High-eld normal-state magnetotransport properties of electron-doped cuprate superconductors

Transcription

1 Walther-Meiÿner-Institut Technische Universität Physik Department für München Lehrstuhl E23 Tieftemperaturforschung High-eld normal-state magnetotransport properties of electron-doped cuprate superconductors Masterarbeit Tzanos Vasileios Themensteller: Prof. Dr. Rudolf Gross Garching, 12. Mai 2014

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3 Contents 1 Introduction Historical review First superconductors High T c superconductors Cuprates Experimental state of the art Conclusion Aim of the experiments Thesis layout Elements of theory Quantum oscillations Landau levels Oscillations in magnetization Smearing factors Oscillations of other quantities Thermoelectric eect Nernst eect Seebeck eect Experimental setup Samples NCCO crystals Crystal Growth Annealing process Sample preparation Experimental equipment High eld experiments Setup preparation Experimental results and discussion High-eld measurements Quantum oscillations Non-oscillatory signal Low eld calibration measurements Conclusion 59 iii

4 Contents 6 Appendix Thermometer calibration Magnetoresistance measurements Insert tail preparation Chromel measurement Acknowledgments 67 iv

5 1 Introduction 1.1 Historical review First superconductors Superconductivity was rst discovered by Kamerlingh Onnes in 1911 when he noticed that the resistance of mercury exhibited a sudden drop to zero around 4:2 K [1]. After 22 years, in 1933, the expulsion of magnetic ux from the superconductor's interior was discovered by Meissner and Ochsenfeld, whose names were given to this eect [2]. While a lot of phenomenological theories were proposed, trying to replicate the experimental results, it took over 40 years from the rst observation of superconductivity, for the Bardeen-Cooper-Schrieer (BCS) theory to be proposed [3]. The BCS theory was the rst theory that explained satisfactorily various experimental results (e.g. isotope effect, specic heat ratio etc), while giving us an in-depth view of how superconductivity arises at the microscopic level. Central to this view is that the formation of Cooper pairs out of two electrons becomes energetically favorable for low enough temperatures. The energy gap created by this pairing prohibits any scattering of the carriers at these temperatures, leading to zero resistance. In this regard the theory needs a coupling mechanism for two electrons which normally repulse each other. In the case of 'conventional' superconductors this mechanism comes from an electron-phonon coupling: an electron distorts the lattice and this leads to the 'capture' of another electron and the formation of a Cooper pair High Tc superconductors The rst high T c superconductor (La 2 x Ba x CuO 4 ) was found in 1986 [4] by Bednorz and Müller earning them a Nobel prize the next year. Following this discovery, the research in this eld was greatly intensied, leading to materials with increasingly high transition temperatures, the highest being around 135 K. One element that immediately distinguishes high T c superconductors from 'conventional' superconductors is their generally higher transition temperatures. In many cases the transition temperatures are signicantly higher than what was expected from a purely electron-phonon coupling. In general, the 'traditional' BCS theory is considered insucient to explain these new types of superconductors, making it necessary to look, even within the frame of BCS theory, for a dierent mechanism that creates Cooper pairs. Such superconductors are labeled 'unconventional' in contrast, and include apart from high T c superconductors, heavy fermion superconductors as well as organic superconductors. However, a theory satisfactorily explaining the pairing mechanism and predicting all the experimental results for these superconductors still eludes us. 1

6 1 Introduction The focus of this work will be on cuprates which, along with iron based superconductors, are the two main families of high T c superconductors. Furthermore, within the cuprate family, we are going to limit ourselves mostly to the electron-doped compounds. 1.2 Cuprates Structurewise, the common feature shared by cuprates are the CuO 2 planes. These conducting planes are separated by insulating layers ("charge reservoir blocks"), making cuprates essentially two-dimensional materials. An overview of the various materials one can build based on the above simplied picture can be found in Fig. 1:1, while a detailed description can be found in [5]. A further common characteristic of cuprates seen in their phase diagram (Fig. 1:2), is that the parent, undoped compounds are Mott insulators. The electronic structure giving rise to the insulating behavior of the "charge reservoir blocks" with the half-lled band, which should, thus, be conducting, is the following. Copper atoms (Cu +2 ) are in d 9 electronic conguration, while oxygen atoms (O 2 ) are in a p 6 conguration. In this case the single 3d hole has a d x 2 y 2 symmetry. The result is a half-lled band consisting of the antibonding combination of the Cu 3d x 2 y 2 orbital with the O p x and p y orbitals, while all other orbitals are lled [6]. It is this half-lled band which, due to strong electronic correlations produces an energy gap of the order of 2 ev, turning these materials into Mott insulators. In terms of the Hubbard model, which can suciently describe the insulating behavior of cuprates, this half-lled band splits, due to Coulomb repulsion between electrons, into one fully lled lower Hubbard band and one empty upper Hubbard band. Inserting charge carriers in the CuO 2 plane, via doping, leads to the destruction of the insulating behavior since the upper Hubbard band becomes also partially occupied. Doping charge carriers is another feature seen among all cuprate superconductors. It divides them into two dierent subgroups, namely electron and hole doped compounds. From a chemical point of view, doping refers to inserting a new element in the "charge reservoir blocks", partly substituting another one in the stoichiometry of the material. It is well known that superconductivity is observed for a doping level of 0:05 < x < 0:27 for hole doped superconductors, while for the electron-doped, the range is approximately 0:12 < x < 0:18. Doping x here, refers to additional charge carriers per Cu atom in the CuO 2 planes. It is worth mentioning, that these values are heavily dependent on the preparation procedure followed, particularly for the electron doped superconductors [7]. As mentioned before, doping destroys the insulating behavior of the parent compound caused by the electron-electron correlations. From a magnetic point of view, doping destroys the long range antiferromagnetic order prevalent in the undoped and far underdoped regime. The destruction happens faster for hole-doped cuprates compared to the electron-doped ones. In the case of hole-doped cuprates the magnetic order ceases to exist at around 3% doping, while for the electron doped ones it exists until superconductivity emerges, or even coexists with superconductivity [8]. Moving to higher doping levels in the phase diagram, one encounters the enigmatic pseudogap region, an area of the phase diagram where a partial energy gap is observed 2

7 1.2 Cuprates Figure 1.1: Classication of cuprate superconductors and their highest T c [5]. at the Fermi surface. Curiously enough, this gap seems to exhibit the same d-wave symmetry as superconductivity in cuprates [9]. The gap is bigger at lower dopings and vanishes at higher dopings [10]. Though it may seem logical to relate the pseudogap 3

8 1 Introduction Figure 1.2: Qualitative phase diagram of cuprates. The exact values can dier depending on the compound. It is clear that the superconducting range is far bigger for hole-doped superconductors. On the other hand, the antiferromagnetic region is considerably more robust in the electron-doped side. with superconductivity, to this day it is still unclear whether pseudogap represents a precursor state of superconductivity [11], or if the energy gap comes from an ordered state in competition with superconductivity [12]. A further, still open, issue is the exact place of the pseudogap in the phase diagram, particularly whether it exists inside the superconducting dome or not. On the hole-doped side of the phase diagram and above the superconducting dome one nds another unusual region, named strange metal region. The name is given in contrast to normal metals, which can be described within the Fermi liquid theory. In this regard the strange metal region violates the Fermi liquid behavior, exhibiting, for example, resistivity with a linear dependence on temperature. This region can, for certain optimally doped samples, reach extremely high temperatures of the order of 1000K [13]. Summing up, cuprate superconductors, compared to conventional ones, have typically higher transition temperatures, while also having a rather rich phase diagram where 4

9 1.2 Cuprates dierent, and perhaps competing, phases are in close proximity to each other. It is clear, at this point, that the traditional electron-phonon coupling of the BCS theory is not sucient. However, as mentioned before, a successful microscopic theory for these materials is not realized yet Experimental state of the art An important part of the current experimental research is focused on investigating the ground state of cuprates in the absence of superconductivity, since we would get information on exactly the state that gives birth to superconductivity, and how it evolves with regards to doping. This would give us more insight on how superconductivity is created in cuprates. Diculties arise from the fact that superconductivity occupies a big enough area in the phase diagram, so that the competition between the dierent phases in proximity to superconductivity, in the phase diagram, is hidden by the superconducting dome. One way of looking inside the dome, is by applying high enough magnetic elds, so that superconductivity is suppressed. Furthermore, a way to investigate the Fermi surface evolution in correlation with the doping level, that has lately proved quite successful, is through magnetotransport measurements, utilizing Magnetic Quantum Oscillations (MQO). More details on the practical aspect of getting information for the Fermi surface from MQO are given in the theoretical chapter of the thesis. The present work will follow this path, reporting on measurements taken mainly in the Laboratoire National des Champs Magnetiques Intenses (LNCMI), Grenoble, but also in the Walther Meissner Institut (WMI), Garching. From an experimental point of view, hole-doped superconductors have been far more studied as compared to electron-doped ones. The main reason for that, apart from being more numerous, is technical diculties in fabrication of suitable samples of electrondoped cuprates. Particularly complicated, even to this day, is the dependence of superconductivity and transition temperature to the annealing conditions of the sample [7, 14]. Relatively small changes in the annealing temperature or the duration of the annealing, can produce substantial dierences in the critical temperature of the sample, or even prohibit the appearance of superconductivity. However, signicant progress in the growing of the samples were made (see chapter on sample preparation for more details) so that now very homogeneous samples with relatively precise doping level 0:25% can be fabricated. It was this progress in fabrication, that gave the opportunity to experiment with electron-doped cuprates more intensively the last years. On the other hand, electron-doped superconductors have also some inherent advantages compared to the hole-doped ones. Firstly, one needs generally much lower magnetic elds to suppress superconductivity, making a large part of the superconducting dome much more readily available for normal-state measurements. Furthermore, electron-doped cuprates have a simpler crystallographic structure and a simpler Fermi surface, which makes them much better testing ground for theoretical results. Finally, their superconducting doping range is considerably more limited, making the investigation easier. For these reasons our main focus during this thesis was on Nd 2 x Ce x CuO 4 (NCCO), an electron-doped cuprate superconductor. Given that there are over 100; 000 publications on cuprates, a detailed presentation 5

10 1 Introduction of experimental results of the last three decades would lie out of the scope of this text. Instead, the discussion here is going to be limited to some results that are relevant to this work. A more detailed insight on past results and new development in the eld of electron-doped cuprates one can see in Ref. [15]. As early as in 1989, resistance measurements of optimally doped NCCO revealed a quadratic temperature dependence of resistivity from T c up to 250 K, consistent with a Fermi liquid description [16]. In contrast, similar measurements for hole-doped cuprates reveal, as mentioned before, a 'strange metal' behavior where resistivity has a linear temperature dependence. Later studies on NCCO crystals for the whole underdoped region showed again an approximately Fermi liquid behavior at higher temperatures (down to x = 0:025) [17], although, in contrast to the optimally doped case, here the resistance showed a minimum at a doping-dependent temperature and an increase at lower temperatures. Undoped samples, on the other hand, showed an insulating behavior for the whole temperature spectrum [17]. Figure 1.3: Fermi surface reconstruction scenario: (right) a big holelike cylindrical surface in the far overdoped region; (middle) reconstructed Fermi surface due to a (=a; =a) superlattice potential; (left) a Fermi surface consisting of electron pockets at the strongly underdoped side. With regards to the cuprate Fermi surface and for the more explored hole-doped side of the phase diagram we have the following information. For the overdoped side, angle-dependent magnetoresistance oscillations measurements on an overdoped Tl 2 Ba 2 CuO 6+, revealed a coherent Fermi surface occupying around 62% of the Brillouin zone [18], in agreement with band structure calculations. Later experiments on the same cuprate, using MQO, revealed similarly a coherent Fermi surface occupying around 65% of the Brillouin zone[19]. On the other hand, results for hole-doped samples in the underdoped region give strikingly dierent results. MQO were reported, for the rst time in a cuprate, in an underdoped YBCO sample (YBa 2 Cu 3 O 6:5 ) (Y123, p=0:10) [20]. The corresponding Fermi surface area of the pockets was calculated at around 2 % of the Brillouin zone. MQO found in underdoped YBa 2 Cu 4 O 8 samples (Y124) revealed an unexpectedly small Fermi surface that occupies around 2:4 % of the Brillouin zone 6

