Abstract
Nematic fluctuations and order play a prominent role in material classes such as the cuprates^{1}, some ruthenates^{2} or the ironbased compounds^{3,4,5,6} and may be interrelated with superconductivity^{7,8,9,10,11}. In ironbased compounds^{12} signatures of nematicity have been observed in a variety of experiments. However, the fundamental question as to the relevance of the related spin^{13}, charge^{9,14} or orbital^{8,15,16} fluctuations remains open. Here, we use inelastic light (Raman) scattering and study Ba(Fe_{1−x}Co_{x})_{2}As_{2} (0 ≤ x ≤ 0.085) for getting direct access to nematicity and the underlying critical fluctuations with finite characteristic wavelengths^{17,18,19,20,21}. We show that the response from fluctuations appears only in B_{1g} (x^{2} − y^{2}) symmetry (1 Fe unit cell). The scattering amplitude increases towards the structural transition at T_{s} but vanishes only below the magnetic ordering transition at T_{SDW} < T_{s}, suggesting a magnetic origin of the fluctuations. The theoretical analysis explains the selection rules and the temperature dependence of the fluctuation response. These results make magnetism the favourite candidate for driving the series of transitions.
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Main
The magnetostructural phase transition is among the most thoroughly studied phenomena in ironbased materials. When Fe is substituted by Co in BaFe_{2}As_{2} the structural transformation at T_{s} precedes the magnetic ordering at T_{SDW} < T_{s} (ref. 22). The nematic phase between T_{s} and T_{SDW} is characterized by broken C_{4} symmetry but preserved O(3) spin rotational symmetry (no magnetic order). Nematic fluctuations are present even above T_{s} in the tetragonal phase, as has been demonstrated by both elastoresistance measurements^{4,6} and studies of the elastic constants^{23,24}. In strained samples, one observes orbital ordering in the photoemission spectra^{25} and electronic nematicity by transport^{4,26}. However, it is rather difficult to derive the dynamics and critical momentum typical for the underlying fluctuations and to identify which of the ordering phenomena drives the instabilities.
Raman scattering provides experimental access to all types of dynamic nematicity although only the charge sector has been studied in more detail^{9,14,27,28}. However, also in the case of spindriven nematic order the technique can play a prominent role for coupling to a twospin operator whereas a fourspin correlation function is the lowest order contribution to the neutron crosssection^{5}. We exploit this advantage here and study the lowenergy Raman response of Ba(Fe_{1−x}Co_{x})_{2}As_{2} experimentally and interpret the results in terms of a microscopic model for a spindriven nematic phase. In addition to earlier work^{9,14,20} we study the temperature dependence of the response in the crucial range between T_{s} and T_{SDW} in great detail and address the spectral shape and the selection rules enabling us to explain the structural and magnetic transitions in a unified microscopic picture.
We study Ba(Fe_{1−x}Co_{x})_{2}As_{2} single crystals, having x = 0, x = 0.025 and x = 0.051 (see Methods), as a function of photon polarization in the temperature range 4.2 < T ≤ 300 K. For the symmetry assignment we use the 1 Fe unit cell in which the fluctuations appear in B_{1g} symmetry. We use the appearance of twin boundaries and of the As A_{1g} (x^{2} + y^{2}) phonon line as internal thermometers for the structural and the magnetic phase transitions, respectively. In this way, T_{s} and T_{SDW} can be determined with a precision of typically ±0.2 and ±1 K, respectively.
