Fermi surface arcs and the infrared conductivity of underdoped YBa₂Cu₃O_6.50

J. Hwang^1,3, J. P. Carbotte^1,2 and T. Timusk^1,2

¹ Department of Physics and Astronomy, McMaster University - Hamilton, Ontario L8S 4M1, Canada
² The Canadian Institute for Advanced Research - Toronto, Ontario M5G 1Z8, Canada

³ Present address: Department of Physics, University of Florida - Gainesville FL 32611, USA

E-mail: jhwang@phys.ufl.edu

Received 22 January 2008, accepted for publication 21 February 2008
Published 26 March 2008

In going from the overdoped side of the cuprate phase diagram to the underdoped side, a new phenomenon known generally as the formation of a pseudogap is seen [1]. The pseudogap manifests itself in many properties and generally has the signature of a decrease in the electronic density of state (DOS) around the Fermi energy. There is no consensus as to its exact microscopic origin. Some observations are consistent with a preformed pair model [2] with the pseudogap associated with the binding energy of the pairs at the pseudogap temperature T* and the superconducting temperature T_c the point at which phases lock in and long range order results. Another influential model is a possible competing order such as d-density [3–8] wave which set in at T* and coexists with the superconducting order parameter below T_c. In this model it may be reasonable to expect two distinct gap scales and some recent data [9, 10] have pointed to this possibility.

In the absence of consensus on microscopic origin, phenomenological models have played an important role in understanding and correlating data. An example is the specific-heat data [11, 12]. The observed trend from overdoped to underdoped can be understood [11, 12] over several families of cuprates by simply introducing a gap Δ_pg in the DOS which obeys a mean-field temperature dependence. More recently, analysis of angle-resolved photoemission spectroscopy (ARPES) data have given a more detailed picture [13] of pseudogap formation on the Fermi surface as a function of temperature. A finite pseudogap first opens at T* but forms only in a region around the antinodal direction leaving an ungapped Fermi surface arc around the nodal direction of length proportional to the reduced temperature t = T/T*. Here we confine our attention to the normal-state region (above T_c) and study how the formation of Fermi surface arcs modifies optical properties.

Optical-conductivity measurements have given a wealth of information on electron dynamics in the cuprates. While early on the pseudogap was seen clearly as a real gap in the real part of the c-axis [14, 15] optical conductivity, its signature in the ab-plane data was less clear and largely confined to a general observation that the optical scattering rate seem to show an additional decrease with decreasing temperature at low frequency as compared with overdoped samples [16]. Only recently has the situation been further clarified when the real part of the optical self-energy was analyzed in more detail [17]. Underdoped samples show a characteristic evolution with decreasing temperature quite distinct from the overdoped case [18–20]. A hat-like peak structure develops at an energy of the order of the pseudogap energy, clearly seen above a smooth structureless background. It was recognized that this easily identified [17] feature could be understood in a simple DOS model with a gap below Δ_pg and the missing states piled up in the energy region just above the pseudogap which we will refer to here as the recovery region. These modification of the DOS show up very directly in the real and imaginary parts of the optical self-energy Σ^op(ω) [21–24].

Consider coupling to a single Einstein mode of energy ω_E. Without a pseudogap (constant electronic DOS) the optical scattering rate 1/τ^op(ω) will be zero till ω = ω_E after which it rises from zero according to the law (ω − ω_E)/ω [25]. With a pseudogap it will be zero till ω = ω_E + Δ_pg and rise after this according to (ω − ω_E − Δ_pg)/ω [20]. On the other hand, if a recovery region is included above ω = Δ_pg to conserve electronic states, there will be additional scattering in the recovery region due to the extra available states. This will make 1/τ^op(ω) rise more rapidly than it would have without the extra states. This rapid rise in 1/τ^op(ω) translates into the formation of a logarithmic-like structure in Kramers-Kronig transform. Monitoring the growth of this structure in the real part of the optical self-energy Σ^op₁(ω) gives us a means to trace pseudogap formation as does the steepness of 1/τ^op(ω) above the energy of the gap plus boson.

