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Low-energy excitations and stripes in superconducting cuprate La1.87Sr0.13CuO4
Solid State Communications 151 (2011) 1681–1685
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Low-energy excitations and stripes in superconducting cuprate La1.87 Sr0.13 CuO4 B.P. Gorshunov a,b,∗ , A.A. Voronkov a,b , V.S. Nozdrin a , E.S. Zhukova a,b,∗∗ , T. Matsuoka c , K. Tanaka c , S. Miyasaka c , S. Tajima c , M. Dressel d a
A.M. Prokhorov Institute of General Physics, Russian Academy of Sciences, Vavilov str. 38, 119991 Moscow, Russia
b
Moscow Institute of Physics and Technology (State University), Institutskii lane 9, 141700, Dolgoprudnyi, Russia
c
Department of Physics, Osaka University, Osaka 560-0043, Japan
d
1. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany
article
info
Article history: Received 29 June 2011 Accepted 2 August 2011 by Y.E. Lozovik Available online 7 August 2011 Keywords: A. Cuprates D. Superconductivity E. Terahertz spectroscopy
abstract The conductivity and dielectric permittivity spectra of single-crystalline La1.87 Sr0.13 CuO4 are directly measured with the electric field polarized perpendicular to the CuO planes (E ‖ c) covering the frequency range 10–40 cm−1 and temperatures 5–300 K. We observe in the superconducting state a well pronounced excitation with strongly temperature dependent parameters. We suggest that the excitation is caused by the transverse Josephson plasma mode that appears due to the different strengths of Josephson coupling between the superconducting charge stripes in the neighboring and next-nearest neighboring copper–oxygen planes of La1.87 Sr0.13 CuO4 . A strongly enhanced low-frequency (below 15 cm−1 ) absorption is seen in the superconducting state that is assigned to delocalized quasiparticles of as yet unknown origin. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Enormous activity is devoted currently to the study of the inplane electronic properties of the high-Tc superconductors (Cu and Fe based), but a lot of attention is also paid to the properties observed along the crystallographic axis c. Investigating the peculiar behaviors of the out-of-plane response of carriers in the normal and superconducting (SC) states delivers information on the mechanism of high-temperature superconductivity, on specific magnetic and charge-ordering phenomena within the layers and on new kinds of excitations, like the transverse Josephson plasma mode in bi-layer superconductors [1,2]. It turned out that the charge dynamics across the SC copper–oxygen (CuO) layers of cuprates is effectively probed by c-axis reflection measurements. In the spectra a characteristic plasma edge is observed in the farinfrared (FIR) range that results from a plasma-like longitudinal optical (LO) response of the Cooper pairs; they become delocalized along the c-axis due to Josephson coupling between the CuO layers [3–7]. It is interesting that in La2−x (Sr, Ba)x CuO4 samples
∗ Corresponding author at: A.M. Prokhorov Institute of General Physics, Russian Academy of Sciences, Vavilov str. 38, 119991 Moscow, Russia. Tel.: +7 499 503 82 12. ∗∗ Corresponding author. E-mail addresses: [email protected] (B.P. Gorshunov), [email protected] (E.S. Zhukova), [email protected] (S. Tajima), [email protected] (M. Dressel). 0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.08.003
the c-axis coherence can be most easily disturbed for concentrations x around 1/8 (Refs. [3–6,8]). This phenomenon is associated with the formation of quasi-static or fluctuating stripes, i.e. spatially alternating within the CuO planes elongated areas of excessive charges and ordered spins [9]. The stripe structure strongly influences the interlayer response. Steady-state FIR reflectivity and time-domain measurements on the La2−x Srx CuO4 family [3–6,8] indicate a strongly temperature, frequency and composition dependent c-axis response of the compounds at terahertz and subterahertz (THz, sub-THz) frequencies. In this work we concentrate on the origin of the corresponding low-energy excitations by directly measuring the c-axis spectra of the conductivity σ (ν) and permittivity ϵ(ν) of single-crystalline La1.87 Sr0.13 CuO4 at frequencies ν = 10–40 cm−1 and at temperatures 5–300 K. 2. Experimental results and analysis La1.87 Sr0.13 CuO4 high-quality single crystals with the onset temperature of the SC transition Tc = 36 K were prepared using a travelling solvent floating zone method. We prepared a planeparallel plate of thickness of 25 µm and an area of about 5 × 5 mm2 with the c axis lying in the plane of the sample. For optical experiments we used a spectrometer based on monochromatic and continuously frequency tunable radiation sources—backwardwave oscillators [10]. In Fig. 1(a) and (b) we present the measured σ (ν) and ϵ(ν) spectra. In the normal state, σ (ν) and ϵ(ν), together with the reflectivity R(ν) (evaluated from measured σ (ν) and
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B.P. Gorshunov et al. / Solid State Communications 151 (2011) 1681–1685
a
b
c
Fig. 1. Spectra of (a) the conductivity σ (ν), (b) the permittivity ϵ(ν) and (c) the reflection coefficient R (calculated on the basis of measured σ (ν) and ϵ(ν))) of La1.87 Sr0.13 CuO4 single crystal measured at different temperatures for polarization E ‖ c. Lines represent the results of fittings as described in the text. The inset in panel (b) shows the temperature dependences of the normal state parameters of the charge carriers: the plasma frequency νpl and scattering rate γ .
