Introduction

In cuprate high temperature superconductors1 and heavy fermion2 compounds, the superconductivity is accompanied by the normal-state 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 condensed-matter physics in the last decades. In iron–pnictide high-temperature superconductors5,6, searching for magnetic fluctuations has also become an important subject7. Quantum critical fluctuations of order parameters take place not only in spatial domain, but also in imaginary time domain8,9. The correlation time τ0 and correlation length ξ are scaled to each other, through a dynamical exponent z, τ0ξz. Several physical quantities, such as the electrical resistivity and spin-lattice relaxation rate (1/T1), can be used to probe the quantum critical phenomena. For the quasiparticles scattering dominated by the quantum critical fluctuations, the resistivity scales as ρTn. For a two-dimensional (2D) antiferromagnetic spin-density wave (SDW) QCP, the exponent n=1 is often observed1,2,10. On the other hand, for a 2D order with q=(0,0) where z=3, n=4/3 at the QCP11,12. Around a QCP, 1/T1 also shows a characteristic T-scaling11.

BaFe2−xNixAs2 is an electron-doped system13 where every Ni donates two electrons in contrast to Co doping that contributes only one electron14. 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 xc1=0.10 and xc2=0.14, respectively. By NMR measurements, we identify xc1 to be a magnetic QCP and suggest that xc2 is a QCP associated with the tetragonal-to-orthorhombic structural phase transition. The highest Tc is found around xc1, which suggests that the superconductivity in the carrier-doped BaFe2As2 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 in-plane electrical resistivity data in BaFe2−xNixAs2 for various x, which are fitted by the equation ρab=ρ0+ATn (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 non-Fermi liquid behaviour. Most remarkably, a T-linear behaviour (n=1) is observed for xc1=0.10 and persists up to T=100 K. Another minimal n=1.1 is found for xc2=0.14, which is in good agreement with a previous transport measurement15. An equally interesting feature is that both the residual resistivity ρ0 and the coefficient A show a maximum at xc1 and xc2 as seen in Fig. 2a. The evolution of the exponent n with Ni content is shown in Fig. 2b.

Figure 1: The in-plane electrical resistivity.
figure 1

The in-plane electrical resistivity ρab for BaFe2−xNixAs2. The data in the normal state are fitted by the equation ρab=ρ0+ATn (red curve).

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Figure 2: The obtained phase diagram of BaFe2−xNixAs2.
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(a) The residual resistivity ρ0 and the coefficient A as a function of Ni-doping x. (b) The obtained phase diagram of BaFe2−xNixAs2. The TN and θ are obtained from NMR spectra and 1/T1T, respectively. AF and SC denote the antiferromagnetically ordered and the superconducting states, respectively. The solid and dashed lines are the guides to the eyes. Colours in the normal state represent the evolution of the exponent n in the resistivity fitted by ρab=ρ0+ATn. The light yellow region at x=0.10 shows that the resistivity is T-linear. The error bar for θ is the s.d. in the fitting of 1/T1T. The error bar for Ts represents the temperature interval in measuring the NMR spectra.

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NMR measurements

We use NMR to investigate the nature of xc1 and xc2. We identify xc1 to be a SDW QCP and suggest that xc2 is a tetragonal-to-orthorhombic structural phase transition QCP. Figure 3a,b display the frequency-swept spectra of 75As NMR for x=0.05 and 0.07, respectively. The very narrow central transition peak above the antiferromagnetic transition temperature TN 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 Hint. For x=0.05, we observed two split broad peaks below TN=74 K. This is due to a distribution of Hint, which results in a broadening of each pair of the peaks. Upon further doping, Hint is reduced and its distribution becomes larger, so that only one broad peak can be seen below TN=48 K for x=0.07. Similar broadening of the spectra was also observed previously in a very lightly-doped sample BaFe1.934Ni0.066As2 (ref. 16). For both x, the spectra can be reproduced by assuming a Gaussian distribution of Hint as shown by the red curves. We obtain the averaged internal field Hint=0.75 T at T=15 K for x=0.05, and Hint=0.39 T at T=25 K for x=0.07. By using a hyperfine coupling constant of 1.88 T per μB obtained in the undoped parent compound17, the averaged ordered magnetic moment S is deduced. As seen in Fig. 4, the ordered magnetic moment develops continuously below TN, being consistent with the second-order nature of the phase transition. It saturates to 0.43 μB for x=0.05, and 0.24 μB for x=0.07, respectively. The ordered moment is smaller than that in the hole-doped Ba1−xKxFe2As2 (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.

