Superconductivity in the Ferromagnet URhGe under uniaxial pressure
SSuperconductivity in the Ferromagnet URhGe under uniaxial pressure
V.P.Mineev
Univ. Grenoble Alpes, CEA, INAC, PHELIQS, GT, F-38000 Grenoble (Dated: November 13, 2018)Uniaxial pressure applied in the b crystallographic direction perpendicular to spontaneous mag-netization in heavy fermion ferromagnet URhGe strongly stimulates superconductivity in this com-pound. The phenomenological approach allows point out two mechanisms of superconducting tem-perature raising. They originates from stimulation by the uniaxial stress both intraband and inter-band amplitudes of triplet Cooper pairing. The phenomenon of reentrant superconductivity undermagnetic field along b-axis is also strongly sensitive to the uniaxial stress in the same direction. Theuniaxial stress accelerates suppression the Curie temperature by the transversal magnetic field. Theemergence of the first order transition to the paramagnetic state occurs at much lower field than inthe absence of uniaxial stress.
PACS numbers: 74.20.De, 74.25.Dw, 74.25.Ha, 74.20.Rp, 74.70.Tx
I. INTRODUCTION
The coexistence of superconductivity and ferromag-netism is the hallmark of heavy fermion uranium com-pounds UGe , URhGe and UCoGe (see the recent ex-perimental [1] and theoretical [2] reviews and referencestherein). The emergence of the superconducting state attemperatures far below the Curie temperature and veryhigh upper critical field strongly indicate on the spin-triplet Cooper pairing in these materials. One of the mostpeculiar observations is the phenomenon of reentrant su-perconductivity in URhGe [3] which is an orthorhom-bic ferromagnet with spontaneous magnetization alongc-axis. At low enough temperature the magnetic fieldabout 1.3 Tesla directed along the b -axis suppresses thesuperconducting state [4] but at much higher field about10 Tesla the superconductivity is recreated and exists tillthe field about 13 Tesla [3]. The maximum of the super-conducting critical temperature in this field interval is ≈ . K . In the same field interval the material transfersfrom the ferromagnet to the paramagnet state by meansof the first-order type transition. The superconductingstate exists not only inside of the ferromagnetic state butalso in the paramagnetic state separated from the ferro-magnetic state by the phase transition of the first order.The theoretical treatment of this phenomenon has beenproposed in Ref.5.There was shown experimentally that a hydrostaticpressure applied to URhGe crystals stimulates ferromag-netism causing an increase of the Curie temperature T c ( P ) and, at the same time, suppresses the supercon-ducting state decreasing the critical temperature of su-perconducting phase transition T sc ( P ) [6], as well as themaximum of the superconducting critical temperature ofthe reentrant superconducting state [7]. The latter is alsoshifted to a bit higher field interval. Quite the oppositebehavior has been registered recently [8] under uniax-ial stress P y in b-direction. In what follows we shalluse the x, y, z coordinate axes pinned correspondinglyto the a, b, c crystallographic directions. Namely, theuniaxial stress suppresses the ferromagnetism decreas- ing the Curie temperature T c ( P y ) and stimulate the su-perconducting state such that the temperature of super-conducting transition increases so strongly that leads tothe coalescence the superconducting and reentrant super-conducting regions in ( H y , T ) phase diagram already atquite moderate uniaxial stress values. The comparison of( H y , T ) phase diagrams at ambient pressure and at someuniaxial stress is presented in Fig.1.Here, I show that the stimulation of superconductingstate originates from two mechanisms: the Curie tem-perature suppression stimulating the intraband pairingand the increase of the magnetic