ϵ(ν)), exhibit the behavior typical for a (poor) conductor in the low-frequency (Hagen–Rubens) regime. We detect a noticeable decrease of the conductivity towards higher frequencies, as seen in the σ (ν, T = 50 K) spectrum of Fig. 1(a). Relying on this dispersion, we estimate the parameters of the carrier (hole) condensate in the normal state by fitting σ (ν), ϵ(ν) and R(ν) with the corresponding expressions of the Drude model, where the complex conductivity is expressed as [11]
σ ∗ = σ (ν) + iσ2 (ν) =
σ0 νγ σ0 γ 2 +i 2 . γ 2 + ν2 γ + ν2
(1)
Here σ2 = ν(ϵ∞ − ϵ)/2 is the imaginary part of the conductivity, ϵ∞ is the high-frequency dielectric constant, σ0 = νp2 /2γ is the dc
conductivity, νp = (Ne2 /π mc 2 )1/2 denotes the plasma frequency of the quasi-free carriers whose concentration is N, with charge e and effective mass m. The temperature variations obtained for the scattering rate γ and the plasma frequency νp of carriers are plotted in the inset of Fig. 1(b). Both quantities slightly decrease upon cooling. The scattering rate is more than two times smaller than the plasma frequency, indicating that in the non-SC state of La1.87 Sr0.13 CuO4 the charge transport across the CuO planes is not overdamped. According to Fig. 1, all spectra significantly change upon entering the SC state of La1.87 Sr0.13 CuO4 . In the permittivity spectrum a strong dispersion of the type
ϵ(ν) = −(νpSC /ν)2
(2)
is observed that is mainly caused by an inductive response of the SC δ -function at ν = 0 in the conductivity spectrum [11]. The spectral strength of this response is given by the value of the plasma frequency νpSC of the SC condensate. In the conductivity spectra plotted in Fig. 1(a), a well defined peak is observed below Tc . At T < 10 K the peak moves outside of our frequency interval, to approximately 47 cm−1 at T = 5 K as
seen in the time-domain spectroscopic data obtained for a crystal from the same batch [12]. Another observation from Fig. 1(a) is that in the SC state the c-axis conductivity of La1.87 Sr0.13 CuO4 significantly increases at the lowest frequencies. This excess conductivity (absorption) component exhibits a notable temperature evolution that reflects the sub-gap quasiparticle dynamics in La1.87 Sr0.13 CuO4 . The component has a Drude-like spectral shape and acquires larger amplitude at lower temperatures, as seen in Fig. 1(a). In this regard, the increase of the sub-gap ac conductivity resembles the behavior of the Drude-like component that is well-known from the in-plane conductivity spectra of the cuprates [13,14]. We have fitted the ϵ(ν) and σ (ν) spectra with the peak in the conductivity spectra modeled by a Lorentzian, the strength of the SC δ -function given by its inductive response (Eq. (2)) and the quasiparticle below-gap absorption described by the Drude term. The quasiparticles produce a contribution to the permittivity of the same type as the SC δ -function, 1ϵ QP ∼ −(νpQP /ν)2 , that can become comparable to the dielectric contribution of the SC δ function, given by Eq. (1). (We use (νpQP )2 to denote the spectral weight (strength) of the sub-gap absorption.) This means that a fit of the permittivity spectrum with expression (2) has to consider also the contribution 1ϵ QP . Analogous situations occur also with the in-plane optical response of cuprates [15,14]. The role of this absorption is crucial for determining the SC plasma frequency in the temperature range from 28–30 K up to Tc . At lower temperatures, the SC δ -function acquires its full strength: νpSC ≫ νpQP ; the value of the magnetic field penetration depth at the lowest temperatures, λL (T → 0) = 8.5 µm, is practically unaffected by the quasiparticle term and is in agreement with the literature data (see Ref. [14] and references therein). 3. Discussion To understand the origin of the low-frequency conductivity peak, we recall the idea proposed in Ref. [16]: in the layered superconductors with different Josephson-coupled layers in a unit cell there should exist more than one LO Josephson plasma excitations that propagate in the direction perpendicular to the layers and a transverse optical (TO) plasma mode that has its wavevector parallel to the layers [17–20]. The transverse Josephson plasma resonance is seen as a well defined mode in the c-axis THz conductivity spectra of SmLa2−x Srx CuO4 [2,21–23]. Several publications report the observation of similar low-energy TO modes in the c-axis conductivity spectra also in a singlelayer compound La2−x Srx CuO4 [6,24]; the origin of the peak was assigned as a transverse Josephson resonance, although no corresponding microscopic mechanism was suggested. We believe that the low-energy resonances seen in the c-axis conductivity spectra of our La1.87 Sr0.13 CuO4 sample and in the La2−x Srx CuO4 crystals with other strontium concentrations and dopings [6,12,25,24] are common in origin and come from transverse excitations of the Josephson plasma of Cooper pairs. We suggest that the conditions for appearance of these TO modes in the single-layer La2−x Srx CuO4 are provided by the charge and magnetic stripes within the crystallographic CuO planes [6,7,26–32]. Since the sizes of magnetically ordered ‘‘islands’’, that are of the order of 100–600 Å according to Refs. [6,29,31], exceed by far the interlayer lattice constant and the SC correlation length in the La2−x Srx CuO4 family, one readily accepts interaction between magnetic stripes in different CuO layers. Similar interactions should exist also between the charge stripes, as has been confirmed by X-ray experiments on NdLSCO [32]. Under conditions when the direction of stripes changes by 90° with respect to the neighboring CuO layers [26], we expect noticeable spatial modulation of the Josephson coupling strength between the SC charge stripes, as is
B.P. Gorshunov et al. / Solid State Communications 151 (2011) 1681–1685
a
b
c-axis
CuO – planes
Coupling strength, tunnel current
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c Fig. 2. Sketch of the charge stripes of La1.87 Sr0.13 CuO4 crystal, cut in the c (a–b) plane. The stripes, whose direction changes by 90° in the neighboring CuO layers [26], are shown by black lines and dots, correspondingly. The spatial modulation of the strength of the Josephson coupling and of the corresponding interlayer supercurrent is indicated by different shadings.
indicated schematically in Fig. 2: the amplitude has to be larger for the coupling between the stripes in the neighboring planes (the area shown with solid-line oval in Fig. 2) and smaller for the coupling between the stripes in the next-nearest planes (the dashedline oval). As a result, we obtain differently Josephson-coupled SC layers within the unit cell of La2−x Srx CuO4 —the condition needed for the appearance of the TO Josephson resonance [16]. Our assignment of the low-frequency mode in the c-axis conductivity spectra of La1.87 Sr0.13 CuO4 to the transverse Josephson plasma resonance is confirmed by detailed inspection of the temperature dependent spectra of the conductivity, loss function and reflection coefficient presented in Fig. 3. It is seen that at temperatures T ≈ 29–30 K and up to Tc , there exists only one broad peak in the loss function that can be well fitted (the solid line in the T = 30 K panel of Fig. 3) with the model of Ref. [16], assuming identical Josephson junctions in the unit cell. At lower temperatures, however, it becomes impossible to describe the spectrum of the loss function with a single longitudinal excitation. As demonstrated by the T = 26 K spectra, two LO modes should be involved now, corresponding to the weight factors of 0.36 and 0.64 (Ref. [16]) for the two Josephson junctions. At those temperatures the maximum in the conductivity spectra is also clearly pronounced and is well fitted with the model of Ref. [16] (solid line). We see that as in the case of SmLa2−x Srx CuO4 , the transverse resonance is found between the two LO plasmons: νLO1 ≈ 29 cm−1 < νTO ≈ 32 cm−1 < νLO2 ≈ 32 cm−1 (T = 26 K). Also the temperature behaviors of the transverse mode parameters in La1.87 Sr0.13 CuO4 and in SmLa2−x Srx CuO4 are quite similar (Fig. 4). Considerably higher values of the eigenfrequency and oscillator strength of the mode in La1.87 Sr0.13 CuO4 are simply due to the larger c-axis superfluid density and correspondingly smaller London penetration depth: λL (T → 0) ≈ 8.5 µm in La1.87 Sr0.13 CuO4 compared to λL (T → 0) ≈ 38 µm in SmLa0.85 Sr0.15 CuO4 [2]. When comparing the TO plasma modes in our La1.87 Sr0.13 CuO4 and in SmLa2−x Srx CuO4 [2], another distinction becomes noticeable. The LO and TO peaks in SmLSCO are pronounced very clearly [2,33], while in LSCO the corresponding structures in the conductivity, loss function and reflectivity spectra are seriously smeared (Fig. 3); also the damping parameter of the TO mode in LSCO is several times larger than that in SmLSCO. We believe that the reasons are of fundamental importance: while in SmLSCO the double-layered structure of the array of Josephson junctions is clearly provided ‘‘on a crystallographic level’’ by periodically sandwiched insulating fluorite and a rocksalt blocks, an appreciable
Fig. 3. Temperature dependences of the parameters of the transverse optical resonance seen in the conductivity spectra of La1.87 Sr0.13 CuO4 for polarization E ‖ c: 2 frequency position νTO (a), damping γTO (b) and spectral weight fTO = 1ϵ · νTO (1ϵ : dielectric contribution of the resonance). The 5 K points are obtained with the time-domain spectrometer for a crystal from the same batch. Open dots present the corresponding data for the SmLa2−x Srx CuO4 crystal from [2]. Lines are guides to the eye.