Figure 3: NMR spectra with doping and temperature dependence of 1/T1 at optimal doping.
figure 3

75As-NMR spectra obtained by sweeping the frequency at a fixed external field H0=11.998 T applied along the c axis. (a) The spectra above and below TN=74 K for x=0.05. Solid, dashed and dotted lines indicate, respectively, the position of the central transition, left- and right-satellites, when an internal magnetic field develops. (b) The spectra above and below TN=48 K for x=0.07. The red curve represents the simulations by assuming a Gaussian distribution of the internal magnetic field. The black arrow indicates the position at which T1 is measured below TN. (c) The central transition line at and below Tc for x=0.10. The spectrum shift at T=1.89 K is due to a reduction of the Knight shift in the superconducting state. (d) The temperature dependence of 1/T1 for x=0.07. The solid and dashed lines show the T3- and T-variation, respectively. The solid and dashed arrows indicate Tc and TN, respectively. The error bar in 1/T1 is the s.d. in fitting the nuclear magnetization recovery curve.

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Figure 4: The temperature dependence of the internal magnetic field and the ordered moment for x=0.07.
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Temperature dependence of the averaged internal magnetic field Hint (left vertical axis) and the averaged ordered moment S (right vertical axis) for x=0.07. The solid and dashed arrows indicate Tc and TN, respectively. The error bar represents the absolute maxima (minima) that can fit a spectrum.

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The onset of the magnetic order in the underdoped regime is also clearly seen in the spin-lattice relaxation. Figure 3d shows 1/T1 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 TN=48 K due to a critical slowing down of the magnetic moments. Below TN, 1/T1 decreases down to Tc. Most remarkably, 1/T1 shows a further rapid decrease below Tc, exhibiting a T3 behaviour down to T=5 K. Such a significant decrease of 1/T1 just below Tc 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 Tc. For x=0.10, as mentioned already, no antiferromagnetic transition was found. Namely TN=0. This is consistent with the extrapolation of the TN versus x relation that gives a critical point that coincides with xc1=0.10, which is further supported by the spin dynamics as elaborated below.

Figure 5a shows the quantity 1/T1T for 0.05≤x≤0.14. The 1/T1T decreases with increasing T down to around 150 K, but starts to increase towards TN or Tc. 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(Fe1−xCox)2As2 (ref. 19). We analyse the (1/T1T)AF part by the self-consistent renormalization theory for a 2D itinerant electron system near a QCP11, which predicts that 1/T1T is proportional to the staggered magnetic susceptibility χ″(Q). As χ″(Q) follows a Curie–Weiss law11, one has . The intraband contribution is due to the density of state at the Fermi level, which is related to the spin Knight shift (Ks) through the Korringa relation =constant20. The Knight shift was found to follow a T-dependence of K=K0+Ksexp(−Eg/kBT) as seen in Fig. 5b, where K0 is T-independent, while the second is due to the band that sinks below the Fermi level21,22. 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/T1 above Tc 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 T-linear resistivity11, but those by other parts of the Fermi surface will not23,24. 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).

Figure 5: The quantity 1/T1T and the Knight shift.
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(a) Temperature dependence of 1/T1T and (b) the Knight shift K for various x. The solid line is a fitting of . The dashed curve is a fit to K=K0+Ks × exp(−Eg/kBT). The obtained parameters are Eg/kB=620±40, 510±40, 370±30, 365±30, 350±30 and 330±30 K for x=0.05, 0.07, 0.09, 0.10, 0.12 and 0.14, respectively. The solid and dashed arrows indicate Tc and TN, respectively. The error bar for 1/T1T is the s.d. in fitting the nuclear magnetization recovery curve. The error bar for K was estimated by assuming that the spectrum-peak uncertainty equals the point (frequency) interval in measuring the NMR spectra.

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Figure 6: The 1/T1 for all the superconducting samples.
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The temperature dependence of the spin-lattice relaxation rate 1/T1 for various x of BaFe2−xNixAs2. The solid and dashed arrows indicate Tc and TN, respectively. The error bar is the s.d. in fitting the nuclear magnetization recovery curve.

