BULK PROPERTIES OF UNCONVENTIONAL SUPERCONDUCTORS

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1 IJRRAS 6 (4) March 0.arpapress.cm/Vlumes/Vl6Issue4/IJRRAS_6_4_3.pdf BULK PROPERTIES OF UNCONVENTIONAL SUPERCONDUCTORS Arthur Ekpekp Departmet f Physics, Delta State Uiversity, Abraka, Nigeria. ABSTRACT I a cvetial s-ave supercductr, the rder parameter is ttally symmetric. Therefre the l eergy excitatis have a gap except fr the case f a gapless supercductr ith magetic impurities. Fr the s-ave state, there is mrever l-lyig cllective mde, sice i the case f charged particles the cllective desity fluctuatis are thig but Plasma mdes. The existece f the gap i the excitati spectrum aturally leads t the expetial temperature depedece f varius physical qualities, such as the specific heat, relaxati rate f uclear magetic resace, (NMR) ad kight shift.. INTRODUCTION I a ucvetial supercductr, the rder parameter ca have pit r lie ers. Due t the excitati spectrum acrss these pits r lies, the excitati spectrum starts frm er eergy. The desity f states f the quasi-particles is defied by () p ( ) ( ) k E k here E k± is the quasi-particles eergy. I this article, e shall restrict urselves t the case f Uitary States, E k E Let us csider several typical examples. I a rdiary s-ave supercductr, the desity f states is 0( ) () ( ) ( ) here N (0) is the desity f states at the Fermi eergy i the rmal phase ad is the magitude f the gap fucti. The desity f states has a gap f ad it diverges at =. N e tur t examples f the P-ave states i rtatially symmetric space. I 3 H e t superfluid phases exists uder differet pressures. The l pressure B phase is the s-called Balia-Werthaer (BW) state f p- ave pairig ad the high pressure. A phase is the s-called Aders-Brikma Mrel (ABM r axial) state. The gap fucti i the BW state. x i y (3) ss( k), ˆ x ik y has a cstat prduct + =. Thus, desity f states has the same frm as the rdiary s-ave state. Csequetly, the equilibrium thermdyamic prperties f the BW state ad the s-ave state are idetical. This shuld t be misuderstd that all their prperties are idetical. N-equilibrium prperties f the BW state like spi susceptibility certai differeces frm thse f the Bardee, Cper, Schriffer (BCS) state.. THEORETICAL CONSIDERATIONS AND CALCULATIONS The gap fucti becmes er at t pits i the ABM r axial state, here the gap fucti has the frm. x i y O (4) ss' O x i y Here the desity f states is give by X ( ) (0) N (5) lg It varies as at l eergies ad has a lgarithmic divergece at. The third example f the p-ave state is the plar state t realied i 3 H e, here the gap fucti is O (6) ss' O Obviusly, the gap has lie ers the equatr. Fr the desity f states e btai 455

2 IJRRAS 6 (4) March 0 Ekpekp Bulk Prperties f Ucvetial Supercductrs ( ), ( ) arcsi It has a frm liear i fr l eergies ( < ) ad is fiite at =. The three states f p-ave pairig csidered here are represetative f examples ffered i rder t discuss the desity f states f quasi-particles i ucvetial supercductrs. The imprtat pit is that the geeric frm f () at l eergies depeds slely the tplgy f the gap er. If they are lie ers, the () ad if they are pit ers, the (). The differece i the eergy depedece f () is reflected i the temperature depedece f varius physical qualities at l temperatures. As a first example, e csider the specific heat. At l temperatures here the T depedece f the rder parameter ca be reflected, the specific heat is give by C de T df ( E) (8) P( E) E ( T T c ) de 0 here f(e) is the Fermi distributi fucti. Therefre, it is bvius that the T depedece f the specific heat depeds the tplgy f the gap structure i the fllig ay. T gapless (9) C T lie er, 3 pit ers T Ather example is the NMR relaxati rate, hich is give by (Mriya, 963) (0) YN Am KBT mx q, / T, q here is the