Figure 1 shows the Raman response Rχ′′(Ω, T) for Ba(Fe_{0.975}Co_{0.025})_{2}As_{2} for various temperatures in A_{1g} and B_{1g} (1 Fe per unit cell) symmetry. B_{2g} spectra were measured only at a few temperatures and found to be nearly temperature independent in agreement with previous data^{14}. Results for other doping levels x are shown in Supplementary Figs 2 and 3. The spectra comprise a superposition of several types of excitation including narrow phonon lines and slowly varying continua arising from electron–hole (e–h) pairs; hence, the continuum reflects the dynamical twoparticle behaviour. The A_{1g} and B_{1g} spectra predominantly weigh out contributions from the central hole bands and the electron bands, respectively^{29}. The symmetrydependent initial slope τ_{0, μ}(T) (μ = A_{1g}, B_{1g}, B_{2g}) (see Fig. 1a, c) can be compared to transport data. [τ_{0, μ}(T)]^{−1} corresponds to the static transport relaxation rate Γ_{0, μ}(T) of the conduction electrons. The memory function method facilitates the quantitative determination of the dynamic relaxation Γ(Ω, T) in absolute energy units. The static limit can be obtained by extrapolation, Γ_{0, μ}(T) = Γ_{μ}(Ω → 0, T) (see Methods and Supplementary Information III). In Fig. 1d we show the result for x = 0.025 corresponding to the spectra of Fig. 1a–c. The results for all doping levels studied are compiled in Supplementary Fig. 4 and compared to the scattering rates derived from the resistivities^{22}.
Figure 1d shows one of the central results: above approximately 200 K Γ_{0, μ}(T) varies slowly and similarly in both symmetries. The more rapid decrease of Γ_{0, B1g}(T) below 200 K is accompanied by a strong intensity gain in the range 20–200 cm^{−1} (see Fig. 1a) as observed before in similar samples^{9,14,28}. The intensity gain indicates that there is an additional contribution superposed on the e–h continuum that, as will be shown below, arises from fluctuations. Therefore, the kink in Γ_{0, B1g}(T) is labelled T_{f} and marks the crossover temperature below which nematic fluctuations can be observed by Raman scattering. At least for low doping, T_{f} is relatively well defined. The kink allows us to separate the two regimes of the lowenergy response above and below T_{f} as being dominated by carrier excitations and fluctuations, respectively.
The additional B_{1g} signal below T_{f} has to be treated in a way different from that in A_{1g} symmetry and in B_{1g} above T_{f}. As it is rather strong it can be separated out with little uncertainty by subtracting the e–h continuum. We approximate the continuum at T_{f} by an analytic function that is then determined for each temperature according to the variation of the resistivity and the A_{1g} spectra and subtracted from all spectra at lower temperatures. The details are explained in Supplementary Information VII. The results of the subtraction procedure are shown in Fig. 2. The response increases rapidly towards T_{s} without however diverging, and the maximum moves to lower energies.
As a surprise, the fluctuations do not disappear directly below T_{s} (Fig. 2b) as one would expect if longranged order would be established. Rather, the intensity decreases continuously and the maximum stays approximately pinned implying that the correlation length does not change substantially between the two transitions at T_{s} = 102.8 ± 0.2 K and T_{SDW} = 98 ± 1 K. The persistence of the fluctuations down to T_{SDW} strongly favours their magnetic origin.
We first compare the data to the theoretical model for thermally driven spin fluctuations associated with the striped magnetic phase ordering along Q_{x} = (π, 0) or Q_{y} = (0, π). In leading order two noninteracting fluctuations carrying momenta Q and −Q are exchanged. Electronic loops (see Fig. 3 and Supplementary Fig. 5) connect the photons and the fluctuations and entail Qdependent selection rules, which were derived along with the spectral response R_{0, μ}(Ω) in ref. 17 and are summarized in Supplementary Information IV. In brief, because the response results from a sum over all electronic momenta close to the Fermi surface cancellation effects may occur if Q connects parts on different Fermi surface sheets having form factors γ_{μ}(k) with opposite sign. For the ordering vectors (π, 0) and (0, π) the resulting selection rules explain the enhancement of the signal in B_{1g} symmetry and its absence in the A_{1g} and B_{2g} channels.
However, the lowestorder diagrams alone can account only for the spectral shape whereas the variation of the intensity around T_{s} remains unexplained. To describe this aspect, we consider the interaction of fluctuations among themselves and with the lattice, all of which becomes crucial in the vicinity of the nematic transition^{19,20}.