disorder is naturally expected in the system of charge and magnetic stripes of LSCO that should lead to correspondingly disordered Josephson tunneling amplitudes. Finally, we discuss possible origins of the below-gap quasiparticle absorption observed in the c-axis response of superconducting La1.87 Sr0.13 CuO4 . The way that it shows up in our conductivity spectra (Fig. 1 is reminiscent of that for the so-called below-gap Drude term seen in the in-plane conductivity spectra in all high-Tc cuprates [13]. The in-plane quasiparticle absorption originates from the d-wave character of the in-plane order parameter, when, due to the zero SC gap values along certain directions in the reciprocal space, the thermal energy or the energy of the probing electromagnetic radiation can generate quasiparticles even far below Tc . One could think of the same mechanism as can produce low-frequency absorption also in the c-direction in La1.87 Sr0.13 CuO4 . In this case one would expect a reduction of both the scattering rate γ QP and the plasma frequency νpQP of the quasiparticle peak on lowering the temperature (see, for example, [14]). According to the estimates we made from our data, however, both quantities are at least unchanged: γ QP ≈ 7 cm−1 at QP T = 5–35 K and νpl ≈ 100 cm−1 at T = 20–33 K. We thus have to assume some other source of quasiparticles to be present in non-superconducting regions in the superconducting state of La1.87 Sr0.13 CuO4 . The interlayer transport of these quasiparticles can be realized via interplane channels that could occur, for example, as a result of c-axis elongated chains of dipoles formed by ordered strontium ions and holes ‘‘ascribed’’ to them, as described in [34]. To learn more about the origin of quasiparticles, and the interplay between the c-directional correlations in the stripe subsystem and the low-frequency c-axis excitations in the La2−x Srx CuO4 family, further THz and sub-THz measurements on crystals with other compositions and dopings are needed.
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Fig. 4. Spectra of the conductivity (gray; yellow online; thick lines are guides to the eye), loss function (closed dots) and reflection coefficient (open dots; Y -scale from 0 to 1.1) of La1.87 Sr0.13 CuO4 single crystal for polarization E ‖ c at different temperatures. Lines shown for T = 26 K and 30 K are calculations according to the model of Ref. [16]. Dashed lines shown for T = 26 K represent two longitudinal plasmons needed to describe the loss function with the model of Ref. [16]. The bottom right panel shows the conductivity and reflectivity spectra combined with the spectra (dashed lines) obtained with the time-domain spectrometer for the La1.87 Sr0.13 CuO4 crystal from the same batch [24].
4. Conclusion Direct measurements of the terahertz and subterahertz
(10–40 cm−1 ) c-axis conductivity and dielectric permittivity of single-crystalline superconducting cuprate La1.87 Sr0.13 CuO4 reveal a transverse optical excitation in the superconducting state that is ascribed to the transverse Josephson plasma mode. We suggest that the mode originates from the interplane interaction of the charge and magnetic stripes that leads to modulation of the strength of the Josephson coupling between the copper–oxygen planes along the c-axis within the unit cell of La1.87 Sr0.13 CuO4 . For the superconducting state, a strong quasiparticle absorption is detected below 15 cm−1 that is assigned to the c-axis transport of normal charge carriers through the channels that connect the nonsuperconducting areas of as yet unknown origin. Acknowledgments The work was supported by the Russian Academy of Sciences Program for Fundamental Research ‘‘Strongly correlated electrons
in solids and solid structures’’, and Federal Program ‘‘Scientific and Educational Human Resources of Innovative Russia’’. Part of the work was supported by the Deutsche Forschungsgemeinschaft (DFG). References [1] D. van der Marel, A. Tsvetkov, Czech. J. Phys. 46 (1996) 3165. [2] T. Kakeshita, S. Uchida, K.M. Kojima, S. Adachi, S. Tajima, B. Gorshunov, M. Dressel, Phys. Rev. Lett. 86 (2001) 4140. [3] S.V. Dordevic, S. Komiya, Y. Ando, Y.J. Wang, D.N. Basov, Phys. Rev. B 71 (2005) 054503. [4] S. Tajima, T. Noda, H. Eisaki, S. Uchida, Phys. Rev. Lett. 86 (2001) 500. [5] A.A. Schafgans, A.D. LaForge, S.V. Dordevic, M.M. Qazilbash, W.J. Padilla, K.S. Burch, Z.Q. Li, S. Kimiya, Y. Ando, D.N. Basov, Phys. Rev. Lett. 104 (2010) 157002. [6] S.V. Dordevic, S. Komiya, Y. Ando, D.N. Basov, Phys. Rev. Lett. 91 (2003) 167401. [7] J. Fink, E. Schierle, E. Weschke, J. Geck, D. Hawthorn, V. Soltwisch, H. Wadati, Hsueh-Hung Wu, H.A. Dürr, N. Wizent, B. Büchner, G.A. Sawatzky, Phys. Rev. B 79 (2009) 100502R. [8] T. Valla, A.V. Fedorov, J. Lee, J.C. Davis, G.D. Gu, Science 314 (2006) 1914. [9] J. Orenstein, A.J. Millis, Science 288 (2000) 468. [10] B. Gorshunov, A. Volkov, I. Spektor, A. Prokhorov, A. Mukhin, M. Dressel, S. Uchida, A. Loidl, Int. J. Infrared Millim. Waves 26 (2005) 1217.
