Fermi Surface of the Electron-doped Cuprate Superconductor Nd_{2-x}Ce_xCuO_{4} Probed by High-Field Magnetotransport

We report on the study of the Fermi surface of the electron-doped cuprate superconductor Nd$_{2-x}$Ce$_x$CuO$_{4}$ by measuring the interlayer magnetoresistance as a function of the strength and orientation of the applied magnetic field. We performed experiments in both steady and pulsed magnetic fields on high-quality single crystals with Ce concentrations of $x=0.13$ to 0.17. In the overdoped regime of $x>0.15$ we found both semiclassical angle-dependent magnetoresistance oscillations (AMRO) and Shubnikov-de Haas (SdH) oscillations. The combined AMRO and SdH data clearly show that the appearance of fast SdH oscillations in strongly overdoped samples is caused by magnetic breakdown. This observation provides clear evidence for a reconstructed multiply-connected Fermi surface up to the very end of the overdoped regime at $x\simeq 0.17$. The strength of the superlattice potential responsible for the reconstructed Fermi surface is found to decrease with increasing doping level and likely vanishes at the same carrier concentration as superconductivity, suggesting a close relation between translational symmetry breaking and superconducting pairing. A detailed analysis of the high-resolution SdH data allowed us to determine the effective cyclotron mass and Dingle temperature, as well as to estimate the magnetic breakdown field in the overdoped regime.


INTRODUCTION
Electronic correlations and the resulting ordering instabilities are central issues in the long-standing problem of high-temperature superconductivity in copper oxides. To elucidate them, the exact knowledge of the Fermi surface and its evolution with doping is of crucial importance. High-field magnetotransport is known as one of the most powerful tools for studying Fermi surfaces of conventional metals [1,2]. It has recently proved very efficient also in the case of cuprate superconductors. A breakthrough in the Fermiology of hole-doped cuprates was made by the observation of semiclassical angle-dependent magnetoresistance oscillations (AMRO) [3,4] and quantum oscillations of the resistance, the Shubnikov-de Haas (SdH) effect [5][6][7][8].
On the strongly overdoped side of the phase diagram of hole-doped cuprate superconductors, both the semiclassical AMRO [3] and quantum oscillations of the resistance [8] have been found for the compound Tl 2 Ba 2 CuO 6+δ . These experiments provided evidence for a large cylindrical Fermi surface, as expected from band-structure calculations [9,10] and angle-resolved photoemission spectroscopy (ARPES) [11]. In contrast, for underdoped YBa 2 Cu 3 O 6.5 [5,12,13] and YBa 2 Cu 4 O 8 [6,7] slow SdH and de Haas-van Alphen (dHvA) oscillations were found, indicating a reconstruction of the Fermi surface. These observations reveal substantial disagreements with ARPES results [14] and are controversially interpreted at present, see e.g. Refs. [12,[15][16][17][18][19][20][21][22][23][24][25]. The electron-doped cuprates Ln 2−x Ce x CuO 4 (Ln = Nd, Pr, Sm) have a number of advantages for high-field Fermi surface studies, as compared to hole-doped cuprates. Due to their lower critical fields, superconductivity can easily be suppressed and the normal state is accessed for any doping level even at the lowest temperatures by applying a magnetic field B 10 T (perpendicular to CuO 2 layers). Moreover, the Fermi surface is expected to be simple: there are neither CuO chains nor any bilayer potential and the magnetic superstructure, established in underdoped compounds, is commensurate. Another important advantage is that, in contrast to most of the hole-doped cuprates, the entire doping range, from the undoped insulating up to the strongly overdoped metallic (superconducting) phase, can be covered using one and the same compound with just slightly different Ce concentrations [see Fig. 1(a)].
