Magneto-elastic coupling model of deformable anisotropic superconductors

Transcription

1 Magnto-last ouplng modl of dformabl ansotrop suprondutors Yngu L, Guohng Kang, and Yuanwn Gao,4 ppld Mhans and Strutur Safty Ky Laboratory of Shuan Provn, Shool of Mhans and Engnrng, Southwst Jaotong Unvrsty, Chngdu, Shuan 6, PR Chna Stat Ky Laboratory of Traton Powr, Southwst Jaotong Unvrsty, Chngdu, Shuan 6, PR Chna Ky Laboratory of Mhans on Envronmnt and Dsastr n Wstrn Chna, Th Mnstry of Eduaton of Chna, Lanhou, Gansu 7, PR Chna 4 Dpartmnt of Mhans and Engnrng Sn, Collg of Cvl Engnrng and Mhans, Lanhou Unvrsty, Lanhou, Gansu 7, PR Chna Corrspondng authors: yngul@swtu.du.n (Y. L) bstrat: W dvlop a magnto-last (ME) ouplng modl for th ntraton btwn th vort latt and rystal lastty. Th thory tnds th Kogan-Clm s ansotrop Gnburg-Landau (GL) modl to nlud th lastty fft. Th ansotrops n suprondutvty and lastty ar smultanously onsdrd n th GL thory fram. W ompar th fld and angular dpndns of th magntaton to th rlvant prmnts. Th ontrbuton of th ME ntraton to th magntaton s omparabl to th vort-latt nrgy, n matrals wth rlatvly strong prssur dpndn of th rtal tmpratur. Th thory an gv th approprat slop of th fld dpndn of magntaton nar th uppr rtal fld. Th magntaton rato along dffrnt vort fram as s ndpndnt wth th ME ntraton. Th thortal dsrpton of th magntaton rato s applabl only f th appld fld modratly los to th uppr rtal fld. Kywords: Magnto-last ntraton, unaal suprondutor, ansotrop GL thory, magntaton. Introduton It s wll known that, th ntraton btwn th dft-ndud stran and th flu vorts n suprondutors gvs rs to th flu-pnnng bhavors [, ]. Smlarly, th last dformaton ausd by vorts, th so alld magnto-last (ME) ntraton, affts th nrgy of th vort latt. Th vort-ndud stran s du to dffrnt spf volums btwn th normal phas (vort or) and th suprondutng phas (suprondutng mattr around th vort). Th normal-stat vort ors ats as homognous stran sours, gnratng loal dformatons n

