- Email: [email protected]
Doping dependence of Bi2Sr2CaCu2O8+σ in the normal state
PHYSICA ELSEVIER
Physica C 263 (1996) 208-213
Doping dependence of BiaSr2CaCu2Os+ in the normal state A.G. Loeser, D.S. Dessau ~, Z.-X. Shen * Department of Applied Physics, and Stanford Synchrotron Radiation Laboratory, Stanford University, StanJbrd, CA 94305-4055, USA
Abstract We have probed the electronic structure of Bi2Sr2CaCu2Os+ ~ with angle-resolved photoemission spectroscopy, observing changes in the electronic structure with doping variation. At cartier dopings below the optimal for superconductivity, we observe an excitation gap in the normal state. The momentum dependence and the magnitude of the gap are very similar to those of the dx2_y: gap observed in the superconducting state.
One of the reasons why the problem of high-temperature superconductivity is so difficult to crack is that the normal state of these materials is not well understood. The optimally doped superconductor sits in the middle of a complicated phase diagram; at higher dopings it becomes a typical metal, while at dopings near half-filling it is an antiferromagnetic (AF) insulator. If these two cases represent the limits of small and large electron correlations, then the superconducting dopings are in the middle of the transition between the two. There, the normal state is frequently called a "strange metal," as it displays many unusual properties which have not yet been explained within the context of any simple theory [1-4]. The thrust of our photoemission experiments presented here has been to probe this transition in the normal state electronic structure, by varying the hole concentration from the overdoped to the underdoped regime.
* Corresponding author. Fax: + 1 415 725 5457; e-mail: [email protected]. Current address: Department of Physics, University of Colorado, Boulder, CO, 80309-0390, USA.
In the overdoped case, many features of the electronic structure can be described with local density approximation (LDA) band theory. Angle-resolved photoemission experiments show a Fermi surface which is in good agreement with the LDA calculations [5-12]. A typical Fermi surface of overdoped metal is depicted in Fig. l(b). Small variations in the doping level indicate that the Fermi surface obeys Luttinger's sum rule, with a volume proportional to the number of carriers. There is, however, still evidence of correlations in the photoemission data. For example, the bands which cross the Fermi level are flatter than predicted by LDA [7,11,12], indicating mass enhancement due to electron-electron interactions. On the other side, at half-filling, the so-called undoped parent compounds are Mott-Hubbard insulators. Already, to explain this, one must abandon band theory in favor of localized models of the electronic structure. At this doping, the only clear photoemission study published thus far is on Sr2CuOzCI 2 [13]. In this study, Wells et al. observed a filled band with maxima in the two-dimensional Brillouin zone at (~r/2, w / 2 ) in units of k/a in the C u - O plane. In the metallic case, the maximum is at
0921-4534/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0921 - 4 5 3 4 ( 9 6 ) 0 0 0 7 4 - 3
A.G. Loeser et al./ Physica C 263 (1996) 208-213
""
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Fig. 1. This sketch depicts possible Fermi surfaces for (a) doping close to half-filling, and (b) doping above optimal Tc. While (b) agrees well with LDA calculations, (a) has pockets centered at ( + aT/2, + "rr/2). The dotted line in (a) is the AF Brillouin zone boundary, and the narrow parts of the Fermi surface have weak ARPES intensity.
(Tr, ,r). This change appears to be caused by the AF order, which halves the size of the Brillouin zone. The zone edge for the AF ordered state includes ('rr/2, at/2), and a gap opens up there. There is also a large drop in photoemission intensity as one follows a dispersive feature from the first into the second AF Brillouin zone, possibly due to the weak scattering potential of the AF lattice. Starting with these data and a rigid band doping picture, one might expect the Fermi surfaces at very low doping to be pockets, as depicted in Fig. l(a). This is a reasonable inference, since the AF correlation length is still long and the rigid band picture may be valid at very low doping. The question becomes, What is the nature of the normal state around the optimal superconducting carrier doping? Is it a correlated band structure, or does one need to employ the Hubbard model or related theories in order to accurately describe the physics? At first glance, the photoemission data at these doping levels seem closest to the metal. Mainly, the apparent Fermi surface agrees well with the LDA calculation, and there are no other obvious differences. In this work, however, we report observations of a gap in the normal state electronic structure, very close to the doping of optimal T~. This observation motivates consideration of the localized approaches as very important for describing the normal state physics. In a photoemission experiment, electrons in a solid are excited above the vacuum level by incident, monochromatic photons. While inelastic scattering
