Topological change of the Fermi surface in ternary iron-pnictides with reduced c/a ratio: A dHvA study of CaFe2P2
Amalia I. Coldea, C.M.J. Andrew, J.G. Analytis, R.D. McDonald, A.F. Bangura, J.-H. Chu, I.R. Fisher, A. Carrington
aa r X i v : . [ c ond - m a t . s up r- c on ] M a y Topological change of the Fermi surface in ternary iron-pnictideswith reduced c/a ratio: A dHvA study of CaFe P Amalia I. Coldea, C.M.J. Andrew, J.G. Analytis,
2, 3
R.D. McDonald, A.F. Bangura, J.-H. Chu,
2, 3
I.R. Fisher,
2, 3 and A. Carrington H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK Stanford Institute for Materials and Energy Sciences,SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA Geballe Laboratory for Advanced Materials and Department of Applied Physics, Stanford University, USA Los Alamos National Laboratory, Los Alamos, NM 87545, USA (Dated: November 2, 2018)We report a de Haas-van Alphen effect study of the Fermi surface of CaFe P using low temperature torquemagnetometry up to 45 T. This system is a close structural analogue of the collapsed tetragonal non-magneticphase of CaFe As . We find the Fermi surface of CaFe P to differ from other related ternary phosphides inthat its topology is highly dispersive in the c -axis, being three-dimensional in character and with identical massenhancement on both electron and hole pockets ( ∼ . ). The dramatic change in topology of the Fermi surfacesuggests that in a state with reduced ( c/a ) ratio, when bonding between pnictogen layers becomes important,the Fermi surface sheets are unlikely to be nested. The nature of the Fermi surface dimensionality and its prox-imity to nesting in the iron-pnictide superconductors is atthe core of understanding the mechanism of superconductiv-ity. An important finding is that under hydrostatic pressure,BaFe As and SrFe As [1] superconduct whereas CaFe As does not [2]. Instead it undergoes a structural transition to aphase with a much reduced c -axis length which is known asthe ‘collapsed tetragonal’ (cT) state. If pressure is applied us-ing a non-hydrostatic medium CaFe As does become super-conducting [1, 3]. This suggests that small uniaxial pressurecomponents stabilize the superconductivity in CaFe As . Un-derstanding why this happens in these materials and whetherthe Fermi surface nesting or the strong coupling with the lat-tice are the relevant parameters will be important for under-standing the origin of superconductivity in iron pnictides.CaFe As has three distinct magnetic and structural phases.Like many other iron-arsenides, at zero pressure, the high tem-perature tetragonal state becomes orthorhombic and antiferro-magnetically ordered below T N ≃ K. At low tempera-tures under a pressure of 0.35 GPa there is a transition to thecT state in which there is a ∼ % decrease in the c -axis lat-tice parameter and a ∼ % increase in the a -axis [4]. Recentbandstructure calculations [5, 6] suggest that these phase tran-sitions have a dramatic effect on the Fermi surface. In the hightemperature tetragonal phase the Fermi surface is predicted tobe similar to the other 122 pnictide consisting of two stronglywarped electron cylinders at the Brillouin zone corners andhole cylinders at the center of the zone ( Γ ). Calculations sug-gest that these hole pockets are sensitive to structural detailsof the As position and the c -lattice constant, whereas the elec-tron pockets are less affected [5]. However, in the cT state thecalculated Fermi surface suffers a major topological change;the two electron cylinders at the zone corners transform intoa single warped cylinder whereas the hole cylinders become alarge three dimensional sheet.The Fermi surface of many iron-pnictides have been exten-sively studied by angle resolved photoemission spectroscopy, however so-far the cT phase of CaFe As has been inaccessi-ble to this technique because of the high pressures involved.Quantum oscillation (QO) studies have the advantage of prob-ing precisely the bulk three dimensional Fermi surface but sofar have only been possible in the magnetically ordered phaseof the iron-arsenides [7] as the tetragonal superconductingphase cannot be accessed because of the high values of the up-per critical field ( ∼ T). Although QO studies are possibleunder high pressure, these measurements are technically verychallenging. One route to provide an insight into the Fermisurface properties of the iron-arsenides is to study the analo-gous non-magnetic phosphide materials. These materials havealmost