Pressure dependence of the single particle excitation in the charge-density-wave CeTe system
We present new data on the pressure dependence at 300 K of the optical reflectivity of CeTe, which undergoes a charge-density-wave (CDW) phase transition well above room temperature. The collected data cover an unprecedented broad spectral range from the infrared up to the ultraviolet, which allows a robust determination of the gap as well as of the fraction of the Fermi surface affected by the formation of the CDW condensate. Upon compressing the lattice there is a progressive closing of the gap inducing a transfer of spectral weight from the gap feature into the Drude component. At frequencies above the CDW gap we also identify a power-law behavior, consistent with findings along the Te series (i.e., chemical pressure) and suggestive of a Tomonaga-Luttinger liquid scenario at high energy scales. This newest set of data is placed in the context of our previous investigations of this class of materials and allows us to revisit important concepts for the physics of CDW state in layered-like two-dimensional systems.
In a strictly one-dimensional (1D) interacting electron system, the Fermi-liquid (FL) state is replaced by a state where interactions play a crucial role, and which is generally referred to as a Tomonoga-Luttinger liquid (TLL) Luttinger ; Tomonaga . According to the predictions of the TLL theoretical framework Schulz , the 1D state is characterized by features such as spin-charge separation and the breakdown of the quasi-particle concepts. The non-FL nature of the TLL is also manifested by the non-universal decay of the various correlation functions. This originates from the power-law behavior () of the density of states (DOS) close to the Fermi level , as immediate consequence of the absence of the discontinuity at in the momentum distribution function. The exponent reflects the nature and strength of the interaction. It is also worthwhile to emphasize that the TLL, which in principle describes so-called gapless 1D fermion systems, may be unstable towards the formation of a spin or a charge gap Voit ; Giamarchi . Spin gaps are usually considered in microscopic 1D models including electron-phonon coupling, and are relevant to the description of the normal state of superconductors and Peierls insulators Voit . Charge gaps, on the other hand, are a more typical consequence of electronic correlations Schulz .
Recently, a great deal of interest has been devoted to the possible breakdown of the FL framework in quasi one-dimensional materials. Of paramount importance in this regard, are the linear-chain organic Bechgaard salts, which were indeed intensively investigated and on which the very first optical signature of a TLL behavior was recognized Vescoli ; VescoliEPJ . Upon compressing the lattice by chemical substitution of the organic molecules and/or counter-ions, the Bechgaard salts display a dimensionality-driven crossover from a 1D Mott-insulator to an incipient 2D Fermi-liquid Vescoli ; Jerome . These findings were later confirmed by optical experiment on the representative (TMTTF)AsF system under externally applied pressure Pashkin . This is associated to a confinement-deconfinement crossover induced by the enhancement of the interchain coupling (i.e., the charge transfer integral, ), which, besides inducing a warping of the Fermi surface (FS), theoretically implies a self-doping process in these materials Giamarchi . Interestingly enough, we identified in our optical data at low temperatures on linear-chain organic materials two relevant energy intervals: a low one (i.e., , being the Mott-Hubbard correlation gap) dominated by the Drude term due to the itinerant charge carriers as consequence of the self-doping, and a high one (, ) for which an asymptotic one-dimensional limit is achieved and a characteristic non-universal power-law behavior in the real part of the optical conductivity () was recognized Vescoli ; VescoliEPJ ; Jerome . It is worth warning the reader, that the TLL theory, applied here, emerged from a truly 1D scenario Luttinger ; Tomonaga ; Schulz , since a rigorous theoretical approach, accounting for the dimensionality crossover, is still missing. Thus, caution should be placed in addressing situations approaching a 2D limit.
