Nodal Quasiparticle Dynamics in the Heavy Fermion Superconductor Cecoin5 Revealed by Precision Microwave Spectroscopy

ARTICLE

Received 18 Oct 2012 | Accepted 21 Aug 2013 | Published 20 Sep 2013 DOI: 10.1038/ncomms3477 Nodal quasiparticle dynamics in the heavy fermion superconductor CeCoIn5 revealed by precision microwave spectroscopy

C.J.S. Truncik1, W.A. Huttema1, P.J. Turner1,S.O¨ zcan2, N.C. Murphy1, P.R. Carrie`re1, E. Thewalt1, K.J. Morse1, A.J. Koenig1, J.L. Sarrao3 & D.M. Broun1

CeCoIn5 is a heavy fermion superconductor with strong similarities to the high-Tc cuprates, including quasi-two-dimensionality, proximity to antiferromagnetism and probable d-wave pairing arising from a non-Fermi-liquid normal state. Experiments allowing detailed com- parisons of their electronic properties are of particular interest, but in most cases are difﬁcult to realize, due to their very different transition temperatures. Here we use low-temperature microwave spectroscopy to study the charge dynamics of the CeCoIn5 superconducting state.

The similarities to cuprates, in particular to ultra-clean YBa2Cu3Oy, are striking: the frequency and temperature dependence of the quasiparticle conductivity are instantly recognizable, a consequence of rapid suppression of quasiparticle scattering below Tc; and penetration-depth data, when properly treated, reveal a clean, linear temperature dependence of the quasiparticle contribution to superﬂuid density. The measurements also expose key differences, including prominent multiband effects and a temperature-dependent renormalization of the quasiparticle mass.

1 Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6. 2 Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK. 3 Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. Correspondence and requests for materials should be addressed to D.M.B. (email: [email protected]).

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n CeCoIn5, evidence for d-wave pairing comes predominantly superconductor is quantum mechanical in origin, and the from experiments that infer the presence and location of nodes fundamental electrodynamic relation is between the current Iin the superconducting energy gap. This includes power laws density and the vector potential13. We therefore imagine a metal in zero-field heat capacity1–3 and thermal conductivity2,3, and the or superconductor perturbed by the sudden application of a field-angle dependence of heat capacity4,5, thermal conductivity6 vector potential dA. As a result, all carriers experience an impulse and quantum oscillations in the superconducting state7. These À qdA, where q is the charge of the carriers. The impulse sets the experiments are supported by evidence for spin–singlet pairing electron assembly into motion with average velocity v ¼ÀqdA/ 8,9 (decreasing Knight shift below Tc and Pauli-limited upper m*, where m* is the effective mass of the carriers. This is sketched critical field10) and by the nature of the spin-resonance peak11. in the centre column of Fig. 1 and corresponds to a current However, the emerging picture of dx2 À y2 pairing symmetry in response that opposes the applied field. A measurement of CeCoIn5 is complicated by observations on the isoelectronic current density immediately after the application of the field homologue CeIrIn5, which suggest a hybrid order parameter with reveals a diamagnetic contribution both line nodes and point nodes12. 2 Measurements of London penetration depth, lL, should nq jd ¼À dA ; ð1Þ provide a particularly clean test of nodal structure, as lL is a mÃ thermodynamic probe that couples preferentially to itinerant electronic degrees of freedom13–16. However, penetration-depth where n is the carrier density. Note that the strength of the data on CeCoIn5 remain surprisingly unclear. Instead of the diamagnetic contribution is proportional to n/m* andpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is closely 2 Ã linear temperature dependence expected for line nodes, all related to the plasma frequency of the carriers, op ¼ nq =m E0. penetration-depth measurements to date17–19 report temperature Care must be taken with the definition of ‘sudden’: if the carriers power laws ranging from T1.2 to T1.5.Thispresentsa are excited with an arbitrarily sharp impulse, n will be the total conundrum—mechanisms such as impurity pair-breaking20,21 electron density, including core electrons, and m* the bare and nonlocal electrodynamics22 should cause a crossover to electron mass, devoid of interaction effects. For practical quadratic temperature dependence, but attempts to understand the purposes, a time scale is chosen that excites free carriers but behaviour in terms of impurity physics require unrealistically high avoids inter-band transitions. levels of disorder23. Measurements of the frequency-dependent In addition to inducing a diamagnetic current, dA changes the superfluid density offer the potential for new insights into this energy of the electron states, tilting the energy dispersion in k- problem. space by an amount dEk ¼Àq:/m* Â k Á dA. Immediately after To properly understand what microwave properties can tell us excitation, the electron assembly is therefore in a non-equilibrium about a material14,24, it is helpful to visualize the measurement configuration. Equilibrium is subsequently restored by scattering process in the time domain. The Meissner response of a processes that transfer momentum to the crystal lattice. By