209
captures some of the electrons, the rest are emitted from the solid, and their energy and momentum is measured by a detector. The resulting spectrum is made up of primary counts, or electrons which exit the solid without loss of energy, and secondaries, which have experienced some inelastic scattering. The secondaries' contribution to the spectrum usually has a characteristic shape, and can be subtracted from the spectrum fairly accurately. Along the energy axis, the resulting spectrum includes convolutions with the photon energy distribution, as well as the electron analyzer's instrument response function. Typically, experimentalists report the net effect in terms of a single Gaussian's full width at half maximum. Momentum resolution is mainly determined by geometrical factors (sample size and analyzer aperture) [14], and is typically reported as + the half width of the resulting angular acceptance function. Typical angle-resolved photoemission (ARPES) studies use + 1 deg. In addition, if the electronic states in the solid are two-dimensional in nature, then conservation laws imply a simple 1-1 mapping between the electron's angle of emission and its momentum inside the solid. This is the case for the high-To superconductors, facilitating our detailed studies of the Brillouin zone. The data reported in this paper were obtained with a VSW analyzer (angular resolution ± 1, energy resolution -- 30 meV), at the undulator beamline 5 - 3 of the Stanford Synchrotron Radiation Laboratory, with a base pressure of roughly 4 × 10-11 Torr. We have described this in more detail elsewhere [15]. Theoretically, the resulting primary spectrum is extremely complex until one makes the so-called sudden approximation, namely, that the photoelectron is ejected from the solid faster than the time scale of various interactions which would alter its momentum and energy. (It is not obvious in an ultraviolet photoemission (LIPS) experiment that this is the case, but the assumption agrees fairly well with results. The lineshape analysis in Claessen et al.'s ongoing study of the two-dimensional metal TiTe 2 can be thought of as testing this approximation, along with other ideas [16].) With the sudden approximation, the photoemission spectrum becomes the electron removal excitation function, which is how we will analyze the data below. Despite the variety of materials available,
210
A.G. Loeser et aL / Physica C 263 (1996) 208-213
Bi2Sr2CaCu2Os+ 8 (Bi2212) remains the most fruitful for high-resolution ARPES studies. Steadily improving sample quality, sample preparation techniques, and instrument resolution allow for new results on this material, even after many ARPES papers have been published. We used Bi2212 single crystals which were grown with a directional solidification technique [17]. The carrier concentrations were adjusted by annealing at 600°C in various mixtures of argon and oxygen, and characterized by T~ measurement in a SQUID magnetometer. In a complimentary study, we achieved much higher variation in doping of thin films by substituting the divalent calcium with trivalent dysprosium during the growth [18]. The analysis of the doping dependence of these samples supports the single crystal results, and will be published soon [19]. As the doping is reduced from the metallic regime down towards the insulator, there appears to be a significant change in the Fermi surface topology. In an ARPES study, the Fermi surface is drawn by making cuts in k-space, showing dispersive features (quasiparticles) as they move through the Fermi energy and lose weight. The k-space locations where the quasiparticle band crosses the Fermi level are identified as points on the Fermi surface, and a line connecting them is the full, measured surface. As stated earlier, the main Fermi surface of overdoped Bi2212 is sketched in Fig. l(b). In the slightly underdoped Bi2212, however, we will show that some of those crossings disappear. This leaves us without a clear, connected Fermi surface. Instead, we obtain something like the dark lines in Fig. l(a). There are a few effects which could cause this change, which will be discussed in more detail below. One possibility is that the Fermi surface has changed from the large, LDA surface to four pockets surrounding the ( + 7r/2, _ 7r/2) points on the AF zone boundary, as one would get by doping the half-filled parent compound with a small number of holes. This would imply the sort of Fermi surface evolution with doping presented in Fig. 1. Another possibility is that the Fermi surface is unchanged, but that there is an anisotropic gap similar to the dx2_y2 one, even in the normal state. Finally, the data presented here do not rule out a combination of both pictures, as the pocket Fermi surfaces may accommodate a small gap. The distinction between a Fermi
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BindingEnergy(meV) Fig. 2. Cuts showing the doping evolution of Bi2212. Panels (1) are the cut from (0, 0) to (xr, xr), and panels (2) are the cut from (at, 0) to ('it, "rr). Sample 1 is underdoped, Tc = 84 K. Sample 2 is overdoped, Tc = 85 K. Sample 3 is overdoped, Tc = 80 K.
