Nematicity and Glassy Behavior Probed by Nuclear Magnetic Resonance in Iron-Based Superconductors

Abstract

Nuclear magnetic resonance provides a wealth of information about the magnetic and nematic degrees of freedom in the iron-based superconductors. A striking observation is that the spin lattice relaxation rate is inhomogeneous with a standard deviation that correlates with the nematic susceptibility. Moreover, the spin lattice relaxation is strongly affected by uniaxial strain, and in doped samples it depends sensitively upon the history of the applied strain. These observations suggest that quenched strain fields associated with doping atoms induce a nematic glass in the iron pnictide materials.

1 Introduction

Electronic nematic order, in which low energy electronic degrees of freedom drive a crystal to spontaneously break discrete rotational symmetry without simultaneously breaking translational symmetry, has attracted broad interest in correlated electron physics [1]. Much of this interest stems from observations that unconventional superconductivity tends to emerge in materials exhibiting competing ground states with different broken symmetries, including nematic, charge, and/or spin density wave fluctuations [2–6]. Measurements of the nematic susceptibility in the iron based superconductors indicated the presence of a putative nematic quantum critical point under the dome of superconductivity in these materials [7, 8]. These observations have suggested a connection between quantum critical nematic fluctuations and the pairing mechanism for superconductivity [9–13]. It is thus important to understand the nature of the nematic fluctuations and their relationship to the coexisting antiferromagnetism throughout the phase diagram.

Nuclear magnetic resonance (NMR) provides a wealth of information about both the magnetic and nematic degrees of freedom in the iron-based superconductors [14]. Studies under uniaxial strain offer an important new direction in thermodynamic phase space to explore the physics of nematicity. Such experiments can broadly be separated into those that probe the static properties of the magnetic and nematic order, and those that probe the dynamical properties. The latter set of experiments have yielded some important surprises, such as a change in the spin anisotropy as a function of strain [15], and the presence of dynamical inhomogeneity when the materials are doped [16]. Below we discuss how NMR probes these broken symmetries and their fluctuations, and summarize results of experiments under uniaxial strain, both in the undoped and doped materials. These observations are consistent with the emergence of a nematic glass in doped samples, possibly driven by the presence of random strain fields from quenched disorder.

2 Coupling to Nuclear Spins

The ⁷⁵As nuclei, with 100% abundance and spin I = 3/2, are excellent sensors of both the nematic and magnetic degrees of freedom in the iron pnictide superconductors. Figure 1 shows the unit cell and indicates the hyperfine coupling between the As nuclei and the four nearest neighbor Fe electronic moments. The As also has a significant quadrupolar moment, Q = 3.14 × 10⁻²⁹ m², which couples to the surrounding electric field gradient (EFG) [17]. The hyperfine and quadrupolar couplings enable microscopic studies of both the magnetic and nematic susceptibilities of the electronic degrees of freedom.

FIGURE 1

2.1 Hyperfine Interaction

The hyperfine coupling is given by:where are the nuclear spin operators, are the Fe spins on the four nearest neighbors to the As nucleus, and the hyperfine coupling tensors, , have dipolar symmetry [18, 19]. In BaFe₂As₂, the eigenvalues of the tensor are A_aa = A_bb = 6.6 kOe/μ_B, and A_cc = 4.7 kOe/μ_B, but these values depend on details of the electronic structure and can vary between compounds. They can change as a function of pressure or doping, but the general symmetry of the does not. The magnitude of these values are greater than those for the direct dipole interaction, which reflects the electronic hybridization between the As 4p and Fe 3d orbitals. The tensor nature of this coupling gives rise to a hyperfine field H_hf ∥c when the Fe moments lie in the ab plane, as illustrated in top row of Figure 2. Alternatively, if the Fe moments ordered parallel to c, then H_hf lies in the plane, as shown in the middle row of Figure 2.

FIGURE 2

Other magnetic orderings can be present in the pnictides, beyond the stripe-type spin density wave illustrated in Figure 2. In particular, the spin-vortex crystal configuration, in which the Fe spins form either loop or hedgehog structures, can be stabilized when a glide symmetry across the Fe planes is broken [20]. This type of order was recently identified in Ni-doped and Co-doped CaKFe₄As₄ [21]. There are two crystallographically distinct As sites in this structure, and the hyperfine field either vanishes at both sites for loop spin-vortex crystal order, or is aligned along the c axis for one As site and vanishes for the other As site for hedgehog spin-vortex crystal order. In this system, doping induces the latter, which was identified by NMR.

