In-plane magnetic field phase diagram of superconducting Sr2RuO4
aa r X i v : . [ c ond - m a t . s up r- c o n ] D ec In-plane magnetic field phase diagram of superconducting Sr RuO V.P. Mineev
Commissariat `a l’Energie Atomique, DSM/DRFMC/SPSMS 38054 Grenoble, France (Dated: October 26, 2018)We develop the Ginzburg - Landau theory of the upper critical field in the basal plane of atetragonal multiband metal in two-component superconducting state. It is shown that typical forthe two component superconducting state the upper critical field basal plane anisotropy and thephase transition splitting still exist in a multiband case. However, the value of anisotropy can beeffectively smaller than in the single band case. The results are discussed in the application to thesuperconducting Sr RuO . PACS numbers: 74.20.De, 74.20.Rp, 74.25.Dw, 74.70.Pq
I. INTRODUCTION
The tetragonal compound Sr RuO is an unconven-tional superconductor (see review ). It reveals propertiestypical for non-s-wave Cooper pairing: the suppression ofsuperconducting state by disorder , the presence of ze-ros in the superconducting gap discovered by the magne-tothermalconductivity measurements , the odd parityof the superconducting state in respect of reflections in( a, c ) plane established by the Josephson interferometrymethod . All these properties are equally possible for sin-gle or multi component order parameter superconductingstate.Other important observations demonstrate the appear-ance of spontaneous magnetization or time reversal sym-metry breaking in the superconducting state of this mate-rial. These are: (i) the increasing of µ SR zero-field relax-ation rate , (ii) the hysteresis observed in field sweeps ofthe critical Josephson current and (iii) the Kerr rotationof the polarization direction of reflected light from thesurface of a superconductor (for the theoretical treat-ment see ).A superconducting state possessing spontaneousmagnetization is described by multicomponent orderparameter . In a tetragonal crystal the supercon-ducting states with two-component order parameters( η x , η y ) corresponding to singlet or to triplet pairing areadmissible . In application to Sr RuO the time re-versal symmetry breaking form of the order parameter( η x , η y ) ∝ (1 , i ) has been proposed first in the paper .The specific properties for the superconducting state withtwo-component order parameter in a tetragonal crystalunder magnetic field in basal plane are (i) the anisotropyof the upper critical field and (ii) the splitting of thephase transition to superconducting state in two sub-sequent transitions . Both of these properties shouldmanifest themselves starting from the Ginzburg-Landautemperature region T ≈ T c but till now there is no ex-perimental evidence for that. The in-plane anisotropyof the upper critical field has been observed only atlow temperatures where it is quite well known phe-nomenon for any type of superconductivity originatingfrom the Fermi surface anisotropy. Theoretically in ap-plication to Sr RuO these properties have been investi- gated by Kaur, Agterberg, and Kusunose . They havefound that one particular choice of the basis functions oftwo-dimensional irreducible representation for a tetrago-nal point group symmetry is appropriate for eliminationof basal plane upper critical field anisotropy but at thesame time the considerable phase transition splitting oc-curs. Vice versa, another particular choice of the basisfunctions almost eliminates the phase transition splittingfor the the one particular field direction but keeps thebasal plane upper critical field anisotropy. Thus the basalplane upper critical field properties look as incompatiblewith multicomponent order parameter structure dictatedby the experimental observations manifesting the spon-taneous time-reversal breaking.All the mentioned theoretical treatments of H c prob-lem have been undertaken for the two component super-conducting state in a single band superconductor. Onthe other hand, in Sr RuO we deal with three bandsof charge carriers . Hence, the formation of multibandsuperconducting state is quite probable. A microscopictheory of such a state was proposed in the paper .Here, in the application to the problem of the uppercritical field in the basal plane of a tetragonal crystalwe shall develop a phenomenological theory of multibandmulticomponent superconducting state. It will be shownthat both properties: H c basal plane anisotropy and thephase transition splitting still exist in a multiband case.However, quantitatively, the value of anisotropy can besmaller than in the single band case. II. UPPER CRITICAL FIELD
