We report low-temperature specific heat measurements in magnetic fields up to
12 T applied parallel and perpendicular to the tetragonal c-axis of the heavy
fermion superconductor Ce$_2$PdIn$_8$. In contrast to its quasi-two-dimensional
(2D) relative CeCoIn$_5$, the system displays an almost isotropic upper
critical field. While there is no indication for a FFLO phase in
Ce$_2$PdIn$_8$, the data suggest a smeared weak first-order superconducting
transition close to $H_{c2}\approx 2$ T. The normal state electronic specific
heat coefficient displays logarithmically divergent behavior, comparable to
CeCoIn$_5$ and in agreement with 2D quantum criticality of spin-density-wave
type

Gegenwart, P.

Gnida, D.

Kaczorowski, D.

Tokiwa, Y.

Physical Review B

English

arXiv

Y. Tokiwa

P. Gegenwart

D. Gnida

D. Kaczorowski

Crossref

Quantum criticality near the upper critical field of Ce<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow /><mml:mn>2</mml:mn></mml:msub></mml:math>PdIn<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow /><mml:mn>8</mml:mn></mml:msub></mml:math>

We report low-temperature specific-heat measurements in magnetic fields up to 12 T applied parallel and perpendicular to the tetragonal c axis of the heavy fermion superconductor Ce2PdIn8. In contrast to its quasi-two-dimensional (2D) relative CeCoIn5, the system displays an almost isotropic upper critical field. While there is no indication for a high-field and low-temperature phase in Ce2PdIn8, the data suggest a smeared weak first-order superconducting transition close to H-c2 approximate to 2 T. The normal-state electronic specific-heat coefficient displays logarithmically divergent behavior, comparable to CeCoIn5 and in agreement with 2D quantum criticality of spin-density-wave type.German Science Foundation [FOR 960

Tokiwa, Yoshi

Gegenwart, Philipp

GRO.publications (Univ. Göttingen)