susceptibility along b-direction stimulating interband pairing. Then, makinguse the phenomenological approach developed in Re.5, Iconsider the ( H y , T ) phase diagram modification causedby uniaxial stress along b-direction. It is demonstratedthat the uniaxial stress strongly accelerates the processof Curie temperature suppression by magnetic field alongb direction. Also the field induced transformation of thesecond to the first order ferro-para phase transition oc-curs at much lower field H cry ( P y ) values than in a stressabsence. This leads to the coalescence of the supercon-ducting and reentrant superconducting states. II. FREE ENERGY
The Landau free energy density of an orthorhombicferromagnet in magnetic field H ( r ) under an externalpressure consists of magnetic, elastic and magneto-elasticparts F = F M + F el + F Mel , (1)where in the magnetic part [2] F M = α z M z + β z M z + δ z M z + α y M y + α x M x + β xy M x M y + β yz M z M y + β xz M z M x − MH , (2)we bear in mind the orthorhombic anisotropy and alsothe term of the sixth order in powers of M z . Here, x, y, z a r X i v : . [ c ond - m a t . s up r- c on ] J un are the coordinates pinned to the a, b, c crystallographicdirections correspondingly, α z = α z ( T − T c ) , α x > , α y > . (3)The elastic part of free energy in an orthorhombic crystalis [9] F el = 12 λ x u xx + 12 λ y u yy + 12 λ z u zz + λ xy u xx u yy + λ xz u xx u zz + λ yz u yy u zz + 12 µ xy u xy + 12 µ xz u xz + 12 µ yz u yz , (4)where u xx , u yy , u zz , u xy , u xz , u yz are the components ofdeformation tensor.In constant magnetic field H = H y ˆ y the x -projection ofmagnetization M x = 0, hence, one can take into accountjust y and z magneto-elastic terms F Mel = ( p x u xx + p y u yy + p z u zz ) M z +( q x u xx + q y u yy + q z u zz ) M y + ru yz M y M z . (5)To find the deformaition arising under infuence of uni-axial stress P y applied along b -axis one must solve thelinear equations ∂ ( F el + F Mel ) ∂u xx = 0 , ∂ ( F el + F Mel ) ∂u zz = 0 ,∂ ( F el + F Mel ) ∂u yz = 0 ∂ ( F el + F Mel ) ∂u yy = − P y (6)in respect of components of deformation tensor. Whereasin the case of hydrostatic pressure P the correspondingsystem of equations is ∂ ( F el + F Mel ) ∂u xx = − P, ∂ ( F el + F Mel ) ∂u zz = − P,∂ ( F el + F Mel ) ∂u yz = 0 ∂ ( F el + F Mel ) ∂u yy = − P (7)Solving the equations (6) and substituting the solutionback to the sum F el + F Mel given by equations (4) and (5)we obtain the magnetic field H y and the uniaxial pressure P y dependent magnetic part of free energy density F M = ( α z + A z P y ) M z + β z M z + δ z M z +( α y + A y P y ) M y + β yz M z M y − M y H y . (8)The coefficients β z and β yz in this expression differ fromcorresponding coefficients in Eq. (2). However, thisdifference is pressure independent so long the magneto-elastic coupling has the simple form given by eq.(5).Hence, in what follows, we keep for these coefficients thesame notations as in Eq.(2).The coefficient at M z now is α z ( P ) = α z ( T − T c ) + A z P y , (9) and the Curie temperature acquires the pressure depen-dence T c ( P y ) = T c − A z P y α z . (10)The expression of coefficients A z and A y through elasticand magneto-elastic moduli (see Appendix) don’t giveup hopes to get a theoretical estimation of their value.But in fact this is not necessary. As we shall see the im-portant role play the sign of them which is determinedexperimentally. The Curie temperature decreases withuniaxial pressure [8] what corresponds to positive A z co-efficient.The coefficient at M y also acquires the pressure depen-dence α y ( P y ) = α y + A y P y . (11)The equilibrium magnetization projection along the y di-rection is obtained by minimization of free energy (8) inrespect of M y M y = H y α y ( P y ) + β yz M z ] . (12)The measurements [8] shows that the susceptibility