FL theory has been thoroughly tested on a variety of materials and is usually valid in higher than one dimension. Nevertheless, this notion seems to break down in several notable exceptions, like in several correlated metals and in the copper oxide-based high-temperature superconductors (HTSC) Proceedings . Focusing on the 2D layered-like HTSC, this was beautifully shown from the perspective of the optical response by the recent investigation reported in Ref. Basov, . Indeed, a similar behavior of at high energies, as in the quasi-1D organic materials, was revealed. The high frequency tail of the mid-infrared feature in HTSC decays in a power-law like fashion. This leads to the question, to which extent the effective dimensionality of the interacting electron gas affects the electronic properties in those metals. There is then the quest for alternative layered-like systems, not affected by strong correlations and thus allowing a broader and more general perspective on these issues.
The two-dimensional rare-earth tri-telluride compounds Te (=La, Ce, Pr, Nd, Sm, Gd and Dy) are well suited to that purpose Dimasi . Their crystal structure is made up of square planar Te sheets and insulating corrugated Te layers which act as charge reservoirs for the Te planes. They exhibit an incommensurate CDW, residing within the Te planes and stable across the available rare-earth series Dimasi ; Ru1 ; Ru2 . In our first optical investigations on this family of compounds, we established the excitation across the CDW gap and discovered that this gap is progressively reduced upon compressing the lattice either with chemical substitution (i.e., by changing ) or with externally applied pressure Sacchettiprb ; Sacchettiprl . For energies larger than the CDW gap, we also observed power-laws in (i.e., ), again consistent with the typical behavior of a Tomonaga-Luttinger liquid system. This finding anticipates that interactions could play a significant role in shaping the electronic properties of these CDW materials at such high frequency scales. The direct interaction between electrons can directly produce an umklapp scattering, allowing the transfer of two particles from one sheet of the FS to the other and thus transferring to the lattice. The advocated wave vector Brouet ; BrouetPRB for the CDW modulation is indeed giving a value for close to a reciprocal lattice vector, so that such umklapp processes are effective Sacchettiprb . Nonetheless, it was recognized, that more studies both theoretically and experimentally would of course be useful in order to ascertain the respective roles of the interactions and of the electron-phonon coupling with respect to the CDW formation. As for the latter coupling, we recently addressed the issue of the interplay between electronic and phononic degrees of freedom in Te by Raman scattering investigations LavagniniRaman . We found that the lattice dynamics is quite of relevance towards the formation of the CDW condensate and primarily affects the low energy scales.
In this contribution, we will focus again our attention on the electronic excitation spectrum (i.e., the CDW gap) and on the impact of the direct interaction between electrons, which is supposed to govern the high-frequency properties of Te and therefore the shape of above the CDW gap Sacchettiprb . Emphasis will be also placed on the possibility to systematically tune such materials so that a dimensionality as well as a confinement-deconfinement crossover may be induced. The prerequisite for a reliable study of the high-frequency behavior in is the capability to achieve the optical response over a very broad spectral range, which extends well above the energy scale of the gap excitation. This was possible so far for the rare-earth Te series, where the physical properties can be varied by lattice compression, achieved through chemical substitution. A more direct and cleaner way to study the electronic properties upon lattice compression is the application of external pressure, since the changes in the electrodynamic response can be monitored while the CDW state is continuously suppressed by tuning the interchain coupling and altering the nesting conditions upon decreasing the lattice constants. Our first investigations of the optical response as a function of applied pressure on CeTe Sacchettiprl allowed us to address yet the pressure dependence of the CDW gap but were covering a too small energy interval to allow addressing the high-frequency decay of .
Here, we present new optical data collected as a function of pressure in CeTe over a broad spectral range going well beyond the CDW gap and extending up to 1.5x10 cm. We have two major targets in mind. Besides revisiting the evolution of the CDW gap upon hydrostatically compressing the lattice and its implication on the electronic properties, we want to achieve a robust determination of the frequency dependence of the optical conductivity at high energy scales, looking for the pressure dependence of its possible power-law behavior. This is supposed to supply a fingerprint on the evolution of the intrinsic dimensionality. We will first introduce the experimental technique and present the data. This will be followed by a short description of the analysis procedure and by a discussion of the resulting electrodynamic response, emphasizing particularly a scenario within the TLL framework. Our final goal is the comparison of the high-frequency optical response in Te as a function of both chemical and applied pressure.