11 1.2 Cuprates [21]. These results bring up the question, whether these small Fermi surface pockets found are unique to these YBCO samples, or are common for underdoped, hole-doped cuprates. The results on the (Y123) point to the latter scenario, since small pockets were not predicted by band structure calculations for this compound. A newer study on YBCO samples (YBa 2 Cu 3 O 6:51 ) revealed two dierent oscillations, corresponding to two dierent carrier pockets [22], further strengthening the Fermi surface reconstruction scenario. Apart from MQO measurements, evidence of a Fermi surface consisting of small electron pockets, for underdoped samples, is given by Seebeck and Nernst coecient measurements [23]. Here for low temperatures a large and negative normal-state signal was observed. Authors attributed this signal to the electron pockets, found in other hole-doped compounds, which dominated the holelike portions of the Fermi surface. Finally, a more recent experiment on HgBa 2 CuO 4+, where Nernst, Seebeck and Hall coecients were measured [24], showed similar results to YBCO samples. Again, these new results support the view, that the reconstruction of the Fermi surface described above is not limited to YBCO but is universal to hole-doped cuprate superconductors. On the electron-doped side there are Hall eect measurements on Pr 2 x Ce x CuO 4 (PCCO) samples, where a change of sign is observed around optimal doping. This a- gain suggests a change of the Fermi surface [25] from a large holelike cylindrical surface at the far overdoped region, to small electron and hole pockets in the optimally doped region. Moreover, Hall eect measurements on NCCO single crystals for doping levels of x = 0:13; 0:15; 0:16; 0:17, provide some hints of a further Fermi surface reconstruction occurring between x = 0:13 and x = 0:15 [14]. More specically, Hall resistivity measurements show negative values for the underdoped sample. On the other hand for optimally doped and overdoped samples, the values can be either positive or negative depending on temperature. These results are consistent with a Fermi surface evolution from electron pockets in the underdoped samples to electron and hole pockets at the optimally doped and overdoped samples. The Fermi surface reconstruction picture agrees with angle resolved photoemission spectroscopy (ARPES) measurements, showing a clear transformation of the Fermi surface with doping for PCCO [27] and NCCO samples [28]. In 2009, for the rst time, MQO were found in NCCO samples [29]. In this case, a large change in the oscillation frequency between x = 0:16 and x = 0:17 doping levels was observed (Fig.(1:5)). Again, the interpretation given to the results is that a Fermi surface reconstruction is taking place, from a big cylindrical Fermi surface to a small Fermi surface occupying around 1 % of the Brillouin zone, for this doping level. These results disagree with previous ARPES measurements in NCCO [30] which support a Fermi arc scenario (see Fig 1:4). The issue is still being debated, although, it should be noted that ARPES measures surface properties of the materials which can dier for the bulk properties, intrinsic to the material. Further measurements on NCCO samples [31, 32] revealed a slightly dierent picture. Here the reconstructed Fermi surface observed at doping levels of x = 0:15 and x = 0:16 seems to extend at least to x = 0:17. More specically, angular-dependent magnetoresistance oscillations (AMRO) showed similar positions for both x = 0:16 and x = 0:17. 7

12 1 Introduction Figure 1.4: Results of ARPES experiments on NCCO samples for various dopings, that shows the evolution of Fermi surface from small pockets to a hole like Fermi surface at optimal doping. Figure taken by [26] Figure 1.5: Fourier transformation of the observed SdH oscillations for NCCO single crystals of doping levels x = 0:15, x = 0:16, x = 0:17. For the rst two doping levels there is a clear peak at around 280 T, while for the most overdoped sample the peak is around 10 kt. Note the sudden change in the peaks between doping levels x = 0:16 and x = 0:17 [29]. Since the position of AMRO is determined by the Fermi surface shape, these results point to a similar Fermi surface for these doping levels. Moreover, further Shubnikovde Haas (SdH) oscillations measurements on NCCO crystals of doping level x = 0:17 revealed a second, slow oscillation [32], along with the fast one, attributed to a big cylin- 8

13 1.2 Cuprates Figure 1.6: MQO observed in NCCO single crystals for doping levels x = 0:16; 0:165; 0:17. For easier observation, the curves have been normalized. One can see the two frequencies, one slow attributed to hole pockets and a fast one attributed to a large orbit due to magnetic breakdown. Note that as doping gets lower, one needs higher elds to observe the fast oscillations [31]. drical Fermi surface (Fig 1:6). These new results favor a magnetic breakdown scenario, where the reconstructed Fermi surface persists at least until a doping level of x = 0:17. In this case the fast oscillations, previously attributed to a big cylindrical Fermi surface, are actually the result of large orbits made possible due to tunneling from electron-like to hole-like bands in the presence of magnetic eld. Further measurements showed both oscillation frequencies down to x = 0:16 [31]. Moreover, one needs progressively higher magnetic elds to reveal the fast oscillations, as the doping level decreases. A possible explanation here is that the energy gap between hole and electron pockets increases at lower doping levels. An important issue that is still unclear, is the mechanism that drives this reconstruction of the Fermi surface. In general, the Fermi surface is expected to change when a new periodicity, apart from the crystal lattice one, is introduced in the system. A scenario proposed in the NCCO case is of a superlattice potential caused by the introduction of an antiferromagnetic (AF) order in the system, see Fig. (1:3). Such a potential is known to exist in undoped and underdoped samples [26, 30]. Recent experiments [33] on NCCO crystals suggest that the superlattice potential persists until x = 0:175, (see 9

14 1 Introduction Fig. 1:7). This is approximately the same doping level where superconductivity seems to end, suggesting a potential connection. A second critical point at optimal doping x = 0:145 is suggested from Hall eect measurements. Here a remarkable change in the behavior between x = 0:142 and x = 0:145 is observed. This feature is attributed to a step like change of the magnetic eld required for a magnetic breakdown, that occurs between these two doping levels. With regards to the mentioned Fermi surface reconstruction scenario, there are some open issues that require further investigation. The rst issue is that the expected reconstructed Fermi surface consists of electron and hole pockets. Therefore, one would expect to observe MQO attributed to electron pockets. However, up to now, such oscillation frequencies have not been seen. It is possible that magnetic breakdown suppresses orbits around electron pockets more strongly than orbits around hole pockets. Experiments on samples with lower doping levels, where the magnetic eld required for magnetic breakdown is higher, could be able to clarify this issue. A second issue is, that the long range AF order, required in this scenario, has not been observed in overdoped samples. A possible explanation could be uctuations in the AF order, slow enough that the reconstructed Fermi surface lasts at least as long as a carrier needs to complete an orbit around the Fermi surface pockets [33]. In terms of Nernst eect measurements, contrary to what is predicted by the Boltzmann theory [34], measurements above the superconducting transition temperature, have shown an unexpectedly large signal. In the case of hole-doped cuprates, this is attributed to superconducting uctuations [35, 36], which are present at temperatures considerably above T C, although alternative suggestions have been made, which attribute it to a Fermi surface reconstruction due to stripe order [37]. On the other hand, for electron-doped cuprates the large normal state Nernst signal has been attributed to two kinds of carriers [38], although for underdoped samples it has been also attributed to superconducting uctuations as in the case of hole-doped cuprates [39]. As mentioned before, this second type of carrier has not been directly observed thus far. 1.3 Conclusion Aim of the experiments One of the primary aims of this thesis was to search for the MQO corresponding to the electron pockets, one expects to nd, if the above mentioned reconstruction scenario is valid. Since the magnetic elds required are rather high, the experiments were not performed in WMI but in LNCMI, Grenoble. A further aim, regarding these high eld measurements, was to acquire information regarding the normal state behavior of Seebeck and Nernst signals. This information was expected to oer valuable knowledge on the ground state of superconductors in the absence of superconductivity. Samples used in this case were NCCO single crystals with dopings x = 0:142 and x = 0:15. This choice follows the idea mentioned in the previous section, that the superlattice potential, assumed to be the reason for the Fermi surface reconstruction, is stronger for lower doping levels and thus magnetic breakdown is no more relevant for the applied magnetic elds. Samples used, were thin NCCO single crystals, whose 10

15 1.3 Conclusion Figure 1.7: (a) Evolution of energy gap MB and breakdown eld B 0 with doping. The superconducting dome and the energy gap appear to end at approximately the same point. (b) Evolution of cyclotron mass and SdH oscillations amplitude with doping. Apart from the Dingle temperature, the amplitude is determined by two parameters (i) the magnetic breakdown eld, that determines the magnetic breakdown probability (ii) the cyclotron mass, which determines the damping factors (see Chapter 2). The peak of the oscillation amplitude is found at x = 0:15. For lower doping level the oscillation amplitude seems to decrease rapidly [33]. in-plane properties we were interested to measure. The reason for measuring in-plane properties was, that the oscillations due to electron pockets might be stronger. A possible reasoning behind this is that due to the symmetry of NCCO crystals, one 11

16 1 Introduction expects nodes in the k x;y -dependent interlayer transfer integral, exactly at the points where the electron pockets are centered. This would mean, that their contribution to out-of-plane conductivity is reduced and subsequently the SdH oscillations would be weaker. In this experiment we tried to search for MQO in the thermoelectric eect. A full theory behind these oscillations is not available, nevertheless their origin lies in the same physical mechanism as SdH oscillations and their frequency is the same. The advantage of this approach is that one can largely avoid inhomogeneities in the in-plane electric conduction. Since NCCO is highly anisotropic with regards to its in-plane (a b plane) and out-of-plane (c axis) electric conductivity, even a minor conduction in the c axis, due to a minor insulating defect in the crystal, would produce quite a big signal. In the case of thermoelectric eect measurements, the inuence of such defects are much less pronounced, since the temperature gradient is much more homogeneous, compared to the electric current, in the presence of insulating defects. Further measurements were performed in LNCMI to measure the background signal of the setup. For this reason optimally doped YBCO samples were tested. At the temperatures and magnetic elds the experiments were performed, these samples were expected to be at the zero resistance superconducting state. As a result their Nernst and Seebeck signal is expected to be zero. Every signal we pick up in such a case is a background signal, attributed to the setup, and should be subtracted from the measurements. The nal experimental part, and the largest in terms of time devoted to it, involved building and testing a thermoelectric eect measuring setup, prepared entirely in W- MI. The purpose of this setup was, primarily to be able to perform experiments more frequently, although at lower magnetic elds. This, by itself, is quite important since normal state thermoelectric eects are sensitive to changes in the Fermi surface and can give us valuable information, particularly in the underdoped regime where no MQO are detected. A second reason for building the setup was, to be able to use it for high eld experiments. Such a possibility is quite advantageous, since we could measure any background signal beforehand. Moreover, optimization of the setup for our needs is much easier Thesis layout In the next chapter a short introduction on relevant theoretical topics is given. This includes some theory behind MQO and presentation of the damping factors for these oscillations. In addition, the basic expressions for the thermoelectric eects, measured during the experiments, are presented. Chapter 3 includes a discussion on sample fabrication and preparation. Moreover the experimental setup is explained, with an emphasis in the setup built in WMI. In chapter 4 the experimental results and their analysis are presented along with a discussion on the results and comparison with previous measurements. Lastly, in chapter 5 there is a conclusion. In the end, there is an Appendix, where some supplementary material is presented. 12