along y direction χ y ( P y ) = M y H y increases with pressure en-hancement. This is owing to both T c ( P y ) and muchstronger α ( P y ) decrease with uniaxial pressure. So, thecoefficient A y is proved to be negative, thus α y ( P y ) = α y − | A y | P y . (13)We shall see in the next Section that both dependencies(10) and (13) cause the enhancement of superconductingcritical temperature. Then I will demonstrate that b-direction magnetic softening makes easier the suppressionof ferromagnetic state by magnetic field in y direction andstimulates the emergency of reentrant superconductingstate.The hydrostatic pressure creates the coefficients depen-dencies of the same form α z ( P ) = α z ( T − T c ) + A hz P, (14) α y ( P ) = α y + A hy P. (15)This case, however, A hz < , A hy > III. SUPERCONDUCTING CRITICALTEMPERATURE
The superconducting state in two band (spin-up andspin-down) orthorhombic ferromagnet is described in itssimplest form in terms of two complex order parameteramplitudes [10],[2]∆ ↑ ( k , r ) = ˆ k x η ↑ ( r ) , ∆ ↓ ( k , r ) = ˆ k x η ↓ ( r ) (17)depending on the Cooper pair centre of gravity coordi-nate r and the momentum k of pairing electrons. Thisparticular order parameter structure allows to explainthe specific temperature dependence of the upper criticalfield anisotropy in URhGe [4], [2].The corresponding critical temperature of transition tothe superconducting state is determined by the BCS-typeformula T = ε exp (cid:18) − g (cid:19) , (18)where the constant of interaction g = g ↑ x + g ↓ x (cid:115) ( g ↑ x − g ↓ x ) g ↑ x g ↓ x (19)is expressed through the constants of intra-band pairing g ↑ x , g ↓ x and the constants of inter band pairing g ↑ x , g ↓ x .They are functions of temperature, pressure and mag-netic field. Thereby the formula (18) is, in fact, an equa-tion for the determination of the critical temperature ofthe transition to the superconducting state. The constants of intra-band pairing interaction forspin-up and spin-down bands are proportional to the av-eraged over the Fermi surface density of states and theodd-part of susceptibility along the direction of sponta-neous magnetization ( see [2] Eq.(103)) g ↑ x ∝ (cid:104) ˆ k x N ↑ (cid:105) (2 β z M z + γ z k F ) , (20) g ↓ x ∝ (cid:104) ˆ k x N ↓ (cid:105) (2 β z M z + γ z k F ) . (21)The magnetization below the Curie temperature is givenby 2 β z M z = α z ( T c ( P y ) − T ) . (22)Assuming that at temperatures far below from T c ( P y )this formula is still qualitatively valid we obtain usingEq.(10) 2 β z M z ≈ α z ( T c − A z P y ) . (23)Thus, the magnetization decreases with uniaxial pres-sure, what causes in its turn the increase of supercon-ducting interaction constant.On the other hand, the constants of inter-band pair-ing interaction are determined by the difference of theodd parts susceptibilities in x and y directions (see [2]Eq.(104)) g ↑ x ∝ (cid:104) ˆ k x N ↑ (cid:105) (cid:26) γ xxx ( α x + β xz M z + 2 γ x k F ) − γ xyy ( α y ( P ) + β yz M z + 2 γ y k F ) (cid:27) ≈ − γ xyy (cid:104) ˆ k x N ↑ (cid:105) ( α y ( P ) + β yz M z + 2 γ y k F ) , (24) g ↓ x ∝ (cid:104) ˆ k x N ↓ (cid:105) (cid:26) γ xxx ( α x + β xz M z + 2 γ x k F ) − γ xyy ( α y ( P y ) + β yz M z + 2 γ y k F ) (cid:27) ≈ − γ xyy (cid:104) ˆ k x N ↓ (cid:105) ( α y ( P ) + β yz M z + 2 γ y k F ) , (25)where we have used the smallness of susceptibility along x -direction in respect of susceptibility along y -axis [11].According to Eqs. (13) and (23) the denominators inthese expressions are decreasing functions of pressure.Thus, the absolute values of the constant of interbandpairing increase with uniaxial pressure.Thus, all the terms in Eq.(19) increase with the uni-axial pressure increase, what results in the growing upthe temperature of superconducting transition shown inFig.1. IV. PHASE TRANSITION IN MAGNETICFIELD PERPENDICULAR TO EASYMAGNETIZATION AXIS
