π Phase Superconductivity and Magnetism in Ferromagnet/Superconductor/Ferromagnet Trilayers

Transcription

1 Solid State Phenomena Vol (2009 pp Online: (2009 Tran Tech Publication, Switzerland doi:104028/wwwcientiicnet/ssp π Phae Superconductivity and Magnetim in Ferromagnet/Superconductor/Ferromagnet Trilayer Manur G Khuainov 1,2, a, Yurii N Prohin 1 1 Kazan State Univerity,Kremlevkaya, 18, Kazan , Ruia 2 Votok Branch, Kazan State Tupolev Technical Univerity, Chitopol , Ruia a mgkhkgtu@mailru Keyword: uperconductivity; erromagnetim; proximity eect; mutual accommodation; boundary value problem; electronic correlation Abtract On the bae o new boundary-value problem or the Eilenberger unction we invetigate the uperconducting and magnetic tate in erromagnet/uperconductor (FM/S nanotructure, where uperconductivity i a uperpoition o the BCS pairing with zero total momentum in the S layer and the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO pairing with nonzero 3D coherent momentum k in the FM layer We originally tudy the interplay between the BCS and FFLO tate in the pure thin FM/S/FM trilayer and two novel π-phae uperconducting tate with electronelectron repulion in the FM layer are predicted The modulated FFLO tate are poible in uch trilayer only in preence o external magnetic ield at uitable parameter o the FM and S layer In the FM/S uperlattice there are alo two π-phae magnetic tate (0π and ππ with compenation o the exchange ield paramagnetic eect Thi act allow u to explain a urpriingly high T c ~ 5K in the hort period Gd/La uperlattice and to predict the ign and value o the electron-electron interaction in the erromagnetic Gd metal Introduction In recent year there i a pronounced interet in unconventional uperconductivity with uperconducting electronic correlation dierent rom the uual inglet BCS pairing with zero total momentum One o uch example i the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO type pairing with a nonzero momentum o pair [1,2], which may be realized not only in erromagnetic uperconductor but alo in the layered erromagnetic metal/uperconductor (FM/S tructure (ee review [3,4] and reerence therein The competition o uperconducting and magnetic tate in the layered FM/S tructure lead to the pronounced non-monotonou dependence o the critical temperature T c on the thickne o the FM layer d [3,4] The exiting theorie o the proximity eect [3,4] connect the nature o non-monotonou behavior o T c (d with vibration o the Cooper pair lux at the FM/S boundary, which arie due to one-dimenional (1D ocillation o the pair amplitude acro the FM layer However, thee theorie predict multiple ocillation o T c (d or even periodically reentrant uperconductivity Wherea, only one local maximum (or minimum in the T c (d dependence in the majority o experiment with real 3D multilayer Fe/V or Gd/Nb i realized The reaon i that the ormer 1D theorie [3] neglect by patial change o pair amplitude along the FM/S boundary Moreover, the recently oberved abence o the 3D uperconductivity uppreion in the hort period Gd/La uperlattice [5] i a real phyical challenge to the exiting proximity eect theory Uually the uperconductivity in analogou FM/S layered tructure take place i only the FM layer thickne d i much le than the S layer critical thickne d c, which mut be o order o the coherence length ξ due to the trong paramagnetic eect o exchange ield I in the erromagnet [3,4] The typical example o above-mentioned behavior are the Gd/Nb and Fe/V uperlattice Wherea in the cae o the Gd/La uperlattice [5] d << ξ and moreover d << d neverthele uperconductivity not only arie but it arie at the critical temperature T c correponding to the "bulk" lanthanum value that i urpriingly high or heterogeneou ytem All right reerved No part o content o thi paper may be reproduced or tranmitted in any orm or by any mean without the written permiion o Tran Tech Publication, wwwttpnet (ID: , Pennylvania State Univerity, Univerity Park, USA-19/09/15,03:40:48

