Frustrated spin order and stripe fluctuations in FeSe A. Baum,1, 2 H. N. Ruiz,3, 4 N. Lazarevi´c,5 Yao Wang,3, 6 T. B¨ohm,1, 2 R. Hosseinian Ahangharnejhad,1, 2, † P. Adelmann,7 T. Wolf,7 Z. V. Popovi´c,5, 8 B. Moritz,3 T. P. Devereaux,3, 9 and R. Hackl1, ∗
arXiv:1709.08998v1 [cond-mat.str-el] 26 Sep 2017
1
Walther Meissner Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany 2 Fakult¨ at f¨ ur Physik E23, Technische Universit¨ at M¨ unchen, 85748 Garching, Germany 3 Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA 4 Department of Physics, Stanford University, California 94305, USA 5 Center for Solid State Physics and New Materials, Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia 6 Department of Applied Physics, Stanford University, California 94305, USA 7 Karlsruher Institut f¨ ur Technologie, Institut f¨ ur Festk¨ orperphysik, 76021 Karlsruhe, Germany 8 Serbian Academy of Sciences and Arts, Knez Mihailova 35, 11000 Belgrade, Serbia 9 Geballe Laboratory for Advanced Materials, Stanford University, California 94305, USA (Dated: September 27, 2017)
The charge and spin dynamics of the structurally simplest iron-based superconductors, FeSe, may hold the key to understanding the physics of high temperature superconductors in general. Unlike in the iron pnictides, long range magnetic order is not found in FeSe in spite of the observation of the same structural transition around 90 K. In this study we report results of Raman scattering experiments as a function of temperature and polarization and and compare them to exact diagonalization data of a frustrated Heisenberg model. Both in experiment and theory we find a persistent low energy peak in B1g symmetry, which softens slightly around 100 K, and assign this feature to spin excitations. Combined with related simulations of neutron scattering data, this study provides evidence for nearly frustrated stripe order and fluctuations of spin stripes with a critical ordering vector close to or at (π, 0). PACS numbers: 74.70.Xa, 75.10.Jm 74.20.Mn, 74.25.nd
Fe-based pnictides and chalcogenides, similarly as cuprates, manganites or some heavy fermion compounds, are characterized by the proximity and competition of various phases including magnetism, charge order and superconductivity. Specifically the magnetism of Febased systems has various puzzling aspects which do not straightforwardly follow from the Fe valence or changes in the Fermi surface topology [1, 2]. Some systems have a nearly ordered localized moment close to 2 µB such as FeTe or rare-earth iron selenides, whereas others do not order down to the lowest temperatures such as FeSe [3] or LaFePO [4]. In contrast, the moments of AFe2 As2 -based compounds (A = Ba, Sr, Eu or Ca) are slightly below 1 µB and display aspects of itinerant spin-density-wave (SDW) magnetism with a gap in the electronic excitation spectrum. The material specific differences are a matter of intense discussion and low- as well as high-energy electronic and structural properties may contribute [1, 5–8]. In fact the main fraction of the electronic density of states at the Fermi energy EF originates in t2g Fe orbitals, and a substantial part of the Fe-Fe hopping occurs via the pnictogen or chalcogen atoms, hence via the xz and yz orbitals. For geometrical reasons the resulting exchange coupling
†
∗
School of Solar and Advanced Renewable Energy, Department of Physics and Astronomy, University of Toledo, Toledo, Ohio 43606, USA
[email protected]
energies between nearest (J1 ) and next nearest neighbour (J2 ) iron atoms have the same order of magnitude, and small changes in the pnictogen (chalcogen) height above the Fe plane influence the ratio J2 /J1 in a way that various types of order are energetically very close [8]. Due to the reduced overlap of the in-plane xy orbitals the hopping integral is reduced and the correlation energy U , although only in the range 1-2 eV, and the Hund’s rule interactions gain influence. Thus the electrons in the xy orbitals have a considerably higher effective mass m∗ and smaller quasiparticle weight Z than those of the xz and yz orbitals. This effect was coined orbital selective Mottness [9–11] and observed by photoemission spectroscopy in Fe-based chalcogenides [12]. It is similar in spirit to what was found by Raman scattering in the cuprates as a function of momentum [13]. Here we address the question as to whether some systematics can be found across the families of the Fe-based systems and also in relationship to other highly correlated systems. Using Raman scattering we specifically study the type of magnetic ordering in FeSe and find substantial differences to other families of Fe-based systems but similarities with the cuprates. To set the stage we show Raman spectra of YBa2 Cu3 O6.05 (YBCO) in Fig. 1 (a) as an example of a N´eel-type antiferromagnet with localized moments [14]. A well-defined peak is observed in B1g (x2 −y 2 ) symmetry (see Methods) at approximately 2.84 J1 as predicted theoretically [15, 16]. The A1g (x2 + y 2 ) maximum appears at a similar position. The response in the two other sym-
Rχ'' (Ω,T) (counts s-1 mW -1)
2
60 50
a
YBa2Cu3O6.05 A1g A2g
40
B1g
30
B2g
1.0
b
0.8
BaFe2As2 B1g
0.6 0.4
T = 300 K T = 150 K 0.2 T = 130 K T = 50 K 0 4000 8000 0 1000 2000 Raman Shift Ω (cm-1)
20
T = 200 K
10 0 0 c
d
K
J2
J3
J1
FIG. 1. Localized and itinerant magnets. (a) Raman spectra of YBa2 Cu3 O6.05 at all four symmetries accessible with light polarizations in the CuO2 plane. From Ref. [22]. (b) B1g spectra of BaFe2 As2 at four characteristic temperatures as indicated. (c) A 4 × 4 cluster used for the simulations. The red spheres represent the Fe atoms, each of which carries localized spin Si , with S = 1. The dashed lines represent the exchange interactions considered in this study, where K is the coefficient of a biquadratic term proportional to (Si · Sj )2 . (d) J2 − J3 phase diagram as obtained from our simulations at T = 0. The black point shows the parameters at which temperature-dependent simulations have been performed.