Ii Experiment and Results
The CeTe single crystal was grown by slow cooling a binary melt, as already described elsewhere Ru1 . At ambient pressure we first collected optical reflectivity data on a characterized CeTe sample over the broad spectral range, extending from the far infrared (FIR) up to the ultraviolet (UV) and obtaining results fully equivalent to our previous investigations Sacchettiprb . A small piece (i.e., 50x50 m) of that specimen was cut and placed on the top surface of NaCl, acting as pressure medium in the stainless steel gasket hole of the pressure cell. We made use of the same experimental set-up, already described in Ref. Sacchettiprl, . A clamp-screw diamond anvil cell (DAC) equipped with high-quality type IIa diamonds (400 m culet diameter) was employed for generating high pressures up to 9 GPa. Pressure was measured with the standard ruby-fluorecence technique Mao .
We carried out optical reflectivity () measurements as a function of pressure exploiting the high brilliance of the SISSI infrared beam-line at ELETTRA synchrotron in Trieste SISSI . We explore the spectral range between 3200 and 1.5x10 cm comment1 . The investigated frequency interval was covered with a Bruker Michelson interferometer equipped with a CaF beamsplitter as well as HgCdTe and Si detector for the energy intervals 3200-11000 and 9000-16000 cm, respectively.
An important issue when measuring the reflectivity inside the DAC concerns the precise determination of the reference signal, which has important implications on the correct shape and absolute value of the resulting . Similarly to our first investigation Sacchettiprl , we measured at each pressure the light intensity reflected by the sample and by the external face of the diamond window , thus obtaining the quantity . At the end of the pressure run, we also measured the light intensity reflected by a gold mirror () placed between the diamonds at zero pressure and again , acting as a reference. One achieves , which is assumed to be pressure independent. This procedure allows us to finally obtain the sample reflectivity at each pressure, which takes into account the variations in the light intensity due to the smooth depletion of the current in the storage ring. As we have shown previously Sacchettiprl , there is a difference between the absolute value of the collected data and that of the expected calculated from the complex refractive index at zero pressure Sacchettiprb and assuming the sample inside the DAC Wooten ; Dressel ; simulation . Such a difference can be ascribed to diffraction effects induced by the non-perfectly flat shape of the sample. This latter issue was taken into account by defining a smooth (pressure independent) correction function which is then applied to all spectra Sacchettiprl .
Figure 1a shows the collected data on CeTe while Fig. 1b displays the resulting (see below for the detailed procedure) as a function of pressure at 300 K in the spectral range covered by the measurement with the sample inside the DAC. Besides extending to much higher frequencies than in our first investigation comment1 , the quality of the present data is also very much improved and the annoying interference pattern due to diffused light in our previous spectra Sacchettiprl are no longer present. The overall trend agrees, however, with the early findings and establishes once more an analogy between the optical responses upon compressing the lattice either by externally applied pressure Sacchettiprl or by chemical substitution Sacchettiprb . at low pressures is characterized by the depletion at about 4000 cm, at the onset of the broad bump which extends into the UV spectral range. The minimum of at about 4000 cm progressively fills in under pressure. The broad bump is ascribed to the excitation across the CDW gap into a single particle (SP) state Sacchettiprb ; Sacchettiprl . Its disappearance with pressure signals the closing of the CDW gap and is accompanied by an overall enhancement of , indicating an enhancement of the metallicity of the system. These latter features and behaviors are by now well established and are common experimental facts in the Te series. An alternative way to emphasize the trend of in the gap region is to consider the ratio between data at various pressures with respect to those at the highest pressure. Figure 2 shows such a ratio for CeTe as well as for previously collected data on NdTe, further underlying the progressive suppression of the CDW gap upon compressing the lattice. Furthermore, we can appreciate that the depletion of in NdTe is, as expected, less pronounced in this spectral range than in CeTe. This is the direct consequence of the smaller gap in the NdTe than in the CeTe compound, as achieved through chemical pressure Sacchettiprb .
