Optical self-energy of superconducting Pb in the terahertz region

Transcription

1 Otical self-energy of suerconducting Pb in the terahertz region T. Mori, 1 E. J. Nicol, 2, * S. Shiizuka, 1 K. Kuniyasu, 3 T. Nojima, 3 N. Toyota, 1 and J. P. Carbotte 4 1 Physics Deartment, Graduate School of Science, Tohoku University, Sendai , Jaan 2 Deartment of Physics, University of Guelh, Guelh, Ontario, Canada N1G 2W1 3 Institute for Materials Research and Center for Low Temerature Science, Tohoku University, Sendai , Jaan 4 Deartment of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1 Received 5 March 2008; revised manuscrit received 6 May 2008; ublished 28 May 2008 New terahertz data on the otical conductivity of Pb are resented, as well as a detailed Eliashberg analysis with articular emhasis on honon-assisted rocesses not included in a BCS aroach. Consideration of the otical self-energy instead of the conductivity itself hels highlight the differences with BCS redictions. Predicted coherence eaks are observed in the otical scattering rates. Imurities enhance the otical effective mass at zero frequency by an order of magnitude and induce a large eak at twice the ga in agreement with theory. This work illustrates the usefulness of the otical self-energy for the analysis of data. DOI: /PhysRevB PACS number s : Gz, Fg, Nf, Kc I. INTRODUCTION Otical roerties of suerconducting Pb have been measured in the ast by several grous, including Palmer and Tinkham, 1 who comared their data with BCS redictions. Joyce and Richards 2 saw honon structure in the absortivity and Farnworth and Timusk 3 analyzed similar data to recover the electron-honon sectral density 2 F. This latter work was based on aroximate equations for the conductivity derived by Allen 4 from Fermi golden rule considerations. Good agreement with the tunneling-derived electron-honon sectral density was found. 5 More elaborate theories of the conductivity based on the Eliashberg equations and a Kubo formula for the current-current correlation function were formulated by Nam 6 and further elaborated uon in many laces 7 including Scalaino 8 and Marsiglio and Carbotte. 9,10 In this aer, we reort new high quality terahertz measurements on Pb. In addition to the new data, we rovide a detailed theoretical analysis based on numerical solutions of the Eliashberg equations with the known 2 F. 5,11 We emhasize articularly the honon-assisted rocesses not included in BCS and describe deviations that these rocesses introduce. BCS deals only with the coherent art of the single article Green s function. In Pb, the electron-honon mass renormalization factor =1.55. In this case, only 1/ 1+ =1/2.55 of the sectral weight resides in the coherent art and / 1+ =1.55/2.55 is found in the incoherent art which describes the honon-assisted rocesses. II. METHODS The Pb films are deosited onto the 0001 surface of a single-crystalline sahire substrate held at room temerature. Four-terminal resistance measurements on films of 30, 50, and 480 nm in thickness show suerconducting transitions around 7.2 K with a width of 50 mk, indicating little degradation in T c from the bulk. The resent time domain terahertz sectroscoy TDTS measurements 12 are carried out with the use of a commercial sectrometer RT-20000, Tochigi Nikon Co. Ltd based on standard transmission techniques, covering the frequency range of THz. The real art 1 and the imaginary art 2 of the otical conductivity = 1 +i 2, where is the hoton energy, are directly evaluated by numerically analyzing both the amlitude and hase of transmitted terawaves through the substrate 0.5 mm in thickness with and without the film. The data will be resented for the 50 nm film most systematically studied. Figure 1 a shows the time-deendent electric field E film t and the Fourier transform E film see inset transmitted through the Pb film in the normal and suerconducting states, where the temeratures T are labeled by reduced temerature t=t/t c. The comlex transmission coefficient t is defined by E film /E ref, where E ref is the reference sectrum without the film. Figure 1 b shows the E(t) [a.u.] T s (ω)/t N (ω) (a) (b) E(ω) [a.u.] t [s] ω/2π [THz] Pb ω/2π [THz] FIG. 1. Color online a The time-deendent electric-field E t and the Fourier transform E inset and b the ower transmission sectrum ratio T S /T N, normalized at t=t/t c =1.02, just above T c. See the text for details /2008/77 17 / The American Physical Society

