Field-Orientation Dependence of Low-Energy Quasiparticle Excitations in the Heavy-Electron Superconductor UBe13
Yusei Shimizu, Shunichiro Kittaka, Toshiro Sakakibara, Yoshinori Haga, Etsuji Yamamoto, Hiroshi Amitsuka, Yasumasa Tsutsumi, Kazushige Machida
aa r X i v : . [ c ond - m a t . s up r- c on ] N ov Field-Orientation Dependence of Low-Energy Quasiparticle Excitationsin the Heavy-Electron Superconductor UBe Yusei Shimizu, ∗ Shunichiro Kittaka, Toshiro Sakakibara, Yoshinori Haga, EtsujiYamamoto, Hiroshi Amitsuka, Yasumasa Tsutsumi, and Kazushige Machida Institute for Solid State Physics (ISSP), University of Tokyo, Kashiwa, Chiba 277-8581, Japan Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki, 319-1195, Japan. Graduate School of Science, Hokkaido University, Sapporo, Hokkaido, 060-0810, Japan. Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama, 351-0198, Japan. Department of Physics, Okayama University, Okayama, 700-8530, Japan. (Dated: August 10, 2018)Low-energy quasiparticle excitations in the superconducting (SC) state of UBe were studied by means ofspecific-heat ( C ) measurements in a rotating field. Quite unexpectedly, the magnetic-field dependence of C ( H ) is linear in H with no angular dependence at low fields in the SC state, implying that the gap is fully open overthe Fermi surfaces, in stark contrast to the previous expectation. In addition, a characteristic cubic anisotropy of C ( H ) was observed above 2 T with a maximum (minimum) for H || [001] ( [111] ) within the (1¯10) plane, bothin the normal as well as in the SC states. This oscillation possibly originates from the anisotropic response ofthe heavy quasiparticle bands, and might be a key to understand the unusual properties of UBe . PACS numbers: 74.70.Tx, 71.27.+a, 74.20.Rp, 75.30.Mb
Three decades have passed since the discoveries of super-conductivity in CeCu Si [1] and UBe [2]. It is widely ac-cepted that their pairing mechanisms are unconventional, fun-damentally because the effective Fermi temperatures of thesesystems ( ∼
10 K) are, like other heavy-electron superconduc-tors, much lower than the Debye temperatures [1]. Determi-nation of the gap symmetry of the heavy-electron supercon-ductors is, however, by no means an easy task, and there areonly a few heavy-electron superconductors whose supercon-ducting (SC) gap structures are fully elucidated. For instance,the SC pairing symmetry of CeCu Si has not been clarifieduntil very recently [3].In this Letter, we focus on a cubic heavy-electron super-conductor UBe . Despite extensive studies over 30 years,the nature of superconductivity in UBe is still elusive. The Be-NMR-Knight shift has been reported to be invariant be-low the superconducting transition temperature T c ≈ . K[4, 5], suggesting an odd-parity pairing. However, µ + SR-Knight-shift experiment indicates a significant decrease of thestatic susceptibility below T c [6], conflicting with the NMRresults. Regarding the gap structure, whereas the specific-heat C ( T ) [7] and the magnetic penetration depth [8] experimentssuggest the presence of point nodes, the NMR spin-relaxationrate [9] and the ultrasound attenuation [10] are rather indica-tive of line nodes. Although there is compelling evidence forunconventional pairing, the SC gap symmetry in UBe thusremains undetermined.Another important issue with respect to the SC state inUBe is a feature observed in thermodynamic quantities suchas C ( H ) [11–13], dc magnetization M ( H ) [14] as well as thethermal expansion [11] at fields below ∼ B ∗ anomaly”) in the H − T phase diagram.Whereas the origin of this anomaly is still unresolved yet, ithas been discussed as a precursor [12] of the second phasetransition below T c observed in U − x Th x Be (0.019 < x < is also highly unusual. It