Critical quadrupole fluctuations and collective modes in iron pnictide superconductors

The multiband nature of iron pnictides gives rise to a rich temperature-doping phase diagram of competing orders and a plethora of collective phenomena. At low dopings, the tetragonal-to-orthorhombic structural transition is closely followed by a spin-density-wave transition both being in close proximity to the superconducting phase. A key question is the nature of high-$T_c$ superconductivity and its relation to orbital ordering and magnetism. Here we study the ${\mathrm{NaFe}}_{1-x}{\mathrm{Co}}_x\mathrm{As}$ superconductor using polarization-resolved Raman spectroscopy. The Raman susceptibility displays critical enhancement of nonsymmetric charge fluctuations across the entire phase diagram, which are precursors to a $d$-wave Pomeranchuk instability at temperature $\theta \left(x\right)$. The charge fluctuations are interpreted in terms of quadrupole interorbital excitations in which the electron and hole Fermi surfaces breathe in-phase. Below $T_c$, the critical fluctuations acquire coherence and undergo a metamorphosis into a coherent in-gap mode of extraordinary strength.

Figure 1

${\mathrm{NaFe}}_{1-x}{\mathrm{Co}}_x\mathrm{As}$ crystal and electronic structure, and $XY$-quadrupole mode. (a) Crystal structure of NaFeAs in the tetragonal phase. (b) Top view of the FeAs layer in the tetragonal phase shown with $d_{xz}-d_{yz}$ orbitals (left) and $d_{xy}$ orbitals (right). Dashed lines represent the two (four) -Fe unit cell in the tetragonal (orthorhombic) phase. (c) and (d) The effect of Co doping is illustrated on the schematic Fermi surfaces (FSs) for ${\mathrm{NaFe}}_{1-x}{\mathrm{Co}}_x\mathrm{As}$ in the tetragonal nonmagnetic BZ for doping $x=0$ (c) and $x>0$ (d). Below is shown a band-dispersion cut along the $\mathrm{\Gamma }\text{−}M$ high-symmetry line. $d_{xy},d_{xz}$, and $d_{yz}$ orbitals are shown with red, blue, and green, respectively. The holelike pockets $\alpha ,\beta$, and $\gamma$ surround the $\mathrm{\Gamma }$ point, and the electronlike pockets $\varepsilon /\delta$ surround the $M$ point. (e) Momentum- and frequency-resolved spectra $A(\mathbf{\text{k}},\omega )$ along the $\mathrm{\Gamma }\text{−}M$ high-symmetry line calculated by first-principles calculations including spin-orbit coupling (see Appendix pp4). (f) Pomeranchuk fluctuations in $B_{2g}$ symmetry, which is sustained by charge transfers between degenerate $d_{xz}$ and $d_{yz}$ Fe orbitals. (See the text and Appendix pp5.) (g) Monoclinic 2-Fe unit cell in the Pomeranchuk phase. (h) Quadrupole ground state in the orthorhombic phase with orthorhombic structural distortion, doubled unit cell, and two neighboring stripes having different orbital occupation. The pluses and minuses indicate a buckling-like modulation effect along the $c$ axis. (i) Phase of the superconducting OPs for the $\gamma$ band at the $\mathrm{\Gamma }$ point and the $\delta /\varepsilon$ bands at the $M$ point for $s^{++},d^{++},s^{\pm }$, and $d^{\pm }$ symmetry. Different colors indicate the opposite sign of the gap function.

Figure 2

Raman susceptibility ${\chi }^{\prime \prime }\left(\omega \right)$ in the $A_{1g},B_{2g}$, and $B_{1g}$ symmetry channels at representative temperature and dopings. (a)–(c) ${\chi }_{A_{1g}}^{\prime \prime }\left(\omega \right)$ showing superconducting features highlighted with blue shading below ≃$200{\mathrm{cm}}^{-1}$ $\left(x=0.0175,x=0.05\right)$. (d)–(f) ${\chi }_{B_{2g}}^{\prime \prime }\left(\omega \right)$ presenting a quasielastic scattering relaxational mode above $T_S\left(x\right)$ and $T_c\left(x\right)$ highlighted with green shading, a density wave suppression, and coherence peak highlighted with light blue shading below $T_S\left(x\right)$ $(x=0$, 5 K), and a low-temperature collective resonance highlighted with blue shading $(x=0.0175,x=0.05$, 5 K). (g)–(i) ${\chi }_{B_{1g}}^{\prime \prime }\left(\omega \right)$ featuring mainly a $B_{1g}$ phonon.

