Mass enhancement in multiple bands approaching optimal doping in a high-temperature superconductor

Pnictides provide an opportunity to study the effects of quantum criticality in a multi-band high temperature superconductor. Quasiparticle mass divergence near optimal doping, observed in two major classes of high-temperature superconductors, pnictides and cuprates, is a direct experimental indicator of enhanced electronic interactions that accompany quantum criticality. Whether quasiparticles on all Fermi surface pockets in BaFe2(As1-xPx)2 are affected by quantum criticality is an open question, which specific heat measurements at high magnetic fields can directly address. Here we report specific heat measurements up to 35T in BaFe2(As1-xPx)2 over a broad doping range, 0.44<= x<= 0.6. We observe saturation of C/T in the normal state at all dopings where superconductivity is fully suppressed. Our measurements demonstrate that quasiparticle mass increases towards optimal doping in multiple pockets, some of which exhibit even stronger mass enhancement than previously reported from quantum oscillations of a single pocket.

cillation measurements at high magnetic fields up to 90T provides direct evidence for a quantum critical point at critical doping in YBa 2 Cu 3 O δ [1]. Mass divergence has also been reported in quantum oscillations studies of a high-temperature pnictide superconductor, BaFe 2 (As 1−x P x ) 2 [2][3][4] which, coupled with with elastoresistity measurements in BaFe 2 (As 1−x P x ) 2 , and elastic moduli and specific heat studies of Ba(Fe 1−x Co x )As 2 [7][8][9][10], support the picture that the physics underlying all high temperature superconductor phase diagrams are driven by quantum criticality. Pnictides provide an exciting opportunity to study the effects of quantum criticality in a system with multiple pockets distributed throughout the Brillouin zone.
Quantum oscillation studies of BaFe 2 (As 1−x P x ) 2 to date are only able to resolve the evolution of mass over a broad range of doping for a single pocket [2][3][4]. Specific heat determines the sum of the masses of all Fermi pockets and thereby can determine the evolution of the total mass of all Fermi surface pockets as one approaches optimal doping. In this study, we utilize high magnetic fields to suppress superconductivity and reveal the doping dependence of the sum of quasiparticle masses through the measurement of the specific heat, and by extension electronic density of states at the Fermi surface in the normal state.
The measurement of specific heat up to magnetic fields as high as 35T provides direct information about the normal state and the nodal structure of the superconducting gap. Assuming the validity of BCS phenomenology in high-temperature superconductors, the jump in specific heat at the superconducting transition temperature, T c , has been previously used to deduce the density of states in the normal state, and to probe the enhancement of the effective mass at T c [2,7,[9][10][11][12][13][14]. However, since there is no solid evidence of the validity of BCS phenomenology in unconventional superconductors, such as BaFe 2 (As 1−x P x ) 2 , it is desired to explore a different way to determine the normal state density [15,16]. The specific heat measurements at low temperatures as a function of magnetic field presented in this work provide a model-independent determination of the doping evolution of the total mass of all electron quasiparticles near a quantum critical point in a multi-band superconductor. Figure 1a shows the magnetic field dependence of specific heat divided by temperature, C/T, of BaFe 2 (As 1−x P x ) 2 for x = 0.46 ( T c = 19.5K) at 1.5K. Two striking features are apparent: √ H behavior at low magnetic fields, followed by saturation above a field we denote H sat . In a normal metallic state, one expects no field dependence of C/T. Therefore, we interpret the value of C/T above H sat , (C/T) sat , as the specific heat of BaFe 2 (As 1−x P x ) 2 in the normal state where superconductivity is fully suppressed (See SI) [16,17]. The √ H behavior of C/T is a hallmark of a nodal superconducting gap, for which independent evidence exists in BaFe 2 (As 1−x P x ) 2 [18][19][20][21][22][23][24][25][26]. Note that the slope of the √ H behavior increases slightly with increasing temperature (Figure 1b) and that at finite temperature the √ H behavior does not extend to very low fields. Instead, at very low fields C/T exhibits field dependence much weaker than √ H. Both of these observations are consistent with the Volovik phenomenology, which requires a monotonic increase of the slope of √ H with increasing temperature and C/T ∝ H at very low fields (see SI) [24][25][26]. Importantly, within Volovik phenomenology the low-field deviation from √ H behavior must disappear in the zero-temperature limit because it originates from the excitation of quasiparticles across the superconducting gap at finite temperature near the gap nodes [24][25][26].