We have recently reported on SdH oscillations in Nd 2−x Ce x CuO 4 (NCCO) single crystals with Ce concentrations corresponding to nearly optimal doping (x = 0.15) and to over- The Fermi surface, reconstructed due to a (π/a, π/a) superlattice potential, consists of electron (blue) and hole (dark red) pockets. The size of the small hole pockets is consistent with the frequency of slow oscillations observed for the optimal, x = 0.15, and slightly overdoped, x = 0.16, compositions [26].
doped compositions (x = 0. 16  thin films [31], it is obviously inconsistent with the conclusions of ARPES [28,29] and neutron-scattering studies [29]. An important issue to be clarified is the exact origin of the Fermi surface reconstruction which was revealed by the SdH oscillations in our overdoped NCCO samples. It was proposed [26,32] to be caused by a (π/a, π/a) superlattice potential (a is the lattice constant within the CuO 2 plane) which is known to exist in undoped and underdoped NCCO [28,33].
Indeed, the frequency of the slow SdH oscillations, F slow ≈ 300 T, is consistent with the size of small hole pockets, which should be formed around (±π/2a, ±π/2a) due to such ordering [see Fig. 1(c)]. On the other hand, no indication of electron pockets, required by the proposed reconstructed Fermi surface topology, were found in the SdH spectra. Obviously, additional work is needed for further verifying the reconstruction scenario. For example, it would be desirable to determine not only the size but also the shape of the Fermi pockets.
Towards this end, the study of semiclassical AMRO is a very efficient tool. AMRO have been widely used for mapping in-plane Fermi surfaces of organic conductors [34] and other layered systems such as Sr 2 RuO 4 [35] and intercalated graphite [36]. This effect has a geometrical origin and is directly related to the shape of a weakly warped cylindrical Fermi surface [3,[37][38][39]. As mentioned above, AMRO already have been observed in hole-overdoped Tl 2 Ba 2 CuO 6+δ (Tl2201) samples [3,4] and successfully used for extracting the shape of the three-dimensional Fermi surface as well as for evaluating the scattering anisotropy.
Very recently, we have also found features characteristic of AMRO in the angle-dependent interlayer magnetoresistance of overdoped NCCO [40]. Although the magnitude of these features observed in applied magnetic fields up to 28 T was too low for a reliable quantitative analysis, they provided an important argument for the existence of magnetic-breakdown orbits on the Fermi surface. This magnetic breakdown scenario was further supported by the observation of two frequencies in the SdH spectrum obtained for strongly overdoped (x = 0.17) NCCO crystals [40].
Here, we present new data on the interlayer magnetoresistance of NCCO single crystals studied as a function of the orientation and strength of the applied magnetic field at Ce concentrations between x = 0.16 and 0.17, corresponding to the overdoped regime of the phase diagram [see Fig. 1(a)]. Our studies confirm the existence of AMRO and reveal magnetic-breakdown quantum oscillations at compositions down to x = 0.16.

EXPERIMENTAL TECHNIQUES
Single crystals of NCCO were grown by the traveling solvent floating zone (TSFZ) method and thermally treated in pure argon to remove interstitial oxygen and release internal strain.
The crystals were thoroughly tested by x-ray diffraction, magnetic and resistive measurements. The best samples, with the lowest doping inhomogeneity (typically within 0.25%) and the largest (high-to-low temperature) resistance ratios, see Fig. 2(a), were selected for the high-field experiments. Details of crystal preparation and characterization are presented elsewhere [41].
The resistance of the samples was measured out-of-plane, that is in the direction perpendicular to the conducting CuO 2 layers, for two reasons. First, for the angle-dependent magnetoresistance studies: the AMRO phenomenon mentioned in Section is an inherent property of the interlayer magnetoresistance and should be much more pronounced in this configuration [34,42]. Second, due to the high resistivity anisotropy, ∼ 10 3 , the interlayer resistance value is usually much higher than the in-plane resistance and, hence, easier to measure. Taking the advantage of the TSFZ technique[? ], we were able to further increase the signal by optimizing the shape of the crystal cut out of the as-grown rod. Given the room-temperature resistivity of NCCO in the range 3 to 6 Ω cm, depending on the doping level, and the typical sample dimensions, about 0.3 × 0.3 × 1 mm 3 with the largest dimension along the c-axis, we could obtain a resistance of 300 − 600 Ω at room temperature and 30 − 100 Ω at low temperatures. For some batches we had to decrease the sample thickness to avoid crystal inhomogeneity. In those cases the low-temperature resistance values were ∼ 1 − 10 Ω. These values are still quite convenient and can be measured accurately.   voltage and current leads, annealed 20 µm Pt wires were attached using either Dupont 4929 silver paste or Epo-Tek H20E conducting epoxy. The latter provided the most reliable contacts with the lowest resistance. Due to their highly anisotropic susceptibility caused by the magnetic moments of the Nd 3+ ions, NCCO crystals experience a very strong torque in a tilted magnetic field. Therefore, for our experiments we firmly glued the samples to a sapphire plate by Stycast 2860 FT (blue color), as shown in Fig. 2(b). The plate was then 7 fixed by the same epoxy to an appropriate sample holder.
