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Doping dependent evolution of infrared spectra in underdoped YBa2Cu3Oy in terms of two-component optical conductivity
Solid State Communications 133 (2005) 277–282 www.elsevier.com/locate/ssc
Doping dependent evolution of infrared spectra in underdoped YBa2Cu3Oy in terms of two-component optical conductivity Y.S. Lee* Department of Physics, University of California at San Diego, La Jolla, CA 92093-0319, USA Received 1 September 2004; received in revised form 31 October 2004; accepted 17 November 2004 by H. von Lo¨hneysen Available online 2 December 2004
Abstract In access to optical spectroscopy of heavily underdoped detwinned YBa2Cu3Oy (YBCO) crystals, we re-examine the doping dependent evolution of infrared spectra in the CuO2 plane of underdoped YBCO in terms of two-component optical conductivity. The extended Drude model analysis is applied to the two-component conductivity, and the results are compared with experimental data in the pseudogap state. We demonstrate that a model consisting of a Drude and Lorentz oscillator components reproduces characteristics of infrared spectra in underdoped YBCO. q 2004 Elsevier Ltd. All rights reserved. PACS: 74.25 Gz; 74.72 Bk Keywords: A. High-Tc superconductors; D. Optical properties
1. Introduction The generic property of high-Tc superconductivity evolving from an antiferromagnetic Mott insulating state by charge carrier doping has attracted much attention to the physics of doped Mott insulators [1]. To elucidate the origin of the unconventional superconductivity [2], it is essential to understand the nature of a conducting state derived from the Mott insulator and its evolution into the high-Tc superconducting state with doping. While optical spectroscopy has provided invaluable insights into the doping dependence of the electronic band structure and elementary excitations, one of intriguing properties in the ab-plane optical conductivity is a strong mid-infrared (IR) spectral feature [3]. To understand this enigmatic feature, two competitive approaches have been used. First is the ‘single-component picture’ viewing that the spectral feature below a charge transfer (CT) excitation is due to one kind of heavily * Tel.: C1 858 822 0149; fax: C1 858 534 0173. E-mail address: [email protected]. 0038-1098/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2004.11.023
damped carriers. In this picture, the strong mid-IR feature is attributed to the frequency dependence of the scattering rate 1/t (u) in the extended Drude model (EDM) description [4– 7], which is suggestive of peculiar interactions in cuprates, e.g. the electron–boson interaction [8–10] or quantum critical uctuation [11]. Especially in the underdoped region, the low frequency suppression in 1/t(u) has been considered as evidence of the pseudogap formation [12,13]. The second approach is the ‘two-component picture’ with a Drude-like coherent mode and a localized mid-IR absorption. While the former is responsible for the transport, the latter accounts for the mid-IR spectral weight. The origin of the mid-IR absorption might be an interband transition related to the midgap state [14], an incoherent mode due to electron–phonon or electron–spin interaction [15,16], inhomogeneity [17], etc. Very recently, it has been reported that for heavily underdoped detwinned YBa2Cu3Oy (YBCO) the optical conductivity in the CuO2 plane is comprised of the wellseparated Drude-like coherent mode and mid-IR absorption [18]. This two-component optical conductivity is also observed in other cuprates and is a general characteristic
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of heavily underdoped cuprates [19–21]. Starting from this finding, in this paper we review the doping dependent evolution of in-plane optical spectra in YBCO in terms of two-component optical conductivity. We apply the EDM analysis to the two-component conductivity feature in the heavily underdoped YBCO. Using a model consisting of two components, we examine the 1/t(u) behavior according to the position of the mid-IR absorption. The twocomponent analysis is extended to the pseudogap state. Finally, we discuss the doping dependence in the loss function.
2. Experiments and analysis method We investigated detwinned YBCO single crystals with oxygen content yZ6.35 (non-superconducting), 6.43 (Tcw13 K), and 6.65 (Tcw60 K) grown by a conventional flux method and detwinned under uniaxial pressure [22]. Reflectivity spectra R(u) at nearly normal incidence were measured with polarized light at frequencies from 20 to 48,000 cmK1 and at temperatures from 10 to 293 K. ~ The complex optical conductivity spectra sðuÞZ s1 ðuÞC is2 ðuÞZ ðiu=4pÞ½~3ðuÞK 3N were obtained from the measured R(u), using the Kramers-Kronig (KK) transformation. To investigate the physical properties of the CuO2 plane without any disturbance by the optical response of chain structures along the b-axis [22–25], we restrict our analysis to the a-axis properties in this paper. ~ According to the EDM [4,5], sðuÞZ ðu2p =4pÞ! ð1=½1=tðuÞK ium ðuÞÞ, where 1/t(u) and m*(u) are frequency dependent scattering rate and effective mass, respectively. The total plasma frequency up is estimated Ð from the sum rule u2p Z 8 0u s1 ðu 0 Þdu 0 at uZ10,000 cmK1. For the two-component optical conductivity modeling, we used a Drude mode s~ D ðuÞZ ðu2p;D =4pÞð1=½GD K iuÞÞ and Lorentz oscillator s~ D ðuÞZ ðu2p;L =4pÞðu=ð½iðu2c K u2 ÞC uGL Þ for the coherent and the mid-IR absorption modes, respectively, and thus the modeled conductivity takes the form s~ M ðuÞZ s~ D ðuÞC s~ L ðuÞZ ðu2p;M =4pÞð1=½1=tM ðuÞK iumM ðuÞÞ. Here, up,D and GD are plasma frequency and scattering rate of charge carriers, respectively. up,L, GL, and uc are the strength, the width, and the position of a Lorentz oscillator, respectively. Subscript M is for modeled quantities. The up,M value is estimated from the sum rule with s1,M(u). To account for the temperature dependence GD is reduced at low T with the other parameters unchanged.