The interactions between spin fluctuations can be represented by a series of quaternion paramagnetic couplings mediated by fermions inserted into the leadingorder Aslamazov–Larkin diagrams as shown in Fig. 3. The inserted fermionic boxes effectively resemble the dynamic nematic coupling constant g of the theory.
We have analysed the problem by extending SU(2) → SU(N) and taking the large N limit. For small frequencies Ω and in the largeN limit, after resumming an infinite number of such boxlike Aslamazov–Larkin diagrams, the Raman response function reads:
Equation (1) states that the Raman response is proportional to the electronic contribution to the susceptibility of the nematic order parameter,
χ_{mag}(q) represents the magnetic susceptibility, which diverges at T_{SDW}. For g ≠ 0 χ_{nem}^{el}(0) has a Curielike T − T^{∗}^{−1} divergence at T^{∗} ≥ T_{SDW}.
If the spins (or charges) couple to the lattice the susceptibility of the nematic order parameter is given by^{4,16,20}
where λ_{sl} denotes the magnetoelastic coupling, and c_{0}^{s} is the bare elastic constant. Obviously, χ_{nem}(0) diverges at higher temperature than χ_{nem}^{el}(0). We identify T_{s} ≥ T^{∗} with the structural transition and conclude that the Raman response (equation (1)) develops only a maximum rather than a divergence at T_{s} in agreement with the experiment here and recent theoretical work^{19,20}.
Close to T_{s}, we expect equation (1) to hold qualitatively also inside the nematic phase, T_{SDW} < T < T_{s}. We argue^{13} that χ_{nem}^{el}(0) and, according to equation (1), the Raman amplitude are smaller than in the disordered (tetragonal) state but different from zero. This explains the continuous reduction of the Raman response of spin fluctuations on entering the nematic state. One can also show that the A_{1g} response gets even further suppressed if one includes collisions between the fluctuations^{20}.
As shown in equation (1) the full Raman response is proportional to the bare response R_{0, μ}(Ω) and to the electronic nematic susceptibility χ_{nem}^{el}(0). Hence, the spectral shape is essentially given by R_{0, μ}(Ω), which is therefore used in Fig. 2 to fit the data, whereas the intensity is dominated by the prefactor T − T^{∗}^{−1}. As the theoretical model is valid only in the limit of small frequencies we argue that the initial slope reflects the temperature dependence of the intensity and is proportional to χ_{nem}^{el}(0), at least close to the transition. For generally reflecting the spectral shape above T_{SDW} (equation (1)), R_{0, B1g}(Ω, T) enables us to directly extract the initial slope of the experimental spectra by plotting R_{0, B1g}(Ω, T)/Ω for all temperatures (Supplementary Information VI). These results are compiled in Fig. 4 along with the variation of χ_{nem}^{el}(0, T) expected from meanfield theory. For low doping, we find qualitative agreement in the ranges T > T_{SDW}. For higher doping the interactions between fluctuations become dominant and the meanfield prediction breaks down (Fig. 4c).
The Raman response was also studied at various other doping levels in the range 0 ≤ x ≤ 0.085. Up to 6.1% Co substitution fluctuations were observed. In contrast to other publications^{14} we were not able to clearly identify and isolate the response of fluctuations at 8.5% although the kink in the B_{1g} relaxation rate used to define T_{f} is clearly observed (Supplementary Fig. 4). The results for χ_{nem}^{el}(0) up to 5.1% are unambiguous and are represented as a colour scale on the phase diagram in Fig. 5. Our phase diagram compares rather well to that derived from the elastic constant m_{66} (ref. 6). In addition, we show T_{f} up to x = 0.085. The fluctuations can be observed over a temperature range of approximately 70–100 K. This is more than in most of the other experiments on unstrained samples and comparable to what is found in the cuprates^{18,30}.
Methods
Samples.