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Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Low-energy excitations and stripes in superconducting cuprate La1.87 Sr0.13 CuO4 B.P. Gorshunov a,b,∗ , A.A. Voronkov a,b , V.S. Nozdrin a , E.S. Zhukova a,b,∗∗ , T. Matsuoka c , K. Tanaka c , S. Miyasaka c , S. Tajima c , M. Dressel d a
A.M. Prokhorov Institute of General Physics, Russian Academy of Sciences, Vavilov str. 38, 119991 Moscow, Russia
b
Moscow Institute of Physics and Technology (State University), Institutskii lane 9, 141700, Dolgoprudnyi, Russia
c
Department of Physics, Osaka University, Osaka 560-0043, Japan
d
1. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany
article
info
Article history: Received 29 June 2011 Accepted 2 August 2011 by Y.E. Lozovik Available online 7 August 2011 Keywords: A. Cuprates D. Superconductivity E. Terahertz spectroscopy
abstract The conductivity and dielectric permittivity spectra of single-crystalline La1.87 Sr0.13 CuO4 are directly measured with the electric field polarized perpendicular to the CuO planes (E ‖ c) covering the frequency range 10–40 cm−1 and temperatures 5–300 K. We observe in the superconducting state a well pronounced excitation with strongly temperature dependent parameters. We suggest that the excitation is caused by the transverse Josephson plasma mode that appears due to the different strengths of Josephson coupling between the superconducting charge stripes in the neighboring and next-nearest neighboring copper–oxygen planes of La1.87 Sr0.13 CuO4 . A strongly enhanced low-frequency (below 15 cm−1 ) absorption is seen in the superconducting state that is assigned to delocalized quasiparticles of as yet unknown origin. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Enormous activity is devoted currently to the study of the inplane electronic properties of the high-Tc superconductors (Cu and Fe based), but a lot of attention is also paid to the properties observed along the crystallographic axis c. Investigating the peculiar behaviors of the out-of-plane response of carriers in the normal and superconducting (SC) states delivers information on the mechanism of high-temperature superconductivity, on specific magnetic and charge-ordering phenomena within the layers and on new kinds of excitations, like the transverse Josephson plasma mode in bi-layer superconductors [1,2]. It turned out that the charge dynamics across the SC copper–oxygen (CuO) layers of cuprates is effectively probed by c-axis reflection measurements. In the spectra a characteristic plasma edge is observed in the farinfrared (FIR) range that results from a plasma-like longitudinal optical (LO) response of the Cooper pairs; they become delocalized along the c-axis due to Josephson coupling between the CuO layers [3–7]. It is interesting that in La2−x (Sr, Ba)x CuO4 samples
∗ Corresponding author at: A.M. Prokhorov Institute of General Physics, Russian Academy of Sciences, Vavilov str. 38, 119991 Moscow, Russia. Tel.: +7 499 503 82 12. ∗∗ Corresponding author. E-mail addresses: [email protected] (B.P. Gorshunov), [email protected] (E.S. Zhukova), [email protected] (S. Tajima), [email protected] (M. Dressel). 0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.08.003
the c-axis coherence can be most easily disturbed for concentrations x around 1/8 (Refs. [3–6,8]). This phenomenon is associated with the formation of quasi-static or fluctuating stripes, i.e. spatially alternating within the CuO planes elongated areas of excessive charges and ordered spins [9]. The stripe structure strongly influences the interlayer response. Steady-state FIR reflectivity and time-domain measurements on the La2−x Srx CuO4 family [3–6,8] indicate a strongly temperature, frequency and composition dependent c-axis response of the compounds at terahertz and subterahertz (THz, sub-THz) frequencies. In this work we concentrate on the origin of the corresponding low-energy excitations by directly measuring the c-axis spectra of the conductivity σ (ν) and permittivity ϵ(ν) of single-crystalline La1.87 Sr0.13 CuO4 at frequencies ν = 10–40 cm−1 and at temperatures 5–300 K. 2. Experimental results and analysis La1.87 Sr0.13 CuO4 high-quality single crystals with the onset temperature of the SC transition Tc = 36 K were prepared using a travelling solvent floating zone method. We prepared a planeparallel plate of thickness of 25 µm and an area of about 5 × 5 mm2 with the c axis lying in the plane of the sample. For optical experiments we used a spectrometer based on monochromatic and continuously frequency tunable radiation sources—backwardwave oscillators [10]. In Fig. 1(a) and (b) we present the measured σ (ν) and ϵ(ν) spectra. In the normal state, σ (ν) and ϵ(ν), together with the reflectivity R(ν) (evaluated from measured σ (ν) and
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B.P. Gorshunov et al. / Solid State Communications 151 (2011) 1681–1685
a
b
c
