Symmetry change of d-wave superconductivity in κ-type organic superconductors
SSymmetry change of d -wave superconductivityin κ -type organic superconductors S. Imajo , , ∗ , K. Kindo , and Y. Nakazawa Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan The Institute for Solid State Physics, the University of Tokyo, Kashiwa, Chiba 277-8581, Japan (Dated: February 24, 2021)Magnetic-field-angle-resolved heat capacity of dimer-Mott organic superconductors κ -(BEDT-TTF) X is reported. Temperature and field dependence of heat capacity indicates that the super-conductivity has line nodes on the Fermi surface, which is a typical feature of d -wave superconduc-tivity. In-plane field-angle dependence exhibits fourfold oscillations as well as twofold oscillationsdue to anisotropy of superconducting gap functions. From analyses of the fourfold term, the gapsymmetry is determined as d x − y + s ± for X =Cu[N(CN) ]Br and d xy for X =Ag(CN) H O. We sug-gest that the symmetry difference comes from the competition of d xy and d x − y antiferromagneticfluctuations depending on lattice geometry. κ -(BEDT-TTF) X organic conductors (BEDT-TTFand X represent bis(ethylenedithio)tetrathiafulvaleneand a monovalent anion, respectively) have been ex-tensively studied as proto-type dimer-Mott compounds.These salts have a quasi-two-dimensional (quasi-2D) elec-tronic system arising from the alternating layered struc-ture of the organic donor BEDT-TTF and the counter an-ion X . As shown in Fig. 1(a), the BEDT-TTF moleculesform dimer units that are arranged in the distorted tri-angular lattice structure. Depending on the lattice ge-ometry and the ratio of the inter/intra dimer transfers,the ground states of κ -type compounds vary betweenmetal, superconductivity, and antiferromagnetic insula-tor as well as quantum spin liquid. In a typical elec- k x k y dimer(a) (b)BEDT-TTF conductingplane φ H ( φ ) a θ H ( θ )(c) o ca FIG. 1. (a) Molecular arrangement of BEDT-TTF inthe conducting plane of κ -(BEDT-TTF) Cu[N(CN) ]Br. (b)Fermi surface of κ -(BEDT-TTF) X (red) in the first Brillouinzone (solid lines) and the extended Brillouin zone (dashedlines). As indicated, the axes k x and k y are defined accordingto the axes of the first Brillouin zone ( k b , k a ) for κ -Ag and( k a , k c ) for κ -Br, respectively. (c)The definition of magneticfield directions against the two-dimensional conducting planeof the sample. The polar angle θ signifies the inclination fromthe plane while the azimuthal angle φ denotes the in-planeangle from the a axis. tronic phase diagram of κ -type salts[1], a superconduct-ing phase neighbors an antiferromagnetic Mott insulat-ing phase. The appearance of the superconducting phaseby suppressing the antiferromagnetic state with the ap-plication of external/chemical pressures implies that theorigin of the superconductivity is closely related to theantiferromagnetic spin fluctuations enhanced at the vergeof the antiferromagnetic phase. When the superconduc-tivity is mediated by the antiferromagnetic fluctuations,the superconducting energy gap function is anticipatedto be anisotropic in the momentum space. In the caseof well-known high- T c cuprates whose superconductiv-ity presumably arises from the antiferromagnetic spinfluctuations, their square lattice structure and almost1/2 filling of the energy band make the d x − y -wave su-perconductivity stable. Based on a situation similar tocuprates, κ -(BEDT-TTF) X would have d -wave symme-try, or, more