Superconductivity Theory and Applications Part 7 docx

Effects of Impurities on a Noncentrosymmetric Superconductor - Application to CePt 3 Si 139 Application of these results to real noncentrosymmetric materials is complicated by the lack of definite information about the superconducting gap symmetry and the distribution of the pairing strength between the bands. As far as the pairing symmetry is concerned, there is strong experimental evidence that the superconducting order parameter in CePt 3 Si has lines of gap nodes (Yasuda et al., 2004; Izawa et al., 2005; Bonalde et al., 2005). The lines of nodes are required by symmetry for all nontrivial one-dimensional representations of 4v C ( 2 A , 1 B , and 2 B ), so that the superconductivity in CePt 3 Si is most likely unconventional. This can be verified using the measurements of the dependence of c T on the impurity concentration: For all types of unconventional pairing, the suppression of the critical temperature is described by the universal Abrikosov-Gor’kov function, see Eq. (32). It should be mentioned that the lines of gap nodes can exist also for conventional pairing ( 1 A representation), in which case they are purely accidental. While the accidental nodes would be consistent with the power-law behavior of physical properties observed experimentally, the impurity effect on c T in this case is qualitatively different from the unconventional case. In this case in the absence of magnetic impurities one obtains the following equation for the critical temperature: 0 11 1 ln 24 2 c ccn T TT                    (43) In the low   1 nc T   and dirty   0 1 nc T   limit of impurity concentration one has 00 1 8 cc c n TT T       (44) 1 0 00 1 nc cc c C T TT T e          (45) This means that anisotropy of the conventional order parameter increases the rate at which c T is suppressed by impurities. Unlike the unconventional case, however, the superconductivity is never completely destroyed, even at strong disorder. 4. Low temperature magnetic penetration depth of a superconductor without inversion symmetry To determine the penetration depth or superfluid density in asuperconductor without inversion symmetry one calculates the electromagnetic response tensor   ,, s K q vT   , relating the current density J  to an applied vector potential A        ,, s Jq KqvTAq     (46) The expression for the response function can be obtained as    2 2 , 2 ˆ ,,, 1 , , , sm n nm nk ne KqvT T k k k mc m                (47) Superconductivity – Theory and Applications 140 where 2 q kk   , 2 ˆ k  is the direction of the supercurrent and represents a Fermi surface average. By using the expression of Green`s function into Eq. (47) one obtains    2 2 2 ,, 2 2 22 22 ,, ,,, . 2 ˆ 1 2 22 sn nimpsF kk kk n nimps k nimps k ne KqvT mc ivk Tdk k m qq ivkEivkE                                                                        (48) Now we separate out the response function as       ,, 0,0,0 ,, ss K q vT K K q vT     (49) where   2 0,0,0 40 c K   (  1 2 2 2 0 4 mc ne        is the zero temperature London penetration depth). Doing the summation over Matsubara frequencies for each band one gets          2 2 , 2 2 22 0 2 22 ,, 2 2 , 2 2 22 ,, 2 ˆ ,, Re . 2 2 2 ˆ Re . 2 sF sF k s F imp k k k imp sF sF k F imp k k fvkf vk ne KqvT k d mc qk igi m fvkf vk ne kd mc qk i m                                                           , 2 2 0 2 1 2 , 2 22 ,,, 2 2 2 , 2 2 . sinh 2 ˆ 12 2 1 22 2 1 22 ˆ ln k kimp F k FFk kkk FF k gi qk m ne k mc qk qk g mm qk qk g mm k                                                                , , 22 2 ,22 ,,, 22 2 2 ,, , 4 1 22 . 2 . 1 22 2 k k kk kFF imp imp kkk F FFk F imp kk k mg qk qk mm qk qk qk g qk mm m                                                                 , 2 2 ,22 , 2 2 22 2 22 ,, 4 . 1 2 . ˆ 2Re . 2 2 k kk F imp k F sF sF F imp k k k imp mg qk m qk fvkfvk kd qk igi m                                                                                 (50) Effects of Impurities on a Noncentrosymmetric Superconductor - Application to CePt 3 Si 141 , 1 2 , 2 22 ,,, 22 2 2 ,, 2 . sinh 2 ˆ 12 2 1 22 2 . 1 22 ˆ ln k F k FFk kkk FFk imp kk qk m ne k mc qk qk g mm qk qk g qk mm k                                                           , , 2 2 ,22 ,, 22 2 2 2 ,, , , 4 . 1 22 . 4 2 11 22 2 2 k k kk FF imp kk F k FFk FF imp kk k k mg qk mm qk m qk qk g qk qk mm m m                                                              , 2 ,22 2 2 22 2 22 ,, . ˆ 2Re . 