The single crystals of undoped^{31} and Cosubstituted Ba(Fe_{1−x}Co_{x})_{2}As_{2} were grown using a selfflux technique and have been characterized elsewhere^{22}. The cobalt concentration was determined by microprobe analysis. T_{s} and T_{SDW} are close to 134 K in the undoped sample and cannot be distinguished. At nominally x = 0.025 we find T_{s} = 102.8 ± 0.1 K and T_{SDW} = 98 ± 1 K by directly observing the appearance of twin boundaries and a symmetryforbidden phonon line, respectively (see Supplementary Information I for details). The extremely sharp transition at T_{s} having ΔT_{s} ≍ 0.2 K indicates that the sample is very homogeneous in the area of the laser spot.
Raman experiment.
The experiments were performed with standard light scattering equipment. For excitation either a solidstate laser (Coherent, Sapphire SF 532155 CW) or an Ar ion laser (Coherent, Innova 300) was used emitting at 532 or 514.5 nm, respectively. The samples were mounted on the cold finger of a Heflow cryostat in a cryogenically pumped vacuum. The laserinduced heating was determined experimentally (see Supplementary Information I) to be close to 1 K per milliwatt of absorbed power. The spectra represent the response Rχ_{μ}′′(Ω, T) (μ = A_{1g}, B_{1g}, A_{2g} and B_{2g}), which is obtained by dividing the measured (symmetry resolved) spectra by the Bose–Einstein factor {1 + n(T, Ω)} = [1 − exp(−ℏΩ/k_{B}T)]^{−1}. χ_{μ}′′(Ω, T) is the imaginary part of the response function, and R is an experimental constant that connects the observed photon count rates with the crosssection and the van Hove function and accounts for units. For simplicity, the symmetry index μ is dropped in most of the cases. The symmetry selection rules refer to the 1 Fe unit cell (see Supplementary Fig. 1), which is more appropriate for electronic and spin excitations.
Static relaxation rates.
Static relaxation rates Γ_{0, μ}(T), with μ denoting the symmetry, are used in various places (see, for example, Fig. 1b),
Γ_{0, μ}(T) can be considered a symmetryresolved ‘Raman resistivity’ that can be compared to, for example, transport data^{32}. As the scattering intensity and consequently R are not known in absolute units Γ_{0, μ}(T) cannot directly be derived. The problem was solved a while ago by adopting the memory function method^{33,34} for Raman scattering^{35}. Then Γ_{0, μ}(T) can be derived by extrapolating the dynamic Raman relaxation rates Γ_{μ}(Ω, T) to zero energy. More details can be found in Supplementary Information III.
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Acknowledgements
We acknowledge useful discussions with T. P. Devereaux, Y. Gallais, S. A. Kivelson, B. Moritz and I. Paul. Financial support for the work came from the German Research Foundation DFG through the Priority Program SPP 1458 (project nos HA 2071/7 and SCHM 1035/5), from the Bavarian Californian Technology Center BaCaTeC (project no. A5 [20122]), and from the Transregional Collaborative Research Center TRR 80. U.K. and J.S. were supported by the Helmholtz Association, through the Helmholtz postdoctoral grant PD075 ‘Unconventional order and superconductivity in pnictides’. R.H. thanks the Stanford Institute for Materials and Energy Sciences (SIMES) at Stanford University and SLAC National Accelerator Laboratory for hospitality. Work in the SIMES at Stanford University and SLAC was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Contract No. DEAC0276SF00515.
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F.K., T.B., B.M., A.B. and D.J. contributed approximately equally to the experiments. U.K., J.S., S.C., M.G. and C.D.C. developed the theory. J.H.C., J.G.A. and I.R.F. prepared and characterized the samples. F.K., T.B. and R.H. conceived the study. U.K., F.K., T.B. and R.H. prepared the manuscript.
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Kretzschmar, F., Böhm, T., Karahasanović, U. et al. Critical spin fluctuations and the origin of nematic order in Ba(Fe_{1−x}Co_{x})_{2}As_{2}. Nature Phys 12, 560–563 (2016). https://doi.org/10.1038/nphys3634
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DOI: https://doi.org/10.1038/nphys3634
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