specifically, d xy -wave, symmetry in the ex-tended Brillouin zone for the unfolded one-band dimermodel, as shown by the outer dashed lines in Fig. 1(b).For organics, however, it is necessary to consider whetherthe strength of the dimerization is sufficient to applythe dimerization approximation or not. Also, the ef-fect that the dimers form the distorted triangular lat-tice must be taken into account. Numerous experimen-tal works for determining its pairing symmetry have beenreported. However, three contradictory conclusions havebeen suggested: s -wave[2–4], d x − y -wave (= d x − y + s ± -wave[5])[6–9], and d xy -wave [10, 11] symmetry. Theoret-ical studies have also proposed d x − y + s ± -[12] and d xy -wave[13–15] symmetry depending on their models. Somerecent theories based on a more realistic model[16–19]suggest that the two symmetries compete and the emer-gent symmetry is determined according to the ratio oftransfer integrals. Nevertheless, the experimental evi-dence of the competition of the emergent symmetry isstill absent.To discuss the pairing symmetry of the dimer-Mott organic superconductors, in this work, wedetermine the superconducting gap structureof κ -(BEDT-TTF) Ag(CN) H O and κ -(BEDT-TTF) Cu[N(CN) ]Br (hereafter, we abbreviate them as a r X i v : . [ c ond - m a t . s up r- c on ] F e b κ -Ag and κ -Br, respectively) by employing angle-resolvedheat capacity measurements capable of detecting theanisotropy of the gap functions[20–22]. From the analy-ses of quasiparticle excitation over the superconductinggap, we classify the superconductivity into quasi-2Dline-nodal superconductivity. However, the detectedanisotropy suggests that the detailed symmetry of thetwo salts is given as d xy for κ -Ag and d x − y + s ± for κ -Br.The single crystals measured in this study were synthe-sized by electrochemical oxidation of BEDT-TTF donorswith electrolytes of counter anions. Heat capacity mea-surements were performed by using a high-resolution re-laxation calorimeter[23] in rotating magnetic fields simi-lar to the reported method[24] and by using a calorime-ter for pulsed magnetic fields[25] in a He cryostat whichcan cool the samples down to ∼ C p / T vs T plot in Fig. 2(a). Asindicated by the arrow, the anomalies associated withthe superconducting transition are observed at ∼ κ -Ag and ∼ κ -Br. Despite the largephonon contribution coming from the soft lattice of theorganics, the anomalies can be seen in this plot. Theinset enlarges the plot below T =4 K . The electronicheat capacity coefficient γ N is estimated from the fittingof the data of the normal state to C p / T = γ N + βT with the lattice heat capacity coefficient β . The valuesof γ N and β are given as 26.6 ± − mol − and 12.4 ± − mol − for κ -Ag and22.5 ± − mol − and 10.0 ± − mol − for κ -Br, respectively. These values agree with thereported values[10, 26–30]. The scaled electronic heatcapacity C ele / γ N T at 0 T is shown as a function oftemperature in Fig. 2(b). It is obtained by subtractingthe lattice heat capacity estimated from the data of thenormal state above H c2 , which is described by the totalof the lattice heat capacity and γ N T . As reported in ear-lier studies[26–30, 32], the low-temperature C ele shows ∼ aT behavior, which is evidence that line nodes existin the two-dimensional Fermi surface. The coefficient ofthe quadratic term a is obtained as 7.4 mJK − mol − for κ -Ag and 2.0 mJK − mol − for κ -Br. Assumingthe gap function is a simple d wave, it is known