jtot = jd jp

m*v m*v m*v Metal =+

m*v m*v m*v s-wave =+ superconductor

m*v m*v m*v d-wave =+ superconductor

Figure 1 | Superfluid density and the paramagnetic back-flow of excitations. A momentum-space picture of the electron assembly in a superfluid density experiment, with electron-like excitations shown as filled circles and hole-like excitations as open circles. Measurements of superfluid density probe the total non-decaying current in thermal equilibrium (left-hand figures), which can be decomposed into the sum of two terms13. Immediately after the application of a vector potential dA, the Fermi sea is displaced by an amount m*v ¼ÀqdA (central figures) giving rise to a diamagnetic current. The vector potential also tilts the energy dispersion (dEk ¼Àq:/m* Â k Á dA) and the resulting redistribution of electrons on the approach to equilibrium produces a paramagnetic back-flow current (right-hand figures). (a) The equilibrium current density in a metal is zero: diamagnetic and paramagnetic currents are equal and opposite, and cancel. (b) The paramagnetic back-flow current in an s-wave superconductor is strongly suppressed by the opening of an isotropic energy gap. The diamagnetic current is little changed from that in the normal state and a net current therefore flows in equilibrium, giving rise to a Meissner effect. (c)Inad-wave superconductor, the paramagnetic back-flow consists predominantly of nodal quasiparticles, which make a linear-in-temperature contribution to superfluid density. However, in CeCoIn5 the diamagnetic contribution also has temperature dependence, probably due to the material’s proximity to a quantum critical point. Isolating the nodal quasiparticle contribution requires that the current response be measured over a wide frequency range, with high frequencies probing the diamagnetic response and low frequencies the total current in equilibrium. Measurements at intermediate microwave frequencies probe the transient processes that lead to the formation of the paramagnetic back-flow current, and provide a wealth of information on the charge dynamics of the nodal quasiparticles14.

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studying the current response in this regime, we learn a great deal CeCoIn5, and is the reason why anomalous temperature power about electronic relaxation mechanisms in the material. laws have been obtained in earlier experiments. Our measure- At times long enough for the electron assembly to have ments reveal that the diamagnetic response (the plasma returned to equilibrium, it is useful to define the total current frequency) of CeCoIn5 weakens on cooling, in a manner density as the sum of diamagnetic and paramagnetic pieces13, corresponding to an increase in quasiparticle effective mass. That jtot ¼ jd þ jp. As discussed above, jd represents the instantaneous this occurs in CeCoIn5 is not too surprising, as it is suggestive of diamagnetic response intrinsic to the electronic states, with the proximity to a quantum critical point25,31. In addition, the tacit understanding that jd has been measured slowly enough to microwave measurements provide a detailed picture of the include only free carriers. The paramagnetic part, jp, is of a very quasiparticle charge dynamics in CeCoIn5, revealing strong different character—it captures the change in current resulting similarities to the cuprate high-temperature superconductors from the reorganization of electrons in the new equilibrium state. and pointing to a common scattering mechanism. jp is therefore very sensitive to the details of the electronic energy spectrum, making it a powerful probe of pairing symmetry in a Results superconductor. The way this process plays out is illustrated in Surface impedance. Measurements of surface impedance, Fig. 1, for a metal and for s-wave and d-wave superconductors. Zs ¼ Rs þ iXs, have been made using resonator perturba- For the metal in equilibrium, jtot ¼ 0: the paramagnetic 14,24,32–36 tion . The sample, a small single crystal of CeCoIn5,is redistribution of electronic occupation results in a back-flow placed inside a dielectric resonator at a maximum in the current of equal but opposite magnitude to the initial diamagnetic microwave magnetic field, as shown in Fig. 2. Screening cur- shift. For a superconductor, in contrast, the equilibrium current rents are induced to flow near the sample surface and penetrate a density in the presence of a magnetic field is non-zero—there is a skin depth d. In the penetrated region, energy is stored both as Meissner effect. The strength of the diamagnetic contribution is field energy and as the kinetic energy of the superelectrons14— unchanged by the onset of superconductivity. Instead, the this leads to a surface reactance XsEom0d. Field penetration opening of a superconducting gap dramatically weakens the changes the effective volume of the resonator and hence its paramagnetic response. The paramagnetic term is strongly temperature dependent, in principle going to zero in a clean d superconductor at zero temperature. The form of this tempera- Recondensing ture dependence is highly sensitive to the structure of the energy Sample cryocooler loading c gap, in particular to the presence of gap nodes. For CeCoIn5, Microwave interlock which is thought to be a d-wave superconductor with line nodes network in the energy gap, the expected behaviour is a linear temperature analyser dependence of jp. However, a complication now arises: the experimentally accessible quantity in a penetration-depth experi- b ment is not the paramagnetic current density jp, but the total TE011 current density jtot. Most experiments skirt this issue by assuming 2.91 GHz that the diamagnetic response, jd, is temperature independent: it is difficult to measure directly, and in most superconductors has Transverse electric TE013 little temperature dependence anyway. resonant 4.82 GHz Although a time-domain picture provides a useful means of modes understanding electrodynamic measurements, the experiments themselves are usually carried out in the frequency domain, TE021 in our case using a set of discrete frequencies ranging from 5.57 GHz o/2p ¼ 0.13 to 19.6 GHz. Low frequencies measure the long-time behaviour and are sensitive to the equilibrium supercurrent a density. High frequencies probe the short-time behaviour and, if carried out in a regime in which o is greater than the electronic Removable t Single- sample relaxation rate 1/ , probe the instantaneous diamagnetic response crystal thermal 3He–4He sample stage and therefore the plasma frequency of the entire electron dilution assembly. At intermediate frequencies, much information can refrigerator be obtained on the scattering dynamics of the thermally excited 14 quasiparticles . This is of particular interest in CeCoIn5 because normal-state transport measurements reveal strong inelastic scattering and non-Fermi-liquid behaviour25,26. In the cuprates, where similarly strong scattering is observed in the normal Dielectric resonator state27, electrodynamic measurements show a rapid collapse in 28–30 quasiparticle scattering on cooling through Tc , indicating Figure 2 | Millikelvin microwave spectroscopy. (a) A platelet single- that the charge carriers couple to a spectrum of fluctuations of crystal of CeCoIn5 is mounted on a removable thermal stage and introduced electronic origin, in contrast to the phonons of a conventional into a dielectric resonator. (b) The resonator is excited in multiple metal. Early measurements on CeCoIn5 are suggestive of similar transverse electric (TE) modes at different frequencies, all characterized by behaviour2,17. a local maximum of the microwave magnetic field (red lines) at the centre Here we solve the puzzle of the anomalous temperature power of the resonator. This induces in-plane screening currents that flow across laws in penetration depth using comprehensive measurements of the broad faces of the CeCoIn5 crystal. (c) The resonator is mounted below the frequency-dependent superfluid density. These allow us to the mixing chamber of a 3He–4He dilution refrigerator. The sample stage is isolate the nodal quasiparticle contribution to London pene- loaded from room temperature through a vacuum interlock, and can be tration depth, revealing that it is accurately linear in temperature. cooled to 0.08 K. A recondensing cryocooler eliminates helium boil off. In the process, we find that the fundamental assumption of a (d) Shifts in sample surface impedance cause changes in resonance line temperature-independent diamagnetic response breaks down in shape that are read out by a low-noise microwave network analyser.