2 Solid State Phenomena Vol Below we will how that ull boundary-value problem have olution which explain the Gd/La urpriing behavior π phae uperconductivity in FM/S/FM trilayer The boundary-value problem or the Eilenberger unction in the FM/S layered tructure have been derived in our previou paper [6] Here we conider the layer o pure BCS uperconductor S, which occupie the region 0 < z < d and it i andwiched between two pure FFLO uperconductor FM and FM occupying the region d < z < 0 and d < z < d + d, correpondingly For thi purpoe we write the el-conitency equation and dierential equation eparately or the layer S, FM and FM For the S layer we get dω ( q, z = 2λ πt Re Φ, z, ω ; (1 p ω> 0 4π 2 2 ω% vzξz Φ (,,, 2 (,, 0 2 z ω = z < z < d p q q, (2 where the ollowing notation or the correlation length ξ z i ued vz ξ = ; 2 % ω = 2ω + i v q 2 % ω ( z (3 Analogouly we get or the FM layer dω ( q, z = 2λ πt Re Φ, z, ω ; (4 p ω> 0 4π 2 2 ω% v zξ z Φ (,,, 2 (,, 0 2 z ω = z d < z < p q q (5 v z ξ z = ; 2 % ω = 2ω + i 2 I + ( 2ω v q % (6 Here and below the index ( denote the belonging o repective parameter and unction to the S(FM metal; the 2D component o the FFLO pair momentum q and q decribe the poible ocillation o order parameter in the x-y plane o the FM/S boundary In the FM layer, occupying the region d < z < d + d, the correponding equation or the primed value Φ look like to Eq (4, (5 and the correlation length ξ' z i deined by ormula v z ξ z = ; 2 % ω = 2 ω ± i(2 I + v q (7 2 % ω Here the upper (lower ign correpond to parallel (antiparallel orientation o the FM and FM layer magnetization, ie 0 (π phae magnetic tate, repectively The boundary condition in the cae o ideal tranparency on the irt FM/S interace (z = 0 look like, z, ω Φ, z, ω Φ Φ, + 0, ω = Φ, 0, ω ; ξ =ξ z z On the econd S/FM interace (z = d we have z=+ 0 z= 0 (8

3 514 Magnetim and Magnetic Material, z, ω ξ Φ, z, ω z Φ Φ, d 0, ω = ± Φ, d + 0, ω ; = ± ξ z= d 0 z z= d + 0 where the upper ign repond or the uual 0-phae uperconductivity with coinciding phae o the order parameter and ' in the FM and FM' layer Wherea the lower ign correpond to the unconventional π-phae uperconductivity at which the ign o the order parameter change on the oppoite one when the S interlayer croing Up to now thi π-phae uperconductivity tate or the FM/S/FM trilayer wa conidered a impoible in principle [3,4] The reaon i that in the previou theorie (ee Re in [3, 4] the electron-electron interaction in the FM layer wa neglected, ie thee theorie were retricted by the partial cae λ =0 only We take into account the own electron-electron interaction in the FM layer Thu, we admit the preence o it own uperconductivity in erromagnet (λ 0 On the other hand, due to the exchange ield I 0 preence in the FM layer not only the BCS but alo the FFLO type o pairing i poible We will earch or the olution o boundary value problem (2, (5, (8, (9 in the orm which exclude the luxe through the outer boundarie (z = -d and z = d + d o the FM/S/FM trilayer, ie, (9 coh[( z d / 2 / ξ ] inh[( z d / 2 / ξ ] Φ ω = + + < < 0 z z, z, A C, 0 z d; % ω coh( d / 2 ξz inh( d / 2 ξz coh[( z + d / ξ ] Φ p q ω = + < < 0 z (,, z, B, d z 0; % ω coh( d / ξ z 0 coh[( z d d / ξ z ] Φ, z, ω = + B, d < z < d + d % ω coh( d / ξ z (10 Here 0 and 0 are the initial value o the order parameter in the thin S and FM, FM ilm The mutual inluence o the FM and S metal i epecially igniicant in the Cooper limit, where the layer thicknee are mall: d ( << ξ ( In thi cae the order parameter and the Eilenberger unction Φ(p,ρ,z,ω are practically contant along the z axi inide the FM and S layer, and the patial variation are poible only in ρ (along the FM/S boundary plane Thu, unlike the FM/S bilayer [6], in the FM/S/FM' trilayer cae the our dierent tate (φχ are poible Thee tate are ditinguihed by phae φ o the uperconducting ( and phae χ o magnetic (I order parameter in the neighboring FM layer In the both π magnetic tate (0π and ππ the neighboring FM layer magnetization are antiparallel to one another and their paramagnetic eect i mutually compenated not only in the S interlayer but alo in thee FM layer interior due to ideal tranparency and the Cooper limit Actually our calculation or the implet cae o metal FM and S with identical electronic tructure how that the Eilenberger unction in the 0π and ππ tate look like c + c Φ ω = Φ ω = Φ ω == + = ω c 2d Φ ω = Φ ω = Φ ω = = 0π 0π 0π 0 0 ( p,, ', ; c c 1; ππ ππ ππ 0 ( p,, ', ; c ω 2d + d (11 where c and c are the relative weight o the FM and S layer, repectively, in the FM/S/FM trilayer The pair-breaking actor o the neighbor FM and FM layer are complex conjugated and mutually compenate each other Thereore the T c o uch 0π and ππ trilayer depend only upon the

