Emergence of the nematic electronic state in FeSe

We present a comprehensive study of the evolution of the nematic electronic structure of FeSe using high-resolution angle-resolved photoemission spectroscopy (ARPES), quantum oscillations in the normal state, and elastoresistance measurements. Our high-resolution ARPES allows us to track the Fermi surface deformation from fourfold to twofold symmetry across the structural transition at $\sim 87\mathrm{K}$, which is stabilized as a result of the dramatic splitting of bands associated with $d_{xz}$ and $d_{yz}$ character in the presence of strong electronic interactions. The low-temperature Fermi surface is that of a compensated metal consisting of one hole and two electron bands and is fully determined by combining the knowledge from ARPES and quantum oscillations. A manifestation of the nematic state is the significant increase in the nematic susceptibility approaching the structural transition that we detect from our elastoresistance measurements on FeSe. The dramatic changes in electronic structure cannot be explained by the small lattice distortion and, in the absence of magnetic fluctuations above the structural transition, point clearly towards an electronically driven transition in FeSe, stabilized by orbital-charge ordering.

Figure 1

Temperature dependence of the hole bands in FeSe. (a) and (b) Band-structure calculation of the Fermi surface of FeSe in the tetragonal phase, projected into the $k_z=0$ plane and colored by the dominant orbital character; the high-symmetry cuts are indicated. Calculations predict three hole pockets around the $\Gamma$ point and two electron pockets around the $M$ point. (c) The high-symmetry $M\text{−}\Gamma \text{−}M$ cut (cut 1) at low temperature (10 K). (d)–(f) The temperature dependence of the $A\text{−}Z\text{−}A$ cut at the $Z$ point ($k_z=\pi /c$ plane). (g) and (h) Extracted band dispersion from constrained multiple Lorentzian fits to momentum-distribution curves (MDC, solid symbols) or peaks in the energy dispersive cut (EDC, open symbols) from the data in (d) and (f). The solid lines in (g) are the renormalized band structure from (a). The insets show a schematic Fermi surface. (i) MDC for the hole bands ($\alpha$ and $\beta$) at high and low temperatures. At 10 K, the $\alpha$ splits near the Fermi level, resulting in two crossings due to formation of twin domains. (j) Fermi surface map at 10 K indicating the elongation of the hole pocket combined with a duplicate rotated by ${90}^{\circ }$ caused by twin domains. (k) Photon-energy dependence of the MDC at $E_F$ (equivalent to scanning $k_z$ in the $\Gamma \text{−}Z$ direction) that shows a quasi-two-dimensional hole band at 10 K.

Figure 2

Development of a strongly elongated electron Fermi surface through the structural transition. (a) Temperature dependence of high-symmetry cut through $M$, showing the development of a splitting of intensity, which results from strong orbital-dependent shifts of the band structure with onset at $T_s=87$ K. (b) The calculated band dispersions in the tetragonal structure. (c) The experimental band structure at 120 K (solid dots) renormalized to the high-temperature data (solid lines); the outer $d_{xy}$ electron band ($\delta$) is not observed at all at the $M$ point. (d) The lifting of $d_{xz/yz}$ degeneracy at 10 K and the consequent band shifting for two different domains, ${\epsilon }_1$ and ${\epsilon }_2$, indicated by the solid and dashed lines, respectively. The insets show the schematic band dispersion for different domains and directions, as detailed in Fig. 6. (e) Temperature dependence of the EDC at the $M$ point, showing the development of the splitting of bands with $d_{xz}$ and $d_{yz}$ character; the splitting $\Delta$ is plotted in Fig. 4. (f) Fermi surface map (integrated within 2 meV of the Fermi level) showing the cross-shape arising from strongly elongated Fermi surfaces at $M$ in the two domains, shown schematically in the inset.

Figure 3

Quantum oscillations in FeSe. (a) Magnetoresistance of FeSe for a field applied along the $c$ axis ($\theta =0$) with the irreversibility field, $B_{\mathrm{irr}}$ around 14 T at 0.36 K. (b) The oscillatory part of the resistivity $\Delta {\rho }_{\mathrm{osc}}$, obtained by subtracting a polynomial background from the raw data. (c) Fourier transform (FFT) of the data in (b) identifying four different frequencies $F_{1-4}$, corresponding to $k$-dependent extremal areas on the Fermi surface. (d) Temperature dependence of the amplitude of oscillation, from which the effective masses may be extracted. (e) Quantum oscillations and (f) the corresponding Fourier transforms as a function of angle $\theta$ of applied magnetic field with respect to the $c$ axis at 0.36 K. By plotting $Fcos\theta$ against $\theta$ one can identify if orbits correspond to a maximum or minimum of a quasi-two-dimensional Fermi surface. Color bar corresponds to the Fourier transform amplitude.