17 2 Elements of theory The aim of this chapter is to present some key theoretical results related to the experiments presented here, particularly MQO. However by no means is this theoretical part intended to be a detailed and sucient introduction to these topics. For more details the reader should refer to any good solid-state physics book. Particularly insightful in the case of MQO is Ref. 40, which was the main source for the material presented here. 2.1 Quantum oscillations MQO were rst discovered by Shubnikov and de Haas in the resistance of bismuth crystals in 1930 [41]. Later in this year, MQO were also observed in the magnetization of the same material [42]. Figure 2.1: The rst experimental observation of MQO by Shubnikov and de Haas [41]. 13

18 2 Elements of theory Central to this quantum phenomenon is the quantization of the energy spectra of the conduction electrons in the presence of a magnetic eld. It is shown that the only permitted states lie on tubes (Landau tubes) in k-space. The quantization formula species the energy levels allowed in terms of certain other parameters (to be presented later), including the external magnetic eld. With increasing magnetic eld the energy of these levels passes through the Fermi level, meaning that at exactly this point the occupied states of this energy level will change innitely fast. It should be expected, that at each such discontinuity the total energy E should undergo some anomaly. Moreover, the passing of the Landau tubes through the Fermi level is periodic in 1=B, where B is the applied magnetic eld. The result is that oscillations should be periodic in 1=B. In the next pages some key results on MQO will be presented, without delving too much into algebraic details Landau levels Changes in the motion of an electron, in the presence of magnetic eld B, are determined by the Lorentz force given by ~ _ k = e(v B); (2.1) where k is the wavevector of the electron and v its velocity. Integration of the above equation over some time interval produces In turn this gives ~(k k 0 ) = e(r R 0 ) B: (2.2) jk k 0 j = njr R 0 j? ; (2.3) where n = eb=~ and the right-hand side is a projection on a plane normal to the magnetic eld direction. It should be noted that this is an equation of only the magnitude of the vectors. There is still a dierence of 90 degrees in their phase. From this point, it is straightforward to calculate quantities describing the motion of the electron like dt, the time element needed for a dk change in the k orbit, or (dr) jj the spatial change element in the direction of magnetic eld. We can extract in this way the cyclotron frequency! c = 2=T, simply by integrating dt over a complete orbit. The result is! c = 2B=~ 2 (@s=@) k=const ; (2.4) where (@s=@) k=const is the innitesimal change of the area, normal to magnetic eld, in the k-space, enclosed by the orbit, that is produced by an innitesimal change in the energy of the orbit. Introducing cyclotron mass m c one can rewrite Eq. (2:4) as: where m c is dened as:! c = eb=m c ; (2.5) 14

19 2.1 Quantum oscillations m c = ~2 2 (@s=@) k=const (2.6) To explain the de Haas-van Alphen eect we must take into consideration the quantization of the electron motion. We start from the semi-classical Bohr-Sommerfeld quantization: I pdq = (i + )2~: (2.7) Here, p, q are canonical variables of the system, i is an integer and is the phase. Moreover the integral should be calculated over a closed orbit of the phase space. Substituting for the suitable canonical variables in the case of a motion in a magnetic eld, the quantization condition transforms to: I (~k ea)dr? = (i + )2~; (2.8) where A is the vector potential of B and R? the projection of R on a plane normal to B. Making use of Eq. (2.2) in the rst term, while using Stoke's theorem for the second term we have: I Z B (R dr? ) BdS = (i + )2~=e: (2.9) S Here S is the area of real space, enclosed by the orbit. Further simplication is done in the following way. We express R as follows: R = R? + RB and note that the component RB, parallel to magnetic eld, makes no contribution, since the term involving it, H is by denition perpendicular to B. Therefore the rst term of Eq. (2:8) becomes B (R? dr? ), which is by denition 2SB. The second term of Eq. (2:8) gives a SB contribution, meaning that we arrive at the following equation: SB = (i + )2~=e; (2.10) This equation, in itself, is a quantization equation that limits the allowed energy levels, since the allowed ux through the area enclosed by the orbit should be quantized in units of 2~=e. To express this quantization law in terms of the corresponding area s of the k-space, we only need to use Eq. (2.3), giving us the following result: and plugging this into Eq (2:7) we get: S = (~=eb) 2 s; (2.11) s(; ) = (i + )2eB=~; (2.12) with being the component of k in the direction of B. This is the Onsager relation, also found, independently by Lifshitz. 15

20 2 Elements of theory For given B,, i and the band structure (k), the allowed energy levels are determined implicitly by the above relation. From this point on we shall put = 1=2. It can be dierent but such cases lie out of the scope of this presentation. It should be noted that the number of quantum levels i is often quite big. Inserting realistic values in, either of the two previous equations, one can make a rough estimate of i in the order of , depending on the magnetic eld applied, in the case of ordinary metals. We can, thus, consider the dierences between two neighboring levels as innitesimal from now on. However, the above requirement could be removed when the energy steps are much smaller than the Fermi energy. In such a case, even a small number of steps is sucient. It would be insightful, at this point, to calculate the separation of the energy levels. which yields: ( ) i=1 ) =const( s) i=1 ; (2.13) ( ) i=1 = B = ~! c ; (2.14) where = e~=m c and m c = ebt stands for the cyclotron mass and T for the period of the orbit. Finally, we calculate the degeneracy of the energy levels. The number of states per unit volume in k-space is V=4 3 (V being the real space volume), taking into account that there are two electrons of dierent spins in each state. In the presence of magnetic eld the allowed states are conned in an area s dened by Eq. (2.12), implying that the number of states of each Landau tube is dened by: D = sdv=4 3 ; (2.15) with d being the innitesimal change of k in the direction of the magnetic eld and s = 2eB=hba Oscillations in magnetization The thermodynamic potential can be dened as = F N, where F is the Helmholtz free energy, is the chemical potential and N is the number of particles. For systems obeying the Fermi-Dirac statistic, while at the same time the allowed states of the system are determined by Eq. (2:12) and the degeneracy of the states is the one given by Eq. (2.15), the thermodynamic potential is given by the following expression: Z 1 = kt d(ebv )2 2 ~ 1 X i ln(1 + e ( i)=kt ) (2.16) Reducing even further the parameter freedom, we are going to assume temperature is 0 and that we are dealing with a two-dimensional slab in the k-space. In this case the dierence, produced by a change, of Eq. (2.16) reduces to: 16

21 2.1 Quantum oscillations X i=n = D i=0 ( i ); (2.17) for D = (ebv=2 2 ~) It turns out, that there is an oscillatory part in the above equation, which is the following: oscil = 2 4 eb2 V 4 2 ~ 1X p= p cos(2p(x 2 1=2)) 5 ; (2.18) where X = s(; )=(2eB=~) Oscillations in magnetization, for example, can be calculated from the M jj = : The result, in this case, is : M osciljj = =const s(; )V 4 3 [X (n + 1)] = N 0 [X (n + 1)]; (2.20) where N 0 is the number of electrons in the volume dened by the area s(; ) and height. We would like now to compute the relevant quantities in all three dimensions. To achieve this, we need to integrate over the values, since until now we assumed a d height for the energy levels. Integrating Eq.(2:18) over we get: oscil = eb2 V 4 2 ~ Z d 1X p=1 which, with further simplications can be expressed as: e V B5=2 1X 1 F oscil = ( )3=2 2~ 2 (A 00 ) 1=2 p 5=2 cos 2p B p=1 where F is the de Haas-van Alphen frequency, dened by: 1 2 cos [2p(X() 1=2)] ; (2.21) p (2.22) F = (~=2e)A: (2.23) Here A is the extremal area of the Fermi surface and A 00 is the second derivative of A with respect to, i.e. A 00 = 2 ] =0. The sign in Eq.(2:22) is the eect the integration over has on each harmonic, which is to shift it by =4. The specic sign depends on X 0, which is the minimum or maximum of X, dened before. If it is a minimum the ( ) should be used and vice versa. With a similar integration over, the parallel to magnetic eld component of magnetic moment can be calculated. This results in the following expressions: 17

22 2 Elements of theory (M jj ) oscil = ( e ~ )3=2 F B 1=2 V 2 1=2 5=2 (A 00 ) 1=2 1X p=1 1 p 3=2 sin[2p F B 1 2 ]: (2.24) 4 We end this section by noting that MQO can be also observed in the density of states. One can nd that the oscillatory part is given by: D oscil () = (2eB~) 1=2 mv 3=2 ~ 2 (A 00 ) 1=2 1X p=1 1 p 1=2 [2p(F B 1 2 ) =4]: (2.25) Smearing factors The above derivation presupposed various simplications. To study realistic examples, one needs to remove these simplications. The result is that the oscillations are smeared, while the amplitude of the oscillations is reduced by a damping factor. Some of these modications are shown below, so that we can acquire the Lifshitz-Kosevich formula. Others, like sample inhomogeneities, are not addressed quantitatively here. Finite temperature For nite temperatures it is well known that the distribution function for energy is f () = 1 (1 + e ( )=k BT ) : (2.26) Practically one can interpret it as a superposition of dierent metals, all at T = 0 K. The net result is, as mentioned before, a phase smearing. In this case the damping factor, for the p-th harmonic of the MQO is given by: R T = 42 pk B T B e( 22 pk B T=B) (2.27) Finite relaxation time The introduction of a nite electron relaxation time means that, due to the uncertainty principle, there will be a broadening in otherwise sharp energy levels. Again, there will be a damping of the oscillations. The damping factor in this case is given by: where T D = ~=2k B, is the Dingle temperature. R D = e ( 22 pk B T D =B) ; (2.28) 18

23 2.1 Quantum oscillations Electron spin It is known, that in the presence of a magnetic eld, the degeneracy of energy levels of electrons with dierent spin is lifted. Thus, each energy level splits into two dierent energy levels, where = g 0 B=2, 0 = e~=m e, and g is the spin-splitting factor. The net eect here is, as if there were two oscillations with dierent phase, meaning that there will be a damping of the oscillations. The damping factor in this case is: R S = cos(p =B): (2.29) Magnetic breakdown In the presence of a strong enough magnetic eld, electrons can jump from one orbit to another despite an existing energy gap. This phenomenon is called magnetic breakdown. The theory behind the magnetic breakdown is not within the scope of this theoretical introduction. The reader is referred to [40] and further works cited there. What will be needed for this work is the damping factor induced on the oscillations, due to magnetic breakdown. This is given by the following expression: R MB = (ip) n 1 q n 2 : (2.30) Here p 2 = P, where P is the probability of a magnetic breakdown. This in turn is given by the expression P = e B 0=B, for B 0 being the breakdown eld. Furthermore, q 2 = Q, where Q is the probability of Bragg reection. The two probabilities should add to unity, that is P + Q = 1. Exponents n 1 and n 2, are the number of points in the orbit that include a magnetic breakdown or a Bragg reection, respectively. Taking all of the above into account, we arrive in the Lifshitz-Kosevich formulas for and M, which are the following: e V B5=2 oscil = ( )3=2 2~ 2 (A 00 ) 1=2 1X p=1 R T R D R S R MB 1 p 5=2 cos[2p(f B 1 2 ) 4 ] (2.31) (M jj ) oscil = ( e ~ )3=2 F B 1=2 V 2 1=2 5=2 (A 00 ) 1=2 1X p=1 R T R D R S R MB 1 p 3=2 sin 2p( F B 1 2 ) 4 (2.32) 19

