Fermi surface of superconducting LaFePO determined by quantum oscillations
A.I. Coldea, J.D. Fletcher, A. Carrington, J.G. Analytis, A.F. Bangura, J.-H. Chu, A.S. Erickson, I.R. Fisher, N.E. Hussey, R.D. McDonald
aa r X i v : . [ c ond - m a t . s up r- c on ] O c t Fermi surface of superconducting LaFePO determined from quantum oscillations
A.I. Coldea , J.D. Fletcher , A. Carrington , J.G. Analytis , A.F. Bangura ,J.-H. Chu , A. S. Erickson , I.R. Fisher , N.E. Hussey , and R.D. McDonald H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, United Kingdom. Geballe Laboratory for Advanced Materials and Department of Applied Physics,Stanford University, Stanford, California 94305-4045 and National High Magnetic Field Laboratory, Los Alamos National Laboratory,MS E536, Los Alamos, New Mexico 87545, USA (Dated: October 29, 2018)We report extensive measurements of quantum oscillations in the normal state of the Fe-basedsuperconductor LaFePO, ( T c ∼ ∼
2. The quasi-twodimensional Fermi surface consist of nearly-nested electron and hole pockets, suggesting proximityto a spin/charge density wave instability.
PACS numbers: 71.18.+y, 74.25.Jb, 74.70.-b
The recent discovery of superconductivity in ferro-oxypnictides [1, 2], has generated huge interest as as an-other possible route towards achieving high T c supercon-ductivity. LaFePO was among the first Fe-based super-conductor to be discovered and has a transition temper-ature of up to T c ≈ T c superconductor ( T c ≈
25 K)when electron doped [3]. By changing the rare-earth ion, T c reaches 55 K in SmFeAsO − x F x [Ref. 2]. Theoret-ical models suggest that the pairing mechanism in theFe-based superconductors may be mediated by magneticfluctuations due to the proximity to a SDW [5, 6, 7].Knowing the fine details of the Fermi surface topology,its tendency towards instabilities as well as the strengthof the coupling of the quasiparticles to excitations is im-portant for understanding the superconductivity.Quantum oscillations provide a bulk probe of the elec-tronic structure, giving detailed information about theFermi surface (FS) topology and mass renormalization.To observe quantum oscillations samples must be ex-tremely clean and the upper critical field must be lowenough for the normal state to be accessed; LaFePO isa material which fulfils both these requirements. Thetetragonal layered structure of LaFePO is made of alter-nating highly conductive FeP layers and poorly conduct-ing LaO layers stacked along the c axis [1], hence theFermi surface is expected to be quasi-two dimensional[8]. Here we report extensive measurements of quantumoscillations in torque and transport data. We find thatthe Fermi surface of LaFePO is composed of quasi two-dimensional nearly-nested electron and hole pockets withmoderate enhancement of the quasiparticle masses.Single crystals of LaFePO, with dimensions up to0 . × . × .
04 mm , and residual resistance ratios ρ (300 K)/ ρ (10 K) up to 85, were grown from a tin flux[9]. Single crystal x-ray diffraction gives lattice param-eters a = 3 . c = 8 . z La = 0 . z P = 0 . T c ≈ T = 0 . θ between the magnetic field direction and the c axis[12]. At low fields we observe behavior typical of a bulkanisotropic type-II superconductor in the vortex state[11]. The signal is reversible, indicating weak pinningof vortices. The upper critical field (Fig. 1a) is stronglyanisotropic, varying between µ H c2 = 7 . ab plane and µ H c2 =0.68 T when B || c . For magnetic fields above ∼ F , which are related to the extremalcross-sectional areas A k , of the FS orbits via the Onsagerrelation, F = (¯ h/ πe ) A k . From the evolution of thesefrequencies as the magnetic field is rotated from beingalong the c axis ( θ = 0 ◦ ) towards the a axis ( θ = 90 ◦ )[12], we can construct a detailed three dimensional pic-ture of the shape and size of the Fermi surface. Fourfrequencies have significantly larger amplitudes than theothers in both samples A and B as shown in Figs. 1d and2a. We label α and α the two closely split frequenciesat F ≈ F ≈
35 T) and the two higher frequen-cies, β and β . Besides these intense features (and theirharmonics [10]), in sample A (which was measured to FIG. 1: (color online) a) Torque measurements on LaFePOmeasured at T = 0 .