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Next, we use 75As NMR to study the structural phase transition. A tetragonal-to-orthorhombic structural transition was found in the parent compound25, but no direct evidence for such structural transition was obtained in the doped BaFe2−xNixAs2 thus far. In BaFe2−xCoxAs2, a structural transition was detected in the low-doping region26, but it is unclear how the transition temperature Ts would evolve as doping level increases. The 75As nucleus has a nuclear quadrupole moment that couples to the electric field-gradient (EFG) Vxx (α=x,y,z). Therefore, the 75As-NMR spectrum is sensitive to a structural phase transition, as below Ts the EFG will change appreciably. Such change was indeed confirmed in the parent compounds BaFe2As2 (ref. 17) and LaFeAsO (ref. 27). When a magnetic field H0 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 H0 and the a axis, νQ 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 H0|| a axis (φ=0°) and H0|| b axis (φ=90°) will give a different fm↔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 on-site 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, Ts, 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 Ts=90 K for x=0.05 and Ts=70 K for x=0.07 agrees well with the resistivity data where an upturn at Ts 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 Ts=~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 Tc. 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 Ts=0 around xc2=0.14. The Ts results obtained by NMR are summarized in the phase diagram as shown in Fig. 2b.

Figure 7: The satellite peaks corresponding to the and transitions.
figure 7

(a–d) The satellite peaks with the external field H0 applied along the a axis or b axis above and below Ts for x=0.05, 0.07, 0.10 and 0.12. The peaks split below Ts due to a change in EFG. All the spectra are fitted by a single Gaussian function above Ts, but by two Gaussian functions below Ts. (e) The satellite peaks for x=0.14, which only shift to the same direction due to a reduction of the Knight shift as in Fig. 3c.

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Figure 8: The enlarged part of the electrical resistivity and its derivative for the underdoped samples.
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(a) Enlarged part of the in-plane electrical resistivity ρab (right vertical axis) and its derivative (left vertical axis) for x=0.05. (b) Data for x=0.07. The solid and dashed arrows indicate the structural phase transition temperature Ts and the antiferromagnetic transition temperature TN determined by NMR, respectively.

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Therefore, the minimal n=1.1 of the resistivity exponent around xc2 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 fluctuations28. 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 xc2 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 iron-pnictides has been discussed in relation to magnetic fluctuations30,31,32 as well as structural/orbital fluctuations33. In this context, it is worthwhile noting that Tc is the highest around xc1. This suggests that the superconductivity in BaFe2−xNixAs2 is more closely related to the antiferromagnetic QCP rather than the structural QCP. Also, previous results suggestive of a magnetic QCP was reported for BaFe2(As1−xPx)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 BaFe2−xNixAs2 is a new material that provides a unique opportunity to study the issues of quantum criticality. In particular, our result suggests that xc2 is a new type of QCP at which the exponent n cannot be explained by existing theories. Although the importance of the tetragonal-to-orthorhombic structural transition, which is often associated with electronic nematicity36,37, has been pointed out in the pnictides33,38,39, the physics of a QCP associated with it is a much less-explored frontier, which deserves more investigations in the future.

Methods

Sample preparation and characterization

The single crystal samples of BaFe2−xNixAs2 used for the measurements were grown by the self-flux method40. Here the Ni content x was determined by energy-dispersive x-ray spectroscopy. The Tc was determined by DC susceptibility measured by a superconducting quantum interference device with the applied field 50 Oe parallel to the ab plane. The Tc 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 four-probe 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 T1

The NMR spectra were obtained by integrating the spin echo as a function of the RF frequency at a constant external magnetic field H0=11.998 T. The nucleus 75As 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 75As, h is Planck constant, Hint is the internal magnetic field at the As nuclear spin site resulting from the hyperfine coupling to the neighbouring Fe electrons. The T1 was determined by using the saturation-recovery method and the nuclear magnetization is fitted to , where M(t) is the nuclear magnetization at time t after the saturation pulse41. The curve is fitted very well and T1 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, Hσ is the distribution of the internal magnetic field Hint, and νc 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 TN. For n=0 (central transition), A0=1 and is the full-width at half maximum (FWHM) of the central peak above TN. For n=±1 (satellites), A±1=3/4 and is the FWHM of satellite lines above TN. The red solid curve in Fig. 3a is a fit with Hσ=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 electron-doped BaFe2−xNixAs2. Nat. Commun. 4:2265 doi: 10.1038/ncomms3265 (2013).