Study of the optical gap in novel superconductors by coherent THz radiation

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Infrared Physics & Technology 51 (2008) 429–432 www.elsevier.com/locate/infrared

Study of the optical gap in novel superconductors by coherent THz radiation P. Calvani a,*, S. Lupi a, M. Ortolani b, L. Baldassarre a, C. Mirri a, R. Sopracase a, U. Schade b, Y. Takano c, T. Tamegai d b

a CNR-INFM-Coherentia and Dipartimento di Fisica, Universita´ La Sapienza, Roma, Italy Berliner Elektronenspeicherring-Gesellschaft fu¨r Synchrotronstrahlung m.b.H., Albert-Einstein Strasse 15, D-12489 Berlin, Germany c National Institute for Materials Science, 1-2-1 Sengen, Tsukuba 305-0047, Japan d Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

Available online 23 December 2007

Abstract We show how coherent synchrotron radiation (CSR) allows one to measure the slight increase in the reflectivity of a BCS superconductor as it is cooled below T c . In fact, by CSR coupled to a conventional interferometric apparatus, one can obtain a signal-to-noise ratio 103 in the sub-THz range. We apply this technique to the measurement of the optical gap in boron-doped diamond and in CaAlSi, a superconductor isostructural to MgB2. In the latter compound we are also able to determine a slight anisotropy between the gap in the hexagonal planes and that along the orthogonal c axis. Ó 2007 Elsevier B.V. All rights reserved. PACS: 74.78.Db; 78.30.j Keywords: Diamond; Superconductor; THz spectroscopy; Coherent synchrotron radiation

1. Introduction Following the discovery of high-T c cuprates in 1986, many other superconducting materials have been found in the last two decades. Among them one may cite K-doped C60, NaxCoO2  nH2O, and MgB2. In this framework, it has been discovered recently that also boron-doped diamond can become a superconductor [1] below critical temperatures T c well above the liquid helium temperature [2], if the doping level is J 2.5%. Meanwhile, other superconductors isostructural to MgB2 have been found, like for example CaAlSi with a T c of 7.7 K. When a new superconductor is found, one wonders whether it will follow a conventional Bardeen–Cooper– Schrieffer (BCS) theory, or present exotic properties. For example, strongly covalent bonds, high concentration of *

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1350-4495/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.infrared.2007.12.022

impurities, and high phonon frequencies make B-doped diamond much different from the conventional metals where the BCS model holds. In turn CaAlSi, with its hexagonal planes stacked along the c axis, may have anisotropic properties that are not considered by the conventional models of superconductivity. Infrared spectroscopy is a powerful tool to characterize both the normal and the superconducting state in such solids, as it probes directly, and with the highest spectral resolution, the lowenergy electrodynamics of solids. In particular, when at T < T c a gap opens in the electronic density of states at the Fermi level, if the Cooper pairs are in a spherically symmetric s state the reflectivity becomes Rs ðxÞ ¼ 1 for any x 6 2DðT Þ, the optical gap. Above T c and in the same low-frequency range, the reflectivity is instead given by the Hagen–Rubens formula 1 Rn ðxÞ ¼ 1  ½8xCðT Þ=x2p 2 , where CðT Þ is the relaxation rate of the carriers and xp their plasma frequency. Therefore, if the metal is in the ‘‘dirty” regime defined by