2 MORI et al. ower transmission sectrum T = t 2 normalized at t =1.02, just above T c. Here, t is written as t = t vft fs ex i n f 1 d f /c t vs 1 r fs r fv ex i2n f d f /c, 1 where t ij =2n i / n i +n j and r ij = n i n j / n i +n j are the comlex Fresnel s transmission and reflection coefficients at the interface between regions i and j, resectively. The subscrits i and j stand for v, f, and s, reresenting the vacuum, Pb film, and sahire substrate, resectively, and n i is the comlex refractive index of region i, d j is the thickness of region j, and c is the seed of light. The magnitude of n s is obtained from a searate measurement on the sahire substrate only, while n f is obtained by numerically solving Eq. 1. The comlex conductivity is finally evaluated from the relation = n 2 f i For a correlated electron system, secifically here, a couled, electron-honon system, it has become a standard ractice to reresent the otical conductivity T, in terms of an otical self-energy o, in analogy to the quasiarticle self-energy which is introduced to include interactions in the one-article Green s function. By definition, 13,14 = i o, 3 where is the lasma frequency. The imaginary art of the otical self-energy gives the otical scattering rate 1/ o T, = 2 o 2 T, and its real art gives the otical effective mass m o T, according to m o T, /m 1 = 2 o T,. Further, m o T, /m 1 is the otical mass renormalization o T,. Here, m is the bare electron mass. Traditionally, it has been common to dislay in otical aers the real and imaginary art of T,, but in the recent literature, it has been recognized that much can be learned from the scattering rate and effective mass renormalization, articularly for comarison with quasiarticle roerties. In terms of T,, Re 1 o = 2 = 4 T, , 4 2 m o T, m Im = 2 4 = T, The Drude conductivity is recovered when all interactions are neglected and only the imurity scattering is included through a frequency and temerature indeendent scattering rate 1/ im. In this case, the real art of o T, which must be related to its imaginary art by Kramers Kronig is zero. For couling to a single oscillator of energy E in the normal state at zero temerature, 15 1 o T =0, = 2 A E E, 6 where A is the couling between electrons and honons and 2 o = 2A ln E + E + E ln 2 E 2 E 2. 7 For 0, o =2A/ E, the quasiarticle electronhonon mass renormalization. In this model, has two arts. One art is a delta function at =0 coming from the region E in Eq. 6, where there is no scattering. If imurity scattering is included, the function broadens into a Lorentzian L = i 2 4 m o /m + i/ im. The second art, which is finite only for E, is a bosonassisted art: B = i 2 4 m o + i 1 m im + 2 A 1 E. Further, if we aroximate in Eq. 8 m o /m by 1+, its zero frequency limit which is exact for small, we obtain a coherent Drude art D = i i/ im, 10 with an otical sectral weight of 2 / 8 1+ and an effective scattering rate of 1/ im 1+. The remaining sectral weight is to be found in the incoherent iece Eq. 9 and is 2 / 8 1+, which follows on alication of the otical sum rule d = III. RESULTS In Fig. 2, we resent our exerimental oints and theoretical lines results for the conductivity. The theoretical results are based on full finite temerature numerical solutions of the isotroic s-wave Eliashberg equations 6 9 and the numerical evaluation of the Kubo formula for the conductivity. The inut arameters to the Eliashberg equations are the electron-honon sectral function 2 F and Coulomb reulsion, both taken from the inversion of tunneling data on Pb according to the method of McMillan and Rowell. 5,11 The inut arameters for Pb, along with a considerable discussion about studies on Pb and the Eliashberg equations used for our calculations in this aer, are given in Ref. 16. The formula for the conductivity used in our work, which was derived by Lee et al., 17 can be found in Ref. 18. Inthe to frame of Fig. 2, we show the real art of the conductivity 1 in 1 cm 1 versus u to a frequency of Hz, while in the lower frame, we show the imaginary art 2. For the real art, we note a kink in both data and theory at an intermediate frequency which decreases in frequency, with increasing temerature. This arises because there are two searate absortion rocesses ossible. At the