The order parameter in multiband tetragonal super-conductor with singlet pairing is∆( r , ˆ k ) = X λ X i η λi ( r ) ψ λi (ˆ k ) . (1)Here the lower latin index i = x, y numerates the com-ponents of the order parameter, the upper greek indexis the band number, and ψ λi (ˆ k ) are the functions of theirreducible representation dimensionality 2 of the pointsymmetry group D h of the crystal in the normal state.Similar decomposition takes place for vectorial order pa-rameter function in triplet state d ( r , ˆ k ) = X λ X i η λi ( r ) ψ λi (ˆ k ) . (2)Although our theory is applicable to the superconductorwith arbitrary number of bands we shall write all theconcrete results for the two band situation.Following derivation given in the paper one can eas-ily obtain the generalization of the G-L equations fora two-component superconducting state in a tetragonalcrystal for multiband case g λµ (cid:2) K µ D i η µj + K µ D j D i η µi + K µ D i D j η µi + K µ D z η µj + K µ ( δ xj D x η µx + δ yj D y η µy ) − Λ( T ) η µj (cid:3) + η λj = 0 . (3)Here D i = − i ∂∂r i + 2 ec A i ( r )is the operator of covariant differentiation, the Planckconstant ~ is taken equal to unity throughout the paper,and the function Λ( T ) isΛ = ln 2 γǫπT , where ln γ = 0 , ... is the Euler constant, ǫ is an energycutoff for the pairing interaction. We assume here thatit has the same value for the different bands. The matrix g λµ is g λµ = V λµ h| ψ µi (ˆ k ) | N µ (ˆ k ) i , here V λµ is the matrix of the constants of pairing interac-tion. The angular brackets mean the averaging over theFermi surface, N µ (ˆ k ) is the angular dependent densityof electronic states at the Fermi surface of the band µ .The gradient terms coefficients are K µ = h| ψ µx (ˆ k ) v µF y (ˆ k ) | N µ (ˆ k ) ih| ψ µi (ˆ k ) | N µ (ˆ k ) i πT X n ≥ | ω n | ,K µ = h ψ µx (ˆ k )( ψ µy (ˆ k )) ∗ v µF x (ˆ k ) v µF y (ˆ k ) N µ (ˆ k ) ih| ψ µi (ˆ k ) | N µ (ˆ k ) i πT X n ≥ | ω n | ,K µ = K µ ,K µ = h| ψ µx (ˆ k ) v µF z (ˆ k ) | N µ (ˆ k ) ih| ψ µi (ˆ k ) | N µ (ˆ k ) i πT X n ≥ | ω n | ,K µ = h| ψ µx (ˆ k ) v µF x (ˆ k ) | N µ (ˆ k ) ih| ψ µi (ˆ k ) | N µ (ˆ k ) i πT X n ≥ | ω n | , where ω n = πT (2 n + 1) is the Matsubara frequency, andthe components of the Fermi velocity of the band µ aregiven by v µF x (ˆ k ) , v µF y (ˆ k ) , v µF z (ˆ k ). The definition of K µ coefficients accepted here differs from the trational one by the terms in the denominators.Neglecting the gradient terms and taking the determi-nant of the system (3) equal to zero we obtain the criticaltemperature T c = 2 γǫπ exp ( − /g ) , (4)where g is defined by g = ( g + g ) / p ( g − g ) / g g . (5)The matrix g λµ in tetragonal cristal has the commonvalue for x and y components of the order parameter.Hence the phase transition to superconducting state oc-curs at the same critical temperature for all the compo-nent of the order parameter in all the bands.In the case of a magnetic field in the basal plane, H = H (cos ϕ, sin ϕ, , A = Hz (sin ϕ, − cos ϕ, g λµ [ − ( K µ ∂ z + Λ) δ ij + h z (cid:18) K µ + K µ sin ϕ − K µ sin 2 ϕ − K µ sin 2 ϕ K µ + K µ cos ϕ (cid:19) ij η µj + η λj = 0 . (6)Here we have introduced notations h = 2 πH/ Φ , K µ = K µ + K µ , K µ = K µ + K µ + K µ .Making use the orthogonal transformation˜ η µp = (cid:18) cos β µ sin β µ − sin β µ cos β µ (cid:19) pl η µl , tan 2 β µ = K µ K µ tan 2 ϕ (7)we come to g λµ " − ( K µ ∂ z + Λ) δ ij + h z (cid:18) b µx b µy (cid:19) ij ˜ η µj + ˜ η λj = 0 . (8)Here b µx,y = K µ + K µ ± q ( K µ cos 2 ϕ ) + ( K µ sin 2 ϕ ) x or y order parameter components areindependent, hence they have independent and non-equaleigen values. The corresponding upper critical fields canbe found only numerically. Here we solve this problemfollowing a variational approach, which is known to givea good accuracy in similar cases . So, we look for asolution for the x component of the order parameter inthe form˜ η µx = (cid:18) ˜ η x ˜ η x (cid:19) = (cid:18) λ x π (cid:19) / (cid:18) C x C x (cid:19) e − λxz . (10)The similar formula and the following calculations arevalid for the y -component of the order parameter.After substitution of (10) in Eqn. (8), multiplicationof it by exp( − λ x z /