P. Calvani et al. / Infrared Physics & Technology 51 (2008) 429–432

2. The optical gap of superconducting diamond Previous studies indicate that B-doped diamond films are in the dirty limit and display a highly symmetric wave function [6]. The optical gap can therefore be measured, and compared with the BCS prediction 2hcDð0Þ= k B T c ¼ 3:53. Moreover, from the optical conductivity rðxÞ that one extracts from RðxÞ, one can obtain the field penetration depth k. In the clean limit CðT c Þ  2Dð0Þ, k coincides with the London penetration depth 1kL , while [7] in the dirty limit CðT c Þ  2Dð0Þ, k  kL ðC=DÞ2 . The sample was a film about 3 lm thick, 2:5  2:5 mm wide, grown by CVD and deposited on pure CVD diamond [2]. X-ray diffraction patterns collected just after the growth showed that the whole film surface has a (1 1 1) orientation, with no appreciable spurious contributions, as already reported for similar samples [2]. The boron concentration was estimated to be 6  1021 cm3. The sample magnetic moment MðT Þ, reported in the inset of Fig. 1, shows the superconducting transition with an onset at T c ¼ 6 K. Below 6.3 K the zero-resistance regime is already established by a sharp transition, which confirms the good homogeneity of the film (see Rdc in the inset). However we assume here T c ¼ 6 K, by considering that the magnetization is a bulk quantity like the infrared conductivity. In order to measure the gap, the sample was illuminated by the CSR extracted from BESSY. An automatic device turned the mirrors of the interferometer sample compartment in such a way that the intensity of the incident beam was tested after every measurement of the intensity I R reflected by the sample. In such a way, I R could be corrected in real time for the incident radiation intensity, which decreased slowly with time. Nothing else in the apparatus was moved while the sample temperature

ν (THz) 0.0

0.2

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M (10-4 e.m.u)

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Tc

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R dc(T)/R dc (12 K)

CðT c Þ > 2Dð0Þ, the ratio Rs =Rn exhibits a peak [3] at 2D. Both the value of 2D and its temperature dependence can be easily compared with the BCS predictions. For example, in the weak coupling limit of the original BCS model, 2D ¼ 3:53 T c . In real experiments, however, one may encounter serious difficulties to measure the small difference between Rs and Rn , as Rn in a good metal may be as high as 0.99 in the range of frequencies of the gap. Therefore, a signal-to-noise ratio on the order of 103 is often needed in the sub-Terahertz region ð1 THz ¼ 33 cm1 Þ to measure the gap. Nowadays, this strong requirement can be fulfilled with a conventional interferometric apparatus, provided that it is coupled to coherent synchrotron radiation (CSR) [4]. A CSR source is routinely open to users at the beamline ‘‘IRS spectroscopy” of the storage ring BESSY, when this latter is working in the so-called lowalpha mode [5]. In such a mode, the electron bunch follows an ideal trajectory along the ring while the current is kept at a low value, typically between 10 and 20 mA. In such conditions the N electrons of a bunch do emit in phase and the resulting intensity grows proportionally to N 2 in the wavelength range of the bunch length.

R(T) / R(10K)

430

0

12

8

T (K) 1.00 T=2.6 K 3.4 K 4.6 K 7.2 K 15 K

0

10

20

ω (cm-1)

30

Fig. 1. Reflectivity of a strongly B-doped diamond film in the sub-THz region, normalized to its values at 10 K. The lines are fits obtained by assuming a BCS reflectivity below T c and a Hagen–Rubens model at 10 K. The inset shows on the left scale the magnetic moment of the sample, as cooled either in a 10 Oe field (FC) or in zero field (ZFC), on the right scale its resistance normalized to its value at 12 K. The FC values are multiplied by 10.

