Quantum Oscillations in the Underdoped Cuprate YBa2Cu4O8
E. A. Yelland, J. Singleton, C. H. Mielke, N. Harrison, F. F. Balakirev, B. Dabrowski, J. R. Cooper
aa r X i v : . [ c ond - m a t . s up r- c on ] F e b Quantum Oscillations in the Underdoped Cuprate YBa Cu O E. A. Yelland , ∗ , J. Singleton , C. H. Mielke , N. Harrison , F. F. Balakirev , B. Dabrowski , J. R. Cooper H.H. Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK. National High Magnetic Field Laboratory, MS-E536,Los Alamos National Laboratory, Los Alamos NM 87545, USA. Department of Physics, Northern Illinois University, De Kalb, IL 60115, USA. Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 0HE, UK. (Dated: October 26, 2018)We report the observation of quantum oscillations in the underdoped cuprate superconductorYBa Cu O using a tunnel-diode oscillator technique in pulsed magnetic fields up to 85 T. There is aclear signal, periodic in inverse field, with frequency 660 ±
15 T and possible evidence for the presenceof two components of slightly different frequency. The quasiparticle mass is m ∗ = 3 . ± . m e . Inconjunction with the results of Doiron-Leyraud et al. for YBa Cu O . [1], the present measurementssuggest that Fermi surface pockets are a general feature of underdoped copper oxide planes andprovide information about the doping dependence of the Fermi surface. PACS numbers: 71.18.+y, 74.20.Mn, 74.20.-z, 74.25.Jb
The mechanism for high-temperature superconductiv-ity in the layered copper oxide superconductors has re-mained elusive for more than twenty years. At the heartof the problem is the evolution of the ground state from aMott-Hubbard insulator to a superconductor as the num-ber of doped holes p per planar CuO unit is increased.In particular, there is no agreement as to how the under-doped region should be described. The recent observa-tion of quantum oscillations in the oxygen-ordered ortho-II phase of YBa Cu O . (O-II Y123) with T c = 57 . p = 0 . Cu O (Y124) with T c = 80 K, and p = 0 .
125 [2] atfields up to 85 T, suggesting that they could be a generalfeature of underdoped cuprates. Our data for Y124 showthat the FS pockets expand as p is increased and give ahigher quasiparticle mass m ∗ than for O-II Y123.The Y124 crystal was grown from flux in a ZrO cru-cible under 600 bar of O at 1100 ◦ C. Other crystals fromthe same batch were of high quality with a residual Cu-Ochain resistivity ≤ µ Ω cm, and a low- T thermal conduc-tivity peak κ b (20 K) = 120 Wm − K − [3]. Pulsed mag-netic fields up to 85 T were provided by the Los Alamos85 T multi-shot magnet [4]. Measurements were madeusing a tunnel-diode oscillator (TDO) technique [5, 6] inwhich two small counter-wound coils form the inductanceof a resonant circuit. The crystal was cut into four pieces,each measuring up to 0 . × . × .
12 mm , which werestacked with their c -axis directions aligned within 2 ◦ ofeach other, and placed in one coil with the c -axis parallelto B and the axis of the coil. The resonant frequency, inour case 47 MHz, can depend on both the skin-depth (or,in the superconducting state, the penetration depth) andthe differential magnetic susceptibility of the sample [7].The sample and coil were immersed in He liquid or Heexchange gas, temperatures ( T ) being measured with aCernox thermometer 5 mm away from the sample. Fig. 1(a) shows the TDO frequency f versus B at T = 0 .
53 K. At B ≈
45 T, f falls substantially indicatingan increase in the penetration of the rf field as the su-perconductivity is suppressed. In the expanded view ofthe raw data taken during the falling part of the pulse,oscillations are visible for fields B >
55 T. The solid linesin Fig. 1(b) show the second derivative d f / d B of datataken at 1.6 K and reveal a clear oscillatory signal. Thefrequency and phase are nearly the same during the rising(36 T to 85 T in 5 ms) and falling (85 T to 36 T in 10 ms)parts of the pulse, ruling out spurious heating and elec-trical interference effects.The standard Lifshitz-Kosevich (LK) form for the os-cillatory magnetization is M ∝ B / R D R T sin(2 πF/B + φ ) [8], where φ is a phase, and in conventional metals theoscillation frequency F is related to a zero-field extremalFS cross-section A by the Onsager relation F = ( ~ / πe ) A [8]; the scattering and temperature damping factors arerespectively R D = exp( − π ~ k F /eℓB ) where k F is theFermi wave vector, ℓ is the mean free path and R T =(14 . m ∗ T /m e B ) / sinh(14 . m ∗ T /m e B ). The dashedline in Fig. 1(b) shows d M/ d B [7] calculated from theLK formula with F = 660 T, φ = π/ m ∗ = 3 . m e , ℓ = 400 ˚A and a suitable scale factor. Note that this es-timate of ℓ assumes pure de Haas-van Alphen oscillations;any Shubnikov-de Haas component would imply a highervalue. The model describes the data well, the decreasein amplitude with B arising from the weak B depen-dence of R D at B ∼
70 T and the factor of 1 /B in thethird derivative. Note however that the non-monotonic B -dependence of the oscillation amplitude at T = 0 .