Note that when the Fe is substituted by another transition metal, the symmetry among the hyperfine couplings of the four nearest neighbors is broken, and the hyperfine field vector can acquire a component perpendicular to the c axis [22]. This effect contributes to inhomogeneous broadening in doped samples. In the presence of nematic order, the C₄ symmetry of the lattice is broken, and in principle A_aa ≠ A_bb. This asymmetry has not been observed directly. In FeSe, the Knight shift tensor becomes anisotropic, with K_aa ≠ K_bb, where K_αα = A_ααχ_αα, and χ_αα is the static magnetic susceptibility [5]. However, in this case the anisotropy is believed to reflect that of the susceptibility, i.e., χ_aa ≠ χ_bb in the nematic phase, rather than an asymmetry in the hyperfine coupling [23]. Curiously, χ_aa > χ_bb in FeSe, but χ_bb > χ_aa in BaFe₂As₂, LaFeAsO, and NaFeAs [24–26], which may reflect different natures of the nematic order in these materials [27]. In particular, the nematic order parameter in the latter can be understood in terms of differential occupations between the d_yz and d_xz orbitals, whereas in the former the nematic order may involve more complex superpositions between 3d orbitals, giving rise to bond-centered nematic order [23, 28, 29]. The sign of the resistivity anisotropies in BaFe₂As₂ and FeSe match those of the magnetic anisotropies [24, 30], suggesting a common origin for the two.

2.2 Quadrupolar Interaction

For nuclei with spin I > 1/2, the quadrupolar coupling is given by:where the EFG tensor is given by:and V is the electrostatic potential at the As site. This quantity is dominated by the occupation of the As 4p orbitals, which in turn are hybridized with the d_xz,yz-orbitals of the neighboring Fe atoms [31]. The EFG asymmetry parameter is given by:

This quantity vanishes in the tetragonal phase because the As 4p_x and 4p_y orbitals are degenerate, hence ν_xx = ν_yy [32]. The EFG tensor is always traceless, thus in this case it can be characterized by a single quantity, ν_zz. The magnitude of ν_zz depends on the degree of hybridization of the As 4p_z orbital, and is sensitive to the c-axis length of the unit cell [33–35] and also varies with temperature [18, 36–38].

The EFG is strongly affected by doping. Replacing the Fe by another transition metal, or replacing the As by P, will give rise to spatially-varying strain fields that will distort the electrostatic potential and alter the EFG tensor. This effect can lower the symmetry such that ν_xx ≠ ν_yy, and create non-zero off-diagonal terms of the tensor. In some cases the effect is sufficiently large that separate resonances can be detected for sites that are adjacent to the substitutional site [31, 39]. This sensitivity to disorder tends to significantly broaden the NMR resonance frequencies in doped samples relative to the parent compound. In doped Ba(Fe,Co)₂As₂ the width of the quadrupolar satellites is dominated by a variation of η that exhibits a Curie-Weiss temperature dependence, which has been associated with the growth of nematic correlations surrounding the dopant atoms [40]. A similar effect has been observed in FeSe where the magnetic linewidth is broadened by the presence of crystal defects (there is no quadrupolar interaction for Se) [41]. In this case, the Knight shift anisotropy reflects the growth of an Edwards-Anderson order parameter of the nematicity [42].

In the presence of long range nematic order, the C₄ symmetry of the EFG tensor is broken and ν_xx ≠ ν_yy [43]. In CaFe₂As₂η reaches 0.4 at low temperature, but in BaFe₂As₂ and SrFe₂As₂, η reaches 1.2 and 1.4, respectively, reflecting the fact that ν_xx and ν_yy have opposite signs [33]. Note that it is sometimes customary to define the principal axes of the EFG such that the eigenvalues |V_zz|≥|V_yy|≥|V_xx|, so that 0 ≤ η ≤ 1. Here we do not follow this convention, maintaining the V_zz associated with the crystal c axis as in the tetragonal phase. This choice is the reason that η > 1, which would correspond to a confusing rotation of orientation of the principal axes under the customary definition. These changes reflect a dramatic rearrangement of the charge distribution around the As nucleus below the structural transition, despite an orthorhombicity of only approximately 1%. The EFG tensor is dominated by the occupations of the As 4p orbitals, which in turn are hybridized with the Fe 3d_yz and 3d_xz orbitals [31, 32]. These orbitals are degenerate in the tetragonal phase, but develop a splitting on the order of 40 meV in the nematic phase [44]. As a result, the relative occupations of the As p_x and p_y orbitals change and alter the EFG.