2 th surroundng suprondutng mattr (ths physal phnomnon s alld th vort-latt nrgy. V fft), suh that affts th Th vort-ndud last dformaton s mportant for th marosop magnt proprts. Gnrally spakng, th vort-ndud stran s rlatvly wak wth rspt to th pnnng ntraton. owvr, n som partular suprondutng matrals, th vort-ndud stran an sgnfantly hang th marosop magntaton prforman, and may aus som anomalous magntaton [, 4]. Th rtal tmpratur T n ths matrals strongly dpnds on th dformaton [5, 6]. In ron pntds systm, partularly Ca(F - Co ) s, th hang rat of stran dpndn n T s on or two ordrs of magntuds hghr than onvntonal suprondutors. Ths maks th ME fft outstandng n F-basd suprondutors. Th ME ntraton n suprondutors s a possbl rason for th prmntally obsrvd ntratons btwn th vort latt and rystal latt. Th obsrvatons [7-9] of th flu-ln latt struturs n NbS sngl rystal ar rmarkably dffrnt from th prdtons wth London thory. Othr fator s proposd to afft th vort ntratons. owvr, t has not bn nludd n th lass London thory and Gnburg-Landau (GL) modl. Ths fator s th ME ntraton, whh plays a rual rol n th formaton of vort latt []. Vorts an ntrat on anothr through th last dformaton fld, and th ME ntraton nrgy dpnds on th vort-latt strutur. Som rlvant prmntal rsults of NbS an b dsrbd [] by ntrodung th ME ntraton n th London thory. Consdrng th suprondutng ansotropy and lastty ansotropy n matrals (as thy hav n natur), th ME ntraton affts ssntally th rlatonshp of th vorts and rystal latt []. s hghly ansotrop suprondutors ar wdly usd n prmnts and ngnrng prat, t s partularly manngful to nvstgat th ntraton btwn th vort-latt and rystal undr onsdratons of ansotrop suprondutvty and lastty. For nstan, th last ntraton btwn vorts s of rmarkabl ansotrop haratrst []: for a st of partular last modulus, th two vorts loatd at [] or [] attrat ah othr lastally; n ontrary, thr s rpulson btwn th vorts at []. In a rlatvly larg magnt fld, th ME ntraton rmovs th drtonal dgradaton of standard 6 trangular vort latt, and auss th dstorton of ths vort latt. Ths qualtatvly onsstnt wth th prmntal data of KF s []. n fftv mthod [, -5] to quanttatvly valuat th ME ntraton s ntrodung th vort-ndud last dformaton n th GL thory, smlar to th approah [] of valuatng th stran fld ndud by th pnnng dfts. In th stran-dpndnt GL modl, th V fft (dffrn n th spf volum btwn normal phas and suprondutng phas) s takn as th man last stran sours. nothr thortal modl [5, ] rgards th vorts as on-dmnsonal stran sours n an nfnt rystal, smplfyng th orgnal problm nto a plan last problm. Takng vort ors as th pont sours of strss, th ME problm s analogy to a thrmal dffuson problm n ansotrop obts subt to pont hat sour. Ths mthod s a London-typ tratmnt, and only mts th rqurmnts of qualtatv alulaton. mor rgorous onsdraton [6] of th ME ntraton nluds both th ffts of th vort-or rgon and th non-or rgon n th vort-latt nrgy. lthough th ordr paramtr hangs slowly n th non-or rgon, n hgh- suprondutors th non-or rgon s domnant n produng last

3 dformaton [6]. Bsds, basd upon th stran-dpndnt GL modl, som analytal soluton mthods [7, 8] for th ME ntraton n suprondutors wth spf shap ar proposd. In ths papr, w dvlop a ME modl aountng for th ntraton of th vort latt and rystal lastty. Th thory tnds th Kogan-Clm s ansotrop GL modl [9] to onsdr th ME ntraton. Th suprondutng ansotropy and last ansotropy ar smultanously nludd n th modl, n ontrast to th prvous thors only onsdrng on typ of ansotropy. Th thortal rsults ar ompard wth som magntaton prmnts, and th ffts of th ME ntraton as wll as th thory applablty ar dsussd. Th papr s struturd as: n ston, w stablsh th ME modl and gv th solutons nar th uppr rtal fld; n ston, w apply th thory to alulat th last nrgy and magntaton; th thortal rsults ar ompard wth th rlvant prmnts n ston 4; at last, w gv th onlusons of th papr.. Magnto-last ntraton n ansotrop suprondutors. nsotrop Gnburg-Landau quatons wth stran fft Th GL thory wth a phnomnologal mass tnsor M dsrbs rasonably th maor bhavors of ansotrop suprondutors nar th rtal tmpratur T. nw fatur n dformabl suprondutors s th vort-ndud stran n trms of th last rspons of th rystal n th prsn of vort latts. Th fr nrgy s [] 4 * * F f dv ( M kluukl ) dv 8. () r, Π ( ), s th vtor potntal of th loal magnt fld, s th ordr paramtr, kl ar last offnts and u s th stran tnsor. Th nvrs mass tnsor M has th prnpal valus M (,, ). and ar th offnts n th GL funtonal panson. In trms of th wll-stablshd stran dpndn of T, th vort-ndud stran ntrs n th fr nrgy va th matral haratrst []: wth ( T T ) u, () M T T M T u (a) [Th man mass M ( M M M ), T s T at ro stran and s th GL ohrn lngth at tmpratur K, / /.. ( t) wth t T T ]. To mak th fr nrgy dmnsonlss, ( ) and ( 4 ) ar takn as th unts of th ordr paramtr and magnt fld, rsptvly. s usual, on taks th avrag London pntraton dpth