In Figure 1b, we extrapolate the √ H dependence to zero field and define (C/T) extrap as the value of C/T at the intercept. We then define γ H = (C/T) sat -(C/T) extrap , which is temperature-independent for sufficiently low temperatures as shown in Figure 1b. γ H represents the electronic specific heat recovered by suppressing superconductivity: it is the component of C/T directly associated with quasiparticles on Fermi surface pockets that superconduct, because all other contributions, including phonons, are magnetic-fieldindependent at low temperatures, as discussed above. In Figure 1c, we determine γ bg , the background electronic C/T at zero-field and zero-temperature, by extrapolating C/T to zero temperature.
Using the physical picture discussed in connection with Figure 1 as a blueprint, we examine the behavior of the electronic specific heat for several chemical compositions in the range x = 0.44 to x = 0.6 (as color-coded in Figure 2a). All exhibit both √ H dependece at low field and saturation behavior at high field (Figure 2b). We can read the value of γ H directly from the panels of Figure 2b corresponding to each doping. The value of γ bg can also be directly read from the plots of C/T versus T 2 at zero magnetic field for each doping (Figure 2c,e). Note that both γ H and γ bg show strong, but opposite, doping-dependences on doping (Figure 2d,e).
The doping dependence of γ H is shown in Figure 3. Clear enhancement of the total quasiparticle density of states on the Fermi surface is observed approaching optimal doping in the 0.44 ≤ x ≤ 0.6 doping range. This observation is an independent evidence for the enhancement of electron-electron interactions that are responsible for quasiparticle mass increase near a quantum critical point. The doping evolution of the Fermi surface density of states can be compared with mass enhancement as determined by quantum oscillation measurements, which is an independent indicator of enhanced interactions in the critical region [2][3][4][5][6].
To convert the density of states on the Fermi surface to an equivalent quasiparticle mass we will use the expression for an effective 2D (cylinder-shaped) Fermi surface, γ = 1.   [2,3]. It is plausible that pockets for which a quantum oscillation mass is not resolved by these measurements over a broad doping range have stronger mass enhancement, thus accounting for this difference.
Our data therefore suggest that some pockets must have an even stronger mass enhancement than that reported for the β pocket.
Further information can be obtained by considering the absolute value of γ H . The total mass of all pockets from quantum oscillation measurements at x = 0.63 (triangle) [6] and in the parent compound (x = 1) (arrow) [5] are also shown in Figure 3. Both are higher than the total quasiparticle mass deduced from γ H , even at our highest doping, x = 0.6.
Rather than considering non-monotonic doping dependence of the total mass, we note that γ total = γ H + γ bg (red circles) matches better the quantum oscillation data available for the total mass of all pockets. This suggests that γ total , not γ H , corresponds to the Fermi surface density of states of all pockets. Such interpretation implies that γ bg corresponds to the density of states of a non-superconducting pocket or pockets. Since γ bg decreases approaching optimal doping, such a non-superconducting pocket would experience a mass decrease by a factor two in the same doping range over which the superconducting pockets double their mass. However, ARPES measurements report comparable superconducting gap energies for all pockets [18]. The observed doping dependence of γ bg might alternatively suggest a non-Fermi surface origin of that component of the specific heat, arising perhaps from localized electrons. Another possibility is that γ bg arises from non-Fermionic modes associated with quantum criticality [27,28]. We emphasize that these questions can now be discussed based on a model-independent analysis of magnetic-field-dependent specific heat as presented in this work. Aside from discussion of whether all pockets superconduct at low temperature, our measurements demonstrate unambiguously that quasiparticle mass increases towards optimal doping, and that some pockets must have even stronger mass enhancement than the β-pocket discussed in quantum oscillation studies.