The reentrant superconducting state in URhGe arisesin high magnetic field along b-axis in vicinity of the firstorder transition from the ferromagnetic to the param-agnetic state. The change of the phase transition typefrom the second to the first order has been describedphenomenologically in Ref.5. With aim to establish theuniaxial pressure dependence of the transtion transfor-mation we reproduce this derivation.The equilibrium magnetization projection along the y direction is given by (12). Substituting this formula toEq.(8) we obtain F M = α z ( P y ) M z + β z M z + δ z M z − H y α y ( P y ) + β yz M z , (26)that gives after expansion of the denominator in the lastterm, F M = − H y α y ( P y ) + ˜ α z M z + ˜ β z M z + ˜ δ z M z + . . . , (27)where ˜ α z = α z ( T − T c ( P y )) + β yz H y α y ( P y )) , (28)˜ β z = β z − β yz α y ( P y ) β yz H y α y ( P y )) (29)˜ δ z = δ z + β yz ( α y ( P )) β yz H y α y ( P y )) (30)We see that under a magnetic field perpendicular to thedirection of spontaneous magnetization the Curie tem-perature decreases as T c = T c ( P y , H y ) = T c ( P y ) − β yz H y α z ( α y ( P y )) = T c − A z P y α z − β yz H y α z ( α y − | A y | P y ) . (31)Thus, at a finite field H y the Curie temperature suppres-sion by uniaxial pressure occurs much faster than in thefield absence.The coefficient ˜ β z also decreases with H y and reacheszero at H cry = 2( α y ( P y )) / β / z β yz . (32)At this field under fulfillment the condition, α z β yz T c α y ( P ) β z > H y > H cry ( P )phase transition from a paramagnetic to a ferromagneticstate becomes the transition of the first order (see Fig.9in the Ref.2). The point ( H cry , T c ( H cry )) on the lineparamagnet-ferromagnet phase transition is a tricriticalpoint. The pressure dependence H cry ( P y ) ∝ (cid:18) − | A y | α y P y (cid:19) / (34)roughly corresponds to the observed experimentally [8]pressure dependence H R ( P y ). An uniaxial pressure en-hancement decreases the field of the first order phasetransition from ferromagnetic to paramagnetic state.The minimization of the free energy Eq. (27) gives thevalue of the order parameter in the ferromagnetic state, M z = 13˜ δ z [ − ˜ β z + (cid:113) ˜ β z − α z ˜ δ z ] . (35) The minimization of the free energy in the paramagneticstate, F para = α y ( P y ) M y − H y M y (36)in respect M y gives the equilibrium value of magnetiza-tion projection on axis y in paramagnetic state, M y = H y α y ( P ) . (37)Substitution back in Eq. (36) yields the equilibrium valueof free energy in the paramagnetic state, F para = − H y α y ( P y ) . (38)On the line of the phase transition of the first order fromthe paramagnetic to ferromagnetic state determined bythe equations [12] F M = F para , ∂F M ∂M z = 0 (39)the order parameter M z has the jump (see Fig.10 in theRef.2) from M (cid:63) z = − ˜ β z δ z . (40)in the ferromagnetic state to zero in the paramagneticstate. Its substitution back in equation F M = F para givesthe equation of the first-order transition line,4˜ α z ˜ δ z = ˜ β z , (41)that is T (cid:63) = T (cid:63) ( H y ) = T c − β yz H y α z ( α y ( P y )) + ˜ β z α z ˜ δ z . (42)The corresponding jump of M y (see Fig.10 in Ref.1) isgiven by M (cid:63)y = M ferroy − M paray = H y α y ( P y ) + β yz M (cid:63) z ) − H y α y ( P y ) . (43) V. CONCLUDING REMARKS