NATURE COMMUNICATIONS | 4:2477 | DOI: 10.1038/ncomms3477 | www.nature.com/naturecommunications 3 & 2013 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms3477 resonant frequency14. Although superconductors have perfect effect of the quasiparticles becomes indistinguishable from that of zero-frequency conductivity, the finite inertia of the electrons the superfluid. This leads to a two-fluid model of the microwave means that accelerating them at high frequencies requires conductivity14 significant electric field at the sample surface: the strength of the field is determined by Faraday’s law and grows in proportion to both the frequency o and the depth of field penetration. 1 The electric field couples to quasiparticle excitations in the sðo; TÞs1 À is2 ¼ 2 þ sqpðo; TÞ : ð2Þ 14 iom0lLðTÞ superconductor producing a surface resistance, Rs, proportional to the power absorption. This grows as the square of electric field, and therefore approximately as o2. This dissipation is measured by monitoring the quality factor of the resonator14,24,36. The In our experiments, the microwave conductivity is obtained from the surface impedance assuming the local electrodynamic complete set of surface impedance data is presented in Fig. 3. 14,24 2 relation s ¼ iom0=Zs .

2 Microwave conductivity. In the frequency domain, current Superfluid density. The static superfluid density, 1=lL,is density j is related to electric field E by a complex-valued micro- obtained from the complex conductivity in the zero-frequency wave conductivity14,24: j(o) ¼ s(o)E(o). In a superconductor, the limit: dominant contribution to the complex conductivity is a purely imaginary response associated with the superfluid density: s ¼ 1=iom l2. There is an additional contribution to the 2 s 0 L 1=l ¼ lim om s2 : ð3Þ L o 0 conductivity, sqp, arising from the non-equilibrium response of !0 the quasiparticles as they relax back to equilibrium—this derives from the transient response to the applied field and contains important information on relaxation mechanisms. sqp is in general At finite frequencies, Im{sqp(o,T)} makes a significant con- complex, but is predominantly real for low frequencies, o oo 1=t, tribution to s2: this screening effect is particularly prominent in where it represents microwave power absorption, becoming ima- CeCoIn5 due to its long quasiparticle lifetimes. To separate these ginary at high frequencies, o441=t, where the field-screening two components, we define a frequency-dependent superfluid

a b 50.0 50

40 10.0 ) ) 5.0 Ω 30 Ω (m R (m S S S Z 20 R 1.0 XS 0.5 10

0 0.1 0 5 10 15 20 25 30 35 0.0 0.5 1.0 1.5 2.0 2.5 Temperature (K) Temperature (K) c 0.4 f (GHz) 0.13 0.3 2.25

Hz) 2.91 / 4.82 μΩ 0.2 (

f 5.57 /

S 6.94 X 0.1 12.28 15.92 19.63 0.0 0.0 0.5 1.0 1.5 2.0 2.5 Temperature (K) Figure 3 | Microwave surface impedance. (a) Surface impedance at 2.91 GHz, showing the results of the normal-state matching technique used to determine absolute reactance: Rs(T) and Xs(T) track well between T ¼ 10 and 35 K, a range in which the quasiparticle relaxation rate is much greater than the measurement frequency. (b) Surface resistance at frequencies from 2.91 to 19.6 GHz, on a logarithmic scale. Absolute surface resistance is determined by a combination of cavity perturbation and in situ, resonator-based bolometry. (c) Surface reactance, at all frequencies measured. For clarity, pﬃﬃ Xs(T) is scaled by 1= f, to factor-out the expected frequency dependence well above Tc.