decreased below T c , driving it to the superconducting phase. By this procedure we obtained in the sub-THz range an error DI R =I R ¼ 0:2%. The ratio I R ðT Þ=I R ð10 KÞ ¼ Rs ðT Þ=Rn ð10 KÞ is reported in Fig. 1. The three curves at T < T c exhibit a strong frequency dependence in the sub- THz region, with the predicted peak at x ’ 2DðT Þ. As a cross-check, the data for T > T c do not show any peak and are equal to 1 within the noise. From a first inspection to our sub-THz data, at T ¼ 2:6 K the peak value is ’12 cm1, which gives 2hcDð2:6 KÞ=k B T c ’ 3. This result motivated us to fit the data using a BCS approach [8], which indeed well describes the data. The main output of the fit, however, is the gap value, which at 4.6, 3.4, and 2.6 K is found to be 2D ¼ 9:5, 10.5, and 11:5 cm1 , respectively. This leads to an extrapolated value [7] 2Dð0Þ ¼ 12:5 cm1 , or 2hcDð0Þ= k B T c ¼ 3:0 0:5, in satisfactory agreement with the BCS prediction of 3.53. Afterwards, we obtained the absolute reflectivity Rn ð10 KÞ up to 20; 000 cm1 (inset of Fig. 2a) by extending the measuring range and taking as reference the film itself, coated with a gold or silver layer evaporated in situ. The reflectivity in the superconducting phase was then reconstructed as Rs ðT Þ ¼ ½I R ðT Þ=I R ð10 KÞRn ð10 KÞ (see Fig. 2a), and used to obtain rðxÞ ¼ r1 ðxÞ þ ir2 ðxÞ by standard Kramers–Kronig transformations. The real part r1 ðxÞ (Fig. 2b) decreases in the sub-THz range for T < T c showing the opening of the gap. At 4.6 K, a residual quasi-particle contribution can still be distinguished at the lowest frequencies. At 2.6 K, zero absorption is attained below 10 cm1, in agreement with the fits of Fig. 1. In Fig. 2b, the conductivity in the superconducting phase rs1 ðxÞ and that in the normal phase rn1 ðxÞ coincide for x J 35 cm1  6D, as expected for a BCS superconductor in the dirty limit [9]. According to the Ferrell–Glover– Tinkham sum rule the area A removed at T < T c below

P. Calvani et al. / Infrared Physics & Technology 51 (2008) 429–432

0.95

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λ (0) / λ (T)

1 / λ (T) (10 cm )

400

0 0 0.0 0.5 1.0 1.5 2.0

T / Tc

Fig. 2. Optical response of superconducting diamond: (a) absolute reflectivity obtained from the ratios of Fig. 1 and the RðxÞ at 10 K shown in the inset; (b) real part of the optical conductivity; (c) inverse square of the penetration depth (squares), compared with its behavior for a dirty BCS superconductor (grey line). In (a), the points at T < T c and x < 2D are replaced by those of the fits in Fig. 1. This allows one to discard unphysical values R > 1, due to residual noise, which would affect the Kramers–Kronig transformations.

r1 ðx; T Þ, builds up the collective mode at x ¼ 0. The spectral weight condensed into this peak, Z 6D A¼ ðrn1  rs1 Þdx ð1Þ 0 1

may be used to extract the penetration depth k ¼ 1=ð8AÞ2 [7]. We thus found k ’ 1 lm at 2.6 K. Using the relation reported above for the dirty limit, we estimated kL  50 nm for our film at 2.6 K. The large difference between k and kL can be explained in terms of a large impurity scattering due to the disorder produced by the dopant. Finally, 1=k2 was plotted in Fig. 2c vs. T =T c , and found in good agreement with the BCS prediction for the dirty limit [7]. The extension of the spectra to higher frequencies, not reported here, allowed us to identify also the phonon excitations which most strongly couple to the carriers and which, therefore, are expected to be responsible for the formation of Cooper pairs in diamond [10]. 3. The anisotropic gap of CaAlSi CaAlSi, a novel superconductor with [11] a T c of 7.7 K, has attracted wide interest for its hexagonal crystal structure which is similar to that of MgB2, where however the carriers are negative. Like MgB2 it has two Fermi-surface sheets. However, presently it is not clear whether in CaAlSi one should expect one SC gap or two gaps like in MgB2. In any case it is reported to have a s-wave anisotropic symmetry [12], but the available experimental data are not conclu-