Before applying the Kramers-Kronig transformations in order to get the optical conductivity one first needs to extend beyond the measured energy interval. We made use of the same analysis successfully applied in our recent work on LaTe Lavagniniprbpressure , based on the well established procedure described in Ref. Pashkin, . For the sake of completeness, we just briefly summarize the essential steps, referring to the quoted literature for more details.
First of all, it is worth recalling that the complete electrodynamic response of CeTe at ambient pressure over the whole FIR-UV spectral range can be consistently fitted within the Lorentz-Drude (LD) approach Wooten ; Dressel ; namely, with one Drude term for the metallic contribution, three Lorentz harmonic oscillators (h.o.) for the broad CDW gap feature and two additional h.o.’s for the onset of the electronic interband transitions Sacchettiprb . With these fit components it is possible to reproduce under pressure as well, by fitting the data in the measured energy interval only and obviously accounting for the sample inside the DAC simulation . The quality of the resulting fit is nevertheless satisfactory despite of the limited spectral range effectively fitted, as exemplified in Fig. 3a for the data taken at 0.7 GPa. This allows us to extrapolate the spectra beyond the experimentally available energy interval (Fig. 3a). Figure 3b displays the optical conductivity calculated within the same Lorentz-Drude approach applied to (Fig. 3a) and shows the related single components of the fit procedure, as well.
The precise frequency dependence of is sensitively governed by subtle changes of the measured . In order to reconstruct free from any constrains imposed by the Lorentz h.o.’s, we perform reliable Kramers-Kronig (KK) transformation of the extended (Fig. 3a). For the sample-diamond interface the KK relation for the phase of is given by Plaskett ; McDonald :
where is the position of the reflectivity pole on the imaginary axis in the complex frequency plane. Analogous to Ref. Lavagniniprbpressure, , is chosen so that from KK and from the Lorentz-Drude fit agree Pashkin . Table I summarizes at all investigated pressures.
Figure 3b compares from KK and from the direct Lorentz-Drude fit. The astonishingly good agreement between the two procedures well testifies for the self-consistency of the applied analysis. Similarly to our previous work reported in Ref. Lavagniniprbpressure, , we took good care to check the impact of the high frequency extrapolations of on the KK results. There is no noticeable effect of the different extrapolations of on in the measured spectral range. We are then confident that the data and particularly their analysis within the KK procedure sketched here are robust in the spectral range covered experimentally and can be trusted all the way up to the experimental limit of approximately 15000 cm.
Figure 1b presents , obtained through the KK procedure sketched above (Fig. 3) and in the corresponding spectral range where the spectra (Fig. 1a) were originally collected. We immediately recognized two features: the single particle peak excitation due to the CDW gap and the finite at low frequencies which defines the onset of the effective metallic contribution. Upon applying pressure it is evident that spectral weight is removed from the CDW gap excitation and moves into the Drude term at low frequencies. This is coincident with the already quoted filling-in of the depletion at about 4000 cm (Fig. 1a). Furthermore, one can qualitatively appreciate the shift of the CDW gap excitation towards low frequencies, indicating its progressive closing upon compressing the lattice. We like to recall that such a behavior of is totally consistent with the findings in the chemical pressure experiment Sacchettiprb as well as in the related LaTe compound under externally applied pressure Lavagniniprbpressure . In contrast to our first set of data on CeTe under pressure Sacchettiprl , the present data extend to high enough frequencies comment1 , so that the high-frequency tail of the CDW gap absorption can be esteemed over the measured energy interval. The upturn in at the high frequency limit of our spectra (i.e, at about 2x10 cm, Fig. 3b) signals the onset of the electronic interband transitions.