identical calculated Fermi surfaces to the arsenides intheir non-magnetic state but are either non-superconductingor have a low T c (and low H c ) and single crystals can begrown with very high purity making them ideal for QO stud-ies. Quantum oscillations have been measured in supercon-ducting LaFePO [8] (which is analogous to the non-magnetic‘1111’ arsenides e.g., LaFeAsO) and SrFe P [9] (which is astructural analogue to the non-magnetic tetragonal ‘122’ ar-senides).Here we report a de Haas-van Alphen effect study of theFermi surface of CaFe P which is a very close structuralanalogue of the cT phase of CaFe As . The isoelectric sub-stitution of As with P does not change the number of Fe 3delectrons, but enhances P-P hybridization causing the latticeto contract in the c direction. Consequently the interlayer P-Pdistance approaches the molecular bond length [10], just asthe As-As distance does in the cT phase [11]. We show exper-imentally that the Fermi surface of CaFe P is different fromthe multiband electron-hole structure found in SrFe P [9]and other pcnitides, and is formed of a single warped cylinderand a large three dimensional hole sheet. We find a isotropicmass enhancements on the electron and hole sheets ( ∼ . ).High quality single crystals of CaFe P were grown from aSn flux similar to previous reports [9]. The residual resistivityratio ρ (300 K)/ ρ (1.8 K) was measured to be greater than 120. D (cid:0) ✁ ✂ ✄ ☎ ✆ ✄ ✝ bbb b ✞ aaa a ✟ ddd dggg g aaa a ✠ q ✡ ☛☞ ✌ ✍ ☛☞ ✞ ✞ ☛☞ ✍ ✎ ☛✏ aaa a ✟ aaa a ✑ aaa a ✟ bbb b F (kT) b)a) t o sc ( a . u . ) FF T A m p li t ud e ( a . u . )
30 35 40 45 -1x10 -2 -2 q ✒ ✓ ✔ ✕ ✖ ab CaFe P B (T) FIG. 1: (color online) a) Oscillatory part of torque at T = 0 . Kfor a single crystal of CaFe P (sample A). The inset shows sampleA attached to a piezocantilever. b) The Fourier transform spectrashowing the evolution of the extremal areas of the Fermi surface withthe field angle, θ . The positions of different pockets, α , α , β , γ , δ and harmonics ( α , α ) as well as a weak peak, α ’ are indicatedby arrows. Torque magnetometry was performed using piezoresistive mi-crocantilevers in high fields 18 T in Bristol (sample B) and45 T (sample A) at the NHFML, Tallahassee. We compare ourdata with the predictions of bandstructure calculations whichwe performed using an augmented plane wave plus local or-bital method as implemented in the WIEN2K code [12]. Theexperimentally determined lattice parameters and internal po-sitions were used for these calculations [13].Fig. 1(a) shows the oscillatory part of raw torque signalfor a CaFe P crystal up to 45 T. The fast Fourier transform(FFT) for several other orientations as the field is rotated from B || c ( θ = 0 ) towards B || a ( θ = 90 ◦ ) is shown in Fig.1b.The FFT frequencies F of oscillatory torque data (in the1/ B domain) are related to extremal Fermi surface areas by F = ( ~ / πe ) A k . Close to B || c the strongest amplitude peak( α ) is at 4.35 kT and we also observe several harmonics of α (see Fig.1b); we can distinguish a frequency α ∼ . kTand a tiny feature α ’ at 6 kT [14]. A small pocket γ is ob-served at ∼ T; at higher angles ( θ = 15 ◦ ) the amplitudeof another frequency, β , becomes significant and the positionof this peak varies strongly with increasing angle. When werotate close to B || a a strong amplitude signal, δ , correspond-ing to an extremal area of 3.2 kT is found.The angular dependence of the observed dHvA frequencies allows us to identify the shape of the Fermi surface sheetsfrom which they originate (see Fig. 2). Rotating away from B || c the α and α orbits display a much stronger angularvariation compared to extremal orbits on a simple cylinderwith a weak k z dispersion ( F ( θ ) ∼ / cos θ ), [8]. This sug-gests that these orbits originate from a strongly warped cylin-drical Fermi surface [14]. The size of the β orbit changesdramatically with θ suggesting that this Fermi surface sheethas an prolate ellipsoidal shape with a maximum at θ = 0 ◦ .The α orbits are well reproduced in both samples, but the β orbit was only observed in sample A in much higher fields (upto 45 T). This can be explained as the impurity damping of thedHvA signal is proportional to exp( − k F / ( ℓB )) so that higherfield are needed to see the larger β orbit ( k F ∝ √ F ) for thesame mean-free-path ( ℓ ).We now compared the experimental data in CaFe P to thepredictions of the band structure calculations. The Fermi sur-face (see Fig. 2 and Fig. 3) is quasi-three dimensional