2 2 k k imp F sF sF F imp k k k imp g qk fvkfvk kd qk igi m                                                                   (50) The factor k g  characterizes and quantifies the absence of an inversion center in a crystal lattice. This is the main result of my work i.e. nonlocality, nonlineary, impurity and nonsentrosymmetry are involved in the response function. The first two terms in Eq. (50) represent the nonlocal correction to the London penetration depth and the third represents the nonlocal and impure renormalization of the response while the forth combined nonlocal, nonlinear, and impure corrections to the temperature dependence. I consider a system in which a uniform supercurrent flows with the velocity s v  , so all quasiparticles Matsubara energies modified by the semiclassical Doppler shift . sF vk   . The specular boundary scattering in terms of response function can be written as (Kosztin & Leggett, 1997)      2 2 0 0 ,, 2 1 spec s T K q vT dq q             (51) In the pure case there are four relevant energy scales in the low energy sector in the Meissner state: T, nonlin E , nonloc E , and k g  . The first two are experimentally controlled parameters while the last two are intrinsic one. In low temperatures limit the contribution of the fully gap ( 0 sin   ) Fermi surface I decrease and the effect of the gap 0 sin   Fermi surface II is enhanced. I consider geometry where the magnetic field is parallel to c axis and thus s v  and the penetration direction q  are in the ab plane, and in general, s v  makes an angle  with the axis. There are two effective nonlinear energy scales 1 nonlin s F l Evku    and 2 nonlin s F l Evku    .where cos sin l ul     and 12 ,1ll   . In the nonlocal   0q  , linear   0 s v  limit, i.e., in the range of temperature where nonlin nonloc ETE one gets    2 0 0 3 2 222 0 00 2ln2 4 ,0, 3 3 42 4 l l l ll l cT wT Kq T w cT uwT w                                     (52) Superconductivity – Theory and Applications 142 where sin cos l wl     , sin l ucosl     , and 2 22 F k qv g     . Depending on the effective nonlocal energy scales 12 12 00 ,,,1 Fl Fl nonloc nonloc vu vu EEll        one obtains  0 2 , , nonloc nonloc spec nonloc nonloc nonloc nonloc TEET T TETE TEET                  (53) For CePt 3 Si superconductor with 0.75 c TK , the linear temperature dependence would crossover to a quadratic dependence below 0.015 nonloc TK   . Magnetic penetration depth measurements in CePt3Si did not find a 2 T law as expected for line nodes. I argue that it may be due to the fact that such measurements were performed above 0.015K. On the other hand, it is note that CePt3Si is an extreme type-II superconductor with the Ginzburg-Landau parameter, 140K  , and the nonlocal effect can be safely neglected, and because this system is a clean superconductor, neglect the impurity effect can be neglected (Bauer et al., 2004; Bauer et al., 2005). In the local, clean, and nonlinear limit   0, 0 s qv the penetration depth is given by   1 2 40,, loc spec s c Kq v T          (54) Where    1 2 , 22 2 2 22 22 2 ,, ,, 1 2 2 ,, . sinh 2 ˆˆ 12 2 Re 2 . 2 1 2 . sinh ˆ 12 s F sF sF k kk k F F kk F KqvT qk fvkfvk m ne kkd mc g qk k m qk ne k mc                                                                         , 2 2 2 22 2 2 ,, ,, 2 ˆ 2Re 2 . 2 1 2 sF sF k kk k F F kk fvkfvk m kd g qk k m                                                       (55) Thus by considering only the second term in the right hand side of Eq. (55) into Eq. (51) one gets Effects of Impurities on a Noncentrosymmetric Superconductor - Application to CePt 3 Si 143  1 2 2 0 1 3 2 0000 3 4 00 2 1 ln2 , 2 2 1 ln2 22 22 2 1 22 , 2 sF k l nonlin nonlin l spec l k sF l nonlin nonlin k sF vk g l l T nonlin nonlin T uEET u g vk T uETE g vk u oTe E E T                                                     (56) The linear temperature dependence of penetration depth is in agreement with Bonalde et al's result (Bonalde et al., 2005). Thus the T behavior at low temperatures of the penetration depth in Eq. (56) is due to nonlineary indicating the existence of line nodes in the gap parameter in CePt 3 Si compound. A T linear dependence of the penetration depth in the low temperature region is expected for clean, local and nonlinear superconductors with line nodes in the gap function. Now the effect of impurities when both s-wave and p-wave Cooper pairings coexist is considered. I assume that the superconductivity in CePt 3 Si is unconventional and is affected only by nonmagnetic impurities. The equation of motion for self-energy can be written as      ,,, nn n imp n pi nT pp i       (57) where the T