Figure 3

Static Raman susceptibility ${\chi }_0^{XY}(T,x)$ in the $B_{2g}$ symmetry channel. (a) Evolution of ${\chi }_0^{XY}(T,x)=2/\pi {\int }_0^{\infty }[{\chi }_{XY}^{\prime \prime }\left(\omega \right)/\omega ]d\omega$ as a function of temperature and doping. The structural transition $T_S\left(x\right)$, the magnetic transition $T_{\text{SDW}}\left(x\right)$, and the superconducting transition temperature $T_c\left(x\right)$ (from Ref. [39]) are indicated by blue triangles, purple squares, and red circles, respectively. $\theta \left(x\right)$ is the mean-field transition temperature associated with the critical behavior of ${\chi }_0^{XY}(T,x)$. (b) ${\chi }_0^{XY}(T,x)$ is shown with a universal fit to $A/[T-\theta (x\left)\right]$, where the temperature axis for each doping $x$ is shifted by $\theta \left(x\right)$. The inset shows the inverse of ${\chi }_0^{XY}(T,x)$ vs temperature $T-\theta \left(x\right)$ with a fit to a universal straight line.

Figure 4

Two-component fit of the NMR relaxation rate $1{/}^{75}{T}_{1}T$ for As in ${\mathrm{NaFe}}_{1-x}{\mathrm{Co}}_x\mathrm{As}$ compared to the static Raman susceptibility ${\chi }_{XY}^{\prime }(0,T)$. Temperature dependence of the NMR relaxation rate $1{/}^{75}{T}_{1}T$ for As (black triangles) compared to the static Raman susceptibility ${\chi }_{XY}^{\prime }(0,T)$ (red circles). $1{/}^{75}{T}_{1}T$ is decomposed into two contributions, $1/T_{1}T={(1/T_{1}T)}_{\text{intra}}+{(1/T_{1}T)}_{\text{inter}}$, where ${(1/T_{1}T)}_{\text{intra}}=C/\left(T\text{−}\theta \right)$ (yellow shades) and ${(1/T_{1}T)}_{\text{inter}}=\tilde{\alpha }+\tilde{\beta }exp(-\tilde{\mathrm{\Delta }}/k_{B}T)$ (blue squares). ${\chi }_{XY}^{\prime }(0,T)$ scales to ${(1/T_{1}T)}_{\text{intra}}$ for all three dopings $x=0$, 0.025, and 0.06, and we attribute $\theta$ to correspond to the Pomeranchuk transition temperature $\theta$ described in the main text, and where the red line is the Curie-Weiss fit. For the fits of $1/T_{1}T$, we have used $\tilde{\mathrm{\Delta }}/k_B=350$ or $\tilde{\mathrm{\Delta }}=240$ ${\mathrm{cm}}^{-1}$. This value of $\mathrm{\Delta }$ corresponds to the minor mode that is present in ${\chi }_{XY}^{\prime \prime }(\omega ,T)$ for all dopings and temperatures above $T_S\left(x\right)$. (a) NMR data for $x=0$ are from Ref. [20]. (b) NMR data for $x=0.025$ are from Ref. [78]. (c) NMR data for $x=0.06$ are from Ref. [79].

