{\omega}/T scaling and magnetic quantum criticality in BaFe2(As0.7P0.3)2

TL;DR

This study reveals ω/T scaling and magnetic quantum criticality in optimally doped BaFe2(As0.7P0.3)2, linking strange-metal behavior with a nearby quantum critical point and the weakly first-order transition out of antiferromagnetic order.

Contribution

It demonstrates ω/T scaling in spin susceptibility and resistivity extending up to high temperatures, indicating magnetic quantum criticality in a clean iron pnictide superconductor.

Findings

Resistivity shows linear temperature dependence up to ~500 K.

Spin susceptibility exhibits ω/T scaling over a wide energy range.

Results suggest a magnetic quantum critical point near optimal doping.

Abstract

We used transport and inelastic neutron scattering to study the optimally phosphorus-doped BaFe$_2$(As$_{0.7}$P$_{0.3}$)$_2$ superconductor ($T_c = 30$ K). In the normal state, we find that the previously reported linear temperature dependence of the resistivity below room temperature extends to $\sim$ 500 K. Our analysis of the temperature and energy ($E=\hbar\omega$) dependence of spin dynamical susceptibility at the antiferromagnetic (AF) ordering wave vector $\chi^{\prime\prime}({\bf Q}_{\rm AF},\omega)$ reveal an $\omega / T$ scaling within $1.1<E/k_BT<110$. These results suggest that the linear temperature dependence of the resistivity is due to the presence of a magnetic quantum critical point in the cleanest iron pnictides near optimal superconductivity. Moreover, the results reconcile the strange-metal temperature dependences with the weakly first-order nature of the quantum transition out of the AF and nematic orders.

Full text

One of the hallmarks of unconventional superconductivity in copper oxide superconductors is the linear temperature dependence of the resistivity in the normal state below about 1000 K palee . First discovered in La2-xSrxCuO4 and YBa2Cu3O7 nearly optimal superconductivity Cava1987 ; Gurvitch ; Takagi ; Ando ; Louis2010 ; Hussey , the linear temperature dependence of the resistivity is incompatible with Landau’s Fermi-liquid theory of metals, where temperature dependence of the resistivity is expected to be quadratic ($T^{2}$) Hilbert2007 , and suggests the presence of a quantum critical point (QCP) responsible for the breakdown of Fermi-liquid behavior and development of strange-metal properties Sachdev1995 ; Qimiao2001 ; Fradkin .

In the case of iron pnictide superconductors Stewart2001 ; dai ; Qimiao2016 , a linear temperature dependence of the resistivity has been found in different families of materials near optimal superconductivity, suggesting the presence of a QCP Chu2009 ; Zhou2013 ; Zhaoyu2016 ; Hosoi2016 ; Kuo2016 ; Fisher2012 . In particular, experimental evidence for a QCP in phosphorus-doped BaFe2(As1-xPx)2, envisioned in early theoretical studies Dai09 , has been mounting SJiang ; Shibauchi2014 . This includes the linear temperature dependence of the resistivity kasa10 , a peak in the effective electron mass Shishido2010 , magnetic penetration depth Hashimoto , heat capacity Walmsley and nuclear magnetic resonance (NMR) Nakai . The phosphorus doping does not involve the Fe-sites. As a result, this series is especially clean, as demonstrated by the relatively small residual resistivity and the observation of quantum oscillations Shishido2010 . Because the minimal disorder will allow for a clear understanding of the implications of the inelastic neutron scattering measurements (see below), here we focus on BaFe2(As1-xPx)2 near its optimal superconductivity.

In the undoped state, BaFe2As2 exhibits a tetragonal-to-orthorhombic structural transition at $T_{s}$, where a nematic phase with in-plane electronic anisotropy is established CFang08 ; CXu ; Dai09 ; Fernandes , followed by a collinear antiferromagnetic (AF) order below $T_{N}\approx 140$ K ($\leq T_{s}$) at ${\bf Q}_{\rm AF}\approx(0.5,0.5)$ [Figs. 1(a), 1(b)] Stewart2001 ; dai . When phosphorus is doped into BaFe2As2 to form BaFe2(As1-xPx)2 SJiang , a QCP is found in BaFe2(As0.7P0.3)2 near optimal superconductivity with suppressed orthorhombic lattice distortion and static AF order [Fig. 1(c)]. Increasing P-doping in BaFe2(As1-xPx)2 suppresses both $T_{s}$ and $T_{N}$, which are associated nematic and AF phase transitions, respectively. If both magnetic and nematic QCPs occur, one would expect to find gradually suppressed second order structural and magnetic phase transitions with increasing P-doping in BaFe2(As1-xPx)2. However, systematic neutron diffraction and NMR experiments on powder and single crystal samples of BaFe2(As1-xPx)2 reveal that the structural and AF phase transitions are coupled at all $x$, and AF phase transition around $x\approx 0.3$ become weakly first order near optimal superconductivity, thus suggesting an avoided magnetic QCP Allred2014 ; DHu2015 ; Dioguardi2016 .