matrix is given by   3 3 ,, 1,, n n u Tppi uppi            (58) here 3  is the third Pauli-spin operator. By using the expression of the Green’s function in Eq. (58) one can write   0 2 0 2 00 ,, 1 n NuI Tppi NuI        (59) where      2 1 2 2 0 4 imp n imp n i d I i                  (60) and 0 u is a single s-wave matrix element of scattering potential u .Small 0 u puts us in the limit where the Born approximation is valid, where large   00 uu , puts us in the unitarity limit. Superconductivity – Theory and Applications 144 Theoretically it is known that the nodal gap structure is very sensitive to the impurities. If the spin-singlet and triplet components are mixed, the latter might be suppressed by the impurity scattering and the system would behave like a BCS superconductor. For p-wave gap function the polar and axial states have angular structures,     0 cos kk TT   and     0 sin kk TT   respectively. The electromagnetic response now depends on the mutual orientation of the vector potential A and ˆ I (unit vector of gap symmetry), which itself may be oriented by surfaces, fields and superflow. A detailed experimental and theoretical study for the axial and polar states was presented in Ref. (Einzel, 1986). In the clean limit and in the absence of Fermi-Liquid effects the following low-temperature asymptotic were obtained for axial and polar states   , , , 0 0 n B T kT a               (61) where in the axial state   24n  and 4 2 7 15 a        , and in the polar state  31n  and  27 3 3ln2 42 a       , for the orientations     . The influence of nonmagnetic impurities on the penetration depth of a p-wave superconductor was discussed in detail in Ref (Gross et al., 1986). At very low temperatures, the main contribution will originated from the eigenvalue with the lower temperature exponent n, i.e., for the axial state (point nodes) with 2 T low, and for the polar state (line nodes) the dominating contribution with a linear T . The quadratic dependence in axial state may arise from nonlocality. The low temperature dependence of penetration depth in polar and axial states used by Einzel et al., (Einzel et al. 1986) to analyze the   2 TT   behavior of Ube 13 at low temperatures. The axial ˆ AI   case seems to be the proper state to analyze the experiment because it was favored by orientation effects and was the only one with 2 T dependence. Meanwhile, it has turned out that 2 T behavior is introduced immediately by T-matrix impurity scattering and also by weak scattering in the polar case. The axial sate., and according to the Andersons theorem the s-wave value of the London penetration depth are not at all affected by small concentration of nonmagnetic impurities. Thus, for the polar state, Eq. (60) can be written as      2 1 2 2 0 22 0 2 imp n imp n i d I icos                 (62) Doing the angular integration in Eq. (62) and using Eqs. (57) and (59) one obtains Effects of Impurities on a Noncentrosymmetric Superconductor - Application to CePt 3 Si 145    2 222 0 00 22 0 2 222 2 2 2 0 00 22 0 20 / 14(0) / n imp n Nnu K Nu K                             (63) here K is the elliptic integral and  im p n i    . We note that in the impurity dominated gapless regime, the normalized frequency   takes the limiting form i     , where  is a constant depending on impurity concentration and scattering strength. In the low temperature limit we can replace the normalized frequency   everywhere by its low frequency limiting form and after integration over frequency one gets   2 222 2 1 5 22 2 4 ˆ ,, 3 k s k Ne T KqvT k mc                   (64) As in the case of d-wave order parameter, from Eqs. (64) and (51) one finds    2 0 00 4 ln 04 24 T T          (65) In p-wave cuprates, scattering fills in electronic states at the gap nodes, thereby suppressing the penetration depth at low temperatures and changing T -linear to 2 T behavior. 