There have been many studies of the boson spectra of the cuprates as revealed by tunneling [26], ARPES [27] and optics [20, 28–33] in which a spectral density function I²χ(ω) is recovered, where I is a coupling constant between electrons and a boson and χ(ω) is the bosonic spectral density. For optimally and overdoped cases, coupling to the sharp resonance at an energy corresponding to the spin-1 collective mode measured in neutron scattering is seen. This resonance starts to exist above T_c in the underdoped case and its intensity (area under the peak) increases with decreasing temperature. For the specific case of orthoII YBCO_6.50 this has been studied by Hwang et al. [20].

In this paper we reanalyze the data on orthoII YBCO_6.50 allowing for the opening of a pseudogap which covers more of the Fermi surface, as the temperature is lowered below T*. We ask whether or not the data is consistent with a linear decrease in the length of the remaining arcs as a function of the reduced temperature t = T/T* as it is seen in the ARPES data in underdoped Bi-2212 [13]. Our analysis of the optical data proceeds through the optical scattering rate 1/τ^op(T, ω). For a system with an energy-dependent self-consistent electronic density of state N(ω) which is related to the imaginary part of the Green's function by the optical scattering rate can be written approximately [22] as

Here n(Ω) = 1/[e^βΩ − 1] is the Bose-Einstein thermal distribution, f() = 1/[e^β(μ−) + 1] the Fermi-Dirac with β the inverse temperature and μ the chemical potential, and I²χ(ω) the electron-boson spectral density. In eq. (1) is the symmetrized electronic density of states [N() + N(−)]/2. Equation (1) reduces properly to that of Shulga et al. [34] in the limit of a constant density of states. It further corresponds to the formula of Allen [35] at zero temperature. The data for 1/τ^op(T, ω) at eight temperatures in the normal state of orthoII YBCO_6.50 are reproduced [20] in fig. 1 (top and bottom frames) as (dark) solid lines. In all cases the data is fit (solid diamonds of different colors as indicated in the figure) with an electron-boson spectral density I²χ(ω) consisting of a broad Millis-Monien-Pines (MMP) background [36] of the form I₀ω/(ω² + ω_sf²), where I₀ is a coupling between spin fluctuation and charge carriers and ω_sf is a typical spin fluctuation energy. In addition a sharp resonance peak modelled by (where I_p is the area under the peak, ω_p is its center frequency, and d is its width) is included and its position in energy fixed at 31 meV chosen to agree with the energy of the measured spin-1 resonance of inelastic neutron scattering [37]. In addition a pseudogap is included with Δ_pg = 350 cm⁻¹ fixed from our previous analysis [20] and assumed temperature independent. The length of the remaining ungapped arc is modelled through the self-consistent DOS which we take to be equal to for ω ≤ Δ_pg and for ω  (Δ_pg, 2Δ_pg) and 1 beyond this. The value of is temperature dependent and assumed to be proportional to the reduced temperature t = T/T*. This agrees with the idea that the arc closes linearly in reduced temperature as T is lowered below the pseudogap temperature T*. Further the lost states below ω = Δ_pg due to the opening of the pseudogap are taken to reappear above Δ_pg in the interval up to 2Δ_pg. The fitting parameters to the data are area under the resonance peak (I_p) and the background (I₀).

Figure 1. (Color online) Top frame: optical scattering rate 1/τop(ω) in cm-1 as a function of ω also in cm-1 for the eight temperatures shown (normal state). The heavy solid squares are fits to the data (solid lines). The inset shows the frequency dependence of for t = 0.5, with quadratic frequency dependence below Δpg and a constant one above it for ω  (Δpg, 2Δpg). Electronic states are conserved. Bottom frame: same as top but previous fit to the data obtained in ref. [20] without including a recovery region above the pseudogap energy which is chosen to conserve state. Middle frame: difference between top frame and scattering rates obtained when the recovery region is excluded, keeping other parameters the same (as top frame fits).