metries is weak and nearly featureless. The temperature dependence is weak for kB T < 0.25 J1 and continuous above [17]. The origin of the scattering in YBCO can be traced back to Heisenberg-type physics of localized moments including only the nearest-neighbour exchange interaction J1 . In contrast, the iron-based compounds are metallic antiferromagnets in the parent state and exhibit rather different Raman signatures. In BaFe2 As2 [Fig. 1 (b)] abrupt changes are observed in B1g symmetry upon entering the SDW state: the fluctuation peak below 100 cm−1 vanishes, a gap develops below some 500-600 cm−1 , and intensity piles up in the range 600-1,500 cm−1 [18, 19]. This behaviour is typical for a SDW or charge density wave (CDW) [20]. Here, the physics of the observed scattering is that of weak-coupling density-wave order resulting from Fermi surface nesting. Yet, for itinerant systems such as these, longer range exchange interactions become relevant and can lead to magnetic frustration[21]. There is no consensus on the limit which best describes the Fe-based systems. Early work considered Heisenbergtype magnetism the most appropriate approach [23], and
the high-energy maxima observed by Raman scattering in AFe2 As2 -based compounds were interpreted in terms of localized spins [19, 24]. The low-energy spectra are reminiscent of CDW or SDW formation [19, 20, 25, 26]. In principle, both effects can coexist if the strength of the correlations varies for electrons from different orbitals. The itinerant electrons form an SDW while those on localized orbitals give rise to a Heisenberg-like response. In contrast to the AFe2 As2 -based compounds, FeSe seems to be closer to localized order. Apart from low lying charge excitations, the remaining, presumably spin, degrees of freedom in FeSe may be adequately described by a spin-1 J1 -J2 -J3 -K Heisenberg model [8] (see Fig. 1 c). This model allows for the presence of different spin stripe orders as indicated in the phase diagram in Fig. 1d [8]. Since various types of spin order are energetically in close proximity [8, 27, 28], frustration may quench longrange order down to the lowest temperatures [3]. Large energy scales were found in neutron scattering experiments in FeSe for both the exchange energies and the fluctuations [28, 29]. Raman scattering can contribute here for covering an energy range of approximately 1 meV to 1 eV and being sensitive to spin excitations [30]. In a recent Raman experiment, the response of FeSe was studied, but the analysis was limited to the results in B1g symmetry and low energies, focusing on charge excitations [31]. In the study presented here, we obtain similar experimental results at low energies, where the data can be compared. In addition, we include higher energies and analyze all symmetries, which will turn out to be crucial. By comparing experimental and simulated Raman data we find a persistent low energy peak in B1g symmetry, which softens slightly around 100 K. We assign the B1g maximum and the related structures in A1g and B2g symmetry to spin excitations. This along with comparisons to neutron scattering data provide evidence for nearly frustrated stripe order and fluctuations of spin stripes with a critical ordering vector close to or at (π, 0). We arrive at the conclusion that frustrated order of localized spins dominates the physics in FeSe.
RESULTS
Symmetry-resolved Raman spectra of single-crystalline FeSe (see Methods) at 40, 90 and 300 K in the energy range up to 0.45 eV (3,600 cm−1 ) are shown in Fig. 2. The data are linear combinations of the polarization dependent raw data (see Methods and Appendix A). Out of the four symmetries shown in the figure the A1g , B1g , and B2g spectra display Raman active phonons, magnons or electron-hole excitations. Intensity in A2g symmetry appears only if chiral or crystal-field excitations contribute or if the off-diagonal elements of the Raman tensor αij are different, αxy 6= αyx , for instance because of resonances. Since the A2g spectra are weaker than the other spectra and vanish below 500-1,000 cm−1 , we ignore them
3
-1 -1
4 0 K 9 0 K T = 3 0 0 K
0 .5 B
1 g
c
0 A
2 g
0 .2 0 d
0 .2 0
B 0
1 k
2 k 3 k 0 1 k R a m a n S h if t Ω( c m
2 k -1
3 k
2 g
a
th e o ry T = 0 K
0 .5 0
0 .5 b
0 .4 1 g
B
1 g
B
2 g
x 0 .0 2
A
0 .2 5
0 .3 0 .2
0
0 .1 0
1 k 2 k 3 k R a m a n S h if t Ω( c m
0
-1
1
2
3 ) R a m a n S h if t Ω( J 1 )
χ' ' ( a r b . u n i t s )
m W -1
R χ' ' ( Ω, T ) ( c o u n t s s
T = T =
F e S e T = 4 0 K
1 g
0 .2 1 .0
0 .7 5
)
0 .4 A
-1
)
b
m W
F e S e a
R χ' ' ( Ω, T ) ( c o u n t s s
1 .5
0
0
)
FIG. 2. Symmetry-resolved Raman spectra of FeSe at various temperatures for large energy transfers. (a) B1g spectra. (b) - (d) A1g , A2g and B2g spectra as indicated. In A1g and B2g symmetry particle-hole excitations dominate the response. In agreement with the simulations weak additional peaks from spin excitations appear at low temperature (shaded areas). The narrow lines close to 200 cm−1 are the A1g and B1g phonons. In the 1 Fe unit cell used here the B1g phonon appears in B2g symmetry since the axes are rotated by 45◦ with respect to the crystallographic (2 Fe) cell. (c) The A2g cross section is completely temperature independent.
