Nature of superconducting fluctuation in photo-excited systems

Transcription

1 Naure of superconducing flucuaion in phoo-excied sysems Ryua Iwazaki, Naoo suji and Shinaro Hoshino Deparmen of Physics, Saiama Universiy, Shimo-Okubo, Saiama , Japan RIKEN ener for Emergen Maer Science EMS, Wako, Saiama 35-98, Japan Daed: April, 9 When we apply he srong elecric-field pulse, here are wo characerisic inverse-ime scales γ and γ. γ originaes from he reduced emperaure measured from he criical emperaure c, and deermines a ime consan for he relaxaion ino he euilibrium sae. he oher one γ comes from he srengh of he elecric-field pulse. We show ha for, he sysem relaxes o he euiarxiv:94.58v cond-ma.supr-con] Apr 9 he phoo-excied sae associaed wih superconducing flucuaion above he superconducing criical emperaure c is sudied based on he ime-dependen Ginzburg-Laundau approach. he excied sae is creaed by an elecric-field pulse and is probed by a weak secondary exernal field, which is reaed by he linear response heory mimicking pump-probe specroscopy experimens. he behavior is basically conrolled by wo relaxaion raes: one is γ proporional o he emperaure measured from he criical poin c and he oher is γ proporional o he exciaion inensiy of he pump pulse. he excied sae approaches he euilibrium sae exponenially in a long ime γ, while in he inermediae ime domain we find a power-law or logarihmic decay wih differen exponens for γ and γ γ, even hough he sysem is locaed away from he criical poin. his is inerpreed as he criical poin in euilibrium being exended o a finie region in he excied siuaion. he parameer dependences on boh he pump and probe currens are also sysemaically sudied in all dimensions. I. INRODUION Exernally excied sysems can show various exoic saes ha are no realizable as euilibrium saes bu neverheless survive for a long ime in non-euilibrium. Such a possibiliy has been inensively sudied by pumpprobe experimens using a laser pulse, and he inriguing phoo-exciaed saes have been creaed and measured 8]. For superconducing maerials in srongly correlaed sysems, recen experimens have shown giganic enhancemen of superconduciviy in a phoo-induced sae 4, 9 5], which has considerable aenion in he physics communiy. While he change in he elecronphonon coupling hrough he phonon drive has been sudied as he origin of he phoo-induced superconducing sae 6 6], phoons also pump he elecrons direcly. Some of he heoreical sudies have proposed scenarios based on phoo-excied elecron sysems 7, 8]. he superconduciviy-like behavior above he ransiion emperaure c is reminiscen of paraconduciviy originaing from he flucuaion effecs: he elecron conducion near he superconducing ransiion poin becomes much higher han in he normal sae. Usually, he superconducing flucuaion is so small in hree dimensional sysems ha i has been mainly discussed in low-dimensional sysems 9]. Wih exciaions, however, he flucuaion effecs may change. Here, we sudy such a phoo-induced phenomenon associaed wih he superconducing flucuaion based on he ime-dependen D Ginzburg-Landau GL approach. he GL heory has been successfully applied o unconvenional superconducors 3]. In he srongly correlaed sysems, since he coherence lengh can be an order of a laice consan, he applicaion of GL analysis may no be jusified bu sill well describes heir ualiaive behaviors. his indicaes ha exrapolaion from he long wavelengh coherence lengh limi works well for exising superconducors. As for he ime-dependen case, i is argued ha he DGL euaion is no microscopically jusified for gapped superconducors, because he relaxaion ime of uasiparicles can become longer han he ime scale of he order parameer variaion 3]. On he oher hand, once he DGL euaion is empirically inroduced as a phenomenological heory, i has been recognized as a powerful ool for imedependen phenomena such as vorex dynamics and flucuaion paraconduciviy/diamagneism 9, 3]. Furhermore, specifically for he paraconduciviy defined in > c, here is no gap in he exciaion specrum and he DGL approach can be jusified. he DGL heory has been applied o he pumpprobe specroscopy below he ransiion emperaure 7]. Wih a fas pump process in real ime, a broad range of freuency componens are excied. he DGL euaion can accoun for he dynamics of small-freuency componens, which should dominae he long-ime behavior. In his paper, we consider he superconducing flucuaion effecs in he phoo-excied sae using he DGL euaion above he ransiion emperaure, and he behaviors in he whole ime scale can be uniuely deermined. he uench dynamics of superconducing flucuaions wih a sudden change of an aracive ineracion has been discussed in Ref. 33. We here consider purely elecronic effecs in phoo-excied sysems. he original paper by Schmid has derived he flucuaioninduced curren for arbirary srengh of elecric fields 34]. Also, recenly he non-linear response is considered in relaion o he nonreciprocal ranspor phenomena in noncenrosymmeric sysems 35 37]. hus he DGL heory has he abiliy o ackle he problem beyond he usual linear-response regime.

