We combine the results of terahertz time-domain spectroscopy with far-infrared transmission and reflectivity to obtain the conductivity of SrRuO 3 over an unprecedented continuous range in frequency, allowing us to characterize the approach to zero frequency as a function of temperature. We show that the conductivity follows a simple phenomenological form, with an analytic structure fundamentally different from that predicted by the standard theory of metals

Beasley, M. R.

Corson, J.

Dodge, J. S.

Orenstein, J.

Reiner, J. W.

Schlesinger, Z.

Weber, Christopher P.

English

Scholar Commons - Santa Clara University

Santa Clara UniversityScholar CommonsPhysics College of Arts & Sciences12-4-2000Low-Frequency Crossover of the Fractional Power-Law Conductivity in SrRuO3J. S. DodgeChristopher P. WeberSanta Clara University, cweber@scu.eduJ. CorsonJ. OrensteinZ. SchlesingerSee next page for additional authorsFollow this and additional works at: https://scholarcommons.scu.edu/physicsPart of the Condensed Matter Physics CommonsCopyright © 2000 American Physical Society. Reprinted with permission.This Article is brought to you for free and open access by the College of Arts & Sciences at Scholar Commons. It has been accepted for inclusion inPhysics by an authorized administrator of Scholar Commons. For more information, please contact rscroggin@scu.edu.Recommended CitationDodge, J. S., Weber, C. P., Corson, J., Orenstein, J., Schlesinger, Z., Reiner, J. W., & Beasley, M. R. (2000). Low-Frequency Crossover ofthe Fractional Power-Law Conductivity in SrRuO3. Physical Review Letters, 85(23), 4932–4935. https://doi.org/10.1103/PhysRevLett.85.4932AuthorsJ. S. Dodge, Christopher P. Weber, J. Corson, J. Orenstein, Z. Schlesinger, J. W. Reiner, and M. R. BeasleyThis article is available at Scholar Commons: https://scholarcommons.scu.edu/physics/23VOLUME 85, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 4 DECEMBER 2000Low-Frequency Crossover of the Fractional Power-Law Conductivity in SrRuO3J. S. Dodge,1,2,* C. P. Weber,1,2 J. Corson,1,2 J. Orenstein,1,2 Z. Schlesinger,3 J. W. Reiner,4 and M. R. Beasley41Department of Physics, University of California at Berkeley, Berkeley, California 947202Materials Sciences Division, E. O. Lawrence Berkeley National Laboratory, Berkeley, California 947203Department of Physics, University of California at Santa Cruz, Santa Cruz, California 950644Edward L. Ginzton Laboratories, Stanford University, Stanford, California 94305(Received 19 June 2000)We combine the results of terahertz time-domain spectroscopy with far-infrared transmission and re-flectivity to obtain the conductivity of SrRuO3 over an unprecedented continuous range in frequency,allowing us to characterize the approach to zero frequency as a function of temperature. We show that theconductivity follows a simple phenomenological form, with an analytic structure fundamentally differentfrom that predicted by the standard theory of metals.PACS numbers: 72.15.Qm, 74.20.Mn, 75.20.En, 78.66.BzOne of the most exciting proposals to emerge from thestudy of high-Tc superconductors is that Landau’s Fermiliquid theory (FLT) breaks down in the metallic state aboveTc [1]. This would have profound implications, since FLTprovides the foundation for our current understanding ofmetals, together with systems as diverse as liquid 3He andnuclear matter [2]. Evidence for its breakdown in high-Tcsuperconductors comes from a variety of experiments,including photoemission, electrical transport, and optics[3]. More recently similar evidence has been found inother compounds [4]. Here we show that the complexoptical conductivity sv, T  of one such material, the fer-romagnetic metal SrRuO3 [5–8], behaves according to aremarkably simple power-law form, which deviatessharply from the prediction of FLT. This observation pro-vides valuable insight into the nature of charge scatteringin unconventional metals.According to FLT, the qualitative properties of an in-teracting electron gas are the same as those of a non-interacting gas, if probed on a sufficiently low-energyscale. The