24 2 Elements of theory Oscillations of other quantities Thermodynamic potential, magnetization and density of states are not the only physical quantities that oscillate with the change of magnetic eld. In general these quantities can be separated into thermodynamic quantities whose oscillatory behavior can be calculated directly from the thermodynamic potential, and other, non-equilibrium properties. The rst category includes the de Haas-van Alphen eect, discussed in the previous section, oscillations in thermal properties (e.g. temperature, specic heat), oscillations in mechanical properties (e.g. elastic properties) and also oscillations in the chemical potential. In the second category one nds, for example, oscillations in the resistance (Shubnikov-de Haas eect), the optical properties or the nuclear magnetic resonance. Ultimately, however, all these eects are of the same etiology, that is the passing of the Landau tubes through the Fermi surface with increasing eld. Relevant to this work are mainly oscillations in the thermoelectric eect and, partially, oscillations in resistance. The theory behind oscillations of the thermoelectric eect is rather complicated and not fully evolved, particularly with regards to the amplitude of the oscillations. However, results show that the periodicity of the oscillations is identical to the one calculated for the de Haas-van Alphen eect [43]. The theory of the Shubnikov-de Haas eect by Adams and Holstein [44] is also quite complicated, and out of the scope of this brief theoretical part, since it involves a detailed study of electron scattering in magnetic eld. A qualitative explanation was given later by Pippard [45]. His argument was that the probability of a scattering event is proportional to the available states where the electrons can be scattered. This probability, in turn, oscillates in the same way as the density of states at the Fermi energy. The end result is, as mentioned, oscillations in the resistivity. Going back to the initial theoretical approach by Adams and Holstein they argue that the oscillatory part of the conductivity normalized by the non-oscillatory component of conductivity, is given by the following expression in the case of phonon scattering: 1 oscil R T = D() oscil 2D() [D() oscil D 0 () ]2 : (2.33) Moreover, according to the authors, the scattering mechanism doesn't seem to play a crucial role in the results, since if one assumes that scattering is caused by ionized impurities, a completely dierent mechanism, the results are within the same order of magnitude. 2.2 Thermoelectric eect The thermoelectric eect refers to the creation of voltage dierence by an applied temperature dierence and viceversa. The thermoelecric eects that are relevant for this work are the Seebeck eect and the Nernst eect. 20

25 2.2 Thermoelectric eect Nernst eect Nernst eect refers to the transverse electric eld E y created from a longitudinal temperature dierence r x T in the presence of a magnetic eld perpendicular to the xy plane. The Nernst signal is dened as: and can be expressed as: N = E y r x T ; (2.34) N xy xx xx xy = : (2.35) xx xy Here and are the electric and thermoelectric conductivity tensors. Using the Boltzmann equation, we get the following expression relating these tensors: 2 kb 2 = 3 j = F : (2.36) Combining Eq (2:35) and (2:36) we can express the Nernst signal in terms of the Hall angle, in the following way [46]: 2 kb 2 N H = j =F : (2.37) 3 If we assume a single band structure, the Nernst signal should be zero, unless the Hall angle depends on the energy at the Fermi level. In the latter case the Nernst signal could be written in the form: = N=B = 2 3 k 2 B T e F ; (2.38) where = e=m is the carrier mobility and m is the eective mass. In the case of a two band structure, as in the case of optimally doped NCCO crystals, we have the following expression for N: N = (a+ xy + a xy )(+ xx + xx ) (a+ xx + a xx )(+ xy + xy ) ( + xx + xx ) 2 + ( + xy + xy ) 2 ; (2.39) where ( ) and (+) refer to electrons and holes respectively Seebeck eect Seebeck eect corresponds to the creation of a longitudinal electric eld E x due to a temperature gradient r x T. If, as before, we assume that there is no charge ow, we acquire the following equation for the Seeback coecient: For temperatures lower than T F one can express S as follows: rv = SrT: (2.40) 21

26 2 Elements of theory S = 2 3 k B T et F : (2.41) Finally from eq (2:38) we can acquire an expression linking the Nernst and Seeback coecients. where H is the Hall angle. B = 2S tan( H )=3; (2.42) 22

27 3 Experimental setup In this chapter, a presentation of the experimental setups used for high eld measurements in LNCMI and calibration measurements in WMI, will be given. In addition, information on the sample preparation will be included. 3.1 Samples All samples used for experiments were manufactured at the crystal laboratory in WMI. For the MQO experiments, NCCO single crystals were used. The doping levels selected for testing were x = 0:142 and x = 0:15. Moreover, optimally doped YBCO samples were used as a way for characterizing the setup. In the next section, a brief description of NCCO crystals is given and a short discussion on the crystal growing method and the contacting procedure is included NCCO crystals Figure (3:1) shows the structure of NCCO crystal. Figure 3.1: Crystal structure of NCCO and LCCO. The construction is similar in the sense that there are innite CuO 2 layers, separated by Re 2 O 2 layers, where Re is the appropriate rare earth for each compound. NCCO crystals have a body-centered tetragonal structure. As mentioned, like all 23

28 3 Experimental setup cuprates, NCCO is highly anisotropic, in the sense that there is large dierence between their interlayer and intralayer resistivity. This anisotropy is found to be of the order of 10 4 [47]. The spin structure of undoped NCCO crystal is shown in Fig (3:2). Here, one can see the 2 dierent ways in which the long rang AF order can be realized. Figure 3.2: Spin structure of undoped NCCO crystals in the absence of an applied magnetic eld. (a) Phase 1 (75 < T < 275K and T < 30K); (b) Phase 2 (30 < T < 75K) [48] Crystal Growth The method of choice for the crystal growth of electron-doped cuprates, a task considerably harder than growing hole-doped cuprates, is the traveling solvent oating zone technique. Generally speaking, critical parameters that are of the utmost importance, with regards to experiments, are high crystal purity, dopant homogeneity and ability to cover the relevant part of the phase diagram. Furthermore, precise orientation of the samples is a necessity, since they are highly anisotropic materials. Problems with dierent growing methods include the existence of spatial inhomogeneities with regards to the doping level and impurities in the nal crystal that can potentially suppress superconductivity. Moreover, the temperature range allowed for growing can be restricting. Finally, there are also considerations with regards to achieving the desired shape and size. A brief description of the method is given here. For more details the reader is advised to refer to the appropriate literature, for example [49], [14]. For the crystal growth, one needs a polycrystalline feed material. This is made of powder that has exactly the same chemical composition one wants to achieve. Then by 24

29 3.1 Samples applying pressure, one creates two rods out of this compressed powder, the feed and the seed rod. These are vertically placed inside the growth chamber. The chamber contains four heating lamps, whose power can be focused by mirrors. Between the two rods a CuO-rich pellet is placed. Fig. (3:3) is indicative of how the method works. Figure 3.3: On the left side a graph of the growing process for LCCO crystals is shown. The initial compound has components with dierent melting temperatures. After T p these components are decomposed. These are then dissolved in the upper liquid-solid boundary. The blue line between point P, where he components are separated due to dierent melting temperature, and E, the point where the compound solidies simultaneously, indicates the ux composition. At the lower solid-liquid boundary the compound solidies and by grain preference growth, a single crystal can be produced. On the right side one can see a schematic of the method. The area around the pellet and where the tips of the two rods meet, is where the focal point of the mirrors is. This part will become the 'oating zone'. Due to the heat applied, this region starts to melt, however due to surface tension it continues to lie between the two rods. At these temperatures the polycrystalline 214 phase is dissolved into the melted part of the feed rod. After that, it moves to lower, colder parts of the feed rod, as shown in the graph, and it crystallizes in a monocrystalline structure. By moving the stage upwards the feed rod continues to melt into the ux and the crystal can continue to grow in the upper direction. During the growing process the whole system of the rods is rotating, so that there is better mixing. The monocrystalline structure in 25

30 3 Experimental setup the end results from a preferred crystallographic axis during the crystallization process. Figure 3.4: Test done utilizing a superconducting quantum interference device (SQUID) in WMI. Magnetic susceptibilty versus temperature is measured for underdoped samples (x = 0:13). It is clear that the eect of the annealing temperature on the transition temperature is quite signicant [50] Annealing process It is a well known fact that the as-grown crystals either are not superconducting at all or exhibit quite lower transition temperature compared to crystals that have been thermally treated [51], [14]. The main feature of this treatment is annealing of the as-grown crystals at temperatures of the order of o C, depending on the doping level, in an inert argon atmosphere, for several hours or even days. In terms of stoichiometry the end result is the removal of apical oxygen atoms in the order of 0:1 2 %. It is not understood completely, how the removal of apical oxygen correlates with superconductivity. It is, thus, often the case that this annealing procedure involves a lot of trial and error and it is hard to optimize for dierent doping levels. Small dierences in the annealing temperatures, within the above mentioned range, can result in substantial dierences in critical temperatures, sometimes even destroying superconductivity [7]. In Fig. (3:4), one can see the results of tests done at WMI. Here, magnetic susceptibility versus temperature was measured. It is clear that relatively small changes in the annealing temperature or duration can have signicant eect on the transition temperature. 26

31 3.1 Samples Sample preparation After the crystals are grown, annealed and characterized, they still must be manually contacted, a task often challenging in the case of small and at crystals. The location of the contacts depends on the quantities one wishes to measure and whether one is interested in in-plane or out-of-plane properties. As an example, Fig. (3:5) shows an optimally doped NCCO sample prepared for in-plane Seebeck and Nernst coecient measurements. Figure 3.5: x = 0:15 doped NCCO sample prepared for Nernst (the two opposing contacts at the centre of the sample) and thermopower measurements (the pair of contacts neighboring to one of the previous). Gold wires have a diameter of 50 m, while platinum ones are 10 m. The single gold wire at the right edge is used to heat the sample, while the three gold wires on the other side are used for heat removal, creating a temperature dierence across the sample. When contacting the sample, there are various parameters, one should take into account. Firstly, the most suitable wires should be used, based on the intended use. As an example in Fig. (3:5) gold wires are used for heat application/removal, since gold is a good heat conductor. On the other hand mechanical stability of the contacts often limits the diameter of these wires. It is often the case that due to a slight error while adjusting the wires, the wires are removed. The heavier the wires, the more dicult they are to be safely glued. One ought, in this case, to nd a balance between, on one hand, the need for as high as possible heating and heat removal and, on the other hand, durability of the contacts. For the contacts seen in the photo, and most other contacts made, the material of 27

32 3 Experimental setup choice was EPOTEK H2OE epoxy. The procedure involved applying a small amount of the epoxy on the sample and then attaching the wires on top. The next step was adding some more epoxy to cover the wires at the point of contact and heating the sample at 120 K for 30 min. This was the suggested treatment from the manufacturer. However tests showed that the contact resistance was quite high, in the order of 1 k. What was tried, was to heat the sample for two more hours at 400 o C and reapply some epoxy, since at these temperatures it disintegrates. The end result was that the contact were still mechanically as strong, but the contact resistance dropped to around 10. Apart from the time consuming procedure, a disadvantage of this epoxy is the diculty in removing it from the sample. The technique employed here is applying mechanical force on the contact via some thin metallic tools. Basically the contact is slowly scratched o the surface, with the added danger of simply breaking the thin at samples. A lot of times it was deemed necessary to prolong wires or glue two wires with each other. In that case, the glue of choice was the DuPont silver paste. One can see it in Fig (3:5) at the points that the wires are glued to the black, made of vespel, platform. The mechanical strength of this glue is worse than the epoxy mentioned before, however it has the advantage of not requiring any post-application treatment apart from letting it dry for around 5 min. Hence, in the cases where time was limited, or a strong mechanical contact was not required, the Du Pont paste was preferred. Finally, there is the issue of where and how to x the sample in the designated space. One obvious parameter, one should take into account, is, that the sample should be electrically insulated from the insert, so the platform, where the sample is placed, should be an insulator. On the other hand, thermal conductivity between the sample and the insert depends on the setup design. As an example for the experiments in LNCMI we used a vespel platform, oering thermal and electrical insulation, while in some calibration measurements, in WMI, the sample was placed on a copper screw where a layer of insulating BF glue was applied. However in this case thermal conduction was still present. It is often the case that, in high magnetic eld experiments, there is a strong magnetic torque exercised on the samples, creating the danger that the sample is completely removed from the platform. In these cases Stycast 2850 FT is used, which is the strongest of the glues used. The drawback in this case is that it needs to be left one day in room temperature until it is suciently hardened. Moreover the way of removing it involves heating the sample at 500 K for two hours, meaning that apart from the time lost all contacts should be redone. One can see this blue glue, in Fig. (3:5), underneath the sample. 3.2 Experimental equipment The aim of this section is to mention, and sometimes give a brief description, of the equipment, devices and inserts used during the experiments. Since the experiments took part in two dierent laboratories, in LNCMI and in WMI, each using dierent equipment, there will be two separate subsections devoted to them. 28