3. Heavily underdoped state and two component modeling We first discuss optical responses in the heavily underdoped YBCO, exemplified by the yZ6.43 data (Fig. 1). In Fig. 1(a), apart from sharp peaks below 0.1 eV due to transverse optical phonon modes, the real part of optical
conductivity s1 (u) below the CT excitation near 14,000 cmK1 is composed of the two separate absorption features: a coherent mode (u!600 cmK1) followed by a mid-IR absorption at 0.5 eV [18]. The coherent mode significantly narrows with the decreased T, while the mid-IR absorption is virtually independent of T. Due to the existence of the coherent mode, the imaginary part of optical conductivity s2(u) is positive below a zero-crossing s2(u0)Z0 (Fig. 1(c)). At lower T, the zero-crossing behavior occurs at a higher frequency with the far-IR spectral weight enhanced. (According to the Drude model, ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u0 Z u2p;D =3N K G2 ) We apply the EDM analysis to these two-component conductivity features. As shown in Fig. 1(e), 1/t(u) at room temperature shows a broad feature with a peak around 1300 cmK1. With the decreased T, the far-IR 1/t(u) is significantly suppressed, while the mid-IR 1/t(u) is enhanced with the peak shifting to a higher frequency. The suppression of the far-IR 1/t(u) can be associated with the development of the coherent mode. However, the peak structure is inconsistent with the single-component description of the electromagnetic response in the context that the scattering incidence should increase with frequency and temperature. The non-monotonic dependence of 1/t(u) indicates the inapplicability of the EDM analysis to the heavily underdoped samples. The peak positions in 1/t(u) correspond nearly to the zero-crossing frequency u0 in s2(u). Because 1=tðu0 ÞZ ðu2p =4pÞð1=s1 ðu0 ÞÞ, the peak enhancement is attributed to the decrease in s1(u0) at lower T [26]. It is noted that the mid-IR absorption is featureless in 1/t(u). We demonstrate that these optical responses in the heavily underdoped YBCO are simply reproduced by the ~ two-component modeling. We fit the sðuÞ for yZ6.43 with a modeled conductivity s~ M ðuÞ consisting of Drude mode s~ D ðuÞ and Lorentz oscillator s~ L ðuÞ (Model II), which are represented by broken lines in Fig. 1(a). The fitting parameters are summarized in Table 1. Then, we apply the EDM analysis to the fitted s~ M ðuÞ and plot 1/tM(u). As shown in the right panel of Fig. 1, s~ M ðuÞ and 1/tM(u) are quite consistent with experimental results for yZ6.43. (Small differences at 500–2000 cmK1 are probably due to the imperfection of a single Lorentz oscillator for the mid-IR absorption in low frequency side) This agreement is not surprising given the experimental fact that the twocomponent optical conductivity feature is quite clear in the heavily underdoped region. To get more insight into the EDM description for the two-component optical conductivity, we examine the change of 1/tM(u) with respect to modification of fitting parameters for Model II. First, we consider the change of the spectral weight distribution. With the other parameters fixed, we increase the Drude weight by 90% and decrease the weight of the Lorentz oscillator by the same amount, preserving the total spectral weight unchanged (Model III). Results are displayed in the top panels of Fig. 2. While no
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Fig. 1. T-dependent s1(u) (top), s2(u) (middle), 1/t(u) (bottom) for yZ6.43. Left panels are for experimental results, and right panels are for results fitted by the two component modeling. Fitting parameters (Model II) are summarized in Table 1. In (a), the dotted and the dot-dashed lines represent a Drude mode and a Lorentz oscillator, respectively. In all panels, the thick (thin) lines represent the low (high) T data.
qualitative change is observed, the peak structure in 1/ tM(u) is weaker and shifts to a higher frequency. The hardening of the peak is attributed to the shift of u0 in s2,M(u0)Z0 to higher frequencies due to the increase in the Drude weight (inset of Fig. 2(a)). Because limu/0 1=tM ðuÞZ ðGD Þðu2p;M =u2p;D Þ, the limu/0 1=tM ðuÞ is smaller with the larger Drude weight. Second, we change the position of the mid-IR absorption. We shift the position of Lorentz oscillator from 3800 to 2000 cmK1 (Model IV).
As shown in the bottom panels of Fig. 2, notably, the peak structure in 1/tM(u) is significantly suppressed, and the slope at high frequencies becomes positive. The softening of the mid-IR absorption pushes the zero-crossing behavior in s2,M(u) above 4000 cmK1 (inset of Fig. 2(c)), which leads to the disappearance of the peak structure in 1/tM(u). The gross feature of 1/tM(u) in Model IV, including the far-IR suppression, is reminiscent of that in the pseudogap state of YBCO [12,13]. It is noted that even the two-component
Table 1 Summary of fitting parameters for two component conductivity s~ M ðuÞ; the plasma frequency up,D and the scattering rate GD at low (high) T for s~ D ðuÞ, the strength up,L, the width GL, and the position uc for s~ L ðuÞ, and the total plasma frequency up,M for 1/tM(u). The unit is cmK1 Model
up,D
GD
up,L
uc
GL
up,M
y
I II III IV V
4100 5250 7250 5250 8000
280(950) 135(750) 135 135 60(700)
10,850 13,550 11,850 13,550 15,000
4200 3800 3800 2000 1600
7000 7000 7000 7000 5000
7000 12,000 12,000 12,000 19,000
6.35 6.43
6.65
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Fig. 2. Left panels: s1,M(u) for (a) Model III and (c) Model IV. In Model III, the Drude plasma frequency up,D is increased to be 7250 cmK1. In Model IV, the Lorentz oscillator is shifted to 2000 cmK1. The dotted and the dot-dashed lines represent the Drude and the Lorentz oscillator modes, respectively. Right panels: 1/tM(u) for (b) Model III and (d) Model IV. Insets show s2,M(u). Model III and IV are representing by the thick lines. For comparison, Model II is displayed (thin lines).
conductivity can exhibit phenomenologically reasonable 1/ t(u) with appropriately low frequency of the mid-IR absorption.