4 Solid State Phenomena Vol ign and magnitude o the electron-electron interaction λ in the FM layer Summing over the requency ω in the Gor kov el-conitency Eq (1, (4, we get the ollowing equation or the reduced critical temperature t 0π = T c 0π /T c and t ππ = T c ππ /T c c ( 0 λ λ c π ππ ln t =, λ > 0; ln t =, λ < 0 (12 λ ( c λ + c λ c λ It i intereting to note, that T c o the ππ tate due to π-phae uperconductivity realization doe not depend on the value o λ It i eay to ee, the critical temperature T c ππ i higher than T c 0π one, i λ < 0 In the cae λ = 0 thee 0π and ππ tate coincide with each other The poibility o two 0π and ππ tate which are dierent in uperconductivity i very urpriing act or the FM/S/FM trilayer ince it i uually conidered that the π phae uperconductivity in uch ytem i principally impoible Exitence o the π phae uperconductivity in the FM/S/FM trilayer mean that coupling between the neighboring FM layer i provided by the uperconducting correlation, ie the role o true order parameter i played by the pair amplitude F(ρ,z,ω rather than (ρ,z On the other hand, it i known [3,4], that the tate o the uperlattice include the tate o the trilayer due to ymmetry reaon Thereore we can qualitatively analyze the experimental data by Go et al [5] Indeed, the exitence o uch π magnetic tate (11, (12 allow u to explain the unexpected weak depreion o uperconductivity which ha been experimentally ound in the hort period Gd/La uperlattice The meaured T c o thi Gd/La uperlattice wa 5 K that practically coincide with critical temperature o bulk lanthanum ample! Thi mean that in the Gd/La uperlattice the 0π type tate i realized rather than ππ one and λ λ ince T c 0π T c a it ollow rom the irt o Eq (12 That i, the electron-electron interaction in the iolated Gd ilm correpond to attraction but it own uperconductivity i depreed by the trong exchange ield I >> T c However, due to compenation o the exchange ield in the 0π tate o the Gd/La uperlattice the uperconductivity o the Gd ilm i retored again The order parameter (z and (z proile correponding to the 0π and ππ tate in the FM/S/FM ytem with thin layer are chematically depicted in the Fig 1 and 2 The phyical meaning o qualitatively dierent behavior o the uperconducting order parameter in thee 0π and ππ tate become clear, i we rewrite the Eq (12 or the critical temperature T c 0π and T c ππ in the orm (ee paper [7] or comparion 2γω 1 2γω 1 ln =, λ > 0; ln =, λ < 0, πt c λ c λ πt c λ D D 0π ππ c + c (13 where γ =1781 i the Euler contant We can ee that in the cae o 0 phae uperconducting tate with λ > 0, the T 0π c i deined by the eective electron-electron attraction λ 0π e = c λ + c λ averaged over the FM/S/FM trilayer a thi i hown in Fig 1 Wherea in the cae o novel π-phae uperconducting tate with λ < 0, the T ππ c i deined by the electron-electron attraction λ ππ e = c λ o the S layer only, ince the contribution o the poitive and negative area under the - and line in Fig 2 are mutually compenate each other