33 3.2 Experimental equipment High eld experiments The high eld experiments were performed at the LNCMI facilities in Grenoble, where some of the strongest resistive magnets are available. The magnet used for the main measurement was the strongest one, able to reach 35T, while producing a homogeneous eld to the order of /cm 3. To achieve this kind of elds it consumes 24 MW and requires water owing rate of 1000 m 3 /h for cooling purposes. Figure 3.6: A view of the magnet from the cellar. LNCMI facilities, Grenoble For the experiments in LNCMI, an insert prepared at the LNCMI was used. The sample holder is shown in Fig. (3:7). The gradient thermometers are placed in the elevations close to the right end, so that they are decoupled from the silver platform. The sample along with its platform is placed on the end of the silver platform (see Fig. (3:5)), meaning that the direction of magnetic eld is perpendicular to the ab plane. As thermometers, measuring the temperature gradient across the sample, RuOx resistive chips with a room temperature resistance of 10 k were used. Their magnetoresistance was measured in advance and found to be 0:09 % at 2K and 19T. Moreover, magnetoresistance was found to be the same for both chips, meaning that no further recalculation of the temperature gradient, due to magnetoresistance, was needed. To apply heat to the sample RuO 2 chips were used. Their room temperature resis- 29

34 3 Experimental setup tance was measured at 10 k. Figure 3.7: Photo of the sample holder used in high eld measurements in LNCMI. For picking up the Nernst and thermopower signals, analogue nanovoltmeters were used (EM nanoelectronics N11 and N31). These were then read by digital Keithley 2000 multimeters. Wires picking up the thermopower signal were properly shielded and they were mechanically attached to the nanovoltmeter input, so as to have the best possible signal. Finally, with regards to temperature control, a Lakeshore 340 temperature controller was used for temperature reading and regulating. In general, the whole set up proved to be quite satisfactory and noise levels were acceptable. However problems with the temperature control and measuring occurred at various points during the experiments (see experimental results chapter). 30

35 3.2 Experimental equipment Setup preparation To be able to perform experiments much more frequently, an additional thermopower measuring setup was prepared for use in WMI. Although, the magnet available in WMI is a superconducting 15 T magnet, meaning that no MQO can be observed, the setup can be utilized for measuring the normal state properties of electron doped superconductors. An additional aim was that this setup would be suitable for high eld measurements in LNCMI. This is important, since the time available for measurements in LNCMI is always limited. Building a setup of our own, would enable us to perform calibration and other testing measurements at WMI. Thermopower insert An already existing thermopower insert was used as the base for the new setup. The lower part of the insert exhibits a platform and can be stabilized at temperatures between T = 4:2 K and T = 1:4 K by pumping on the helium bath. The measuring unit has to be kept under vacuum. For this reason, a vacuum pot is screwed to this lower part of the insert surrounding the measuring tail. The pot is sealed with vacuum grease. The liquid helium platform of the insert is separated from the measuring tail by a plastic, 5 cm long, cylindrical piece. Its aim is to drastically reduce the heat transfer from the part of the insert that is within the helium bath. Since this conguration means that the heat link is too weak, a silver piece was placed in a manner, in which it provided thermal contact between the parts of the insert, separated by the plastic, insulating piece. A further issue, regarding temperature control, is the thermal transfer from the rest of the insert to the copper platform seen in Fig. (3:8). If the thermal link is too strong, achieving temperatures of the order of K, might be a challenge. On the other hand, if the link is too weak, then reaching low temperatures can be impossible without the introduction of exchange gas in the sample space. In this case, the main heat transfer comes from the wires that are soldered at the top of the insert and are at room temperature. To limit this heat transfer from the wires, all the wires coming to the copper platform are thermalized. This is done by wrapping them around a rod, placed inside the insert. During measurements the lowest temperature we were able to reach was 2:6 K. This should be compared to a value of T = 1:4 K, which is he lowest theoretically possible temperature we could achieve. Typical values for heating power required during the experiments were around 20 mw, for temperatures around 10 K. With regards to the setup wiring, all wires were twisted to avoid loops. Brass wires of diameter d = 0:1 mm were used, in most cases, to reduce thermal transfer from the warm, upper part of the insert. For the wires connecting the heaters with the upper part of the insert, copper wires of a diameter d = 0:1 mm were used. Copper wires were also used for transferring the thermopower signal, so that we could avoid any added signal. In the latter case, we should mention that, in order to avoid parasitic signal from any soldering, long wires (around 7 m) connected the measuring device directly to the sample area. These wires were placed into a plastic tube to avoid any potential 31

36 3 Experimental setup Figure 3.8: View of the insert tail from above. damage caused. Subsequently the wires were shielded via a metallic tube. All the wires coming from the upper part of the insert, were soldered on the platform at the tail of the insert. A diculty we faced, with regards to keeping the insert vacuum tight, was related to the wires carrying the thermopower signal. As mentioned, these are long wires transferring the signal directly from the sample tail to the measuring device. All the other wires are soldered at pins lying at the inner side of the upper part of the insert. Then, cables are attached to the outer side to transfer the signal to the measuring device. Since we decided to avoid any further soldering, which could give an additional signal, we prepared a ange, made of brass, which had a small hole at its side. A metallic tube was soldered at this point and the thermopower wires came through this tube. To hermetically seal the tube, a small brass cup full of pizein was used. A photo of this construction and some more details can be found in the Appendix. 32

37 3.2 Experimental equipment Temperature sensors and heaters There are four temperature sensors in use in the insert. Two of them are RuOx chips of room temperature resistance of 10 k. These are used to measure the temperature gradient on the sample. They are calibrated up to around 200 K, although above 40 K they should not be considered reliable. On the other hand, they are suitable for low temperature measurements and have a low relative magnetoresistance. Their calibration graphs and magnetoresistance measurements can be found in the Appendix. A third temperature sensor is used for measuring the temperature of the copper tail of the insert. In this case, since having a sensor that is reliable up to room temperature, a LakeShore CX-1030-SD-HT sensor was preferred. Finally, we utilize a capacitance sensor for measurements in magnetic eld. Such a sensor is not suited for temperature measurement, since its capacitance changes with time. However it is useful for stabilizing the temperature in magnetic elds, since it is not aected by them. With regards to heaters, there are two heating chips used, both of them made of NiCr. The rst is a resistive chip of 100, room temperature, resistance. This is used to control the base temperature of the copper tail of the insert. A second chip, with a room temperature resistance of 500 is used to create the temperature gradient on the sample. Both chips show a small magnetoresistance, which is crucial for keeping a stable gradient during a eld sweep. Layout of the copper measuring tail For the needs of the new setup a new measuring tail, made of pure Copper, was prepared in the workshop of WMI. The reader can nd a drawing of the piece, with all the relevant lengths, in the Appendix. A photo of this copper piece, prepared for measurements, is shown in Fig (3:8), while a schematic of it, is shown in Fig. (3:9). Looking at this tail and starting from the left side, one can see a cavity where a capacitance sensor is placed. Moving to the right, one can see a rectangular piece. On the side that it touches the platform it is covered by an insulating layer. On the other side there is a copper surface, where lines have been carved, dividing it into 26 smaller rectangular surfaces. Here, soldering tin was placed to enable the wiring of the insert. Between the capacitance sensor and the mentioned piece, one can see the heater responsible for regulating the base temperature of the sample space. Further to the right, three plastic screws are visible. These serve as thermal insulation for the sample heater and the two sample thermometers, mounted on top of them. It is worth mentioning, that the wires attached to these RuOx thermometers are made of platinum-tungsten alloy, since it showed the least magnetoresistance of the wires measured. The results of these measurements can be found in the Appendix. Finally, at the upper corner of the platform, at the right side, one nds the thermometer responsible for measuring the temperature of the copper platform. The sample itself is placed on the small copper cylindrical platform, at the right side, that is easily removable. A photo of this platform is shown in Fig. 3:10. The sample can be placed either on a copper screw or in a vespel platform. After the sample is placed, the wires picking up the Seebeck voltage, are contacted to the gold 33

38 3 Experimental setup Figure 3.9: Drawing of the sample holder. Locations, where thermometers and heaters will be placed are indicated. wires lying on the black insulating piece. These wires, in turn, are connected to the gold wires going to the thermometers but also to manganin wires to pick up the voltage signal. Manganin was chosen in this case for its small heat conductivity. For the rst 3 4 cm these manganin wires are untwisted to avoid an added thermal contact between hot and cold parts of the sample. A further parameter one has to take into account, is the minimization of loops perpendicular to the magnetic eld, since this would cause induced voltages to appear. Measuring devices In the measurements in WMI, a Lakeshore 340 temperature controller was used for reading the two sample thermometers and the capacitance sensor. Moreover, the resistance of the third thermometer was measured by a 4-point measurement with the help of a Keithley 2000 multimeter and a Keithley 2400 sourcemeter. Parasitical signal was removed by changing the polarities of the measurement current. With regards to picking up the Seebeck signal, originally a nanovoltmeter Keithley 2182 was used. Later, a custom circuit employing an EM amplier (A10) was utilized. The circuit is shown in Fig.(3:11). The amplied signal was read through a Keithley 2000 multimeter connected to its output. All the handling and monitoring of the whole set up was done via a Labview custom programm. 34

39 3.2 Experimental equipment Figure 3.10: Zoomed in photo of the sample platform. The sample is placed on a vespel substrate. Thermopower wires are connected to gold wires lying on the plastic insulating piece. From there, they are connected to manganin wires, to reduce heat transfer, which are then soldered to the copper wires that transfer the signal to the measuring device. 35

40 3 Experimental setup Figure 3.11: Top side view of the custom made circuit used to incorporate the amplier. One can see the amplier at the center. At the left side are the batteries, which were chosen for the reduced noise level they oer, compared to the conventional power supply line. 36

41 4 Experimental results and discussion Experimental results presented here, include measurements taken in high magnetic elds in LNCMI, as well as measurements done in WMI. As mentioned in the introductory part, the former measurements were done with the aim of observing MQO coming from the electron pockets as well as measuring the normal state Nernst and Seebeck eects. The latter measurements were made with the purpose of testing the new thermopower setup. 4.1 High-eld measurements NCCO single crystal samples of doping levels x = 0:15 and x = 0:142 were tested. Typical temperature dierence applied to the sample was 0:4 0:5 K. Typical sample size for in-plain measurements is 4 0:5 0:05 mm. Measurements were taken for high steady elds up to T. An experimental problem, one has to face, is, that the voltage measured via the nanovoltmeters is the sum of various parasitic contributions, apart from the actual voltage one wants to measure. Firstly, due to the changing magnetic eld, there are sweep-induced voltages in the signal we pick up. These are easily seen, since there is a step-like change in the voltage when the eld sweep stops or starts. Moreover, it is often the case that, one picks up a signal even with zero temperature gradient, when thermopower should be zero. The exact nature of this signal is not exactly known. Finally, an unavoidable thermopower signal comes from the wiring, when there is a gradient active on the sample. To eliminate these unwanted voltages we took the following steps. Firstly, during the downsweeps of the magnetic eld we stopped the eld sweep every 5T and recorded the voltage dierence due to the elimination of the sweep-induced voltage. Furthermore, we switched o the gradient and let the sample acquire a uniform temperature. Ideally, at this point the signal should be zero. Since this was rarely the case, it was necessary to remove this additional voltage. In order to do this, we made an interpolation of these points. Then, we subtracted this curve, along with the sweep-induced voltage, from our raw data, eectively rezeroing the setup. Finally, one has to take into account the parasitic signal that depends on temperature gradient. Although at the superconducting state it is easy to remove its contribution, this parasitic voltage is not easy to be estimated during the normal state. To solve this problem, we performed further measurements using optimally-doped YBCO samples. These hole-doped cuprate superconductors have quite high critical temperatures and upper critical elds. For the samples we tested the critical temperature was measured around 92K, while the upper critical eld is a least 140T [52]. Moreover, the zero resistance state extends up to 60 T, at T = 4:2 K [53]. At the temperatures and elds, used 37