optical conductivity of a system of noninter-acting, charge e carriers obeys the Drude form, sv ne2m1t 2 iv, where n is the carrier density andm and t are the effective mass and scattering time of thecarriers. FLT predicts that s at low v remains of the Drudeform in the presence of interactions, with a spectral weightwhich decreases as interactions increase. The f -sum ruledictates that the total spectral weight is conserved, so thatspectral weight must shift to higher energies. This addi-tional component to sv is known as the incoherent partof the intraband conductivity.Infrared reflectivity studies indicate that both high-Tcsuperconductors and SrRuO3 exhibit conductivitywith an anomalous power-law dependence on frequency,s1v ~ v2a , with a  0.5 for SrRuO3 [8] and a  0.7in the high-Tc materials [9,10]. The Drude form yieldss1 ~ v22 at comparable frequencies. If FLT is valid, theconductivity in excess of Drude must be identified withinterband transitions or the incoherent component ofthe spectrum, and there must be a crossover at lowerfrequency to the renormalized Drude conductivity [11,12].Recently, Ioffe and Millis suggested that the entire con-ductivity spectrum of the high-Tc materials could be un-derstood as a single component, rather than two, as inFLT [13]. The spectrum they derived was generalized byvan der Marel to the following useful form [14]:sv A1t 2 iva, (1)where A is a parameter with units that depend on thevalue of a. Ioffe and Millis obtained a  12 by assum-ing that the carrier lifetime depends strongly on its direc-tion of motion [15–18]. In the limit 1t ! 0, Eq. (1) issimilar to one derived by Anderson [19], under differentassumptions. Note that Eq. (1) includes the Drude formas a special case, with a  1. In allowing a to deviatefrom unity, we obtain the observed power-law dependenceof the conductivity on frequency. However, we also sub-vert a standard assumption of transport physics that theconductivity at v  0 is proportional to a scattering time.Instead, the dc conductivity is proportional to a fractionalpower of a scattering time, that is, sdc  Ata . Moreover,the analytical structure of sv changes, from having asimple pole at v  2it, to being multiple valued witha branch point there.Although the models described above are distinct, theymay be difficult to distinguish experimentally if the Drudecomponent predicted by FLT is masked by the incoher-ent conductivity. The most stringent test of FLT is at lowtemperature, where the Drude component, if one exists,would be sharpest. In the cuprates, the temperature rangeover which the normal state conductivity can be studied islimited by the onset of superconductivity. The absence ofsuperconductivity in SrRuO3 permits a much more mean-ingful test of FLT than previously available.We find that sv, T  of SrRuO3 at low temperatureis described well by Eq. (1) over nearly three decades inv (6 2400 cm21) with a  0.4, in strong disagreement4932 0031-90070085(23)4932(4)$15.00 © 2000 The American Physical SocietyVOLUME 85, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 4 DECEMBER 2000  0 0.2 0.4 0.6 0.8   1 1.2 1.4   0.0            0.5 0  10  20   [THz]φ σ(ω,T) [rad]|σ(ω,T)| [103 Ω-1cm-1]  0 0.5   1 01020αχ2~ω/2πFIG. 1. Measured amplitude (upper panel) and phase (lowerpanel) of sv, T in SrRuO3 at four representative temperatures.: T  8 K; : 40 K; : 60 K; and : 80 K. Lines arefits to the data using Eq. (1), with a  0.4. The inset showsthe reduced x2 error associated with the phase (solid line) andamplitude (dashed line) fits to Eq. (1), as a function of a.with FLT. Our results at frequencies below 100 cm21were obtained from transmission measurements on thinfilm samples of SrRuO3 grown epitaxially on NdGaO3substrates [20,21]. To probe the region where v  1t,we used conventional Fourier-transform infrared spec-troscopy to measure the transmittance (T ). For the rangev ø 1t, we used time-domain terahertz spectroscopy(TDTHz) to measure the complex transmission amplitudetv, T  in the millimeter wave region of the spectrum. Theresidual resistivity in these films is