Fig. 1. Spectra of (a) the conductivity σ (ν), (b) the permittivity ϵ(ν) and (c) the reflection coefficient R (calculated on the basis of measured σ (ν) and ϵ(ν))) of La1.87 Sr0.13 CuO4 single crystal measured at different temperatures for polarization E ‖ c. Lines represent the results of fittings as described in the text. The inset in panel (b) shows the temperature dependences of the normal state parameters of the charge carriers: the plasma frequency νpl and scattering rate γ .
ϵ(ν)), exhibit the behavior typical for a (poor) conductor in the low-frequency (Hagen–Rubens) regime. We detect a noticeable decrease of the conductivity towards higher frequencies, as seen in the σ (ν, T = 50 K) spectrum of Fig. 1(a). Relying on this dispersion, we estimate the parameters of the carrier (hole) condensate in the normal state by fitting σ (ν), ϵ(ν) and R(ν) with the corresponding expressions of the Drude model, where the complex conductivity is expressed as [11]
σ ∗ = σ (ν) + iσ2 (ν) =
σ0 νγ σ0 γ 2 +i 2 . γ 2 + ν2 γ + ν2
(1)
Here σ2 = ν(ϵ∞ − ϵ)/2 is the imaginary part of the conductivity, ϵ∞ is the high-frequency dielectric constant, σ0 = νp2 /2γ is the dc
conductivity, νp = (Ne2 /π mc 2 )1/2 denotes the plasma frequency of the quasi-free carriers whose concentration is N, with charge e and effective mass m. The temperature variations obtained for the scattering rate γ and the plasma frequency νp of carriers are plotted in the inset of Fig. 1(b). Both quantities slightly decrease upon cooling. The scattering rate is more than two times smaller than the plasma frequency, indicating that in the non-SC state of La1.87 Sr0.13 CuO4 the charge transport across the CuO planes is not overdamped. According to Fig. 1, all spectra significantly change upon entering the SC state of La1.87 Sr0.13 CuO4 . In the permittivity spectrum a strong dispersion of the type
ϵ(ν) = −(νpSC /ν)2
(2)
is observed that is mainly caused by an inductive response of the SC δ -function at ν = 0 in the conductivity spectrum [11]. The spectral strength of this response is given by the value of the plasma frequency νpSC of the SC condensate. In the conductivity spectra plotted in Fig. 1(a), a well defined peak is observed below Tc . At T < 10 K the peak moves outside of our frequency interval, to approximately 47 cm−1 at T = 5 K as
seen in the time-domain spectroscopic data obtained for a crystal from the same batch [12]. Another observation from Fig. 1(a) is that in the SC state the c-axis conductivity of La1.87 Sr0.13 CuO4 significantly increases at the lowest frequencies. This excess conductivity (absorption) component exhibits a notable temperature evolution that reflects the sub-gap quasiparticle dynamics in La1.87 Sr0.13 CuO4 . The component has a Drude-like spectral shape and acquires larger amplitude at lower temperatures, as seen in Fig. 1(a). In this regard, the increase of the sub-gap ac conductivity resembles the behavior of the Drude-like component that is well-known from the in-plane conductivity spectra of the cuprates [13,14]. We have fitted the ϵ(ν) and σ (ν) spectra with the peak in the conductivity spectra modeled by a Lorentzian, the strength of the SC δ -function given by its inductive response (Eq. (2)) and the quasiparticle below-gap absorption described by the Drude term. The quasiparticles produce a contribution to the permittivity of the same type as the SC δ -function, 1ϵ QP ∼ −(νpQP /ν)2 , that can become comparable to the dielectric contribution of the SC δ function, given by Eq. (1). (We use (νpQP )2 to denote the spectral weight (strength) of the sub-gap absorption.) This means that a fit of the permittivity spectrum with expression (2) has to consider also the contribution 1ϵ QP . Analogous situations occur also with the in-plane optical response of cuprates [15,14]. The role of this absorption is crucial for determining the SC plasma frequency in the temperature range from 28–30 K up to Tc . At lower temperatures, the SC δ -function acquires its full strength: νpSC ≫ νpQP ; the value of the magnetic field penetration depth at the lowest temperatures, λL (T → 0) = 8.5 µm, is practically unaffected by the quasiparticle term and is in agreement with the literature data (see Ref. [14] and references therein). 3. Discussion To understand the origin of the low-frequency conductivity peak, we recall the idea proposed in Ref. [16]: in the layered superconductors with different Josephson-coupled layers in a unit cell there should exist more than one LO Josephson plasma excitations that propagate in the direction perpendicular to the layers and a transverse optical (TO) plasma mode that has its wavevector parallel to the layers [17–20]. The transverse Josephson plasma resonance is seen as a well defined mode in the c-axis THz conductivity spectra of SmLa2−x Srx CuO4 [2,21–23]. Several publications report the observation of similar low-energy TO modes in the c-axis conductivity spectra also in a singlelayer compound La2−x Srx CuO4 [6,24]; the origin of the peak was assigned as a transverse Josephson resonance, although no corresponding microscopic mechanism was