thatthe coefficient is expected to be ∼ k B γ N /∆ [29, 32],where ∆ is the gap amplitude at 0 K. This equationgives ∆ for κ -Ag and κ -Br as 2.3 k B T c and 3.2 k B T c ,respectively. Comparing with ∆ =2.14 k B T c for weak-coupling d -wave superconductors, it is found that κ -Agis in a weak-coupling regime while the Cooper pairingof κ -Br is rather strong. Figure 2(c) presents C p / T as afunction of external field reduced by H c2 in parallel and C p / T ( m J K - m o l - ) -90 -60 -30 0 30 θ (deg.) ε = 11.8 n = 0.5 ε = 10 n = 1 µ H = 0.5 T T = 0.8 K C p / T ( m J K - m o l - ) -90 -60 -30 0 30 θ (deg.) ε = 9.7 n = 0.5 ε = 10 n = 1 µ H = 0.2 T T = 1.0 K C e l e / γ N T T (K) C p / T ( m J K - m o l- ) T (K )0 T κ -Ag κ -Br H ⊥ =2 T0 T H ⊥ =7 T T c T c (a) (b)(d)(c) κ -Ag κ -Br C ele ~ aT C p / T ( m J K - m o l- ) γ / γ N H || / H c2|| , H ⊥ / H c2 ⊥ H ⊥ (0 K)Ref. 10Ref. 32 H || (0.9 K) κ -Br κ -Ag H || (0.8 K) H ⊥ (0.8 K) C p / T ( m J K - m o l- ) √ H √ H C ele = g N T κ -Br κ -Ag C p / T ( m J K - m o l- ) κ -Ag κ -Br FIG. 2. (a) C p / T of κ -Ag and κ -Br as a function of squaredtemperature T . For clarity, an offset is added to the dataof κ -Ag. The red squares denote the 0 T data, while theblue circles represent the data of the normal state in mag-netic fields perpendicular to the conducting plane H ⊥ . Theinset is an enlarged graph of (a) below T =4 K . The blacksolid lines are the linear fit to the data of the normal state.The open and solid symbols denote the data for κ -Ag and κ -Br, respectively. (b) Temperature dependence of the super-conducting electronic heat capacity C ele / γ N T . The dashedline indicates the electronic heat capacity of the normal state.The dotted lines indicate the quadratic temperature depen-dence of C ele . (c) Recovery of the low-temperature electronicheat capacity in fields. The top and bottom panels show thedata for κ -Ag and κ -Br, respectively. The red circles (theleft scale) represent the data in parallel fields whereas theblue squares (the right scale) indicate those in perpendicularfields. The data of κ -Br are taken from Ref. [25] (red circles),Ref. [10] (blue squares), and Ref. [32] (blue triangles). Theblack dashed curves signify the √ H dependence, and the col-ored thick curves are guides for the eye. (d) Polar-angle θ dependence of heat capacity of κ -Ag and κ -Br. The solid andopen symbols represent the obtained data and their mirroredreflection at θ =0 ◦ to confirm symmetric features. The solidand dashed curves are the fits to Eq. (1) with the displayedparameters (cid:15) and n . perpendicular directions[30, 31] with the reported γ / γ N of κ -Br in perpendicular fields (the blue squares andtriangles in the bottom panel)[29, 32]. In the low-fieldregion, the data clearly show √ H dependence, indicativeof linear dispersion of the energy spectrum at the Fermilevel in the superconducting state. This behavior canbe observed in the polar-angle dependence of C p shownin Fig. 2(d). The dip-type behavior toward the paralleldirection (0 ◦ ) can be described by the following formulabased on the anisotropic effective mass model: C p ( θ ) = C p (0 ◦ ) + ∆ C p [1 − ( (cid:112) cos θ + (cid:15) sin θ ) n ] / (1 − (cid:15) n ) , ∆ C p = C p (90 ◦ ) − C p (0 ◦ ) . (1)In this model, we assume that the angle dependence orig-inates from the anisotropy of the critical field H c asexpressed by the anisotropic factor (cid:15) . In the case of or-ganic superconductors, the paramagnetic pair-breakingeffect is