2.3 NMR Spectra

In order to resolve the in-plane asymmetry that emerges in the Knight shift or EFG tensors, it is important to measure the NMR spectra of carefully oriented single crystals. In an external magnetic field, the As nuclear spins experience the sum of three interactions: , where is the Zeeman interaction. The spectrum will consist of three resonances. For sufficiently large Zeeman interaction (γH₀ ≫ ν_αα), the frequencies are given to first order by:where f_sat1,2 and f_cen are the frequencies of the two satellites and the central transition, θ is the angle between H₀ and c, and ϕ is the azimuthal angle of the projection of H₀ in the ab plane. The Knight shift is given by:

The second order correction to the central transition is given by:

Measurements of the two satellite transitions as a function of field orientation can be used to extract the Knight shift and EFG tensors.

In the tetragonal phase, there are three distinct resonances for each As in the unit cell (e.g., one As site in AFe₂As₂). However, in the nematic phase there are multiple domains in an untwinned crystal, giving rise to extra resonances. If the magnetic field is oriented in the plane, then there may be two sets of resonances associated with these two domains because the azimuthal angle, ϕ, differs for each domain, as observed in Figure 3. This behavior is illustrated in Figure 4. In general, there are five distinct transitions (six if the Knight shift is also anisotropic), but in certain directions the spectra from the two domains overlap and three resonances are recovered. Large values of |η| cause the satellites to switch positions from below to above the central transition as the in-plane angle ϕ is rotated.

FIGURE 3

FIGURE 4

When η is non-zero the nuclear spin Hamiltonian does not commute with , and the eigenstates, |i⟩, are superpositions of the |I_z⟩ states. In such cases it may be possible to observe hidden transitions when the matrix element does not vanish. This effect can arise when ν_zz ≳ γH, but in almost all cases there is an internal field, H_int, that develops in the antiferromagnetic state that accompanies the orthorhombic distortion. Typically γH_int ≫ ν_zz is sufficiently large that such hidden transitions should not be evident. For CaFe₂As₂, however, ν_zz/γH ∼ 0.7 thus it may be possible to observe these transitions for small external fields H₀ applied in the plane, as illustrated in Figure 5. To date no such transitions have been reported.

FIGURE 5

2.4 Relaxation Rate

The hyperfine field, H_hyp, and the EFG not only give rise to static spectral signatures, but their fluctuations can also drive relaxation of the nuclear spins. The nuclear spin energies are generally several order of magnitude smaller than the relevant electron energies, and thus the nuclear spin ensemble can easily be manipulated to have a non-equilibrium distribution among the energy levels. The nuclear spins will return to thermal equilibrium with the “lattice,” or the electronic spin system, over a time scale, T₁, known as the spin-lattice relaxation time. This rate, , can be measured with high precision and is determined by the couplings described above in Section 2.1, Section 2.2. In essence, the nuclear spins act as quantum sensors that probe the noise spectra of the electronic spin and charge degrees of freedom. There are three important spectral densities that may play a role:where is an ensemble average. Here h_± = H_hyp,x ± iH_hyp,y, V_{± 1} = V_xz ± iV_yz, and V_{± 2} = (V_xx − V_yy)/2 ± iV_xy [45]. J(ω) and J^(1,2)(ω) are the spectral densities of the hyperfine and EFG fluctuations, respectively. The measurements probe these spectral functions at the nuclear resonance frequency, ω_N, typically on the order of 100 MHz (eV). This frequency is nearly vanishing in comparison to the relevant electronic energies, typically on the order of meV. If there is only one type of fluctuation, then is directly related to the appropriate spectral density. For example, for pure magnetic fluctuations, (e.g., V_{± 1,2} = 0), then , or if h_± = 0 and V_{± 1} = 0, then , or if h_± = 0 and V_{± 2} = 0 then . On the other hand, if multiple types of fluctuations are present, then the nuclear spin relaxation is a complex function of multiple spectral functions and it is not straightforward to disentangle the different contributions to [46–48]. It is common for the magnetic fluctuations to dominate, in which case the contribution from quadrupolar contributions can be ignored. However, as we discuss below in Section 2.4.1, there is a significant quadrupolar contribution to the of the As in the pnictides.

When the electronic system is close to an thermodynamic instability, critical fluctuations will slow down and may have a dramatic effect on . As an example, consider the case where the magnetic fluctuations exhibit Lorentzian autocorrelation function: , where τ_c is the correlation time. In this case:

Usually ω_Nτ_c ≪ 1, but close to a phase transition where there is critical slowing down, τ_c can approach infinity as a power law. As a result, exhibits a peak at the phase transition temperature, as illustrated in Figure 6. This property has been exploited to investigate the critical dynamics of many correlated electron systems, especially the pnictides [39, 49].

