Infrared spectroscopy study of the in-plane response of YBa2Cu3O6.6 in magnetic fields up to 30 Tesla

With Terahertz and Infrared spectroscopy we studied the in-plane response of an underdoped YBa2Cu3O6.6 single crystal with Tc=58(1) K in high magnetic fields up to B=30 Tesla applied along the c-axis. Our goal was to investigate the field-induced suppression of superconductivity and to observe the signatures of the three dimensional (3d) incommensurate copper charge density wave (Cu-CDW) which was previously shown to develop at such high magnetic fields. Our study confirms that a B-field in excess of 20 Tesla gives rise to a full suppression of the macroscopic response of the superconducting condensate. However, it reveals surprisingly weak signatures of the 3d Cu-CDW at high magnetic fields. At 30 Tesla there is only a weak reduction of the spectral weight of the Drude-response (by about 3%) that is accompanied by an enhancement of two narrow electronic modes around 90 and 240 cm-1, that are interpreted in terms of pinned phase modes of the CDW along the a- and b-direction, respectively, and of the so-called mid-infrared (MIR) band. The pinned phased modes and the MIR band are strong features already without magnetic field which suggests that prominent but short-ranged and slowly fluctuating (compared to the picosecond IR-time scale) CDW correlations exist all along, i.e., even at zero magnetic field.


I. Introduction
The cuprate high-Tc superconductors (HTSC) that were discovered in 1986 [1] still hold the record TC value [2] for materials at atmospheric pressure with TC=135 K in Hg-1223 [3]. These cuprates have a rich phase diagram with various charge or spin ordered states that coexist or compete with superconductivity (SC). Recently, the observation of a two-dimensional charge density wave order (2d-CDW) in the CuO2 planes of underdoped YBa2Cu3O6+x (YBCO) with NMR [4] and x-ray diffraction techniques [5][6][7] has obtained great attention. The 2d-CDW has a maximal strength for a hole doping level of p0.11-0.12 (equivalent to an oxygen content of x0.5-0.6), with an in-plane wave vector of about q0.3 r.l.u. (reciprocal lattice units) and typically a short correlation length of less than ξ 10 nm [8]. It develops below about 150 K and its strength increases gradually with decreasing temperature until it sharply decreases below TC [6,8], presumably due to the competition with superconductivity.
When applying a large magnetic field along the c-axis [4,[9][10][11][12][13] or uniaxial pressure along the a-axis [14], this 2d-CDW can be enhanced, such that its strength keeps increasing toward low temperature. Notably, even a long-ranged, three dimensional charge density-wave order (3d-CDW) can be induced in underdoped YBCO with a hole doping of p  0.11 -0.12 and TC  55 -60 K by applying a magnetic field in excess of 15-20 Tesla [10,11] or likewise by applying uniaxial pressure along the a-axis [14]. Meanwhile, it was shown that the short-ranged quasi-2d charge density correlations exist in large parts of the temperature and doping phase diagram of YBCO [2] as well as in other compounds like La2-xSrxCuO4 [15], Bi-2212 [16], Bi-2201 [17] and Hg-1201 [18]. These observations raise important questions about the role of CDW fluctuations in the superconducting pairing interaction [19] and in the so-called pseudogap phenomenon which leads to a severe suppression of the low-energy electronic excitations already well above TC in the underdoped part of the phase diagram [20][21][22].
Infrared spectroscopy (IR) is a well-suited technique to study the gap formation, collective modes and pair breaking excitations of correlated quantum states [23] as well as infraredactive phonon modes that can be renormalized or even activated by the coupling to the electronic excitations [24]. This technique has already provided valuable information about the superconducting state of various superconductors [24,25]. For conventional BCS superconductors [26] or the unconventional iron-based high-TC pnictides [27] it was used successfully to determine the energy gap, Δ SC , as well as the density of the superconducting condensate, ns. For the case of an isotropic superconducting gap (in the so-called dirty limit for which the superconducting coherence length, ξSC, is smaller than the mean free path of the carriers) the real part of the optical conductivity at T << TC is fully suppressed up to a threshold energy of hω=2Δ SC above which the conductivity rises steeply and gradually approaches the normal state value. The corresponding missing spectral weight (defined as the frequency integral of the conductivity difference spectrum of σ1(TC)-σ1(T << TC)) is shifted to a δ-function at zero frequency that accounts for the inductive and loss-free response of the superconducting condensate. The response of the condensate is also seen at finite frequency in the imaginary part of the optical conductivity, or the real part of the dielectric function, ε1= 1 -Z0/2π *1/ω *σ2, where Z0=377 Ω is the vacuum impedance (and σ2 is in units of (Ω cm) -1 ).