Although the AF phase transition in BaFe2(As1-xPx)2 is weakly first order near optimal superconductivity Allred2014 ; DHu2015 ; Dioguardi2016 , the small ordered moment and low ordering temperature do not exclude the possibility of an extended temperature and energy range where quantum criticality underlies the linear resistivity. This is consistent with the fact that when superconductivity in BaFe2(As0.7P0.3)2 is suppressed by a magnetic field, the temperature dependence of the resistivity is quadratic below the zero field $T_{c}=30$ K, deviating from the linear temperature dependence above 30 K [see inset of Fig. 1(d)] fisher2014 . Figure 1(d) shows our measurement of the resistivity for BaFe2(As0.7P0.3)2 from 2 K to 790 K. In addition to confirming the linear temperature dependence of the resistivity from 30 K to 300 K kasa10 , the data reveal that it extends all the way up to $\sim$ 500 K, above which a clear deviation from the linear behavior is seen; this exemplifies the extended temperature range for the strange-metal behavior. The nematic QCP Kuo2016 alone is unlikely to be responsible for the observed linear temperature dependent resistivity Shibauchi2014 , given that fluctuations at small wavevectors are inefficient in degrading an electrical current. If linear temperature dependence of the resistivity in BaFe2(As0.7P0.3)2 in Fig. 1(d) is associated with a magnetic quantum critical fluctuations, one would expect that spin dynamics at ${\bf Q}_{\rm AF}$ to follow $\omega/T$ scaling within a finite energy ($E=\hbar\omega$, where $\omega$ is frequency) and temperature range stockert2011 ; dai2005 ; dai2006 ; Kim2015 .

To test this hypothesis, we use inelastic neutron scattering to study BaFe2(As0.7P0.3)2 ($T_{c}\approx 30$ K), focusing on temperature and energy dependence of spin fluctuations near ${\bf Q}_{\rm AF}$. In previous inelastic neutron scattering experiments on BaFe2(As0.7P0.3)2 CHLee13 ; ding2016 ; ding2017 , a $c$-axis dispersive neutron spin resonance coupled to superconductivity has been identified. By using neutron triple-axis and time-of-flight spectroscopy, we find that energy and temperature dependence of the imaginary part of dynamic susceptibility at ${\bf Q}_{\rm AF}$ $\chi^{\prime\prime}({\bf Q}_{\rm AF},\omega)$, which is related to magnetic scattering $S({\bf Q}_{\rm AF},\omega)$ via $\chi^{\prime\prime}({\bf Q}_{\rm AF},\omega)\propto(1-e^{-\hbar\omega/k_{B}T})S({\bf Q}_{\rm AF},\omega)$, follows the $\omega/T$ scaling for $9<E<61$ meV and 5 K$<T<200$ K. For energies less than about $E=10$ meV, the $\omega/T$ scaling fails to describe the data. These results suggest that the observed linear temperature dependence of the resistivity in Fig. 1(d) may arise from magnetic quantum critical fluctuations important for controlling the transport and nematic properties of BaFe2(As0.7P0.3)2.

We used the standard four-probe method to measure the resistivity of BaFe2(As0.3P0.7)2 from 10 K to 790 K in a Janis 4 K closed cycle refrigerator with high temperature capability. As we can see from Fig. 1(d), the remarkable linear temperature dependence of the resistivity extends up to 500 K. Combined with previous transport measurements below 30 K when superconductivity is suppressed by a high magnetic field [see inset of Fig1(d)] fisher2014 , we find that the temperature range for linear resistivity is from 30 K to 500 K.

Our inelastic neutron scattering experiments were carried out on the Wide Angular-Range Chopper Spectrometer (ARCS) at the Spallation Neutron Source and HB-3 triple axis spectrometer at the High-Flux Isotope Reactor, both at Oak Ridge National Laboratory. For the time-of-flight measurement on ARCS, we used $E_{i}=80$ meV with $k_{i}$ parallel to the $c$ axis. Total mass 17 g high-quality BaFe2(As0.3P0.7)2 single crystals were co-aligned in the $[H,H,L]$ scattering plane with an in-plane mosaic $<$ 5.5∘ ding2017 . We define $Q=(H,K,L)=\frac{2\pi}{a}H\hat{i}+\frac{2\pi}{a}K\hat{j}+\frac{2\pi}{a}L\hat{k}$, where the tetragonal lattice constants are $a=b\approx 3.96\AA$ and $c\approx 12.87\AA$.