FIGURE 6

As discussed above in Section 2.3, the nuclear spin eigenstates become superpositions of the I_z states for non-zero values of η. In this case, it is important to consider spectral densities other than those described in Eqs 6, 7 to account for fluctuations in other directions. For example, hyperfine field fluctuations, H_hyp,z, may play a role in relaxing the nuclear spins. In such cases the form of the relaxation function is more complicated, and Eq. 8 will contain more terms.

2.4.1 Quadrupolar Relaxation From Nematic Fluctuations

Although magnetic relaxation is usually dominant, quadrupolar relaxation should also be present in the pnictides because the EFG changes dramatically at the nematic phase transition. This effect has been investigated in BaFe₂(As_1−xP_x)₂ by comparing the relaxation of the ⁷⁵As to ³¹P [43]. The phosphorus isotope has spin I = 1/2, and does not have a quadrupolar moment. It therefore is only sensitive to magnetic fluctuations, whereas the As probes both magnetic and quadrupolar fluctuations. Importantly, both the As and the P are located in the same crystallographic site, and experience similar hyperfine interactions. In fact, the ⁷⁵As exhibits a higher relaxation rate, indicating the presence of a second relaxation channel that is, not probed by the ³¹P. Moreover, the extra relaxation increases with decreasing temperature, and exhibits a peak at the nematic ordering temperature. An analysis of the extra relaxation rate suggests that the nematic susceptibility diverges below T_N for the lightly doped system, and below T_c for the optimally doped system. It was not possible to quantitatively disentangle the contributions from the magnetic and quadrupolar channels separately, primarily because the doping introduced inhomogeneous relaxation, as discussed below in Section 4. The three relaxation channels (one magnetic and two quadrupolar) may also be coupled to one another. For example, a nematic fluctuation can give rise spectral densities J (ω_N), J⁽¹⁾(ω_N), and J⁽²⁾(ω_N). Nevertheless, this work provided direct proof of nematic fluctuations that diverge near the superconducting transition temperature, T_c. Theoretical work has suggested that nematic fluctuations may play a role in the superconducting pairing in these materials [9, 11].

3 Response to Uniaxial Strain

In the iron based superconductors, the electronic nematic degrees of freedom couple to lattice strain with B_2g symmetry: ɛ_B2g = (ɛ_xx − ɛ_yy)/2, where x and y are defined with respect to the tetragonal unit cell as illustrated in Figure 3 [4]. This coupling gives rise to a linear response of the electronic nematicity to external strain on a crystal. Uniaxial strain can be applied via different techniques, however in recent years piezoelectric-based strain cells have demonstrated excellent properties that enable one to carefully control the level of strain even at cryogenic temperatures [50]. Even though piezoelectric actuators do not have linear response to control voltages, the strain can be controlled through feedback control algorithms for the long periods of time necessary for NMR experiments [51]. The strain is measured by a capacitive dilatometer. Note that even though the device applies a uniaxial stress, σ₀, along the x direction, the response of the crystal is not a pure uniaxial strain. Rather, the non-zero stress tensor elements become: ɛ_xx = σ₀/E and ɛ_yy = ɛ_zz = −νσ₀/E, where E is the Young’s modulus and ν is the Poisson’s ratio of the material. This gives rise to strain in two symmetry channels:

On the other hand, the electronic nematicity has a much stronger response to B_2g strain than A_1g, thus uniaxial stress is sufficient to probe the intrinsic B_2g nematic response [7].

3.1 Spectra

In BaFe₂As₂, uniaxial strain dramatically alters the EFG tensor and η ∝ɛ_xx, as illustrated in Figure 7 [32]. In the absence of strain, the crystal remains tetragonal down to the nematic transition temperature T_s = 135 K, in which case the EFG tensor spontaneously develops an asymmetry such that ν_xx ≠ ν_yy. However, in the presence of a finite strain field, η becomes finite above T_s. In fact, η ∝ɛ_xx and the constant of proportionality is the nematic susceptibility. As seen in Figure 7, the response is largest just above T_s, where the nematic susceptibility diverges. This behavior is illustrated in Figure 8, which reveals that dη/dɛ_xx ≈ 300 at T_s.

FIGURE 7

FIGURE 8

Uniaxial stress has also been utilized to study the NMR spectra in LaFeAsO and FeSe [23]. In this case, compressive stress on the order of 10–20 MPa was applied by tightening a screw. A similar approach in which a crystal is suspended between the prongs of a horseshoe-shaped device was also utilized for NMR studies under tensile strain [52]. Neither approach directly measures the strain, precluding the possibility for feedback control. Moreover, differential thermal contraction may also give rise to finite strain fields even without applied stresses via the tightening screw. Nevertheless, such approaches can be used to mechanically detwin the crystal in the nematic phase [53]. If the crystal is cooled in a finite strain field, then a single nematic domain should nucleate below the transition. For a twinned crystal, there should be two sets of resonances as discussed above in Section 2.3. In detwinned or partially de-twinned crystals, the relative intensity changes. In FeSe, this approach enabled the identification of which resonance corresponds to which domain [23].