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30 f (GHz) 1.0 10 )

0.13 T 0.8 ( 25 2.25 2

0.6 )

–2 2.91 8 0.4 s–wave (0)/ m

4.82 2 ) μ 20 0.2 ( 5.57 –2 2 0.0 m 0 6.94 6 0.0 0.2 0.4 0.6 0.8 1.0 μ (

15 12.28 & T/T 2 c ≡ 15.92 / ) 1 19.63 % 4 ( 10 Δ 2 1/2 L 1/ 2 /2 5 1 p a + bT Δ % /2& 1.25 1 L a T 0 0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Temperature (K) Temperature (K) 2 Figure 4 | Superfluid density of CeCoIn5. (a) Frequency-dependent superfluid density, 1/d (o,T)om0s2(o,T), plotted as a function of temperature, 2 2 for frequencies from 0.13 to 19.6 GHz. 1=lL ðTÞ, the zero-frequency limit of 1/d (o,T), is obtained from fits to complex conductivity spectra and lies on top of 2 the 0.13-GHz data. lL(T-0) ¼ 1,960 Å. The inset shows a close-up of the low-temperature region, in which the temperature slope of 1/d (o,T) changes 2 2 2 1.25 sign with increasing frequency. (b) The temperature-dependent part of the total superfluid density, Dð1=lL Þ1=lL ðT ! 0ÞÀ1=lL ðTÞ, follows a T 2 2 2 power law. The paramagnetic part of the superfluid density, 1=lp 1=d ð19:6 GHz; TÞÀ1=lL ðTÞ, isolates the contribution from nodal quasiparticles and follows a linear temperature dependence. Its zero-temperature intercept indicates a residual, uncondensed spectral weight of 7%. Inset: the normalized superfluid density of an s-wave superconductor, calculated using Mattis–Bardeen theory37 for the same set of reduced frequencies (and same colour scheme) as the CeCoIn5 experiment. In the s-wave case, the isotropic energy gap leads to exponentially activated behaviour at low temperatures.

density, for a conventional metal, the temperature slope of om0s2(o,T ) =l2 1 changes sign at high frequencies, a clear indication that 1 d has 2 om0s2ðo; TÞ temperature dependence in CeCoIn5. The observed behaviour d ðo; TÞ corresponds to an effective mass that increases on cooling, which 1 ÈÉ was raised as a possibility in earlier work19. This is probably a ¼ 2 À om0Im sqpðo; TÞ : ð4Þ l ðTÞ consequence of the proximity of CeCoIn5 to quantum L 25 2 criticality . The quasiparticle relaxation contribution to 1/d (o) vanishes Interestingly, de Haas-van Alphen measurements made at low in the static limit, but is non-zero at all finite frequencies. At high fields (6–7 T) on CeCoIn show an extreme departure from the - 2 5 frequencies, oIm{sqp} nqpe /m*, providing a measure of the standard Fermi-liquid, Lifshitz–Kosevich model, but are well quasiparticle density, nqp, and therefore the uncondensed described by a non-Fermi-liquid theory based on antiferromag- oscillator strength in the quasiparticle spectrum. Equation 4 netic quantum criticality38. Such behaviour can also be allows an unambiguous separation of equilibrium-superfluid and interpreted as a temperature-dependent quasiparticle mass. At quasiparticle-relaxation effects, as long as data are taken over a higher fields (13–15 T), where the material is tuned away from wide frequency range. The frequency-dependent superfluid quantum criticality, the quantum oscillations revert to the density of CeCoIn5 is plotted in Fig. 4a for frequencies ranging standard Lifshitz–Kosevich form, in which quasiparticle mass is from 0.13 to 19.6 GHz. We see that the frequency dependence temperature independent. of 1/d2 is indeed very strong and note the difficulty of isolating 2 2 The paramagnetic contribution to the superfluid density, 1=lp, 1=lLðTÞ from a measurement at any single microwave frequency. 2 which is sensitive to the nodal structure of the order parameter, 1=lLðTÞ has a strong temperature dependence across the whole can now be isolated via the relation temperature range, in contrast to an s-wave superconductor15, but is not strictly linear at low temperatures, as reported in 1 1 1 ¼ À : ð6Þ previous studiesÀÁ17–19. This can be seen clearly in Fig. 4b, where l2 l2 l2 2 1.25 L d p we show that D 1=lLðTÞ is well described by a T power law. 2 2 Note that our experiment directly measures the absolute In Fig. 4b we plot 1=lpðTÞ, using measurements of 1/d (o,T)at penetration depth, and therefore the curvature is not the result 19.6 GHz as a proxy for 1=l2 below 0.6 K. The paramagnetic - d of uncertainties in the absolute value of lL(T 0). back-flow term, when properly isolated, is linear in temperature, To resolve this puzzle, we allow for the possibility that the providing direct evidence that the low-energy quasiparticles have instantaneous diamagnetic response of CeCoIn5 is temperature a nodal spectrum and giving strong support for a d-wave pairing dependent, as discussed in the Introduction. In analogy with the state. 2 superfluid density 1=lL, which is obtained from the static limit of 2 2 om0s2, we define a diamagnetic contribution 1=ld.1=ld is Microwave spectroscopy. At finite frequencies, the real part of proportional to the conduction electron density, n, and a Fermi the microwave conductivity, s1, is entirely due to quasiparticle surface average of the inverse of the effective mass, m*, and can be 14 relaxation . Data for s1 are presented in Fig. 5a as a function of accessed experimentally as the high-frequency limit of om s : 0 2 temperature. At all frequencies measured, s1(T) shows an initial rise on cooling through T ¼ 2.25 K and a broad peak at inter- 1 2 1 c 2 ¼ m0ne ¼ lim om0s2ðoÞ : ð5Þ mediate temperatures. Note that the form of this peak is very l mÃ o441=t d FS different from the conductivity coherence peak in an s-wave In our experiment, the condition that frequency be much larger superconductor15. Theoretical curves37 for the conductivity of an than the quasiparticle relaxation rate, 1/t, is satisfied at the s-wave superconductor are shown in the inset of Fig. 5b; this lowest temperatures and highest frequencies. In Fig. 4a we see behaviour has been confirmed by a number of classic experiments that instead of becoming temperature independent, as expected on conventional superconductors such as Al (refs 39, 40) and Pb