almost isotropic, and therefore, the origin of theanisotropy can be attributed to the orbital pair-breakingeffect. This means that the anisotropic factor (cid:15) can bedescribed as (cid:15) = H orb (cid:107) /H orb ⊥ = (cid:112) m ⊥ /m (cid:107) , where H orb and m are the orbital limit and the effective mass, respec-tively. The obtained values of (cid:15) about 10 ( m ⊥ / m (cid:107) ∼ n reflects the low-energy excitation,which corresponds to the magnetic field dependence ofthe electronic heat capacity, and therefore, n =1/2 ex-actly agrees with the observed √ H dependence. The √ H dependence and the value n =1/2 are consistent with thequasiparticle excitation observed as the T dependence.It should be noticed that κ -Br exhibits non-monotonicfield dependence (a kink around H / H c2 ∼ κ -Ag as described by the thick curves thatare guides for the eye. The different behaviors betweenthe two imply the difference in the gap structure.To examine the nodal structures, we discuss the in-plane anisotropy for determining the positions of theline nodes. In Fig. 3(a), the heat capacity at vari-ous conditions as a function of the azimuthal angle φ from the a -axis direction is displayed. The data indi-cate that the anisotropy has twofold and fourfold compo-nents, as reproduced by C p ( φ )/ T = C / T + C / T + C / T ,where C and C indicate the twofold and fourfoldterms. Although the anisotropy may not be expressedin a simple sinusoidal form, here we simply use theequations C = C (cid:48) cos[2( φ - φ )] and C = C (cid:48) | cos(2 φ ) | or C = C (cid:48) cos(4 φ ) to evaluate the amplitude of the compo-nents. The origin of C can be assigned to the anisotropyof the d -wave gap function because the crystal structuredoes not have fourfold rotational symmetry. In Fig. 3(b),we extract the fourfold components C / T from the totalheat capacity by subtracting the other terms. It is ob-viously seen that the amplitude and sign of the fourfoldterms depend on temperature and magnetic field. Fig-ure 4(a) displays the thermal variation of the percentageof the fourfold amplitude A = C (cid:48) /[ C p ( µ H ) − C p (0 T)],scaled by the field-dependent parts of C p at each tem-perature. The theories and experiments for angle-resolved heat capacity measurements[20–22] suggest thatthe Doppler effect for the d -wave superconductivity givesa fourfold oscillation in the field-angle dependence of thedensity of states. When considering only the zero-energydensity of states, the minima of the fourfold term is par-allel to the directions of the gap nodes. Nevertheless,we need to take care that the oscillations show the sign C p / T ( m J K - m o l - ) φ (deg.) κ -Br ( µ H = 2 T, T = 0.7 K) -0.30.00.3 C p / T ( m J K - m o l - ) φ (degree) κ -Ag ( µ H = 0.2 T, T = 0.7 K) C p / T ( m J K - m o l - ) µ H = 2 T, T = 1.2 K κ -Br ( µ H = 2 T, T = 1.2 K) C p / T ( m J K - m o l - ) φ (deg.) κ -Br ( µ H = 2 T, T = 0.7 K) C p / T ( m J K - m o l - ) µ H = 2 T, T = 1.2 K κ -Ag ( µ H = 2 T, T = 1.2 K) (b)(a) -0.30.00.3 C / T ( m J K - m o l - ) µ H = 0.5 T, T = 1.0 K C p / T ( m J K - m o l - ) µ H = 0.5 T, T = 1.0 K C p / T ( m J K - m o l - ) κ -Ag ( µ H = 0.2 T, T = 0.7 K) -202 C p / T ( m J K - m o l - ) µ H = 2 T, T = 1.2 K κ -Ag ( µ H = 2 T, T = 1.2 K) -0.0300.03 C p / T ( m J K - m o l - ) µ H = 2 T, T = 1.2 K κ -Br ( µ H = 2 T, T = 1.2 K) C p / T ( m J K - m o l - ) κ -Ag ( µ H = 0.5 T, T = 1.0 K) -0.30.00.3 C / T ( m J K - m o l - ) κ -Ag ( µ H = 0.5 T, T = 1.0 K) C p / T ( m