uclear resace frequecy, ϒ N the gyrmagetic rati f the uclear spi, Am the hyperfie cuplig cstat, ad X (q, ) the dyamical susceptibility trasverse t the magetic field at the ucleus. It is straightfrard t exted the stadard BCS result (Hebel ad Shchter, 959) t iclude the ucvetial case, T N (E) T 0 de ( E) A( ( ) ) E df, () E de here /T N is the relaxati rate i the rmal state ad (K) detes the average f the rder parameter the Fermi surface. This average vaishes fr the ucvetial supercductrs, hich 4 belg t ther represetati tha The resace frequecy is geerally small cmpared ith eergy scales f electrs. Therefre, e may take the limit f 0. If the itegral cverges. Hever, fr the s-ave r the BW state, the itegral diverges lgarithmically if is set equal t er. Fr supercductig states ith pit r lie ers, the itegral cverges ad the temperature depedece f /T at l temperature is give as T gapless () 3 T T lie er, 5 pit ers T A imprtat csequece f grup thery is that, ith spi rbit cuplig, lie ers are t alled fr dd-parity supercductrs Vl Vik ad Gr Kv, 984, 985, Blut, 985, Ueda ad Rice, 985a, 985b. (Aders, 984). Blut i particular, gave a geeral prf f this. Therefre at very l temperatures pure samples shuld bey per las crrespdig t the pit ers he they are dd-parity supercductrs. Varius per la behaviur are reprted i heavy-fermi materials fr may prperties, icludig the specific heat ad the NMR relaxati rate discussed here. The accumulated bdy f data frm this type f experimet idicates clearly that there are may l-lyig excitatis assciated ith des f gap fuctis i heavy-fermi supercductrs. Hever, i sme cases, there is csistecy abut the gap structures amg the results fr differet quatities amg the results fr differet quatities experimetally bserved, if e assume expets fr (7) 456

3 IJRRAS 6 (4) March 0 Ekpekp Bulk Prperties f Ucvetial Supercductrs pure material. Oe pssible explaati fr this kid f icsistecy is that the temperature rage experimetally accessible is t yet sufficietly l t derive the geuie expets. Ather, prbably mre plausible, explaati is that this discrepacy ca be reslved he e iclude the effect f impurity scatterigs. The mst atural framerk i hich t discuss the effects f impurity scatterig supercductivity is the Abriksv Gr kv thery (Abriksv ad Gr kv, 960). The essetial prperties f the impurity scatterig may be see i the simple example f s-ave scatterig. It is cveiet t simplify the calculati by eglectig spirbit cuplig. I the Abriksv Gr kv thery, the gap fucti is give by (3) ss' TV k, k' Fss( K' ) V ' here F is the amalus Gree s fucti. I this frmati the gap fucti ca be csidered as the (amalus) self eergy due t the pairig ptetial. Impurity scatterig gives additial ctributrs t the self-eergy. First e treat this prblem i the Br apprximati (Gr kv ad Kalug, 985, Ueda ad Rice, 985b). Oe ctributi t the self-eergy is f the rmal type, fig.(a). () (4) ( i ) iu G( K', ) k here Ƞ i is the impurity ccetrati ad U characteries the s-ave scatterig ptetial [G ss (K, i ) = G(K, i ) ss ]. There is als a ctributi f the amalus type fig. (b), () ' (5) ( i ), ss iu Fss Kss k' I the case f -magetic impurities i s-ave supercductrs, Aders s therem maifests itself i the fllig fact. Fr the s-ave state () is simply prprtial t i ad () is Prprtial t, ad their prprtiality cstats are the same uder the assumpti f a cstat desity f states ear the Fermi-eergy. Therefre, the effect f impurity scatterig ca be take as a simple rermaliati f the eergy scale ithut ay ifluece thermdyamic prperties. I ctrast, fr ay ucvetial state () is er, sice the summati f equati (5) vaishes. Therefre, simple scalig lger rks i this case. Geerally, a differece betee the t prprtiality cstats leads t despairig effects. (a) (b) u u u u G(K, i ) F(K, i ) x x x (c) Fig.: The t types f self-eergies f impurity scatterig rmal ad amalus. I the Br apprximati, the rmal type f self-eergy is expressed by the diagram (a) ad the amalus e by (b). i the T-matrix apprximati, multiple scatterig prcesses (c) are als take it accut. With the self-eergies due t impurity scatterig, the Gr kv equati, fr the Gree s fucti are mdified as () G ( k, ) ss ss( k) Fs ' s ( K, ) (6) () i K) ( i ) F ( K, i ) G( K, i ) 0 (7) ( s' s ss' 457