typically 50 mV cm.SrTiO3 substrates produce SrRuO3 films with lowerresistance, but this substrate material has such a largetemperature-dependent dielectric constant that an accuratedetermination of the conductivity from transmissionmeasurements is prohibitively difficult. At frequenciesabove 100 cm21, we measure the reflectivity from a thickfilm of SrRuO3 deposited on a SrTiO3 substrate [8]. Wehave derived sv, T  from each of these measurements,as described below.We use TDTHz to measure the complex transmissionamplitude tv of SrRuO3 in the range 0.2–1.2 THz(6 36 cm21) [22]. We compared the SrRuO3 film on itssubstrate to a bare NdGaO3 substrate at each temperature,using a vapor flow cryostat with a translating samplemount. The ratio of the two complex transmission ampli-tudes, tsr v  tsamplevtrefv, is a simple functionof the substrate index n, sv of the film, and the filmthickness d:tsrv n 1 1n 1 1 1 svZ0d. (2)Z0 is the impedance of free space. We inverted Eq. (2) toobtain both the amplitude and the phase of the complexconductivity as a function of temperature and frequency,which we show in Fig. 1 for our most thoroughly stud-ied SrRuO3 film. Also shown are the results of the bestfit to Eq. (1) with a  0.4, following the procedure de-scribed below. The statistical error on a typical measure-ment of the conductivity phase fsv, T   argsv, T is 60.02 rad at 1 THz, 2 orders of magnitude better thanthat obtained in typical reflectivity measurements at thisfrequency. The temperature dependence of the effectivepath length in the substrate and cryostat windows providesthe largest source of systematic error, which we estimateto be less than 60.03 rad at 1 THz. Errors in the mea-surement of the conductivity amplitude are dominated bystatistical uncertainty in the transmission amplitude, whichis typically 63%.To obtain an objective best fit of Eq. (1) to the TDTHzmeasurements, we first perform a least-squares fit of themeasured conductivity phase to fv  a tan21vt ateach temperature, for several values of a. This allowsus to determine tT ; a, the best fit value of t for eachtemperature, with different assumed values of a. Unlike aglobal fit to sv, this procedure is independent of theconductivity amplitude. Next we performed a separateleast-squares fit of the amplitude to jsvj  A1t2 1v2a2, using the t  tT ; a from the phase fits andallowing only A to vary. The quality of these fits [23]are shown as a function of a in the inset of Fig. 1. Thephase fits exhibit a weak optimum at a  0.4, with a sharpdecrease in quality below a  0.3 but with relatively littlechange in quality as a increases from 0.4 to unity. Theamplitude fits, on the other hand, worsen dramatically as achanges from 0.3 to unity. Taken together, these fits allowus to limit the range of acceptable values for a to 0.2–0.5.We develop this analysis further in Fig. 2, which showsa logarithmic plot of the conductivity amplitude at ourlowest frequency, v12p  0.2 THz, versus tT ; aa ,with t in femtoseconds. The plot is parametric in tem-perature for several different values of a. In Eq. (1),sv ! 0  Ata ; if A remains constant with tempera-ture, Eq. (1) will yield a straight line with unity slope onthis plot, indicated by the dotted line. The best fit to thisslope is obtained for a  0.4, and already at a  0.6 aclear deviation is observed. At higher values of a the slopedecreases yet further, so the case a  1 corresponding toDrude conductivity requires A to increase strongly withtemperature. Thus, a  0.4 6 0.1 provides not only thebest fit to Eq. (1), but also the most compact descriptionof the data.In Fig. 3 we show s1v of SrRuO3 over two and a halfdecades in frequency, obtained in three separate measure-ments. Data in the lowest frequency range is taken from4933VOLUME 85, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 4 DECEMBER 2000 2 10 20 51050{τ(T;α) [fs]}α|σ(ω1,T)| [103 Ω-1cm-1]FIG. 2. Logarithmic plot demonstrating the scaling relation-ship of sv1, T, v12p  0.2 THz, and tT ; a obtained byfitting to Eq. (1). The plot is parametric in temperature from5–92.5 K in steps of 2.5 K, for