here. In the high-energy limit the intensity is smaller than that in other Fe-based systems such as BaFe2 As2 (Fig. 1b and 8). However, in the energy range up to approximately 3,000 cm−1 there is a huge additional contribution to the B1g cross section in FeSe [Fig. 2a]. The response is strongly temperature dependent and peaks at 530 cm−1 in the low-temperature limit. Upon cooling from 300 to 40 K the intensity increases in the range between 2,000 and 3,000 cm−1 . Between 90 and 40 K the A1g and B2g spectra increase slightly in the range around 700 and 3,000 cm−1 , respectively [indicated as shaded areas in Fig. 2b and d]. The overall intensity gain in the A1g and B2g spectra in the shaded range is a fraction of approximately 5% of the B1g intensity. The B2g spectra indicate the existence of a pseudogap in the range up to 2,000 cm−1 which is already fully developed at the structural transition at Ts = 89.1 K. Currently we do not have an explanation but it is likely to be in the particle-hole channel since our simulations do not indicate any changes in the spin channel in this energy range. In Fig. 3 we show the low-temperature data along with the results of the finite cluster zero temperature numerical simulations for Raman scattering in the spin-1 frustrated Heisenberg model as described in the methods section. The energy scale for the simulations [Fig. 3b] is given in units of J1 which has been derived to be 123 meV or 990 cm−1 [8], allowing a semi-quantitative comparison with the experiment. As mentioned before the experimental A1g and B2g spectra are dominated by
FIG. 3. Symmetry-resolved Raman spectra of FeSe over a range of large energy shifts at low temperature. (a) Experimental results for symmetries as indicated at 40 K. The B1g peak at 500 cm−1 dominates the spectrum. In A1g and B2g symmetry the electron-hole continuum dominates the response, and the magnetic contributions yield only small additional contributions at approximately 700 and 3,000 cm−1 , respectively. (b) Simulated Raman spectra at T = 0 K including only magnetic contributions. A1g and B1g symmetries have peaks solely at low energies whereas the B2g contributions are at high energies only. The B1g response shown was multiplied by a factor of 0.02.
electron-hole excitations, and the spin contributions are weak. The opposite is true for B1g symmetry. This difference is also borne out in the simulations which indicate a factor of 20 to 100 difference in the respective spectral weights. For the selected values of J1 = 123 meV, J2 = 0.528 J1 , J3 = 0, and K = 0.1 J1 , the positions of the spin excitation in the three symmetries are qualitatively reproduced. The choice of parameters is motivated by the previous use of the J1 -J2 Heisenberg model, with J1 = J2 to describe the stripe phase of iron pnictides [23]. Here we use a value of J2 smaller than J1 to enhance competition between N´eel and stripe orders when describing FeSe. At finite temperature between the structural transition at Ts = 89.1 K and approximately 200 K there is an additional structure in the B1g spectra in the range below 200 cm−1 [Fig. 2a] which develops in a fashion very similar to what was found in Ba(Fe1−x Cox )2 As2 [32–34]. The feature in FeSe was observed before, modeled by a Drude-like response as derived by Gallais and coworkers [35], and interpreted in terms of quadrupolar orbital fluctuations [31]. The comparison of the different scattering symmetries and our simulations indicate that this excitation is at least partly an additional scattering channel superimposed on the particle-hole and the spin response as shown in Fig. 4. For modeling the entire response phenomenologically we first use the response χ00AL (Ω) expected for the exchange of two stripe-like fluctuations with critical wave vectors Q = (π, 0) (and equivalent directions) upon evaluating Aslamazov-Larkin-type of diagrams [34, 36]. This response is an additional independent contribution
0
4 0 0 8 0 0 R a m a n S h ift Ω
0 .5
0
2 5 0
3 0 a
F e S e
b
m W
0 .1 0
0
1 .5
5 0 0 R a m a n S h ift Ω ( c m
-1
7 5 0
1 0 0 0
th e o ry 2 5
T (J 1) 0 .0 8 0 .1 2 0 .2 0 .2 5
0 .5
T (K ) 2 1 6 1 9 1 0 0 2 5 0 5 0 0 7 5 R a m a n S h if t Ω(
2 0 0
1 .0
1 5 1 3 0 1 0 c m
-1
1 5 1 0
χ' ' ( a r b . u n i t s )
1 .0
2 g
-1
1 g
T = 9 0 K
m W -1
R χ' ' ( Ω, T ) ( c o u n t s s
R χ' ' ( Ω, T )
-1
)
B
B
-1
0 .2
F e S e
R χ' ' ( Ω, T ) ( c o u n t s s
1 .5
)
4
5
)
0
0 .2 5 0 .5 0 0 .7 5 1 R a m a n S h if t Ω( J 1 )
0
)
FIG. 4. Phenomenological description of the data at 90 K using the AL response [36] for the fluctuation peak at low energy. Black lines show Raman spectra. The blue line is the electron-hole continuum taken from the B2g spectrum shown in the inset. The two magnon contribution (green) is obtained by subtracting the red AL response and blue continuum from the black B1g spectrum.
on top of the dominating response of the two-magnon scattering centred at 530 cm−1 . This contribution could come from charge or spin stripe-like fluctuations. After subtracting χ00AL (Ω) (red) and an analytic approximation to the particle-hole continuum (blue) we recover the approximate shape of the response of the localized spins (green). We then compare the experiments with numerical simulation for the temperature dependence of the Raman susceptibility in Fig. 5. For the simulations we use the same parameters as at T = 0 (Fig. 3). At zero temperature the simulations show a single low energy B1g peak around 0.3 J1 . As temperature is increased a shoulder forms on the low energy side of the peak and the whole peak softens slightly for temperatures up to kB T = 0.12 J1 , corresponding to approximately 150 K. With the exception of intensity at low energies seen in the experiment at temperatures near the structural transition, there is good agreement in the dominant features of the Raman susceptibility between the theory and experiment. To support our explanation of the Raman data, we simulated the dynamical spin structure factor and compared our findings to results of neutron scattering [28]. While clearly finding no long-range order, above the structural transition neutron scattering finds similar intensity at finite energy for several wave vectors along the line (π, 0) − (π, π). Upon cooling, the spectral weight at these wave vectors shifts away from (π, π) to directions along (π, 0), although the respective peaks remain relatively broad. In Fig. 6(a) and (b) we show the results of the simulations for two characteristic temperatures. As temperature decreases, spectral weight shifts from (π, π) towards (π, 0) in agreement with experiment. In Fig. 6(c)
FIG. 5. Temperature dependence of the B1g response. (a) Experimental spectra at selected temperatures as indicated. The spectra include several excitations the decomposition of which is shown in Fig. 4. (b) Simulated Raman response at temperatures as indicated. Only magnetic excitations are included. The coupling constant was derived as J1 = 123 meV in Ref. [8].