4 / L ( M 6 ) as th unt lngth. Th unts of th nrgy dnsty f and last modul ar of Eq. () aqurs th dmnsonlss form whr th dmnsonlss T has th unt T T. Th fr nrgy now rads 4. T, (b) u F u u u dv 4 * * ( kl kl ). (4) r, all quantts ar dmnsonlss, wth th sam notatons as thr dmnsonal ountrparts. ( ) whr th GL paramtr L. Th nvrs dmnsonlss mass tnsor MM has th gnvalus MM. Fg.. (a) Th rystal oordnat systm ( X, Y, Z ) onds wth th prnpal as of th rystal a, b and n a unaal layrd suprondutor. (b) Th vort oordnat fram (, y, ) s obtand by rotatng ( X, Y, Z ) wth an angl around as Y. Th as gvs th vort drton. Introdung th oblqu oordnat systm th ansotrop ME problm transforms nto a quas-sotrop on. (,, ), Varyng th fr nrgy (4) wth rspt to ) an thn b prsntd as [, 6, 9]: *, and, th quatons of qulbrum n th vort fram (Fg. ( u ), (5a) l * kl k R( k) k, (5b) 4

5 u kl kl Th lastty quaton whr F u gvs, (5) ( klukl ). (5d) Eqs. (5a)-(5d) dsrbs th ME bhavor of dformabl suprondutors n th prsn of vort-ndud stran. Th solutons for th GL quatons nar th uppr rtal fld hav bn wll stablshd for th sotrop as []. Basd on ths onsdraton, w ntrodu a nw oordnat systm (,, ) (Fg. ) whr th nvrs mass tnsor has th unt matr form,.. (s ppnd for th dtals). In th nw fram, Eqs. (5a)-(5d) rad ( u ), (6a) kl l k * R( ), (6b) kl u kl, (6) kl ( u ) kl, (6d). Solutons at th uppr rtal fld t, th stran fft u and hgh-ordr an b ngltd [9], and Eq. (6a) now rads ( ). (7) Ths looks atly lk th sotrop as of brkosov s lassal tratmnt []. Th solutons of Eq. (7) ar and. Not that / / s dfnd n th nw fram ( ) ( ) [Eq. (6)]. Th uppr rtal fld thn rads / ( ) Th angular dpndn os sn [Eq. ()] ylds (,, ). Whl n th vort fram (, y, ),. (8a) ab so. (8b) sn os sn os sn os ab whr m m for layrd matral,, and ab so. Now, w turn to Eq. (6b). If w prss th ordr paramtr as p( ), thn usng th gaug-nvarant suprmomntum Q and th urrnt dnsty J : 5

6 Eq. (6b) s rwrttn as Q, J J Q kl l k, (9). () t, th ansotropy dos not hang th faturs of Q n th sotrop as drastally [9]: Q and Q rmans larg, howvr, Q, J. (a) Gong bak to th orgnal oordnat fram (, y, ), w obtan wth th hlp of Eq. (6): J J. (b) Usng th angular dpndn of and [Eq. ()], w hav J J ( )sn os. () os sn Ths rvals an mportant dstnt fatur n ansotrop suprondutors at th mmdat vnty of : th aal urrnts do st n an array of vorts, and ths urrnts vanshs whn th vorts drt along on of th prnpal rystal as a and. In sotrop matrals ( ), thr s no aal urrnts at any vort drtons.. Solutons nar : brkosov dntts n ansotrop as wth magnto-last fft s n orgnal brkosov approah whr th wll-known brkosov dntts ar drvd for th sotrop as wthout stran fft, w ntrodu th oprators and thn Eq. (6a) rads (s ppnd B for th dtald drvatons) u Q ( ). () ountng for Eq. (a), Q has th ordr of ( ) nar. ll trms on th RS (rght hand sd) of Eq. () an b ngltd n th lnar appromaton. Instad of solvng Eq. (), w hav a mor smplfd form,, whh s furthr smplfd to Usng, p( ) and Q. () n Eq. (), w hav Q Q. Combnng ths wth Q and Q [Eqs. (9) and ()], w obtan th frst brkosov dntty n ansotrop as: onst, (4a) or n th vort fram, and 6

