that only one broad peak can be seen below T_{N}=48 K for x=0.07. Similar broadening of the spectra was also observed previously in a very lightlydoped sample BaFe_{1.934}Ni_{0.066}As_{2} (ref. 16). For both x, the spectra can be reproduced by assuming a Gaussian distribution of H_{int} as shown by the red curves. We obtain the averaged internal field H_{int}=0.75 T at T=15 K for x=0.05, and H_{int}=0.39 T at T=25 K for x=0.07. By using a hyperfine coupling constant of 1.88 T per μ_{B} obtained in the undoped parent compound^{17}, the averaged ordered magnetic moment S is deduced. As seen in Fig. 4, the ordered magnetic moment develops continuously below T_{N}, being consistent with the secondorder nature of the phase transition. It saturates to 0.43 μ_{B} for x=0.05, and 0.24 μ_{B} for x=0.07, respectively. The ordered moment is smaller than that in the holedoped Ba_{1−x}K_{x}Fe_{2}As_{2} (ref. 18), which is probably due to the fact that Ni goes directly into the Fe site and is more effective in suppressing the magnetic order. Upon further doping, at x=0.09 and x=0.10, however, no antiferromagnetic transition was found, as demonstrated in Fig. 3c, which shows no splitting or broadening ascribable to a magnetic ordering.
The onset of the magnetic order in the underdoped regime is also clearly seen in the spinlattice relaxation. Figure 3d shows 1/T_{1} for x=0.07, which was measured at the position indicated by the arrow in Fig. 3b, in order to avoid any influence from possible remnant paramagnetic phase, if any. As seen in Fig. 3d, a clear peak is found at T_{N}=48 K due to a critical slowing down of the magnetic moments. Below T_{N}, 1/T_{1} decreases down to T_{c}. Most remarkably, 1/T_{1} shows a further rapid decrease below T_{c}, exhibiting a T^{3} behaviour down to T=5 K. Such a significant decrease of 1/T_{1} just below T_{c} is due to the superconducting gap opening in the antiferromagnetically ordered state. This is clear and direct evidence for a microscopic coexistence of superconductivity and antiferromagnetism, as the nuclei being measured experience an internal magnetic field yet the relaxation rate is suppressed rapidly below T_{c}. For x=0.10, as mentioned already, no antiferromagnetic transition was found. Namely T_{N}=0. This is consistent with the extrapolation of the T_{N} versus x relation that gives a critical point that coincides with x_{c1}=0.10, which is further supported by the spin dynamics as elaborated below.
Figure 5a shows the quantity 1/T_{1}T for 0.05≤x≤0.14. The 1/T_{1}T decreases with increasing T down to around 150 K, but starts to increase towards T_{N} or T_{c}. The increase at low T is due to the antiferromagnetic spin fluctuation , and the decrease at high T is due to an intraband effect . Namely, . A similar behaviour was also seen in Ba(Fe_{1−x}Co_{x})_{2}As_{2} (ref. 19). We analyse the (1/T_{1}T)_{AF} part by the selfconsistent renormalization theory for a 2D itinerant electron system near a QCP^{11}, which predicts that 1/T_{1}T is proportional to the staggered magnetic susceptibility χ″(Q). As χ″(Q) follows a Curie–Weiss law^{11}, one has . The intraband contribution is due to the density of state at the Fermi level, which is related to the spin Knight shift (K_{s}) through the Korringa relation =constant^{20}. The Knight shift was found to follow a Tdependence of K=K_{0}+K_{s}exp(−E_{g}/k_{B}T) as seen in Fig. 5b, where K_{0} is Tindependent, while the second is due to the band that sinks below the Fermi level^{21,22}. Correspondingly, we can write . The resulting θ is plotted in Fig. 2b. Note that θ is almost zero at x=0.10, which yields a constant 1/T_{1} above T_{c} as seen in Fig. 6. The result of θ=0 means that the staggered magnetic susceptibility diverges at T=0, indicating that x=0.10 is a magnetic QCP. Therefore, the exponent n=1 in the resistivity is due to the magnetic QCP (Generally speaking, scatterings due to the magnetic hot spots give a Tlinear resistivity^{11}, but those by other parts of the Fermi surface will not^{23,24}. In real materials, however, there always exist some extent of impurity scatterings that can connect the magnetic hot spots and other parts of the Fermi surface as to restore the n=1 behaviour at a magnetic QCP).