we show the evolution of the spectral weights around (π, π) and (π, 0) in an energy window of (0.4 ± 0.1) J1 with temperature, similar to the integration intervals in Ref. [28]. In the experiment, the temperature where the integrated dynamical spin structure factor changes most dramatically occurs near the structural transition. From our simulations, the temperature where similar changes occur in comparison to neutron scattering corresponds to the temperature at which the simulated B1g response shows the most pronounced shoulder and the overall intensity begins to decrease. Not surprisingly, this also occurs near the structural transition in the Raman scattering experiment. The agreement of the experiment with the theory in both neutron and Raman scattering suggests that the dominant effect in FeSe is frustrated magnetism of essentially local spins. The differences between the classes of ferro-pnictides and -chalcogenides, in particular the different degrees of itineracy, may then originate in a subtle orbital differentiation across the families [1]. If FeSe is near a phase boundary then it is consistent with the sensitivity of FeSe to intercalation [37, 38] and layer thickness [39], which could affect the exchange interactions through the hopping. The exception to this agreement is the missing intensity in fluctuations below 200 cm−1 found in the B1g symmetry. This missing intensity is likely a combination of spin and orbital physics. In the case of orbital fluctuations the occupation of the xz and yz orbitals fluctuates, and charges are redistributed between the (π, 0) and (0, π) electron pockets in a quadrupolar fashion [31, 40]. In the spin sector two fluctuations having wave vectors (±π, 0) and (0, ±π) are exchanged simultaneously between the central hole and the electron pockets [34, 36]. Both scattering processes are compatible with B1g selection rules. The temperature dependence
5 T = 0.25 J1
K
1.5 1
0.5 0 0 2
0.5
1
b
1.5
2
T = 0.0 J1
1
0.5 0 0
0.00 0.01 Intensity (arb. units)
K
1.5
0.01
0.5
1
H
1.5
2
0.00
30
4.0
c
25
3.0
20
2.0
15
1.0
10
E = (0.4 ± 0.1) J1 0
0.1
T (J1)
0.2
Intensity x10-3 (arb. units)
a
Intensity (arb. units)
2
0.0
FIG. 6. Simulations of the dynamical structure factor of spin excitations. (a) and (b) Cuts through the first Brillouin zone for an energy window of 0.4 ± 0.1 J1 at T = 0.25 and 0 J1 , respectively. At high temperature there is intensity at (π, π) indicating the tendency towards N´eel order. At low temperature the intensity at (π, π) is reduced and the stripe-like antiferromagnetism with π, 0 ordering wave vector becomes stronger. (c) Integrated (π, π) and (π, 0) intensities as a function of temperature.
may pave the way towards distinction, since the fluctuations are expected to disappear in the ordered phase they precede [34]. Hence critical spin fluctuations may survive in FeSe down to the lowest temperatures [3]. One may argue that the (π, 0) fluctuations are strong above the structural transition, where orthogonal Fe-Fe bonds become inequivalent (corresponding to a distortion with (π, 0) wave vector), and gradually die out below Ts similarly as in the case of Ba(Fe1−x Cox )2 As2 . The reduction of the spectral weight of (π, 0) fluctuations once (π, 0) order starts to prevail explains the gap-like behaviour below the maximum in B1g symmetry. The appearance of superconducting gap excitations below Tc [31] (see also Fig. 9) demonstrates that the response from both local spins and fluctuations is superimposed on a particle-hole continuum. In summary, we have measured the Raman response of FeSe in all symmetries and compared to simulations of a frustrated spin-1 system on a finite size cluster. We decompose the experimental data in order to determine which parts of the spectra originate in particle-hole excitations, spin fluctuations, and low energy fluctuations described by Aslamazov-Larkin type of diagrams. Comparison of the experimental data with simulations gives evidence that the dominant contribution of the Raman spectra comes from a frustrated spin-1 system with competition between (π, 0) and (π, π) ordering vectors. These features of the Raman spectra, which agree qualitatively with a spin only model, consist of a dominant low energy peak in B1g symmetry along with a peak at similar energy but lower intensity in A1g and a higher energy peak
in B2g symmetry. This agreement suggests that frustrated magnetism is a dominant feature found in FeSe across a wide range of temperatures. This understanding will likely help with understanding the mechanism behind the superconducting phase found in FeSe.
ACKNOWLEDGEMENT
The work was supported by the German Research Foundation (DFG) via the Priority Program SPP 1458 (grant-no. Ha2071/7) and the Transregional Collaborative Research Center TRR80 and by the Serbian Ministry of Education, Science and Technological Development under Projects ON171032 and III45018. We acknowledge support by the DAAD through the bilateral project between Serbia and Germany (grant numbers 56267076 and 57142964). The collaboration with Stanford University was supported by the Bavarian Californian Technology Center BaCaTeC (grant-no. A5 [2012-2]). Work in the SIMES at Stanford University and SLAC was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Contract No. DE-AC02-76SF00515. Computational work was performed using the resources of the National Energy Research Scientific Computing Center supported by the U.S. Department of Energy, Office of Science, under Contract No. DE-AC02-05CH11231
AUTHOR CONTRIBUTIONS
A.B., T.B., and R.H. conceived the experiment. B.M. and T.P.D. conceived the ED analysis. P.A. and T.W. synthesized and characterized the samples. A.B., N.L., T.B., and R.H.A. performed the Raman scattering experiment. H.N.R. and Y.W. coded and performed the ED calculations. A.B., H.N.R., N.L., B.M., and R.H. analyzed and discussed the data. A.B., H.N.R., N.L., Z.P., B.M., T.P.D., and R.H. wrote the paper. All authors commented on the manuscript.