2 librium sae wih he ime consan γ, bu in he inermediae ime domain, he power-law or logarihmically decaying regimes are found. he exponens are differen beween he ime regions γ and γ. As for he conduciviy calculaed from he probe curren, he enhancemen by he pulse is no observed. On he oher hand, he anisoropy shows a nonrivial behavior: alhough he sysem is spaially anisoropic due o he phoo-induced curren, he excied curren uickly damps compared o he superconducing flucuaion. Hence we find an isoropic bu sill non-euilibrium sae, which is confirmed hrough he properies of probe currens. We also perform a sysemaic sudy on he parameer dependences on he pump and probe elecric currens in all dimensions. his paper is organized as follows. Secion II provides he heoreical formalism for pump-probe procedure based on he DGL approach. he numerical resuls are given in Sec. III and Sec. IV where differen pump procedures are aken. We summarize hese resuls in Sec. V. Appendices A and B are devoed o he deailed explanaion for our numerical calculaion, and Appendix summarizes he euilibrium propery realized for < before he pump and. he addiional daa for hree and one dimensions are also shown in Appendix D. II. PUMP-PROBE PROEDURE IN DGL EQUAION A. General Formulaion Le us begin wih he DGL euaion wih a random force 34] Γ A] ψr, = δfψ, δψ r, Fψ, A] = drψ r, a + fr,, iear, ] ψr,, 4m δfψ, A] jr, = δar,, 3 where F and j are he GL free energy and superconducing curren densiy, respecively, and Γ is a ime developmen facor. We have aken = k B = unless oherwise saed explicily. he classical complex field ψr, is he superconducing order parameer or macroscopic wave funcion. We consider he spaially uniform elecric field and he vecor poenial can be chosen as he one dependen only on ime: Ar, = A. We noe ha he fourh-order erm ψr, 4 is irrelevan here, since we consider he sysem above he ransiion emperaure > c. he phoo-exciaions in he superconducing sae below c has been discussed in Ref. 7. he coefficien of he second-order erm behaves as a = a c > wih an order-of-uniy consan a. he random force fr, is also inroduced o represen he hermal flucuaion. Performing he Fourier ransformaion, we ge Γ ψ = ea] 4m + a ψ + f. 4 he soluion for he macroscopic wave funcion is hen wrien as ψ = d Γ f d ea ] ] exp + a. Γ 4m 5 We assume ha f is spaially and emporally uncorrelaed such ha he random average wrien by a bracke symbol is given by f f = V δ δ, 6 where V is a sysem volume. he consan is deermined so ha he random average is idenical o he hermal average 34], which resuls in = ΓV. he mos fundamenal physical uaniy in our sysem is he superconducing flucuaion ψ. Wih he random force, he effec of he superconducing flucuaion can be aken ino accoun and he uaniy ψ becomes finie. Since he uaniy wih he wavevecor is gauge-dependen, we consider he real-space uaniy ψr, = V ψ, 7 which is now a gauge-invarian uaniy, and is also spaially uniform in our presen seup. In he analysis, we separae he conribuions from he euilibrium sae and he non-euilibrium one, and define he deviaion by δ ψ = ψ ψ e, 8 where e sands for he euilibrium value. In he following, we use δo as he deviaion of he uaniy O from ha in euilibrium sae. We separae he vecor poenial ino wo pars as A = A + A where A and A represen pump and probe lighs, respecively. he vecor poenial A for he probe is reaed by he linear-response heory. he flucuaion conribuion o he elecric curren densiy is given by j = e mv ea] ψ = j + j, 9

