Metamagnetic phase transition in ferromagnetic superconductor URhGe
MMetamagnetic phase transition in ferromagnetic superconductor URhGe
V.P.Mineev , ∗ Universite Grenoble Alpes, CEA, IRIG, PHELIQS, F-38000 Grenoble, France Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia (Dated: February 17, 2021)Ferromagnetic superconductor URhGe has orthorhombic structure and possesses spontaneousmagnetisation along c-axis. Magnetic field directed along b -axis suppresses ferromagnetism in c -direction and leads to the metamagnetic transition to polarised paramagnetic state in b -direction.The theory of these phenomena based on the specific magnetic anisotropy of this material in ( b, c )plane is given. Line of the first order metamagnetic transition ends at a critical point. The Van derWaals - type description of behaviour of physical properties near this point is developed. The tripletsuperconducting state destroyed by orbital effect is recreated in vicinity of the transition. It shownthat the reentrance of superconductivity is caused by the sharp increase of magnetic susceptibilityin b direction near the metamagnetic transition. I. INTRODUCTION
Investigations of uranium superconducting ferromag-nets UGe , URhGe and UCoGe continue attract atten-tion mostly due to the quite unusual nature of its super-conducting states created by the magnetic fluctuations(see the recent experimental [1] and theoretical [2] re-views and references therein). They have orthorhombiccrystal structure and the anisotropic magnetic properties.The spontaneous magnetisation is directed along a axis inUGe and along c -axis in URhGe and UCoGe. The ferro-magnetic state in the two last materials is suppressed bythe external magnetic field H y directed along b crystallo-graphic direction. In URhGe at field H y = H cr ≈
12T the second order phase transition to ferromagneticstate is transformed to the transition of the first order[3]. The superconducting state suppressed [4] in muchsmaller fields H y ≈ , c -direction is the paramagneticstate.There was established, however, [3, 6, 7] that in fieldsabove H cr the magnetisation along b direction looks likeit has field independent ”spontaneous” component M y = M y + χ y H y . (1)This state is called polarised paramagnetic state. Theformation of this state is related with so called metam-agnetic transition observed in several heavy-fermion com-pounds (see the paper [8] and the more recent publication[9] and references therein). To take into account the for-mation of polarised paramagnetic state one must intro-duce definite modifications in the treatment performedin [5]. Here I present the corresponding derivation. ∗ E-mail: [email protected]
The paper is organized as follows. In the ChapterII after the brief reminder of results of the paper [5]the description of the metamagnetic transition is pre-sented. It is based on the specific phenomenon of mag-netic anisotropy in URhGe obtained with a local spin-density approximation calculations by Alexander Shick[10]. After the general consideration of the metamag-netic transition the modifications introduced by the uni-axial stress are considered. Then the Van der Waals -type theory of phenomena near the metamagnetic criti-cal point is developed and some physical properties arediscussed.The phenomenon of the reentrant superconductingstate is the subject of Chapter III. It is shown that therecreation of superconductivity is caused by the sharpincrease in the magnetic susceptibility [7] in b directionnear the metamagnetic transition. The Conclusion con-tains the summary of the results. II. METAMAGNETIC TRANSITION IN URhGe
As in the previous publications (eg [2]) I shall use x, y, z as the coordinates pinned to the corresponding crystal-lographic directions a, b, c . The Landau free energy of anorthorhombic ferromagnet in magnetic field H ( r ) = H y ˆ y is F = α z M z + β z M z + δ z M z + α y M y + β y M y + δ y M y + β yz M z M y − H y M y , (2)Here α z = α z ( T − T cc ) , α y > , (3)and I bear in mind the terms of the sixth order in powersof M z , M y with the coeffucients δ z > , δ y > x -direction themagnetisation along hard x -direction M x = 0. a r X i v : . [ c ond - m a t . s up r- c on ] F e b A. Transition ferro-para