4. Underdoped (pseudogap) state We now turn to the pseudogap state of the underdoped YBCO. Fig. 3 shows optical responses for yZ6.65, where the pseudogap onset temperature T* is estimated to be above 300 K from the dc resistivity measurement [22]. With the decreased T, the s1(u) shows the development of a narrow peak at zero frequency from a broad single component-like feature at high T. It is most evident that at TRTc a peak structure near 0.2 eV is separated from the well-developed Drude-like far-IR conductivity. Coincidently, the far-IR 1/ t(u) is suppressed deviating from the nearly linear udependence at high frequencies, which has been considered as evidence of the pseudogap formation (Fig. 3(b)) [12,13]. On the basis of the electron–boson scattering theory [27], this gap-like feature in 1/t(u) could be suggestive of a strong coupling to the corresponding bosonic mode. The bosonic spectral density a2F(u) is estimated to be a2 FðuÞ z WðuÞZ ð1=2pÞðd2 =du2 ½u=tðuÞÞ [28,29]. Actually, W(u) at 65 K for yZ6.65 exhibits a distinct resonance, as shown in Fig. 3(c). It has been suggested that the resonance in W(u) should be associated with the magnetic resonance observed in inelastic neutron scattering experiments [8–10].
It is noted that all characteristic features in the pseudogap state of YBCO can be well-described with the twocomponent modeling. We fit the yZ6.65 YBCO data with s~ M ðuÞ (Model V). We use the Lorentz oscillator at 1600 cmK1 for the peak structure in the mid-IR. As shown in Fig. 3(b), the T-dependent 1/tM(u) (open circles) are quite consistent with experimental results. Naturally, the low frequency suppression in 1/t(u) is associated with the development of the coherent mode, while no significant Tdependence at higher frequencies is suggestive of the existence of the mid-IR mode which is essentially independent of temperature [30]. Even the WM(u) estimated from 1/tM(u) is comparable to that obtained from the experiment, which might imply that the resonance in W(u) is just an artificial by-product from the blended optical conductivity. These findings suggest that the optical response in the pseudogap state of YBCO be interpreted with the two-component optical conductivity and the pseudogap formation be accompanied by the emergence of the mid-IR absorption from the single band optical response. Interestingly, these behaviors have been commonly observed in systems with partial gap opening [31]. It is worthwhile to note that the Fermi surface structure is quite similar in the heavily underdoped and the pseudogap states. According to the recent angle resolved photoemission spectroscopy results on La2KxSrxCuO4, in the heavily underdoped region the Fermi surface is confined to the nodal region (diagonals of Brillouin-zone) with a large
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Im½K1=~3ðuÞ spectra for yZ6.35, 6.43 and 6.65 at lowest temperature (or TRTc). In heavily underdoped region (yZ 6.35 and 6.43) one can find two peak structure; peak I near 2000 cmK1 and peak II above 5000 cmK1. A Drude or Lorentz oscillator mode is represented by a peak in Im½K1=~3ðuÞ, whose position is associated with the spectral weight and the position of the mode: In condition that pffiffiffiffiffi u[G D (or GL), qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u zup;D = 3N for the Drude mode and u z u2c C u2p;L =3N for the Lorentz oscillator [34]. Naturally, two peak structure in Im½K1=~3ðuÞ are related to the twocomponent optical conductivity; Peak I and Peak II correspond to the Drude mode and the Lorentz oscillator, respectively. With the increased doping Peak I is significantly suppressed and finally disappears for yZ 6.65, whereas peak II is enhanced and shifted to higher frequencies [20]. The peak near 1.0 eV in Im½K1=~3ðuÞ has been considered to be related to the screened plasma frequency for charge carriers for some cuprates [3]. However, the gradual evolution of Im½K1=~3ðuÞ with doping suggests that the single peak structure for yZ6.65 might be associated with the mid-IR absorption. To obtain more understanding, we plot the modeled Im½K1=~3M ðuÞ spectra with the results of twocomponent modeling (Model I for yZ6.35, II, and V). As shown in Fig. 4(b), surprisingly, the two-component modeling conductivity reproduces the doping dependence
Fig. 3. (a) s1(u), (b) 1/t(u), and (c) W(u) at 65 K for yZ6.65. In (a), the dotted and the dot-dashed lines represent the Drude and the Lorentz oscillator modes, respectively. In (b) and (c), the open circle symbols represent results of the two component modeling (Model V).
portion of (p, 0) of Brillouin-zone gapped [32]. Similarly, the pseudogap is open near (p, 0) of Brillouin-zone [33]. These experimental observations indicate that in both states only the small patch of Fermi surface in the nodal region contributes to the conduction. It is likely that this topological similarity in the Fermi surface could lead to a general behavior of the two-component spectral feature in underdoped YBCO.