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sha_base64="u5bnpeknissx9cjjjp4laici=">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</laexi> probe E <laexi sha_base64="kpzybymzpniwwjkbvyjywrro=">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</laexi> <laexi sha_base64="kpzybymzpniwwjkbvyjywrro=">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</laexi> <laexi sha_base64="kpzybymzpniwwjkbvyjywrro=">aaajuniclvzla9rqfd59doly3ipviufeo5u4rfbepyjupa/7wm5snlbmuweq5kon+ggvbhsgoubb/hh/g76e8rlbufc+njkzebqzjjn3u+d8ofc79xg9azl+imrex//a4impwrfodpdz86oji6dw/hdbc+qy4zrud6arvnsmh5hjibjdeanrs3zkreuou6l9sc83xedrsnouvvuro+awawgbomf3kraveoygujjel+brvncxual4mnfhbi9shbjjyoyszjdgwwldlix+dyisoaxkiaplsn9kohnheel4aeabunfrwo9rb3f6xogbdy+huihkcj8u8fpvii/gkvos/vwhc6hcdvduojzri8uldkzfkxjogjwdrgre6vnphppo9a3yqaafzepswcudvnw8pcfarxa+yzrzekzjinkpspku/x8f7srcwj8ip4l35a3di3ygck7rp/fhqs6+znyhmu9zzzvh7auixa/zrznawysnwkvhlqmebhwr7lc43jdhoykyglcrzgxc5xbup7iwc3nmnsdulo7wonpkboud+gik6ilkfapln4prghdghdzmvwaabpx3zrsvnghbi+nvmmonvmjzgzjncj9xudmg8f5hejpy/beex3od5jxiv6rnw9ul7ofxnf4kkuphbre+quvbkbw6k8nsclxavav8i6vlffae7mrealji33jzxzdmk8vhln4j6dmdoz3rajvfpb5r8zjppzwxepevzbvfjuwe9k3zdpbshrmuus/lrjb4jarxrxufjbjiwapd/kkafpm3e+zunsq9n89mxenmkxjr55lpa8kffvgubrdpup4yih3fpgivuxy9dlm7px9ablfl7lplukezwf3zixmhhpjb+h4wvhd7gvob59vffmefpv6s4/bcjpum</laexi> <laexi sha_base64="kpzybymzpniwwjkbvyjywrro=">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</laexi> probe <laexi sha_base64="v4vgaba6bfr+povongrphxpqq=">aaajsxiclvzla9rqfd59np6aksbxuxiucciioiuovupa/7gm6jonnexejoloecp6fzx4urbhfgz3aiuxfqniupoiglv3ussxumjzbdjod+95wv537npj7lhmequz9pbnzh5jqweprmxndwypnlwj3zfkoufarr+ia4gecuhmzoyrxpl5w3jzb9x/cn6qem6zwkdzxzs7w6y6ahhymhprwuxifga7yivgm5jpr/ovuzuywdskmsq6fs3skmbvlsokyanwowiyd8vkfkkgksnrwkpdsgddzrakmio4gw4sbblpx9ri7smpgi3o98ul8ef/fnyuidlulj46ngs9nahnlc+fkyzemzzbve6edu/smal+h3yddpjrz7m6m/dyovudb8/zvf8l7fgpnqzgnewfeeefvpfud/flvb8u36pzg7impum5p/jdezcn58zuioyxzzub3unqiejjm3ujoxozxgn/gegosinsryf5zjhaoyxvni4/doz7vywfdowufeaan3u4dvxs8acpygdoyb9lbvbzvbdixwdzxbbunubunuqay+casryqvxmk+rijmex/ijze5vwfpboyhvbrzge7zkje7cja4rwj9rqwbmj/9dsfh6fhij3benu7ld8djw6nyvwkdjk+gjmpj6nwdwuyda5wi3fhnd7xlfjke78wmkbunxvqymyzlyvcdwnuzm3hsglcr6p7y+kzlrf3kgexsfxxwu8nhvpz5jjpmywsd6iiz/wumk9zfmzg86enflsebpbe3+7uo4f/ozsvjpfiu6lwke5wgkejlejscqxxmvzm+kbmnpcie+5gewo6+d4oww6gl3qjdprlharian9ijelj6xvpd+lx7hrr9scwf+u8a6pslrc3ima==</laexi> pump / <laexi sha_base64="u5bnpeknissx9cjjjp4laici=">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</laexi> <laexi sha_base64="u5bnpeknissx9cjjjp4laici=">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</laexi> <laexi sha_base64="u5bnpeknissx9cjjjp4laici=">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</laexi> <laexi 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poenial and b elecric field in he pump-probe process considered in he single pump-sho model. c,d he same plos for he double pump-sho model. where j = j = e mv e mv e mv ea ] ψ, ea ] ψ ψ A. Here j is a phoo-excied curren densiy and j is a small curren induced by he probe ligh. he subscrips and represen he perurbaion order of A. If we consider he euilibrium case, we reproduce he sandard paraconduciviy formula 9, 34], which are summarized in Appendix based on our formalism. We can also consider he ime-dependen GL free energy F, which is modified from he euilibrium by he phoo-exciaion. Afer aking he random average, we ge Fr, = ea] a + ψ, 4m F fluc + F kin. 3 he firs erm F fluc = V a ψr, represens he flucuaion conribuion o he free energy, and he second erm represens he kineic energy of ooper pairs. In order o consider he anisoropy in he kineic energy, we divide he free energy as F kin = µ F µ kin where F µ kin = µ ea µ ] ψ. 4 4m In he nex wo subsecions, we specify he shape of he pump/probe lighs more concreely, and derive he deailed expressions for he above physical uaniies. B. Single pump-sho model As he simples form of he pump process, we consider he elecric field wih a single dela funcion single pumpsho model, see Figs. a and b]. he ime dependence of he vecor poenial is A = A θ + A θ sin Ω, 5 where θ is Heaviside s sep funcion. While he realisic seup in he experimens involves an oscillaing Gaussian shape, our choice of elecric fields makes he euaions simple enough o evaluae decaying properies precisely in numerical calculaions. More specifically, he inegraion wih respec o ime is evaluaed analyically, and hence we can focus on he implemenaion of -inegrals. Wih his seup, he superconducing flucuaion is explicily wrien as δ ψ = V Γ e g ea g V Γ e g ea g ea e ea A θ cos Ω ] e ea A θ cos Ω ] V Γ e ea A θ g ea g ea + Ω e g ea + OA, 6