In the latter it leads to a downturn to negative values at low frequency as described by the equation: ε δ 1 ∼ 1-ω 2 ρ,SC /ω 2 .
Similar to the SC state, the CDW order gives rise to a gap-like suppression of the optical conductivity below a threshold energy that corresponds to twice the energy gap of the CDW, 2Δ CDW . In contrast to the SC case, the missing spectral weight (SW) below 2 CDW is shifted to higher energy where it gives rise to a broad band above the gap edge that originates from the excitations across the CDW gap. The collective phase mode of the CDW is typically coupled to the lattice and, accordingly, has a strongly reduced spectral weight and is shifted away from the origin (zero frequency) to finite frequency due to defects on which the CDW is pinned.
The IR-response of the cuprate HTSC has been intensively investigated [23,37] but the interpretation of the superconducting gap features remains controversial. The expected characteristics in terms of a sharp gap edge at 2Δ SC and a full suppression of the optical conductivity at ω < 2Δ SC are not observed here. Instead, there is only a partial suppression of the low frequency optical conductivity without a clear gap feature and typically only a relatively small fraction of the free carrier spectral weight condenses and contributes to the superconducting condensate [38][39][40]. The nature of the rather large amount of residual lowenergy SW is still debated with the conflicting interpretations ranging from a gapless SC state due to disorder and pair-breaking effects to competing orders due to charge-and/or spin density wave correlations and fluctuations therefore that are slow on the infrared spectroscopy time scale [32,33,35]. The latter interpretation has obtained renewed attention due to the observation of a static CDW order in the underdoped cuprates and by recent reports of fluctuating CDW correlations that persist in a wide doping range and at elevated temperatures [41].
This calls for a study of the magnetic field effect on the in-plane infrared response of underdoped cuprates for which a static and long-range ordered CDW state is established in the range above 15 to 20 Tesla. To our best knowledge, previous magneto-optical studies of the in-plane response of YBCO single crystals (with the B-field applied along the c-axis) are limited to 7 Tesla and show hardly any change of the free carrier response [42]. Corresponding studies of the c-axis response (perpendicular to the CuO2 planes) revealed only a weak suppression of the superconducting condensate density [43].
Here we present a study of the IR response of an underdoped YBCO crystal in high magnetic fields up to 30 Tesla which has been reported to suppress superconductivity and induce a 3d-CDW [44,10]. We observe indeed a full suppression of the superconducting condensate above 20 Tesla but only weak changes of spectroscopic features that can be associated with the 3d-CDW. In particular, the magnetic field leads to a weak reduction of the spectral weight of the Drude-response due to the free carriers (by about 3%) and a corresponding, moderate enhancement of two electronic modes around 90 and 240 cm -1 and of the so-called midinfrared (MIR) band. The latter features are interpreted in terms of pinned phase modes (along the a-and b-axis directions, respectively) and the excitations across the CDW gap. Notably, these characteristic CDW features are prominent even in zero magnetic field. In return, our data suggest that fairly strong, but likely short-ranged and slowly-fluctuating CDW correlations exist already in zero magnetic field.

II. Experiments
A single crystal of YBa2Cu3O6.6 was synthesized using a flux-based growth technique with Ystabilized Zr2O crucibles [45] and post-annealing in air at 650°C for 1 day with subsequent rapid quenching into liquid nitrogen. The twinned crystal had a flat and shiny ab-plane with a size of about 3.5x3.5 mm 2 that was mechanically polished to optical grade using oil-based solutions of diamond powder with diameters of first 3 μm and then 1 μm. Its superconducting transition temperature of TC=58(1) K has been determined with dc magnetization in fieldcooling mode in 30 Oersted applied parallel to the sample surface using the vibrating sample magnetometer (VSM) option of a physical property measurement system (PPMS) from Quantum Design.