Let us remind first the treatment developed in Ref.5undertaken in the assumption β y >
0. Then in constantmagnetic field H = H y ˆ y the equilibrium magnetisationprojection along the y direction M y ≈ H y α y + β yz M z ) (4)is obtained by minimisation of free energy (2) in respectof M y neglecting the higher order terms. Substitutingthis expression back to (2) we obtain F = α z M z + β z M z + δ z M z − H y α y + β yz M z , (5)that gives after expansion of the denominator in the lastterm, F = − H y α y + ˜ α z M z + ˜ β z M z + ˜ δ z M z + . . . , (6)where ˜ α z = α z ( T − T c ) + β yz H y α y , (7)˜ β z = β z − β yz α y β yz H y α y , (8)˜ δ z = δ z + β yz α y β yz H y α y . (9)Thus, in a magnetic field perpendicular to the directionof spontaneous magnetization the Curie temperature de-creases as T c = T c ( H y ) = T c − β yz H y α y α z . (10)The coefficient ˜ β z also decreases with H y and reacheszero at H y = H (cid:63) = 2 α / y β / z β yz . (11)At this field under fulfilment the condition, α z β yz T c α y β z > H (cid:63) , T c ( H (cid:63) )) on the line paramagnet-ferromagnet phase transition is a tricritical point. Thequalitative field dependences of the normalised Curietemperature t c ( H y ) = T c ( H y ) T c and b ( H y ) = ˜ β z β z are plottedin Fig 1a.On the line of the first order phase transition from theferromagnet to the paramagnet state the M z component of magnetisation drops from M (cid:63)z to zero [2]. The M y component jumps from M y ≈ H (cid:63) α y + β yz M (cid:63) z ) to M y ≈ H (cid:63) α y .Then at fields H y > H (cid:63) M y ≈ H y α y (13)proportional to the external field. This contradicts ex-perimental observations [3, 6, 7] which demonstrate thepresence of a ”spontaneous” part of magnetization in thefield above the transition in accordance with eq. (1). B. Transition ferro-polarised para
The part of free energy depending from M y F y = α y M y + β y M y + δ y M y + β yz M z M y − H y M y , (14)is valid also far from the transition to the ferromag-netic state in the temperature region where M z is notsmall. The important fact obtained with the local spin-density approximation calculations [10] is that the coef-ficient β y < b, c ) plane is hardly related with a naiveitinerant electron picture. In the paper [12] there wasevaluated the magnetic dipole moment (cid:104) T z (cid:105) ”related tothe anisotropy of the local magnetic field produced by thespin when the valence cloud is distorted either by spin-orbit and/or crystal field interaction. The high value ofthe (cid:104) T z (cid:105) / (cid:104) S z (cid:105) ratio found in URhGe could be expectedsince this compound presents a high magnetocrystallineanisotropy, so it should not be too close to the itinerantlimit, where (cid:104) T z (cid:105) is expected to be strongly suppressed.”The M y component of magnetisation is determined bythe equation2˜ α y M y + 4 β y M y + 6 δ y M y , = H y , (15)where ˜ α y = α y + β yz M z . (16)Taking into account the third order term we obtain M y ≈ H y α y − β y H y α y . (17)The coefficient β y < M y ( H y ) depends from the temperatureand pressure dependence of coefficients α y , β y , δ y . Attemperature decrease the field dependence of M y cantransfer from the monotonous growth taking place at β y < ˜ α y δ y to the S-shape dependence. This transfor-mation occurs at some temperature T cr such that in thedependence H y ( M y ) appears an inflection point. It isdetermined by the equations ∂H y ∂M y = 0 , ∂ H y ∂M y = 0 (18)having common solution M cr = − β y δ y , (19)at β y = ˜ α y δ y . The corresponding critical field is H cr = H y ( M cr ) = 165 √ α / y | β y | / . (20)At T < T cr the inequality β y >