suggested. We believe that the low-energy resonances seen in the c-axis conductivity spectra of our La1.87 Sr0.13 CuO4 sample and in the La2−x Srx CuO4 crystals with other strontium concentrations and dopings [6,12,25,24] are common in origin and come from transverse excitations of the Josephson plasma of Cooper pairs. We suggest that the conditions for appearance of these TO modes in the single-layer La2−x Srx CuO4 are provided by the charge and magnetic stripes within the crystallographic CuO planes [6,7,26–32]. Since the sizes of magnetically ordered ‘‘islands’’, that are of the order of 100–600 Å according to Refs. [6,29,31], exceed by far the interlayer lattice constant and the SC correlation length in the La2−x Srx CuO4 family, one readily accepts interaction between magnetic stripes in different CuO layers. Similar interactions should exist also between the charge stripes, as has been confirmed by X-ray experiments on NdLSCO [32]. Under conditions when the direction of stripes changes by 90° with respect to the neighboring CuO layers [26], we expect noticeable spatial modulation of the Josephson coupling strength between the SC charge stripes, as is
B.P. Gorshunov et al. / Solid State Communications 151 (2011) 1681–1685
a
b
c-axis
CuO – planes
Coupling strength, tunnel current
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c Fig. 2. Sketch of the charge stripes of La1.87 Sr0.13 CuO4 crystal, cut in the c (a–b) plane. The stripes, whose direction changes by 90° in the neighboring CuO layers [26], are shown by black lines and dots, correspondingly. The spatial modulation of the strength of the Josephson coupling and of the corresponding interlayer supercurrent is indicated by different shadings.
indicated schematically in Fig. 2: the amplitude has to be larger for the coupling between the stripes in the neighboring planes (the area shown with solid-line oval in Fig. 2) and smaller for the coupling between the stripes in the next-nearest planes (the dashedline oval). As a result, we obtain differently Josephson-coupled SC layers within the unit cell of La2−x Srx CuO4 —the condition needed for the appearance of the TO Josephson resonance [16]. Our assignment of the low-frequency mode in the c-axis conductivity spectra of La1.87 Sr0.13 CuO4 to the transverse Josephson plasma resonance is confirmed by detailed inspection of the temperature dependent spectra of the conductivity, loss function and reflection coefficient presented in Fig. 3. It is seen that at temperatures T ≈ 29–30 K and up to Tc , there exists only one broad peak in the loss function that can be well fitted (the solid line in the T = 30 K panel of Fig. 3) with the model of Ref. [16], assuming identical Josephson junctions in the unit cell. At lower temperatures, however, it becomes impossible to describe the spectrum of the loss function with a single longitudinal excitation. As demonstrated by the T = 26 K spectra, two LO modes should be involved now, corresponding to the weight factors of 0.36 and 0.64 (Ref. [16]) for the two Josephson junctions. At those temperatures the maximum in the conductivity spectra is also clearly pronounced and is well fitted with the model of Ref. [16] (solid line). We see that as in the case of SmLa2−x Srx CuO4 , the transverse resonance is found between the two LO plasmons: νLO1 ≈ 29 cm−1 < νTO ≈ 32 cm−1 < νLO2 ≈ 32 cm−1 (T = 26 K). Also the temperature behaviors of the transverse mode parameters in La1.87 Sr0.13 CuO4 and in SmLa2−x Srx CuO4 are quite similar (Fig. 4). Considerably higher values of the eigenfrequency and oscillator strength of the mode in La1.87 Sr0.13 CuO4 are simply due to the larger c-axis superfluid density and correspondingly smaller London penetration depth: λL (T → 0) ≈ 8.5 µm in La1.87 Sr0.13 CuO4 compared to λL (T → 0) ≈ 38 µm in SmLa0.85 Sr0.15 CuO4 [2]. When comparing the TO plasma modes in our La1.87 Sr0.13 CuO4 and in SmLa2−x Srx CuO4 [2], another distinction becomes noticeable. The LO and TO peaks in SmLSCO are pronounced very clearly [2,33], while in LSCO the corresponding structures in the conductivity, loss function and reflectivity spectra are seriously smeared (Fig. 3); also the damping parameter of the TO mode in LSCO is several times larger than that in SmLSCO. We believe that the reasons are of fundamental importance: while in SmLSCO the double-layered structure of the array of Josephson junctions is clearly provided ‘‘on a crystallographic level’’ by periodically sandwiched insulating fluorite and a rocksalt blocks, an appreciable
Fig. 3. Temperature dependences of the parameters of the transverse optical resonance seen in the conductivity spectra of La1.87 Sr0.13 CuO4 for polarization E ‖ c: 2 frequency position νTO (a), damping γTO (b) and spectral weight fTO = 1ϵ · νTO (1ϵ : dielectric contribution of the resonance). The 5 K points are obtained with the time-domain spectrometer for a crystal from the same batch. Open dots present the corresponding data for the SmLa2−x Srx CuO4 crystal from [2]. Lines are guides to the eye.