surface pocket picture and a dx2_y2 gap is even more difficult if we consider impurity scattering and the resulting dirty d-wave picture. In this case, the gap is suppressed for momentums close to the (1, 1) direction, which coincides with a large part of the pocket Fermi surfaces [20]. The cuts used to demonstrate these changes at the Fermi energy are shown in Fig. 2. We examine the (0, 0) to ('rr, ~r) and the (-rr, 0) to (~r, ~ ) apparent Fermi surface crossings for three samples with a range of dopings. One general change which is apparent is that all of the dispersive features become broader as the hole doping is reduced. This is apparently due to an increased electronic damping of the quasiparticles, since changes in the quality of the single crystals around the C u - O planes are unlikely. The quality of the reduced samples and their surfaces is verified by the observation that, in the superconducting state, the sharp quasiparticle peak at (Tr, 0) is roughly as narrow as that of the optimally doped sample. On the other hand, we cannot suggest a more specific mechanism for this broadening, a notion consistent with the material being a "strange metal." More importantly, a gap opens up in the
A.G. Loeser et a l . / Physica C 263 (1996) 208-213
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exhibits the normal crossing behavior described above. As the peak loses its intensity, its leading edge moves to negative binding energy. In the 80 K overdoped case, the ('rr, 0) to (~, w) crossing has this characteristic as well. At lower dopings, however, the leading edge behavior looks less like a Fermi surface crossing. For the underdoped sample in particular, the dispersive feature clearly does not reach the Fermi level, while the situation is unclear for the 85 K overdoped sample. By gap, we essentially mean that a dispersive feature no longer crosses the Fermi level. If the Fermi surface evolution is as depicted in Fig. l, we stress that the presence (absence) of a Fermi crossing from (w, 0) to (w, ~r) in the overdoped (underdoped) sample is a clean way to distinguish between Figs. l(a) and (b). Otherwise, the Fermi surface clearly has a gap in the (~, 0) region of k-space; this is demonstrated here in the
Comparison of a spectrum from the most overdoped to the tmderdoped sample, in the region of the Fermi crossing from (~, 0) to ('rr, "rr). Note the difference in edges of the two samples.
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~ -10 underdoped sample. Fig. 3 shows two spectra taken from roughly the same point in the Brillouin zone, in the vicinity of the apparent Fermi surface crossing. For the underdoped curve, the excitation spectrum clearly does not extend to as low a binding energy, nor even to the Fermi level itself, indicating the creation of an excitation gap. Because there is no successful model to describe the single-particle excitation spectrum in these materials theoretically, the experimentalist must rely on simpler, quantitative measurements to characterize the spectra. One such quantity is the energy of the leading edge of the spectrum, which we report, as a binding energy, as the position on the curve at which the intensity is half (above the background) of the peak intensity. From simple models of dispersive features, we see that the leading edge should pass the Fermi energy, i.e., go to negative binding energy in a usual Fermi level crossing. This is mainly caused by finite instrumental resolution that shifts the spectral weight of a sharp feature from below to above E F. In Fig. 4, we show the leading edge binding energies for the spectra in Fig. 2, in order to emphasize the leading edge shifts with doping. The normal state (0, 0) to (q¢, ~r) crossing for any doping level
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212
A.G. Loeser et al./Physicc C 263 (1996) 208-213
same manner used originally to identify the superconducting gap. The nature of the gap near (~, 0) in the underdoped sample is unknown at this point. There are two alternative ways to explain the data, based on a weak-coupling SDW picture, and a strong-coupling picture. The weak-coupling picture can be understood in a phenomenological way. Owing to shortrange AF correlations, a local 45 ° rotated ~/2 × ~/2 periodicity imposes a new AF Brillouin zone that makes (0, 0) and ('rr, "rr) equivalent. This halving of the Brillouin zone creates additional bands which cross the original ones near ('rr, 0) and ('rr/2, rr/2). If the new AF bands and the original bands interact, an SDW gap will open at (~r, 0), as estimated by band calculations [21,22]. In this scenario, the Fermi surface of the underdoped sample loses its ARPES intensity at the edge of the first AF Brillouin zone (dashed boundary of Fig. 1), and is completed by the gray line in Fig. l(a), in the second AF Brillouin zone. This results in Fermi surface pockets centered at ( _ ~r/2, 5- ~r/2). We have observed some weak features that may correspond to the dashed Fermi surface, but we have not determined whether this is intrinsic or due to a structural distortion. In the context of strong coupling theory, our data may be a consequence of dx-~_y2 pairing in the normal state. The opening of a spin-channel dx2_y2 gap in the normal state of the underdoped sample is already a prediction in a phase diagram [23,24], extending the original RVB idea [25]. In fact, Fig. 4 shows that a normal state gap opens in underdoped samples of Bi2212 with T~ as high as 84 K (optimal Tc of the system is 91 K) [26]. At this doping level, where the AF correlation virtually vanishes, it is tempting to interpret those data as due to the opening of a dx~_y:-like gap at the LDA Fermi surface, similar in size and symmetry to the dx2_yZ gap in the superconducting state, as supported by many other experiments [26]. That interpretation yields a larger gap around the (~r, 0) point, and a node, or small gap, near the (1, 1) direction, consistent with our observations in Fig. 4. As we have stated earlier, the Fermi surface points of the underdoped sample can accommodate a small gap because of their error bars, especially since impurity scattering can cause the gap to be small over an extended region. The corresponding experimental phase diagram is shown in
? I I
100
0 Half
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Fig. 5. Rough ARPES phase diagram of Bi2212 around the region of optimal superconductivity doping. The regions are (N) normal state, (SC) superconducting state, and (G) state with a gap. The curved line represents the superconducting phase transition.