The ab-plane reflectivity spectra R(ω) in zero magnetic field were measured in Fribourg, at a near-normal angle of incidence using an ARS-Helitran flow-cryostat attached to a Bruker VERTEX 70v Fourier transform infrared spectrometer. Spectra from 40 to 8 000 cm -1 were collected at different temperatures ranging from 300 to 12 K. The absolute reflectivity values have been obtained with a self-referencing technique for which the sample is measured with an overfilling technique, first with the bare surface and subsequently with a thin gold coating (that is in situ evaporated) [46,47]. In addition, for each spectrum the intensity has been normalized by performing an additional measurement on a reference mirror made of polished steel. In the near-infrared to ultraviolet range (5 000 -50 000 cm -1 ) the complex dielectric function has been obtained with a commercial ellipsometer (Woollam VASE) for each temperature and at an angle of incidence of φ= 70°. The ellipsometric spectra have been obtained for two different geometries with the plane of incidence either along the ab-plane or the c-axis. The latter was only measured at room temperature assuming that it has just a weak temperature dependence. The obtained spectra have been corrected for anisotropy effects using the standard Woollam software to obtain the true ab-and c-axis components of the complex dielectric function. The optical conductivity was obtained by performing a Kramers-Kronig analysis of R(ω) [25]. Below 40 cm -1 , we used a superconducting extrapolation (R = 1 -Aω 4 ) for T < TC or a Hagen-Rubens one (R = 1 -A√ ) for T > TC. On the high-frequency side, we assumed a constant reflectivity up to 28.5 eV that is followed by a free-electron (ω -4 ) response.
The corresponding magnetic field dependent reflectivity measurements have been performed at the LNCMI in Grenoble, with a Bruker VERTEX 80v Fourier transform infrared spectrometer attached to the experimental setup to create magnetic fields up to 30 Tesla. The sample was placed in a sealed volume with low-pressure helium exchange gas that was inserted into a liquid helium bath with T=4.2 K (or a nitrogen bath with T=77 K). The measurements were carried out starting from zero field in different magnetic fields up to 30 Tesla, by taking the intensity ratio between the sample and a gold reference mirror, I(B). Prior to these measurements, the magnetic field was ramped up and down two times to settle any field induced movement of the optical components. In addition, to ensure reproducibility, the measurement sequence was repeated at least 4 times. From the measured sample/reference intensity ratio at a given field I(B) the corresponding reflectivity spectrum R(B) has been obtained using the relationship With R(0T) as obtained with the setup in Fribourg (see description above) we thus derived an

a. Temperature dependent optical response in zero magnetic field
The optical response of the YBa2Cu3O6.6 crystal in zero magnetic field at selected temperatures above and below the transition superconducting transition temperature TC is summarized in crystal. The spectra in the normal state are governed by a Drude-peak with an anomalously strong tail towards the high frequency side. With decreasing temperature the Drude-peak becomes narrower and electronic spectral weight is redistributed from the tail towards the head of the Drude-peak. In addition, there is a band around 240 cm -1 that becomes narrower and more pronounced with decreasing temperature that is apparently of electronic origin (since its oscillator strength is way too strong for an infrared-active phonon mode) and was previously interpreted in terms of the pinned CDW mode along the b-axis [32,33].
Superimposed on this electronic background are also several infrared-active phonon modes that give rise to comparably much weaker and narrower peaks.
In the superconducting state at 12 K << TC=58 K, there is only a partial suppression of the lowfrequency optical conductivity due to the formation of a superconducting energy gap below 2Δ SC . The so-called missing spectral weight, which is transferred to a -function at zero frequency and contributes to the loss-free response of the superconducting condensate, amounts to a fairly small portion of the available low-energy electronic spectral weight (SW).