METHODS Experiment
The FeSe crystals were prepared by the vapor transport technique. Details of the crystal growth and characterization are described elsewhere [41]. Before the experiment the samples were cleaved in air. The surfaces obtained in this way have several atomically flat regions allowing us to measure spectra down to 5 cm−1 . At the tetragonal-to-orthorhombic transition Ts twin boundaries appear and become clearly visible in the observation optics. As described in detail by Kretzschmar et al. [34] the appearance of stripes can be used to determine the
6 laser heating ∆TL and Ts to be (0.5 ± 0.1) K/mW and (89.1 ± 0.2) K, respectively. Calibrated Raman scattering equipment was used for the experiment. The samples were attached to the cold finger of a He-flow cryostat having a vacuum of approximately 5 · 10−5 Pa (5 · 10−7 mbar). For excitation we used a diode-pumped solid state laser emitting at 575 nm (Coherent GENESIS) and various lines of an Ar ion laser (Coherent Innova 304). The angle of incidence was close to 66◦ for reducing the elastic stray light entering the collection optics. Polarization and power of the incoming light were adjusted in a way that the light inside the sample had the proper polarization state and, respectively, a power of typically Pa = 4 mW independent of polarization. For the symmetry assignment we use the 1 Fe unit cell (axes x and y parallel to the Fe-Fe bonds) which has the same orientation as the magnetic unit cell in the cases of N´eel or single-stripe order (4 Fe cell). The orthorhombic distortion is along these axes whereas the crystallographic cell assumes a diamond shape with the length of the tetragonal axes preserved. Because of the rotated axes in the 1 Fe unit cell the Fe B1g phonon appears in the B2g spectra.
Simulations
sum over nn is over nearest neighbours, the sum over 2nn is over next nearest neighbours, and the sum over 3nn is over next next nearest neighbours. We determine the dominant order according to the largest static spin structure factor, given by the following. S(q) =
(2)
l
Due to the possible spontaneous symmetry breaking we adjust the structure factor by the degeneracy of the momentum. To characterize the relative strength of the dominant fluctuations we project the relative intensity of the dominant static structure factor onto the range [0,1] using the following intensity = 1 −
dqnext S(qnext ) dqmax S(qmax )
(3)
where dq is the degeneracy of momentum q, qmax is the momentum with the largest dq Sq , and qnext is the momentum with the second largest dq Sq . The Raman susceptibility for B1g , B2g , and A1g symmetries for non-zero temperatures were calculated using the Fleury Loudon scattering operator[23] given by ˆ= O
We use exact diagonalization to study a Heisenberg Model on a 16 site square lattice, which contains the necessary momentum points and is small enough that exact diagonalization can reach high enough temperatures to find agreement with the temperature dependence in the experiment. This was solved using the parallel Arnoldi method [42]. The Hamiltonian is given by the following
1 X iq · Rl X e exp SRi +Rl · SRi N i
X
ˆ ij )(ˆ ˆ ij )Si · Sj Jij (ˆ ein · d eout · d
(4)
i,j
where Jij are the exchange interaction values used in ˆ ij is a unit vector connecting sites i the Hamiltonian, d ˆin/out are the polarization vectors. For the and j and e symmetries calculated we use the polarization vectors
(1)
1 1 ˆout = √ (ˆ ˆin = √ (ˆ x + yˆ), e x + yˆ) for A1g ⊕ B2g , e 2 2 ˆin = x ˆout = yˆ for B2g , e ˆ, e (5) 1 1 ˆin = √ (ˆ ˆout = √ (ˆ e x + yˆ), e x − yˆ) for B1g , 2 2
where Si is a spin-1 operator because the local moments of iron chalcogenides has been found to give 2 µB [43], the
(where x and y point along the Fe-Fe directions). We use this operator to calculate the Raman response R(ω) using the continued fraction expansion [44] where R(ω) is given by
H=
X [J1 Si · Sj + K(Si · Sj )2 ] nn
+
X 2nn
J2 Si · Sj +
X
J3 Si · Sj
3nn
1 X −βEn 1 ˆ † ˆ R(ω) = − e Im hΨn |O O|Ψn i ˆ πZ n ω + En + i − H
where Z is the partition function and the sum over n represents a sum over the eigenstates of the Hamiltonian with energy less than E0 + 2J1 , where E0 is the ground state energy. From this the elastic peak is removed by taking the Raman susceptibility given by
(6)
χ00 (ω) = 12 [R(ω) − R(−ω)]. The dynamical spin structure factor was calculated using the same method with ˆ the following equation and P also using equation 6 with O replaced with Sqz = √1N l eiq · Rl Slz .