35 K and in magnetic fields up to 18 Tfor different magnetic field directions. The arrows indicate theposition of H c2 . b) Oscillatory part of the torque (dHvA) andc) resistance (SdH) in magnetic fields up to 45 T. d) Fouriertransform spectra (field range 15-45 T) for two different mag-netic field directions. e) The angle dependence of the Fouriertransform spectra for the field range 10-18 T (sample B). much higher fields) we see several other frequencies withsmaller amplitudes; these are labeled γ , δ , ε and havefrequencies in the range 0.7–1.7 kT. An extremely weaksignal, η , was observed only in sample B (measured up to18 T) which we believe originates from a small misalignedcrystallite [12]. The observed frequencies correspond toa fraction varying between 2.8% to 9% of the basal planearea of the Brillouin zone. They are significantly largerthan those observed in the double-layer Fe-As compound,SrFe As [13], which is non-superconducting, and is likelyto have a reconstructed Fermi surface at low tempera-tures due to a spin-density-wave ground state.Figs. 1e and 2a show the angular dependence of themain frequencies and their amplitudes. For a purelytwo-dimensional cylindrical Fermi surface the dHvA fre-quency varies like 1 / cos θ and deviations from this indi-cate the degree of warping for a quasi-two-dimensionalcylinder. As shown in Fig. 2b, the orbits β and β bothoriginate from sections of FS which have significant warp-ing, but with opposite curvature (i.e., maximal and min- FIG. 2: (color online) a) Angle dependence of all observedfrequencies. Different symbols correspond to sample A (opencircles), sample B (filled circles) and sample C (diamonds).Possible harmonics of the main frequencies are shown by tri-angles and the dashed lines indicate their calculated location.Solid lines are fits to a 1/cos θ dependence. b) Angle de-pendence of F cos θ . Solid lines are calculations for a simplecosine warped cylinder. c) The temperature dependence ofthe Fourier amplitude for θ = 32 ◦ (data are offset for clarity).Solid lines are fits to the Lifshitz-Kosevich formula [15]. Theobtained values of the effective mass are listed in Table I. imal areas respectively). The angle dependence of theseorbits is well described by Yamaji’s analysis of simplecosine warping of a two-dimensional cylinder [14], as isthe large increase in the amplitude of the oscillations at θ ∼ ◦ , where the frequencies cross (see Fig. 1e). Theseobservations indicate that β and β arise from the sameFS sheet. For the α orbits the warping of the FS sheetis very small, but the almost identical amplitudes, effec-tive masses (see below) and frequencies strongly suggeststhat these orbits arise from a single, weakly warped, FSsheet.The effective mass of the quasiparticles on the variousorbits were determined by fitting the temperature de-pendent amplitude of the oscillations to the conventionalLifshitz-Kosevich formula[15], as shown in Fig. 2c. Theobtained masses range between 1.7 m e and 2.1 m e and arelisted in Table I ( m e is the free electron mass).We now compare our experimental observations withpredictions of the density functional theory calculationsof the electronic structure of LaFePO. Our calculationswere made using the WIEN2K code with the experimen-tal lattice parameters and atomic positions and includingspin-orbit interactions [16]. The resulting band struc-ture is in good agreement with that reported by Leb´egue[8]. The density of states at the Fermi level are derivedmainly from Fe and P bands suggesting that the carri-ers flow mainly in the 2D FeP layer. The Fermi surface TABLE I: Effective masses ( m ∗ ) and frequencies from dHvAdata for samples A and B. Calculated band masses ( m b ) inLaFePO for shifted and unshifted bands. Orbits are labeledby band number and the location of their center.Branch F (kT) m ∗ /m e Orbit m b /m e (exp.) Sample A Sample B (calc.) unshifted shifted α α β β δ γ ǫ Q ≈ [ π, π, mainly comprises small, slightly warped tubular sectionsrunning along the c direction. There are two hole cylin-ders centered on the Brillouin zone centre (Γ) and twoelectron cylinders centered on the zone corner (M) (seeFig. 