4 IJRRAS 6 (4) March 0 Ekpekp Bulk Prperties f Ucvetial Supercductrs Fr the uitary states they are easy t slve (8) GK, ( i ) ( K) F ' ( K, i ) ss (9) ss ( ) ( K) here i = i - () (i ). By substitutig equatis (8) ad (9) it equatis (3) ad (4), e btai self csistecy equatis fr ss (K) ad () (i ). We sh the self-csistecy equatis fr the BW, ABM, ad Plar states as typical examples i a rtatially ivariat system. (a) BW State: (0) N (0) VKBT, () (b) ABM state: i 3 N (0) VKBT lg () i (3) lg i (c) Plar state: 3 (0) N VK BT Lg (4) Lg I the abve expressis V is the stregth f the pairig iteracti defied by i ( 0) u 458 (5) V ( K, K) 3VK ˆ. (rtatially symmetric frm) ad N is the stregth f the impurity scatterig i.e. half f the scatterig rate, T N. Equatis (0) ad () have the same frm as fr magetic impurities i a rdiary s-ave supercductr. The three sets f equatis reduce t the same set f equatis he they are liearied. The trasiti temperature btaied by the liearied equati decreases as a fucti f i the same ay as i rdiary gapless supercductrs ith magetic impurities. I this frmalism, the desity f states f quasi particles is give by ( ) m( ) i (6) I fig. (a) e sh the desity f states fr the BW state. Fr a eak impurity scatterig, there is a gap i the desity f states give by 3 ( ) 3 (7) g Whe, the system is i a gapless regime. This behaviur i the BW state is the same as fr the usual paramagetic impurity effect i a s-ave supercductr. The desity f states fr the ABM state is sh i fig.(b). The desity f states at l eergies is give by 3 (8) (0) N ( ) ( ) (0) ( N Ct )

5 IJRRAS 6 (4) March 0 Ekpekp Bulk Prperties f Ucvetial Supercductrs The mst remarkable result is btaied fr the plar state (fig.c). I this case, he there are impurities, ereergy excitatis alays exist ad their desity f states at the Fermi level is give by (0) Sih (9) The mai cclusi f the Br apprximati is that the mst serius effect uld be ay plar state, i.e., a state ith lie ers, sice i this case the l temperature behaviur is mdified by ay ccetrati f impurities. I ctrast, a state ith pit er has a critical ccetrati befre a essetial mdificati f the per las sets i. Althugh the aalysis as carried thrugh ly fr the simplest frm f p-ave states, the results deped merely the geeric frm f the desity f states are therefre shuld be applicable ith slight mdificati t ay state ith the same geeric frm. I a sigle site Kd prblem, resistivity becmes a cstat at T = 0 after a lgarithmic icrease. The cstat crrespds t the phase shift f π/, the uitary limit. I may heavy-fermi systems, resistivity icreases as temperature is lered, reaches a maximum ad the decreases rapidly. The value f the resistivity at the maximum, i may cases, is csistet ith the value f the uitarity limit. Therefre, it uld t be surprisig if scatterers i heavy-fermi systems had large phase shifts, as Pethick ad Pies (986) pited ut. T treat scatterig ith a large phase shift, the Br apprximati is t sufficiet, ad multiple scatterig prcesses shuld be icluded. Multiple scatterig f electrs by magetic impurities i rdiary supercductrs as studied by Shiba (968), usig a T-matrix apprximati. He fud that there exists a lcalied excited state arud a classical impurity spi, hich at fiite ccetrati frms a impurity bad. Fr the ivestigati f the impurity effect i the BW state metied befre, Buchhlt ad Zickagl als emplyed the T-matrix apprximati, as did Schmitt Rik, Miyake, ad Vama (986) ad Hirschfeld, Vlliardt, ad Wlfle (986), idepedetly, i studyig the csequeces f resat impurity scatterig i heavy-fermi systems. FIG.. Desity f states f quasiparticles btaied by the br apprximati: (a) fr the BW state; (b) fr the ABM state; (c) fr the plar state (Ueda ad Rice, 985b). 459