sive. While angle resolved photoemission (ARPES) [13], within the energy resolution, distinguished in CaAlSi one isotropic gap of about 1:2 meV ¼ 4:2 k B T c , muon spin relaxation (lSR) data [14] suggested a highly anisotropic gap, or possibly two distinct gaps like in MgB2. Penetration depth measurements [12] support weakly anisotropic swave gap, but not two distinct gaps. Infrared spectroscopy, which allows one to measure the gap directly and with a resolution in energy higher than in ARPES, may help to solve this issue, provided that one attains a suitable signal-to-noise ratio. The present sample of CaAlSi was a single crystal with a 2  4:5  3 mm3 . It was grown by the floating-zone method and characterized as described in Ref. [15]. Its magnetic moment MðT Þ (see the inset of Fig. 3) shows the SC transition with an onset at 6.7 K. The ratio RðT Þ=Rð10 KÞ is reported in Fig. 3 for the radiation polarized in the hexagonal sheets at T, both below and above T c . As Rn ð10 KÞ ’ 0:99 at low x, the total variation of RðT Þ=Rn ð10 KÞ between 8 and 3.3 K is much lower than in diamond, which is a poor metal, and just slightly more than 1%. In Fig. 3 the curves at T < T c exhibit the expected peak at 2Dab ðT Þ, while for T > T c (12 K), the above ratio is equal to 1 at any x within the noise. At 3.3 K the peak frequency is about 17 cm1, which gives 2hcD=k B T c ’ 3:8. We modeled the optical conductivity rðxÞ, for the hexagonal planes in the normal state, by a conventional Drude model with the plasma frequency xp;ab and the scattering rate C determined by a fit to the normal state (T = 10 K) optical conductivity (see below). In the SC state we used the Mattis– Bardeen model with a fixed T c ¼ 6:7 K and Dab as a free parameter. The curves Rðx; T Þ=Rðx; 10 KÞ calculated in this way are also reported in Fig. 3. The fit is good at the three temperatures and provides 2Dab ¼ 15 and 17:5 cm1 at 4.5 and 3.3 K, respectively. This leads to an extrapolated value [7] 2Dab ð0Þ ¼ 19:0 1:5 cm1 , or 2hcDð0Þ=k B T c ¼ 4:1 0:4, a value which confirms – with the higher

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3.3 K 4.5 5.6 12 0

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Fig. 3. Ratio between the sub-THz reflectivity of the ab planes below T c and in the normal phase at 10 K. The lines are fits based on a BCS reflectivity below T c and on a Hagen–Rubens model at 10 K. The inset shows the magnetic moment measured by cooling the sample in a field of 10 Oe, showing the superconducting transition at T c ¼ 6:7 K.

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P. Calvani et al. / Infrared Physics & Technology 51 (2008) 429–432 ν (THz) 0.0

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R(T)/R(10 K)

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Fig. 4. Ratio between the sub-THz reflectivity of the c axis below T c and that in the normal phase at 10 K. The lines are fits to Eq. (2).

resolution provided by infrared spectroscopy – a previous determination by ARPES [13]. This result suggests that CaAlSi is a BCS superconductor with moderately strong electron–phonon coupling. On the basis of our fits, a single gap well describes the superconducting transition in the hexagonal planes. Let us now consider the optical response of CaAlSi along the c axis. Fig. 4 shows the ratio Rs ðT Þ=Rn (10 K), as measured with the radiation polarized along the c direction. Here also the total variation of the reflectivity between 10 and 3.3 K is barely 1%, due to the extremely high reflectivity in the normal phase. A peak appears below T c , as in Fig. 3, but with a different shape. Such shape can be fitted by assuming that, along c, only a fraction 1  a of the quasiparticles condenses into the SC state. By using a two-fluid model, one can then write the optical conductivity of the c axis below T c as rsc ðxÞ ¼ arD ðxÞ þ ð1  aÞrMB ðxÞ

ð2Þ

where rD ðxÞ is a Drude conductivity for the normal fraction a and rMB ðxÞ is a Mattis–Bardeen conductivity for the SC fraction. aðT Þ and 2Dc ðT Þ are the free parameters. This model is then used to fit (solid lines) the reflectivity ratios in Fig. 4. One obtains 2Dc ðT Þ ¼ 18 and 20 cm1 at 4.5 and 3.3 K, respectively, and a ¼ 0:7 at both T ’s. The extrapolation to T ¼ 0 gives 2Dc ð0Þ ¼ 22 1 cm1 2hcDc ð0Þ=k B T c ¼ 4:5 0:5. Therefore, Dab ð0Þ K Dc ð0Þ and a slight anisotropy is found, which may reconcile the penetration depth results of Ref. [12] with the ARPES determination of the gap [13], thanks to the higher resolution in energy of the infrared spectroscopy. 4. Summary In conclusion, in the experiments reported here we have determined the optical gap of two superconductors discovered in the last few years, by exploiting both the high res-