5. Doping dependence in loss function We finally discuss the doping dependence in the loss function Im½K1=~3ðuÞ. Fig. 4(a) displays the experimental
Fig. 4. Doping dependent loss functions Im½K1=~3ðuÞ from experiments (top) and two component modeling (bottom) at TZ 10 K (65 K for yZ6.65). Model I, II, and V correspond to the yZ 6.35, 6.43, and 6.65 YBCO, respectively.
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in Im½K1=~3M ðuÞ and the single peak structure for yZ6.65 as well. In the heavily underdoped region where the mid-IR absorption is well separated from the coherent mode, the position of peak I is nearly consistent with u0 in s2(u0)Z0, while peak II is identified without the corresponding zerocrossing. As the mid-IR absorption moves to lower frequencies with the increased doping, the optical constants of two components are significantly blended and the zerocrossing behavior is removed in spite of the existence of the coherent mode. This leads to the disappearance of peak I. Instead, the zero-crossing behavior by the mid-IR absorption occurs, so that peak II develops significantly. From this, one can say that the zero-crossing in 31(u) or the peak structure in Im½K1=~3ðuÞ can not be simply related to the screened plasma frequency in cuprates. Furthermore, the weak doping dependence in the position of Peak II could be attributed to the compensation between the increase in u2p;L and the decrease in u2c with doping.
6. Summary We demonstrated that optical responses from the heavily underdoped to the pseudogap states in YBCO can be successfully described with the two-component optical conductivity. According to the two-component description, the mid-IR absorption gradually shifts to low frequencies with the increased doping, while the coherent component below 600 cmK1 is rather common. While the low frequency suppression of 1/t(u) merely reflects narrowing of the coherent mode, its high frequency behavior depends strongly on the position of the mid-IR absorption. The peaks in Im½K1=~3ðuÞ near 1.0 eV in underdoped region are associated with the mid-IR absorption, not with the screened plasma frequency of charge carriers.
Acknowledgements We thank D.N. Basov for allowing us to use spectrometers at UCSD and discussing experimental results. We are grateful to Kouji Segawa and Yoichi Ando for providing us with a high quality of detwinned YBCO single crystals. This research was supported by the US department of Energy Grant and the NSF.
References [1] M. Imada, et al., Rev. Mod. Phys. 70 (1998) 1039. [2] J. Orenstein, et al., Science 288 (2000) 468 (and references therein). [3] T. Timusk, et al., in: D.M. Ginsberg (Ed.), Physical Properties of High Temperature Superconductors III. World Scientific, Singapore, 1992 (chapter 5). [4] J.W. Allen, et al., Phys. Rev. B 15 (1977) 2952. [5] B.C. Webb, et al., Phys. Rev. Lett. 57 (1986) 1951. [6] L.D. Rotter, et al., Phys. Rev. Lett. 67 (1991) 2741. [7] S.L. Cooper, et al., Phys. Rev. B 47 (1993) 8233. [8] Ar. Abanov, et al., Phys. Rev. Lett. 89 (2002) 177002. [9] E. Schachinger, et al., Phys. Rev. B 67 (2003) 214508. [10] J. Hwang, et al., Nature 427 (2004) 714. [11] D. van der Marel, et al., Nature 425 (2003) 271. [12] A.V. Puchkov, et al., J. Phys.: Condens. Matter 8 (1996) 10049. [13] D.N. Basov, et al., Phys. Rev. Lett. 77 (1996) 4090. [14] A. Ino, et al., Phys. Rev. B 65 (2002) 094504. [15] (a) D. Mihailovic, et al., Phys. Rev. B 57 (1998) 6116; (b) S. Lupi, et al., Phys. Rev. Lett. 83 (1999) 4852. [16] (a) D. Munzar, et al., Physica C 312 (1999) 121; (b) J.-H. Eom, et al., Phys. Rev. B 70 (2004) 024522. [17] S.A. Kivelson, et al., private communications. [18] Y.S. Lee, et al., Phys. Rev. B 70 (2004) 014518. [19] W.J. Padilla, et al., unpublished. [20] S. Uchida, et al., Phys. Rev. B 43 (1991) 7942. [21] M. Dumm, et al., Phys. Rev. Lett. 91 (2003) 077004. [22] K. Segawa, et al., Phys. Rev. Lett. 86 (2001) 4907. [23] P. Wachter, et al., Phys. Rev. B 49 (1994) 13164. [24] Z. Schlesinger, et al., Phys. Rev. Lett. 65 (1990) 801. [25] C.C. Homes, et al., Phys. Rev. B 60 (1999) 9782. [26] For the same reason, the peaks in 1/tM(u) are more prominent than those in experimental 1/t(u). [27] S.V. Shulga, et al., Physica C 178 (1991) 266. [28] P.B. Allen, Phys. Rev. B 3 (1971) 305. [29] J.P. Carbotte, et al., Nature 401 (1999) 354. [30] The parallel offset behavior in the 1/t(u) with T at optimal dopings (or above T*) is reproduced in a complicated way where the coherent mode strength increases but the mid-IR mode strength decreases at lower T. [31] (a) A.W. McConnel, et al., Phys. Rev. B 58 (1998) 13565; (b) Y.S. Lee, et al., Europhys. Lett. 55 (2001) 280; (c) Y.S. Lee, et al., Phys. Rev. B 64 (2001) 165109; (d) Y. Onose, et al., Phys. Rev. B 87 (2004) 217001. [32] T. Yoshida, et al., Phys. Rev. Lett. 91 (2003) 027001. [33] T. Timusk, B. Statt, Rep. Prog. Phys. 62 (1999) 61. [34] F. Wooten, Optical Properties of Solids, Academic Press, New York, 1972.