42 4 Experimental results and discussion for these experiments, these samples should be fully superconducting and thus, produce zero thermopower voltage. Any signal picked up should be attributed to parasitic reasons and thus, should be subtracted. What we did, was to take measurements with YBCO samples, at similar temperatures as the measurements we took with the NCCO samples and then subtract the voltage. The assumption here is that, although the two experiments were performed with a four month dierence and using a dierent magnet, the parasitic signal should be the same. In this regard, to get as reliable results as possible, the setup, wiring and sample contacting were the same for both experiments Quantum oscillations MQO were observed only at the Seebeck signal of the sample with x = 0:15. Example of these oscillations are shown in Figs. 4:1, 4:2. S e b e c k s i g n a l ( µ V / K ) T = 3 K M a g n e t i c f i e l d ( T ) Figure 4.1: Seebeck signal for a NCCO single crystal of a doping level of x = 0:15 and temperature T = 3 K. Note the oscillations, visible with the eye, at high elds. The background signal, estimated by the YBCO measurements, is not subtracted. This is done so that oscillation are more easily discernible, since YBCO measurements were quite noisy. The frequencies of the oscillations, for various temperatures, are shown in Table 4:1, whereas an example of a fast Fourier transformation (FFT) is shown in Fig. 4:3. The frequency was extracted by performing a FFT on the signal curves after we had subtracted a low order polynomial. An example of the background-subtracted oscillatory 38

43 4.1 High-eld measurements O s c i l l a t o r y c o m p o n e n t ( µ V / T = 3 K M a g n e t i c f i e l d ( T ) Figure 4.2: Oscillatory component of the Seebeck shown in Fig. 4:1. From the original signal a sixth order polynomial was subtracted to make the oscillations clearer. The frequency of these oscillations is T. signal is shown in Fig 4:2. The frequencies observed are in agreement with previous experiments [29]. There, oscillations frequencies of 290 T were reported for x = 0:15 samples. These frequencies have been attributed to orbits around small hole pockets, created by a reconstruction of the Fermi surface. For more details on why such a frequency should correspond to hole, and not electron, pockets see Ref. 54. Briey, if one is to accept the model, with regards to the reconstruction of the Fermi surface, mentioned in the introductory part, had these oscillations come as a result of orbits around the electron pockets, it would imply that the superstructure gap in overdoped or optimally doped samples would be comparable to the undoped, which is highly unlikely. Moreover, in such case, the energy gap would be big enough that no magnetic breakdown should be observed, which is contrary to the observation of oscillations coming from a large orbit [31]. Table 4.1: Oscillation frequencies Temperature[K] Frequency[T] One can calculate the cyclotron mass from the temperature dependence of the oscil- 39

44 4 Experimental results and discussion F F T a m p l i t u d e T = 3 K F r e q u e n c y ( T ) Figure 4.3: FFT of the data for T = 3K. Frequency curve has a clear peak around 290 T. lations amplitude. Such a dependence comes from the temperature dependence of the damping factor R T, introduced in the theoretical part (Eq. (2:27)). For the calculation we used the expression R T = (KT=B)= sinh(kt=b), where K = 2 2 k B m e =~e and is the cyclotron mass, normalized to the free electron mass m e. Plotting the amplitude versus temperature, as seen in Fig 4:3, one can t the data points using the above expression, inserting the mass as a tting parameter. As can be seen in Fig 4:4, the normalized cyclotron mass is estimated at = mc m = 0:980 e 0:018. Comparing our result with previous results [54], we notice that there is an acceptable agreement. It is worth mentioning here, that this calculation has more uncertainties than the calculations of the oscillation frequency. The main reason is, the signal noise which makes the exact oscillation amplitude dicult to extract precisely. In this regard, the actual experimental error is higher, compared to the error shown in Fig. 4:4, which is only the tting error. We estimate the total error at = 0:05. The conclusion from our high eld measurements of MQO is that they appear to conrm the previous measurement, discussed above. Such a reproduction of previous results is, in itself, important. However, there are some questions left unanswered, along with some new questions that arise from these measurements. The rst question, and in relation to the aims of this experiment, is why no oscillations, that could be attributed to electron pockets, were observed. A possible reason 40

45 4.1 High-eld measurements O s c i l l a t i o n A m p l i t u d e ( µ V / K ) µ = δµ= T e m p e r a t u r e ( K ) Figure 4.4: Cyclotron mass t of the oscillation amplitude values for dierent temperature. Since R T is the only damping factor dependent on temperature, we can insert the respective equation in the tting program, with being a parameter to be calculated. is that such pockets do not exist. However, such an assumption comes in contrast to Hall [33] and Nernst [38] eect measurements, where, the sign change in the observed Hall resistivity and the large normal-state Nernst signal, point to a two band structure. Although a big shift to our views (i.e. the non existence of electron pockets) should not be excluded, it would be helpful to look for less radical solutions to reconcile the experimental data. In this direction it would be insightful to have an approximate estimation of the expected amplitude of the oscillations from electron pockets. It was mentioned in the theoretical part, that no complete theory exists for the amplitude of MQO of the thermoelectric eect. To proceed, we are going to assume that the amplitude should depend, among other things, on the same damping factors that all other MQO depend, namely R T ; R D ; R s ; R MB (see section 2:1:3). A further factor to take into account, is a dierence in contributions from these pockets to the total electrical and thermal conductivity. We are going to address each point separately. Starting from the last point, we would like to estimate the non-oscillatory in-plane conductivity for electron and hole pockets. We will also assume that a relative dierence, between the electron and hole pockets, in the amplitude of their respective MQO, 41

46 4 Experimental results and discussion reects a relative dierence in the non-oscillatory conductivity. If one assumes the same relaxation time, the relative dierence between the conductivities should reduce to r = ne = e where n is the number of carriers for each pocket and the normalized n h = h cyclotron mass. Starting from the number of carriers, we can estimate the relative difference, by estimating the relative dierence of the Fermi surface areas for each pocket. Such an estimation can be done in the following way. One can calculate the number of carriers per layer in an area s of the Fermi surface by the following expression: n = 2s (2=L) 2 ; (4.1) where L is the lattice constant. In reduced Brillouin zone of the reconstructed Fermi surface this equation takes the form n = 4a2 (2) 2 (se 2s h ): (4.2) Here,2 1=2a is the lattice constant, s e, s h are the areas of each electron and hole pocket respectively. Further, there are 2 hole pockets versus 1 electron pocket equivalents in the reduced Brillouin zone. In the case of the measured sample, where x = 0:15, n should also be set n = 0:30 since there are 2 Cu atoms per unit cell. Based on the above expression the area per electron pocket is given by: s e = 2s h + 2 0:30: (4.3) a2 If we use the experimentally obtained [29] s h = 0:011s B:Z:, where s B:Z: = 42 a 2, we can estimate the electron pockets area at s e = 0:097s B:Z:, where s B:Z: is the area of the large, unreconstructed Brillouin zone. The ratio for the carriers is given by: n e =n h = S e =2S h = 4:4. The cyclotron mass for electrons can be estimated as 1:04 (see next paragraph). This leads to an estimation of a ratio r = A e =A h 4:1 between the expected amplitudes for oscillations due to electron and hole pockets respectively. Going back to the damping factors of MQO and looking at the exact expressions given for each reduction factor by the Eq (2:27) (2:30), it is clear that any dierence in R T ; R D ; R s between electron and hole pockets, could only come from dierent cyclotron masses. Since, cyclotron mass can be written, using Eq (2:5) as m c = qbt, where T is the period of the orbit, we can argue in the following way. Suppose a large orbit, due to magnetic breakdown, in the reconstructed Fermi surface, seen in Fig. 1:3. The period corresponding to such an orbit would be T MB = 2T h + T e, which implies that during the orbit carriers would trace four times, one half of the circumference of a hole pocket and four times a quarter of the electron pocket's circumference. In turn this should give us the following equation for the normalized cyclotron masses. MB 2 h + e ) e MB 2 h ; (4.4) where all the cyclotron masses have been normalized by the free electron mass. 42

47 4.1 High-eld measurements Experimentally MB has been calculated before and found to be MB = 3:0 0:3 [55]. Based on our previous calculation of the cyclotron mass of the holes ( h = 0:98), we estimate the normalized cyclotron mass for the electron pockets carriers as e = 1:04. We are now going to examine the change in amplitudes, that such a dierence in the cyclotron mass induces to the oscillation amplitude. From the damping factors mentioned before, one can assume that R s does not play much role in this case. Indeed, the term involving the cyclotron mass is a cosinusoidal term, where the small mass dierence is not expected to inuence this rough estimation considerably. From the other terms the term R D should give a ratio Re D RD h = e 0:35 0:70. Here we set T D = 12 K, as found in Ref. 31 and B = 30 T. For the term R T we get a result RT e =Rh T 0:96, where we inserted the values T = 3 K and B = 30T. The bigger contribution to the damping factor imbalance is made by the magnetic breakdown term. Based on Eq. (2:30) and the fact that orbits around electron pockets include 4 Bragg reection points, while orbits around hole pockets involve 2 Bragg reection points, the ratio of the damping factors is: R e MB=R h MB = 1 P = 1 e B 0=B : (4.5) Setting B 0 = 12 T (see Ref [33]) and B= 30 we obtain R e MBN=R h MBN = 0:33. If we sum up all the mentioned reduction factors, we can estimate the expected amplitude ratio as: A e 0:92; (4.6) Ah i.e. the expected amplitude of the oscillations from the electron pockets is almost the same as that from the hole pockets. Typically, the oscillations due to hole pockets have an amplitude 20 times higher than the noise level. In that sense, we would expect to observe MQO from electron pockets. However, there is a large amount of uncertainty in the above calculation, particularly due to a very high sensitivity of the amplitude ratio Ae to small variations in the A h cyclotron masses. In Fig. 4:5, the expected amplitude ratio is plotted as a function of the cyclotron mass of the electron pockets carriers. If we assume a noise level 20 times smaller than the amplitude of the oscillations coming from the hole pockets, one would need a normalized cyclotron mass e 1:43, in order for the respective oscillation to be indistinguishable from the noise. Taking into account an error of MB 0:3 for MB and an error h = 0:05 for h, the highest possible value for e would be e = 1:44, which would result in the amplitude A e just below the resolution level. Taking into account the error in the evaluation of the magnetic breakdown eld, the amplitude ratio A eh may further decrease by around 15 %. It is, thus, possible, although not very likely, that the reason for not observing MQO, coming from orbits around electron pockets, could, possibly, be not good enough resolution. In this regard, further experiments, with an improved, by a factor of 2 or 3, signal-to noise ratio, would be able to fully clarify this issue. In contrast to the Seebeck signal, Nernst signal did not show any MQO in our experiments. The reasons are presently unclear. Possible reasons could be the smaller 43