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f (GHz) 2.5 f (GHz) 3.0 2.91 1.1 1.5

4.82 2.2 n 2.0 s–wave / 1.0

2.5 13.4 1

5.57 ) ) –1 –1 6.94 22.7 0.5 2.0 12.28 1.5 75.3 cm cm 15.92 0.0 –1 –1 0.0 0.2 0.4 0.6 0.8 1.0 1.5 19.63 YBa Cu O μΩ

μΩ 1.0 2 2 6.993 T/Tc

dc ( ( b –axis 1 1

1.0 bgnd 0.5 0.5

0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 0 20406080 Temperature (K) Temperature (K)

Figure 5 | Quasiparticle conductivity of CeCoIn5 and YBa2Cu3O6.993. (a) Real part of the conductivity, s1, of CeCoIn5 as a function of temperature, for discrete frequencies from 2.91 to 19.63 GHz. Also plotted are parameters from ﬁtting to conductivity spectra: sbgnd(T); and sdc(T) ¼ s0(T) þ sbgnd(T). Shaded conﬁdence bands denote standard errors in these parameters. (b) For comparison, the real part of the b-axis conductivity of Tc ¼ 89 K 42 YBa2Cu3O6.993, as a function of temperature, at frequencies from 1.1 to 75.3 GHz (data from Harris et al. ). Inset: the normalized quasiparticle conductivity of an s-wave superconductor, calculated using the Mattis–Bardeen theory37 for the same set of reduced frequencies (and same colour scheme) as the

CeCoIn5 experiment: a prominent BCS coherence peak is observed immediately below Tc, with exponential freeze-out at low temperatures.

3.0 T (K) T (K) 0.15 2.5 2.5 1.00 ) 0.20 )

–1 –1 1.25 0.30 2.0 2.0 1.50 cm 0.50 cm

–1 –1 1.75 0.75 1.5 1.5 2.00 μΩ 1.00 μΩ ( ( 1 1 1.0 1.0

0.5 0.5

0.0 0.0 0 5 10 15 20 0 5 10 15 20 Frequency (GHz) Frequency (GHz)

30 35 2.0 35 1.8 ) 30 1.6 y

–2 25 30 1.4

m 25 1.2 μ

( 20 25 20 1.0 2 0.0 0.5 1.0 1.5 2.0 15 0 T (K)

20 15 (GHz) 10 ≡ 15 5 ) 1/2

10 0

( 10 0 2468 2 3 3 T (K )

1/ 5 5

0 0 0 5 10 15 20 0.0 0.5 1.0 1.5 2.0 Frequency (GHz) Temperature (K)

Figure 6 | Complex conductivity spectra and model ﬁts. (a,b) Real part of the conductivity as a function of frequency, at discrete temperatures.

(c) Frequency-dependent superfluid density at the same set of temperatures as in a and b. The curves in a, b and c denote simultaneous fits to s1(o) and 1/d2(o) at each temperature, using the three-component conductivity model described in Methods. (d) The relaxation rate 1/t(T) obtained from the 3 fits, as a function of temperature. The primary inset shows 1/t(T)p asffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a function of T . Vertical bars indicate standard errors. The red lines are a fit to the 3 3 2 function h=tðTÞ¼h=t0 þ AkBT =D ðTÞ, with DðTÞ¼D0 tanhð2:4 Tc=T À 1Þ, D0 ¼ 3 kBTc and A ¼ 3.36. The secondary inset shows y(T), the best-fit frequency exponent of the modified Drude spectrum in equation 8. The shaded band denotes the 1–s confidence interval. y(T) is constrained to lie in the range 1oyr 2.

(ref. 41). Instead of arising from BCS coherence factors, the connection in the underlying charge dynamics. However, the fact conductivity peak in CeCoIn5 is the result of a sharp collapse in that the frequency scales are similar is somewhat puzzling, con- quasiparticle scattering below Tc that outpaces the gradual sidering the large difference in energy scales such as Tc: the reason condensation of quasiparticles into the superﬂuid. For for this is that while inelastic scattering rates scale as Tc in the two comparison, we plot the b-axis conductivity of ultra-pure materials, elastic scattering rates are determined by disorder and YBa2Cu3O6.993 (ref. 42) in Fig. 5b. The qualitative similarities of do not follow the same scaling. Nevertheless, the similarities in s1(o,T) in the two materials are striking, revealing a deep s1(o,T) provide strong support to the conjecture that these