The angle-dependent magnetoresistance was measured in steady fields up to 28 and 34 T provided by the 20 MW resistive magnet at the LNCMI-Grenoble. The samples were mounted onto a home-made two-axes sample rotator allowing an in situ rotation at a fixed B, at temperatures between 50 and 1.4 K. The resistance was measured as a function of polar angle θ between the field direction and the crystallographic c-axis for different fixed azimuthal angles ϕ, as shown in Fig. 2(c). SdH oscillations were studied with the magnetic field applied parallel to the c-axis. These experiments were done in pulsed fields provided by the Dresden and Toulouse high-field facilities as well as in steady fields at Grenoble. The anomalous θ-dependence obtained for NCCO is almost insensitive to the azimuthal orientation of the applied field, that is, to the angle ϕ. This suggests that it is determined solely by the out-of-plane field component. Such behavior is similar to what was observed in some organic layered metals and associated with incoherent interlayer charge transfer (for a recent discussion, see [43]). In addition, the R(θ) dependence taken at ϕ = 0 • exhibits a clear hysteresis with respect to the rotation direction at |θ| ≥ 6 • . This hysteretic behavior is most likely related to a field-induced reorientation of ordered spins. Indeed, recent reports on the in-plane field rotation [44,45]

ANGLE-DEPENDENT SEMICLASSICAL MAGNETORESISTANCE
ig. 3 ordering are present even in the superconducting compound with x = 0.13. Further work is necessary for clarifying the detailed mechanism responsible for coupling the interlayer magnetoresistance to electron spins.
As we increase the Ce concentration to the optimal and, further on, to the overdoped regime, the anomalous contribution to the magnetoresistance weakens, giving way to the conventional behavior associated with the orbital effect of the applied magnetic field on the charge carriers. This causes a positive, ϕ-dependent slope dR/d|θ| > 0 of the angular dependence R(θ) over an extended angular range 30 • |θ| 80 • , as shown in Fig. 3 azimuthal orientation and its independence of temperature and the magnetic field strength, as is shown in Fig. 4, clearly point to the geometrical origin of this feature. Therefore, we attribute this hump-like feature to the AMRO effect.
The R(θ) curves in Fig. 3(c),(d) are qualitatively similar to the angle-dependent magnetoresistance of the hole-overdoped cuprate Tl2201 [3,4], which also exhibits a local maximum around θ = 0 • and a shoulder at |θ| 30 − 40 • . In Tl2201 having the same body-centered tetragonal crystal symmetry, the entire angular dependence can be nicely described within the semiclassical Boltzmann transport model with a k-dependent scattering time [4,46]. In particular, both the central hump and the side feature come from the AMRO effect of the slightly warped large Fermi cylinder centered at the corner of the Brillouin zone. By analogy, it is tempting to consider our data on NCCO as a manifestation of the unreconstructed large Fermi cylinder, as in Fig. 1(b). However, the situation seems to be more intricate. By contrast to the case of Tl2201, the central (θ = 0 • ) hump is much more pronounced, as compared to the side features. Furthermore, at odds with what is expected from an AMRO peak, it does not diminish at increasing the temperature. Instead, it develops in size and becomes a dominating feature in the angular dependence above ∼ 20 K, as shown, e.g., in Fig. 4(a) for a sample with x = 0.165. The 21 and 23 K curves in this Figure rather resemble the R(θ) dependence in the underdoped, x = 0.13 NCCO, c.f. Fig. 3(a). It is, therefore, likely [40] that, at least at elevated temperatures, the central hump originates from a remnant of the anomalous, presumably incoherent, interlayer transport channel rather than from the conventional AMRO effect.