3.2 Relaxation Rate

The spin-lattice-relaxation rate varies strongly with uniaxial strain in BaFe₂As₂, as shown in Figure 9 [15]. Strain enhances the relaxation rate and the effect is anisotropic: the enhancement is stronger for in-plane fields than for out-of-plane fields. The enhancement can be understood qualitatively by realizing that strain enhances the antiferromagnetic ordering temperature, T_N, thus for T > T_N, the correlation length should grow with increasing strain. The correlation time τ_c grows with increasing correlation length, thus enhancing as illustrated in Figure 6.

FIGURE 9

A more complete picture of the nematicity and the antiferromagnetism can be obtained by analyzing the reciprocal-space structure of the spin susceptibility tensor, χ_αβ(q). In the disordered tetragonal phase, χ_xx(q) = χ_yy(q), but in the presence of nematicity these components are no longer equal. In the ordered phase, the susceptibility diverges at the antiferromagnetic ordering wavevectors, Q₁ = (π, 0) in one domain, and Q₂ = (0, π) in the other domain. A nematic order parameter can defined as [15]. By symmetry, the only non-vanishing components are ϕ_xy, ϕ_yx and ϕ_zz. In the ordered state, only ϕ_xy condenses, which corresponds to the spins oriented parallel to the antiferromagnetic ordering vector, as shown in the upper row of Figure 2. ϕ_zz order is illustrated in the second row of Figure 2. In this case, the spins are oriented along the c direction, and the hyperfine field is in-plane. Although this channel does not condense, ϕ_zz fluctuations are present and contribute to . ϕ_yx is illustrated in the bottom row, where the spins are oriented in-plane and are perpendicular to the antiferromagnetic ordering vector. For ϕ_yx order the hyperfine field vanishes, so is insensitive to these types of fluctuations.

The anisotropy of contains key information about these two nematic channels, ϕ_xy and ϕ_zz. The fluctuations h_α(τ) of the hyperfine field that contribute to the spin-lattice-relaxation rate as described in Eq. 10 are perpendicular to the applied field, H₀. If H₀ ∥c, then the in-plane fluctuations driven by ϕ_zz fluctuations will dominate . If H₀ ⊥c, then both ϕ_zz and ϕ_xy fluctuations will contribute. As seen in Figure 9, measured for H₀ ⊥c has a much stronger variation with applied strain. This suggests that ϕ_zz fluctuations are enhanced under strain more than ϕ_xy fluctuations. In fact, a detailed analysis of the nematic susceptibilities, and indicates that the latter is largest, despite the fact that ϕ_xy condenses at zero strain [15]. These results imply that for sufficient strain, the system would order with the spins oriented along the c-axis. This interpretation was recently confirmed by polarized inelastic neutron scattering results [54].

4 Glassy Behavior

4.1 Relaxation

Measurements of the doped pnictides under strain are challenging because the doping introduces both static and dynamic inhomogeneities. The quadrupolar linewidths of the satellites broaden significantly with doping, thus it is more difficult to discern static changes in the resonances due to strain. Measurements of the spin-lattice-relaxation rate are more straightforward and do not require sharp resonances, but doping introduces a distribution of local hyperfine fields and autocorrelation times, τ_c, so that there is no longer a homogeneous value throughout the bulk of the materials [16].

4.1.1 Stretched Relaxation

Heterogeneous spin lattice relaxation is directly manifest in the behavior of the magnetization and leads to a stretching exponent. In this case, the nuclear magnetization is described by:where M₀ is the initial magnetization, M_eq is the thermal equilibrium magnetization, and 0 < β ≤ 1 is a stretching exponent [55]. β = 1 corresponds to homogeneous relaxation, but in several doped pnictides β ∼ 0.4–0.9 depending on the doping level and temperature [16, 56–60]. A typical temperature dependence is shown in Figure 10 for x = 0.06. For temperatures greater than approximately 100 K, the relaxation is homogeneous, but below this temperature β decreases from unity to approximately 0.4 at T_N. The exponent β is a measure of the width of the distribution of local relaxation rates, , where W₁ is the relaxation rate. , is the median of this distribution and β is related to the logarithmic FWHM of [55]. When β = 1, the distribution is a delta function, and as β decreases the width of increases exponentially. Figure 10 indicates that for x = 0.06 just above T_N the distribution of relaxation rates is at least two to three orders of magnitude in width. Additionally, the average correlation time, τ_c, does not have a power law divergence, but rather an activated or Vogel-Fulcher behavior with temperature, which tends to broaden the peak in with temperature, as illustrated in Figure 6.