53 ˜ α y δ y (21)is realised and the equation ∂H y ∂M y = 0 acquires two realsolutions, hence, the field dependence of M y acquires theS-shape plotted at Fig.1b. Equilibrium transition fromthe lower to the upper part of the curve M y ( H y ) corre-sponds to a vertical line connecting the points M and M defined by the Maxwell rule (cid:82) M ( H ) dH = 0. Theintegration is performed along the curve M y ( H y ). The M y component of magnetisation jumps from M to M (see Fig1b). At the same time the Curie temperaturedrops from T c ( H cr ) = T c − β yz M α z (22)to zero or even to negative value and the ferromagneticorder along z -direction disappears. Here we assume thatthe Curie temperature given by Eq.(22) exceeds the crit-ical temperature T cr .The described jump-like transition is realised in thecylindrical specimen in the magnetic field parallel to thecylinder axis. In specimens of the arbitrary shape withdemagnetisation factor n the transition occurs in somefield interval where the specimen is filled by the domainswith different magnetisation.Thus, at T < T cr and H y = H cr we have the phasetransition of the first order from the ferromagnetic statewith spontaneous magnetisation along z -direction to thepolarised paramagnetic state with induced magnetisationalong y -direction.When the critical field H cr is smaller than the criticalfield of transition ferro-para H (cid:63) , the ferro-para transi-tion, discussed in the previous section does not occurs.At T < T cr in fields H y exceeding H cr the field de-pendence of M y component of magnetisation behaves inaccordance with Eq.(1) corresponding to the experimen-tal observations. C. Uniaxial stress effects
It is known that a hydrostatic pressure applied toURhGe crystals stimulate ferromagnetism and at thesame time suppresses the superconducting state [13] andthe reentrant superconducting state [14] as well. Thelater is also shifted to a bit higher field interval. On thecontrary, the uniaxial stress along b -direction suppressesthe ferromagnetism decreasing the Curie temperatureand stimulates the superconducting state so strongly thatit leads to the coalescence of the superconducting andreentrant superconducting regions in the ( H y , T ) phasediagram [15]. The phenomelogical description of thesephenomena was undertaken in the paper[16]. There wasshown that both coefficients α z and α y in the Landaufree energy Eq.(2) acquire the linear uniaxial pressuredependence α z ( P y ) = α z ( T − T c ) + A z P y , (23) α y ( P y ) = α y − | A y | P y (24)corresponding to the moderate uniaxial pressure suppres-sion of the Curie temperature T c ( P y ) = T c − A z P y α z , (25)reported in [15] in the absence of an external field. How-ever, under the external field along y -direction the dropof the Curie temperature Eq.(10) is accelerated T c ( H y , P y )) ≈ T c − A z P y α z − β yz H y α y ( P y )) α z (26)in correspondence with the observed behaviour. More-over, the uniaxial stress causes strong decrease of thecritical field Eq.(20) H cr = H y ( M cr ) = 165 √ α y ( P y )) / | β y | / . (27) D. Van der Waals-type theory near the criticalpoint
The critical end point temperature for the first ordertransition in URhGe is T cr = 4 K and the critical hieldis H cr = 12 T . Let us expand the function H y ( M y ) attemperature slightly deviating from critical temperature T = T cr + t and the magnetisation near its critical value M y = M cr + m . We have h = H y − H cr = bt + (cid:20) ∂H y ∂M y | t =0 + 2 at (cid:21) m + 12 ∂ H y ∂M y | t =0 m + 16 ∂ H y ∂M y | t =0 m , (28)Here, we neglected by the temperature dependence of thesecond and the third order terms. Taking into accountthat ∂H y ∂M y | t =0 = ∂ H y ∂M y | t =0 = 0 we obtain h = bt + 2 atm + 4 Bm , (29)which obviously corresponds to the expansion of pressure p = P − P cr in powers of density η = n − n cr near theVan der Waals critical point [17].At t < m = − m = (cid:114) − at B . (30)The line of phase equilibrium between the two phasesbelow and above the transition is given by the equa-tion h = bt, t <
0. Hence, according to the Clausius-Clapeyron relation b = dhdt = s − s m − m = T cr qm − m , (31)where q is the transition latent heat. Near the criticalpoint the coefficient b is positive and finite, q ∝ √− t . At T → b →