3c-f). In addition, there is a small three-dimensional(3D) hole pocket centered on Z. The spin-orbit interac-tion makes small but significant changes to the Fermisurfaces; most notably it breaks the degeneracy of thebands crossing the Fermi level along the XM line suchthat the two cylindrical Fermi surfaces no longer touchand it also increases the separation of the elliptical holepocket and the two cylindrical hole surfaces (Fig. 3c).The frequencies of the extremal dHvA orbits obtainedfrom the calculated band structure are compared withthe experimental data in Fig. 3a. The calculation pre-dicts that there should be 9 frequencies (two for each tubeplus one for the 3D hole pocket) in the range 0 . − θ ≃ ◦ ), which is broadly similar to what is observedexperimentally. In particular orbits β and β closely re-semble those expected from the larger electron cylinderin both frequency and curvature. The shape and splittingof orbits α and α are similar to that of the smaller elec-tron cylinder. The larger amplitude of these oscillationsindicates that they both suffer significantly less dampingthan the other orbits (we estimate a mean free path of ≈ α and β orbits, respectively).This may be a natural consequence of the fact that bothof these electron orbits originate from the same piece ofFermi surface in the larger unfolded Brillouin zone (cor-responding to the Fe-sublattice [5]). Assuming the aboveassignment for the α and β frequencies, the three remain-ing frequencies ( γ , δ , ε ) would then correspond to holeorbits, although their exact assignment is less clear. Forthese hole orbits the scattering is a factor ∼ α and β are shifted by −
83 meV and −
30 meV, respec-tively and the hole bands by +53 meV (Fig. 3b). Theband that gives rise to the 3D pocket (which we do notobserve experimentally) influences significantly the de-gree of warping along the c -axis of the hole cylinders; ifthis band was absent we would have a better agreementbetween our data and calculations (Figs. 3a and 3b).A two-dimensional cut in the ΓXM plane for our cal-culated unshifted and shifted Fermi surface (see Figs. 3eand 3f) shows that the electron pockets at the corner ofthe Brillouin zone (M) have almost similar shapes andsizes to the hole pockets at the centre of the zone (Γ).Nesting requires a perfect match between the size andthe Fermi surface topology of the electron and hole pock-ets and this could stabilize a spin density wave (magneticorder) or charge density wave (structural order) with awave vector Q ≈ [ π, π,
0] [6, 7, 22]. LaFePO is non-magnetic [18] but importantly our results suggest thatsize and the shape of the electron and hole pockets are in-deed very similar (Figs. 3e and f) implying that LaFePOmay be close to nesting and hence to a spin/charge den-sity wave instability.For an undoped LaFePO the volume of the electronand hole sheets should be equal (compensated metal) butshifting the bands (as described above) creates an imbal-ance of ∼ .
03 electrons per formula unit. A possibleexplanation for this imbalance can be related to a smallamount of electron doping in our crystals due to oxy-gen non-stoichiometry ( ∼ .
5% oxygen deficiency whichis below the resolution of our x-ray data).The band structure calculation allows us to estimatethe many-body (electron-phonon and electron-electron)enhancements of the quasiparticle masses over their bandvalues. Table I shows the band masses for all the pre-dicted orbits as calculated, and after application of bandshifts. For the identified bands a moderate renormal-isation is found, with m ∗ /m b = (1 + λ ) ≈
2. Forthe hole orbits the calculated frequencies do not corre-spond exactly to the experimental ones, however, as theband masses for most of the hole orbits are in the range0 . − . m e and the measured m ∗ values are ≈ m e ,a similar level of enhancement is likely. Reported mea-surements of the electronic specific heat coefficient ( γ )vary between 7-12 mJ/mol K [9, 17, 18]. In 2D ma-terials γ can be estimated from the renormalised dHvAmasses, summed over all of the observed FS sheets. Fourquasi-2D FS sheets with an effective mass ∼ m e wouldgive γ dHvA ∼ , close to the lower end of theexperimentally observed values, without including