various choices of a: , 0.2;, 0.3; , 0.4; , 0.5; and , 0.6. The dotted line is given bysv  0 ~ ta .TDTHz measurements at the same four representative tem-peratures shown in Fig. 1, and extends from 6 36 cm21.In the intermediate frequency range 26 80 cm21, we havemeasured T v, T , which is a real quantity and thereforeincapable of providing the complex conductivity withoutfurther analysis. Using the tT  obtained from TDTHzfor a  0.4, we have calculated the conductivity phaseexpected in this frequency range at each temperature, thenused these phase values to calculate the conductivity ampli-tude directly from T v, T . The results of this procedureare shown for the same four temperatures as the TDTHzdata. The continuity of the results at the crossover fre-quency of these two distinct measurements may be takenas an indication of the high accuracy with which we havedetermined the conductivity. At frequencies probed by in-frared reflectivity, the conductivity is relatively tempera-ture independent below 100 K, and we show only onemeasurement taken at 40 K [8].With the parameters obtained from the TDTHz data,Eq. (1) may be used to predict the behavior of the con-ductivity at all frequencies and temperatures. The dot-ted lines in Fig. 3 show the conductivity calculated fromEq. (1), using the parameters obtained from TDTHz. Witha single global parameter A and a single temperature-dependent parameter tT , this model fits the data excep-tionally well, at frequencies 2 orders of magnitude higherthan those at which the parameters were obtained. As weincrease the temperature above 95 K, both our measure-ments and earlier reflectivity measurements begin to devi-ate from the form discussed here, and develop a pseudogapstructure which may be related to the transition from ferro-100101102103103104ω/2π [cm-1]σ 1(ω,T) [Ω-1cm-1]0 100 200 3000100200τ [fs]T [°K]FIG. 3. Logarithmic plot of the conductivity obtained by threemethods, in three ranges of frequency. The conductivity ob-tained from the infrared reflectivity at 40 K is indicated by thelong dashed line. Results from far-infrared transmission mea-surements, as described in the text, are indicated by solid lines,and TDTHz measurements by short dashed lines, with both setsordered in temperature from top to bottom with T  8, 40, 60,and 80 K. Least-squared fits to Eq. (1) using only TDTHz dataare shown by dotted lines. Inset: temperature dependence oft obtained for a  0.4 (closed circles), compared to Eq. (3)(solid line).magnetism to paramagnetism [8]. We leave a detailed dis-cussion of this behavior to a later publication, limiting ourdiscussion here to temperatures below 95 K, deep withinthe ferromagnetic state.As shown in Fig. 3, 1t sets the frequency scale atwhich the v2a divergence is cut off, forcing s1 ~ tain the dc limit. Thus the ubiquitous practice of inferringscattering times from electrical transport via sdc ~ t iserroneous whenever the conductivity behaves as Eq. (1)with a fi 1. The resistivity of SrRuO3 for 25 & T &120 K exhibits approximately linear temperature depen-dence, which then crosses over at lower temperatures tobecome constant as impurity scattering dominates [7]. Inour analysis, this implies a scattering rate with at least aquadratic dependence on temperature.A simple form which approximates the observed behav-ior ish̄tTh̄t01kBT2T0, (3)with a temperature independent, or elastic, scattering timet0 and a characteristic temperature T0 [13]. Here h̄ isPlanck’s constant and kB Boltzmann’s constant. This givesa temperature-dependent resistivity, rT   A 1 BT2,observed in the highest quality films at low temperatures4934VOLUME 85, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 4 DECEMBER 2000[24]. The inset of Fig. 3 shows tT ; 0.4 together withthe best fit to Eq. (3), with t0  198 fs and T0  40 K.The agreement is quite good over the entire temperaturerange, although in the region T . 