0. In whole temperature interval (0 , T cr ) theline of the phase equilibrium is almost vertical.
1. Specific heat
The specific heat at fixed external field(see [17]) is C h ∝ T (cid:0) ∂h∂t (cid:1) m (cid:0) ∂h∂m (cid:1) t . (32)Then, using Eq.(29) we obtain C h ∝ b T at + 12 Bm . (33)Thus, the contribution to heat capacity according to theequation of state (29) near the critical point grows solong m decreases till to m and then begins to fall when m increases starting from m (see Fig1.c). This is thecontribution to the specific heat of the whole system andcannot be directly attributed to the specific heat of itiner-ant electrons proportional to the electron effective mass.The low temperature behaviour of the URhGe specificheat in magnetic field has not been established by a di-rect measurement but was derived [6] by the applicationof the Maxwell relation (cid:16) ∂S∂H y (cid:17) T = (cid:16) ∂M y ∂T (cid:17) H y from thetemperature dependence of the magnetisation M y ( T, H y )in the fixed field. The changes of the ratio C ( T ) /T havebeen ascribed to the the electron effective mass depen-dence from magnetic field [6, 18]. This was done in theassumption that URhGe is a weak itinerant ferromag-net, in other words, all the low temperature degrees of freedom in this material belong to the itinerant electronsubsystem. As we already mentioned above, the strongmagnetic anisotropy of this material [12] points on theimportance of the magnetic degrees of freedom localisedon the uranium ions and related with crystal field levels.See also the papers [2, 19, 20].
2. Resistivity
The magnetic field dependence of effective mass wasalso found [18, 21] by the aplication the Kadowaki-Woodsrelation A ( H y ) ∝ ( m (cid:63) ) where coefficient A is a pre-factor in the low-temperature dependence of resistivity ρ = ρ + AT .The A ( H y ) behaviour is determined by the processes ofinelastic electron-electron scattering which in the multi-band metals interfere with scattering on impurities (see fi[22–26]) and on magnetic excitations with field dependentspectrum. The non-spherical shape of the Fermi surfacesheets and the screening of el-el Coulomb interaction canintroduce deviations from T resistivity dependence. So,the physical meaning of the coefficient A ( H y ) behaviouris not so transparent and its relationship with the elec-tron effective mass is questionable.One can also note, that the temperature fit of the ex-perimental data was done in very narrow temperatureinterval and the T temperature dependence claimed in[21] seems somewhat unreliable. Compare with the re-sults reported in [13, 27].
3. Correlation function
The correlation function of fluctuations of the magneti-sation density m near the critical point at t < ϕ ( k ) = T at + 6 Bm + γ ij k i k j ) . (34)This is in correspondence with a marked increase of theNMR relaxation rate 1 /T with field H y increasing to-ward 12 T reported in [28, 29]. III. REENTRANT SUPERCONDUCTIVITY
The superconducting state in URhGe is completelysuppressed by the magnetic field H c ( T = 0) ≈ y -direction due to the orbital depairing effect. Then su-perconductivity recovers in the field interval 9 −
13 Taround the critical field H cr ≈
12 T of the transition ofthe first order from the ferromagnetic state with sponta-neous magnetization along z-direction to the state withinduced magnetization along y -direction. Evidently suchtype behaviour is possible if the magnetic field somehowstimulates the pairing interaction surmounting the or-bital depairing effect.In numerous publications starting from the paper byA.Miyake et al [18] the treatment of this phenomenonwas related with the assumption of an enhancement ofelectron effective mass m (cid:63) = m (1 + λ ) leading to theenhancement of pairing interaction and consequently ofthe temperature of transition to superconducting stateaccording to the Mc-Millan-like formula [30] T sc ≈ (cid:15) exp (cid:18) − λλ (cid:19) (35)derived in the paper [31] for the