3 OPTICAL SELF-ENERGY OF SUPERCONDUCTING Pb IN FIG. 2. Color online Real and imaginary arts of the otical conductivity, 1 T, and 2 T,, resectively, versus at various reduced temeratures t=t/t c. The oints are the data and the lines are theory. lowest frequencies, there can be a direct absortion from the thermally excited quasiarticles. This contribution rovides a Drude-like eak to 1. At frequencies above twice the ga, direct absortion through the creation of two excitations out of the condensate becomes ossible. In addition, there is another incoherent honon-assisted contribution to the absortion and this is the art of most interest in this work. However, it cannot easily be searated out of the real art of the conductivity. We will see that consideration of the otical self-energy o defined in Eq. 3 hels in this regard. First, however, we note that the fit between the theory and the data is very good at low frequency in all cases. We have used the data in the normal state t=1.02 to determine that the elastic scattering rate 1/ im =22 mev and the lasma frequency =4.3 ev. The value of the lasma frequency is in aroximate agreement with that estimated from microwave data for Pb. 18 Turning next to the imaginary art of shown in the lower frame of Fig. 2, we note the same level of agreement and the oints for t=1.02, which are in the normal state, also show good agreement with the theory. In general, the suerconducting roerties of Pb, while very different from those redicted by BCS, have been remarkably well described by Eliashberg theory 16 over a broad range of exeriments and our new data shown here oses no excetion. This is imortant, given that in the ast, it is has been argued that vertex corrections 19,20 and other asects of electromagnetic theory might be needed to understand the data in Pb and yet this aears not to be the case here. With this level of agreement between theory and data, we should now be able to roceed to test for more subtle effects due to the hononassisted rocesses in the absortion. FIG. 3. Color online The otical scattering rate 1/ o T, and mass renormalization 1+ o T, vs. The oints are the exeriment and the lines are the theory. The data for t=1.02 are in the normal state just above T c and the other two data sets are below T c in the suerconducting state. To see searately, more clearly, the role of the coherent absortion rocesses included in BCS theory and described by the coherent art of the electron Green s function G, and the incoherent honon-assisted rocesses described by the incoherent art of G, it is convenient to deal with the real and imaginary arts of the otical self-energy defined in Eq. 3. Equations 4 and 5 are for the otical scattering rate and the effective mass, resectively. These are shown in Fig. 3. The to frame gives 1/ o and the bottom 1+ o at three temeratures. To agree with the exerimental data, it was necessary to include an imurity scattering rate of 22 mev. This is the only scattering that would be included in a BCS theory for which the normal state conductivity would be a ure Drude form. The otical scattering rate corresonding to this Drude is a constant flat line indeendent of frequency and temerature. We see clearly in the curve for the normal state that significant deviations from this constant occur. These are due to the boson-assisted rocesses which become even more imortant as the frequency is increased beyond the range of the data shown. This is shown in the to frame of Fig. 4, where we resent our theoretical results over an extended frequency range u to 36 mev, well beyond the end of the honon sectrum. For clarity, only the t=0.52 suerconducting state and t=1.02 normal state curves are shown. Starting with the normal state thick blue curve, we note a large deviation from the horizontal dashed black line at 22 mev for elastic imurity scattering. The additional