J K - m o l - ) µ H = 2 T, T = 1.2 K κ -Br ( µ H = 4 T, T = 0.7 K) -0.040.000.04 C p / T ( m J K - m o l - ) µ H = 2 T, T = 1.2 K κ -Br ( µ H = 4 T, T = 0.7 K) FIG. 3. (a) Raw data of in-plane field-angle dependenceof the heat capacity. The black curves are fits to the formulaintroduced in the text. (b) The fourfold oscillatory componentderived from the data shown in (a) by subtracting the othercomponents. The black curves for κ -Ag (top panel) are fits to C = C (cid:48) | cos(2 φ ) | , while those for κ -Br (bottom panel) are fitsto C = C (cid:48) cos(4 φ ). Since the C term should not be expressedby such simple forms, the fittings are just approximationsto describe the observed fourfold symmetry, leading to thedifference of the fitting functions. change depending on temperature and field due to thecontribution of the finite-energy density of states. Thecorrespondence of the positions between the gap nodesand the oscillation minima holds in the low-temperatureand low-field region, as schematically shown by the redarea in Fig. 4(b)[20, 22]. At the dotted curves, the signof the C term reverses due to the contribution of thefinite-energy density of states. Even though the size andshape of the area in which the zero-energy Doppler effectis predominant are greatly influenced by the structure ofthe Fermi surface[21], the positions of the gap nodes canbe determined by the sign in the low-energy limit whenthe sign crossover is observed. Thus, the negative signat µ H =0.2 T (=0.03 H c (cid:107) ), T =0.7 K (=0.13 T c ) for κ -Agand the positive sign at µ H =2 T (=0.06 H c (cid:107) ), T =0.7 K(=0.07 T c ) for κ -Br indicate that the gap nodes of κ -Agare located at the k x and k y directions whereas thoseof κ -Br are positioned at φ =45 ◦ apart from the crystalaxes, as schematically depicted in Fig. 4(c). These po-sitions mean that their fourfold symmetry is assigned to d xy -wave for κ -Ag and d x − y -wave for κ -Br.Next, we give consideration to the origin of the twofoldterm C . Detailed analyses of this C componentare shown in Supplemental Material.[33]. If the two-dimensional plane of the measured sample was slightlymisaligned from the magnetic field, the quasiparticle ex-citation by perpendicular fields also contributes to the -2-1012 A ( % ) T / T c κ -Br µ H = 4 T ( H/H c2|| ~0.12) µ H = 2 T ( H/H c2|| ~0.06) A ( % ) T / T c κ -Ag µ H = 0.2 T ( H/H c2|| ~0.03) µ H = 0.5 T ( H/H c2|| ~0.06) µ H = 2.0 T ( H/H c2|| ~0.26) (a) T c T H || d -waveARHC oscillation0.1~0.2 T c ~0.5 T c . ~ . H c H c antinode node (b) k x k y κ -Ag κ -Br (c) FIG. 4. (a) The amplitude of the fourfold oscillatory term invarious fields as a function of reduced temperature T / T c . Thepositive (negative) value indicates that the maxima (minima)of the C term are located at the directions of the crystalaxes. (b) Schematic phase diagram of angle-resolved heat ca-pacity for d -wave superconductivity. In the red areas, thenode positions are located at the directions of the minima ofangle-resolved heat capacity. In the low-energy limit, namely,the low- T and low- H region, the node location correspondsto the minima of the C term due to the predominant zero-energy Doppler effect[20–22]. The dotted curves indicate theregion at which the sign of the C term reverses and the am-plitude of that becomes almost zero. (c) Determined positionsof the gap nodes in the Fermi surface for κ -Ag and κ -Br. Thegreen dots signify the position of the nodes. angle dependence. However, the