and iscomposed of a large hole sheet in the form of a flat pillow atthe top of the zone whereas the electron sheets are stronglydistorted quasi-two-dimensional tubes centered on the zonecorners. There are also two tiny hole pockets centered at Z(see the bottom panel of Fig.3) containing only ∼ . holescompared with the large hole and electron sheets which eachcontains ∼ . holes/electrons; if the P position is optimizedin the calculation (by minimising the total energy, z P =0.3890)they disappear.Fig. 2 shows good agreement between data and calculationin the case of the β frequency which corresponds to cross sec-tions on the hole sheet (band 3) and the α and α frequen-cies which correspond to the minimum and maximal extremalareas of the strongly warped electron cylinder (band 4) (thedegree of warping is roughly ∆ F/F ∼ in CaFe P ascompared with ∼ in SrFe P [9]). The complex in-planeand interplane corrugations of the Fermi surface could giverise to additional branches not predicted by our calculations.This could explain the origin of the δ branch, which is ob-served within ∼ ◦ of θ = 90 ◦ ( B || a ) (Fig. 2) and mayaccount for the small shift found for α as well as the pres-ence of the weak feature at α ’. It is worth emphasizing thatthe agreement between data and calculations ( ± meV) forthe main sheets is good in contrast to our findings for LaFePOand SrFe P [8, 9] where rigid band shift of up to 100 meVwere needed to bring the bandstructure into agreement withexperiment.Any small band shifts would mainly affect the small pock-ets of the Fermi surface centered at the Z point, as stated ear-lier. For example, by shifting the hole bands down the tinypockets at the Z point disappear and the large hole pillowtransforms into a large torroid [15]; when B || a we could ex-pected orbits from extremal areas on such torroid which maygive rise to the δ branch (which is about half the size of the β orbit close to B || a ) (see bottom panel of Fig. 3). The γ frequency increases to about θ = 42(3) ◦ and then decreasesuggesting that it has a short closed cylinder shape (see Fig.2). By shifting the hole bands up (by ∼ meV) the small 3D TABLE I: Experimental and calculated Fermi surface parametersof CaFe P close to θ = 0 ◦ ( B k c ) similar to those predicted forCaFe As in the cT phase [6].Experiment Calculations F (kT) m ∗ m e ℓ (nm) Orbit F (kT) m b m e m ∗ m b α ( e ) min α ( e ) max β ( h ) min pocket centered at the Z point (band 2) could be assigned tothe γ branch (Fig.3). Alternatively, γ which has a low effec-tive mass ( m ∗ = 0 . m e ) could originate from an Sn impurityphase which has orbits with similar frequencies and masses[16], but x-ray diffraction measurements does not identify anySn impurities at the level of ∼ . In any case this pocket, γ , accounts for only a tiny fraction of holes ( ∼ . ) and webelieve it is unlikely to play any major role.Fig. 3 shows a comparison between the Fermi surfaceof CaFe As in the tetragonal phase ( c/a =2.98), CaFe As in the cT phase ( c/a =2.66 at P =0.63 GPa) and CaFe P ( c/a =2.59). As mentioned before, it is clear that there is aremarkable similarity between CaFe P and the cT phase ofCaFe As . Thus in CaFe As applying chemical pressure (by the isoelectronic substitution of As by P) is equivalent to applied hydrostatic pressure , as found in other ternary pnic-tides [17]. This state of reduced c/a ratio has a differentFermi surface topology compared to CaFe As or SrFe P ( c/a =3.03). Yildirim [11] has argued that the cT phase ofCaFe As occurs when, by reducing the Fe moment, the Fe-As bonding weakens and the (inter and intra-planar) As-Asbonding gets stronger causing the significant strong reductionin the c axis [4]. Similarly, in non-magnetic phosphides, thereduction in the c -axis (or the c/a ratio) results in an increaseP-P hybridization between pnictogen ions along the c direc-tion (close to the single bond distance) [10]. The spacer be-tween the iron layers (Sr or Ba) limits the degree of this hy-bridization between layers and such a state with strong pnic-togen bonding is unlikely to occur [5, 17].The effective masses, m ∗ , of CaFe P for each Fermi sur-face orbit extracted from the temperature dependence of thedHvA signals, using the conventional Lifshitz-Kosevich for-mula [18], are shown in Table I and compared to the corre-sponding bandstructure values. The masses for the electronand hole sheets are enhanced by the same factor of ∼ . incontrast to the sheet dependent variation of the enhancementobserved