Volovik effect in a highly anisotropic multiband superconductor: Experiment and theory

We present measurements of the specific heat coefficient $\gamma$ ($\equiv C/T)$ in the low-temperature limit as a function of an applied magnetic field for the Fe-based superconductor BaFe${}_2$(As${}_{0.7}$P${}_{0.3}$)${}_2$. We find both a linear regime at higher fields and a limiting square root $H$ behavior at very low fields. The crossover from a Volovik-like $\sqrt{H}$ to a linear field dependence can be understood from a multiband calculation in the quasiclassical approximation assuming gaps with different momentum dependence on the hole- and electronlike Fermi surface sheets.

Figure 1

The original specific heat data (Ref.13) on BaFe${}_2$(As${}_{0.7}$P${}_{0.3}$)${}_2$ as a function of field up to 15 T (solid symbols) with data from present work between 15 and 35 T (open symbols). Note the agreement between the linear, $C/T\propto H$, extrapolation of the 15 T [colored lines (Ref.13)] and 35 T (black lines, present work) results. We extract $\gamma$ from the data using two (equivalent) methods: (a) by making an extrapolation $C/T=\gamma +\beta T^2+\delta T^4$ from 2 K and above or (b) by taking the smoothed value of $C/T$ at 1.5 and 2 K found by fitting approximately ten data points around these temperatures to obtain $C/T$ (1.5 K) and $C/T$ (2 K) with decreased scatter. The temperature restriction eliminates both the influence of the anomaly and the field-induced nuclear contribution (see text), negligible for $H⩽4$ T above 1 K. The absolute accuracy of these data is $\pm 5\%$ (error bars are not shown at low fields since they are approximately the same size as a data point), while the precision of the data is approximately $\pm 2\%$. In addition, additional data with finer gradations in the measured fields up to 4 T were taken to explore the low-field nonlinear behavior. These data are shown on an expanded scale in Fig. 2.

Figure 2

Low-field $\gamma$ data up to 4 T from Fig. 1 on an expanded scale for $T$ $=$ 2 K (red dots) and 1.5 K (blue up triangles). Green down triangles show asymptotic ${\lim }_{T\to 0}C/T$ determined over the range $1.5<T<5$ K. The fitting functions of the data are labeled beside the curves. Best power-law fits to field dependence are shown in each case.

Figure 3

The spatially averaged ZDOS, normalized to the normal-state value $N(\omega =0)/N_0$ for a nodeless (lower two data sets, orange) and a nodal (upper two data sets, blue) single-band superconductor. The dashed lines show the idealized linear $H$ and $\sqrt{H}$ behavior for a clean $s$-wave (lower line) and $d$-wave (upper line) SC, respectively. The symbols are numerical results for a single-band SC with an isotropic $s$-wave gap (circles) and a strongly anisotropic nodal gap (triangles). Additionally we compare results that do (solid symbols) and do not (open symbols) take into account the vortex core reduction due to the Kramer-Pesch effect. Here we have ignored the field dependence of the superconducting gap, i.e., $\Delta \left(H\right)={\Delta }_0$.

Figure 4

Results of quasiclassical calculations for the parameters in Table . (a) Magnetic field dependence of the gaps in the two-band model calculated within the Pesch approximation (Refs.41 and 43) for cases 1–4. We assume ${\Delta }^e(H=0)={\Delta }^h(H=0)$ here. The four sets of coupling constants ${\lambda }_{ij}$ are listed in Table . (b) Field dependence of the space-averaged ZDOS $N^e\left(H\right)$ on the electron pocket for the four cases with anisotropic gap with angular variation ${\Phi }_e\left(\theta \right)=(1+r\cos 2\theta )/\sqrt{1+r^2/2}$. (c) Field dependence of the space-averaged ZDOS $N^h\left(H\right)$ for the four cases with isotropic gap along the hole pocket.

Figure 5

Comparison of the experimentally measured normalized specific heat coefficient (pink circles) to different theoretical results for the spatially averaged ZDOS. The dotted violet and solid orange curves are the predictions for the spatially averaged ZDOS for a clean $s$-wave and a $d$-wave SC. The blue squares (case Qa) and green diamonds (case Qb) are the differently weighted sums of ${\overline{N}}^e\left(H\right)$ and ${\overline{N}}^h\left(H\right)$ evaluated for case 4 of Figs. 4b and 4c. The black line (no symbols) (case Qc) is obtained using the formula ${\gamma }^{\mathrm{tot}}=a_1{\overline{N}}_e\left(H\right)+a_2{\overline{N}}_h\left(H\right)$, where $a_1=3.2$ mJ/(mol K${}^2$), $a_2=10.3$ mJ/(mol K${}^2$) are determined with a least-squares fit to experimental data below 30 T. Note that the “$d$-wave” and “$s$-wave” curves represent simple extrapolations of the low-field $\sqrt{H}$ and $H$ terms up to $H_{c2}$. The error bar shown corresponds to the absolute accuracy of the data discussed above in Fig. 1.

Figure 6

${\gamma }^{\mathrm{tot}}$ plotted with experimental data in absolute units. Case Qc is the black solid line similar to that in Fig. 5. Case Dc is obtained using the formula ${\gamma }^{\mathrm{tot}}=a_1{\overline{N}}_e\left(H\right)+a_2{\overline{N}}_h\left(H\right)$, where ${\overline{N}}_e\left(H\right)$ and ${\overline{N}}_h\left(H\right)$ are represented by the open squares and circles in Fig. 8 and $a_1=1.50$ mJ/(mol K${}^2$), $a_2=65.6$ mJ/(mol K${}^2$) for case Dc are determined with a least-squares fit to the experimental data below 15 T.

Figure 7

Experimentally measured specific heat coefficient (pink circles) compared to calculations with deep gap minima (case 1, $r=0.9$, blue triangles) and accidental nodes (case 4, $r=1.3$, green squares). In both cases the weight of electron and hole pocket contributions has been chosen in agreement with case Qb.

Figure 8

Normalized specific heat coefficient determined within the quasiclassical approach (blue triangles and blue dashed curve) and within the Doppler-shift method (open squares and circles). Total normalized specific heat coefficients obtained within the Doppler-shift method are shown by magenta dotted line, olive dash-dotted line, and black solid line for cases Da, Db, and Dc, corresponding to Qa, Qb, and Qc, respectively. The dotted violet and solid orange curves are the power laws predicted for the spatially averaged ZDOSs of an idealized $s$-wave and a $d$-wave SC. Pink circles show the experimentally determined normalized specific heat coefficient.