Figures 2(a)-(d) show images of spin excitations $S({\bf Q},\omega)$ at $E=12\pm 3$, $22\pm 3$, $42\pm 3$, $52\pm 3$ meV, respectively, at $T=120$ K. Consistent with earlier work ding2016 , spin excitations form transversely elongated ellipses centered around ${\bf Q}_{\rm AF}$ that disperse outward with increasing energy. We convert $S({\bf Q},\omega)$ to $\chi^{\prime\prime}({\bf Q},\omega)$ and fit the in-plane dynamical susceptibility with two-dimensional Gaussian function to get the absolute intensity of $\chi^{\prime\prime}({\bf Q},\omega)$ at ${\bf Q}={\bf Q}_{\rm AF}=(0.5,0.5)$. Figures 2(e) and 2(f) show energy dependence of $\chi^{\prime\prime}({\bf Q}_{\rm AF},\omega)$ at $T=32$ and 120 K, respectively. The solid lines in the figures are fits to the data with typical paramagnetic relaxational form $\chi^{\prime\prime}({\bf Q}_{\rm AF},\omega)=I\frac{\Gamma*E}{\Gamma^{2}+E^{2}}$, where $\Gamma$ is related to full width at half maximum of the excitations, corresponding to the relaxation lifetime of the excitations, and $I$ is the peak intensity of the excitations. Figures 2(g) and 2(h) show temperature dependence of $\Gamma$ and $I$, respectively. While $\Gamma$ increases approximately linearly with increasing temperature, $I$ decreases with increasing temperature.

To test if the measured imaginary part of the dynamic susceptibility at ${\bf Q}_{\rm AF}$ follows the $\omega/T$ scaling expected for magnetic quantum critical fluctuations, we consider $\chi^{\prime\prime}({\bf Q}_{\rm AF},\omega)T^{\alpha}=F({\bf Q}_{\rm AF},\omega/T)$, where the scaling exponent $\alpha$ and the scaling function $F({\bf Q}_{\rm AF},\omega/T)$ are determined through the best-observed collapse of the data onto one universal curve dai2006 . We exclude the data under $E=20$ meV below 30 K to eliminate the impact of superconductivity on $\chi^{\prime\prime}({\bf Q}_{\rm AF},\omega)$ due to the appearance of the neutron spin resonance in superconducting state CHLee13 ; ding2016 . By fitting the data with this function, we find $\alpha=0.605$ independent of the functional form of $F({\bf Q}_{\rm AF},\omega/T)$ [Fig. 4 (a),(b)]. For $E/T$ from 1.1 to about 110, the data collapse into a single curve. From neutron time-of-flight measurements on BaFe2(As0.7P0.3)2, we know that spin excitations disperse transversely away from ${\bf Q}_{\rm AF}$ for energies above 100 meV. Figures 4(a) and 4(b) suggest that spin excitations up to 100 meV follow $\omega/T$ scaling. However, low-energy spin excitations ($E/T<1.1$) seem to deviate from the scaling curve, consistent with transport measurements where resistivity deviates from linear temperature dependence below 30 K.

To further explore if the low-energy spin fluctuations follow $\omega/T$ scaling, we carried out inelastic neutron scattering measurements using HB-3 triple axis spectrometer. Figure 3 shows $\chi^{\prime\prime}({\bf Q},\omega)$ for $E=4,6,10$ meV at various temperatures. The spin fluctuations in these energies are fitted to a Gaussian function and give the dynamical susceptibility at ${\bf Q}_{\rm AF}$. The spin fluctuations at $E=4,6,10$ meV are dramatically suppressed with increasing temperature, but still present even at 300 K [Figs. 3(a)-(c)]. By combining results from constant-energy and constant-${\bf Q}$ scans, we deduce the data using the same parameters as in Figs. 4(a) and 4(b) and the outcome is shown in Figs. 4(c) and 4(d) with $\alpha=0.605$. Clearly, the $\omega/T$ scaling fails for low-energy spin fluctuations with energy approximately below 10 meV.