48 4 Experimental results and discussion Nernst signal, compared to the Seebeck, and an, initially, more noisy setup. The setup was improved in the next days, but most measurements at low temperatures, where one expects to observe the oscillations, had already been done. Since there is no full theory predicting the amplitude of these oscillations, it is hard to make any further comments on that. In any case, further experiments, with better resolution, are needed in order to resolve this issue. l o g ( A m p l i t u d e r a t i o ) T = 3 K E µ Figure 4.5: The expected amplitude ratio between the oscillations due to electron pockets and the amplitude of the oscillations due to orbits around the hole pockets carriers as a function of the normalized cyclotron mass e. Finally, regarding the underdoped sample (x = 0:142), no MQO were observed. Regarding our failure to pick up any MQO for the sample with doping level x = 0:142, there is a possibility that such a failure is just a matter of resolution. Looking at the damping factors expressions, it is clear that the strength of the oscillations, with respect to doping, depends on three parameters. These are, normalized cyclotron mass of the respective charge carriers, breakdown eld and the Dingle temperature. Taking into account Fig.1:7 and keeping in mind the expressions for the damping factors, it is obvious that, for increasing doping level, cyclotron mass changes tend to increase the oscillation amplitude, while breakdown eld changes tend to reduce them. Since we do not know these values at x = 0:142, we can not make an exact calculation. As seen in Fig.1:7, if one extrapolates the, experimentally found values, the result is close to zero. However, such an estimation is not considered fully reliable. Since there are hints, that 44

49 4.1 High-eld measurements there is a critical point at 0:142 < x < 0:145 [33], it would prove quite helpful to know, whether the reason of our inability to observe MQO, comes from not enough resolution of the experimental setup or for some other reason. Therefore, additional experiments on this doping level might be needed Non-oscillatory signal Apart from MQO, one can extract valuable information on the ground state of the cuprates by thermoelectric eects. As mentioned, our setup does not allow us to distinguish between thermoelectric signal and parasitic, gradient dependent signal. Such a signal is generated due to the temperature dierence of the wires attached to the hot and to the cold end of the sample. To remove this parasitic signal it was necessary to try to replicate the experiments, using optimally doped YBCO samples, which are expected to have a zero thermoelectric signal at these temperatures and elds. The parasitic, gradient-dependent, signal was then subtracted from the signal measured in the NCCO samples. In the gures presented in this section, the measured voltage signal is plotted versus the applied magnetic eld for various temperatures. Furthermore, all curves have been vertically shifted, so that all curves start with a zero voltage signal. The temperatures noted for each curve represent the average of the temperatures shown by the two sample thermometers. Before proceeding with the results, one should keep in mind the eect a magnetic eld B has on the superconducting state. For 0 < B < B C1 the magnetic eld does not penetrate the superconductor and the resistance is 0. For B C1 < B < B C2, the superconductor is penetrated by the magnetic eld but it is still supeconducting. The areas, where magnetic eld can penetrate are called vortices. For a high enough magnetic eld B C1 < B 0 < B C2, these vortices can be unpinned and start moving, creating a nite resistance. We refer to this regime as the ux ow regime. Finally, for B > B C2, superconductivity is destroyed and normal state is reached. In the ux ow regime of the superconductors, one can observe a Nernst and Seebeck signal. It can be shown that vortices moving with a velocity v create an electric eld perpendicular to B and v. The electric eld is calculated by the expression [56]: E = v B: (4.7) Under the inuence of a temperature gradient both vortices and quasiparticles move from the hot end to the cold end of the sample. The movement of the vortices is responsible for a transverse electric eld. On the other hand, due to the quasiparticle thermal diusion, compensating supercurrent is creating at the superconducting regions. The supercurrent interacts with the vortices resulting in a force perpendicular to the thermal-diusion current. The transverse movement of the vortices results according to Eq.(4:7) in a longitudinal electric eld. Results on Seebeck eect The results of the Seebeck eect measurements are plotted in Figs. 4:6 4:7. In both gures, the lower graph shows the signal measured for the NCCO samples, after 45

50 4 Experimental results and discussion subtracting the gradient-independent parasitic signal and the oset, while on the upper graph, the gradient dependent parasitic signal, found from the measurements on YBCO samples, has been also subtracted. It is clear that YBCO signal adds signicant noise in the curves. This is more evident at lower temperatures, particularly for the lower doped sample (Fig. 4:7). For this reason, we consider the results unsuitable for extracting quantitative conclusions. In this regard, further measurements using the newly prepared thermopower setup will be needed in order to get fully reliable measurements. However, we will attempt a qualitative discussion of the acquired results. Starting from the sample with doping level x = 0:15 (Fig 4:6), one can see a clear increase in the Seebeck signal with a peak around B = 4 T, for T = 6:6 K and T = 8:9 K. This peak is clear for both unsubtracted and subtracted signals. Therefore we consider it a genuine signal and not some artifact due to a parasitic signal. We attribute this peak in the Seebeck signal to vortices that move in a direction perpendicular to the voltage drop, due to the Magnus force. Comparing the curves for the two highest temperatures we note a clear temperature dependence of the height of the peaks. For T = 3 K this peak is no longer discernible. What is seen is a linear increase in the signal at low elds, up to B = 5 T. Since the prole of this signal increase is not similar to what is observed for higher temperatures, we consider it non-intrinsic to the sample, but some artifact due to parasitic signal. An additional reason for this conclusion is seen at the resistance measurements for this temperature (see Appendix). In these measurements it is clear that up to B = 5 T the sample has zero resistance. Since moving vortices would create a resistance, we conclude that the genuine signal of the sample below B = 5 T is 0. In Fig. 4:8 the Seebeck signal is plotted as a function of temperature at B = 15 T. The points of this graph are taken from the subtracted curves (Fig. 4:6(a)). At this eld the sample is clearly in the normal state. We observe an almost linear dependence on temperature. In addition, we see a sign change in the signal, from negative, at lower temperatures, to positive at higher temperatures. The existence of two dierent carriers could be a possible explanation. In this scenario, a dierent temperature dependence of the thermoelectric coecients, for holes and electrons, could lead to such behavior. With regards to the eld dependence of the normal state Seebeck signal shown in Fig. 4:6, it seems to increase linearly with eld, until at higher magnetic elds it saturates. Moreover, the saturation point seems to depend on the temperature. Qualitatively similar behavior is shown by YBCO samples [57]. There, for the lower temperatures measured, there seems to be a magnetic eld threshold after which the signal saturates. Furthermore the saturation eld is decreasing with temperature. It is also worth noting that for temperatures T = 3 K and T = 6:6 K, a sign change is observed, which could also point to a two band structure. As a last comment, we note that the magnetic eld dependence of the Seebeck signal is similar to what has been observed on Hall eect measurements in optimally doped NCCO samples [14, 33]. In the former experiment, for low temperatures, a negative sign was observed, with a minimum around B = 10 T. For higher elds the signal increased approaching 0. In the latter experiment, where higher magnetic eld was used the signal turned positive above a certain eld. In addition, for higher temperatures the signal becomes purely positive [14]. Such a behavior is similar to the behavior we observe in 46

51 4.1 High-eld measurements S e e b e c k s i g n a l ( µ V / K ) ( a ) 3 K 6. 6 K 8. 9 K S e e b e c k s i g n a l ( µ V / K ) ( b ) M a g n e t i c f i e l d ( T ) Figure 4.6: Seebeck signal for the NCCO sample of doping level x = 0:15. The initial point for all curves has been vertically shifted to zero. (a) Seebeck signal acquired after subtracting the gradient-dependent YBCO signal from the curves seen in (b); (b) Seebeck signal of NCCO samples where the gradientindependent oset and the signal due to eld sweep have been subtracted. our measurements. 47

52 4 Experimental results and discussion S e e b e c k s i g n a l ( µ V / K ) ( a ) 2. 7 K 4. 6 K 1 2 K S e e b e c k s i g n a l ( µ V / K ) ( b ) M a g n e t i c f i e l d ( T ) Figure 4.7: Seebeck signal for sample of doping level x = 0:142. The initial point for all curves has been vertically shifted to zero. (a) Seebeck signal acquired after subtracting the gradient-dependent YBCO signal from the curves seen in (b); (b) Seebeck signal of NCCO samples where the gradient-independent oset and the signal due to eld sweep have been subtracted. The lower doped sample (Fig. 4:7), seems to be considerably more sensitive to the noise caused by the subtraction of the YBCO signal. For T = 12 K we observe a clear signal increase at low elds, starting from B 3 T, which is attributed to a voltage created by vortex motion. For T = 2:7 K, the unsubtracted curve (Fig. 4:7(b)) is at 48

53 4.1 High-eld measurements for the most part, while the subtracted curve (Fig. 4:7(a)) shows a linear increase with applied eld. We consider this linear increase as an artifact, since the prole of the signal increase is dierent from T = 12 K. For T = 4:6 K we observe a feature, with a small peak at B = 5 T, at the unsubtracted curve, which could be generated by vortex movement, however in the subtracted signal the feature is lost. The normal state signal is negative for T = 12 K and T = 4:7 K. For T = 3 K the signal is negative until B 20 T, where it turns to positive and saturates above 25 T(Fig. 4:7(a)). However, this behavior is not seen in (Fig. 4:7(b)). Therefore, more accurate measurements are necessary to reach a denite conclusion. S e e b e c k s i g n a l ( µ V / K ) B = 1 5 T T e m p e r a t u r e ( K ) Figure 4.8: Temperature dependence of the Seebeck signal for the x = 0:15 doped sample and for constant eld B = 15 T. In Fig. 4:9, we have plotted the signal at B = 15 T for dierent temperatures. The values have been taken from the unsubtracted signal. Contrary to the higher doped sample, which showed an almost linear temperature dependence (Fig. 4:8), in the lower doped sample, we observe a lower signal at T = 4:6 K compared to T = 3 K. In this regard we note the dierent behavior for the two samples but since the subracted and unsubtracted curves show a qualitative dierence, further experiments will be required for more reliable results. 49

54 4 Experimental results and discussion S e e b e c k s i g n a l ( µ V / K ) B = 1 5 T T e m p e r a t u r e ( K ) Figure 4.9: Temperature dependence of the Seebeck signal for sample with a doping level x = 0:142 and for constant eld B = 15 T. Results on Nernst eect In Figs. 4:10; 4:11 the results from the measurements of the Nernst signal are plotted versus the applied magnetic eld. All curves have been shifted so that the starting point is at zero voltage. Moreover, for each curve, sweep induced signal and oset signal are removed in the way presented at the beginning of the chapter. We consider these results more reliable than the previous results on Seebeck signal since, by the nature of the calculation, any parasitic signal that is not sensitive to the magnetic eld direction is removed. For the optimally doped sample we observed a vortex-generated signal at low elds. The peaks are at B 3:4 T for T = 8:9 K and B 4 T for T = 6:6 K. Moreover, these peaks are found approximately at the same elds, as the respective peaks at the Seebeck signal. This is expected, since the physical mechanisms generating these signals, are related. For the lowest temperature T = 3 K, we did not observe any peak. This is also in agreement with the results on Seebeck eect. With regards to the normal state, for the higher temperatures (T = 6:6 K, T = 8:9 K) the signal is positive. For the lower temperature (T = 3 K), the signal is negative until B 18 T, where it changes sign and becomes positive. In Fig. 4:12 the Nernst signal at B = 15 T has been plotted for various temperatures. Here, we observe an approximately linear dependence on temperature. Moreover, a sign change is observed close to T = 4 50