7
[1] Z. P. Yin, K. Haule, and G. Kotliar, “Kinetic frustration and the nature of the magnetic and paramagnetic states in iron pnictides and ironchalcogenides,” Nature Mater. 10, 932–935 (2011). [2] Qimiao Si, Rong Yu, and Elihu Abrahams, “Hightemperature superconductivity in iron pnictides and chalcogenides,” Nat. Rev. Mater. (2016), 10.1038/natrevmats.2016.17. [3] S.-H. Baek, D. V. Efremov, J. M. Ok, J. S. Kim, Jeroen van den Brink, and B. B¨ uchner, “Orbital-driven nematicity in FeSe,” Nature Mater. 14, 210–214 (2014). [4] A. E. Taylor, R. A. Ewing, T. G. Perring, R. D. Parker, J. Ollivier, S. J. Clarke, and A. T. Boothroyd, “Absence of strong magnetic fluctuations in FeP-based systems LaFePO and Sr2 ScO3 FeP,” J. Phys.: Condens. Matter 25, 425701 (2013). [5] I. I. Mazin and M. D. Johannes, “A key role for unusual spin dynamics in ferropnictides,” Nature Phys. 5, 141 (2009). [6] Antoine Georges, Luca de’ Medici, and Jernej Mravlje, “Strong correlations from hunds coupling,” Annu. Rev. Cond. Mat. Phys. 4, 137–178 (2013). [7] K. M. Stadler, Z. P. Yin, J. von Delft, G. Kotliar, and A. Weichselbaum, “Dynamical mean-field theory plus numerical renormalization-group study of spin-orbital separation in a three-band hund metal,” Phys. Rev. Lett. 115, 136401 (2015). [8] J. K. Glasbrenner, I. I. Mazin, Harald O. Jeschke, P. J. Hirschfeld, R. M. Fernandes, and Roser Valent, “Effect of magnetic frustration on nematicity and superconductivity in iron chalcogenides,” Nature Phys. 11, 953–958 (2015). [9] Anisimov, V. I., Nekrasov, I. A., Kondakov, D. E., Rice, T. M., and Sigrist, M., “Orbital-selective Mott-insulator transition in Ca2−x Srx RuO4 ,” Eur. Phys. J. B 25, 191– 201 (2002). [10] Luca de’ Medici, S. R. Hassan, Massimo Capone, and Xi Dai, “Orbital-selective mott transition out of band degeneracy lifting,” Phys. Rev. Lett. 102, 126401 (2009). [11] Luca de’ Medici, “Hund’s induced fermi-liquid instabilities and enhanced quasiparticle interactions,” Phys. Rev. Lett. 118, 167003 (2017). [12] M. Yi, Z-K Liu, Y. Zhang, R. Yu, J.-X. Zhu, J.J. Lee, R.G. Moore, F.T. Schmitt, W. Li, S.C. Riggs, J.-H. Chu, B. Lv, J. Hu, M. Hashimoto, S.-K. Mo, Z. Hussain, Z.Q. Mao, C.W. Chu, I.R. Fisher, Q. Si, Z.-X. Shen, and D.H. Lu, “Observation of universal strong orbital-dependent correlation effects in iron chalcogenides,” Nature Commun. 6, 7777 (2015). [13] F. Venturini, M. Opel, T. P. Devereaux, J. K. Freericks, I. T¨ utt˝ o, B. Revaz, E. Walker, H. Berger, L. Forr´ o, and R. Hackl, “Observation of an Unconventional MetalInsulator Transition in Overdoped CuO2 Compounds,” Phys. Rev. Lett. 89, 107003 (2002). [14] P. E. Sulewski, P. A. Fleury, K. B. Lyons, and S-W. Cheong, “Observation of chiral spin fluctuations in insulating planar cuprates,” Phys. Rev. Lett. 67, 3864 (1991). [15] C. M. Canali and S. M. Girvin, “Theory of Raman scattering in layered cuprate materials,” Phys. Rev. B 45, 7127–7160 (1992). [16] Simon Adrian Weidinger and Wilhelm Zwerger, “Higgs
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
mode and magnon interactions in 2D quantum antiferromagnets from Raman scattering,” Eur. Phys. B 88, 237 (2015). P. Knoll, C. Thomsen, M. Cardona, and P. Murugaraj, “Temperature-dependent lifetime of spin excitations in RBa2 Cu3 O6 ( R = Eu, Y),” Phys. Rev. B 42, 4842–4845 (1990). L. Chauvi`ere, Y. Gallais, M. Cazayous, M. A. M´easson, A. Sacuto, D. Colson, and A. Forget, “Impact of the spin-density-wave order on the superconducting gap of Ba(Fe1−x Cox )2 As2 ,” Phys. Rev. B 82, 180521 (2010). Shunji Sugai, Yuki Mizuno, Ryoutarou Watanabe, Takahiko Kawaguchi, Koshi Takenaka, Hiroshi Ikuta, Yasumasa Takayanagi, Naoki Hayamizu, and Yasuhiro Sone, “Spin-Density-Wave Gap with Dirac Nodes and Two-Magnon Raman Scattering in BaFe2 As2 ,” J. Phys. Soc. Japan 81, 024718 (2012). Hans-Martin Eiter, Michela Lavagnini, Rudi Hackl, Elizabeth A. Nowadnick, Alexander F. Kemper, Thomas P. Devereaux, Jiun-Haw Chu, James G. Analytis, Ian R. Fisher, and Leonardo Degiorgi, “Alternative route to charge density wave formation in multiband systems,” Proc. Nat. Acad. Sciences 110, 64–69 (2013). Taner Yildirim, “Frustrated magnetic interactions, giant magneto-elastic coupling, and magnetic phonons in ironpnictides,” Physica C 469, 425 (2009). N. Chelwani, A. Baum, T. B¨ ohm, M. Opel, F. Venturini, A. Erb, H. Berger, L. Forr´ o, and R. Hackl, “Magnetic excitations and amplitude fluctuations in insulating cuprates,” ArXiv e-prints (2017), arXiv:1705.01496. C.-C. Chen, C. J. Jia, A. F. Kemper, R. R. P. Singh, and T. P. Devereaux, “Theory of Two-Magnon Raman Scattering in Iron Pnictides and Chalcogenides,” Phys. Rev. Lett. 106, 067002 (2011). K. Okazaki, S. Sugai, S. Niitaka, and H. Takagi, “Phonon, two-magnon, and electronic Raman scattering of Fe1+y Te1−x Sex ,” Phys. Rev. B 83, 035103 (2011). L. Chauvi`ere, Y. Gallais, M. Cazayous, M. A. M´easson, A. Sacuto, D. Colson, and A. Forget, “Raman scattering study of spin-density-wave order and electronphonon coupling in Ba(Fe1−x Cox )2 As2 ,” Phys. Rev. B 84, 104508 (2011). Y.-X. Yang, Y. Gallais, F. Rullier-Albenque, M.-A. M´easson, M. Cazayous, A. Sacuto, J. Shi, D. Colson, and A. Forget, “Temperature-induced change in the Fermi surface topology in the spin density wave phase of Sr(Fe1−x Cox )2 As2 ,” Phys. Rev. B 89, 125130 (2014). Fa Wang, Steven A. Kivelson, and Dung-Hai Lee, “Nematicity and quantum paramagnetism in FeSe,” Nature Phys. 11, 959–963 (2015). Qisi Wang, Yao Shen, Bingying Pan, Xiaowen Zhang, K. Ikeuchi, K. Iida, A. D. Christianson, H. C. Walker, D. T. Adroja, M. Abdel-Hafiez, Xiaojia Chen, D. A. Chareev, A. N. Vasiliev, and Jun Zhao, “Magnetic ground state of FeSe,” Nature Commun. 7, 12182 (2016). M. C. Rahn, R. A. Ewings, S. J. Sedlmaier, S. J. Clarke, and A. T. Boothroyd, “Strong (π, 0) spin fluctuations in β − FeSe observed by neutron spectroscopy,” Phys. Rev. B 91, 180501 (2015). Thomas P. Devereaux and Rudi Hackl, “Inelastic light scattering from correlated electrons,” Rev. Mod. Phys.