7 whr s an arbtrary onstant, and has bn gvn n Eq. (8a)., (4b) To normal th soluton of th homognous quaton (), on has to fnd th soluton to th at nonlnar quaton (). Substtutng and (), and wth th hlp of Eqs. () and (4b), w hav u ( ) ( ) [s Eq. (6)] n Eq. / [ ( ) ]. (5) whr. W hav omttd Q n th drvaton of Eq. (5), sn t s of hghr ordr of ( ) nar. Th stn of a soluton for th nhomognous lnar quaton (5) rqurs that, th RS of Eq. (5) s orthogonal to th soluton of th orrspondng homognous quaton [9]. Ths lads to u =, (6) th sond brkosov dntty. Eq. (6) gnrals th Kogan-Clm s rsult wthout th ME ntraton [9],.., to th ansotrop matral wth th ME fft..4 Fr nrgy Barng n mnd that from Eq. (4b) th magnt nduton s an b obtand n trms of Eq. (6): B, (7) ( B ) =. (8) u ( ) If w ntrodu th followng notatons u, ( ),, ( ), ( ), (9) (th last on s drvd from th latt strutur [9] wth and y th strutur paramtr), Eq. (8) rads ( B ) =. () 7

8 Th gnral form of th fr nrgy n ansotrop suprondutors wth lastty fft s [6, 9] kl kl Usng Eqs. (4b) (7) and (9) n Eq. (), th fr nrgy () s rwrttn as F u u. () kl F B u ukl, (a) 4 and prssng n trms of Eq. (), on has anothr form of th fr nrgy: ( B) ( ) kl kl ( ) F B u u. (b) Now lt us go furthr wth th last trm n th fr nrgy (a), onsdrng th fft of th vort-ndud stran wth th hlp of lastty quatons (6) and (6d). Th Fourr form of th vort ndud stran u rads [6] q u u [ q u ( ) q u ( )] ρ q q, () q whr u ( q ) s th dsplamnt omponnt n Fourr spa, and u s th homognous stran ndud by th vort. pplyng Eq. () n Eqs. (6) and (6d), w hav kl, (4a) u G ( q) u ( q) S ( q) ( q ), (4b) whr G kl ( q ) q q, S ( q ) q and ( q ) s th Fourr transform of Eq. (). Solvng Eq. (4a) and (4b), k l w hav th stran fld n dformabl suprondutors: kl, (5a) u kl u ( q) S ( q) G ( q) ( q ). (5b) Wth th hlp of Eqs. (5a) and (5b), th last trm n th fr nrgy (a) now an b prssd as (th dtald drvaton an b found n [6]) 4 kl kl uukl uukl G ( q) u ( q) u ( q ) u. (6) q Th fnal form of th fr nrgy an b wrttn as or wth th hlp of Eq. (), 4, (7a) F B ( B ) F B. (7b) 8