FIGURE 10

4.1.2 Inverse Laplace Transform

Rather than fitting the magnetization with a stretched exponent, as in Eq. 13, it is possible to fit the data assuming a particular form of . One such approach is via a log-normal distribution, where:

In this case the median of the distribution () is e^μ, and the standard deviation is σ₁ is roughly inversely proportional to β. The relaxation in the pnictides is well-captured by this form of the distribution [16].

An alternative approach is to extract directly from the data, rather than to assume a particular form. This is known as an inverse Laplace transform (ILT), and has been utilized to investigate a range of correlated electron systems [61–63]. Figure 11 displays representative data set with such a fit, and Figure 12 shows the distribution versus temperature extracted using this approach in Ba(Fe_1−xCo_x)₂As₂ with x = 0.059. It is clear that both the median and width of this distribution change in the vicinity of T_N, and that the width approaches several orders of magnitude. The ILT approach does not make any a priori assumptions about the distribution, and may uncover features that are not captured by Eq. 14. On the other hand, the ILT requires time series data with high a signal-to-noise ratio, a complicated analysis algorithm, and assumptions about the smoothness of the distribution [64]. The distribution shown in Figure 12 exhibits several structures for T ≪ T_N that are likely artifacts due to the poor signal to noise at these temperatures. Ultimately, the most meaningful information about the distribution appear to be the median and width, which are well-captured by the stretched exponential approach. Note that although the ⁷⁵As has I = 3/2 and hence a multi-exponential relaxation form even for homogeneous relaxation, the stretched exponential approach accurately captures the true distribution of relaxation rates, [64].

FIGURE 11

FIGURE 12

4.1.3 Nematicity and Dynamical Heterogeneity

Similar glassy behavior has been observed in the cuprates and been associated with stripe-glass behavior [65–73]. The cuprates, however, are doped Mott insulators, and the glassy behavior was attributed to intrinsic frustration between the competing effects of Coulomb repulsion and charge segregation [74–76]. The iron arsenides do not exhibit charge ordering and thus a different mechanism must be driving the glassy dynamics. This behavior is present for a range of doping levels, and is most pronounced near optimal doping where T_c reaches a maximum. Figure 13 shows a phase diagram for both Cu and Co doping. Like Co, Cu introduces extra electrons but it also suppresses superconductivity [77]. The color scale indicates the standard deviation of a log-normal distribution fit, σ₁. As shown in Figure 13, the distribution exhibits the greatest width in the vicinity of a putative nematic quantum critical point near x ∼ 0.07 if there were no superconductivity. This behavior is striking because it bears a similarity to the divergent nematic susceptibility measured by elastoresistance in these materials [7, 8]. In other words, the width of the distribution of relaxation rates correlates with the magnitude of the nematic susceptibility.

FIGURE 13

What is the origin of this glassy behavior, and is there a physical connection with the nematic susceptibility? A compelling hypothesis is that the dopants introduce quenched random strain fields that couple to the nematic order parameter and give rise to a distribution of local correlation times, τ_c. Local strains can nucleate different nematic domains, even in absence of long range nematic order. The result is a nematic glass, in which a distribution of local nematic domains (with different orientations) exist throughout the sample, each fluctuating with a different correlations times. These nematic fluctuations in turn drive spin fluctuations that are reflected in the distribution .

Another hypothesis for the inhomogeneous dynamics is that static inhomogeneity of the EFG and Knight shift tensors induced by the dopants suppresses spin diffusion among the nuclear spins, so that they do not all relax with a common spin temperature. In principle, dipolar interactions between neighboring nuclear spins I₁ and I₂ may contain terms such as , which can give rise to mutual spin flips [78]. This effect could enable nuclei in regions that exhibit fast relaxation to transfer polarization with those in regions with slow relaxation, giving rise to a more homogeneous relaxation throughout the sample. If the static EFG and Knight shift tensors vary spatially, however, then mutual spin flips between neighbors would not conserve energy. In fact, spin diffusion should behave differently for a spin-1/2 versus a spin-3/2 nucleus. For the latter there are only six out of sixteen possible configurations where the two neighbors can undergo a mutual spin flip while conserving energy, whereas there are two out of four possibilities for the former. Thus spin-diffusion should be suppressed for ⁷⁵As relative to ³¹P, and thus one might expect β to be closer to unity for the spin-1/2 ³¹P. In fact, both sites exhibit similar values of β at all temperatures in BaFe₂(As,P)₂ [43]. These results suggest, therefore, that spin diffusion does not play a significant role in the observed glassy dynamics.