55 4.1 High-eld measurements N e r n s t s i g n a l ( µ V \ K ) x = K 6. 6 K 8. 9 K M a g n e t i c f i e l d ( T ) Figure 4.10: Nernst signal for samples of doping level x = 0:15. The curves have been vertically shifted, so that the initial point is at zero voltage. K. The sign change, observed also in the Seebeck signal, could point to the existence of two types of charge carriers. Furthermore, similarity to the Hall measurements in [14, 33] is observed, as in the case for the Seebeck signal for this sample. Previous measurements of the Nernst signal on optimally doped NCCO samples, in magnetic eld, can be found in [58]. A comparison shows that both signals exhibit a non-linear dependence on magnetic eld in the normal state. This behavior is in disagreement with the linear behavior seen in PCCO samples [59]. Quantitatively, the signal presented in [58] is approximately three to four times higher than what we found. The reason for this discrepancy in the observed signal is unclear at the moment. A further dierence is that in [58], they failed to observe a sign change of the Nernst signal in the eld sweeps. However, since the lowest temperature in their experiment was T = 5 K, it is possible that the sign change happens at lower temperatures than 5 K, since we only observe it at T = 3 K. As a last comment for this sample, we note a saturation of the signal for B > 25 T. Furthermore, for the lowest temperature T = 3 K, the observed signal decreases after 27 T. The results for the lower doped sample (x = 0:142) are shown in Fig. 4:11. Here, Nernst signal versus magnetic eld curves are plotted for various temperatures. For T = 12 K, we observe a clear vortex generated signal with a peak at B = 3:5 T. For the 51

56 4 Experimental results and discussion N e r n s t s i g n a l ( µ V / K ) x = K 4. 6 K 1 2 K M a g n e t i c f i e l d ( T ) Figure 4.11: Nernst signal for samples of doping level x = 0:142. Curves have been vertically shifted so that the initial point shows no signal. lowest temperature (B = 2:7 K) we observe a feature emerging at B 6 T, with a peak at B = 7:5. This is possibly due to vortices motion, although we note that no similar feature was observed in the Seebeck signal. For T = 4:6 K, the feature we observe has a peak at B 7:5 T and could be attributed to vortex movement, since a similar feature was observed in the unsubtracted Seebeck signal (Fig. 4:7). With regards to the normal state signal, for the temperatures T = 2:7 K and T = 4:6 K we notice a positive Nernst signal. For the higher temperature T = 12 K, we notice a negative signal, which at B 15 T becomes positive. In Fig 4:12, the Nernst signal dependence on temperature is shown, for a constant eld B = 15 T. As in the case for the Seebeck signal the results are dierent compared to the x = 0:15 doped sample. Furthermore, the normal state Nernst signal shows a non-linear dependence on the applied eld. However in this case no saturation is observed even for the highest elds. Comments With regards to the non-oscillatory thermopower results, we note again that they are not fully reliable, especially for the Seebeck eect, since the YBCO measurements were quite noisy. In this regards the improvement of the new experimental setup will help in this direction, since it will be continuously available for test measurements and further optimization. With this in mind, we attempted a qualitative description of the results. 52

57 4.1 High-eld measurements S e e b e c k s i g n a l ( µ V / K ) B = 1 5 T T e m p e r a t u r e ( K ) Figure 4.12: Nernst signal, for the higher doped sample, plotted versus temperature at B = 15 T. From the discussion that we attempted, we consider two features the most important, in relation to the results on MQO and other related results on NCCO. The rst is the sign change observed in the optimally doped sample. This is a hint, apart from the large normal state Nernst signal, that there are two dierent types of carriers present at this doping level. It is possible that the sign change could be attributed to a temperature dependent Fermi surface reconstruction as in [24, 57]. However, such a scenario is not likely in this case, since the sign change is not observed for T = 8:9 K. However, the information we have from the MQO measurements do not show any temperature dependence of the frequency of the observed oscillations. A similar argument is harder to make for the lower doped sample (x = 0:142). A sign change is observed in the Nernst signal at T = 12 K (Fig. 4:11), but the shape of the curve seems suspicious, around 10 T, and further measurements need to be done to clarify this issue. A second point we would like to mention is the dierent behavior seen in the two samples. This is seen both by a direct comparison of the signals and by comparison of the temperature dependence of the signals at a xed magnetic eld (B = 15 T), (Fig. 4:8; 4:12 versus Fig. 4:9; 4:13). If this change in behavior is genuine, it might point to the same direction as recent measurements on Hall resistivity [33], were a critical point was identied at 0:142 < x < 0:15. 53

58 4 Experimental results and discussion N e r n s t s i g n a l ( µ V / K ) B = 1 5 T T e m p e r a t u r e ( K ) Figure 4.13: Nernst signal, for the lower doped sample, plotted versus temperature at a constant eld B = 15 T. 4.2 Low eld calibration measurements The last part of the chapter is devoted to measurements taken in WMI using a 15T superconducting magnet. Here only the measurements on NCCO and YBCO samples will be mentioned and discussed. The reader can nd some additional preliminary measurements in the Appendix. For the reasons mentioned in the previous section, optimally doped YBCO samples were measured at various temperatures. The results are shown in the Fig. 4:14 Temperature dierences applied in the sample were around 0:2 0:5 K. These are similar values to the ones used for the thermopower measurements on NCCO crystals. Here the gradient dependent signal was of the same magnitude as the noise. We consider this a positive result, since it means that the parasitic gradient-dependent signal from the wires is quite small in this setup. Moreover, the picked up signal seems be quite small for most temperatures. No particular temperature or eld dependence is noticed. In this regards the setup seems quite promising, showing better results compared to the setup used in LNCMI. A noise level around 5 10 nv is considered quite acceptable, given the equipment used. A feature we noticed is a voltage drift that appears in the Keithley 2182 nanovoltmeter. This drift was not very fast nor very big, but meant that we had to frequently rezero the signal. This means that, there is some uncertainty on what part of the signal is due to drift and what is intrinsic on the setup. However, 54

59 4.2 Low eld calibration measurements - 0, K 7. 6 K K K S i g n a l ( µ V ) - 0, 1 0-0, 1 5-0, 2 0-0, 2 5-0, M a g n e t i c f i e l d ( T ) Figure 4.14: Longitudinal voltage signal from the optimally doped YBCO sample. as mentioned, both the actual signal and the drift are not very big and the induced uncertainty is quite limited. At this point, the same x = 0:15 doped NCCO sample, measured in LNCMI, was measured using the prepared setup. It would be quite helpful to experiment with an already measured sample, so we could compare the results. The original attempt, using the same setup as in the previous measurements, was not succesful. There was quite a big noise (in the order of 1 V), bigger than the signal we wanted to measure, and quite a big drift. Since we could not get any meaningful results, it was decided to change the measuring device. Instead of a Keithley 2182 Nanovoltmeter, we integrated an nanovolt amplier into a custom circuit (see Fig.3:11), with the aim of lowering the noise as much as possible. In Fig 4:10 the results from the measurements are shown. In comparison with the previous measurements there is an improvement. However, as is easily deduced from the graph, there are still a lot of issues, that need to be ironed out. An obvious feature is the strong noise until, roughly, 4T. The origin of this increased noise is still unclear to us. It seems to stop at the superconducting transition. Moreover, the point where the noise disappears seems to move to lower elds for higher temperatures, again a hint that it might correlate with superconductivity. However, as mentioned, whether this is a coincidence or there is a correlation between superconductivity and this feature is still unanswered. In this regard, further tests will be 55

60 4 Experimental results and discussion S e e b e c k s i g n a l ( µ V / K ) M a g n e t i c f i e l d ( T ) 4. 2 K 7 K 8 K Figure 4.15: Seebeck signal for optimally doped NCCO crystal. needed. Regarding the measured signal, there were also a lot of issues that need improving. Firstly, the signal seems to be still noisy, with noise levels in the order of 50 nv being not rare. Moreover, as seen in Fig. 4:15, there are some spikes in the signal. Further tests are required to understand if the main culprit for this behavior lies in the sample contacts or somewhere else. Moreover, a relatively strong drift in the voltage signal was present during most measurements. Again further testing, while changing various parameters of the setup (e.g. measuring device, sample contacts) will be needed. Regarding more technical issues, we faced problems with the thermometers measuring the temperature gradient. It seems that, although they had been calibrated together there is still some dierence in their calibration of the order of (0:2 0:3)K at certain temperatures. Such a discrepancy, while small, is signicant for the calculations. However, in the actual measurements, this did not prove to be a problem, since we controlled the temperature in magnetic eld via a capacitance sensor. In this regard a recalibration of the thermometers might be needed. Despite all these issues needed to be sorted out, the results show a qualitative agreement with the results in high eld experiments. Although, at this point, any quantitative comparison is not possible due to unknown oset and drift, it seems that the setup is working quite well, sometimes, as in the case of the YBCO measurements, outperforming the already established setup in LNCMI. In this regard, further tests and 56

61 4.2 Low eld calibration measurements improvements are needed until we can extract a fully reliable signal. 57

62 58

63 5 Conclusion High eld measurements of the in-plane thermoelectric eects in Nd 2 x Ce x CuO 4 (NC- CO) single crystals were performed at the LNCMI facilities, in Grenoble. Samples of doping level x = 0:15 and x = 0:142 were investigated. We measured the Seebeck and Nernst signal of these samples, at high elds and for various temperatures. MQO were observed at the Seebeck signal of the x = 0:15 doped sample. The results with regards to oscillation frequency and cyclotron mass are in agreement with previous data on Shubnikov-de Haas oscillations in the interlayer resistivity [29]. Like in the interlayer magnetoresistance experiments, no second frequency of MQO, which would correspond to the electron pockets of the reconstructed Fermi surface, has been found. This might be due to the resolution of the setup. Therefore additional measurements with an optimized setup should be able to clarify the issue. With regards to the lower doped sample, no MQO oscillations were observed. However, it is possible that the oscillation amplitude is much lower compared to the x = 0:15 doped sample, making their observation much more challenging. Further measurements could shed more light here. With regards to the non-oscillatory signal, we do not consider the present data suitable for a detailed quantitative analysis. However, the qualitative discussion showed two interesting points. The rst is a sign change, observed in the optimally doped sample, which might point to the existence of two dierent carrier types. Secondly we noted a considerable qualitative dierence in the behavior of the two samples. Since the existence of a critical point has been proposed for a doping level 0:142 < x < 0:145, the results presented here might strengthen this scenario. In both cases further, more accurate measurements should be done. Finally, we prepared a setup for measuring thermoelectric eects, intended for use both at WMI and at LNCMI. The rst tests have been done. The results showed that the setup has the potential of operating with the needed resolution. However, more tests will be needed to fully optimize it. 59

64 60

65 6 Appendix 6.1 Thermometer calibration R u O x s e n s o r R e s i s t a n c e ( O h m ) T e m p e r a t u r e ( K ) Figure 6.1: Typical calibration curve for RuOx sensors. The temperature was measured via a pre-calibrated Cernox sensor. It is obvious that due to the curve shape, these sensors are only suitable for low temperature measurements. 61

66 6 Appendix 6.2 Magnetoresistance measurements R e s i s t a n c e ( O h m ) T = 1 5 K M a g n e t i c f i e l d ( T ) Figure 6.2: Typical magnetoresistance measurement for RuOx chips. The temperature here is T = 15:3 K. 62

67 6.2 Magnetoresistance measurements R e s i s t a n c e ( O h m ) M a n g a n i n T = 1. 4 K M a g n e t i c f i e l d ( T ) Figure 6.3: Magnetoresistance measurements for thin (d = 0:1 mm) manganin wires The temperature here is T = 1:4 K. R e s i s t a n c e ( O h m ) P t W T = 1. 4 K M a g e t i c f i e l d ( T ) Figure 6.4: Magnetoresistance measurements for thin (d = 0:1 mm) PtW wires. The temperature here is T = 1:4 K. We consider the initial jump an artifact due to the start of the eld sweep. 63

68 6 Appendix 6.3 Insert tail preparation Figure 6.5: Drawings used for the construction of the insert tail. All lengths are in mm. 64

69 6.3 Insert tail preparation Figure 6.6: The small tube contains the wires carrying the thermopower signal. To make the construction vacuum tight, we immersed the tube into hot, liquid pizein. At room temperature it solidies and it is vacuum tight. 65