Doping dependent evolution of infrared spectra in underdoped YBa2Cu3Oy in terms of two-component optical conductivity Y.S. Lee* Department of Physics, University of California at San Diego, La Jolla, CA 92093-0319, USA Received 1 September 2004; received in revised form 31 October 2004; accepted 17 November 2004 by H. von Lo¨hneysen Available online 2 December 2004
Abstract In access to optical spectroscopy of heavily underdoped detwinned YBa2Cu3Oy (YBCO) crystals, we re-examine the doping dependent evolution of infrared spectra in the CuO2 plane of underdoped YBCO in terms of two-component optical conductivity. The extended Drude model analysis is applied to the two-component conductivity, and the results are compared with experimental data in the pseudogap state. We demonstrate that a model consisting of a Drude and Lorentz oscillator components reproduces characteristics of infrared spectra in underdoped YBCO. q 2004 Elsevier Ltd. All rights reserved. PACS: 74.25 Gz; 74.72 Bk Keywords: A. High-Tc superconductors; D. Optical properties
1. Introduction The generic property of high-Tc superconductivity evolving from an antiferromagnetic Mott insulating state by charge carrier doping has attracted much attention to the physics of doped Mott insulators [1]. To elucidate the origin of the unconventional superconductivity [2], it is essential to understand the nature of a conducting state derived from the Mott insulator and its evolution into the high-Tc superconducting state with doping. While optical spectroscopy has provided invaluable insights into the doping dependence of the electronic band structure and elementary excitations, one of intriguing properties in the ab-plane optical conductivity is a strong mid-infrared (IR) spectral feature [3]. To understand this enigmatic feature, two competitive approaches have been used. First is the ‘single-component picture’ viewing that the spectral feature below a charge transfer (CT) excitation is due to one kind of heavily * Tel.: C1 858 822 0149; fax: C1 858 534 0173. E-mail address: [email protected]. 0038-1098/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2004.11.023
damped carriers. In this picture, the strong mid-IR feature is attributed to the frequency dependence of the scattering rate 1/t (u) in the extended Drude model (EDM) description [4– 7], which is suggestive of peculiar interactions in cuprates, e.g. the electron–boson interaction [8–10] or quantum critical uctuation [11]. Especially in the underdoped region, the low frequency suppression in 1/t(u) has been considered as evidence of the pseudogap formation [12,13]. The second approach is the ‘two-component picture’ with a Drude-like coherent mode and a localized mid-IR absorption. While the former is responsible for the transport, the latter accounts for the mid-IR spectral weight. The origin of the mid-IR absorption might be an interband transition related to the midgap state [14], an incoherent mode due to electron–phonon or electron–spin interaction [15,16], inhomogeneity [17], etc. Very recently, it has been reported that for heavily underdoped detwinned YBa2Cu3Oy (YBCO) the optical conductivity in the CuO2 plane is comprised of the wellseparated Drude-like coherent mode and mid-IR absorption [18]. This two-component optical conductivity is also observed in other cuprates and is a general characteristic
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of heavily underdoped cuprates [19–21]. Starting from this finding, in this paper we review the doping dependent evolution of in-plane optical spectra in YBCO in terms of two-component optical conductivity. We apply the EDM analysis to the two-component conductivity feature in the heavily underdoped YBCO. Using a model consisting of two components, we examine the 1/t(u) behavior according to the position of the mid-IR absorption. The twocomponent analysis is extended to the pseudogap state. Finally, we discuss the doping dependence in the loss function.
2. Experiments and analysis method We investigated detwinned YBCO single crystals with oxygen content yZ6.35 (non-superconducting), 6.43 (Tcw13 K), and 6.65 (Tcw60 K) grown by a conventional flux method and detwinned under uniaxial pressure [22]. Reflectivity spectra R(u) at nearly normal incidence were measured with polarized light at frequencies from 20 to 48,000 cmK1 and at temperatures from 10 to 293 K. ~ The complex optical conductivity spectra sðuÞZ s1 ðuÞC is2 ðuÞZ ðiu=4pÞ½~3ðuÞK 3N were obtained from the measured R(u), using the Kramers-Kronig (KK) transformation. To investigate the physical properties of the CuO2 plane without any disturbance by the optical response of chain structures along the b-axis [22–25], we restrict our analysis to the a-axis properties in this paper. ~ According to the EDM [4,5], sðuÞZ ðu2p =4pÞ! ð1=½1=tðuÞK ium ðuÞÞ, where 1/t(u) and m*(u) are frequency dependent scattering rate and effective mass, respectively. The total plasma frequency up is estimated Ð from the sum rule u2p Z 8 0u s1 ðu 0 Þdu 0 at uZ10,000 cmK1. For the two-component optical conductivity modeling, we used a Drude mode s~ D ðuÞZ ðu2p;D =4pÞð1=½GD K iuÞÞ and Lorentz oscillator s~ D ðuÞZ ðu2p;L =4pÞðu=ð½iðu2c K u2 ÞC uGL Þ for the coherent and the mid-IR absorption modes, respectively, and thus the modeled conductivity takes the form s~ M ðuÞZ s~ D ðuÞC s~ L ðuÞZ ðu2p;M =4pÞð1=½1=tM ðuÞK iumM ðuÞÞ. Here, up,D and GD are plasma frequency and scattering rate of charge carriers, respectively. up,L, GL, and uc are the strength, the width, and the position of a Lorentz oscillator, respectively. Subscript M is for modeled quantities. The up,M value is estimated from the sum rule with s1,M(u). To account for the temperature dependence GD is reduced at low T with the other parameters unchanged.