We have shown that both the Curie and the supercon-ducting critical temperature are changed with the pres-sure applied to URhGe specimen. The functional pres-sure dependences can be established from general phe-nomenological considerations. Whereas the direction ofchanges must be chosen by comparison with the experi-mental findings according to which uniaxial in b-directionand hydrostatic pressure act in the opposite sense: thefirst suppresses the ferromagnetism and enhances super-conductivity, the second stimulates ferromagnetism andsuppresses superconducting state. The phenomenologicalapproach allows to point out two mechanisms of super-conducting temperature raising. They originates fromstimulation by the uniaxial stress both intraband and in-terband amplitudes of triplet Coper pairing.The magnetic field in b-direction decreases the Curietemperature and leads to the transformation of the ferro-para phase transition from the second to the first order.The pairing interaction in vicinity of the first order tran-sition from ferromagnetic to paramagnetic state causedby field H y is strongly increased in comparison with itszero field value. This effect proves to be stronger thanthe orbital suppression of superconductivity by magneticfield and leads to the re-appearance of superconductingstate at field of the order 10 Tesla. Here we have demon-strated that in presence of an uniaxial pressure along b-axis the process of the ferromagnetism suppression bymagnetic field occurs much faster. The field inducedtransformation of the second to the first order ferro-paraphase transition occurs at much lower field H cry ( P y ) val-ues. The effect is so strong that even small uniaxial stresscauses the coalescence [8] of superconducting and the re-entrantant superconducting area in the ( H y , T ) phase di-agram shown in Fig.1.The shape of superconducting region drown in Fig.1blooks similar to the upper critical field temperature de- pendence H bc ( T ) in the other ferromagnetic supercon-ductor UCoGe. Unlike to URhGe the hydrostatic pres-sure applied to UCoGe suppresses ferromagnetism andstimulate superconductivity [1]. This allows us to spec-ulate that in the case of UCoGe an uniaxial pressure ap-plied along b-axis can transform S -shape H bc ( T ) curvein two separate superconducting and reentrant supercon-ducting regions like it is in URhGe in the absence of uni-axial pressure. Acknowledgments
I am indebted to D.Braithwaite and J.-P. Brison for thehelpful discussions of their recent experimental results.
Appendix A
The explicit expressions for A z and A y coefficientsthrough the elastic and magneto-elastic moduli are A z = p x ( b + B ) + p y ( b + B ) + p z ( b + B ) , (A1) A y = q x ( b + B ) + q y ( b + B ) + q z ( b + B ) , (A2) B = λ x a b + λ y a b + λ z a b + λ xy ( a b + a b ) + λ xz ( a b + a b ) + λ yz ( a b + a b ) ,B = λ x b + λ y b + λ z b + 2 λ xy b b + 2 λ xz b b + 2 λ yz b b , (A3) B = λ x b c + λ y b c + λ z b c + λ xy ( b c + b c ) + λ xz ( b c + b c ) + λ yz ( b c + b c ) , and a = D − ( λ yz − λ y λ z ) , a = D − ( λ xy λ z − λ xz λ yz ) , a = D − ( λ y λ xz − λ xy λ yz ) ,b = D − ( λ xy λ z − λ yz λ xz ) , b = D − ( λ xz − λ x λ z ) , b = D − ( λ x λ yz − λ xz λ xy ) , (A4) c = D − ( λ y λ xz − λ xy λ yz ) , c = D − ( λ x λ yz − λ xy λ xz ) , c = D − ( λ xy − λ x λ y ) ,D = λ x λ y λ z + 2 λ xz λ xy λ yz − λ xz λ y − λ yz λ x − λ xy λ z . (A5) [1] D. Aoki and J. Flouquet, J. Phys. Soc. Jpn. , 061011(2014).[2] V. P.Mineev, Usp. Fiz. Nauk , 129 (2017) [Phys.-Usp. , 121 (2017)], arXiv:1605.07319.[3] Levy F, Sheikin I, Grenier B, Huxley A D Science , 247006(2005).[5] V.P.Mineev, Phys. Rev. B , 063703 (2009).[8] D.Braithwaite, D.Aoki, J.-P.Brison, J.Flouquet,G.Knebel, A.Pourret, to be published (2017).[9] L.D. Landau and E.M. Lifshitz, Theory of Elastic-ity, Course of Theoretical Physics Vol.VII.
Oxford:Butterworth-Heinemann (1986).[10] V.P.Mineev, Phys. Rev. B , 195107 (2011).[12] L.D.Landau and E.M.Lifshitz, Statistical Physics,Course of Theoretical Physics Vol V.
Oxford: Butterworth-Heinemann,1995)
I order
II order PM FM H y T curie T . I order
II order PM FM H y T SC RSC SC T curie a b FIG. 1: (Color online) Schematic phase diagrams of URhGein a magnetic field along b-axis perpendicular to the sponta-neous magnetization direction: ( a ) at ambient pressure and( bb