The superposition of the conventional orbital and anomalous contributions to the R(θ) dependence makes it problematic for a quantitative analysis. However, already at the present stage, we can assert that the Fermi surface giving rise to the AMRO is the same in the doping interval from x = 0.16 to 0.17. This is evidenced by the similarity of the AMRO features in Fig. 3(c),(d) and Fig. 4, as well as by their angular positions shown in Fig. 5. Having in mind the strong difference between the quantum-oscillation spectra reported for x = 0.16 and 0.17 [26], one is lead to the conclusion that the large cyclotron orbit, indistinguishable from that on the original Fermi cylinder shown in Fig. 1(b), and the small orbit corresponding to hole pockets of the reconstructed Fermi surface [see Fig. 1(c)] should coexist at one and the same composition. Indeed, this was confirmed by the observation of both SdH frequencies in the strongly overdoped (x = 0.17) NCCO crystal [40]. In Section , we present further data on quantum oscillations which provides clear evidence for magnetic breakdown to take place also at lower doping down to x = 0.16.
We note that for x = 0.15 the R(θ) dependence presented in Fig. 3    By contrast to hole-underdoped YBa 2 Cu 3 O 6.5±δ , for which several different low SdH frequencies have been reported [15,16,47], the present NCCO compound only shows a single frequency. In particular, so far we have not found any signature of oscillation beating, caused by a slight warping of the Fermi surface in the interlayer direction. This suggests that the distance between adjacent Landau levels, ω c (where ω c = eB/m c is the cyclotron frequency and m c is the relevant effective cyclotron mass), exceeds the interlayer dispersion at fields B 20 T. Therefore, one should take into account the strong two-dimensionality of the system in the analysis of the oscillations. Generally speaking, the quasi-two-dimensional SdH effect is a very complex problem and has not yet received a comprehensive description even within the standard Fermi liquid theory [34,48]. Moreover, it was recently pointed out [49] that the presence of strong electron correlations should dramatically affect the behavior of quantum oscillations, provided ω c significantly exceeds the interlayer transfer integral t ⊥ and scattering-induced broadening of Landau levels ∼ /τ (where τ is the scattering time). However, our case is simplified due to the weakness of the oscillation amplitude and the absence of higher harmonics, which is indicative of a strong Landau level broadening.
Therefore, we adopt the so-called two-dimensional Lifshitz-Kosevich (LK) formula [50] for parison, the effective mass plot obtained from the pulsed-field data is shown by diamonds in Fig. 8(b), giving a somewhat lower value, µ pulse 0.16 = 0.7 ± 0.1. The discrepancy is apparently caused by an uncertainty in the sample temperature in the pulse-field experiment. Although the induced eddy currents do not cause a significant heating effect due to the relatively long pulse duration, 0.2 s, the temperature is affected by the fast orientation of the paramagnetic Nd 3+ ions reducing the entropy of the spin system. Obviously, the effect is enhanced at lower temperatures, which results in a flatter apparent T -dependence and, hence, in a smaller µ. Therefore, the derivation of the cyclotron mass from the steady-field data is considered more reliable. Unfortunately, the fast oscillations have so far been detected only  Fig. 8(c), yielding T D ≈ 10 K. Strictly speaking, when fitting the B-dependent SdH amplitude we should take into account the effect of magnetic breakdown through the gap (which is thus considered as a breakdown junction) between the hole and electron pockets of the reconstructed Brillouin zone. The closed orbit on the small hole pocket responsible for the slow oscillations involves reflections from two breakdown junctions. Therefore, the corresponding damping factor for the oscillation amplitude is [50,51]: , where B 0 is the characteristic breakdown field. Including R MB with a very low breakdown field, B 0 = 1 T, (see below) into the field dependence A osc (B), we obtain the fit shown by the red line in Fig. 8(c), which is almost indistinguishable from the pure [B 0 = ∞, green line in Fig. 8(c)] Dingle fit and gives just a slightly different T D ≈ 12 K. Using the conventional relationship between T D and the scattering time τ , T D = /2πk B τ , we estimate τ ≈ 0.15 ps.
This corresponds to a mean-free path averaged over the cyclotron orbit of 18 nm.