9 .5 Comparsons wth th rsults n th lassal thors Lt us valdat Eq. () by omparng t to th lassal rsults. If th suprondutng matral s sotrop but wth stran dpndn,.., and, Eq. () rdus to th rsult ( B) = by Cano t al. [6]. On th othr hand, f th suprondutng matral s ansotrop but wthout stran dpndn,.., on obtans from Eq. () that ( B ). Th last on onds wth Kogan-Clm s rsult [9]. = Th valdty of Eq. (7) an b hkd by omparng t to th lassal rsults: n th partular as whr th suprondutng ansotropy vanshs,.., and B B, Eq. (7b) rdus to ( B) F B. Th last on ust rprodus th rsult by Cano t al. [6]. If th matral suprondutvty s ndpndnt wth stran,.., Eq. (7b) taks th form of ( B ) F B, whh onds wth th Kogan-Clm s rsult [9].. pplaton of th thory. Paramtrs, and n Fr nrgy Lt us frst dtrmn and n th fr nrgy (7b). From Eq. (9) w hav ( ) whh s of ordr of ( ) [], so that for hgh- suprondutors []. Thus, w an nglt n th fr nrgy (7b)..6 s a gomtr onstant for th trangular latt []. In dformabl suprondutors, th fr nrgy (7b) dpnds on th strans va u wth gvn n Eq. (b). For th unform dformaton, usng Eq. (5a) w hav on nds an plt form of. Rgardng Eq. (b), not that. Now, to alulat kl kl T u s always masurd by th strss dpndns of T. Thus, usng kl p u kl, w hav T u kl T p. Fnally, w prss as kl, (8) kl whr T mn mn p. (9a) 9

10 Sn th nw fram kl kl s an nvarant ndpndnt wth oordnats transformaton, w an alulat (,, ) ] n trms of th rystal fram ( ab,, ). suprondutor, suh that th nonro omponnts ar p p p ab and and kl [n mn p s th unform strss applyng on th p p. raftr, w onsdr th last mdum as ttragonal rystal latt. In th rystal fram, th nonro omponnts of th last modulus tnsor kl [n th Cartsan oordnat systm ( ab,, ) th ovarant omponnts ar qual to th orrspondng ontravarant ons,.. aaaa bbbb C kl kl ] ar [], aabb C, abab C66, C, aa bb C, aa bb C55. Thus, th nonro omponnts of rad (barng n mnd that kl kl lk lk du to th symmtry n th stran tnsor) kl has th gnral symmtry proprts T ( C C ) C aa bb ab pab T p, T C C pab T p, (9b) whr C ar th last modul n th rystal fram. Thus, w hav kl or n an stmaton for th ordr of magntud, kl ab 4 ab. (a) C C C C kl kl T. (b) p whr s of th ordr of th last modulus. Combnng Eq. (8) wth Eq. () dtrmns. So far, th paramtrs bn dtrmnd., and n th fr nrgy hav. Elast nrgy Th last nrgy an b valuatd wth th hlp of Eqs. () (6) (8) and (b): T r,. p F u u ( B ). () kl l kl ( )

11 . Magntaton Followng th approahs n [9], on nds th prssons of th fld to obtan th magntaton M B 4. In th vort fram (, y, ), th fld omponnts Now th magntaton M n th vort fram rads M and ar alulatd by [9] B B, (a) M F B B. (b) B 4 M B, (a) n onvntonal unts. Th rato 4 M B, (b) M M ( ) sn os os sn. (4) s ndpndnt wth th vort strutur paramtr and lastty paramtr. Ths tnd th onluson by Kogan and Clm [9] to th stuaton wth th ME fft: th magntaton rato s ndpndnt wth th ME ntraton. Th magntaton omponnts n th rystal fram ( X, Y, Z ) s obtand by usng th oordnat transformaton rlatons n Eqs. (a) and (b): Th magntaton rato n th rystal fram s B B sn 4M X sn, (5a) os sn B B os 4M Z os. (5b) os sn M M X tan tan. (6) Z 4. Comparsons wth prmnts 4. ngular dpndn of th uppr rtal fld Ghosh t al. [] masur th ansotrop uppr rtal fld n CalS. Th prmntal rsults an b fttd wth Eq. (8b), by hoosng th approprat ansotrop paramtr (Fg. ). Th nrass n th ansotropy paramtr wth tmpratur onds wth th masurmnts n th sam prmnt. Th smlar tmpratur dpndn s