8 79, 175 (2007). [31] Pierre Massat, Donato Farina, Indranil Paul, Sandra Karlsson, Pierre Strobel, Pierre Toulemonde, MarieAude Masson, Maximilien Cazayous, Alain Sacuto, Shigeru Kasahara, Takasada Shibauchi, Yuji Matsuda, and Yann Gallais, “Charge-induced nematicity in FeSe,” Proc. Nat. Acad. Sciences 113, 9177–9181 (2016). [32] K.-Y. Choi, D. Wulferding, P. Lemmens, N. Ni, S. L. Bud’ko, and P. C. Canfield, “Lattice and electronic anomalies of CaFe2 As2 studied by Raman spectroscopy,” Phys. Rev. B 78, 212503 (2008). [33] Y. Gallais, R. M. Fernandes, I. Paul, L. Chauvi`ere, Y.-X. Yang, M.-A. M´easson, M. Cazayous, A. Sacuto, D. Colson, and A. Forget, “Observation of Incipient Charge Nematicity in Ba(Fe1−x Cox )2 As2 ,” Phys. Rev. Lett. 111, 267001 (2013). [34] F. Kretzschmar, T. B¨ ohm, U. Karahasanovi´c, B. Muschler, A. Baum, D. Jost, J. Schmalian, S. Caprara, M. Grilli, C. Di Castro, J. H. Analytis, J.-H. Chu, I. R. Fisher, and R. Hackl, “Critical spin fluctuations and the origin of nematic order in Ba(Fe1−x Cox )2 As2 ,” Nature Phys. 12, 560–563 (2016). [35] Yann Gallais and Indranil Paul, “Charge nematicity and electronic Raman scattering in iron-based superconductors,” C. R. Physique 17, 113 – 139 (2016). [36] S. Caprara, C. Di Castro, M. Grilli, and D. Suppa, “Charge-Fluctuation Contribution to the Raman Response in Superconducting Cuprates,” Phys. Rev. Lett. 95, 117004 (2005). [37] Matthew Burrard-Lucas, David G. Free, Stefan J. Sedlmaier, Jack D. Wright, Simon J. Cassidy, Yoshiaki Hara, Alex J. Corkett, Tom Lancaster, Peter J. Baker, Stephen J. Blundell, and Simon J. Clarke, “Enhancement of the superconducting transition temperature of FeSe by intercalation of a molecular spacer layer,” Nature Mater. 12, 15–19 (2013). [38] An-min Zhang, Tian-long Xia, Kai Liu, Wei Tong, Zhaorong Yang, and Qing-ming Zhang, “Superconductivity at 44 K in K intercalated FeSe system with excess Fe,” Sci. Rep. 3, 1216 (2013). [39] Jian-Feng Ge, Zhi-Long Liu, Canhua Liu, Chun-Lei Gao, Dong Qian, Qi-Kun Xue, Ying Liu, and Jin-Feng Jia, “Superconductivity above 100 K in single-layer FeSe films on doped SrTiO3 ,” Nature Mater. 14, 285–289 (2015). [40] V. K. Thorsmølle, M. Khodas, Z. P. Yin, Chenglin Zhang, S. V. Carr, Pengcheng Dai, and G. Blumberg, “Critical quadrupole fluctuations and collective modes in iron pnictide superconductors,” Phys. Rev. B 93, 054515 (2016). [41] A. E. B¨ ohmer, F. Hardy, F. Eilers, D. Ernst, P. Adelmann, P. Schweiss, T. Wolf, and C. Meingast, “Lack of coupling between superconductivity and orthorhombic distortion in stoichiometric single-crystalline FeSe,” Phys. Rev. B 87, 180505 (2013). [42] D.C. Sorensen, R.B. Lehoucq, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (Siam, Philadelphia, 1998). [43] H. Gretarsson, A. Lupascu, Jungho Kim, D. Casa, T. Gog, W. Wu, S. R. Julian, Z. J. Xu, J. S. Wen, G. D. Gu, R. H. Yuan, Z. G. Chen, N.-L. Wang, S. Khim, K. H. Kim, M. Ishikado, I. Jarrige, S. Shamoto, J.-H. Chu, I. R. Fisher, and Young-June Kim, “Revealing the dual nature of magnetism in iron pnictides and iron chalco-
[44] [45]
[46]
[47]
[48]
genides using x-ray emission spectroscopy,” Phys. Rev. B 84, 100509 (2011). Elbio Dagotto, “Correlated electrons in high-temperature superconductors,” Rev. Mod. Phys. 66, 763 (1994). S. Sugai, Y. Mizuno, K. Kiho, M. Nakajima, C. H. Lee, A. Iyo, H. Eisaki, and S. Uchida, “Pairing symmetry of the multiorbital pnictide superconductor BaFe1.84 Co0.16 As2 from Raman scattering,” Phys. Rev. B 82, 140504 (2010). F. Venturini, Raman Scattering Study of Electronic Correlations in Cuprates: Observation of an Unconventional Metal-Insulator Transition, Ph.D. thesis, TU-M¨ unchen (2003). B. Muschler, W. Prestel, R. Hackl, T. P. Devereaux, J. G. Analytis, Jiun-Haw Chu, and I. R. Fisher, “Bandand momentum-dependent electron dynamics in superconducting Ba(Fe1−x Cox )2 As2 as seen via electronic Raman scattering,” Phys. Rev. B 80, 180510 (2009). T. B¨ ohm, F. Kretzschmar, A. Baum, M. Rehm, D. Jost, R. Hosseinian Ahangharnejhad, R. Thomale, C. Platt, T. A. Maier, W. Hanke, B. Moritz, T. P. Devereaux, D. J. Scalapino, S. Maiti, P. J. Hirschfeld, P. Adelmann, T. Wolf, H.-H. Wen, and R. Hackl, “The case for spin-fluctuation induced superconductivity in Ba1−x Kx Fe2 As2 ,” ArXiv e-prints (2017), arXiv:1703.07749.