The obtained cyclotron mass and scattering time values determine the field strength parameter ω c τ which merely reaches unity at B 35 T (for the slow oscillations; for the fast SdH oscillations it is even lower due to heavier m c ). This justifies the validity of the standard two-dimensional LK formula for fitting the steady field data. However, as mentioned above, one can expect deviations from the LK theory at higher fields, when ω c τ > 1. In particular, in future it would be interesting to discuss the pulsed field data in terms of the recent theory [49] taking into account strong electron interactions.
As follows from the comparison between the two fits in Fig. 8  the oscillations from the small hole pockets at the same field (we recall that the latter require only two reflections from magnetic breakdown junctions). Additional damping comes from the temperature and Dingle factors, since the cyclotron mass corresponding to the electron orbits is, obviously, heavier than that of the small hole orbits. Altogether, we expect the amplitude of the oscillations coming from the electron pockets to be about two orders of magnitude lower than that associated with the hole pockets at x = 0.16 and even further reduced at higher doping levels. It is, therefore, not surprising that no clear signature of the electron pockets has been found in the SdH spectrum so far.
Using the Blount criterion, one can estimate the energy gap ∆ between the electron and hole bands formed by the superlattice potential [52]: ∆  [41]. Furthermore, our data show that the superlattice potential is strongly reduced at the highest doping level. Therefore, it is likely that both the superlattice potential and superconductivity vanish at the same carrier concentration. This would strongly suggest a close relation between the two ordering phenomena.
The existence of the superlattice potential in the overdoped regime distinguishes our electron-doped compound from the hole-doped cuprates. In the latter, a broken symmetry has been observed only below optimal doping so far. The mechanism responsible for the broken translational symmetry is still to be clarified. One possibility is that the commensurate antiferromagnetic ordering persists from the underdoped regime [53,54]. It was very recently proposed [55] that, by contrast to hole-doped cuprates, the mechanism underlying the metal-insulator transition in the electron-doped materials is of magnetic origin rather than due to electronic correlations. In this case the metallic and superconducting state could coexist with antiferromagnetism, provided the magnetic interaction is not strong enough for causing a metal-insulator transition. However, until now no static magnetic ordering has been detected in overdoped NCCO [56]. This seems to favor another scenario associated with a "hidden" d-density-wave ordering [17,57]. On the other hand, it is possible that the ordering only exists at high magnetic fields, i.e. at the conditions of our present experiment.
An in-depth study of the magnetic properties at high fields should elucidate this issue.
Although our experiments already give new and interesting insight, further detailed Fermi surface studies in high fields are highly important for clarifying the origin of the superlattice potential. Angle-dependent magnetoresistance experiments extending up to the highest steady fields of 45 T and covering also a broader doping range, should allow a quantitative characterization of the Fermi surface geometry and its dependence on the carrier concentration. The angle dependence of the SdH oscillations is also potentially important for understanding the spin state of the conduction electrons and the nature of the ordering [15,16,58].
As mentioned above, the suggested reconstruction of the Fermi surface in overdoped NCCO implies the existence of electron pockets, in addition to the hole ones. However, the magnetic breakdown occurring at the highest doping levels suppresses orbits encircling the whole electron pocket more effectively than those around hole pockets. Therefore, lower doping levels with a stronger superlattice potential appear to be more appropriate for searching for a manifestation of the electron pockets in high-field magnetotransport.
Another still open question is how the Fermi surface develops upon entering the underdoped range of the phase diagram. The lowest doping level at which we have so far succeeded in finding SdH oscillations is x = 0.15. The reason why they are not observed at lower doping is not clear. Could it be that the closed hole pockets disappear immediately below the optimal doping? Or is it because the conventional orbital effect on the interlayer conductivity becomes too small, compared to the emerging anomalous incoherent magnetotransport?
In the latter case, it might be more convenient to look for magnetic quantum oscillations in the in-plane resistivity or in thermodynamic properties. Unfortunately, the oscillations of magnetization (de Haas-van Alphen effect) can hardly be detected in NCCO due to an overwhelming magnetic contribution from the Nd 3+ moments. One could, however, search for the oscillations in some other properties such as sound velocity or magnetostriction.