Fig. 5. Pairing of electrons above Tc in underdoped samples has also been proposed on very general grounds. Based on a d-wave model calculation extended from the Ginzburg-Landau theory, Doniach and Inui pointed out that T~ will be determined by the temperature above which phase coherence is destroyed [27]. Similar conclusions were also drawn from the analysis of empirical data such as the low-temperature London penetration depth [28]. Even though the dx2_y2 pairing in the normal state is an attractive scenario in the doping range near optimal T~, our data set suggests that a simple dx2_y2 gap at the LDA Fermi surface is not the whole story if one considers the entire doping range, in particular if the gap size is similar to that of the superconducting state near optimal doping. Rather, a d-wave gap which increases as the doping decreases, mixed with the pocket-like excitation spectra suggested by Sr2CuO2C12, may be a more realistic answer. This general evolution of the low-energy excitations (or approximate Fermi surface) has been discussed in a few recent papers in the strong-coupling limit of the t-J model [29,30]. The physical picture of these discussions can be thought of as a mixture of a d-wave gap and pockets near (_+~r/2, 5--rr/2). A deeper investigation of these issues can be found in our forthcoming paper [26].
A.G. Loeser et al. / Physica C 263 (1996) 208-213
In summary, we have presented results from a doping-dependent study of the normal state electronic structure. A n anomalous excitation gap is observed in the normal state of the underdoped sample. Our current work is on the lineshape of either state, and a continued probing of the doping and temperature dependence of the normal state.
Acknowledgements We thank D.S. Marshall, C.H. Park, A.Y. Matsuura, W.E. Spicer, P. Fournier and A. Kapitulnik for experimental collaboration. We acknowledge stimulating discussions with R.B. Laughlin, P.A. Lee, Xiao-Gang Wen, H. Fukuyama, N. Nagaosa, A.J. Millis, D.J. Scalapino and N. Bulut. Much of the data presented here was obtained from the Stanford Synchrotron Radiation Laboratory, which is operated by the DOE Office of Basic Energy Sciences, Division of Chemical Sciences. The Office's Division of Materials Science has provided support for this research. The Stanford work was also supported by NSF grants DMR-9311566 and DMR-9357507. Beamline 5 of SSRL was built with DARPA, ONR, AFOSR, AOR, DOE and NSF support.
References [1] J. Phys. Chem. Solids, Vol. 52, No. 11/12 (1991). [2] J. Phys. Chem. Solids, Vol. 53, No. 12 (1992).
213
[3] J. Phys. Chem. Solids, Vol. 54, No. 10 (1993). [4] Proc. Stanford Conf. on Spectroscopies in Novel Superconductors, to be published as a special volume of J. Phys. Chem. Solids. [5] C.G. OIson et al., Science 245 (1989) 731. [6] J.C. Campuzanoet al., Phys. Rev. Lett. 64 0990) 2308. [7] D.S. Dessau et al., Phys. Rev. Lett. 71 0993) 2781. [8] D.M. King et al., Phys. Rev. Lett. 70 (1993) 3159. [9] R.O. Andersonet ai., Phys. Rev. Leu. 70 (1993) 3163. [10] R. Liu et al., Phys. Rev. B 45 (1992) 5615; ibid., 46 (1992) 11056. [11] D.M. King et al., Phys. Rev. Lett. 73 (1994) 3298. [12] K. Gofron et al., Phys. Rev. Leu. 73 (1994) 3302. [13] B.O. Wells et al., Phys. Rev. B 46 (1992) 11830. [14] Momentumresolutionwill be effected by elastic scatteringof the photoelectrons.This is a possibilityin the high-Tc materials, especiallyconsideringthe highly imperfectnature of the crystals. [15] D.S. Dessau et al., Phys. Rev. Lett. 66 (1991) 2160. [16] R. Ciaessenet al., preprint, and references within. [17] D.M. Mitzi et al., Phys. Rev. B 41 (1990) 6564. [18] D.S. Marshall et al., Phys. Rev. B, to be published. [19] D.S. Marshall, D.S. Dessau, A.G. Loeser et al., unpublished. [20] L.S. Borkowski and P.J. Hirschfeld, Phys. Rev. B 49 (1994) 15404. [21] O.K. Andersenet al., to be published in Ref. [4]. [22] J.R. Schrieffer and A. Kempt, to be publishedin Ref. [4]. [23] G. Kotliar and J. Liu, Phys. Rev. B 38 (1988) 5142. [24] T. Tanamoto,K. Kohno and H. Fukuyama,J. Phys. Soc. Jpn. 61 (1992) 1886. [25] P.W. Anderson,Science 235 (1987) 1196. [26] A.G. Loeser, Z.-X. Shen et al., to be published. [27] S. Doniach and M. lnui, Phys. Rev. B 41 (1990) 6668. [28] V.J. Emery and S.A. Kivelson,Nature 374 (1995) 434. [29] Xiao-GangWen and P.A. Lee, preprint. [30] A.J. Millis and H. Monien, Phs. Rev. B 50 (1994) 16606.