Figure 4

Symmetry-breaking of the electronic structure of FeSe. (a) Band-structure calculation of the Fermi surface in the high-temperature $P4/nmm$ tetragonal space group. (b) Schematic and (c) the experimental high-temperature in-plane Fermi surface consists of significantly shrunk pockets. (d) Elongated hole and electron pockets below the structural transition. (e) Schematic and (f) experimental low-temperature Fermi surface including the effect of twin domains (the $d_{xy}$ electron pocket is not plotted here). (g)–(i) Three-dimensional representations of Fermi surface of FeSe, as described in a, b, and d, respectively. (j) Resistivity as a function of temperature for the sample used in Fig. 3. (k) The induced change in resistivity to in-plane strain measured by the $m_{66}$ parameter from elastoresistance tensor in FeSe, which provides a direct measure of the electronic nematic order parameter, indicating that the structural transition is electronically-driven, as discussed in the main text. (l) Temperature dependence of the intense $k_{F1}$ for the $\epsilon$ pocket around the $M$ point, showing the Fermi surface deformation which onsets at $T_s$. (m) Energy splitting of the bands with $d_{xz}$ and $d_{yz}$ band character at the $M$ point, extracted from Fig. 2.

Figure 5

The effect of matrix element effects in ARPES experiments on FeSe at low temperatures. Low-temperature cuts at high-symmetry points in the Brillouin zone performed on an additional sample S3. The measurements were performed for two different incident beam polarizations: linear horizontal, LH,($p$) (top row) or linear vertical, LV, ($s$) (bottom row). All samples measured were carefully aligned such that the $\Gamma \text{−}M$ direction corresponded to the scattering plane (equivalently, the local coordinate frame of the Fe atoms is aligned to the scattering plane). In this geometry, matrix-element effects allow us to select predominantly bands with $d_{xz}$ (LH) or $d_{yz}$ (LV) orbital character. Further details can be found in Ref. [18]. As detailed in Fig. SM2, the spin-orbit coupling mixes the orbital characters of different bands near anticrossings and also at the high-symmetry $\Gamma$ point and this effects allows the $d_{xy}$ band to be observed near $\Gamma$($Z$) in LH, otherwise the ARPES signal for $d_{xy}$ band due to the matrix elements effects [18] would be very small at $\Gamma$, or not even observed, as in the case of the $M$ point (Fig. 2). At the $\Gamma$ point, there is a mixing of orbital character due to spin-orbit coupling (as described below), as shown in the LV polarization cuts at $\Gamma$ and $Z$ in (b) and (d). The $Z$* and $A$* cuts are performed at 50 eV, which are close to the next symmetry points at the top of the Brilloiun zone besides those at 23 eV shown in Fig. 1. The photon energy or the $k_z$ dependence reveals the degree of corrugation of the two-dimensional bands; the hole pocket $\alpha$ around $\Gamma$($Z$) in (a) and (b) is more strongly-corrugated than the electron pocket around $M$(A) in (f) and (h).

Figure 6

Schematic representation of the development of the dramatic Fermi surface deformation at the $M$ point. (a)–(c) Schematic high-temperature cuts and Fermi surface around the $M$ point; there is a fourfold degeneracy (even with spin-orbit coupling [19]) of the $d_{xz}$ and $d_{yz}$ bands dispersing along the $k_x$ and $k_y$ directions (similar to a double saddle-point). This degeneracy is lost when the fourfold symmetry of the lattice is broken, and the $d_{xz}$ and $d_{yz}$ bands separate. The $d_{xy}$ band is not observed at the $M$ point at any temperature and is therefore shown as a dashed line only in (a)–(c). (d)–(f) Schematic of the low-temperature Fermi surface and high-symmetry cuts for one domain only when the observed pocket becomes highly elongated. The band with $d_{yz}$ orbital character moves up, whereas the $d_{xz}$ band is pushed down away from the Fermi level as suggested by experiments on detwinned crystals from Ref. [11]. The positions of the band crossings (band minima of the band crossing at the Fermi level) are marked by black dots. Consequently, at the Fermi level, the $d_{yz}$ portion of the Fermi surface contracts, whereas the $d_{xz}$ portion expands. (g)–(h) Schematic of the experimental data at $M$, formed by a superposition of dispersions originating from twin domains that are at ${90}^{\circ }$ with respect to each other (solid and dashed lines). The beam spot is sufficiently large that we expect to always observe the superposition of bands in the two structural domains. The contribution of the second domain to the experimental $k_y$ cut is equivalent to the dispersion along $k_x$ but with orbital characters reversed ($x\leftrightarrow y$). Therefore the observed experimental cuts have four observed dispersions that result in a cross-shapes Fermi surfaces in Figs. 2 and 4 of the main text. Although the $d_{xy}$ band is not observed at the $M$ point in ARPES, we can assign the $F_3$ frequency in quantum oscillations with the heaviest mass to the $d_{xy}$ pocket, which needs to break the rotational symmetry, similar to the other pockets.