The large amount of residual low-frequency spectral weight differs from the predicted behaviour of a BCS-type superconductor, even considering that the SC order parameter has a d-wave symmetry with line nodes on which the gap vanishes [49,50]. It is also in strong contrast with the nearly complete superconducting gap that is typically observed in the infrared spectra of the iron-arsenide superconductors [27,46,[51][52][53][54][55][56]. The inset of Fig. 1(b) shows that the plasma frequency of the superconducting condensate of ΩpS 5350 cm -1 is obtained from the missing spectral weight in the optical conductivity, via the so-called Ferrel-Glover-Tinkham (FGT) sum rule (blue line), as well as from the inductive term in the imaginary part (red line). The inductive term due to the superconducting δ-function at zero frequency, 1,SC, has been derived, as shown in the inset of Fig. 1(c) for the real part of the dielectric function, by subtracting from the measured spectrum the contribution of the regular response at finite frequency, ε1,SC= ε1,measured -ε1,regular. Following the procedure outlined in Ref. [33], ε1,regular has been obtained via a Kramers-Kronig analysis of the spectrum of 1( > 0). Overall, these spectra and the value of the SC plasma frequency compare well with previous reports [32,57,58]. There are also some relatively narrow dip features forming around 500 cm -1 , 240 cm -1 and 90 cm -1 that grow in magnitude with the magnetic field. For comparison, Fig. 2(b) shows that these magnetic field induced changes of the reflectivity do not occur (or are much smaller) when the sample is kept at 77 K where it is in the normal state already at zero magnetic field.  Figure 3 shows the spectra for σ1 and ε1 at 30 Tesla (green lines) as obtained using the extrapolation type 3 (if not explicitly mentioned otherwise) together with the zero field spectra in the SC state at 12 K (blue lines) and in the normal state at 65 K (red lines). Fig. 3(a) reveals that a magnetic field of 30 Tesla (applied at 4.2 K) gives rise to a similar increase of the optical conductivity below about 800 cm -1 (green vs. blue line) as the one that occurs when superconductivity is suppressed upon raising the temperature above Tc in zero magnetic field (red vs. blue line). Fig. 3(b)  cm -1 . Specifically, for the 240 cm -1 mode which can be identified and analysed already in zero magnetic field, the spectral weight increases by about 50 000 Ω -1 cm -2 . The corresponding estimate for the 90 cm -1 mode is less reliable since it is close to the lower limit of the measured spectrum and superimposed on the narrow head of the Drude-response for which the conductivity is steeply rising up towards low frequency. Nevertheless, its enhancement at 30

b. Magnetic field dependence
Tesla is evident from Fig. 3(b) and 3(c) where it gives rise to a resonance feature in ε1 and a maximum in the spectrum of Δσ1, respectively.

Furthermore, the difference plots in Figs. 3(c) and (d) establish that in the normal state at 30
Tesla the Drude-peak has a reduced SW as compared to the one at 65 K and zero Tesla. The light blue line in Fig. 3(c) shows a fit with a Drude-function which yields a SW loss of about 3 % as compared to the total SW of the free carrier response with a plasma frequency of ωpl≈15000 cm -1 . The evolution of the integrated spectral weight of the difference plot, Fig. 3(e) shows that the SW loss of the Drude-peak is compensated by the growth of the peaks at 90 and 240 cm -1 and, at higher energy, by an increase of the MIR band. The magnitude and the energy scale of the latter effect are somewhat uncertain since toward the upper limit of our measurement of 6000 cm -1 the spectra are increasingly affected by the choice of the high energy extrapolation for the KK analysis. Nevertheless, it is evident that some of the SW-loss of the Drude-peak is compensated by a SW-gain of the MIR band. A schematic summary of the above described magnetic-field-induced spectral weight redistribution is displayed in Fig. 3(f).

IV. Discussion
Our infrared data reveal that a magnetic field in excess of 20 Tesla causes a complete suppression of the δ-function at zero frequency that represents the loss-free response of the coherent SC condensate. This finding is consistent with previous reports based on thermal conductivity measurements of YBCO which concluded that the upper critical field, HC2, has a minimum around p0.1-0. 12 where it falls to about 20 Tesla [44].
However, our IR-data do not necessarily imply that a magnetic field above 20 Tesla induces a true normal state. They are likewise consistent with a superconducting state that lacks macroscopic phase coherence but exhibits local superconducting correlations, see e.g. Ref.
[59], that can fluctuate and give rise to dissipation and thus are difficult to distinguish from a Drude-response of normal state carriers.
Apart from the suppression of the coherent superconducting response, we find that the IR spectra exhibit surprisingly weak changes that can be associated with the 3d-CDW that develops above 15-20 Tesla [12]. The only noticeable effect is a weak reduction of the SW of the Drude-response by about 3% that is compensated by the enhancement of two narrow peaks at 90 and 240 cm -1 and of the broad MIR band. In terms of the response of a CDW, the enhanced bands at 90 cm -1 and 240 cm -1 can be assigned to pinned phase modes along the aand b-axis directions, respectively, and the enhanced MIR band can be understood as additional excitations across the CDW gap. The finding that magnetic-field-induced spectral weight changes are rather small implies that the 3d-CDW order is weak and involves only a relatively small fraction of the low-energy electronic states. Moreover, since the pinned phase modes and the MIR band are pronounced features already at zero magnetic field, our data suggest that strong CDW correlations exist irrespective of the magnetic field, even deep in the superconducting state at T << TC. Note that the IR spectroscopy technique is even sensitive to rather short-ranged and fluctuating (on the picosecond time scale) CDW correlations. The above described scenario is therefore not necessarily in disagreement with the very weak, quasi 2d-CDW order that is observed with x-ray diffraction at zero magnetic field and its suppression at T << TC [5,6]. Evidence for such an incipient CDW order has also been obtained in a recent RIXS study in which it was found that very broad (quasi-elastic) Bragg-peaks exist over a wide temperature and doping regime, without magnetic field [41]. The scenario of a slowly fluctuating CDW order that involves a substantial part of the low-energy states already at zero magnetic field can also account for the large residual low-energy spectral weight in the IR spectra that does not condense at T << TC (blue line in Fig. 3(a)).