Physica C 263 (1996) 208-213
Doping dependence of BiaSr2CaCu2Os+ in the normal state A.G. Loeser, D.S. Dessau ~, Z.-X. Shen * Department of Applied Physics, and Stanford Synchrotron Radiation Laboratory, Stanford University, StanJbrd, CA 94305-4055, USA
Abstract We have probed the electronic structure of Bi2Sr2CaCu2Os+ ~ with angle-resolved photoemission spectroscopy, observing changes in the electronic structure with doping variation. At cartier dopings below the optimal for superconductivity, we observe an excitation gap in the normal state. The momentum dependence and the magnitude of the gap are very similar to those of the dx2_y: gap observed in the superconducting state.
One of the reasons why the problem of high-temperature superconductivity is so difficult to crack is that the normal state of these materials is not well understood. The optimally doped superconductor sits in the middle of a complicated phase diagram; at higher dopings it becomes a typical metal, while at dopings near half-filling it is an antiferromagnetic (AF) insulator. If these two cases represent the limits of small and large electron correlations, then the superconducting dopings are in the middle of the transition between the two. There, the normal state is frequently called a "strange metal," as it displays many unusual properties which have not yet been explained within the context of any simple theory [1-4]. The thrust of our photoemission experiments presented here has been to probe this transition in the normal state electronic structure, by varying the hole concentration from the overdoped to the underdoped regime.
* Corresponding author. Fax: + 1 415 725 5457; e-mail: [email protected]. Current address: Department of Physics, University of Colorado, Boulder, CO, 80309-0390, USA.
In the overdoped case, many features of the electronic structure can be described with local density approximation (LDA) band theory. Angle-resolved photoemission experiments show a Fermi surface which is in good agreement with the LDA calculations [5-12]. A typical Fermi surface of overdoped metal is depicted in Fig. l(b). Small variations in the doping level indicate that the Fermi surface obeys Luttinger's sum rule, with a volume proportional to the number of carriers. There is, however, still evidence of correlations in the photoemission data. For example, the bands which cross the Fermi level are flatter than predicted by LDA [7,11,12], indicating mass enhancement due to electron-electron interactions. On the other side, at half-filling, the so-called undoped parent compounds are Mott-Hubbard insulators. Already, to explain this, one must abandon band theory in favor of localized models of the electronic structure. At this doping, the only clear photoemission study published thus far is on Sr2CuOzCI 2 [13]. In this study, Wells et al. observed a filled band with maxima in the two-dimensional Brillouin zone at (~r/2, w / 2 ) in units of k/a in the C u - O plane. In the metallic case, the maximum is at
0921-4534/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0921 - 4 5 3 4 ( 9 6 ) 0 0 0 7 4 - 3
A.G. Loeser et al./ Physica C 263 (1996) 208-213
""
{~,ltJ'
/
%
,, (0,0) "% ,~)" a
',,~¢/
Fig. 1. This sketch depicts possible Fermi surfaces for (a) doping close to half-filling, and (b) doping above optimal Tc. While (b) agrees well with LDA calculations, (a) has pockets centered at ( + aT/2, + "rr/2). The dotted line in (a) is the AF Brillouin zone boundary, and the narrow parts of the Fermi surface have weak ARPES intensity.
(Tr, ,r). This change appears to be caused by the AF order, which halves the size of the Brillouin zone. The zone edge for the AF ordered state includes ('rr/2, at/2), and a gap opens up there. There is also a large drop in photoemission intensity as one follows a dispersive feature from the first into the second AF Brillouin zone, possibly due to the weak scattering potential of the AF lattice. Starting with these data and a rigid band doping picture, one might expect the Fermi surfaces at very low doping to be pockets, as depicted in Fig. l(a). This is a reasonable inference, since the AF correlation length is still long and the rigid band picture may be valid at very low doping. The question becomes, What is the nature of the normal state around the optimal superconducting carrier doping? Is it a correlated band structure, or does one need to employ the Hubbard model or related theories in order to accurately describe the physics? At first glance, the photoemission data at these doping levels seem closest to the metal. Mainly, the apparent Fermi surface agrees well with the LDA calculation, and there are no other obvious differences. In this work, however, we report observations of a gap in the normal state electronic structure, very close to the doping of optimal T~. This observation motivates consideration of the localized approaches as very important for describing the normal state physics. In a photoemission experiment, electrons in a solid are excited above the vacuum level by incident, monochromatic photons. While inelastic scattering
209