6 NATURE COMMUNICATIONS | 4:2477 | DOI: 10.1038/ncomms3477 | www.nature.com/naturecommunications & 2013 Macmillan Publishers Limited. All rights reserved. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms3477 ARTICLE materials are different manifestations of the same correlated superconductor, both regimes are expected to carry signatures electron problem: two-dimensional (2D) d-wave super- of the nodal quasiparticle spectrum, namely, the linear energy conductivity proximate to antiferromagnetism1,43–45. dependence of the density of states47–50. Quantitative insights into the quasiparticle dynamics are Quasiparticle scattering by impurities (elastic scattering) has obtained from conductivity frequency spectra, plotted in Fig. 6. been studied extensively in d-wave superconductors21,47,51,52.In The collapse in scattering inferred from s1(T) can now be seen the strong-scattering regime, impurities have a pair-breaking directly as the narrowing of s(o) on cooling. We simultaneously effect, causing a crossover to T2 behaviour in the low-temperature fit to the real and imaginary parts of s(o) using a multi- superfluid density. We are able to rule out this type of scattering component conductivity model, which builds on equation 2 by in CeCoIn5 on the the basis of the data in Fig. 4a. In general, the introducing a particular form for the quasiparticle conductivity quasiparticle scattering rate is determined by the phase space for spectrum, as described in Methods. The first piece of this recoil—in the weak-scattering (Born) limit, the elastic scattering conductivity model is the superfluid term. Its weight is rate acquires a linear energy dependence (and therefore a linear 2 proportional to 1=lLðTÞ, which is plotted in Fig. 4a and is temperature dependence) that reflects the structure of the clean tightly constrained by the 0.13 GHz data. For sqp(o) we use a d-wave density of states. two-component model consisting of a narrow, Drude-like Inelastic scattering can occur by a number of mechanisms, but spectrum, whose width gives the average quasiparticle relaxation the proximity to antiferromagnetism in CeCoIn5 makes spin- rate 1/t, and a frequency-independent background conductivity, fluctuation scattering an important candidate. Curiously, both sbgnd. The Drude-like spectrum has been modified by the spin-fluctuation scattering and direct quasiparticle–quasiparticle inclusion of a conductivity frequency exponent, y(T), which scattering are expected to give rise to a T3 temperature controls the detailed shape of the spectrum and allows for a dependence in a d-wave superconductor47–50. On closer distribution of quasiparticle relaxation rates42,46. The need for a inspection, this simply reflects the fact that a spin fluctuation is two-component quasiparticle spectrum is most apparent in the a correlated electron–hole pair: in the superconducting state, low-temperature traces in Fig. 6a, which reveal narrow spectra, correlations between electrons and holes weaken, and the spin- 3–4 GHz wide, riding on top of a broad background: a single- fluctuations increasingly resemble a dilute quasiparticle gas47. component spectrum cannot simultaneously capture the narrow, However, the T3 scattering rate should not ordinarily be 47 long-lived part of s1(o) and the frequency-independent beha- observable directly in electrical transport: Walker and Smith viour above 12 GHz. In the absence of higher frequency data, we have noted that charge currents require umklapp processes to simply approximate the broad part of the spectrum as a constant. relax, in order that net momentum be removed from the electron We will later see that such a model is well motivated by the system during scattering. For d-wave quasiparticles, which at low multiband nature of CeCoIn5 (ref. 3). Two of the fit parameters temperatures are confined to the vicinity of the nodal points, are plotted in Fig. 5a: sdc(T), the zero-frequency limit of the momentum conservation leads to a minimum energy threshold, quasiparticle conductivity; and the background conductivity or ‘umklapp gap’, DU, as illustrated in Fig. 7. Below this threshold, sbgnd(T). The quasiparticle relaxation rate, 1/t(T), is plotted in umklapp processes cannot be excited and hence are frozen out, 2 Fig. 6d. The conductivity frequency exponent, y(T), is plotted in with 1=tumklapp T expðÀDU=kBTÞ at low temperatures. the inset of Fig. 6d. We now turn to the data on CeCoIn5. Immediately E above Tc, 1/2ptn 120 GHz (normal-state quasiparticle lifetime, 2 tn ¼ s1m0l0, is obtained by using the zero-temperature penetra- Discussion tion depth as a gauge of plasma frequency.) In temperature units, : The quasiparticle scattering dynamics of a superconductor /kBtn ¼ 6 K, several times larger than Tc, placing CeCoIn5 in a separate broadly into two regimes: low-temperature elastic similar regime of strong inelastic scattering as the cuprates27.On scattering due to static disorder; and inelastic scattering, which cooling into the superconducting state, 1/t(T) quickly drops into becomes important at higher temperatures. In a d-wave the low microwave range, where we can resolve it directly in the

Y Y M M k1' k ' G G q 2 k2 Umklapp k k1 k2 threshold 1 k2 k1

X X k2'

k ' k1' k2' 1 q q+G k2' k1'

k1 k2

Figure 7 | Quasiparticle–quasiparticle scattering and the umklapp gap. (a) Two quasiparticles, of wave vectors k1 and k2, interact and scatter into states 0 0 0 0 k1 and k2, exchanging momentum q. Such a scattering process leaves the net momentum in the electron system unchanged (k1 þ k2 ¼ k1 þ k2) and is therefore ineffective at relaxing an electrical current. (b) In a solid, crystal momentum is only conserved to within a reciprocal lattice vector G: that is, 0 0 a k1 þ k2 ¼ k1 þ k2 þ G. Umklapp processes, for which G 0, transfer momentum from the electron assembly to the crystal lattice and are very effective at relaxing an electrical current. (c) An umklapp process in a single-band d-wave superconductor47. A hole-like Fermi sea, characteristic of a cuprate superconductor, is shown shaded in blue, with nodes (open circles) on the zone diagonals. To conserve crystal momentum, a near-nodal quasiparticle, k1, must be partnered with a second quasiparticle, k2, located well away from a node. The energy threshold for this process is the umklapp gap, and strongly 7,53 suppresses quasiparticle–quasiparticle scattering at low temperatures. (d) The Fermi surface of CeCoIn5 in the kz ¼ 0 plane , illustrating the material’s multiband nature. An inter-band umklapp process is shown, in which a nodal quasiparticle in a quasi-2D band (dark blue) scatters from an electron in one of the light, 3D Fermi pockets (red). Thermodynamic experiments indicate that superconductivity in the light, 3D pockets is weak3 and this should substantially reduce the umklapp threshold.