3. Heavily underdoped state and two component modeling We first discuss optical responses in the heavily underdoped YBCO, exemplified by the yZ6.43 data (Fig. 1). In Fig. 1(a), apart from sharp peaks below 0.1 eV due to transverse optical phonon modes, the real part of optical
conductivity s1 (u) below the CT excitation near 14,000 cmK1 is composed of the two separate absorption features: a coherent mode (u!600 cmK1) followed by a mid-IR absorption at 0.5 eV [18]. The coherent mode significantly narrows with the decreased T, while the mid-IR absorption is virtually independent of T. Due to the existence of the coherent mode, the imaginary part of optical conductivity s2(u) is positive below a zero-crossing s2(u0)Z0 (Fig. 1(c)). At lower T, the zero-crossing behavior occurs at a higher frequency with the far-IR spectral weight enhanced. (According to the Drude model, ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u0 Z u2p;D =3N K G2 ) We apply the EDM analysis to these two-component conductivity features. As shown in Fig. 1(e), 1/t(u) at room temperature shows a broad feature with a peak around 1300 cmK1. With the decreased T, the far-IR 1/t(u) is significantly suppressed, while the mid-IR 1/t(u) is enhanced with the peak shifting to a higher frequency. The suppression of the far-IR 1/t(u) can be associated with the development of the coherent mode. However, the peak structure is inconsistent with the single-component description of the electromagnetic response in the context that the scattering incidence should increase with frequency and temperature. The non-monotonic dependence of 1/t(u) indicates the inapplicability of the EDM analysis to the heavily underdoped samples. The peak positions in 1/t(u) correspond nearly to the zero-crossing frequency u0 in s2(u). Because 1=tðu0 ÞZ ðu2p =4pÞð1=s1 ðu0 ÞÞ, the peak enhancement is attributed to the decrease in s1(u0) at lower T [26]. It is noted that the mid-IR absorption is featureless in 1/t(u). We demonstrate that these optical responses in the heavily underdoped YBCO are simply reproduced by the ~ two-component modeling. We fit the sðuÞ for yZ6.43 with a modeled conductivity s~ M ðuÞ consisting of Drude mode s~ D ðuÞ and Lorentz oscillator s~ L ðuÞ (Model II), which are represented by broken lines in Fig. 1(a). The fitting parameters are summarized in Table 1. Then, we apply the EDM analysis to the fitted s~ M ðuÞ and plot 1/tM(u). As shown in the right panel of Fig. 1, s~ M ðuÞ and 1/tM(u) are quite consistent with experimental results for yZ6.43. (Small differences at 500–2000 cmK1 are probably due to the imperfection of a single Lorentz oscillator for the mid-IR absorption in low frequency side) This agreement is not surprising given the experimental fact that the twocomponent optical conductivity feature is quite clear in the heavily underdoped region. To get more insight into the EDM description for the two-component optical conductivity, we examine the change of 1/tM(u) with respect to modification of fitting parameters for Model II. First, we consider the change of the spectral weight distribution. With the other parameters fixed, we increase the Drude weight by 90% and decrease the weight of the Lorentz oscillator by the same amount, preserving the total spectral weight unchanged (Model III). Results are displayed in the top panels of Fig. 2. While no
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Fig. 1. T-dependent s1(u) (top), s2(u) (middle), 1/t(u) (bottom) for yZ6.43. Left panels are for experimental results, and right panels are for results fitted by the two component modeling. Fitting parameters (Model II) are summarized in Table 1. In (a), the dotted and the dot-dashed lines represent a Drude mode and a Lorentz oscillator, respectively. In all panels, the thick (thin) lines represent the low (high) T data.
qualitative change is observed, the peak structure in 1/ tM(u) is weaker and shifts to a higher frequency. The hardening of the peak is attributed to the shift of u0 in s2,M(u0)Z0 to higher frequencies due to the increase in the Drude weight (inset of Fig. 2(a)). Because limu/0 1=tM ðuÞZ ðGD Þðu2p;M =u2p;D Þ, the limu/0 1=tM ðuÞ is smaller with the larger Drude weight. Second, we change the position of the mid-IR absorption. We shift the position of Lorentz oscillator from 3800 to 2000 cmK1 (Model IV).
As shown in the bottom panels of Fig. 2, notably, the peak structure in 1/tM(u) is significantly suppressed, and the slope at high frequencies becomes positive. The softening of the mid-IR absorption pushes the zero-crossing behavior in s2,M(u) above 4000 cmK1 (inset of Fig. 2(c)), which leads to the disappearance of the peak structure in 1/tM(u). The gross feature of 1/tM(u) in Model IV, including the far-IR suppression, is reminiscent of that in the pseudogap state of YBCO [12,13]. It is noted that even the two-component
Table 1 Summary of fitting parameters for two component conductivity s~ M ðuÞ; the plasma frequency up,D and the scattering rate GD at low (high) T for s~ D ðuÞ, the strength up,L, the width GL, and the position uc for s~ L ðuÞ, and the total plasma frequency up,M for 1/tM(u). The unit is cmK1 Model
up,D
GD
up,L
uc
GL
up,M
y
I II III IV V
4100 5250 7250 5250 8000
280(950) 135(750) 135 135 60(700)
10,850 13,550 11,850 13,550 15,000
4200 3800 3800 2000 1600
7000 7000 7000 7000 5000
7000 12,000 12,000 12,000 19,000
6.35 6.43
6.65
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Fig. 2. Left panels: s1,M(u) for (a) Model III and (c) Model IV. In Model III, the Drude plasma frequency up,D is increased to be 7250 cmK1. In Model IV, the Lorentz oscillator is shifted to 2000 cmK1. The dotted and the dot-dashed lines represent the Drude and the Lorentz oscillator modes, respectively. Right panels: 1/tM(u) for (b) Model III and (d) Model IV. Insets show s2,M(u). Model III and IV are representing by the thick lines. For comparison, Model II is displayed (thin lines).
conductivity can exhibit phenomenologically reasonable 1/ t(u) with appropriately low frequency of the mid-IR absorption.