53 Kand T = 1 . /B factor) may be signs of beating between two closefrequencies.Fig. 2 shows ∆ f , the TDO frequency minus a smoothmonotonic background [9], versus 1 /B , at various tem-peratures. The oscillations are periodic in 1 /B as ex-pected from quantized cyclotron orbits of Fermi-liquid-like quasiparticles. They are also damped rapidly at m H irr (T) T (K) m H m (T) m H hys (T) f ( M H z ) B (T) ×3(a)-0.500.51 55 60 65 70 75 80 85 d f / d B ( k H z / T ) B (T) d f /d B upd f /d B down C *d M /d B (calc) (b) T (K) m H m (T) FIG. 1: (color online) (a) Resonant frequency f of the TDOversus magnetic field B recorded during an 85 T pulse at T =0 .
53 K. Upper inset: µ H hys ( ◦ ) and µ H m (rising - △ , falling- N ) from this work and µ H irr ( • ) from Ref. 12. Lower insetshows expanded view of the µ H m data at low T . The dashedlines are guides to the eye. (b) Second derivative d f/ d B of1.6 K data (solid lines). The dashed line is given by LK theoryassuming ∆ f ∝ d M/ d B [7], and using a suitable scale factor. higher T , consistent with thermal smearing of the FS.The agreement in frequency and phase with Fig. 1(b)shows that the oscillations are not an artefact of back-ground subtraction.In Ref. 1 it was pointed out that no hole pockets arepresent in a band calculation for O-II Y123 [10]. How-ever small hole pockets of mainly chain character canbe formed by allowing small shifts of the Fermi level∆ E F ≈
25 meV [11]. Our observation of quantum os-cillations in Y124, for which calculations find no smallpockets near E F [11], suggests that the FS pockets arelikely to be a general feature of the copper oxide planesof underdoped cuprates.The insets to Fig. 1 show the T -dependence of µ H m ,the field of the well-defined peak in | d f / d B | and µ H hys where the hysteresis between the rising- and falling-fieldcurves ceases to be detectable. H m is where vortex pin-ning becomes weak enough for the rf field to penetratefurther than the London penetration depth (but still lessthan the normal state skin depth). In cuprate super-conductors, vortex pinning becomes very small above anirreversibility field H irr , which is usually much less thanthe estimated upper critical field H c , although these twofields may converge as T →
0. Our values of H m are sim-ilar to H irr determined previously using torque magne- D f ( k H z ) B (T -1 ) 0.53K1.6K4.0K3.6K6.5K F / B FIG. 2: Changes in resonant frequency ∆ f of the tunnel-diode oscillator circuit versus 1 /B recorded during 85 T pulsesat various temperatures. A smooth monotonic backgroundhas been subtracted [9]. The dotted lines are equally spaced in1 /B . The oscillatory signal is periodic in 1 /B with frequency F = 660 ±
15 T. tometry [12] on another crystal from the same batch (up-per inset) and very recently from the resistivity of otherY124 crystals [13]. Somewhat unexpectedly we find thatfor the present crystal µ H hys is 20 T larger than µ H m .The sudden onset of hysteresis at B = 36 T occurs whenthe insert magnet is energized. This and experiments ina faster-sweeping 65 T magnet show that the hysteresisincreases with | d B/ d t | . The lower inset shows our valuesof µ H m ( T ) on a larger scale [14].The frequency determined from LK fits to the data inFig. 2 and the peak positions in the Fast Fourier trans-form (FFT) spectra shown later in Fig. 3(a) both give F = 660 ±
15 T. This corresponds to a FS pocket of only2.4% of the Brillouin zone (BZ) area A BZ ( ~ πe A BZ =27 . p QO = 0 . ± .
005 compared to p =0 . ± .