We have shown that the energy and temperature dependent of $\chi^{\prime\prime}({\bf Q}_{\rm AF},\omega)$ in BaFe2(As0.7P0.3)2 follows $\omega/T$ scaling within an extended temperature and energy range. To appreciate the importance of this material being clean, we recall that, in the case of doped heavy-Fermion materials such as UCu5-xPdx, the observed $\omega/T$ and associated non-Fermi liquid behavior such as linear temperature dependence of the resistivity have been attributed to either quantum criticality of long-range magnetic order or localized moments in a disordered setting Hilbert2007 ; MacL2004 ; Miranda1996 ; Bernal1995 ; Aronson95 . In Cu-doped Ba(Fe1-xCux)2As2, the AF order found in parent compound broadens with increasing Cu-substitution, suggesting the occurrence of a spin-glass state WYWang2017 . The $\omega/T$ scaling found in Ba(Fe1-xCux)2As2 can be understood by the theoretical models for the AF QCP or spin-glass QCP Kim2015 . Recently, the existence of an AF QCP was suggested in Ba(Fe0.97Cr0.03)2(As${}_{1-x}P_{x}$)2 at $x=0.42$ based on $\omega/T$ scaling shiliang2018 . However, Cr doping in BaFe2(As1-xPx)2 induces significant impurity scattering at the Fe position. It is therefore unclear if the observed non-Fermi liquid behavior and $\omega/T$ scaling at $x$ = 0.42, which is clearly different from the behavior at $x=0.3$ without Cr doping, are due to impurity scatterings from disordered moments of Cr spins. By contrast, BaFe2(As1-xPx)2 has negligible disorder Shishido2010 given that the P-substitution does not go into the Fe-plane, and this is consistent with the instrumental resolution limited magnetic Bragg peaks up to $x=0.3$ DHu2015 . Therefore, the observed $\omega/T$ scaling we have observed over an extended temperature and energy range in the $x=0.3$ compound must be attributed to quantum spin fluctuations related to the vanishing weakly first order AF phase transition. The violation of the $\omega/T$ scaling observed below 10 meV adds further support to this interpretation.

Quantum criticality associated with a weakly first-order but concurrent AF and nematic quantum phase transitions were anticipated theoretically for the P-substituted iron arsenides Dai09 ; Wu16 . P-for-As substitution effectively decreases the strength of electron correlations by enhancing the kinetic energy, thereby reducing the magnetic and the concomitant nematic order. This effect can be analyzed by a field theory containing both the magnetic and nematic degrees of freedom. Expressed in terms of ${\vec{m}}_{A/B}$, the staggered magnetization of the spins of the $3d$-electrons on the $A$ and $B$ sublattices of the Fe-square lattice, the $(1/2,1/2)$/$(1/2,-1/2)$ AF degree of freedom is $({\vec{m}}_{A}\pm{\vec{m}}_{B})/2$ while the nematic one is ${\vec{m}}_{A}\cdot{\vec{m}}_{B}$. The effective theory contains an interaction term, $-u_{I}({\vec{m}}_{A}\cdot{\vec{m}}_{B})^{2}$, with the coupling $u_{I}>0$. In the renormalization group sense, this interaction term is relevant with respect to an underlying magnetic QCP, but only marginally so. Consequently, it drives both the AF and nematic quantum phase transitions to be concurrent and weakly first order, resulting in a large dynamical range of quantum criticality Dai09 . The same conclusion follows from a large-$N$ saddle-point calculation Wu16 .

In summary, we have systematically measured the temperature and energy evolution of spin fluctuations in the optimal doped BaFe2(As0.7P0.3)2. We find evidence for the $\omega/T$ scaling in spin fluctuations over extended temperature and energy range, consistent with linear temperature dependence of the resistivity. These results provided strong evidence that the linear temperature dependence of resistivity arises BaFe2(As0.7P0.3)2, which is located near optimal superconductivity, from magnetic quantum criticality. Therefore, the presence of a magnetic QCP may ultimately be responsible for the anomalous transport and magnetic properties of the iron pnictides and strongly influence their superconductivity.

The neutron scattering work at Rice is supported by the U.S. NSF-DMR-1700081 (P.D.). A part of the material synthesis work at Rice is supported by the Robert A. Welch Foundation Grant No. C-1839 (P.D.). The theoretical work at Rice is supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DESC0018197, and the Robert A.Welch Foundation Grant No. C-1411. Q.S. acknowledges the hospitality and the support by a Ulam Scholarship of the Center for Nonlinear Studies at Los Alamos National Laboratory. The transport work at IOP is supported by the Ministry of Science and Technology of China (No. 2017YFA0302903, No. 2016YFA0300502) and the National Natural Science Foundation of China (No. 11674406). This research used resources at the High Flux Isotope Reactor and Spallation Neutron Source, DOE Office of Science User Facilities operated by the Oak Ridge National Laboratory.