superconducting statewith p -pairing in an itinerant isotropic ferromagneticmetal. Similar to the liquid He-3 in this model there aretwo independent phase transition to the superconduct-ing state in the subsystems with spin-up and spin-downelectrons. The constant λ determined by the Hubbardfour-fermion interaction [31, 32] increases as we approachbut not too much close to ferromagnetic instability. Inframe of this model the question of why the growth of themagnetic field H y approaches the ferromagnetic transi-tion remains unanswered.The following development of this type approach hasbeen undertaken by A.Chubukov and co-authors [33].The reentrant superconductivity and mass enhancementhave been associated with the Lifshitz transition [34]which occurs in one of the bands in a finite magnetic fieldstimulating the splitting of spin-up and spin-down bands.There was established modest enhancement of the tran-sition critical temperature in the field about 10 T. Thus,the model can claim to the qualitative explanation of thesuperconducting state reentrance. However, it should benoted that the measured [34] quasiparticle mass in thecorresponding band does not increase but decreases andremains finite, implying that the Fermi velocity vanishesdue to the collapse of the Fermi wave vector. The cross-section of the Fermi surface of this band corresponds to7% of the Brillouin zone area. Thus, the reentrance ofsuperconductivity is hardly could be associated with theobserved Lifshitz transition.The models [31, 33] describe the physics of pure itiner-ant electron subsystem. Such a treatment is approved inapplication to the He Fermi-liquid. The measurementsby x-ray magnetic circular dichroism [12] point to the lo-cal nature of the URhGe ferromagnetism. Namely, thecomparison of the total uranium moment µ Utot to the totalmagnetisation M tot at different magnitude and directionof magnetic field indicates that the uranium ions dom-inate the magnetism of URhGe. The same is true alsoin the parent compound UCoGe [35]. So, the magneticsusceptibility χ ij ( q , ω ) is mostly determined by the lo-calised moments subsystem. Hence, an approach basedon the exchange interaction between conduction electronsand magnetic moments localised on uranium atoms seemsmore appropriate.Using the standard functional-integral representationof the partition function of the system (see fi [36]), we ob-tain the following term in the fermionic action describing an effective two-particle interaction between electrons: S int = − I (cid:90) dx dx (cid:48) S i ( x ) D ij ( x − x (cid:48) ) S j ( x (cid:48) ) , (36)where S ( r ) = ψ † α ( r ) σ αβ ψ β ( r ) is the operator of the elec-tron spin density, x = ( r , τ ) is a shorthand notation forthe coordinates in real space and the Matsubara time, (cid:82) dx ( ... ) = (cid:82) d r¯ (cid:82) β dτ ( ... ), I is the coupling constants ofelectrons with spin fluctuations, D ij ( x − x (cid:48) ) is the spin-fluctuation propagator expressed in terms of the dynam-ical spin susceptibility χ ij ( q , ω ).Making use the interaction (36) one can calculate theelectron self energy and find the dependence of the elec-tron effective mass from magnetic field as well the tem-perature of transition to the superconducting state withtriplet pairing. In application to UCoGe in magneticfield parallel to direction of spontaneous magnetisationthis program has been accomplished in the paper [37]. Inthe simplified case of a single-band (say spin-up) equal-spin pairing superconducting state the critical tempera-ture without including the orbital effect of the field is T sc = (cid:15) exp (cid:18) − λ (cid:104) N ( k ) χ uzz (cid:105) I (cid:19) , (37)where, as in the McMillan formula, 1 + λ corresponds tothe effective mass renormalisation, whereas the pairingamplitude expressed through the odd in momentum partof static susceptibility χ uzz = 12 [ χ zz ( k − k (cid:48) ) − χ zz ( k + k (cid:48) )] , which is the main source of the critical temperature de-pendence from magnetic field. Here, χ zz ( k ) = 1 χ − z + 2 γ ij k i k j , (38)and χ z = χ z ( H z ) is the z -component of susceptibility inthe finite field H z . Its magnitude at H z → χ z . The angular brackets denote averaging overthe Fermi surface and N ( k ) is the angular dependentdensity of electronic states on the Fermi surface, (cid:104) N ( k ) χ uzz ( H z ) (cid:105) ≈ (cid:104) N ( k )ˆ k z (cid:105) k F χ z (2 χ z ) − + 4 γ zz k F . (39)The denominator in the exponent of Eq.