olution of Fourier-transform spectroscopy and the high brilliance of coherent synchrotron radiation. This technique allows one to achieve the signal-to-noise ratio on the order of 103 which is needed to appreciate the tiny increase in the reflectivity across T c . We have thus found that in hole-doped diamond 2D is 3k B T c at T ¼ 0, and the Ferrell–Glover–Tinkham sum rule holds within approximately 6D. This shows that diamond is a BCS superconductor in the dirty limit. In CaAlSi, which has a crystal structure similar to that of MgB2, we have determined the optical gap both in the hexagonal sheets, and along the orthogonal c axis, finding a slight anisotropy which can reconcile previous contradictory results reported by other techniques. Moreover, the behavior of the reflectivity below T c suggests that only about 30% of the carriers propagating along the c axis contribute to the optical spectral weight of the condensate at the lowest temperature here reached (3.3 K). This may be due either to the presence of nodes in the order parameter, or to the existence of two different gaps, of which one is smaller than the lowest energy here attained. References [1] E.A. Ekimov, V.A. Sidorov, E.D. Bauer, N.N. Mel’nik, N.J. Curro, J.D. Thompson, S.M. Stishov, Nature 428 (2004) 542. [2] (a) Y. Takano, M. Nagao, I. Sakaguchi, M. Tachiki, T. Hatano, K. Kobayashi, H. Umezawa, H. Kawarada, Appl. Phys. Lett. 85 (2004) 2851; (b) Y. Takano et al., Diam. Relat. Mater. 14 (2005) 1936. [3] D.N. Basov, S.V. Dordevic, E.J. Singley, W.J. Padilla, K. Burch, J.E. Elenewski, L.H. Greene, J. Morris, R. Schickling, Rev. Sci. Instrum. 74 (2003) 4703. [4] M. Ortolani, P. Calvani, S. Lupi, U. Schade, A. Perla, M. Fujita, K. Yamada, Phys. Rev. B 73 (2006) 184508. [5] M. Abo-Bakr, J. Feikes, K. Holldack, P. Kuske, W.B. Peatman, U. Schade, G. Wustefeld, H.-W. Hubers, Phys. Rev. Lett. 90 (2003) 094801. [6] T. Yokoya, T. Nakamura, T. Matsushita, T. Muro, Y. Takano, M. Nagao, T. Takenouchi, H. Kawarada, T. Oguchi, Nature 438 (2005) 647. [7] M. Dressel, G. Gru¨ner, Electrodynamics of Solids, Cambridge University Press, Cambridge, UK, 2002. [8] W. Zimmermann, E.H. Brandt, M. Bauer, E. Seider, L. Genzel, Physica C 183 (1991) 99. [9] D.C. Mattis, J. Bardeen, Phys. Rev. 111 (1958) 412. [10] M. Ortolani, S. Lupi, L. Baldassarre, U. Schade, P. Calvani, Y. Takano, M. Nagao, T. Takenouchi, H. Karawada, Phys. Rev. Lett. 97 (2006) 097002. [11] M. Imai, K. Nishida, T. Kimura, K. Abe, Appl. Phys. Lett. 80 (2002) 019. [12] R. Prozorov, T.A. Olheiser, R.W. Giannetta, K. Uozato, T. Tamegai, Phys. Rev. B 73 (2006) 184523. [13] S. Tsuda, T. Yokoya, S. Shin, M. Imai, I. Hase, Phys. Rev. B 69 (2004) 00506R. [14] S. Kuroiwa, H. Takagiwa, M. Yamazawa, J. Akimitsu, K. Ohishi, A. Koda, W. Higemoto, R. Kadono, J. Phys. Soc. Jpn. 73 (2004) 2631. [15] A.K. Ghosh, M. Tokunaga, T. Tamegai, Phys. Rev. B 68 (2003) 054507.