6 IJRRAS 6 (4) March 0 Ekpekp Bulk Prperties f Ucvetial Supercductrs FIG. 3. The desity f states f the plar ad the ABM states btaied by the T-Matrix apprximati fr a pairbreakig parameter f / ad differet values f the phase shift. The isert illustrate the resace peaks i the l-eergy, gapless regi. (Hirschfeld et al., 986). I the T-matrix apprximati, the self-eergy due t impurity scatterig is give by fig.(c). ( ) ( i ) (30) iu G( K, ) UG( K, ) k k A self csistet thery is btaied by usig the dd part f the self-eergy, [ () (i ) - () (-i )]/, i equatis (6) ad (7). The eve part f the self-eergy [ () (i ) + () (-i )]/, is just a shift f the chemical ptetial ad ca be eglected. I this thery, impurity scatterig is characteried by t parameters. Oe is the phase shift defied by ta u, (3) ad the secd is the scatterig rate Γ= ½ T N Si i the uitarity limit Schmitt Rik et al ad Hirshfeld et al, assumed that i a Kd lattice, each magetic i leads t a phase shift f cducti electrs. Hever, the et effect is er because f the peridicity; the resistivity f a peridic system is er at er temperature. Therefre, a -magetic i i a such a lattice uld appear t ffer a phase shift π/ ith respect t the backgrud. With this assumpti, the impurity scatterig is characteried agai by a sigle parameter Γ. Figure 3 shs the calculated desity f quasi particle states fr a plar state fr varius scatterig ptetials, settig as the pair breakig parameter Γ/ = 0.0. At l eergies, there is a resace peak ad at higher eergies, the desity f states is almst idetical t its value ithut statemet impurities (ct ). The same statemet ca be made fr the ABM (axial) state. I bth cases the idth f the resace peak icreases as Γ/ gets larger. 3. RESULTS AND DISCUSSION The per la behaviur discussed here are a maifestati f the aistrpy f the gap fucti f ucvetial supercductig states. Hever, these pers give ifrmati ly abut the geeric frm f the gap fucti. Frm the desity f quasi particle states, e ca immediately see the effect f resat impurity scatterigs specific heat. At very l temperature (T/Tc < 0.), it shs a small T liear specific heat due t the appearace f the resace hile at elevated temperature, it flls clsely the per la expected ithut impurities (Hirshfeld et al, 986, 988, Miyake, 986, Ott et al, 987). Similarly, the NMR relaxati rate shs a Krrigalike behaviur at very l temperatures ad flls a per la fr the pure case at higher temperatures (Hirschfeld et al, 988). The situati is very differet fr trasprt prperties i heavy-fermi supercductrs. Fr these quatities, the Br apprximati is iadequate t ly quatitatively but als qualitatively. As a example, e csider thermal cductivity K. I a simple kiematic thery, it is give by K = /3 V F Tc. I the Br apprximati, it ca be 460

7 IJRRAS 6 (4) March 0 Ekpekp Bulk Prperties f Ucvetial Supercductrs sh that the prduct f the relaxati time () ad the desity f states () is almst eergy idepedet (Cffey et al, 985, Pethick ad Pies, 986). I this result, the mdificati f () discussed i this article, is eglected, hich gives ly a mir chage he the impurity ccetrati is small. Therefre, the thermal cductivity i the Br apprximati is almst liear i T ad the cefficiet remais the same rder as its rmal state value, hich ctradicts the experimetally bserved T behaviurs. Pethick ad Pies prpsed that the discrepacy may be reslved he the resat ature f the impurity scatterig i the uitarity limit is take it accut. Calculatis f the thermal