005 estimated from the a -axis thermopower[17, 18]. For O-II Y123, the corresponding values of p QO = 0 . ± .
006 and p = 0 . p QO would be a factor 2 smaller. Thesame reduction in p QO is given by earlier calculationsusing the t - J model [21]. In both cases there is a dis-crepancy between p and p QO but this is not an issue ifboth electron and hole pockets are present [22].FFTs are shown in Fig. 3(a) for all temperatures mea-sured, for the rising and falling parts of the pulse. Theamplitudes of the peak at 660 T were fitted to R T giving m ∗ = 3 . ± . m e as shown in Fig. 3(b). Fig. 3(c) shows a m p lit ud e ( k H z ) T (K) (c)00.20.40.60.811.2 FF T a m p . ( a . u . ) (b)01234560 1 2 3 4 5 FF T a m p . ( a . u . ) F (kT) (a)0.53K1.1K1.6K3.0K3.6K4.0K6.5K FIG. 3: (a) (color online) Fast Fourier transforms in 1 /B of ∆ f ( B ) over the range 60 < B <
85 T. The red (black)lines show data for the rising (falling) part of the pulse. Asingle peak is present in the FFTs with a frequency F =660 ±
15 T. An extra, less reproducible peak near 200 T hasbeen removed from some of the FFTs by subtracting a slowlyvarying background. (b) FFT amplitude versus T . Open(closed) symbols show rising (falling) field data. The solid lineshows the LK damping factor R T with best-fit value m ∗ =3 . ± . m e . (c) amplitude of oscillatory function of the formsin(2 πF/B + φ ) with F = 660 T fitted to ∆ f ( B ) in the range67 < B <
77 T. The best-fit R T curve, shown by a solid line,has m ∗ = 3 . ± . m e . Dashed lines show the LK formula for m ∗ = 2 . m e and m ∗ = 3 . m e . the results of a separate LK analysis of the amplitudes ofa sin curve fitted to the data between 67 and 77 T, giving m ∗ = 3 . ± . m e . Our best value is m ∗ = 3 . ± . m e .Fig. 4a shows the overall variation of m ∗ /m e with p that is obtained by combining the present result withthat of Ref. 1. The value m ∗ /m e for p = 0 was ob-tained from ARPES spectra of the parent Mott insulatorCa CuO Cl [23] for states well below the chemical po-tential. It is an approximate value since there is no FSand the usual Fermi liquid mass enhancement effects aresuppressed. The limited data raise the possibility that m ∗ /m e could become very large as p approaches 0.19,the “special point” where heat capacity and other mea-surements on many hole-doped cuprates suggest that thepseudogap energy scale E G goes to zero. Fig. 4b showsthe p -dependence of E G and the specific heat jump at T c ,for YBa Cu O x [24]. The latter is usually ∼ γT c where γ is the Sommerfeld coefficient, for example for a weakcoupling BCS superconductor it is equal to 1 . γT c .For Y124, every two-dimensional (2D) FSsheet in the BZ will give a contribution to γ of1 . m ∗ /m e mJ mol − K − . This is independent of thenumber of carriers in the sheet and arises because in 2Dboth γ and m ∗ are proportional to the energy derivative E G / k B ( K ) dg ( m J m o l - K - ) p m * / m e (a)(b)CCOC OII-Y123Y124 FIG. 4: (color online) (a) m ∗ /m e values ( ⊡ ) for Y124 (thiswork), O-II Y123 [1] and Ca CuO Cl (CCOC), the latterfrom the dispersion of Cu-O orbital states well below thechemical potential measured by ARPES [23]. CCOC is aparent Mott insulator with p = 0 and no FS. The dashedline is a guide to the eye. (b) Heat capacity anomaly δγ at T c for various YBa Cu O x samples ( ⊙ ) and the pseudogapenergy E G ( ⊡ ) extracted from the same heat capacity datausing a triangular gap model [24]. δγ at T c is also shown( △ )for Y124 [25]. of the FS area [26] multiplied by the same enhancementfactor. Our value m ∗ = 3 . ± . m e thus implies acontribution γ = 4 . ± . − K − for every 2D FSpocket of the observed frequency present in the BZ. Anupper limit obtained from specific heat measurements ofpolycrystalline Y124 [25] is γ = 9 mJ mol − K − . Thisis a “normal state” value at T = 0 K and zero field,obtained by applying an entropy conserving constructionto γ ( T ) from T > T c to T ≪ T c , and is consistent withthe measured jump of δγ = 15 mJ mol − K − at T c . If anestimated chain contribution of 3.5 ± . − K − is subtracted, this leaves a plane contribution γ plane = 5.5 ± . − K − . Hence comparison of heatcapacity data with our results casts doubt on theoriginal model [1] involving four hole pockets near the( ± π/ ± π/