4 MORI et al. FIG. 4. Color online The calculated otical scattering rate 1/ o T, and otical mass renormalization 1+ o T, for a large energy range. Results are for our Pb calculation with imurity scattering rate of 22 mev. The red thin line is in the suerconducting state and the blue thick line is for the normal state, both at t =0.52. We comare with BCS dot-dashed and dashed black curves and ure Pb dotted curves. The vertical arrows indicate the eaks in 2 F shifted by 2. frequency-deendent scattering, due to boson-assisted rocesses, adds on to 1/ im according to Eq. 6 in the case of a delta function sectral density. For, this contribution will saturate at 2 A see Eq. 6 even for a distributed 2 F sectrum. Turning next to the suerconducting state thin red curve, to frame, we see larger boson structures with the vertical arrows indicating the transverse and longitudinal eaks in the honon frequency distribution F of Pb shifted by 2. These are receded by a coherence eak in the region above 2, which reflects the increased density of final states for scattering in the suerconducting as comared with the normal state. This curve is strikingly different from the dotdashed black curve which would aly to a BCS suerconductor. An interesting comarison is offered in the dotted curves which were also calculated with the Pb 2 F sectrum, but without residual scattering, i.e., 1/ im =0. An imortant oint to note is that no coherence eak is seen in the dotted thin red curve because this is an imurity effect, but at higher frequencies, the dotted curve is very close to the solid one once it is dislaced uward by 22 mev. This holds exactly for the normal state data, blue dotted ure and solid with residual scattering curves. Returning to the lower frame of Fig. 3, we show the otical effective mass renormalization 1+ o T,. The solid blue dots aly to the normal state at t=1.02, just above T c. Note that as 0, o 1.7 which is just slightly larger than the zero temerature quasiarticle mass renormalization for Pb, which is This is exected since it is known that the mass renormalization increases slightly with increasing temerature. Furthermore, the normal state effective mass has only a small frequency deendence redicted and this is observed. There is no deendence on imurity scattering as it dros out of this curve. This fact can be understood simly. The Kramers Kronig KK transform of a constant is zero so that a constant scattering rate corresonds to no mass renormalization. This is in striking contrast to the suerconducting state for which imurities rofoundly alter the otical curves. This is seen in the other two curves for t=0.52 red hexagons and t=0.82 black squares. Note first the very large eaks which occur at twice the ga edge 2. These eaks have not been emhasized before. They would not be there in the clean case no imurities. Their origin is easily understood from our results for the scattering rate shown in the to frame. We see the shar, almost vertical rise in 1/ o at 2, with the energy scale of the rise given by the value of 1/ im =22 mev. The KK transform of a ste at E is a logarithm with a singularity at E. The larger the ste, the larger the log singularity. Of course, here, we do not quite have a vertical ste, and hence, the corresonding eak in the effective mass is smeared. Another feature to note is the 0 limit of o. We note that Eq. 5 can be rewritten as m o T, m = , 12 and that in the suerconducting state in the limit of 0, we get m o =0 = 2 m 4 lim 1 = n 0 2 n s T, 13 where n is the electron density in the normal state and n s T is the suerfluid density at temerature T. For T=0, direct strong-couling calculations of n s T=0 show that it is very nearly equal to n/ 1+ which leads to m o T=0, =0 /m 1+, same as for the normal state, and reflects the fact that only the coherent art of the otical conductivity articiates in the suerfluid condensation. Returning to Eq. 12, one can verify that in the normal state, substitution of the Drude form Eq. 10 again gives m o /m=1+ as exected. If imurities are included, as we have done in our work, n s T =0 is no longer n/ 1+, but the dirty limit instead has the aroximate form in renormalized BCS 9 n s T =0 1+ n ln for 1, where =1/ 2 im 1+, which gives 1 + o T=0, =0 =8.3. The exact numerical result Fig. 3, bottom frame is 9.5 at T/T c =0.52. Most of the difference can be assigned to the small reduction in n s T between T =0 and t=t/t c =