Figure 7

Band-structure calculations of FeSe using relaxed and experimental structures. (a) Band structure calculations performed using wien2k [45] based on the relaxed crystal structure of Ref. [16] and including spin-orbit coupling. A comparison to a calculation without spin-orbit coupling is presented in (b) where there are no spin-orbit induced band anti-crossings, and additionally the $d_{xz}$ and $d_{yz}$ orbitals are degenerate at $\Gamma$. (c) Calculation based on the experimental structure reported in Ref. [9], including spin-orbit coupling. The main difference between the two crystallographic structures in (a) and (c) is the Se position, $z_{\mathrm{Se}}$, above the Fe plane, significantly shorter for the relaxed structure, $z_{\mathrm{Se}}=0.24128$ [16], as compared with the experimental one, $z_{\mathrm{Se}}=0.26668$ [9]). (d) Detailed cuts of the relaxed structure near the $\Gamma$ point showing only the $d_{yz}$ weight. As well as lifting the degeneracy at the $\Gamma$ point, spin-orbit coupling mixes the orbital characters of bands near anticrossings and also at the high-symmetry $\Gamma$ point. (e) Calculation of band dispersion in the low-temperature $Cmma$ space group (when the fourfold rotational symmetry is broken) taking the structure of Ref. [46]. A splitting of bands at the $M$ point is now allowed and is calculated to be 17 meV, in the absence of any band renormalization effects.

Figure 8

Comparison between the predicted and measured values of quantum oscillations frequencies. (a) Predictions of quantum oscillation frequencies as a function of angle in FeSe, using the $P4/nmm$ relaxed structure of Ref. [16]. $\theta$ is defined as the angle of rotation from the $c$ axis to the $a$ axis. Black stars are experimental data points. Inset: corresponding Fermi surface. The predicted quantum oscillation frequencies (calculated using the SKEAF algorithm [47]) consists of frequencies up to $\sim 2.6$ kT arising from quasi-2D bands. The calculation are performed for the high-temperature tetragonal phase; in the orthorhombic paramagnetic case, by including the small structural distortion, $2\times {10}^{-3}$, we find very little difference to the shape of the Fermi surface or predicted quantum oscillations frequencies. (b) Calculated quantum oscillation frequencies at $\theta =0$ as a function of rigid band shift. Large shifts of $\sim 200$ meV would be required to bring the calculated frequencies down to the order of the experimental frequencies. Like in other iron-based superconductors, the ground state in DFT of FeSe is not a paramagnetic state but a collinear antiferromagnetic stripe-order, with other possible magnetic orders close in energy [16]. Studies of quantum oscillations of the magnetically ordered phase typically reveal small pockets, originated from a reconstructed Fermi surfaces [48, 49]. A magnetically reconstructed Fermi surface would produced frequencies of comparable sizes to those observed in FeSe, however, FeSe does not undergo any magnetic transition and our ARPES measurements do not show any evidence of band folding to support this scenario.

Figure 9

Elastoresistance measurements of FeSe. Elastoresistivity measurements as a function of temperature at (a) 105 and (b) 40 K, with strain induced by a varying voltage to a piezoelectric stack, as discussed in detail in Ref. [5]. Samples were bonded to the top surface of the piezostack with the crystallographic [110] ${}_{\text{tetragonal}}$ axis (checked with x-ray diffraction) parallel or perpendicular to the piezostack long axis, as shown in the inset. Resistive strain gauges were bonded to the bottom surface of the piezostack to monitor the applied strain by sweeping the voltage applied to the piezostack. Due to the biaxial strain of the piezostack, the fractional change in resistivity, $\Delta \rho /{\rho }_0$ (normalised to the zero-strain value ${\rho }_0$), were made either perpendicular (${\rho }_{xx}$) or parallel (${\rho }_{yy}$) to the piezostack long axis; ${\epsilon }_{xx}$ and ${\epsilon }_{xx}$ is the uniaxial strain component along the perpendicular and the parallel directions, respectively. There is a sign change as a function of temperature for $\Delta \rho /{\rho }_0$ in (a) and (b). (c) The variation of the elastoresistance tensor component, $m_{66}$, as a function of temperature. The $2m_{66}$ parameter is the ratio between the subtracted parallel ($yy$) and perpendicular ($xx$) components of the fractional change in resistivity and the applied strained, being defined as $2m_{66}=(\Delta {\rho }_{xx}/{\rho }_0-\Delta {\rho }_{yy}/{\rho }_0)/({\epsilon }_{xx}-{\epsilon }_{yy})$, as detailed in Ref. [5]. The $m_{66}$ increases dramatically towards the structural transition, which in this experimental set up is at $\sim 92$ K. The solid line corresponds to a fit to the function $m_{66}=A/(T-T^{*})+A_0$, which gives $T^{*}\sim 66\left(1\right)$ K and $A_0=-31\left(3\right)$. There is an unusual sign change of $m_{66}$ as a function of temperature below 65 K, as discussed in the main text.