form of this contribu-tion in these salts should be (cid:112) | cos( φ ) | , different from theobserved C (cid:48) cos[2( φ - φ )]. This difference indicates thatthe twofold term is intrinsic to the present salts. Sincethe point group of κ -Br is orthorhombic D h , the d x − y symmetry is categorized in the A g irreducible represen-tation, and the s -wave symmetry also belongs to the samerepresentation in this point group. This leads to the lin-ear combination of the s -wave and d x − y -wave symme-tries in the A g symmetry as the d x − y + s ± -wave sym-metry. This situation is different from the d xy symmetry( B g ) for the monoclinic C h group of κ -Ag. Some recentstudies[9, 17, 24] point out that the d x − y + s ± -wave gapfunction possesses twofold rotational symmetry due tothe mixing of the s ± -wave gap. The C term observed byexperiments may be the symmetry origin in the case of κ -Br. However, κ -Ag, namely d xy -wave superconductivity,also shows a clear C component. In other angle-resolvedmeasurements for the dime-Mott superconductors[6, 10],the origin has been discussed in terms of the anisotropyof the ellipsoidal Fermi surface because the anisotropyof the Fermi velocity also contributes to the quasiparti-cle excitation outside the vortices[21, 22]. We thereforeconsider that the appearance of the C term observed inboth compounds originates from the combination of theanisotropy of the gap function and the Fermi velocity andit should be a common feature for organic superconduc- tors with ellipsoidal Fermi surfaces.Our results elucidate the symmetries of κ -Ag and κ -Br as d xy and d x − y + s ± , respectively, even thoughthey are classified into the same κ -type dimer-Mott sys-tem. Why do these compounds have different symme-tries? To answer the question, we here compare thepresent results with the theoretical studies for the pairingmechanisms based on the antiferromagnetic spin fluctua-tions. In fact, recent theoretical calculations without thedimer approximation[16–19] pointed out that the d xy -and d x − y + s ± -wave symmetries compete according tothe transfer integrals in the dimer units and the geo-metric frustration of the dimer triangular lattice. Eventhough Guterding, etal .[17] suggested that d x − y + s ± -wave symmetry is stable for all of the discovered κ -typesalts, it should be noticed that κ -Ag is located at theverge of the phase boundary between d xy and d x − y + s ± in the diagram in their calculation. Therefore, it is pos-sibly reasonable that the superconducting symmetry of κ -Ag is d xy because of some differences between the the-ories and our experiments, such as the temperature de-pendence of the transfer integrals. Moreover, we shouldnotice that κ -Br is also near the boundary even thoughthe parameters are different. The previous work on theangle-resolved heat capacity by Malone, etal .[10] indi-cated that the symmetry of κ -Br is attributed to d xy .This apparent disagreement could also come from theproximity to the boundary of the two symmetries since T c of our salt ( ∼ ∼ κ -(BEDT-TTF) X ( X =Cu[N(CN) ]Brand X =Ag(CN) H O) by means of the heat capacitymeasurements. The T temperature and √ H field de-pendence of the heat capacity in the low-energy regionindicates the presence of line nodes in the Fermi sur-face. The scenario is certainly consistent with the re-ported results based on two-dimensional d -wave super-conductivity. From the results of the angle-resolved heatcapacity measurements, we established