2) points where photoemission (ARPES)experiments on underdoped crystals give evidence forFermi arcs [27].Four half-pockets of holes in a reduced BZ still givean electronic heat capacity that is a factor ∼ . / . . ± . γ plane . RecentHall effect measurements [22] suggest that the quantumoscillations may be due to a single electron pocket inthe reduced BZ, centered at ( π ,0). This would be con-sistent with γ plane but implies that the proposed holepockets [22] only make a very small contribution to theheat capacity. In contrast to heavy fermion compounds,where the large heat capacity often suggested that quan-tum oscillations from the heavy electrons were not beingdetected in some of the early experiments, in the presentcase it is the small heat capacity that provides significantconstraints to theoretical models for the FS pockets.The Fermi energy ( E F ) can be calculated if we as-sume that the FS sheets responsible for the oscillationsare nearly 2D, that is, open in the c -axis direction. Fora parabolic energy dispersion, we find E F = 295 K forY124 and 375 K for O-II Y123. Intriguingly these areof the same order as the pseudogap energies E G ob-tained from heat capacity and magnetic susceptibility[24, 25] which are E G = 570 ±
30 K for O-II Y123 and E G = 360 ±
25 K for Y124. Note that these values of E G are consistent with the values of p quoted earlier.If the pockets of carriers are still present at lower fieldsand higher T , these low values of E F would lead to T -dependent diamagnetism, which although small, wouldbe much more anisotropic than the spin susceptibility.This provides another means of testing theoretical modelsand making comparisons with ARPES data. Anomalous T -dependent magnetic anisotropy has been detected inthe normal state of various cuprate superconductors andthe similarity with Landau-Peierls diamagnetism in the organic conductor HMTSF-TCNQ has been noted [28].In summary, we have observed quantum oscillations inthe 80 K cuprate superconductor Y124 that have a largerorbit area than in O-II Y123, with T c of 57 K, and aconsiderably larger effective mass. Comparison with heatcapacity data places strong constraints on the number ofpockets present in the BZ, and supports models with areduced BZ and small FS.After completing the present measurements, we be-came aware of Hall resistivity results for YBa Cu O [13] giving values of F and m ∗ that agree with ours.JRC and EAY thank A. Carrington, S.M. Hayden, N.E.Hussey [29], J.W. Loram and J.L. Tallon for helpful dis-cussions and collaboration and the EPSRC (U.K.) forfinancial support. This work is supported by DoE grantsLDRD-DR-20070085 and BES Fieldwork grant, “Sciencein 100 T”. Work at NHMFL is performed under the aus-pices of the National Science Foundation, DoE and theState of Florida. [*] Present address: School of Physics and Astronomy, Uni-versity of St Andrews, KY16 9SS, United Kingdom.[1] N. Doiron-Leyraud, et al. , Nature , 565 (2007).[2] The a -axis thermopower [17] suggests p = 0 . ± . T -dependence of the a -axis resistivity suggest Y124 hasnearly the same doping as YBa Cu O . [15].[3] E. A. Yelland, PhD Thesis, University of Cambridge, UK(2003).[4] N. Harrison, et al. , Phys. Rev. Lett. , 056401 (2007).[5] T. Coffey, et al. , Rev. Sci. Instr. , 4600 (2000).[6] C. Mielke, et al. , J. Phys. Condens. Matter , 8325(2001).[7] We calculate the amplitude of dHvA oscillations in thedifferential magnetization using the expression for a 2Dslab of k -space given in Fig. 2.11 of [8]. For one orbitof area 0.024 of the 2D BZ, with m ∗ /m e = 3, a slabthickness δk = 2 π/ c (where c is the spacing betweenCuO bi-layers), a coil filling factor of 1/4 and a normal-state a -axis resistivity ρ a = 32 µ Ω cm, the signal ampli-tude should be 2 kHz peak-to-peak. For ρ a & µ Ω cmwe expect dHvA oscillations to dominate, while for ρ a . µ Ω cm, Shubnikov-de Haas oscillations would do so.The ρ a measured in Ref. [13] suggests that we are in theregime where both contributions are significant.[8] D. Shoenberg, Magnetic oscillations in metals (Cam-bridge University Press, Cambridge, UK 1984).[9] The subtracted background is of the form c + c ( B −
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