Figure 5

Decomposition of the Raman susceptibility ${\chi }_{XY}^{\prime \prime }(\omega ,T,x)$ in the $B_{2g}$ symmetry channel. (a) ${\chi }_{XY}^{\prime \prime }(\omega ,T,x)$ (black dots) for the representative doping $x=0.0165$ and temperature 55 K in the tetragonal phase. The red line is a fit to the data, which are decomposed into three components: a continuum background (blue shading), a Lorentz oscillator (magenta shading), and a relaxational mode (green shading), $A\left(x\right){\omega }_P(T,x){[{\omega }_P(T,x)-i\omega ]}^{-1}$. Parts (b)–(d) show the doping dependence of the intensity of the continuum (b), the relaxational mode (c), and the oscillator (d) above $T_S\left(x\right)$ and $T_c$. The dashed red line near $x\simeq 0.025$ indicates the doping at which the intensity in (b) and (d) becomes negligible. (e) Doping dependence of the mean-field transition temperature $\theta \left(x\right)$.

Figure 6

Illustration of symmetry modes in momentum space. (a) Neutral quadrupole charge-density modulation with $d^{\pm }$ symmetry in the $B_{2g}$-symmetry channel. In momentum space, the mode is illustrated by positive and negative regions of the BZ. The mode is sustained by $d_{xz}-d_{yz}$ intraband and interband transitions at the $\mathrm{\Gamma }$ and $M$ point, respectively. The coupled interband electron-hole excitations between participating bands at the $\mathrm{\Gamma }$ and $M$ point help to stabilize the $d^{\pm }$ mode. (b) $s^{\pm }$ breathing mode in $A_{1g}$ symmetry.

Figure 7

Raman susceptibility ${\chi }_{XY}^{\prime \prime }(\omega ,T)$ in the $B_{2g}$ symmetry channel at dopings $0\le x\le 0.0175$. (a)–(c) Temperature and frequency evolution of $\text{Log}\left[{\chi }_{XY}^{\prime \prime }(\omega ,T)\right]$ at dopings $x=0$, 0.015, and 0.0175. The structural transition $T_S\left(x\right)$ is indicated on the temperature axis, and the coherence peak, $2{\mathrm{\Delta }}_{\text{DW}}$, on the frequency axis for $x=0$ and 0.015. (d)–(f) ${\chi }_{XY}^{\prime \prime }\left(\omega \right)$ for $T\le 100$ K displaced vertically for clarity. All dopings show the development of the relaxational mode (RM) in the tetragonal phase described by $A\left(x\right){\omega }_P(T,x){[{\omega }_P(T,x)-i\omega ]}^{-1}$; $x=0$ and 0.015 show the development of the coherence peak and spectral weight suppression in the orthorhombic phase; $x=0.0175$ shows the emergence of the sharp resonance in the superconducting phase. The insets display ${\omega }_P(T,x)$ vs temperature.

Figure 8

Raman susceptibility ${\chi }_{XY}^{\prime \prime }(\omega ,T)$ in the $B_{2g}$-symmetry channel at dopings $0.02\le x\le 0.08$. (a)–(e) ${\chi }_{XY}^{\prime \prime }(\omega ,T)$ at various temperatures displaced vertically for clarity. All dopings show the development of the relaxational mode (RM) in the tetragonal phase described by $A\left(x\right){\omega }_P(T,x){[{\omega }_P(T,x)-i\omega ]}^{-1}$ and the emergence of the sharp resonance in the superconducting phase.

Figure 9

Raman susceptibilities ${\chi }_{XX}^{\prime \prime }\left(\omega \right)-{\chi }_{xy}^{\prime \prime }\left(\omega \right)$ and ${\chi }_{XY}^{\prime \prime }\left(\omega \right)$ in the superconducting state. (a) ${\chi }_{XX}^{\prime \prime }\left(\omega \right)-{\chi }_{xy}^{\prime \prime }\left(\omega \right)$ (top row) and ${\chi }_{XY}^{\prime \prime }\left(\omega \right)$ (bottom row) in the superconducting (5 K) and normal (23 K) states at doping levels as indicated. The vertical dashed line, shown for $x=0.0175$ and 0.05, indicates the lowest superconducting gap determined by ARPES at ≃9 meV [72] and ≃10 meV [73], respectively.

