Fermi surface in BaNi$_2$P$_2$

We report measurements of the de Haas-van Alphen (dHvA) oscillation and a band structure calculation for the pnictide superconductor BaNi$_2$P$_2$, which is isostructural to BaFe$_2$As$_2$, the mother compound of the iron-pnictide high-$T_c$ superconductor (Ba$_{1-x}$K$_x$)Fe$_2$As$_2$. Six dHvA-frequency branches with frequencies up to $\sim$8 kT were observed, and they are in excellent agreement with results of the band-structure calculation. The determined Fermi surface is large, enclosing about one electron and hole per formula unit, and three-dimensional. This is in contrast to the small two-dimensional Fermi surface expected for the iron-pnictide high-$T_c$ superconductors. The mass enhancement is about two.

and a band-structure calculation performed for BaNi 2 P 2 . BaNi 2 P 2 crystallizes in the ThCr 2 Si 2 structure as BaFe 2 As 2 and becomes superconducting below T c ∼ 3 K. 12,13 BaNi 2 P 2 single crystals were prepared by high-pressure synthesis from the constituent elements: the synthesis temperature and pressure were 1250C and 1 GPa, respectively. The typical in-plane residual resistivity, residual resistivity ratio (RRR), and T c of the crystals have been reported to be 5 µΩcm, 18, and 2.51 K, respectively. 14 The dHvA torque measurements were performed on five single-crystal samples (samples #1, #2, #11, WM, and HM) with typical dimensions of 0.4 × 0.2 × 0.02 mm 3 by using piezoresistive microcantilevers. The crystal axes of samples #2 and WM were determined using X-ray diffractometry with a three-circle goniometer combined with a CCD area detector. For the other samples, the axes were identified by the inspection of the crystal shapes and surfaces: crystals are generally elongated along a < 110 > axis, and zigzags, each short segment of which is parallel to a < 100 > axis, run parallel to < 110 > axes on {001} surfaces. Samples #1, #2, and #11 were measured up to a magnetic field B of 17.8 T in a dilution refrigerator, while sample WM and samples #2 and HM were measured in a helium-3 refrigerator up to 25 and 36 T provided by the water-cooled resistive and hybrid magnets, respectively, at the Tsukuba Magnet Laboratory of the NIMS. 15 The field was rotated in the (010) and (110) planes, and the field angles θ (010) and θ (110) were measured from the [001] axis. The six fundamental-frequency branches α, β, γ, δ, , and ζ are clearly visible, among which β is composed of the symmetry-related sub-branches β l , β h , β xl , β xh , and β y , and γ appears as a doublet (γ l and γ h ) for some field directions. The second harmonic of α is also observed in a wide range of field directions. The effective mass m * and Dingle temperature x * D associated with dHvA orbits were estimated for the selected field directions from the temperature and field dependences of dHvA oscillation amplitudes, respectively, 16 and are listed in Table I, where carrier mean free paths l are also derived from F , m * , and The electronic band structure of BaNi 2 P 2 was calculated within the local density approximation (LDA) by using a full potential LAPW (FLAPW) method. We used the program codes TSPACE 17 and KANSAI-06. The following experimental lattice parameters 12 were used for the calculation: a = 3.947Å, c = 11.820Å, and z = 0.3431 for the 4e sites of P. This corresponds to the Sommerfeld coefficient of γ = 8.78 mJ / mol K 2 , but no experimental value has been reported. Approximately one-half of the density of states comes from the Nid states. The calculated Fermi surface is shown in Fig. 2(b). The carrier numbers are 0.06 and 0.88 holes for band-25 and 26 hole surfaces, respectively, and 0.60 and 0.34 electrons for band-27 and 28 electron surfaces, respectively. Note that BaNi 2 P 2 is a compensated metal.
The theoretical dHvA frequencies calculated from this Fermi surface are shown in Fig. 2(a) and compared with the experimental ones.     The last column of Table I shows that the dHvA mass enhancement m * /m band is ∼ 2. It is interesting to note that this value is similar to the enhancement of 2.3 found in the typical

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phonon-mediated strong-coupling superconductor Pb (T c = 7.2 K). 24 In ref. 25, the electronphonon coupling constant λ ep in BaNi 2 As 2 has been evaluated from first-priciples calculations, and the corresponding mass enhancement 1 + λ ep = 1.76 is close to our experimental value for BaNi 2 P 2 . The authors of ref. 25 argue that BaNi 2 As 2 is a phonon-mediated superconductor and suggest that a role played by spin fluctuations, if any, is to suppress T c rather than to promote it.
Finally, we mention reports of dHvA measurements performed on related compounds. For LaFePO, dHvA frequencies in the range F ≈ 1 ∼ 3 kT have been observed. 26,27 All the frequencies can be attributed to warped cylindrical Fermi surfaces and seem to basically be explained by band-structure calculations. However, the quantitative agreement between the observed and calculated frequencies is rather poor, which is probably related to the difficulty in the band-structure calculations of the Fe-based compounds mentioned above. The dHvA mass enhancement is estimated to be about four in ref. 26 and about two in ref. 27. The factor-of-two difference again seems to suggest the difficulty in band-structure calculations.
It has also been reported that dHvA oscillations are observed in SrFe 2 As 2 . 28 However, only small frequencies (F 0.4 kT) are reported, and since measurements are in the orthorhombic antiferromagnetic phase, a straightforward comparison with band-structure calculations or the present data is difficult.
In summary, we have performed dHvA measurements and a band-structure calculation for BaNi 2 P 2 . All the four surfaces of the Fermi surface predicted by the calculation are experimentally observed, and the agreement between the experimental and calculated dHvA frequencies is excellent. The determined Fermi surface is large and three-dimensional, in contrast to the Fermi surface in the Fe-based compounds. The mass enhancement is about two, which is close to the theoretical value estimated for BaNi 2 As 2 . 25