12 found n LaFsO - F thn flms [], rangng from. at K to 4. at 5 K. owvr, for MgB, s drasng wth tmpratur []. Th ltron band strutur and ansotrop natur of th ordr paramtr ar rsponsbl for ths bhavors. Fg.. Th uppr rtal fld as a funton of th angl fts roughly wll th masurd at tmpraturs., 4., 5. and 5.5 K. Eq. (8b) (lns) (dots) n CalS sngl rystal []. Th nsrt shows th tmpratur dpndn of th ansotropy paramtr, whh s takn as th fttng paramtr n Eq. (8b). 4. Fld dpndn of mmagntaton Th thory onsdrs th mdat vnty of B, and hgh- matral, suh that from Eq. () th appld fld M approahs th magnt nduton B,.. M M B and M. Th paramtrs n Eq. () an b rwrttn as (n onvntonal unts) ( ), (7a) () T, (7b) T p ab, (7) sn os sn os for alulaton onvnn. ordng to th magntaton prmnt [4] n La.45 Nd.4 Sr.5 CuO 4 [4], w tak T.5 K, (6K). T, (8K).5 T, (). T, (6K) 4.8 T, (8K) 4.5 T and n Eq. (). Sn th mhanal paramtrs ar not nvolvd n ths prmnt, w tak 9 T p K m rg and rg m aordng to a smlar rystal La.45 Nd.4 Sr.4 CuO 4 as suggstd by [5]. W obtan M( ) shown n Fg. (a). Th alulaton urv dsrbs wll th prmntal data nar, but thr ar rlatvly larg dvatons away from. Ths s vry mportant sn th thory s ssntally stablshd at th ntrmdat vnty

13 .Th ontrbutons to th slop dm db 4 of nlud th magnt part and th ME ntraton. Sn th suprondutvty s snstv to prssur n La.45 Nd.4 Sr.5 CuO 4 [5],.. onsdrabl T p valu, 4. s omparabl to 9.4. Wthout nludng th lastty fft,.. sttng n Eq. (b), th agrmnt around s rlatvly poor [Fg. (b)], sn th slop of M( ) s not proprly valuatd. Fg.. (a) Th fld dpndn of th magntaton. Th lns ar obtand wth Eq. (b), to ompar th prmntal data n La.45 Nd.4 Sr.5 CuO 4 [4]. Th arrows ndat th uppr rtal fld, whr th magntaton nrass to. Th nsrt shows th magnfaton nar. (b) Comparson wth th rsult of,.. th as wthout lastty fft. 4. ngular dpndn of magntaton rato To hk th ansotrop magntaton rato Eq. (4), w ft Eq. (4) to th prmntal magntaton data n YBa Cu O 7- at 7K and 5T [5]. Th ft gvs th ansotrop paramtr 4.5 [Fg. 4(a)]. Ths s onsstnt wth th rportd valus ~ n othr ltraturs [6-8]. Takng (7K) 6 T (roughly orrspondng to th prmntal valu n [9]) and 5.5, w obtan ( ) for ths as. Wthn th rang 6 9, th ft s rlatvly poor. Ths an b attrbutd to th narrow pak rang of ( ), whr th valus ar svral tms largr than th appld fld 5T. Th lattr dvat svrly from th applablty rang.

14 Fg. 4. (a) ngular dpndn of th MX M Z rato n YBCO rystal. Th prmntal data [5] (sold squar) ar masurd at tmpratur 7K and magnt fld 5T. Th ln orrsponds to Eq. (4) usng 5. Th nsrt shows th ( ) at ths valu. (b) Smlar wth (a), but for B- rystal at 7K and T. Th lns orrspond to Eq. (4) wth dffrnt valus. () bd, but for Tl- rystal at K and T. W also ft Eq. (4) to th M M data for B- rystal [5] at T and T 7K, by varyng to obtan th bst ft [Fg. 4(b)]. Th fttng urv of 5 rprodu farly th prmntal data n a wd rang pt for 6. ( ) s obtand by takng (7K) 5. T [] and 5 n Eq. (8b). Compard wth th good-fttng rang, th poor-fttng rang shows svral tms hghr valus of. Smlar wth YBCO, n ths as, th dffrn btwn th appld fld and dtrmns th fttng auray. W also tak 5 to ft th data; ths valu approahs 7 proposd by Tuomnn t al. [5]. Ths basally rsolvs th dvatons n th 4