recently in SrFe P (which has two electron and twohole Fermi surface pockets). By comparing the quasiparticleenhancement on the electron sheets (which have the largestmean free path and often match better to the bandstructurecalculations being less sensitive to structural changes com-pared with the hole pockets) we observe that in LaFePO theaverage enhancement is 2.38 (2.2 for inner and 2.54 for outerpocket) [8] whereas in SrFe P is 1.85 (1.6 for inner and 2.1for the outer electron sheet)[9]. The conventional electron- -90 -45 0 45 900.00.5 g q (degrees) band 2 dd a CaFe P b a a ' band 3 band 4 d H v A F r equen cy F ( k T ) FIG. 2: (color online) Angle dependence of all observed frequenciescompared with the band structure predictions (solid lines). Differentsymbols correspond to sample A (filled circles) and sample B (opencircles). Solid grey points indicate the position of very weak features.The bottom panel shows an expanded low frequency scale for the γ pocket and the dotted line is a guide to the eye. The calculatedFermi surface of CaFe P is also shown. The solid lines delimitsthe Brillouin zone and the Fermi surface sheets are represented in anextended zone. phonon coupling, λ e − ph , in the cT phase of CaFe As is cal-culated to be 0.23 [11] but such calculations for CaFe P arenot yet available. Due to their structural and electronic sim-ilarities it is likely that in CaFe P a mass enhancement of1.23 could be due to a conventional electron-phonon coupling,but other effects could also be important [19]. The furtheranisotropic mass enhancement observed in LaFePO and pos-sibly in SrFe P could have another origin and may be re-lated to the nesting of the Fermi surface. The mean free paths, ℓ , of the minimum electronic orbit is a factor 2 larger thanthat of large hole sheet and of the maximum electronic orbit.This suggests both anisotropy in scattering between electronand holes but also along k z , as also observed in SrFe P andLaFePO [8, 9].In the case of the superconducting LaFePO and SrFe P FIG. 3: (color online) Comparison of the calculated Fermi surfacetopology of CaFe As (tetragonal phase), CaFe As (cT phase) andCaFe P . Slices through the center of the Brillouin zone (solid lines)in the (110) plane are shown. [8, 9] the energies in the band structure were shifted in op-posite direction for the electron and hole pockets to match upthe experimental data. These asymmetric band shifts are sug-gested to result from geometric nesting [22]. In the presentmeasurements on CaFe P no band shifts were required ( ∼± meV) to achieve agreement with experiment and con-sidering this model it would imply the absence of nesting inCaFe P (or the lack of long-range order in the cT phase ofCaFe As [21].)In conclusion, we have experimentally determined theFermi surface of CaFe P which is closely related to the col-lapsed tetragonal phase of CaFe As . We find that the Fermisurface is composed of a single highly dispersive electroncylinder at the zone corners and a large three dimensional pil-low shaped hole surface. The mass enhancement due to many-body interactions is isotropic ( ∼ . ) and may be dominatedby the electron-phonon coupling. The features of this Fermisurface which does not fulfill nesting condition is likely to beshared by the non-magnetic cT phase of CaFe As and otherternary pcnitides with reduced c/a ratio and may explain why superconductivity is absent in such a state.We thank M. Haddow and E. Yelland for technical sup-port. We acknowledge financial support from the Royal Soci-ety, EPSRC, and EU 6th Framework contract RII3-CT-2004-506239. Work at Stanford was supported by the U.S. DOE,Office of Basic Energy Sciences under contract DE-AC02-76SF00515. Work performed at the NHMFL in Tallahassee,Florida, was supported by NSF Cooperative Agreement No.DMR-0654118, by the State of Florida, and by the DOE. [1] P. Alireza, et al. , J. Phys. Cond. Mat. , 012208 (2008).[2] W. Yu, et al. , Phys. Rev. B , 020511 (2009).[3] M. S. Torikachvili, S. L. Bud’ko, N. Ni, and P. C. Canfield,Phys. Rev. Lett. , 057006 (2008).[4] A. Kreyssig, et al. , Phys. Rev. B , 184517 (2008).[5] Y.-Z. Zhang, H. C. Kandpal, I. Opahle, H. O. Jeschke, andR. Valenti p. arXiv:0812.2920 (2008).[6] D. A. Tompsett and G. G. Lonzarich, arXiv:0902.4859.[7] J. G. Analytis, et al. , arXiv:0902.1172.[8] A. I. Coldea, et al. , Phys. Rev. Lett. , 216402 (2008).[9] J. G. Analytis, et al. , arXiv:0904.2405.[10] E. Gustenau, P. 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