4.1.4 Edwards-Anderson Parameter

In a recent paper, Wiecki and collaborators pointed out that the NMR linewidth of ⁷⁷Se in FeSe is directly proportional to the Edwards-Anderson parameter in a random-field Ising model of the nematicity [42]. Crystal defects provide local strain fields, h_i, at site i, and the enhanced nematic susceptibility gives rise to large spatial variations of local nematicity, ϕ_i(h_i) (we drop the distinction between ϕ_xy, ϕ_yx and ϕ_zz for simplicity). In this case, the Knight shift is known to vary linearly with ϕ, and the second moment of the NMR spectrum is directly proportional to the Edwards-Anderson parameter: , where the brackets indicate a thermal average and the overbar indicates an average over disorder configurations.

A similar approach can be utilized to analyze the distribution of relaxation rates, . In undoped BaFe₂As₂, the spin lattice relaxation rate is a function of the nematic order parameter:where α depends the correlation length, and is hence strongly temperature dependent [15]. In the doped system, we can model the distribution of relaxation rates by assuming a distribution of local strain fields, so that:where p_σ(h_j) is the disorder distribution [42]. The second moment of is , thus the second moment of the distribution measures the Edwards-Anderson order parameter.

As shown in Figure 13, σ₁ is a strong function of temperature and doping. The magnitude of this quantity is given in part by the coefficient α, reflecting the growth of antiferromagnetic fluctuations. Note, however, that σ₁ would be zero if q_EA = 0. In other words, σ₁ is finite because the Edwards-Anderson parameter is non-zero, which indicates the presence of a nematic glass. Higher order corrections to the relationship between W₁ and strain, and the presence of other strain channels due the dopant atoms, are likely to modify the quantitative relationship between σ₁ and q_EA, and these issues should be addressed in future theoretical work.

4.2 Response to Strain

In principle, an external strain field of sufficient magnitude might overcome the local strain fields and give rise to more homogeneous relaxation. This hypothesis was tested in Ba(Fe_1−xCo_x)₂As₂ with x = 0.048 under tensile strain [52]. Although the magnitude of changed in response to the strain field, no changes were observed to the stretching exponent. This observation suggests that the intrinsic strains around the dopants exceed the homogeneous external strain, so that remains unchanged under strain. In fact, the lattice strain around Co dopants in Ba(Fe_1−xCo_x)₂As₂ can reach several percent [79]. The applied strain was only a fraction of a percent, thus it is not surprising that no change was observed in the width of the distribution.

4.3 Strain Hysteresis

Although the width of the does not change with externally applied strain, the median of the distribution does, and in a manner that depends on the strain history. Figure 14 illustrates a protocol to investigate such hysteretic effects. A crystal is strain-cooled either under compressive (or tensile) strain from room temperature down to a base temperature. The spin lattice relaxation is then measured for a series of different levels of increasing (or decreasing) strain at constant temperature. The sample is then returned to room temperature (where the relaxation is homogeneous and there is no glassy behavior), and then the process is repeated under the opposite direction. For a fully linear response, the spin lattice relaxation as a function of strain should be independent of whether it was tensile-cooled or compressive-cooled.

FIGURE 14

Figure 15 shows the results of such a study on Ba(Fe_1−xCo_x)₂As₂ for four different base temperatures for x = 0.059. As expected, β decreases as temperature is reduced, but remains independent of the applied strain, and is the same for both tensile- and compressive-cooling. On the other hand, the median of the distribution, , is strain-dependent and is different under the two strain conditions at low temperature.

FIGURE 15

Note that the horizontal axes in Figure 15 are displacement, not strain. Strain is defined as ɛ = (d − d₀)/L, where d is the displacement of the capacitative dilatometer, d₀ is the equilibrium unstrained displacement, and L is the sample length. d₀ can be determined at room temperature, but thermal contractions of both the sample and the strain device make it difficult to determine d₀ under cryogenic conditions. It is best to utilize an independent measure of strain, such as the EFG tensor discussed in Section 3.1, to find the displacement corresponding to ɛ = 0. Such a calibration was not possible for the measurements in Figure 15. It is also important to note that the level of strain during the cooling part of the protocol in Figure 14 is likely temperature-dependent, rather than constant as illustrated. Again, this is due to differential thermal contractions between the sample and the strain device. The displacement measured by the dilatometer is the quantity that is held constant.