(37) can be ex-pressed through its value at H z → (cid:104) N ( k ) χ uzz ( H z ) (cid:105)(cid:104) N ( k ) χ uzz ( H z → (cid:105) = χ z χ z ξ m k F ) χ z χ z + 2( ξ m k F ) . (40)Here the product γ zz k F χ z = ( ξ m k F ) is expressedthrough the magnetic coherence length ξ m which nearthe zero temperature is of the order several interatomicdistances.In assumption ( ξ m k F ) (cid:29) (cid:104) N ( k ) χ uzz ( H z ) (cid:105) ≈ χ z ( H z ) χ z (cid:104) N ( k ) χ uzz ( H z → (cid:105) . (41)This very rough estimation presents the qualitative de-pendence of exponent in equation (37) from magneticfield. The longitudinal susceptibility drops with the aug-mentation of magnetic field parallel to the spontaneousmagnetisation leading to the suppression of the temper-ature of transition to the superconducting state. Thesame mechanism works in the opposite sense in the fieldperpendicular to spontaneous magnetisation.In field perpendicular to the spontaneous magnetisa-tion the similar approach applied to the simplified singleband model in weak coupling approximation yields (seeEq.(169) in the review [2]) the critical temperature T sc ≈ (cid:15) exp (cid:32) − (cid:104) N ( k ) χ uzz (cid:105) cos ϕ + (cid:104) N ( k ) χ uyy (cid:105) sin ϕ ] I (cid:33) , (42)where tan ϕ = H y /h and h is the exchange field actingon the electron spins. This is the critical temperature oftransition to the superconducting state without includingthe orbital effect.The orbital effect suppresses the superconducting stateand near the upper critical field at zero temperature H c y ( T = 0) = H = cT sc (43)the actual critical temperature is T orbsc = a (cid:112) H − H y , (44)where a √ c is the numerical constant of the order of unity.This is the usual square root BCS dependence of the crit-ical temperature from magnetic field in low tempera-ture - high field region such that T orbsc ( H y = H ) = 0.However, in the present case the magnitude H itself isa function of the external field H y . Let us look on itsbehaviour.Similar to Eq.(41) we get (cid:104) N ( k ) χ uzz ( H y ) (cid:105) cos ϕ + (cid:104) N ( k ) χ uyy ( H y ) (cid:105) sin ϕ ≈ χ z ( H y ) χ z (cid:104) N ( k ) χ uzz ( H y → (cid:105) cos ϕ + χ y ( H y ) χ y (cid:104) N ( k ) χ uyy ( H y → (cid:105) sin ϕ ] . (45)Here χ z ( H y ) and χ y ( H y ) are the z and y componentsof susceptibility in finite field H y and χ z and χ y arethe corresponding susceptibilities at H y →
0. Unlike tothe Eq.(41) the field dependence of the Eq.(45) is not sovisible. One can note, however, the different field depen-dence of two summands in the Eq.(45).(i) The susceptibility along z direction χ z ( H y ) increases with magnetic field H y following to the decreasing of theCurie temperature according to Eq.(10). The growth ofsusceptibility along z direction at the approaching thefield H y to H cr is confirmed by the field dependence ofthe NMR scattering rate 1 /T reported in [28, 29]. Atthe same time, the increase of χ z ( H y ) is limited by thedecrease of cos ϕ . We do not know how fast it is becausethe magnitude of the exchange field is not known.(ii) As the field approaches to H cr the low temperaturesusceptibility χ y ( H y ) has a high delta-function-like peak[7] with magnitude more than 10 times greater than itis at H y → ϕ is also in-creased. This indicates that in URhGe the more impor-tant is the second term connected with the metamagnetictransition.Thus, in vicinity of metamagnetic transition one canexpect the increase of the critical temperature estimatedwithout including the orbital effect according to Eq.