2) and spacial integration we obtain g λµ ( E µx − Λ) C µx + C µx = 0 , (11)where E µx = K µ λ x + h b µx λ x . (12)The transition field is determined by condition of vanish-ing of the determinant of the system (11). Particularlywe are interested in upper critical field behavior near thecritical temperature. Obviously, E µx ∝ h , hence it tendsto zero at T → T c . So, in vicinity of critical temperaturewe receive after the simple calculationsln T c T = E x (1 + a ) + E x (1 − a )2 , (13)where T c is determined by Eqn. (4) and a = g − g p ( g − g ) + 4 g g . (14)The maximum of critical temperature at non-zero mag-netic field is accomplished at following λ x value λ x | max = hλ x , λ x = s ˜ b x ˜ K , (15)where ˜ b x = b x (1 + a ) + b x (1 − a ) , (16)˜ K = K (1 + a ) + K (1 − a ) . (17)So, after the substitution of Eqns. (12), (15), (16), (17)into Eqn. (13) we obtain h x,y = 2(1 − T /T c ) q ˜ K ˜ b x,y . (18) In the case of single band superconductivity our varia-tional solution is exact and we obtain from (17) dropingout all the terms with index µ = 2 h x,y = 1 − T /T c q b x,y K . (19)So, the situation for the two-band and one-band su-perconducting states is characterized by the same prop-erties: the basal plane anisotropy of the upper criticalfield corresponding to the largest of two eigenvalues h y and two consecutive phase transitions to the supercon-ducting state, first with y component and when with x and y components of the order parameter.It is worth noting that in the multiband case due tothe compensation of the different bands contribution the actual value of the anisotropy of H c can be smallerthan in one-band situation. The phase transition split-ting, however, still persists in multiband case. The ab-sence of experimental evidence of this phenomenon ar-gues in support of one component superconducting statein Sr RuO III. CONCLUSION
We have demonstrated that for a two-component su-perconducting state in a tetragonal crystal the basalplane anisotropy of the upper critical field and the phasetransition splitting are inherent properties both for theone band and multiband superconductivity. Thus, theexperimentally established absence of these phenomenain Sr RuO says opposite to the possibility of existenceof two component superconducting state in this material. ACKNOWLEDGEMENTS
I am indebted to M.Zhitomirsky, Y. Liu andD.Agterberg for the discussions of the problem of uppercritical field in the mlticomponent multiband supercon-ducting state. A.P.Mackenzie and Y.Maeno, Rev.Mod.Phys. , 657(2003). A.P.Mackenzie, R.K.W.Haselwimmer, A.W.Tyler,G.G.Lonzarich, Y.Mori, S.Nishizaki, and Y.Maeno,
Phys.Rev.Lett. , 161 (1998). M.A.Tanatar, M.Suzuki, S.Nagai, Z.Q.Mao, Y.Maeno, andT.Ishiguro, Phys.Rev.Lett. , 2649 (2001). K.Izawa, H.Takahashi, H.Yamaguchi, Yuji Matsuda,M.Suzuki, T.Sasaki, T.Fukase, Y.Yoshida, R.Settai, andY.Onuki, Phys.Rev.Lett. , 2653 (2001). K.D.Nelson, Z.Q.Mao, Y.Maeno, Y.Liu, Science , 1151(2004). G.M.Luke, Y.Fudamoto, K.M.Kojima, M.I.Larkin,J.Merrin, B.Nachumi, Y.J.Uemura,, Y.Maeno, Z.Q.Mao,Y.Mori, H.Nakamura, and M.Sigrist, Nature , 558(1998). F.Kidwingira, J.D.Strand, D.J.Van Harlingen, Science , 1267 (2006). J.Xia, Y.Maeno, P.T.Beyersdorf, M.M.Fejer, andA.Kapitulnik, Phys.Rev.Lett. , 167002 (2006). V.M.Yakovenko, Phys.Rev.Lett. , 087003 (2006). V.P.Mineev, Phys.Rev. B , 212501 (2007). V.P.Mineev, K.V.Samokhin ”Introduction to unconven- tional Superconductivity”, Gordon and Breach SciencePublishers, 1999. T.M.Rice and M.Sigrist, J.Phys.: Condens. Matter , L643(1995). L.P.Gor’kov, Pis’ma Zh.Eksp.Teor.Fiz. , 351 (1984)[JETP Letters , 1155(1984)]. L.I.Burlachkov Zh. Eksp. Teor. Fiz. , 1138 (1985) [Sov.Phys. JETP , 800 (1985)]. Z.Q.Mao, Y.Maeno, S.NishiZaki, T.Akima, andT.Ishiguro, Phys.Rev.Lett. , 991 (2000). R.P.Kaur, D.F.Agterberg, and H.Kusunose, Phys.Rev. B , 144528 (2005). M.E.Zhitomirsky and T.M.Rice, Phys.Rev.Lett. ,057001 (2001). V.P.Mineev, Int. Journ. Mod.Phys. , 2963 (2004). M.E.Zhitomirsky and V.-H. Dao, Phys.Rev. B , 054508(2004). D.F.Agterberg, Phys.Rev. B64