Next, we use ^{75}As NMR to study the structural phase transition. A tetragonaltoorthorhombic structural transition was found in the parent compound^{25}, but no direct evidence for such structural transition was obtained in the doped BaFe_{2−x}Ni_{x}As_{2} thus far. In BaFe_{2−x}Co_{x}As_{2}, a structural transition was detected in the lowdoping region^{26}, but it is unclear how the transition temperature T_{s} would evolve as doping level increases. The ^{75}As nucleus has a nuclear quadrupole moment that couples to the electric fieldgradient (EFG) V_{xx} (α=x,y,z). Therefore, the ^{75}AsNMR spectrum is sensitive to a structural phase transition, as below T_{s} the EFG will change appreciably. Such change was indeed confirmed in the parent compounds BaFe_{2}As_{2} (ref. 17) and LaFeAsO (ref. 27). When a magnetic field H_{0} is applied in the ab plane, the NMR resonance frequency f is expressed by
where m=3/2, 1/2 and −1/2, φ is the angle between H_{0} and the a axis, ν_{Q} is the nuclear quadrupole resonance frequency, which is proportional to the EFG, and . For a tetragonal crystal structure, η=0. For an orthorhombic structure, however, the a axis and b axis are not identical, which results in an asymmetric EFG so that η>0. Therefore, for a twined single crystal, the field configurations of H_{0} a axis (φ=0°) and H_{0} b axis (φ=90°) will give a different f_{m↔m−1}(φ,η), leading to a splitting of a pair of the satellite peaks into two. The above argument also applies to the case of electronic nematic phase transition such as orbital ordering, as the EFG is also sensitive to a change in the occupation of the onsite electronic orbits.
As shown in Fig. 7a,b, only one pair of satellite peaks is observed at high temperature for both x=0.05 and 0.07. Below certain temperature, T_{s}, however, we observed a splitting of the satellite peaks. This is strong microscopic evidence for a structural transition occurring in the underdoped samples, where the NMR spectra split owing to the formation of the twinned orthorhombic domains. In fact, each satellite peak can be well reproduced by assuming two split peaks. The obtained T_{s}=90 K for x=0.05 and T_{s}=70 K for x=0.07 agrees well with the resistivity data where an upturn at T_{s} is observed, see Fig. 8. This feature in the spectra persists to the higher dopings x=0.10 and 0.12. For the optimal doping x=0.10, the spectrum is nearly unchanged for 40 K≤T≤60 K, but suddenly changes at 20 K. This indicates that a structural transition takes place below T_{s}=~40 K for this doping composition. For x=0.12, the two satellite peaks do not change at high temperatures, but shift to the opposite direction below T=10 K. Each broadened peak at T=6 K can be well fitted by a superposition of two peaks. This is the first observation that a structural phase transition takes place below T_{c}. For x=0.14, however, no broadening of the spectra was found down to T=4.5 K; the spectrum shift to the same direction is due to a reduction of the Knight shift in the superconducting state. This indicates that T_{s}=0 around x_{c2}=0.14. The T_{s} results obtained by NMR are summarized in the phase diagram as shown in Fig. 2b.
Therefore, the minimal n=1.1 of the resistivity exponent around x_{c2} implies quantum critical fluctuations associated with the structural QCP. We hope that our work will stimulate more experimental measurements such as elastic constant, which is also sensitive to such critical fluctuations^{28}. It should be emphasized here that the exponent n=1.1 is smaller than n=4/3 expected for the fluctuations dominated by q=(0,0) (ref. 12). Our result suggests that the fluctuation associated with x_{c2} is local (namely, all wave vectors contribute to the quasiparticles scattering), which would lead to an n=1 (ref. 29).
Discussion
The mechanism for the superconductivity in ironpnictides has been discussed in relation to magnetic fluctuations^{30,31,32} as well as structural/orbital fluctuations^{33}. In this context, it is worthwhile noting that T_{c} is the highest around x_{c1}. This suggests that the superconductivity in BaFe_{2−x}Ni_{x}As_{2} is more closely related to the antiferromagnetic QCP rather than the structural QCP. Also, previous results suggestive of a magnetic QCP was reported for BaFe_{2}(As_{1−x}P_{x})_{2} (refs 34,35), but isovalent P substitution for As does not add carriers there. In the present case, Ni directly donates electrons into the system (chemical doping), so the tuning parameter is totally different here. Thus, our work demonstrates that BaFe_{2−x}Ni_{x}As_{2} is a new material that provides a unique opportunity to study the issues of quantum criticality. In particular, our result suggests that x_{c2} is a new type of QCP at which the exponent n cannot be explained by existing theories. Although the importance of the tetragonaltoorthorhombic structural transition, which is often associated with electronic nematicity^{36,37}, has been pointed out in the pnictides^{33,38,39}, the physics of a QCP associated with it is a much lessexplored frontier, which deserves more investigations in the future.