The wide investigated spectral range allows us a precise determination of the single particle peak excitation , then ascribed to the CDW gap. The Lorentz-Drude fit gives us access to the various phenomenological components pertinent to the energy interval considered here and ultimately allows to disentangle the (re-)distribution of spectral weight among them as a function of pressure. Analogous to our previous work Sacchettiprb ; Sacchettiprl ; Lavagniniprbpressure we define the CDW gap energy as the average weighted energy:
where is the resonance frequency and is the strength of the -Lorentz h.o. The sum is over the first three h.o.’s (Fig. 3b). The values of are summarized in Table I and fairly agrees with our estimations based on the first data set presented in Ref. Sacchettiprl, . The decrease of upon hydrostatically compressing the lattice is quite evident. Nonetheless, we remark that decreases in a slightly slower manner than in Ref. Sacchettiprl, . This is due to the unavoidable uncertainties, particularly in the determination of the low frequency spectral weight, since only the onset of the metallic contribution can be observed in the experiment with applied pressure. Nevertheless, guided by the trend of established in the Te series Sacchettiprb , we can exploit the onset of the metallic component in for a rough estimation of the Drude weight in CeTe under pressure. The plasma frequency values related to the Drude term are also reported in Table I.
Instead of treating the single energy scales separately, it is more appropriate and instructive to discuss them from the perspective of sum rule arguments. We can indeed estimate the widely used ratio:
between the Drude weight in the CDW state (i.e., ) and the total weight of the hypothetical normal state (i.e., when the weight encountered in the CDW gap () merges into the Drude term) Sacchettiprb ; Sacchettiprl ; Lavagniniprbpressure . This latter equation, following well established concepts employed elsewhere Perucchi , tells us how much of the FS survives in the CDW ground state and is not gapped by the formation of the CDW condensate. The estimation of for the Te series (i.e., chemical pressure) Sacchettiprb relies on the very robust Lorentz-Drude analysis of the electrodynamic response over a very broad spectral range, covered with optical experiments at ambient pressure. We established that as little as 2% to 7% of the original FS remains in the CDW state, going across the chemical series from LaTe to DyTe Sacchettiprb . These values are smaller but of the same order of magnitude than those obtained through a de Haas-van Alphen investigation of LaTe RuHvA . Although the agreement among the two techniques is not perfect, these latter data give some confidence in our procedure based on spectral weight arguments of in establishing the un-gapped fraction of the FS. For the optical experiment under applied pressure, caution should be placed on the estimation of , which is obviously affected by the limitations in estimating the Drude weight, as pointed out above (i.e., this is accounted for by the systematically larger error in CeTe under pressure than in the Te series). Values of as a function of pressure are summarized in Table I, and are found to be in agreement with those of the chemical pressure experiment. Figure 4 compares the relationship between and upon compressing the lattice, both chemically Sacchettiprb and hydrostatically. The overall similar trend of versus in both experiments is pretty obvious. On the one hand, this indicates that the analysis of the spectral weight redistribution in terms of is also reliable for the applied pressure experiment and on the other hand, this generally emphasizes the increase of the metallicity in Te upon suppressing the CDW condensate. Closing the gap releases additional charge carriers in the conducting channel so that the fraction of the un-gapped FS increases.
As pointed out above, is totally unaffected by the extrapolation of above the upper experimental limit at about 1.5x10 cm, necessary for the purpose of the KK transformation. This allows a robust assessment of the shape and frequency dependence of as a function of pressure at frequencies larger than the CDW gap. Figure 5 highlights for the spectral range above the CDW gap, using a bi-logarithmic scale representation. Furthermore, has been rescaled by its maximum value and the frequency axis by the frequency where the maximum of occurs. The high energy tail of the CDW gap is well reproduced by a power-law behavior with exponent between 1.6 and 1.3 (Table I) comment2 ; commenteta ; exponent . The overall behavior of is very similar to the data on the linear-chain organic Bechgaard salts Vescoli ; VescoliEPJ ; Jerome and agrees with findings in the rare-earth series Te Sacchettiprb as well as with our most recent data collected on LaTe under pressure Lavagniniprbpressure . Therefore, we are confident that the power-law behavior is here a reliable experimental fact.