exhibitsnon-Fermi-liquid (NFL) behavior down to very close to T c as revealed by electrical resistivity, specific heat [16, 17],and magnetic susceptibility [18]. The origin of NFL be-haviors in UBe remains unclear, and several possibili-ties have been discussed so far. These include quadrupolarKondo effect with Γ -crystalline-electric-field ground statefor f (U , J =4) configuration [19], an antiferromagneticquantum-critical point induced by a magnetic field[17, 20],and a competition between Kondo-Yosida and Γ -crystalline-electric-field singlets for f configuration [21]. Since the SCstate apparently emerges out of the NFL state, its understand-ing is crucial in elucidating the pairing mechanism in UBe .In order to gain more insight into the SC gap symmetryas well as the normal state, we performed specific-heat mea-surements of UBe at low temperatures down to ∼
75 mK inmagnetic fields up to 5 T. The single crystal of UBe , usedin the present study was prepared by an Al-flux method [22].This is the same crystal as used in the previous dc magnetiza-tion study [14]. The specific heat C was measured by a stan-dard quasi-adiabatic heat-pulse method. Field-angular depen-dences C ( H, φ ) were measured with H rotating in the (1¯10) crystal plane that includes three principal directions [001] , [111] , and [110] . The angle φ is measured from the [001] axis.Figure 1(a) shows C ( T ) /T curves measured in variousmagnetic fields up to 5 T. The zero-field data are also plottedin Fig. 1(b) in log-log scale. There is no Schottky contributionfrom Be nuclei, owing to their long nuclear spin-relaxationtime of the order of 10 sec [9], much longer than our measur-ing time (10 sec) of the specific heat. The inset of Fig. 1(a)shows the C ( T ) /T vs T plot; C ( T ) below ∼ T as previously reported [7]. Note that the residual den-sity of states, C ( T ) /T | T → , is very small.Magnetic-field dependence of the specific heat and its !" !" $% & ’ ( ) * + $ , ! - . "/ !" $! $%, ! . " $ $ (8- ! " !" " ! " g $ % % & ’ !" " &’"($!’$ + , -./0012345567389:;: <=, 555>?0 "( a ! " ! !"! &’ ( ) * + , - & . % / m $&’20&&345 && ! "&.*7/ ’ 10’1110 !" f()*+,- !" ! !"! () . / ( ’ -( !"&(4 ! $(467+ "8 ("(90*5 ) "- )"""- )"" - FIG. 1: (Color online) (a) C ( T ) /T of UBe at µ H = 0 , 1, 2,3, 4, and 5 T for H || [001] . The inset shows C ( T ) /T vs T plot for H = 0 . (b) The result of three-band full-gap analysis for C ( T ) /T ( H = 0 ) in log-log scale. The parameters are: α = 1 . , α = 0 . , α = 0 . , and γ : γ : γ = 55 : 38 : 7. (c) C ( H ) /T at T =0 . K for H || [001] (solid circles) and H || [111] (open triangles) asa function of H in the low-field region. The dashed line is a linear fitto the data below ∼ H || [001] . (d) C ( φ ) /T in a field of 1 Trotated in the (1¯10) plane, measured at 0.08, and 0.14 K. anisotropy in low fields reflect quasiparticle excitations withinthe SC gap [23–25]. In the case of line nodes, C ( H ) ∝ ( H/H c2 ) / is expected [23–25], whereas for point nodes, C ( H ) ∝ HH c2 ln HH c2 [26], or C ( H ) ∝ ( H/H c2 ) . [27].In either case, the field dependence of C/T should exhibita convex upward curvature at low fields. For a clean isotropic s -wave superconductor, on the other hand, C ( H ) /T ∝ H at low fields because low-energy quasiparticles are mainlyconfined in vortices whose density increases in proportion to H [28]. Figure 1(c) shows C ( H ) /T of UBe below 1.6 Tfor H || [001] and H || [111] measured at 0.08 K. Surprisingly,the low-field C ( H ) /T curve is rather linear in H , suggest-ing the absence of nodal quasiparticles. Note that there is no anisotropy in this H -linear behavior of C ( H ) below ∼ C ( φ ) /T obtained in a field of 1 T rotated in the (1¯10) crystal plane at T = 0 . and 0.14 K [Fig. 1(d)]; thereis no significant angular variation in C ( φ ) /T , implying that !" !"