Methods
Sample preparation and characterization
The single crystal samples of BaFe_{2−x}Ni_{x}As_{2} used for the measurements were grown by the selfflux method^{40}. Here the Ni content x was determined by energydispersive xray spectroscopy. The T_{c} was determined by DC susceptibility measured by a superconducting quantum interference device with the applied field 50 Oe parallel to the ab plane. The T_{c} is 3.5, 14, 18.5, 18.2, 16.8 and 13.1 K for x=0.05, 0.07, 0.09, 0.10, 0.12 and 0.14, respectively.
The resistivity measurements
Resistivity measurements were performed in quantum design physical properties measurement system by a standard dc fourprobe method. Here we have used the same single crystals used in the NMR measurements. The electrical resistance was measured upon both warming and cooling processes in order to ensure no temperature effect from the electrodes on the samples. Both the warming and the cooling speed is 2 K min^{−1}.
Measurements of NMR spectra and T _{1}
The NMR spectra were obtained by integrating the spin echo as a function of the RF frequency at a constant external magnetic field H_{0}=11.998 T. The nucleus ^{75}As has a nuclear spin I=3/2 and the nuclear spin Hamiltonian can be expressed as
where γ/2π=7.9219 MHz T^{−1} is the gyromagnetic ratio of ^{75}As, h is Planck constant, H_{int} is the internal magnetic field at the As nuclear spin site resulting from the hyperfine coupling to the neighbouring Fe electrons. The T_{1} was determined by using the saturationrecovery method and the nuclear magnetization is fitted to , where M(t) is the nuclear magnetization at time t after the saturation pulse^{41}. The curve is fitted very well and T_{1} is of single component.
Simulation of the spectra
In order to reproduce the spectra, we assume that the distribution of the internal magnetic field is Gaussian as,
where I(f) is the intensity of the spectra, H_{σ} is the distribution of the internal magnetic field H_{int}, and ν_{c} is the nuclear quadrupole resonance frequency tensor along the c axis. Then the spectrum is fitted by convoluting equation 3 with a Gaussian broadening function that already exists above T_{N}. For n=0 (central transition), A_{0}=1 and is the fullwidth at half maximum (FWHM) of the central peak above T_{N}. For n=±1 (satellites), A_{±1}=3/4 and is the FWHM of satellite lines above T_{N}. The red solid curve in Fig. 3a is a fit with H_{σ}=0.37 T, =0.03 MHz and =0.38 MHz, which was taken from the spectrum at T=100 K. The red solid curve in Fig. 3b is a fit with =0.04 MHz and =0.5 MHz, which was taken from the spectrum at T=50 K.
Additional information
How to cite this article: Zhou, R. et al. Quantum criticality in electrondoped BaFe_{2−x}Ni_{x}As_{2}. Nat. Commun. 4:2265 doi: 10.1038/ncomms3265 (2013).
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Acknowledgements
We thank Q. Si, K. Miyake, K. Kuroki, T. Takimoto, R. Fernandes, T. Xiang, Y. Yanase and S. Fujimoto for helpful discussion. This work was partially supported by National Basic Research Program of China (973 Program), Nos. 2011CBA00100 and 2011CBA00109, and by CAS.
Author information
Affiliations
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
 R. Zhou
 , Z. Li
 , J. Yang
 & Guoqing Zheng
Max Planck InstituteHeisenbergstrasse 1, D70569 Stuttgart, Germany
 D. L. Sun
 & C. T. Lin
Department of Physics, Okayama University, Okayama 7008530, Japan
 Guoqing Zheng
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Contributions
The single crystals were grown by D.L.S. and C.T.L. The NMR measurements were performed by R.Z., Z.L., J.Y. and G.q.Z. The electrical resistivity was measured by R.Z., Z.L. and J.Y. G.q.Z. coordinated the whole work and wrote the manuscript, which was supplemented by R.Z. All authors have discussed the results and the interpretation.
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The authors declare no competing financial interests.
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Correspondence to Guoqing Zheng.
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