the possible pair-ing symmetry of the superconductivity as d xy for κ -Agand d x − y + s ± for κ -Br. Comparing our results withthe theoretical studies, we find that the apparently con-tradictory result can be rationalized by the ratio of thetransfer integrals between the dimers. This indicates thatthe symmetry of κ -(BEDT-TTF) X is dominated by thestrength of the dimerization and the distorted triangulardimer lattice even though the superconductivity is causedby the same pairing mechanism, the antiferromagneticspin fluctuations.We thank K. Kawamura (Osaka University) and Dr.H. Akutsu (Osaka University) for kind support in the x-ray structural analysis. [1] K. Kanoda, J. Phys. Soc. Jpn. , 051007 (2006).[2] H. Elsinger, J. Wosnitza, S. Wanka, J. Hagel, D.Schweitzer, and W. Strunz, Phys. Rev. Lett. , 6098(2000).[3] J. M¨uller, M. Lang, R. Helfrich, R. Steglich, and T.Sasaki, Phys. Rev. B , 140509(R) (2002).[4] J. Wosnitza, S. Wanka, J. Hagel, M. Reibelt, D.Schweitzer, and J. A. Schlueter, Synth. Meth. ,201 (2003).[5] From the viewpoint of the point group, d x − y -wave sym-metry is categorized into A g for the D h point group and A g for C h , which also contain s -wave symmetry. Thus,in the case of κ -(BEDT-TTF) X salts, the d x − y waveis equivalent to d x − y + s ± in terms of symmetry classi-fication.[6] K. Izawa, H. Yamaguchi, T. Sasaki, and Y. Matsuda,Phys. Rev. Lett. , 027002 (2001).[7] T. Arai, K. Ichimura, K. Nomura, S. Takasaki, J. Ya-mada, S. Nakatsuji, and H. Anzai, Phys. Rev. B ,104518 (2001).[8] K. Ichimura, M. Takami, and K. Nomura, J. Phys. Soc.Jpn. , 114707 (2008).[9] D. Guterding, S. Diehl, M. Altmeyer, T. Methfessel, U.Tutsch, H. Schubert, M. Lang, J. M¨uller, M. Huth, H.O. Jeschke, R. Valent´i, M. Jourdan, and H.-J. Elmers,Phys. Rev. Lett. , 237001 (2016).[10] L. Malone, O. J. Taylor, J. A. Schlueter, and A. Carring-ton, Phys. Rev. B , 014522 (2010).[11] D. C. Cavanagh and B. J. Powell, Phys. Rev. B ,054505 (2019).[12] A. Benali, Synth. Meth. , 120 (2013).[13] J. Schmalian, Phys. Rev. Lett. , 4232 (1998).[14] H. Kondo and T. Moriya, J. Phys. Soc. Jpn. , 3695(1998).[15] T. Watanabe, H. Yokoyama, Y. Tanaka, and J. Inoue, J.Phys. Soc. Jpn. , 074707 (2006).[16] K. Kuroki, T. Kimura, R. Arita, Y. Tanaka, and Y. Mat-suda, Phys. Rev. B , 100516(R) (2002).[17] D. Guterding, M. Altmeyer, H. O. Jeschke, and R.Valent´i, Phys. Rev. B , 024515 (2016).[18] H. Watanabe, H. Seo, and S. Yunoki, J. Phys. Soc. Jpn , 033703 (2017).[19] H. Watanabe, H. Seo, and S. Yunoki, Nat. Commun. ,3167 (2019).[20] A. B. Vorontsov and I. Vekhter, Phys. Rev. B , 224501(2007).[21] M. Hiragi, K. M. Suzuki, M. Ichioka, and K. Machida, J.Phys. Soc. Jpn. , 094709 (2010).[22] T. Sakakibara, S. Kittaka, and K. Machida, Rep. Prog.Phys. , 094002 (2016).[23] S. Imajo, S. Fukuoka, S. Yamashita, and Y. Nakazawa,J. Therm. Anal. Calorim. , 1871 (2016).[24] S. Imajo, S. Yamashita, H. Akutsu, H. Kumagai, T.Kobayashi, A. Kawamoto, and Y. Nakazawa, J. Phys.Soc. Jpn, , 023702 (2019).[25] S. Imajo, C. Dong, A. Matsuo, K. Kindo, and Y. Ko-hama, arXiv:2012.02411.[26] T. Ishikawa, S. Yamashita, Y. Nakazawa, A. Kawamoto, and M. Oguni, J. Therm. Anal. Calorim. , 435 (2008).[27] S. Imajo, S. Yamashita, H. Akutsu, and Y. Nakazawa,Int. J. Mod. Phys. B , 1642014 (2016).[28] B. Andraka, C. S. Jee, J. S. Kim, G. R. Stewart, K. D.Carlson, H. H. Wang, A. V. S. Crouch, A. M. Kini, andJ. M. Williams, Solid State Commun. , 57 (1991).[29] O. J. Taylor, A. Carrington, and J. A. Schlueter, Phys.Rev. Lett. , 057001 (2007).