cductivity usig the T-matrix apprximati ere carried ut idepedetly by Schmitt Rik, Miyake ad Varma (986) ad Hirschfeld, Vlhardt, ad Wlfle (986). Their results may be summaried as flls. At very l temperatures, K/T ges t a fiite value due t the appearace f the l eergy resace. At higher temperatures, i a ide temperature rage, K flls a T la fr the case f lie ers ad a T 3 la fr the pit ers. This result meas that the prduct f () ad () shs almst the same behaviur as () here is the scatterig rate i the rmal state. This fact cat be uderstd by N N the Br apprximati, as e discussed befre. It shuld als be metied that Vrtex crrectis t the thermal resistivity are discussed by Hirschfeld, Wlfle ad Eiel (988). The T-Matrix apprximati is als applied t the study f ultrasic alterati i heavy-fermis (Hirschfeld et al, 986; Schmitt-Rik et al, 986). The temperature depedece f the sud atteuati depeds its plariati ad prpagati directi. Miyake (986) ad Schmitt-Rik et al (986) have ccluded that the assumpti f a state ith lie ers, tgether ith a scatterig i the impurities limit, leads t results csistet ith the experimetal bservatis i UPt 3, CeCu Si ad UBe 3. A amalus temperature depedece f the Ld peetrati depth i UBe 3 as reprted by Grss et al (986); ( T ) (0) fll T la. The authrs aalyed the temperature depedece by the Br apprximati ad ccluded that the behaviur is csistet ith a eergy gap ith pit des. Recetly, Chi ad Muikar (988, 989b) develped a thery f the superfluid desity tesr hich determies the peetrati depth. They treated the impurity scatterig by the T-matrix apprximati ad pited ut the pssibility that impurity scatterig ehaces the aistrpy f the desity tesr. 4. CONCLUSION I cclusi, impurities mdify the per las, especially at l temperatures. The csequeces f resat scatterig i ucvetial supercductrs fr specific heat, thermal cductivity, ultrasic atteuati, NMR relaxati rate, ad electrmagetic absrpti have bee examied by several authrs as e have see i this article. I particular, Miyake (987) ad Schmitt-Rik et al (986) have pited ut that the experimetally bserved per las are sme csistet ith lie ers tha ith pit ers. Hever, t dra a defiite cclusi abut the gap structure, e eed further experimets that are directly related t the symmetry f the rder parameter. 5. REFERENCES []. Aders, P. W. (984): Phys. Rev. B30, 4000 []. Blut, E. J. (985): Phys. Rev. B3, 935 [3]. Chi, C. H., Muikar, P. (988): Phys. Rev. B37, 5947 [4]. Chi, C. H., Muikar, P. (989b): Phys. Rev. B39, 96 [5]. Cffey, J., Rice, T. M ad Ueda, K. (985): Phys. C8, 839. [6]. Grss, F., Chadrasekhar, B. S., Eiel, D., Adres, P. J., Hirschfeld, H.R., Ott, J., Beers, Fisk, Z ad Smith, J. L. (986): Z. Phys. B64, 75. [7]. Gr kv, L. P. ad Kalug, P. A. (985): Pis ma Zh. Eksp. Ter. Fi. 4, 908 [JETP Lett. 4, 53 (985)]. [8]. Hebel, L. C. ad Slichter, C. P., (959): Phys. Rev. 6, 79. [9]. Hirschfeld, P. P., Vllhardt, D, ad Wlfle, P. (986): Slid State Cmm 59,. [0]. Hirschfeld, P. P., Wlfle, P. ad Eiel, D. (988): Phys. Rev. B37, 83. []. Miyake, K. (987): J. Mag Mag. Mater 63 & 64, 4 []. Miyake, K. ad Varma, C. M. (986): Phys. Rev. Lett. 57, 67. [3]. Mriya, T. (963): J. Phys. Sc. Jp. 8, 56. [4]. Ott, H. R. (987a): I Prgress i l temperature Physics, XI, Edited by Breer, F. (Nrth Hllad, Amsterdam) p.5. [5]. Ott, H. R. (987b): Helv. Phys. Acta. 60, 6. [6]. Pethick, C. J. ad Pies, D. (986): Phys. Rev. Lett. 57, 8. [7]. Schmitt-Rik, Miyake, S. K. ad Varma, C. M. (986): Phys. Rev. Lett. 57, 575. [8]. Shiba, H. (968): Prg. Ther. Phys. 40, 435. [9]. Ueda, K. ad Rice, T. M. (985a): Phys. Rev. B3, 74. [0]. Ueda, K. ad Rice, T. M. (985b): I thery f Heavy Fermi ad Valece Fluctuatis, edited by Kasuya ad Spriger, T. S. []. Vlvik, G. E. ad Gr kv, L. P. (985): Ucvetial Supercductivity, Rev. Md. Phys. Vl. 63, N.. 46