! %& ’ ( ) * + , % - . / $01 m $%&2/ %"!"3%- &"" %%"!0"%- &"" "! 0%-%"! %%)4.% "!5$%- &"" !" d ’ !"! ( ) (* + , ’ - . / ( % ) m $(*31 (’=) >?8 &$ FIG. 2: (Color online) (a) Magnetic-field dependence of C ( H ) /T up to 5 T for H || [001] (solid circles) and H || [111] (open triangles)measured at T = 0 . , 0.14, 0.24, 0.40, and 0.95 K. (b) δ ( C/T ) ≡ ( C [001] − C [111] ) /T as a function of H , obtained at T = 0 . (solidcircles), 0.40 (open squares), and 0.95 K (open circles). the C ( H ) /T ∝ H behavior holds for all directions.We would like to emphasize that the linear slope of C ( H ) /T in Fig. 1(c) is unusually small. In this regard,it has been argued that the line-nodal or point-nodal sub-linear dependences in C ( H ) /T described above would besmeared out at a finite temperature T /T c > p H/H c2 [29].Even in such case, however, the rate of increase of C ( H ) /T should be greater than that governed by the localized-quasiparticle contribution from vortex cores approximated as C ( H ) /T = ( C n /T )( H/H orbc2 (0)) , where µ H orbc2 (=25 T [30])denotes the orbital-limiting field. Assuming the normal-state value C n /T at T ∼ · mol − · K − by tak-ing into account an entropy balance, we estimate the slope ( C n /T ) / ( µ H orbc2 (0)) of the ordinary vortex core contribu-tion to be 0.044 J · mol − · K − T − . The observed initial slopein Fig. 1(c) is 0.02 J · mol − · K − T − , a factor of two smallerthan this; apparently, there is a significant deficiency of quasi-particles. We will come back to this point later.Figure 2(a) shows the field variation of C ( H ) /T up to 5 Tfor H || [001] measured at T = 0 . , 0.14, 0.24, 0.40, and0.95 K (closed symbols). We also plot the data for H || [111] measured at T = 0 . , 0.40, and 0.95 K (open symbols). C ( H ) /T curve for H || [001] in the SC state at T = 0 . K ex-hibits a strong upturn above ∼ T. This behavior is quite rem-iniscent of a superconductor with a strong Pauli paramagneticeffect, as observed for CeCu Si [3]. Note that a weak humpappears in C ( H ) /T above ∼ T . This hump has been known asthe “ B ∗ anomaly” [11–14]. We observe that this anomaly in C ( H ) is clearer for H || [001] than for H || [111] . Accordingly,a substantial anisotropy develops in C ( H ) above this field.An anisotropy has also been observed by dc magnetizationcurves above B ∗ [31, 32]. In order to display the evolution ofthe anisotropy in C ( H ) /T , we plot in Fig. 2(b) the difference δ ( C/T ) ≡ ( C [001] − C [111] ) /T at T = 0 . , 0.40, and 0.95 K,where C [001] ( C [111] ) denotes the specific heat for H || [001] ( H || [111] ). For T = 0 . K, δ ( C/T ) shows a distinct positivepeak around 3.5 T due to the B ∗ anomaly. At 0.40 K, δ ( C/T ) changes the sign and shows a monotonic decrease with in- !" ! !"! $% & ’ ( ) * + $ , - . /
0 1 f$%234/ !1 !7 $, !07$, !- % 0/ %00 /%000/ $-$; (<. % 0/ %000/ %00 / !" ! !"! %& ’ ( ) * + , % - . / f%&2340 ! !11!51!51!61!71!71!81!" $ % )
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FIG. 3: Angular dependences of C ( H, φ ) /T , measured in (a) µ H = 2 T, and (b) 4 T at T = 0 . , 0.14, 0.28, and 0.40 K.Data for T = 0 . K and µ H = 4 T in the normal state are alsoplotted. creasing field above 2 T, reflecting the anisotropy of H c2 . Inthe normal state at 0.95 K, δ ( C/T ) turns positive again andincreases monotonically with increasing field above ∼ C/T is substantiallysuppressed in a field of 5 T for both directions.In Fig. 3, we show the field-angle dependences of
C/T inthe (1¯10) plane measured in a magnetic field of (a) 2 T, and(b) 4 T. For µ H = 2 T, an appreciable angular variation canbe seen at T = 0 . , and 0.14 K, with a maximum (minimum)at [001] ( [111] ) and a local maximum at [110] , i.e. C [111]