The interpretation of this residual spectral weight in terms of collective excitations, rather than normal (unpaired) carriers, furthermore resolves a seeming contradiction with specific heat [60,61] and NMR Knight shift [62] measurements which detect only a very low density of unpaired carriers at T << TC.
Finally, we address the question which role the CDW correlations are playing in the formation of the MIR band. Whereas our IR data show that the MIR band is slightly enhanced when a 3d-CDW order develops at high magnetic fields, it was previously shown that antiferromagnetic (AF) spin-fluctuations are also strongly involved in the formation of the MIR band [63,64]. The latter one is indeed most pronounced close to the Mott-insulator state and its doping dependence has been successfully explained in terms of AF correlations that are enhanced by the electron-phonon interaction [65]. A consistent explanation of the MIR band thus may require taking into account the interplay between the spin and charge correlations as well as their coupling to the lattice.

V. Summary and Conclusion
In summary, we have studied the infrared in-plane response of an underdoped YBa2Cu3O6.6 single crystal with Tc=58(1) K in high magnetic fields up to B=30 Tesla. We found that a B-field in excess of 20 Tesla fully suppresses the coherent response of the superconducting condensate and leads to a response that is similar to the one in zero magnetic field at a temperature slightly above TC. We remarked that a true normal state may not be restored yet, since local superconducting correlations and fluctuations can give rise to a dissipative response that cannot be easily distinguished from the Drude-like response of normal carriers.
Moreover, we found that the 3d-CDW, which develops above about 15-20 Tesla in such underdoped YBCO crystals, gives rise to surprisingly weak changes of the infrared response.
The only noticeable features are due to a weak suppression of the SW of the Drude-response by about 3% and a corresponding spectral weight increase of two narrow electronic modes around 90 and 240 cm -1 and of the MIR band above 1000 cm -1 . The former two modes have been assigned to the pinned phase mode of the CDW along the a-axis and b-axis directions, respectively. The weak enhancement of the MIR band can be understood in terms of the electronic excitations across the CDW gap.
Notably, the pinned phase mode of the CDW is a prominent feature already in zero magnetic field. This suggests that the pronounced CDW correlations exist not only at high magnetic fields, where they are readily seen with x-ray diffraction in terms of sharp Bragg-peaks, but also at zero magnetic field, where only relatively weak and broad CDW Bragg-peaks are typically observed with x-rays. We pointed out that this difference can be explained in terms of the high sensitivity of the IR-spectroscopy technique to short-ranged and slowly fluctuating CDW correlations. The conjecture that strong but short-ranged and slowly fluctuating CDW correlations exist even in absence of the magnetic field and for a wide temperature range is confirmed by a recent RIXS study which revealed that strong but broad and quasi-static Bragg-peaks exist already in zero magnetic field [41] and persists up to elevated temperatures and over an extended doping range. In the IR-response, the pinned phase mode at 240 cm -1 is indeed observed up to rather high temperatures and for a wide doping range up to (at least) optimum doping. Moreover, the strength of the pinned phase mode at 240 cm -1 shows no sign of a suppression in the superconducting state below Tc. This suggests that the relationship between SC and the CDW correlations is not purely competitive, as has been proposed on the observed decrease of the CDW Bragg-peak seen with XRD in zero magnetic field [5,6] but, in fact, may be more intricate and dependent on the correlation length as well as the dynamics of the CDW order. These questions are beyond the scope of our present work and will hopefully stimulate further detailed studies, for example of the evolution of the CDW phase mode(s) as a function of temperature, doping, magnetic field or uniaxial pressure.