anycontribution from a 3D pocket.A recent ARPES study of LaFePO [19] showed anelectron-like FS pocket centered at the M point and twohole sheets at the Γ point. The electron sheet and thesmaller hole orbit have areas close to the experimentallyobserved dHvA frequencies. When compared to bandstructure calculations, the bands found in the ARPESmeasurements are renormalized by a factor two, similarto the mass enhancements we have found. This suggeststhat the renormalization occurs over the whole band, rather than being localized to energies close to the Fermilevel, as found in Sr RuO , [20]. One of the hole pock-ets seen in ARPES is much larger ( >
12 kT) than anydHvA frequencies observed here; its presence creates asignificant charge imbalance of one electron per unit cell,suggesting that this feature is related to surface effects[19].As in the other ferrooxypnictides, in LaFePO the band-structure is very sensitive to the position of the P atom[21, 22]. Considering the P atom in the position op-timized (the ‘relaxed’ structure) results in hole orbitswhich are much too small to explain our data. Muchlarger band energy shifts are needed to bring it into ap-proximate agreement and we find that the band structurecalculated with the experimental P position provides abetter description of the data.In conclusion, we have found experimentally that theFermi surface of the superconductor LaFePO, is in broadagreement with band structure calculations with a massenhancement of about 2 due to many-body interactions.The difference in amplitudes of the dHvA signal betweenthe hole and electron pockets is an indication of differentscattering rates affecting these orbits. The near-perfectmatching between the hole and the electron orbits thatwe observe, suggests that LaFePO may be close to aspin/charge density wave transition and that magneticfluctuations are an important ingredient in the physicsof the Fe-based superconductors.We thank E.A. Yelland, N. Fox, MF Haddow for tech-nical help and I. Mazin for helpful comments. Thiswork was supported financially by EPSRC (UK) and theRoyal Society. AIC is grateful to the Royal Society fora Dorothy Hodgkin Fellowship. Work at Stanford wassupported by the DOE, Office of Basic Energy Sciencesunder contract DE-AC02-76SF00515. Work performedat the NHMFL in Tallahassee, Florida, was supportedby NSF Cooperative Agreement No. DMR-0654118, bythe State of Florida, and by the DOE. [1] Y. Kamihara et al. , J. Am. Chem. Soc. , 10012(2006).[2] Z.-A. Ren et al. , Chin. Phys. Lett. , 2215 (2008).[3] Y. Kamihara et al. , J. Am. Chem. Soc. , 3296 (2008)[4] C. de la Cruz et al. , Nature , 899 (2008).[5] I.I. Mazin et al. , Phys. Rev. Lett. , 057003 (2008)[6] A.V. Chubukov, D. Efremov and I. Eremin, arXiv:0807.37355.[7] V. Chetkovic and Z. Tesanovic, arXiv:0804.4678[8] S. Leb`egue, Phys. Rev. B , 035110 (2007).[9] J. Analytis et al. , (unpublished).[10] Changes in lever resistance were measured with a Wheat-stone bridge circuit. The sensitivity of the levers is ap-proximately 0.1 ◦ per Ohm. At certain angles and at thehighest fields, the deflection of the cantilever is largeenough to generate harmonics of the main dHvA frequen-cies. [11] Z. Hao and J. R. Clem Phys. Rev. B , 7622 (1991).[12] The X-ray laue diffraction indicates that the exact axis ofrotation between is ∼ ◦ (sample A) and ∼ ◦ (sampleB) from the a axis; a small piece of crystal with its c -axis oriented 73 ◦ away from that of the main piece of thecrystal was detected in sample B.[13] S.E. Sebastian et al. , J. Phys.:Cond. Mat. , 1520 (1989).[15] D. Shoenberg, Magnetic oscillations in metals (Cam-bridge University Press,Cambridge 1984).[16] P. Blaha et al. , WIEN2K, an augmented plane wave +local orbitals program for calculating crystal properties ,Karlheinz Schwarz, Technica Universit¨at Wien, Austria,(2001). We used 10 k points (in the full Brillouin zone)for convergence and 10 k points for the FS calculationsand the GGA96 exchange correlation potential. [17] Y. Kohama et al. , arXiv:0806.3139.[18] T.M. McQueen et al. , Phys. Rev. B, et al. , Nature , 81 (2008)[20] N.J.C. Ingle et al. , Phys. Rev. B , 205114 (2005) [21] V. Vildosola et al. , Phys. Rev. B et al.et al.