Constant effective mass across the phase diagram of high-$T_c$ cuprates

We investigate the hole dynamics in two prototypical high temperature superconducting systems: ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ and $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_y$, using a combination of dc transport and infrared spectroscopy. By exploring the effective spectral weight obtained with optics in conjunction with dc Hall results, we find that the transition to the Mott insulating state in these systems is of the “vanishing carrier number” type since we observe no substantial enhancement of the mass as one proceeds to undoped phases. Further, the effective mass remains constant across the entire underdoped regime of the phase diagram. We discuss the implications of these results for the understanding of both transport phenomena and pairing mechanism in high-$T_c$ systems.

Figure 1

(Color online) Optical data for YBCO and both optical and transport data for LSCO discussed in the context of the high-$T_c$ phase diagram. Left panels are data for $x=0.03$ LSCO and right panels depict the low temperature optical conductivity as a function of doping for both LSCO and YBCO. Data in panels (f)–(h) are at $T=7\mathrm{K}$ and at $T$ just above $T_c$ in panels (e). Panel (a): phase diagram with the values shown for LSCO; the yellow area is the long range AF region (denoted by $T_N$), the green area is the superconducting region (marked by $T_c$), and the pseudogap region is denoted by $T^{*}$. To facilitate a direct comparison with the data, temperature is plotted on the horizontal axis and doping on the vertical axis. The Hall coefficient $R_H$ $({10}^{-3}{\mathrm{cm}}^3∕\mathrm{C})$ displayed in panel (b) reveals two characteristic regimes (I and II). This is exemplified for the $x=0.03$ sample but similar behavior is found for all heavily underdoped crystals. These regimes can also be identified with significant regions in the dc resistivity corrected for thermal expansion (c). Panel (d): the effective number of carriers, $N_{\mathrm{eff}}$ $\left({10}^6{\Omega }^{-1}{\mathrm{cm}}^{-2}\right)$, determined from the optical data as described in the text. Right panels: the real part of the conductivity at low temperature [$T=10\mathrm{K}$, except $T=T_c$ for (e) and (f) and $T=150\mathrm{K}$ for $y=6.65$], displayed as black solid curves. Red colors signify fits using the standard Drude model. The blue area is a subtraction of the Drude fit from ${\sigma }_1\left(\omega \right)$ and is referred to as the “mid-infrared contribution” throughout the text. The resonant structure in the latter contribution can be adequately described with a Lorentzian oscillator (dashed lines). The loss function $\left[\mathrm{Im}(-1∕\epsilon )\right]$, panels (i), also clearly depicts two components (shown as red and blue curves) for dopings $x<0.10,y<6.65$, and the progression to a single component, (black solid curve) for higher dopings.

Figure 2

(Color online) Top panels show the optical effective mass as determined by the methods described in the text. Red symbols are the mass determined using a combination of optics and transport (left axis labels), and the black symbols represent the mass determined within the extended Drude model (right labels). All methods produce consistent results and mass values which are doping independent over large regions of the phase diagram. The low temperature long-range AF ordered phase is denoted by the yellow shaded area and SC region by the green area. Note that despite the formation of the AF regime near the Mott transition, there is no noticeable enhancement of the optical mass. Bottom panels show both the effective spectral weight from optics data (red and blue open squares) and the number of holes as determined by transport (red and blue dots).