15 rgon 6. owvr, s narly two ordrs of magntud hghr than th appld fld. for 5 ar srously largr than 5. It must b notd, th smngly good ft by 5 s atually out of sop of th thory, so t masks th physs of th ansotrop magntaton. W mplmnt th ft for th Tl- sampls, whh s supposd to hav rmarkably hgh and valus,.. [] and (K).4T []. In ths as, th agrmnt s farly good for 9, ndpndntly of th valu usd abov ~5 [Fg. 4()]. Wth hghr valu,.. = and 5, th ampltud of th pak approahs th prmntal data at th vnty of 9. owvr, th hgh valus lad to th two or thr ordrs of magntud hghr than th appld fld n ths rgon. So, th thory s not applabl n ths as, vn for th small- rgm. Ths onluson an also b vrfd by omparng Eq. (6) wth th prmntal [] magntaton rato along th rystal as. Equaton (6) stablshs th angular dpndn of MX M Z : a largr orrsponds to a hghr rato valu. In ontrary, howvr, th prmnt [] for Tl- shows an oppost dpndn. 5. Conludng rmarks Basd on th ansotrop GL thory, w modl th ME ntraton n a unaal suprondutor. Th prsnt modl nluds th ansotrop suprondutvty and last dformaton, by ntrodung th ansotrop ltron mass and th last nrgy n th GL fr nrgy. Th GL quatons ar now orrlatd to th ME ntraton. Th solutons ar obtand by usng th ansotrop brkosov dntts undr th onsdratons of th lastty fft. s a valdaton, th formulas of th fr nrgy s ompard wth th lassal rsults. Th lastty paramtr, vort-latt paramtr and ansotropy paramtr ar dtrmnd for prat. W drv th pratal forms of fr nrgy and magntaton. Th formula of th ansotrop magntaton rato rmans ts lassal rprsntaton, vn takng nto aount th ME ouplng fft. W ompar th thory wth th prmnts. By nludng th ME ntraton, th modl gvs a satsfd dsrpton of th fld dpndn of th magntaton nar th uppr rtal fld. In F-basd suprondutors, th strong dpndn T p lads to a rmarkabl ME ntraton, whh s omparabl to th vort nrgy. Th ME ouplng fft s mportant for an approprat dsrpton of th magntaton bhavors. Th fft of th ME ntraton s absnt n th magntaton rato M M along dffrnt vort fram as. Ths nabls th fttng of th rlvant prmnts to dtrmn th ansotrop paramtr substantally. owvr, t should b notd that, ssvly hgh valus lad to unralstally larg, spally whn th fld drts appromatly n th suprondutng layr ( ab plan). On must bar n mnd th applablty rang of th thory:. Th approprat rlaaton of onsdrng th pratal applaton s: and ar n th sam ordr of magntud. 5

16 Th thory s onvnbl only at th vnty of. Etndng th thory to a modrat fld rang s sgnfant for prat. owvr, n ths as, th prs vort strutur and spatal dstrbuton of ordr paramtr should b onsdrd. Ths maks th problm ntrat, not to mnton th ansotrops of lastty and suprondutvty. Th ansotrop ME ntraton n th modrat magnt fld rang wll b th fous of our nt rsarh. knowldgmnts Ths work was prformd wth supports from th Natonal Natural Sn Foundaton of Chna (7, 46, 7 and 9) and Natonal Ky Prot of Magnto-Rstrton Fuson Enrgy Dvlopmnt Program (GB). Th Fundamntal Rsarh Funds for th Cntral Unvrsts ( ) and Chna Postdotoral Sn Foundaton (6M677) ar also aknowldgd. ppnd : Transformaton of oordnats Lt us onsdr a unaal layrd suprondutor (Fg. ). In th rystal oordnat systm ( X, Y, Z ) whr Z as s normal to th layrs, and th omponnts KL ( K, L,,) of th nvrs dmnsonlss mass tnsor KL. () Th uppr nds dtat th ontravarant quantts. Supposng an array of vorts tltd from th rystal as, w ntrodu a nw oordnats (, y, ) of vorts. (, y, ) s rotatd wth rspt to th ( X, Y, Z ) through an angl about th Y as (Fg. ). Whn ( X, Y, Z ) transforms nto (, y, ), th tnsor omponnts transform as th oordnats transformaton rlatons X os X sn, X, X sn X os (ntrodung th notatons X X, X Y, X Z, whr th transformaton offnts, y and for th onvnn of tnsor analyss): os sn os sn KL KL sn os sn os os sn ( )sn os ( )sn os sn os =, () os sn K K X. () sn os 6