captures some of the electrons, the rest are emitted from the solid, and their energy and momentum is measured by a detector. The resulting spectrum is made up of primary counts, or electrons which exit the solid without loss of energy, and secondaries, which have experienced some inelastic scattering. The secondaries' contribution to the spectrum usually has a characteristic shape, and can be subtracted from the spectrum fairly accurately. Along the energy axis, the resulting spectrum includes convolutions with the photon energy distribution, as well as the electron analyzer's instrument response function. Typically, experimentalists report the net effect in terms of a single Gaussian's full width at half maximum. Momentum resolution is mainly determined by geometrical factors (sample size and analyzer aperture) [14], and is typically reported as + the half width of the resulting angular acceptance function. Typical angle-resolved photoemission (ARPES) studies use + 1 deg. In addition, if the electronic states in the solid are two-dimensional in nature, then conservation laws imply a simple 1-1 mapping between the electron's angle of emission and its momentum inside the solid. This is the case for the high-To superconductors, facilitating our detailed studies of the Brillouin zone. The data reported in this paper were obtained with a VSW analyzer (angular resolution ± 1, energy resolution -- 30 meV), at the undulator beamline 5 - 3 of the Stanford Synchrotron Radiation Laboratory, with a base pressure of roughly 4 × 10-11 Torr. We have described this in more detail elsewhere [15]. Theoretically, the resulting primary spectrum is extremely complex until one makes the so-called sudden approximation, namely, that the photoelectron is ejected from the solid faster than the time scale of various interactions which would alter its momentum and energy. (It is not obvious in an ultraviolet photoemission (LIPS) experiment that this is the case, but the assumption agrees fairly well with results. The lineshape analysis in Claessen et al.'s ongoing study of the two-dimensional metal TiTe 2 can be thought of as testing this approximation, along with other ideas [16].) With the sudden approximation, the photoemission spectrum becomes the electron removal excitation function, which is how we will analyze the data below. Despite the variety of materials available,
210
A.G. Loeser et aL / Physica C 263 (1996) 208-213
Bi2Sr2CaCu2Os+ 8 (Bi2212) remains the most fruitful for high-resolution ARPES studies. Steadily improving sample quality, sample preparation techniques, and instrument resolution allow for new results on this material, even after many ARPES papers have been published. We used Bi2212 single crystals which were grown with a directional solidification technique [17]. The carrier concentrations were adjusted by annealing at 600°C in various mixtures of argon and oxygen, and characterized by T~ measurement in a SQUID magnetometer. In a complimentary study, we achieved much higher variation in doping of thin films by substituting the divalent calcium with trivalent dysprosium during the growth [18]. The analysis of the doping dependence of these samples supports the single crystal results, and will be published soon [19]. As the doping is reduced from the metallic regime down towards the insulator, there appears to be a significant change in the Fermi surface topology. In an ARPES study, the Fermi surface is drawn by making cuts in k-space, showing dispersive features (quasiparticles) as they move through the Fermi energy and lose weight. The k-space locations where the quasiparticle band crosses the Fermi level are identified as points on the Fermi surface, and a line connecting them is the full, measured surface. As stated earlier, the main Fermi surface of overdoped Bi2212 is sketched in Fig. l(b). In the slightly underdoped Bi2212, however, we will show that some of those crossings disappear. This leaves us without a clear, connected Fermi surface. Instead, we obtain something like the dark lines in Fig. l(a). There are a few effects which could cause this change, which will be discussed in more detail below. One possibility is that the Fermi surface has changed from the large, LDA surface to four pockets surrounding the ( + 7r/2, _ 7r/2) points on the AF zone boundary, as one would get by doping the half-filled parent compound with a small number of holes. This would imply the sort of Fermi surface evolution with doping presented in Fig. 1. Another possibility is that the Fermi surface is unchanged, but that there is an anisotropic gap similar to the dx2_y2 one, even in the normal state. Finally, the data presented here do not rule out a combination of both pictures, as the pocket Fermi surfaces may accommodate a small gap. The distinction between a Fermi
Ill I I I!
400
200
0
400
200
0
400
200
BindingEnergy(meV) Fig. 2. Cuts showing the doping evolution of Bi2212. Panels (1) are the cut from (0, 0) to (xr, xr), and panels (2) are the cut from (at, 0) to ('it, "rr). Sample 1 is underdoped, Tc = 84 K. Sample 2 is overdoped, Tc = 85 K. Sample 3 is overdoped, Tc = 80 K.