5 OPTICAL SELF-ENERGY OF SUPERCONDUCTING Pb IN In the lower frame of Fig. 4, we show additional theoretical results on the otical effective mass which range over a larger frequency interval beyond that covered by the terahertz data. The dot-dashed black curve is the BCS result for 1/ im =22 mev while the solid thin red curve includes as well the incoherent honon-assisted Holstein rocesses. We see clearly in this curve an image of the Pb honon frequency distribution with transverse and longitudinal honon eaks shifted by 2, indicated by vertical arrows. The thin red dotted lines were obtained for ure Pb i.e., 1/ im is set equal to zero and the only source of scattering comes from the inelastic art described by 2 F. In this case, there is no eak in 1+ o at twice the ga, but at higher frequencies, the dotted and solid curves agree. For the normal state at t=1.02, the solid thick blue curve alies. Note that for larger than a few times the honon energy, normal and suerconducting curves merge and all tend toward 1, i.e., o 0 as. For BCS in the normal state, o =0 for all, as shown by the long-dashed curve. IV. A SUM RULE Recently, there have been several discussions of sum rules on 1/ o in the literature In articular, it was noted that there exists a differential sum rule on the difference in otical scattering rate between the suerconducting and normal states. Defining the artial sum I as 1 I = 0 o S 1 o 15 N d, it can be shown 22 that I =0 for any value of 1/ im and. The energy scale for arriving at the asymtotic limit is set by even in the dirty limit. As defined, in BCS, 22 I is negative and has its maximum absolute value at 2, after which, it begins to decrease toward zero, but always remaining negative. Our results for the case of Pb are shown in Fig. 5 and deviate significantly from the BCS behavior just described. The solid thick blue curve gives the result for 1/ o S 1/ o N at t=0.52 and shows a raid increase through 0 at twice the ga, after which a coherence eak is seen, followed by boson structure beyond which the curve crosses the axis again. The solid thin red curve is the result for the artial sum I. The dashed black curves are the BCS results for comarison. The crossing of the horizontal axis at a frequency slightly above the end of the honon sectrum in the case for Pb thin red curve is due to the inelastic scattering rocesses and is not seen in the BCS curve. Note also the large ositive eak. FIG. 5. Color online The difference of 1/ S o 1/ N o vs shown u to 35 mev and the artial sum, defined as I in the text. The black dashed curves are for the BCS case. In the inset, we comare data with theory. V. CONCLUSIONS We have resented new terahertz data on the otical conductivity of Pb and comared them with Eliashberg results with an emhasis on honon-assisted Holstein rocesses not art of BCS. We find that these show u most clearly in the otical self-energy rather than in the conductivity itself. In articular, the imurity comonent of the total scattering is seen directly in the otical scattering rate as a large constant which can easily be subtracted out to reveal directly the honon-assisted art. In the normal state, imurity scattering dros out of the otical effective mass, leaving only honon renormalizations which show excellent agreement between the theory and the exeriment. By contrast, in the suerconducting state, the imurities rovide a large renormalization of the zero frequency effective mass and cause a eak to aear at twice the ga in the otical mass 1+ o T,, which we redict here and observe in the exeriment. ACKNOWLEDGMENTS We thank S. Goshima, M. Mori, T. Suzuki, and H. Matsui for valuable assistance and advice in TDTS exeriments. The exerimental work at Sendai has been suorted by Grantsin-Aid for Scientific Research and from the Ministry of Education, Culture, Sorts, Science and Technology, Jaan. This work has also been suorted by the Natural Science and Engineering Council of Canada NSERC and the Canadian Institute for Advanced Research CIFAR. *nicol@hysics.uoguelh.ca 1 L. H. Palmer and M. Tinkham, Phys. Rev. 165, R. R. Joyce and P. L. Richards, Phys. Rev. Lett. 24, B. Farnworth and T. Timusk, Phys. Rev. B 10, ; 14, P. B. Allen, Phys. Rev. B 3, W. L. McMillan and J. M. Rowell, in Suerconductivity, edited by R. D. Parks Dekker, New York, 1969, Vol. 1, S. B. Nam, Phys. Rev. 156, J. R. Schrieffer, Theory of Suerconductivity Benjamin, New

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