NATURE COMMUNICATIONS | 4:2477 | DOI: 10.1038/ncomms3477 | www.nature.com/naturecommunications 7 & 2013 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms3477 width of the conductivity spectra. Below 1 K, in the disorder- 1.5 K, the best-fit value of y(T) sits at the Drude limit, y ¼ 2, dominated elastic regime, the relaxation rate reaches a residual implying that the low-energy quasiparticles relax at value of 1/2pt0 ¼ 3 GHz. The observation of a roughly approximately the same rate—this is consistent with the weak temperature-independent relaxation rate implies an energy- temperature dependence of 1/t in this range and reflects the independent phase space for recoil, and is difficult to additional phase space for recoil provided by the multiband Fermi understand in the context of simple d-wave superconductivity. surface. Above 1.5 K, y(T) drops quickly, falling below 1.2 on the 7 However, CeCoIn5 displays prominent multiband effects . approach to Tc. This indicates a rapidly broadening distribution Parts of its Fermi surface have 2D character, with approximately of quasiparticle relaxation rates at higher energies, possibly cylindrical geometry (shown in blue in Fig. 7d). The Fermi associated with the development of hot spots on the Fermi surface surface also contains small, approximately isotropic Fermi due to spin-fluctuation scattering56. pockets, with three-dimensional (3D) character (shown in red To summarize the charge dynamics of CeCoIn5, our data fit in Fig. 7d). Quantum oscillation measurements reveal quite well with a picture of heavy quasiparticles coexisting with different masses for the 2D and 3D Fermi sheets, with the mass of uncondensed light quasiparticles3. The heavy quasiparticles the 3D pockets only weakly enhanced. The coexistence of heavy experience a large decrease in scattering below Tc, and and light-mass electron systems has been used by Tanatar et al.3 participate strongly in forming the superfluid, with only 7% to provide a simultaneous explanation of measurements of heat spectral weight remaining uncondensed as T-0. The light capacity (pm*) and thermal conductivity (p1/m*). In the quasiparticles undergo a much smaller decrease in scattering and microwave measurements, the 3D Fermi pockets provide have significant residual conductivity at low temperatures. This additional phase space for the scattering processes, as well as a suggests that the spectrum of fluctuations responsible for inelastic natural explanation for the broad background conductivity, sbgnd, scattering, mass enhancement and superconducting pairing observed in the s1(o) spectra. A distribution of relaxation rates couples strongly to the heavy parts of the Fermi surface, but naturally arises when there is a strong variation of effective mass much less efficiently to the light band. over the Fermi surface, as occurs in CeCoIn5 (refs 7,54): narrow Our results provide a new window into the low-energy charge conductivity spectra correspond to heavy quasiparticles and dynamics of CeCoIn5 and uncover a complex interplay between broad spectra to light ones55. At the very lowest temperatures, d-wave superconductivity, multiband physics and quantum there is a downturn in 1/t(T) that, although small, seems to be criticality. The phenomena revealed can only be understood statistically significant. This is consistent with the observation of using measurements over a wide frequency range. Many of the temperature dependence of the quasiparticle effective mass. features are strongly reminiscent of the cuprates, confirming a In the intermediate temperature range, the relaxation rate is close connection between these two classes of material. An strongly temperature dependent and, as shown in Fig. 6d, is well important difference is the observation of temperature-dependent described by a sum of a temperature-independent elastic term quasiparticle mass, which not only resolves the issue of and a T3 inelastic term. To facilitate a detailed comparison with anomalous power laws in London penetration depth, but shows spin-fluctuation theory, the functional form we fit to the that quantum criticality is not completely circumvented by the relaxation rate is onset of superconductivity57.