17 Th thr nvarants of th tnsor, tr( μ ), μμ and dt( μ ) gv th followng usful rlatons,. (4) To rndr sotrop ( ), w ntrodu an oblqu angld rtlnar oordnat systm (,, ). Th oordnat transformaton btwn (, y, ) and (,, ) s hosn as [9] Th nvrs transformaton s a, a, b, b, d. (5) d ( ad). (6) Th ovarant and ontravarant transformaton offnts now rad a b, ( ad) d a b. (7) d Sttng th ovarant bas vtors of vtors g n (,, ) : (,, ) as g, g and g g g b b Obvously, th nw oordnat systm s oblqu: th nw whl th nw as s nlnd wth th k, on an dtrmn th ovarant bas a ( ad) a ( ad) k. (8) d d k k and at an angl =artan d as ar along wth th old and (Fg. ). Th ondn of th nw th old as s mportant sn n th vort problm nar, any -ndpndnt quantty s Th tnsor omponnts Th sotrop rqurs that n as, and -ndpndnt:. (9) (,, ) rlats to mn m n through th transformaton rlatons:. () mn m n, n matr form t boms T b b b a a a a ad, () d d a ad d d and on obtans / a, b / / /,, d ( ), () / whr th nvaran rlatons Eq. (4) and (obvously from th dfnton of th dmnsonlss nvrs mass) hav bn usd. Th mtr tnsor g of th nw fram s obtand usng Eq. (8), 7

18 / g g g. () / ( ) Th onstant g mak th ovarant drvatvs rdu to th partal ons, and sn dt g, th Lv-Cvta tnsor k prsrvrs ts, form whn transformd from (, y, ) to wth onstant g and dt g, Eqs. (5a)-(5d) an b wrttn as (,, ). In th nw fram (,, ) ( u ), (4a) kl l k * R( ), (4b) kl u kl, (4) whr omponnt, ( ), u, kl ( u ) kl, (4d), and th ovarant omponnt of th magnt fld l and m mnr r n m l glm wth th ontravarant. (5) kl ar obtand from thr ountrparts n th fram (, y, ) through th oordnat transformaton rlatons: kl,, k l u u, k l kl and m l l m kl k l qrst q r s t. s an ampl, w fgur out th ontravarant and ovarant omponnts of th magnt fld: / a / b y, / / / / d / / a ( ad) / b y. (6) / / d ppnd B: Drvatons of brkosov dntts n ansotrop dformabl suprondutors Introdung th oprators and, w ar allowd to arry out th followng alulatons: 8

19 ( )( ) = ( ) = ( ), (7) whr ( ) [Eq. (5)] has bn usd n th last qualty. If w prss th ordr paramtr as p( ) (as usual), thn [barng n mnd that th modul of s ndpndnt wth (9)] aordng to Eq. [ p( )] p( ) p( ), (8) ( ) ( ) Q r, w hav ntrodud th gaug-nvarant suprmomntum Q,. (9) Q. () Thn, rads [( ) ] [( ) ]( Q) ( Q) ( ) ( Q) Q Q Q, () whr Q [Eq. (9)] has bn usd. Thus, takng togthr Eqs. (7)-(), n Eq. (6a) ylds Substtutng Eq. () nto Eq. (6a), w obtan Q. () u Q ( ). () 9

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