disorder is naturally expected in the system of charge and magnetic stripes of LSCO that should lead to correspondingly disordered Josephson tunneling amplitudes. Finally, we discuss possible origins of the below-gap quasiparticle absorption observed in the c-axis response of superconducting La1.87 Sr0.13 CuO4 . The way that it shows up in our conductivity spectra (Fig. 1 is reminiscent of that for the so-called below-gap Drude term seen in the in-plane conductivity spectra in all high-Tc cuprates [13]. The in-plane quasiparticle absorption originates from the d-wave character of the in-plane order parameter, when, due to the zero SC gap values along certain directions in the reciprocal space, the thermal energy or the energy of the probing electromagnetic radiation can generate quasiparticles even far below Tc . One could think of the same mechanism as can produce low-frequency absorption also in the c-direction in La1.87 Sr0.13 CuO4 . In this case one would expect a reduction of both the scattering rate γ QP and the plasma frequency νpQP of the quasiparticle peak on lowering the temperature (see, for example, [14]). According to the estimates we made from our data, however, both quantities are at least unchanged: γ QP ≈ 7 cm−1 at QP T = 5–35 K and νpl ≈ 100 cm−1 at T = 20–33 K. We thus have to assume some other source of quasiparticles to be present in non-superconducting regions in the superconducting state of La1.87 Sr0.13 CuO4 . The interlayer transport of these quasiparticles can be realized via interplane channels that could occur, for example, as a result of c-axis elongated chains of dipoles formed by ordered strontium ions and holes ‘‘ascribed’’ to them, as described in [34]. To learn more about the origin of quasiparticles, and the interplay between the c-directional correlations in the stripe subsystem and the low-frequency c-axis excitations in the La2−x Srx CuO4 family, further THz and sub-THz measurements on crystals with other compositions and dopings are needed.
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Fig. 4. Spectra of the conductivity (gray; yellow online; thick lines are guides to the eye), loss function (closed dots) and reflection coefficient (open dots; Y -scale from 0 to 1.1) of La1.87 Sr0.13 CuO4 single crystal for polarization E ‖ c at different temperatures. Lines shown for T = 26 K and 30 K are calculations according to the model of Ref. [16]. Dashed lines shown for T = 26 K represent two longitudinal plasmons needed to describe the loss function with the model of Ref. [16]. The bottom right panel shows the conductivity and reflectivity spectra combined with the spectra (dashed lines) obtained with the time-domain spectrometer for the La1.87 Sr0.13 CuO4 crystal from the same batch [24].
4. Conclusion Direct measurements of the terahertz and subterahertz
(10–40 cm−1 ) c-axis conductivity and dielectric permittivity of single-crystalline superconducting cuprate La1.87 Sr0.13 CuO4 reveal a transverse optical excitation in the superconducting state that is ascribed to the transverse Josephson plasma mode. We suggest that the mode originates from the interplane interaction of the charge and magnetic stripes that leads to modulation of the strength of the Josephson coupling between the copper–oxygen planes along the c-axis within the unit cell of La1.87 Sr0.13 CuO4 . For the superconducting state, a strong quasiparticle absorption is detected below 15 cm−1 that is assigned to the c-axis transport of normal charge carriers through the channels that connect the nonsuperconducting areas of as yet unknown origin. Acknowledgments The work was supported by the Russian Academy of Sciences Program for Fundamental Research ‘‘Strongly correlated electrons
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