95 K t is comparableto our measurement accuracy.It is interesting to note that despite the discrepancy ofEq. (1) with FLT, the functional form of Eq. (3) is exactlywhat FLT would predict for t, though with T0 at least2 orders of magnitude larger than 40 K. While evidencefor a low-temperature Fermi liquid state in SrRuO3 hasbeen provided by the observation of Shubnikhov–de Haasoscillations, with amplitudes that display the temperaturedependence of a Fermi liquid [24], the measurements pre-sented here indicate that this behavior breaks down at muchlower energies than would be predicted by conventionalFLT. The origin of this fragility, the nature of the low en-ergy scale T0 which appears to set a crossover scale, andthe way such behavior may coexist with the manifestlynon-Fermi liquid conductivity form of Eq. (1) remain im-portant open questions raised by these measurements.In summary, we have studied in detail the complex con-ductivity of SrRuO3 at low frequencies and temperatures,and shown that it agrees well with the simple phenomeno-logical form given in Eq. (1). The difference betweena  0.4 observed here and the Drude form expected fromFLT, with a  1, is reflected in the interpretation of t,one of the fundamental parameters in the transport theoryof metals. We have described how this quantity influencesthe measured conductivity, but proper interpretation of itsmicroscopic meaning must await further analysis. In par-ticular, careful studies of the dependence of t on substitu-tional impurities or structural disorder would help to revealits physical origin [13].J. S. D. thanks Daniel Chemla for encouragement inpursuing this work. This work was supported by theDirector, Office of Energy Research, Office of BasicEnergy Sciences, Division of Materials Sciences of theU.S. Department of Energy under Contract No. DE-AC03-76SF00098, by the Stanford NSF-MRSEC Pro-gram, and by NSF Grants No. DMR-0071949 andNo. 9870252.*Present address: Physics Department, Simon FraserUniversity, Burnaby, Canada V5A 1S6.Email address: jsdodge@sfu.ca[1] P. W. Anderson, Science 235, 1196 (1987).[2] D. Pines and P. Nozières, The Theory of Quantum Liquids,Vol. 1: Normal Fermi Liquids (Addison-Wesley, Reading,MA, 1989).[3] J. Orenstein and A. J. Millis, Science 288, 468 (2000).[4] M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys.70, 1039 (1998).[5] I. Bozovic et al., Phys. Rev. Lett. 73, 1436 (1994).[6] L. Klein et al., Phys. Rev. Lett. 77, 2774 (1996).[7] L. Klein et al., J. Phys. Condens. Matter 8, 10 111 (1996).[8] P. Kostic et al., Phys. Rev. Lett. 81, 2498 (1998).[9] Z. Schlesinger et al., Phys. Rev. Lett. 65, 801 (1990).[10] A. El Azrak et al., Phys. Rev. B 49, 9846 (1994).[11] D. B. Tanner and T. Timusk, in Physical Propertiesof High Temperature Superconductors III, edited byD. M. Ginsberg (World Scientific, Singapore, 1992),pp. 363–469.[12] A. Georges et al., Rev. Mod. Phys. 68, 13 (1996).[13] L. B. Ioffe and A. J. Millis, Phys. Rev. B 58, 11 631 (1998).[14] D. van der Marel, Phys. Rev. B 60, R765 (1999).[15] A. Carrington et al., Phys. Rev. Lett. 69, 2855 (1992).[16] R. Hlubina and T. M. Rice, Phys. Rev. B 51, 9253(1995).[17] B. Stojković and D. Pines, Phys. Rev. Lett. 76, 811(1996).[18] T. Xiang and W. N. Hardy, cond-mat/0001443.[19] P. W. Anderson, Phys. Rev. B 55, 11 785 (1997).[20] C. H. S. Ahn, Ph.D. thesis, Stanford University, 1996.[21] C. B. Eom et al., Science 258, 1766 (1992).[22] M. C. Nuss and J. Orenstein, in Millimeter and Submil-limeter Wave Spectroscopy of Solids, edited by G. Grüner,Topics in Applied Physics Vol. 74 (Springer, Berlin, 1998),pp. 7–50.[23] The reduced x2 values shown are obtained by fixing a,finding the x2 error on fits to Eq. (1) for all temperaturesfrom 5–92.5 K and all frequencies from 0.2–1.2 THz, thendividing by the total number of degrees of freedom, equalto the number of data points minus the number of freeparameters in the fit.[24] A. P. Mackenzie et al., Phys. Rev. B 58, R13 318(1998).4935

Low-Frequency Crossover of the Fractional Power-Law Conductivity in SrRuO\u3csub\u3e3\u3c/sub\u3e

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Low-Frequency Crossover of the Fractional Power-Law Conductivity in SrRuO\u3csub\u3e3\u3c/sub\u3e

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