5 516 Magnetim and Magnetic Material 0π λ > 0 (z ππ λ < 0 (z FM S FM z -d 0 d d + d S FM -d 0 d d + d FM z Fig 1 {Color online} The chematically pictured uperconducting order parameter (z and (z proile correponding to the 0π tate in the FM/S/FM trilayer Fig 2 {Color online} The chematically pictured uperconducting order parameter (z and (z proile correponding to the ππ tate in the FM/S/FM trilayer It i neceary to note that, in the cae o electron-electron repulion (λ < 0 in the FM layer, the T c ππ > T c 0π and the π-phae uperconducting tate (ππ i energetically avorable in comparion with 0 phae uperconducting tate (0π In the oppoite cae with electron-electron attraction (λ > 0 in the FM layer, on the contrary the T c 0π will be greater than T c ππ (Fig T c /T c 04 λ =001 ππ 00 λ =001 λ =0005 0π Fig 3 The T c (d /d dependencie or all poible tate φχ At λ > 0 the two tate are realized, and the 0π tate win At λ < 0 the ππ tate will poe greater T c than the π0 tate 02 λ = 0005 π d /d Moreover, in the FM/S/FM trilayer there are exit two other tate 00 and π0 in which the neighbor FM layer magnetization are parallel to one another Their critical temperature t 00 and t π0 are a ollow ic (2 I + xq v π ln t = ln t + Ψ Re dx ; Ψ + 2 4πT 1 ct ic (2 I + xq v 1 π0 ππ ln t = ln t + Ψ Re dx π0 2 2 Ψ + 2 4πT 1 ct (14

6 Solid State Phenomena Vol Thee 00 and π0 tate poe inteniied paramagnetic eect o exchange ield I, and being in cloe analogy to the FM/S bilayer cae [6], admit competition between the mooth BCS and ocillating FFLO tate with the FM layer thickne d increaing Thee 00 and π0 tate have lower T c in relation to the previou 0π and ππ one Probably they could be oberved in the preence o the external magnetic ield H > H coer (where H coer i the coercive ield o the FM ilm i, o coure, the averaged exchange ield c I i not too trong By the way, in the ield cooling Gd/La uperlattice with parallel orientation o magnetization in all the Gd layer the uperconducting tranition ha been completely uppreed [5] Thi mean that thicknee o the Gd and La layer in the experiment [5] were not optimal or the 00 type FFLO tate obervation Concluion For the FM/S/FM trilayer we have ound the novel π0 and ππ tate with π-phae uperconductivity at repulive interaction between electron in the FM layer The poibility o the π-phae uperconductivity in thi trilayer i alo a new and intereting act or tudying by the pin polarized tranport and Joephon eect technique [3,4] In the FM/S uperlattice there are 0π and ππ tate with compenation o trong exchange ield I due to antierromagnetic orientation o the neighboring FM layer magnetization Thi act allow u to explain a urpriingly high T c ~ 5K o the hort period Gd/La uperlattice and to predict the magnitude and ign o electron-electron interaction in the gadolinium metal Thu on the bae o our invetigation o the FM/S bilayer [6] and FM/S/FM trilayer we can propoe the method o the proximity eect pectrocopy, which make poible the detection o unknown electronic parameter o contacting metal: or example, one can determine the exchange ield I and the electronic correlation parameter λ in erromagnet FM, i well tudied BCS uperconductor S with known λ ue a a probe The extenion o our reult on the high temperature uperconductor (HTS and the S/HTS and FM/HTS nanotructure tudy could be give good method or a probe o the electronic correlation type and the order parameter ymmetry in HTS material Reerence [1] P Fulde and RA Ferrell: Phy Rev Vol A135 (1964, p 550 [2] AI Larkin and YuN Ovchinnikov: Sov Phy JETP Vol 20 (1964, p 762 [3] YuA Izyumov, YuN Prohin and MG Khuainov: Phyic-Upekhi Vol 45 (2002, p 109 [4] AI Buzdin: Rev Mod Phy Vol 77 (2005, p 935 [5] JP Go, et al: J Magn Magn Mater Vol 240 (2002, p 592 [6] MG Khuainov, MM Khuainov and YuN Prohin, in: Progre in Superconductivity Reearch, edited by OA Chang, Chapter 3, Nova Science Publiher, NY (2008, p 79 [7] PG de Genne: Rev Mod Phy Vol 36 (1964, p 225

7 Magnetim and Magnetic Material /wwwcientiicnet/SSP π Phae Superconductivity and Magnetim in Ferromagnet / Superconductor / Ferromagnet Trilayer /wwwcientiicnet/SSP DOI Reerence [3] YuA Izyumov, YuN Prohin and MG Khuainov: Phyic-Upekhi Vol 45 (2002, p 109 doi:101070/pu2002v045n02abeh001025