3 3 h h kBT Methods ¼ þ A 2 ; ð7Þ tðTÞ t0 D ðTÞ Surface impedance. Phase-sensitive measurements of microwave surface impe- 14,24,32–36 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dance, Zs ¼ Rs þ iXs, were made using resonator perturbation techniques , with temperature-dependent changes in Z obtained from resonator frequency, where the gap DðTÞ¼D0tanhð2:4 Tc=T À 1Þ and D0 ¼ 3kBTc. s The prefactor A is expected to be of order one48: numerical spin- f0, and resonant bandwidth, fB, using the cavity perturbation approximation DZs ¼ G(DfB(T)/2–iDf0(T)). Here G is a resonator constant determined empirically fluctuation calculations obtain A ¼ 2.4 for parameters relevant to from the known DC resistivity of CeCoIn (ref. 58). At the lowest frequency, 50 5 optimally doped cuprates ; fits to our CeCoIn5 data give 0.13 GHz, surface reactance was measured using a tunnel-diode oscillator and was A ¼ 3.36. With these values so close, the charge dynamics of previously published by O¨ zcan et al.19 At all other frequencies, surface impedance was measured using dielectric-resonator techniques, in a dilution-refrigerator- CeCoIn5 seem to be consistent with a spin-fluctuation based variant of the apparatus described in the study by Huttema et al.36 The mechanism. absolute surface resistance was obtained at each frequency using an in situ As noted above, inelastic contributions to electrical relaxation bolometric technique, by detecting the synchronous rise in temperature when the rate in a d-wave superconductor are expected to be suppressed by sample was subjected to a microwave field of known, time-varying intensity59. 3 Thermal expansion effects make a small contribution to the apparent surface an umklapp gap. The observation of T behaviour in CeCoIn5 is th reactance, DXs Àðom0c=2ÞbðTÞ, where c is the thickness of the sample and therefore somewhat surprising, as it implies that the umklapp gap b(T) is the volume coefficient of thermal expansion. This was corrected for using is small. As shown in Fig. 7c, the umklapp gap in a simple d-wave thermal expansion data from Takeuchi et al.60 superconductor is determined by the location of the gap nodes 47 with respect to the reciprocal lattice vectors . In a multiband Absolute surface reactance. The absolute surface reactance was obtained at superconductor, in which superconductivity is weak on parts of 2.91 GHz by matching Rs(T) and Xs(T) between 10 K and 35 K, a temperature range the Fermi surface, the situation is more complex. This is in which thep imaginaryffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi part of the normal-state conductivity is small and om r = Eom l illustrated for the CeCoIn Fermi surface in Fig. 7d, in which Rs ¼ Xs ¼ 0 dc 2. To a first approximation, Xs 0 L is used to obtain the 5 surface reactance at the other frequencies. This estimate is refined by taking into we show how inter-band scattering can reduce the threshold for account the quasiparticle contribution to Xs. We carry this procedure out at umklapp processes. T ¼ 0.1 K, using the following self-consistent method. The quasiparticle contribu- Additional insights into the quasiparticle charge dynamics tion to s2 is initially set to zero, so that s2(o) arises purely from the superfluid s = om l2 l come from the conductivity frequency exponent, y(T), which is conductivity, s ¼ 1 i 0 L, with L obtainedpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi from the 2.91-GHz Xs data. om =s o used in our conductivity model to capture energy- or momen- The local electrodynamic relation, Zs ¼ i 0 , is used to obtain Xs( ) from the measured Rs(o) and the calculated s2(o). This step is carried out tum-dependent scattering. In the Drude limit (y ¼ 2) all without any explicit knowledge of s (o). From X (o) and R (o) we obtain 42,46 1 s s quasiparticles relax at the same rate. Previous studies have s1(o), again using the local electrodynamic relation. A Drude-like spectrum, qp 2 2 s1 ðoÞ¼s0=ð1 þ o t Þþsbgnd,isfittos1(o). The corresponding imaginary part, shown that yo2 works well in capturing the phenomenology of qp 2 2 s2 ðoÞ¼s0ot=ð1 þ o t Þ, provides an estimate of the quasiparticle contribution d-wave superconductors, in which the quasiparticle relaxation qp to s2(o). The total imaginary conductivity is the sum of s2 ðoÞ and a superfluid rate has a strong energy dependence due to the Dirac-cone term of the same form as in step 1, but with l adjusted to make s (o) consistent 21,51,52 L 2 structure of the d-wave quasiparticle spectrum . Below with the directly determined value of Xs at 2.91 GHz. The refined estimate of s2(o)

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Research support for sample preparation was provided by the Division of Materials 54. Hall, D. et al. Fermi surface of the heavy-fermion superconductor CeCoIn5:The de Haas-van Alphen effect in the normal state. Phys. Rev. B 64, 212508 (2001). Science and Engineering of the U.S. Department of Energy Office of Basic Energy 55. Prange, R. E. & Kadanoff, L. P. Transport theory for electron-phonon Sciences. interactions in metals. Phys. Rev. 134, A566–A580 (1964). 56. Rosch, A. Interplay of disorder and spin fluctuations in the resistivity near a quantum critical point. Phys. Rev. Lett. 82, 4280–4283 (1999). Author contributions 57. Laughlin, R. B., Lonzarich, G. G., Monthoux, P. & Pines, D. The quantum P.J.T., W.A.H., C.J.S.T., S.O¨ ., P.R.C., E.T., N.C.M., K.J.M., A.J.K. and D.M.B. designed criticality conundrum. Adv. Phys. 50, 361–365 (2001). and set up the dilution-refrigerator-based systems for microwave spectroscopy. C.J.S.T., ¨ 58. Bauer, E. D. et al. Thermodynamic and transport investigation of CeCoIn5– W.A.H., P.J.T., S.O., N.C.M. and D.M.B. carried out the experiments. C.J.S.T. carried out xSnx. Phys. Rev. B 73, 245109 (2006). the data analysis. J.L.S. prepared the sample of CeCoIn5. D.M.B. wrote the paper and 59. Turner, P. et al. Bolometric technique for high-resolution broadband supervised the project. microwave spectroscopy of ultra-low-loss samples. Rev. Sci. Instrum. 75, 124–135 (2004). 60. Takeuchi, T. et al. Thermal expansion and magnetostriction studies in a heavy- Additional information fermion superconductor, CeCoIn5. J. Phys. Condens. Matter 14, L261–L266 Competing financial interests: The authors declare no competing financial interests. (2002). Reprints and permission information is available online at http://npg.nature.com/ Acknowledgements reprintsandpermissions/ We thank M. Dressel, S.R. Julian and M. Scheffler for discussions and correspondence. How to cite this article: Truncik, C. J. S. et al. Nodal quasiparticle dynamics in the heavy

Research support for the experiments was provided by the Natural Science and Engi- fermion superconductor CeCoIn5 revealed by precision microwave spectroscopy. neering Research Council of Canada and the Canadian Foundation for Innovation. Nat. Commun. 4:2477 doi: 10.1038/ncomms3477 (2013).