(42).The radicand in the equation (44) after being negativein some field interval acquires the positive value as thefield approaches to H cr . The critical temperature Eq.(44)reaches maximum in vicinity of metamagnetic transitionsee Fig2b.Similar arguments in favour of stimulation supercon-ductivity near the metamagnetic transition in field par-allel to b axis can be applied to the discovered recentlyother superconducting compound UTe [39–41] isostruc-tural with URhGe. However, in view of many particularproperties of this material we leave this subject for futurestudies .In the parent compound UCoGe the metamagnetictransition is absent (at least at H y < b axis is probably mostlydetermined by the first term in the Eq.(45).Near H y = H cr at temperatures T < T cr the NMRspectrum is composed of two components indicating thatthe transition is of the first order accompanied by thephase separation [28]. Thus, in almost whole intervalnear H cr the superconductivity is developed in mixtureof ferromagnetic state with polarisation along z directionand the field polarised state with polarisation along y -direction. IV. CONCLUSION
We have demonstrated that in the orthorhombic fer-romagnet URhGe the ferromagnetic ordering along c -axis suppressed in process of increase of magnetizationin the perpendicular b -direction induced by the externalmagnetic field. This process is accelerated by the ten-dency to the metamagnetic transition which occurs at H y = H cr = 12 T. The transition of the first order is ac-companied by the suppression of the ferromagnetic statewith polarization along c -axis and the arising of mag-netic state polarized along b -xis. The line of first orderphase transition is finished at the critical end point withtemperature T = T cr = 4 K.The uniaxial stress along b -axis causing moderate sup-pression of the Curie temperature in the absence of mag-netic field accelerates the Curie temperature drop in fi-nite magnetic field H y and quite effectively decreases thecritical field of metamagnetic transition. As result, thesuperconducting state recovers itself in much smaller fieldand can even merged with superconducting state in the small fields region.The superconducting state suppressed in field H y ≈ −
13) T in the vicinityof the critical field. This phenomenon is related to thestrong increase of the pairing interaction caused mostlyby the strong augmentation of the magnetic susceptibilityalong b -direction in vicinity of the metamagnetic transi-tion. [1] D.Aoki, K.Ishida and J.Flouquet, J. Phys. Soc. Jpn. ,022001 (2019).[2] V.P.Mineev, Usp. Fiz. Nauk , 129 (2017) [Phys.-Usp. , 121 (2017).[3] Levy F, Sheikin I, Grenier B, Huxley A D Science , 247006(2005).[5] V.P.Mineev, Phys. Rev. B , 195107(2011).[7] S.Nakamura, T.Sakakibara, Y.Shimizu, S.Kittaka,Y.Kono, Y.Haga, J.Pospisil, and E.Yamamoto, Phys.Recv. B , 094411 (2017).[8] Y. Aoki, T.D. Matsuda, H. Sugawara, H. Sato, H.Ohkuni, R. Settai, Y. Onuki, E. Yamamot, Y. Haga, A.V.Andreev, V. Sechovsky, L. Havela, H. Ikeda, K. Miyake,Journ.Mag.Magn.Mat. ,271 (1998).[9] D.Aoki, T.Combier, V.Taufour, T.D.Matsuda,G.Knebel, H.Kotegawa, and J.Flouquet, J.Phys.Soc. Jpn. , 094711 (2011).[10] A.B.Shick, Phys.Rev. B , 180509(R) (2002).[11] R.Z.Levitin, A.S.Markosyan, Usp. Fiz. Nauk , 623(1988) [Phys.-Usp. , 730 (1988).[12] F.Wilhelm, J.P.Sanchez, J.-P.Brison, D.Aoki, A.B.Shick,and A.Rogalev, Phys.Rev. , 235147 (2017).[13] F. Hardy, A. Huxley, J. Flouquet, B. Salce, G. Knebel, D.Braithwaite, D. Aoki, M. Uhlarz, C. Pfleiderer, PhysicaB 359-361, 1111 (2005).[14] A.Miyake, D.Aoki, and J. Flouquet, J.Phys. Soc. Jpn. , 063703 (2009).[15] D.Braithwaite, D.Aoki, J.-P.Brison, J.Flouquet,G.Knebel, A.Pourret, Phys. Rev .Lett. (to be pub-lished) (2017).[16] V.P.Mineev, Phys. Rev. B , 104501 (2017).[17] L.D.Landau and E.M.Lifshitz, Statistical Physics,Course of Theoretical Physics Vol V.