[30] S. Imajo, Y. Nakazawa, and K. Kindo, J. Phys. Soc. Jpn, , 123704 (2018).[31] The Pauli-limiting field H P (cid:107) is assumed to be the in-plane H c2 (cid:107) here because it is regarded as the critical fieldof the uniform superconductivity if the possible FFLOstate does not occur above H P (cid:107) .[32] Y. Nakazawa and K. Kanoda, Phys. Rev. B ,R8670(1997).[33] See Supplemental Material for detailed analyses of the C term. Supplemental Material forSymmetry change of d -wavesuperconductivityin κ -type organic superconductors I. TWOFOLD TERM C IN THE IN-PLANEANISOTROPY
In this section, we present the twofold term C , ex-tracted from the data shown in Fig. 3(a). Fig. S5 displaysthe temperature and field dependence of the twofold com-ponents C / T , extracted from the total heat capacityby subtracting the other terms. As discussed in themain text, if C originates from the sample misalign-ment against magnetic fields, it can be described bythe formula (cid:112) | cos( φ ) | . However, the observed twofoldterm C can be a simple twofold sinusoidal fit C (cid:48) cos[2( φ - φ )]. This means that the present twofold componentarises from the in-plane anisotropy of the salts. To eval-uate the twofold contribution, we here scale this com-ponent with the change of the electronic heat capacity as A = C (cid:48) /[ C p ( µ H ) − C p (0 T)]. In Fig. S5(c), we findthat the temperature-dependent behavior can be roughlyscaled by √ H for κ -Ag without any phase shift, whichindicates that the C term simply reflects the quasipar-ticles excited by magnetic fields. In contrast, κ -Br showsthe phase shift (Fig. S5(b)) depending on field, whichleads to the change of the sing as shown in Fig. S5(d).The phase shift of the twofold term indicates that the C term should be composed of some elements, such asthe anisotropy of the Fermi velocity and the twofold ro-tational symmetry of the d x − y + s ± -wave gap function.Indeed, the behavior of κ -Br is roughly similar to thetemperature and field dependence of the C term comingfrom the gap symmetry, which is different from the caseof κ -Ag. The difficulty to decompose these componentsmakes the detailed discussion hard. Nevertheless, thisfact implies that the C term of κ -Ag comes only fromthe anisotropy of the Fermi velocity, whereas that of κ -Br may result from not only the anisotropy of the Fermivelocity but also that of the d x − y + s ± -wave gap sym-metry. This possible viewpoint agrees with the discussionon the C term. -101 C / T ( m J K - m o l - ) κ -Ag ( µ H = 0.5 T, T = 1.0 K) -101 C p / T ( m J K - m o l - ) φ (deg.) µ H = 0.2 T, T = 0.7 K κ -Ag ( µ H = 0.2 T, T = 0.7 K) C p / T ( m J K - m o l - ) µ H = 2 T, T = 1.2 K κ -Ag ( µ H = 2 T, T = 1.2 K) -101 C / T ( m J K - m o l - ) φ (deg.) µ H = 0.2 T, T = 0.7 K κ -Ag ( µ H = 0.2 T, T = 0.7 K) (b)(a) -0.0500.05 C p / T ( m J K - m o l - ) θ (degrees) µ H = 2 T, T = 1.2 K κ -Br ( µ H = 2 T, T = 1.2 K) C p / T ( m J K - m o l - ) µ H = 2 T, T = 1.2 K κ -Br ( µ H = 4 T, T = 0.7 K) C p / T ( m J K - m o l - ) φ (deg.) κ -Br ( µ H = 2 T, T = 0.7 K) A ( % ) T / T c κ - Br µ H = 4 T µ H = 2 T (d)(c) A ( % ) T / T c κ - Ag µ H = 0.2 T µ H = 0.5 T µ H = 2.0 T FIG. S5. (a),(b) Twofold oscillatory component derived fromthe data shown in Fig. 3(a) (the main text) by subtractingthe other components. The solid curves are the fits to theformula C / T = C (cid:48) cos[2( φ - φ )]. (c),(d) A as a function ofreduced temperature T / T cc