Despite these complications, these measurements reveal that , and presumably the distribution of local nematic domains, depends on the past history of applied strain. Hysteresis is commonly observed in ordered ferromagnets, in which the static bulk magnetization is hysteretic with the applied field. If the applied field exceeds the coercive field, the magnetization will saturate, and when the applied field is reduced to zero, a remnant magnetization remains. Similar behavior should be expected in an ordered nematic without quenched disorder, in which nematic domains can reorient in an external strain field [80]. The NMR measurements described here, on the other hand, are in a nematic glass phase with quenched disorder, and probe the distribution of local correlation times of different fluctuating domains. Hysteresis has been studied in magnetic materials with quenched disorder in both the weak-disorder limit in the paramagnetic phase, as well in the strong-disorder limit in the spin glass phase [81]. The particular domain pattern is likely determined by the combination of local strains and the external strain field applied during the cool down. The fluctuation time of the domains is a complicated function of these strain fields as well as the behavior of their neighboring nematic domains. As the external strain field changes, the local correlation times change, but in such a manner that the median of the distribution of correlation times changes, but not the standard deviation. If the system were homogenous and not glassy, then the nematic domains would be mobile and there would be a single average correlation time. The presence of a distribution of correlation times that is altered by external strain provides evidence for the formation of a nematic glass, in which some domains exhibit very large correlation times, whereas others have short correlation times and may be reoriented by small external strain fields.

4.4 Theoretical Models

A quantitative theory to connect the measured distribution of the spin-lattice-relaxation rate, , to the nematic domains and their dynamics in the presence of external strain, particularly in the context of the iron pnictides, is highly desirable. Although no such theory yet exists, there are important models that capture elements of the glassy behavior in these materials. Nematic ordering in the presence of random strain fields can be mapped onto the random field Ising model (RFIM) [80, 82, 83]. Detailed numerical simulations have uncovered a broad distribution of local correlation times, which agrees qualitatively with the observations of stretched relaxation in NMR experiments [16].

On the other hand, the iron pnictides exhibit both antiferromgnetism and nematicity, and there are two types of spin order for each of two possible nematic domains, as illustrated in Figure 2. Recently, a random Baxter-field model was proposed as an alternative to capture both the nematic and antiferromagnetic fluctuations in the presence of quenched random strain fields [84]. In this case, there are two Ising variables at each lattice site associated with the magnetic and the nematic degree of freedom. These two variables are coupled due to the next nearest neighbor coupling, J₂, in the J₁ − J₂ model for the iron pnictides [4]. Random strain fields couple only to the nematic degree of freedom, but the magnetic variable is also affected and consequently the system breaks up into domains of all four configurations illustrated in the top row of Figure 2.

5 Summary and Conclusion

NMR is a powerful tool to investigate nematicity in the iron based superconductors, providing microscopic insight in the local antiferromagnetic and nematic order parameters. Investigations of the NMR response to uniaxial strain in BaFe₂As₂ and FeSe have revealed important information about the anisotropy of the dynamical and static spin susceptibilities. In particular, the temperature dependence of the spin fluctuations under strain reflect an unusual piezomagnetic response in which the antiferromagnetically ordered spins can change their orientation direction under sufficient strain.

The doped pnictides have been less studied under strain because static and dynamical heterogeneity present in these materials render interpretation of results more difficult. Strain fields surrounding dopants create random fields that couple to the nematic order parameter, which may have a large response at low temperature. As a result, the NMR spectra broaden significantly, reflecting a distribution of static EFGs, some of which may indicate static local nematic order. More importantly, the NMR spin lattice relaxation rate reveals a broad distribution of local spin fluctuations that gradually freeze out at lower temperatures. This distribution likely reflects the correlated nematic fluctuations breaking up into spatial domains in a nematic glass. This distribution depends on the history of the strain field, but remains broad even at the highest uniaxial strain levels applied. These results suggest that quenched disorder creates large random strain fields.

Future studies on other doped materials may shed important light on the nature of the nematic glass. For example, hole doping by substituting Ba with K may introduce milder levels of local strain fields, in which case external strain might be able to induce homogeneous relaxation. Studies of static hysteretic behavior with strain in the ordered nematic phase may also shed important light on the mesoscopic physics of the domain formation, the coercive strain fields, and remnant nematicity. Unfortunately, NMR studies under strain require small crystals, and the NMR signal in doped samples tends to be suppressed due to the same glassy physics giving rise to the stretched relaxation. These effects conspire to make such experiments on doped samples particularly challenging.

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