Oxford:Butterworth-Heinemann,1995).[18] A.Miyake, D.Aoki, and J. Flouquet, J.Phys. Soc. Jpn. , 094709 (2008).[19] R.Troc, Z. Gajek, and A.Pikul, Phys. Rev. B , 224403(2012).[20] V.P.Mineev, Phys. Rev. B , 046401(2016).[22] R.W.Keyes, J. Phys. Chem. Solids , 1 (1958). [23] V.F.Gantmakher, I.B.Levinson, Zh. Eksp. Teor. Fiz. ,261 (1978) [Sov. Phys. JETP
133 (1978)].[24] J.Appel and A.W.Overhauser, Phys.Rev.B , 758 1978.[25] S.S.Murzin, S.I.Dorozhkin, A.C.Gossard, Pis’ma Zh.Eksp. Teor.Fiz. , 101 (1998) [JETP Letters , 113(1998)].[26] H.K.Pal, V.I.Yudson, and D.L.Maslov, Lith.J.Phys. ,142 (2012).[27] K.Prokes, T.Tahara, Y.Echizen,T.Takabatake,T.Fujita,I.H.Hagmusa, J.C.P.Klaasse, E.Br¨uck, F.R.deBoer,M.Divis, V.Sechovsky, Physica B , 220 (2002).[28] H.Kotegawa, K.Fukumoto, T.Toyama, H.Tou, H.Harima,A.Harada, Y.Kitaoka, Y.Haga, E.Yamamoto, Y.Onuki,K.M.Itoh, and E.E.Haller, J.Phys. Soc. Jpn. , 054710(2015).[29] Y. Tokunaga, D.Aoki, H.Mayaffre, S. Kr¨amer, M.-H.Julien,C. Berthier, M. Horvati?, H. Sakai, S. Kambe, andS. Araki, Phys. Rev. Lett. , 216401 (2015).[30] W.L.McMillan, Phys.Rev. , 331 (1968).[31] D. Fay and J. Appel: Phys. Rev. B (1980) 3173.[32] W. F. Brinkman and S. Engelsberg, Phys. Rev. , 417(1968).[33] Yu.Sherkunov, A.V. Chubukov, and J.J.Betouras,Phys.Rev.Lett. , 097001 (2018).[34] E.A.Yelland, J.M.Barraclough, W.Wang, K.V.Kamenev, and A.D. Huxley, Nat. Phys. , 890 (2011).[35] M.Taupin, J.P.Sanchez, J.-P.Brison, D.Aoki, G.Lapertot,F.Wilhelm, and A.Rogalev, Phys.Rev. , 035124 (2015).[36] N.Karchev, Phys.Rev.B , 054416, (2003).[37] V.P.Mineev, Annals of Physics (NY), to be published(2020).[38] W.Knafo, T.D.Matsuda, D.Aoki, F.Hardy,G.W.Scheerer, G.Ballon, M.Nardone, A.Zitouni,C.Meingast and J.Flouquet, Phys. Rev. B , 184416(2012).[39] S.Ran, I-Lin Liu, YunSuk Eo, D.J.Campbell, P.M.Neves,W.T.Fuhrman, S.R.Saha, C.Eckberg, H.Kim, D.Graf,F.Balakirev, J.Singleton, J.Paglione and N.Butch, Na-ture Physics ,1250 (2019).[40] G.Knebel, W.Knafo, A.Pourret, Qun Niu, M.Valiska,D.Braithwaite, G.Lapertot, M.Nardone, A.Zitouni,S.Mishra, I.Sheikin, G.Seyfarth, J.-P.Brison, D.Aoki,J.Flouquet, J. Phys. Soc. Jpn , 063707 (2019).[41] Q.Niu, G.Knebel, D.Braithwaite, D.Aoki, G.Lapertot,M.Valiska, G.Seyfarth, W.Knafo, T.Helm, J.-P.Brison,J.Flouquet, and A.Pourret, arXiv:2003.08986 [cond-mat](2020). FIG. 1: (Color online) a) Schematic behaviour of the nor-malised Curie temperature t c ( H y ) = T c ( H y ) T c and coefficient b ( H y ) = ˜ β z β z . b) Schematic dependence M y ( H y ) at T < T cr and H cr < H (cid:63) . c) Schematic behaviour C h /T . FIG. 2: (Color online) a) Schematic field dependence of y component susceptibility χ y ( H y ) at temperature T → T orbscorbsc