The fits obtained are seen in the top frame of fig. 1 and are seen to be excellent. They are better than the earlier fits obtained in ref. [20] where a pseudogap was included but without arcs and recovery region. These fits are reproduced in the lower frame and are clearly not as good. In particular, in the region around ω = 1000 cm⁻¹ the new fits capture the narrow range of variation with temperature observed in the data. This effect is a direct consequence of the temperature dependence assumed for the DOS which itself reflects the temperature dependence of the Fermi surface arcs. When the arcs become smaller with reducing normalized temperature t = T/T* more states are lost in the pseudogap region as a results of a reduction in the value of . But in our model for which is shown in the inset of the upper frame of fig. 1, it is assumed that the lost states are to be found in the region between Δ_pg and 2Δ_pg so as to respect conservation of states in the electronic DOS. The existence of this recovery region directly leads to more scattering i.e. an increase in 1/τ^op(ω) in this energy range than would be the case if no recovery occurred, i.e. was equal to its background value of 1. This effect pushes up the 1/τ^op(ω) curves more at low temperatures than at high temperature where the amount of extra states due to recovery is reduced and finally vanishes in our model at the pseudogap temperature T = T*. It is therefore through the effect on 1/τ^op(T, ω) of the recovery region that we see in optics the arcs and their temperature dependence. The arcs have their greatest effects at low temperature as can be seen in the middle frame of fig. 1 where we show the contribution to the optical scattering rate due to the recovery states in above Δ_pg. At high temperatures this contribution vanishes as becomes equal to 1 everywhere. We note that, with increasing T away from T_c, the effect of the recovery region in 1/τ^op(T, ω) becomes small and we cease to be able to extract from the data any quantitative information on the arcs and on the recovery region above it.

In fig. 2 we show the results of our fits for the electron-boson spectral function I²χ(ω) as a function of frequency and temperature. At each temperature the background was allowed to vary as was the area under the resonance peak fixed at an energy of 31 meV [37]. We see that the peak disappears with increasing temperature and at 250 K and above only the background remains. Relative to the peak, the temperature dependence of the background is small.

In fig. 3 we show results for the area under the peak as a function of temperature above the superconducting T_c = 59 K, i.e. in the normal state. The points are denoted by (blue) solid diamond, they are also compared with our previous results from optics (green)open hexagon [20] and with neutron results (purple) solid pentagon [37]. It is seen that the fit to the neutron data has improved as compared to our previous optically derived estimates.

Data on the optical scattering rate in orthoII YBCO_6.50 show clear evidence for the formation of a pseudogap which opens at T* (the pseudogap temperature). We have used a model of the pseudogap with Fermi arcs with linear variation in reduced temperature t = T/T* as suggested in ARPES experiments on underdoped Bi-2212. The fraction of the Fermi surface which remains ungapped provides a finite but reduced value for the self-consistent density of state at ω = 0 with linear in t. A quadratic dependence of in ω is assumed till ω = Δ_pg after which point we have assumed there is a region of increased DOS which contains all the missing states so that by ω = 2Δ_pg states are conserved. This is a reasonable assumption and certainly supported by our fit for T near T_c. At higher temperatures however, while the data remains consistent with the assumed model, details such as the extent in energy and exact amount of recovery could not be quantitatively pinned down. With this model we calculate the optical scattering time τ^op(ω) at several temperatures in the normal state with electron-boson spectral density consisting of an MMP background and resonance peak set at 31 meV to agree with the available inelastic neutron scattering data on the local magnetic susceptibility [37]. Two parameters, the amplitude of the background and the area under the resonance peak are varied to get a best least-square fit to the data. The quality of the fit obtained in this way is considerably superior to an earlier fit also based on a pseudogap model but without recovery and Fermi surface arcs. We conclude that the data is more consistent with arcs than without. The effect of the arc shows up clearly in the region just above the main rise in scattering rate where all the curves bunch up together. This is properly captured in our fits while it is missing in the previous analysis. Also our new fits agree better with the neutron studies as to the temperature dependence of the area under the resonance peak which is one of the important parameter varied in our least-square fit.

Acknowledgments

This work has been supported by the Natural Science and Engineering Research Council of Canada (NSERC) and the Canadian Institute for Advanced Research (CIAR).

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