surface pocket picture and a dx2_y2 gap is even more difficult if we consider impurity scattering and the resulting dirty d-wave picture. In this case, the gap is suppressed for momentums close to the (1, 1) direction, which coincides with a large part of the pocket Fermi surfaces [20]. The cuts used to demonstrate these changes at the Fermi energy are shown in Fig. 2. We examine the (0, 0) to ('rr, ~r) and the (-rr, 0) to (~r, ~ ) apparent Fermi surface crossings for three samples with a range of dopings. One general change which is apparent is that all of the dispersive features become broader as the hole doping is reduced. This is apparently due to an increased electronic damping of the quasiparticles, since changes in the quality of the single crystals around the C u - O planes are unlikely. The quality of the reduced samples and their surfaces is verified by the observation that, in the superconducting state, the sharp quasiparticle peak at (Tr, 0) is roughly as narrow as that of the optimally doped sample. On the other hand, we cannot suggest a more specific mechanism for this broadening, a notion consistent with the material being a "strange metal." More importantly, a gap opens up in the
A.G. Loeser et a l . / Physica C 263 (1996) 208-213
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exhibits the normal crossing behavior described above. As the peak loses its intensity, its leading edge moves to negative binding energy. In the 80 K overdoped case, the ('rr, 0) to (~, w) crossing has this characteristic as well. At lower dopings, however, the leading edge behavior looks less like a Fermi surface crossing. For the underdoped sample in particular, the dispersive feature clearly does not reach the Fermi level, while the situation is unclear for the 85 K overdoped sample. By gap, we essentially mean that a dispersive feature no longer crosses the Fermi level. If the Fermi surface evolution is as depicted in Fig. l, we stress that the presence (absence) of a Fermi crossing from (w, 0) to (w, ~r) in the overdoped (underdoped) sample is a clean way to distinguish between Figs. l(a) and (b). Otherwise, the Fermi surface clearly has a gap in the (~, 0) region of k-space; this is demonstrated here in the
Comparison of a spectrum from the most overdoped to the tmderdoped sample, in the region of the Fermi crossing from (~, 0) to ('rr, "rr). Note the difference in edges of the two samples.
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~ -10 underdoped sample. Fig. 3 shows two spectra taken from roughly the same point in the Brillouin zone, in the vicinity of the apparent Fermi surface crossing. For the underdoped curve, the excitation spectrum clearly does not extend to as low a binding energy, nor even to the Fermi level itself, indicating the creation of an excitation gap. Because there is no successful model to describe the single-particle excitation spectrum in these materials theoretically, the experimentalist must rely on simpler, quantitative measurements to characterize the spectra. One such quantity is the energy of the leading edge of the spectrum, which we report, as a binding energy, as the position on the curve at which the intensity is half (above the background) of the peak intensity. From simple models of dispersive features, we see that the leading edge should pass the Fermi energy, i.e., go to negative binding energy in a usual Fermi level crossing. This is mainly caused by finite instrumental resolution that shifts the spectral weight of a sharp feature from below to above E F. In Fig. 4, we show the leading edge binding energies for the spectra in Fig. 2, in order to emphasize the leading edge shifts with doping. The normal state (0, 0) to (q¢, ~r) crossing for any doping level
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212
A.G. Loeser et al./Physicc C 263 (1996) 208-213
same manner used originally to identify the superconducting gap. The nature of the gap near (~, 0) in the underdoped sample is unknown at this point. There are two alternative ways to explain the data, based on a weak-coupling SDW picture, and a strong-coupling picture. The weak-coupling picture can be understood in a phenomenological way. Owing to shortrange AF correlations, a local 45 ° rotated ~/2 × ~/2 periodicity imposes a new AF Brillouin zone that makes (0, 0) and ('rr, "rr) equivalent. This halving of the Brillouin zone creates additional bands which cross the original ones near ('rr, 0) and ('rr/2, rr/2). If the new AF bands and the original bands interact, an SDW gap will open at (~r, 0), as estimated by band calculations [21,22]. In this scenario, the Fermi surface of the underdoped sample loses its ARPES intensity at the edge of the first AF Brillouin zone (dashed boundary of Fig. 1), and is completed by the gray line in Fig. l(a), in the second AF Brillouin zone. This results in Fermi surface pockets centered at ( _ ~r/2, 5- ~r/2). We have observed some weak features that may correspond to the dashed Fermi surface, but we have not determined whether this is intrinsic or due to a structural distortion. In the context of strong coupling theory, our data may be a consequence of dx-~_y2 pairing in the normal state. The opening of a spin-channel dx2_y2 gap in the normal state of the underdoped sample is already a prediction in a phase diagram [23,24], extending the original RVB idea [25]. In fact, Fig. 4 shows that a normal state gap opens in underdoped samples of Bi2212 with T~ as high as 84 K (optimal Tc of the system is 91 K) [26]. At this doping level, where the AF correlation virtually vanishes, it is tempting to interpret those data as due to the opening of a dx~_y:-like gap at the LDA Fermi surface, similar in size and symmetry to the dx2_yZ gap in the superconducting state, as supported by many other experiments [26]. That interpretation yields a larger gap around the (~r, 0) point, and a node, or small gap, near the (1, 1) direction, consistent with our observations in Fig. 4. As we have stated earlier, the Fermi surface points of the underdoped sample can accommodate a small gap because of their error bars, especially since impurity scattering can cause the gap to be small over an extended region. The corresponding experimental phase diagram is shown in
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Fig. 5. Rough ARPES phase diagram of Bi2212 around the region of optimal superconductivity doping. The regions are (N) normal state, (SC) superconducting state, and (G) state with a gap. The curved line represents the superconducting phase transition.