Figure 10

Doping dependence of superconducting gaps, in-gap collective modes, and their width. (a) Doping dependence of the superconducting gaps and in-gap collective modes from Fig. 9. (b) Normalized ${\omega }_{B_{2g}}$ at various dopings from Fig. 9.

Figure 11

Raman susceptibility ${\chi }_{XY}^{\prime \prime }\left(\omega \right)$ in the superconducting state. ${\chi }_{XY}^{\prime \prime }\left(\omega \right)$ in the $B_{2g}$-symmetry channel in the superconducting (3 K) and normal (23 K) state for a laser excitation of 647 nm (1.91 eV) at doping levels as indicated.

Figure 12

Energy diagram of the superconducting state. (a) Energy diagram of the superconducting state including the superconducting gap $2\mathrm{\Delta }$ and the in-gap collective modes ${\omega }_{A_{1g}}^{p\text{−}h},{\omega }_{B_{2g}}^{p\text{−}h}$, and ${\omega }_{B_{2g}}^{p\text{−}p}$ shown together with their spectroscopic signatures in the Raman data. The red horizontal dashed line indicating the lowest superconducting gap coincides with that determined by ARPES at ≃9 meV for $x=0.0175$ [72]. The $A_{1g}$ spectrum for $x=0.0175$ (green) was obtained at 5 K with an excitation energy of ${\omega }_L=2.6$ eV, and the $B_{2g}$ spectra for $x=0.0165$ (orange) and $x=0.0175$ (purple) were obtained at 3 K and ${\omega }_L=1.91$ eV. The modes determined from a two-band-model calculation (not included) are shown together with the $A_{1g}$ and $B_{2g}$ spectra for illustration, and they have the area fully colored below the modes of ${\omega }_{A_{1g}}^{p\text{−}h}$ (green), ${\omega }_{B_{2g}}^{p\text{−}h}$ (dark blue), and ${\omega }_{B_{2g}}^{p\text{−}p}$ (light blue). (b) Illustration of the symmetry of the BZ for the ${\omega }_{A_{1g}}^{p\text{−}h}$ and ${\omega }_{B_{2g}}^{p\text{−}h}$ modes in the $p\text{−}h$ channel having, respectively, $s^{\pm }$ and $d^{\pm }$ symmetry (see Appendix pp5).

Figure 13

Correction of Raman spectra illustrated for a doping $x=0.02$ and temperature 5 K in the $XY$ symmetry channel. (a) Raman scattering intensity (black solid line) shown together with a luminescence background $L\left(\omega \right)$ (red dashed line) to be subtracted. (b) Raman susceptibility after background subtraction and conversion to ${\chi }_{XY}^{\prime \prime }(\omega ,T,x)$ (see the text).

Figure 14

Decomposition of the Raman susceptibility ${\chi }_{XY}^{\prime \prime }(\omega ,T,x)$ in the $B_{2g}$-symmetry channel. ${\chi }_{XY}^{\prime \prime }(\omega ,T,x)$ with the three-component fit for $x=0$ in a range of temperatures above the structural transition showing intensity vs frequency and temperature.

Figure 15

Momentum- and frequency-resolved electronic spectra $A(\mathbf{\text{k}},\omega )$ along the $\mathrm{\Gamma }\text{−}M$ high-symmetry line. (a) DFT without spin-orbit coupling (SOC). (b) DFT with SOC. (c) DFT + DMFT without SOC. (d) DFT + DMFT with SOC. Without SOC, the eigenvalues of the electronic states with $xz$ and $yz$ orbital character are degenerate at the zone center $\mathrm{\Gamma }$ point due to the fourfold symmetry of the tetragonal crystal structure in the paramagnetic state. Including SOC lifts this degeneracy, leading to a splitting of the $xz/yz$ states, roughly 70 meV at the $\mathrm{\Gamma }$ point in the static mean-field treatment (DFT+SOC). However, strong electronic correlations strongly renormalize this static mean-field splitting to ≃10 meV in the DFT+DMFT+SOC calculation.