In the case of a 1D material, one would predict different exponents for the optical behavior above the Peierls gap, depending on the hierarchy of the energy scales as well as on the leading interaction involved in the formation of the CDW condensate Giamarchibook . As anticipated in the introduction, the observed power-law behavior in of the Te series would be consistent with predictions based on the Tomonaga-Luttinger liquid scenario, with umklapp scattering processes playing a rather important role Sacchettiprb ; Giamarchi ; exponent . Therefore, the direct electron-electron interaction rather than the electron-phonon one would dominate in this case at such high energy scales above the CDW gap. Identifying the behavior of within the TLL scenario further implies that these systems and specifically their electronic properties are shaped by a hidden one dimensionality, despite their 2D layered-like structure Sacchettiprb . The issue of the effective dimensionality of the interacting electron gas is quite of relevance here. The advocated FS of Te consists of two sheets of open FS’s of a quasi 1D material (associated to the and orbitals, respectively). The measured vector for the CDW modulation is very close to the vector that corresponds to the nesting of the two sides of this quasi-1D FS. Attributing the one dimensionality to the fact that Te have a nearly perfect nested quasi-1D FS plays a decisive role, since the charge transfer integral () along the transverse direction (i.e., describing the hopping between the and orbitals) is not small and is much larger than the temperature of the measurements. Indeed, would normally lead to coherent transverse hopping, so that FS would have significant warping in the transverse direction and the material would not be 1D anymore. The warping of the FS would then loose its relevance only at . However, this is not the appropriate situation for Te, since eV Brouet ; BrouetPRB , while the power-law behavior (Fig. 5) is observed already for frequencies . But if nesting is strong and occurs with a well defined vector, then the system still acts as a 1D system would essentially do. The 1D character, indicated by the high frequency power-law behavior of (Fig. 5), may then persist even for , provided that one looks at phenomena involving the nesting wave-vector.
Finally, it is worth pointing out that the exponent progressively decreases upon compressing the lattice (inset in Fig. 5) which would be quite suggestive of a crossover from a weakly interacting towards a non-interacting electron gas system upon reducing the lattice constant commenteta . Such a trend in is not uncommon and has been most clearly observed in the linear-chain Bechgaard salts Vescoli ; Pashkin . Furthermore, it is worth recalling that, apart from the exception of LaTe, behaves similarly in the chemical pressure experiment across the Te series Sacchettiprb . Nevertheless, the trend of in CeTe (inset of Fig. 5) is different from what has been observed in LaTe under pressure Lavagniniprbpressure . In this latter material, , while being also quite close to 1, remains basically constant as a function of pressure. We may speculate that pressure on the double layered Te is more effective than on the single layered Te. This may imply a more pronounced (1D to 2D) dimensionality crossover upon applying pressure in the Te than in the Te series.
We have reported a comprehensive infrared optical investigation of CeTe under externally applied pressure. We achieved two major results. First, we could revisit the pressure dependence of the CDW gap, confirming its closing upon compressing the lattice. The gap closing is obviously accompanied by the enhancement of the FS fraction accounting for the metallic state. The second major achievement is the clear-cut identification of the power-law behavior at high frequencies, well within the investigated and accessible spectral range. Such a firm experimental finding allows an unprecedented discussion of the pressure dependence of electronic correlation effects. Our data can be consistently explained within the framework of the TLL theory and share several common properties with other prototype 1D systems, like the linear-chain organic materials.
Acknowledgements.The authors wish to thank T. Giamarchi for fruitful discussions. One of us (A.S.) wishes to acknowledge the scholarship of the Della Riccia Foundation. This work has been supported by the Swiss National Foundation for the Scientific Research as well as by the NCCR MaNEP pool and also by the Department of Energy, Office of Basic Energy Sciences under contract DE-AC02-76SF00515.
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