9
Rχ''(Ω,T) (counts s-1 mW-1)
1.5 1.0
a
A1g
0.5
BaFe2As As2 BaFe 2 2
0.0 1.0
b
A2g
B2g
B1g
0.5 c 0.0 0
FIG. 7. (Color online) Raman spectra of FeSe at T = 40 K. (a) Spectra at polarizations in the FeSe plane as indicated. The x and y axes run along the Fe-Fe bonds and x0 and y 0 are rotated by 45◦ . (b) Sums of complementary spectra each yielding the full set of all four accessible symmetries. The spectra are multiplied as given in the legend.
Appendix A: Polarization dependence of the Raman spectra of FeSe
Fig. 7(a) shows the complete set of polarization resolved Raman spectra we measured for FeSe at T = 40 K up to a maximum energy of 0.45 eV. The measured spectra have been corrected for the sensitivity of the instruo n ) . ment and divided by the Bose factor 1 − exp(− k~Ω BT In Fig. 7b sums of corresponding pairs of spectra are shown. Each sum contains the full set of all four symmetries (A1g + A2g + B1g + B2g ) accessible with the light polarizations in the Fe plane. All three sets exhibit the same spectral shape. The spectra measured with linear light polarizations at 45◦ with respect to the Fe-Fe bonds (x0 x0 and x0 y 0 ) were multiplied by a factor of 0.65 to fit the other configurations. The same factor was applied when calculating the sums for extracting the pure symmetries. The reason for this deviation from the expected x0 x0 and x0 y 0 intensities lies in small inaccuracies in determining the optical constants. Since we never observed polarization leakages the main effect pertains obviously on the power absorption and transmission rather than phase shifts between the parallel and perpendicular light polarizations.
Appendix B: Raman spectra of BaFe2 As2
Fig. 8 shows the Raman spectra of BaFe2 As2 as a function of symmetry and temperature. Towards high energies the spectra increase almost monotonically over
2000
4000 2000 Raman Shift Ω (cm-1)
d 4000
FIG. 8. (Color online) Symmetry-resolved spectra of BaFe2 As2 for three temperatures 150 K > TSDW , 130 K . TSDW and 50 K TSDW .
an energy range of approximately 0.7 eV. We could not observe the pronounced nearly polarization-independent maxima in the range 2,000 - 3,000 cm−1 reported in Ref. [45]. At high energies our spectra are temperature independent. At low energies pronounced changes are observed in A1g and B1g symmetry upon entering the striped spin density wave (SDW) state below TSDW = 135 K as described by various authors [18, 19, 25]. In A2g and B2g symmetry the changes are small but probably significant in that polarization leakage is unlikely to be the reason for the weak low-temperature peaks in the range 2,000 cm−1 and the gap-like behaviour below approximately 1,000 cm−1 . The changes are particularly pronounced in B1g symmetry. As shown in Figs. 8 (c) and 1 (b) and in more detail elsewhere [34] the fluctuation peak vanishes very rapidly and the redistribution of spectral weight from low to high energies sets in instantaneously at TSDW . All these observations show that the polarization and temperature dependences here are fundamentally different from those of FeSe (Fig. 2).
Appendix C: Superconductivity
Fig. 9 shows Raman spectra of the FeSe sample at temperatures below (blue line) and above (red line) the superconducting transition temperature Tc , which was determined to be Tc = 8.8 K by measuring the third harmonic of the magnetic susceptibility [46]. Both spectra show a sharp increase towards the laser line which can be attributed to increased elastic scattering due to an accumulation of surface layers at low temperatures. Below Tc a broad peak emerges centred around approximately 28 cm−1 which we identify as pair breaking peak
) -1
2 .0
m W
T
-1
1 .5
T
at 2∆ ≈ (4.5 ± 0.5) kB Tc . Above 50 cm−1 the spectra at T < Tc and T ≥ Tc are identical. We could not resolve the second peak close to 40 cm−1 as observed in Ref. [31]. The gap ratio of (4.5 ± 0.5) kB Tc is comparable to what was found for Ba(Fe0.939 Co0.061 )2 As2 [47] but smaller than that found for Ba1−x Kx Fe2 As2 [48].
0 .5
c
c
0 .0 -0 .5
R χ' ' ( Ω, T ) ( c o u n t s s
≥
T < T
R χ' ' ( Ω)
2 .5
10
-1 .0 1 .0
0
4 0 8 0 R a m a n S h ift Ω
0 .5 0 0
2 0
4 0 6 0 8 0 -1 R a m a n S h ift Ω ( c m )
1 0 0
1 2 0
FIG. 9. (Color online) Raman spectra of FeSe in B1g symmetry above (red) and below (blue) the superconducting transition at Tc = 8.8 K. The inset shows the difference between the two spectra R∆χ00 (Ω) = Rχ00 (Ω, T < Tc ) − Rχ00 (Ω, T ≥ Tc ).