Quantum criticality near the upper critical field of Ce2PdIn8

OPUS Augsburg

RAPID COMMUNICATIONSPHYSICAL REVIEW B 84, 140507(R) (2011)Quantum criticality near the upper critical field of Ce2PdIn8Y. Tokiwa and P. GegenwartI. Physikalisches Institut, Georg-August-Universität Göttingen, D-37077 Göttingen, GermanyD. Gnida and D. KaczorowskiInstitute of Low Temperature and Structure Research, Polish Academy of Sciences, P.O. Box 1410, PL-50-950 Wrocław, Poland(Received 11 August 2011; revised manuscript received 6 October 2011; published 25 October 2011)We report low-temperature specific-heat measurements in magnetic fields up to 12 T applied parallel andperpendicular to the tetragonal c axis of the heavy fermion superconductor Ce2PdIn8. In contrast to its quasi-two-dimensional (2D) relative CeCoIn5, the system displays an almost isotropic upper critical field. While there is noindication for a high-field and low-temperature phase in Ce2PdIn8, the data suggest a smeared weak first-ordersuperconducting transition close to Hc2 ≈ 2 T. The normal-state electronic specific-heat coefficient displayslogarithmically divergent behavior, comparable to CeCoIn5 and in agreement with 2D quantum criticality ofspin-density-wave type.DOI: 10.1103/PhysRevB.84.140507 PACS number(s): 74.70.Tx, 74.40.Kb, 74.20.Mn, 74.25.BtUnconventional superconductivity (SC) often occurs insystems with competing phases near a quantum criticalpoint (QCP). Indicative for quantum criticality is, e.g., theobservation of non-Fermi-liquid (NFL) behavior in the normalstate at T > Tc in zero field or down to the lowest accessibletemperatures at fields above the upper critical magnetic fieldHc2. Heavy fermion (HF) superconductors have sufficientlysmall Hc2, allowing for a detailed thermodynamic study ofthe normal-state properties down to very low temperaturesto investigate NFL behavior. The layered tetragonal CeCoIn5with Tc = 2 K is a well-studied heavy fermion superconductorwith a NFL normal state due to a field-induced QCP slightlybelow Hc2.1–8 Its SC properties are also fascinating. Thetransition at Hc2(T ) becomes first order below 1 K due tostrong Pauli limiting9,10 and clear phase transition anomaliesat the high-field and low-temperature (HFLT) corner in theSC phase diagram have been discovered,11,12 which werethought to be related to a modulated SC Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state.13,14 The origin of the HFLT SCstate is under extensive debate, as subsequent NMR andneutron-diffraction experiments have revealed the existenceof an antiferromagnetic (AF) ordering within the HFLTphase,15–17 while a very recent NMR study suggests spatiallydistributed normal quasiparticle regions due to the formationof a FFLO state.18The recently discovered HF superconductor (Tc = 0.7 K)Ce2PdIn8 (Refs. 19 and 20) belongs to the same class ofCenT In3n+2 (T: transition metal, n = 1, 2, and ∞) systems,21containing a series of CeIn3 and TIn2 layers stacked alongthe c axis. While the n = 1 systems with T = Co, T = Ir(Ref. 22), and T = Rh (Ref. 23) display anisotropic propertiesdue to the layered, quasi-two-dimensional (2D) structure,cubic CeIn3 (n = ∞) is a completely isotropic system. Thebilayer (n = 2) systems are therefore expected to displayintermediate behavior due to the more three-dimensional (3D)character of their crystal structure consisting of a stack oftwo CeIn3 layers separated by one PdIn2 layer. Ce2PdIn8 is anambient-pressure HF superconductor of this class of materials.Its upper critical field Hc2 is strongly Pauli limited with a largeMaki parameter of 2.9.24 Clean single crystals have becomeavailable, with a residual resistivity of ∼2.5 μ cm and a largemean free path of l = 420 Å,24 exceeding the SC coherencelength ξ = 82 Å. Thus, the system may fulfill the necessaryconditions for the formation of the FFLO state, and it isinteresting to investigate the high-field and low-temperaturepart of the SC phase by a thermodynamic probe. Furthermore,recent low-T electrical resistivity measurements at fields abovethe upper critical field have revealed strong similarities toCeCoIn5, i.e., a quasilinear dependence at Hc2 which turnsinto Fermi-liquid behavior only at larger fields,24,25 suggestinga field-induced QCP near Hc2. Because of these commonfeatures, it is highly interesting to study the low-temperaturespecific heat of single-crystalline Ce2PdIn8 in magnetic fieldsapplied both parallel and perpendicular to the tetragonalc axis.Single crystals were grown by the self-flux method,19and we have confirmed by specific-heat and susceptibilitymeasurements that they are not contaminated by CeIn3. Verythin single-crystalline pieces were glued on the sample holder,and the basal plane orientation for the different pieces was notinvestigated. The specific heat at temperatures down to 70 mKand magnetic fields up to 12 T was measured in a dilutionrefrigerator with a SC magnet by employing the quasiadiabaticheat-pulse method.Figure 1 shows the specific heat after subtraction of thenuclear contribution as Cel(T )/T , at magnetic fields up toHc2. The nuclear contribution has been determined by fittinga α(H )T −3 contribution to C/T with α(H ) ∝ H 2. At zerofield, Cel/T increases upon cooling to Tc = 0.68 K, at which atypical mean-field-type anomaly is found. Within the SC state,it decreases linearly down to 0.2 K and saturates at a residualvalue of Cel/T = 0.65 J/mol K2 (0.33 J/Ce mol K2), whichis much larger than the respective value of 0.04 J/mol K2found for CeCoIn5,26 indicating a larger quasiparticle densityof states arising from pair breaking of defects. The specific heatat magnetic fields along different directions is rather similarand displays only a weak anisotropy in Hc2 = 2.0 T and 2.2 Tfor H ⊥ [001] and H ‖ [001], respectively. At high fields,the SC transition changes its shape and becomes a rathersymmetric peak different from the expectation for standardsuperconductors. This may indicate a smeared first-ordertransition. Note that in CeCoIn5 a sharp and large peak of the140507-11098-0121/2011/84(14)/140507(4) ©2011 American Physical SocietyRAPID COMMUNICATIONSY. TOKIWA, P. GEGENWART, D. GNIDA, AND D. KACZOROWSKI PHYSICAL REVIEW B 84, 140507(R) (2011)FIG. 1. (Color online) Electronic specific heat as Cel(T )/T ofCe2PdIn8 at different magnetic fields perpendicular to (a) and along(b) the c axis, respectively. The dotted line displays the raw data at2 T. The inset in (a) displays the entropy contributions, obtained fromthe integral of the electronic specific-heat coefficients at 0 and 2 T,extrapolated down to zero temperature by assuming a constant (forzero field) and logarithmic dependence for 2 T [cf. black solid line inFig. 5(a)], respectively.specific heat is found at low temperatures, which has provento be due to a first-order transition.9In Fig. 2, we compare the zero- and high-field SC transitionanomalies of Ce2PdIn8 and CeCoIn5 (Ref. 27) on a reducedtemperature scale. At H = 1.8 T for the former and 11 T forthe latter system, respectively, Tc is suppressed to 20% of itszero-field value. The peak at the SC to normal transition inFIG. 2. (Color online) Comparison on Cel(T )/T for Ce2PdIn8(circles, H ⊥ c, left-hand axis) and CeCoIn5 (squares, H ‖ c, right-hand axis) (Ref. 27), plotted against the reduced temperature.FIG. 3. (Color online) Electronic specific heat divided by tem-perature Cel/T for Ce2PdIn8 at 0.1 and 0.15 K as a function ofmagnetic field perpendicular to (a) and along (b) the tetragonal caxis, respectively.the latter system is much sharper, indicating a much strongerfirst-order nature of SC transition in CeCoIn5. The peakheight for CeCoIn5 has been reported to be very sensitiveto impurity scattering. Only 0.22% of Cd doping suppressessignificantly the sharp peak in specific heat and leads to a verysimilar C/T dependence as found for 1.8 T in Ce2PdIn8.27We may thus associate the reduced peak height for Ce2PdIn8to the approximately ten-times-shorter electronic mean freepath, compared to undoped CeCoIn5 (4000 Å).28 The HFLTphase in undoped CeCoIn5 causes an additional jump inCel(T )/T of ∼ 0.2 J/mol K2 at 11 T, whereas in Ce2PdIn8 noanomaly could be detected within the measurement error of∼0.02 J/mol K2 at 0.1 K.Measurements of the electronic specific-heat coefficient asa function of magnetic field, displayed in Fig. 3, also show nosign of an anomaly in addition to Hc2. According to the theoret-ical study by Ichioka and Machida,29 the convex shape of resid-ual Cel/T as a function of field, which is also observed in ourexperiments, is an indication of a strong Pauli-limiting effect.Figure 4 shows the phase diagram of Ce2PdIn8 for twoperpendicular magnetic field directions, revealing an almostisotropic field effect on superconductivity. The strongly convexshape of Hc2(T ) indicates a pronounced effect of Paulilimiting, dominating over orbital limiting. In CeCoIn5, Hc2is also strongly Pauli limited and furthermore displays a pro-nounced anisotropy, which could be related to the anisotropicnormal-state spin susceptibility. For Ce2PdIn8, by contrast,FIG. 4. (Color online) SC phase diagram of Ce2PdIn8 for twoperpendicular field orientations. The solid lines are guides to theeyes.140507-2RAPID COMMUNICATIONSQUANTUM CRITICALITY NEAR THE UPPER CRITICAL . . . PHYSICAL REVIEW B 84, 140507(R) (2011)2D3DH     [001]H // [001](a)(b)2D3D1. 51. 0Cel/ T ( J/ moK   l2 )7 8 90.12 3 4 5 6 7 8 912T  ( K )2. 2T3T4T5T6T7T10T12T1. 21. 00. 8γ  (J/mol K2 )10864H  ( T)1. 21. 00. 8γ  (J/mol K2 )1210864H  ( T)1. 51. 0Cel/ T ( J/ moK   l2 )FIG. 5. (Color online) Electronic specific heat divided by tem-perature Cel(T )/T (on a logarithmic scale) of Ce2PdIn8 at magneticfields above Hc2, perpendicular to (a) and along (b) the tetragonalc axis, respectively. The solid lines represent a ln(1/T ) dependence.The insets display the field dependence of the Sommerfeld coefficient(see text). The colored solid lines represent the fitted curves usingtheoretical functions for 2D and 3D AF QCPs (Ref. 31).the upper critical field is almost isotropic, which is consistentwith the much weaker anisotropy of the reported normal-statesusceptibility.30Next, we turn to the signatures of NFL behavior in thenormal state. As shown in Fig. 5, for fields slightly aboveHc2, a logarithmic divergence of the specific-heat coefficientis found over more than one decade in T . The entropy releasein zero field at Tc = 0.68 K due to the formation of theSC state is almost fully balanced at H = 2 T [cf. inset ofFig. 