4. Underdoped (pseudogap) state We now turn to the pseudogap state of the underdoped YBCO. Fig. 3 shows optical responses for yZ6.65, where the pseudogap onset temperature T* is estimated to be above 300 K from the dc resistivity measurement [22]. With the decreased T, the s1(u) shows the development of a narrow peak at zero frequency from a broad single component-like feature at high T. It is most evident that at TRTc a peak structure near 0.2 eV is separated from the well-developed Drude-like far-IR conductivity. Coincidently, the far-IR 1/ t(u) is suppressed deviating from the nearly linear udependence at high frequencies, which has been considered as evidence of the pseudogap formation (Fig. 3(b)) [12,13]. On the basis of the electron–boson scattering theory [27], this gap-like feature in 1/t(u) could be suggestive of a strong coupling to the corresponding bosonic mode. The bosonic spectral density a2F(u) is estimated to be a2 FðuÞ z WðuÞZ ð1=2pÞðd2 =du2 ½u=tðuÞÞ [28,29]. Actually, W(u) at 65 K for yZ6.65 exhibits a distinct resonance, as shown in Fig. 3(c). It has been suggested that the resonance in W(u) should be associated with the magnetic resonance observed in inelastic neutron scattering experiments [8–10].
It is noted that all characteristic features in the pseudogap state of YBCO can be well-described with the twocomponent modeling. We fit the yZ6.65 YBCO data with s~ M ðuÞ (Model V). We use the Lorentz oscillator at 1600 cmK1 for the peak structure in the mid-IR. As shown in Fig. 3(b), the T-dependent 1/tM(u) (open circles) are quite consistent with experimental results. Naturally, the low frequency suppression in 1/t(u) is associated with the development of the coherent mode, while no significant Tdependence at higher frequencies is suggestive of the existence of the mid-IR mode which is essentially independent of temperature [30]. Even the WM(u) estimated from 1/tM(u) is comparable to that obtained from the experiment, which might imply that the resonance in W(u) is just an artificial by-product from the blended optical conductivity. These findings suggest that the optical response in the pseudogap state of YBCO be interpreted with the two-component optical conductivity and the pseudogap formation be accompanied by the emergence of the mid-IR absorption from the single band optical response. Interestingly, these behaviors have been commonly observed in systems with partial gap opening [31]. It is worthwhile to note that the Fermi surface structure is quite similar in the heavily underdoped and the pseudogap states. According to the recent angle resolved photoemission spectroscopy results on La2KxSrxCuO4, in the heavily underdoped region the Fermi surface is confined to the nodal region (diagonals of Brillouin-zone) with a large
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Im½K1=~3ðuÞ spectra for yZ6.35, 6.43 and 6.65 at lowest temperature (or TRTc). In heavily underdoped region (yZ 6.35 and 6.43) one can find two peak structure; peak I near 2000 cmK1 and peak II above 5000 cmK1. A Drude or Lorentz oscillator mode is represented by a peak in Im½K1=~3ðuÞ, whose position is associated with the spectral weight and the position of the mode: In condition that pffiffiffiffiffi u[G D (or GL), qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u zup;D = 3N for the Drude mode and u z u2c C u2p;L =3N for the Lorentz oscillator [34]. Naturally, two peak structure in Im½K1=~3ðuÞ are related to the twocomponent optical conductivity; Peak I and Peak II correspond to the Drude mode and the Lorentz oscillator, respectively. With the increased doping Peak I is significantly suppressed and finally disappears for yZ 6.65, whereas peak II is enhanced and shifted to higher frequencies [20]. The peak near 1.0 eV in Im½K1=~3ðuÞ has been considered to be related to the screened plasma frequency for charge carriers for some cuprates [3]. However, the gradual evolution of Im½K1=~3ðuÞ with doping suggests that the single peak structure for yZ6.65 might be associated with the mid-IR absorption. To obtain more understanding, we plot the modeled Im½K1=~3M ðuÞ spectra with the results of twocomponent modeling (Model I for yZ6.35, II, and V). As shown in Fig. 4(b), surprisingly, the two-component modeling conductivity reproduces the doping dependence
Fig. 3. (a) s1(u), (b) 1/t(u), and (c) W(u) at 65 K for yZ6.65. In (a), the dotted and the dot-dashed lines represent the Drude and the Lorentz oscillator modes, respectively. In (b) and (c), the open circle symbols represent results of the two component modeling (Model V).
portion of (p, 0) of Brillouin-zone gapped [32]. Similarly, the pseudogap is open near (p, 0) of Brillouin-zone [33]. These experimental observations indicate that in both states only the small patch of Fermi surface in the nodal region contributes to the conduction. It is likely that this topological similarity in the Fermi surface could lead to a general behavior of the two-component spectral feature in underdoped YBCO.