Fig. 5. Pairing of electrons above Tc in underdoped samples has also been proposed on very general grounds. Based on a d-wave model calculation extended from the Ginzburg-Landau theory, Doniach and Inui pointed out that T~ will be determined by the temperature above which phase coherence is destroyed [27]. Similar conclusions were also drawn from the analysis of empirical data such as the low-temperature London penetration depth [28]. Even though the dx2_y2 pairing in the normal state is an attractive scenario in the doping range near optimal T~, our data set suggests that a simple dx2_y2 gap at the LDA Fermi surface is not the whole story if one considers the entire doping range, in particular if the gap size is similar to that of the superconducting state near optimal doping. Rather, a d-wave gap which increases as the doping decreases, mixed with the pocket-like excitation spectra suggested by Sr2CuO2C12, may be a more realistic answer. This general evolution of the low-energy excitations (or approximate Fermi surface) has been discussed in a few recent papers in the strong-coupling limit of the t-J model [29,30]. The physical picture of these discussions can be thought of as a mixture of a d-wave gap and pockets near (_+~r/2, 5--rr/2). A deeper investigation of these issues can be found in our forthcoming paper [26].
A.G. Loeser et al. / Physica C 263 (1996) 208-213
In summary, we have presented results from a doping-dependent study of the normal state electronic structure. A n anomalous excitation gap is observed in the normal state of the underdoped sample. Our current work is on the lineshape of either state, and a continued probing of the doping and temperature dependence of the normal state.
Acknowledgements We thank D.S. Marshall, C.H. Park, A.Y. Matsuura, W.E. Spicer, P. Fournier and A. Kapitulnik for experimental collaboration. We acknowledge stimulating discussions with R.B. Laughlin, P.A. Lee, Xiao-Gang Wen, H. Fukuyama, N. Nagaosa, A.J. Millis, D.J. Scalapino and N. Bulut. Much of the data presented here was obtained from the Stanford Synchrotron Radiation Laboratory, which is operated by the DOE Office of Basic Energy Sciences, Division of Chemical Sciences. The Office's Division of Materials Science has provided support for this research. The Stanford work was also supported by NSF grants DMR-9311566 and DMR-9357507. Beamline 5 of SSRL was built with DARPA, ONR, AFOSR, AOR, DOE and NSF support.
References [1] J. Phys. Chem. Solids, Vol. 52, No. 11/12 (1991). [2] J. Phys. Chem. Solids, Vol. 53, No. 12 (1992).
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[3] J. Phys. Chem. Solids, Vol. 54, No. 10 (1993). [4] Proc. Stanford Conf. on Spectroscopies in Novel Superconductors, to be published as a special volume of J. Phys. Chem. Solids. [5] C.G. OIson et al., Science 245 (1989) 731. [6] J.C. Campuzanoet al., Phys. Rev. Lett. 64 0990) 2308. [7] D.S. Dessau et al., Phys. Rev. Lett. 71 0993) 2781. [8] D.M. King et al., Phys. Rev. Lett. 70 (1993) 3159. [9] R.O. Andersonet ai., Phys. Rev. Leu. 70 (1993) 3163. [10] R. Liu et al., Phys. Rev. B 45 (1992) 5615; ibid., 46 (1992) 11056. [11] D.M. King et al., Phys. Rev. Lett. 73 (1994) 3298. [12] K. Gofron et al., Phys. Rev. Leu. 73 (1994) 3302. [13] B.O. Wells et al., Phys. Rev. B 46 (1992) 11830. [14] Momentumresolutionwill be effected by elastic scatteringof the photoelectrons.This is a possibilityin the high-Tc materials, especiallyconsideringthe highly imperfectnature of the crystals. [15] D.S. Dessau et al., Phys. Rev. Lett. 66 (1991) 2160. [16] R. Ciaessenet al., preprint, and references within. [17] D.M. Mitzi et al., Phys. Rev. B 41 (1990) 6564. [18] D.S. Marshall et al., Phys. Rev. B, to be published. [19] D.S. Marshall, D.S. Dessau, A.G. Loeser et al., unpublished. [20] L.S. Borkowski and P.J. Hirschfeld, Phys. Rev. B 49 (1994) 15404. [21] O.K. Andersenet al., to be published in Ref. [4]. [22] J.R. Schrieffer and A. Kempt, to be publishedin Ref. [4]. [23] G. Kotliar and J. Liu, Phys. Rev. B 38 (1988) 5142. [24] T. Tanamoto,K. Kohno and H. Fukuyama,J. Phys. Soc. Jpn. 61 (1992) 1886. [25] P.W. Anderson,Science 235 (1987) 1196. [26] A.G. Loeser, Z.-X. Shen et al., to be published. [27] S. Doniach and M. lnui, Phys. Rev. B 41 (1990) 6668. [28] V.J. Emery and S.A. Kivelson,Nature 374 (1995) 434. [29] Xiao-GangWen and P.A. Lee, preprint. [30] A.J. Millis and H. Monien, Phs. Rev. B 50 (1994) 16606.