1(a)], indicating that the same degrees of freedom areresponsible both for the NFL behavior and the SC state.At H > Hc2 for both magnetic field directions, Fermi-liquidbehavior is gradually recovered, as seen from the evolutiontoward a temperature-independent specific-heat coefficientCel/T . In the insets of Fig. 5, we plot the magnetic fielddependence of the Sommerfeld coefficient γ (H ) obtainedfrom the saturated Cel/T values. Upon reducing the fieldfrom large values down to Hc2, divergent behavior is found inγ (H ). This indicates the existence of a field-induced QCP inclose vicinity to the upper critical field of superconductivity,similar to CeCoIn5.2 In order to obtain information on theposition of the QCP, as well as on the nature of the underlyingquantum critical fluctuations, we have carefully compared thedata with the predictions of the Hertz-Millis theory for an AFQCP in both 2D and 3D. The predicted field dependencesare γ0 + γh ln[1/(H − Hc)] (2D) and γ0 − γh√H − Hc (3D),respectively.31 We have performed least-squares fitting ofour data using γ0, γh, and Hc as free parameters. The fielddependence expected for 3D critical fluctuations appears to betoo weak to explain the experimental γ values at sufficientlylarge fields. A much better description could be obtained overone decade in h = H − Hc, using the expectation for 2Dcritical behavior. The obtained quantum critical fields for the2D description are Hc = 2.0 ± 0.2 T and 1.7 ± 0.3 T for H ⊥and ‖ to [001], respectively. These critical fields are very closeto Hc2. Note that recent electrical resistivity measurementshave also suggested a field-induced QCP very close to Hc2in this system.24 Furthermore, at H ≈ Hc2 both the lineartemperature dependence of the electrical resistivity,24 as wellas the logarithmic temperature dependence of the specific-heatcoefficient, are in agreement with the theoretical prediction fora 2D AF QCP.In summary, we have studied the effect of magnetic fieldson the SC and normal-state behavior in Ce2PdIn8 by measuringthe specific heat at low temperatures and high magnetic fields.Even though the upper critical field Hc2(T ) is strongly Paulilimited, a phase transition into a HFLT phase as seen in therelative CeCoIn5 has not been detected. It has been recentlyreported that an extremely small amount of impurities destroys(or smears out) this phase transition in CeCoIn5. For 0.05%Cd-doped or 0.08% Sn-doped CeCoIn5 it could not be detectedby specific-heat measurements anymore.27,32 It has also beenshown theoretically that a very small amount of impuritiessmears out the FFLO phase.33 Although Ce2PdIn8 is a cleanlimit superconductor, the electronic mean free path of theinvestigated single crystal is approximately ten times smallercompared to CeCoIn5. This seems to be reflected in a muchlarger residual specific-heat coefficient Cel/T at zero field andreduced peak height of the SC transition in high fields. Froma comparison of the respective specific-heat data, we find theinvestigated Ce2PdIn8 single crystals comparable to 0.22%Cd-doped CeCoIn5.27 Thus, a possible HFLT transition inCe2PdIn8 already could be destroyed or smeared out by impu-rity scattering. Furthermore, we note that the upper critical fieldof the latter compound is almost isotropic, whereas it displaysa pronounced anisotropy in CeCoIn5. The observed NFLnormal-state behavior in Ce2PdIn8 appears to be rather similarto that found in CeCoIn5. We have observed a logarithmicdivergence of Cel/T as a function of temperature at fields nearHc2 and a logarithmic field dependence of the Sommerfeldcoefficient, both of which are compatible with 2D AF fluctu-ations due to a QCP near Hc2, in close similarity to CeCoIn5.This work has been supported by the German ScienceFoundation through FOR 960 (quantum phase transitions).1C. Petrovic, P. G. Pagliuso, M. F. Hundley, R. Movshovich, J. L.Sarrao, J. D. Thompson, and Z. Fisk, J. Phys. Condens. Matter 13,L337 (2001).2A. Bianchi, R. Movshovich, I. Vekhter, P. G. Pagliuso,and J. L. Sarrao, Phys. Rev. Lett. 91, 257001(2003).140507-3RAPID COMMUNICATIONSY. TOKIWA, P. GEGENWART, D. GNIDA, AND D. KACZOROWSKI PHYSICAL REVIEW B 84, 140507(R) (2011)3J. Paglione, M. A. Tanatar, D. G. Hawthorn, E. Boaknin, R. W.Hill, F. Ronning, M. Sutherland, L. Taillefer, C. Petrovic, and P. C.Canfield, Phys. Rev. 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Quantum criticality near the upper critical field of Ce$_2$PdIn$_8$

Quantum criticality near the upper critical field of Ce $_2$ PdIn $_8$

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Quantum criticality near the upper critical field of Ce $_2$ PdIn $_8$