5. Doping dependence in loss function We finally discuss the doping dependence in the loss function Im½K1=~3ðuÞ. Fig. 4(a) displays the experimental
Fig. 4. Doping dependent loss functions Im½K1=~3ðuÞ from experiments (top) and two component modeling (bottom) at TZ 10 K (65 K for yZ6.65). Model I, II, and V correspond to the yZ 6.35, 6.43, and 6.65 YBCO, respectively.
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in Im½K1=~3M ðuÞ and the single peak structure for yZ6.65 as well. In the heavily underdoped region where the mid-IR absorption is well separated from the coherent mode, the position of peak I is nearly consistent with u0 in s2(u0)Z0, while peak II is identified without the corresponding zerocrossing. As the mid-IR absorption moves to lower frequencies with the increased doping, the optical constants of two components are significantly blended and the zerocrossing behavior is removed in spite of the existence of the coherent mode. This leads to the disappearance of peak I. Instead, the zero-crossing behavior by the mid-IR absorption occurs, so that peak II develops significantly. From this, one can say that the zero-crossing in 31(u) or the peak structure in Im½K1=~3ðuÞ can not be simply related to the screened plasma frequency in cuprates. Furthermore, the weak doping dependence in the position of Peak II could be attributed to the compensation between the increase in u2p;L and the decrease in u2c with doping.
6. Summary We demonstrated that optical responses from the heavily underdoped to the pseudogap states in YBCO can be successfully described with the two-component optical conductivity. According to the two-component description, the mid-IR absorption gradually shifts to low frequencies with the increased doping, while the coherent component below 600 cmK1 is rather common. While the low frequency suppression of 1/t(u) merely reflects narrowing of the coherent mode, its high frequency behavior depends strongly on the position of the mid-IR absorption. The peaks in Im½K1=~3ðuÞ near 1.0 eV in underdoped region are associated with the mid-IR absorption, not with the screened plasma frequency of charge carriers.
Acknowledgements We thank D.N. Basov for allowing us to use spectrometers at UCSD and discussing experimental results. We are grateful to Kouji Segawa and Yoichi Ando for providing us with a high quality of detwinned YBCO single crystals. This research was supported by the US department of Energy Grant and the NSF.
References [1] M. Imada, et al., Rev. Mod. Phys. 70 (1998) 1039. [2] J. Orenstein, et al., Science 288 (2000) 468 (and references therein). [3] T. Timusk, et al., in: D.M. Ginsberg (Ed.), Physical Properties of High Temperature Superconductors III. World Scientific, Singapore, 1992 (chapter 5). [4] J.W. Allen, et al., Phys. Rev. B 15 (1977) 2952. [5] B.C. Webb, et al., Phys. Rev. Lett. 57 (1986) 1951. [6] L.D. Rotter, et al., Phys. Rev. Lett. 67 (1991) 2741. [7] S.L. Cooper, et al., Phys. Rev. B 47 (1993) 8233. [8] Ar. Abanov, et al., Phys. Rev. Lett. 89 (2002) 177002. [9] E. Schachinger, et al., Phys. Rev. B 67 (2003) 214508. [10] J. Hwang, et al., Nature 427 (2004) 714. [11] D. van der Marel, et al., Nature 425 (2003) 271. [12] A.V. Puchkov, et al., J. Phys.: Condens. Matter 8 (1996) 10049. [13] D.N. Basov, et al., Phys. Rev. Lett. 77 (1996) 4090. [14] A. Ino, et al., Phys. Rev. B 65 (2002) 094504. [15] (a) D. Mihailovic, et al., Phys. Rev. B 57 (1998) 6116; (b) S. Lupi, et al., Phys. Rev. Lett. 83 (1999) 4852. [16] (a) D. Munzar, et al., Physica C 312 (1999) 121; (b) J.-H. Eom, et al., Phys. Rev. B 70 (2004) 024522. [17] S.A. Kivelson, et al., private communications. [18] Y.S. Lee, et al., Phys. Rev. B 70 (2004) 014518. [19] W.J. Padilla, et al., unpublished. [20] S. Uchida, et al., Phys. Rev. B 43 (1991) 7942. [21] M. Dumm, et al., Phys. Rev. Lett. 91 (2003) 077004. [22] K. Segawa, et al., Phys. Rev. Lett. 86 (2001) 4907. [23] P. Wachter, et al., Phys. Rev. B 49 (1994) 13164. [24] Z. Schlesinger, et al., Phys. Rev. Lett. 65 (1990) 801. [25] C.C. Homes, et al., Phys. Rev. B 60 (1999) 9782. [26] For the same reason, the peaks in 1/tM(u) are more prominent than those in experimental 1/t(u). [27] S.V. Shulga, et al., Physica C 178 (1991) 266. [28] P.B. Allen, Phys. Rev. B 3 (1971) 305. [29] J.P. Carbotte, et al., Nature 401 (1999) 354. [30] The parallel offset behavior in the 1/t(u) with T at optimal dopings (or above T*) is reproduced in a complicated way where the coherent mode strength increases but the mid-IR mode strength decreases at lower T. [31] (a) A.W. McConnel, et al., Phys. Rev. B 58 (1998) 13565; (b) Y.S. Lee, et al., Europhys. Lett. 55 (2001) 280; (c) Y.S. Lee, et al., Phys. Rev. B 64 (2001) 165109; (d) Y. Onose, et al., Phys. Rev. B 87 (2004) 217001. [32] T. Yoshida, et al., Phys. Rev. Lett. 91 (2003) 027001. [33] T. Timusk, B. Statt, Rep. Prog. Phys. 62 (1999) 61. [34] F. Wooten, Optical Properties of Solids, Academic Press, New York, 1972.