5. Effect of impurities on the low temperature NMR relaxation rate of a noncentrosymmetric superconductor I consider the NMR spin-lattice relaxation due to the interaction between the nuclear spin magnetic moment n I  ( n  is the nuclear gyro magnetic ratio) and the hyperfine field h, created at the nucleus by the conduction electrons. Thus the system Hamiltonian is 0intso n HH H H H   (66) where 0 H and so H are defined by Eqs. (1) and (2), nn HIH   is the Zeeman coupling of the nuclear spin with the external field H  , and int n HIh   is the hyperfine interaction. The spin-lattice relaxation rate due to the hyperfine contact interaction of the nucleus with the band electron is given by  2 0 1 Im 1 lim 2 R J R TT        (67) where  is the NMR frequency, 8 3 ne J     ( e  is the electron geomagnetic ratio) is the hyperfine coupling constant, and   R    , the Fourier transform of the retarded correlation Superconductivity – Theory and Applications 146 function of the electron spin densities at the nuclear site, in the Matsubara formalism is given by (in our units 1 B k   )       0 R TS S       (68) here T  is the time order operator,  is the imaginary time,   HH SeSe       , and       † Sr r r              † Sr r r        (69) with  † r    and   r    being the electron field operators. The Fourier transform of the correlation function is given by   0 n i RR n ide         (70) The retarded correlation function is obtained by analytical continuation of the Matsubara correlation function   n RR n ii i        . From Eqs. (66)- (70), one gets     2 0 1 ,, 11 lim 2 Im , , , , ii n n nm n nm n pp J TT T Trpii pi TrFpiiFpi                           (71) where 2 m mT   are the bosonic Matsubara frequencies. By using Eqs. (11) and (12) into Eq. (71), the final result for the relaxation rate is      2 1 0 1 f Jd N N M M TT                  (72) where  1 1 T fe        is the Fermi Function.,   N   and   M   defined by the retarded Green’s factions as     , Im , R p Np          (73)     , Im , R p MFp         (74) In low temperatures limit the contribution of the fully gap ( 0 sin   ) Fermi surface I decrease and the effect of the gap 0 sin   Fermi surface II is enhanced. Effects of Impurities on a Noncentrosymmetric Superconductor - Application to CePt 3 Si 147 As I mentioned above, the experimental data for CePt3Si at low temperature seem to point to the presence of lines of the gap nodes in gap parameter (In our gap model for 0 , 0 sin   has line nodes). Symmetry imposed gap nodes exist only for the order parameters which transform according to one of the nonunity representations of the point group. For all such order parameters 0M   .Thus, Eq. (72) can be written as    2 2 1 0 1 4 cos 2 Jd NN TT T h T            (75) In the clean limit the density of state can be calculated from BCS expression  0 22 ReNN        (76) For the gap parameter with line nodes from Eq. (76) one gets  0 0 2 NN      (77) Thus from Eq. (75) one has 22 23 0 2 1 0 1 2 JNT T    (78) Therefore, line nodes on the Fermi surface II lead to the low-temperature 3 T law in 1 1 T  which is in qualitative agreement with the experimental results. In the dirty limit the density of state can be written as     22 0 , 1, BCS BCS imp N Nd uN        (79) In the limit, 0  where 0 imp n N NV      ( N V is the electron density) the density of state is     2 0 imp NNac    (80) where 0 cotcg   ( 0  is the s-wave scattering phase shift), a is a constant, and   0N the zero energy   0   quasi-particle density of state is given by Superconductivity – Theory and Applications 148  1 2 2 0 11 1 42 o NN           (81) where     . In the unitary limit   0 u  ,0c    0 2   , from Eqs. (75) and (80) one obtains  2 2 1 1 0 JN T T  (82) Thus the power-low temperature dependence of 1 1 T  is affected by impurities and it changes to linear temperature dependence characteristic of the normal state Koringa relation again is in agreement with the experimental results. 6. Conclusion In this chapter I have studied theoretically the effect of both magnetic and nonmagnetic impurities on the superconducting properties of a non-centrosymmetric superconductor and also I have discussed the application of my results to a model of superconductivity in CePt3Si. First, the critical temperature is obtained for a superconductor with an arbitrary of impurity concentration (magnetic and nonmagnetic) and an arbitrary degree of anisotropy of the superconducting order parameter, ranging from isotropic s wave to p wave and mixed (s+p) wave as particular cases. The critical temperature is found to be suppressed by disorder, both for conventional and unconventional pairings, in the latter case according to the universal Abrikosov-Gor’kov function. In the case of nonsentrosymmetrical superconductor CePt3Si with conventional pairing ( 1 A representation with purely accidental line nodes), I have found that the anisotropy of the conventional order parameter increases the rate at which c T is suppressed by impurities. Unlike the unconventional case, however, the superconductivity is never completely destroyed, even at strong disorder. In section 4, I have calculated the appropriate correlation function to evaluate the magnetic penetration depth. Besides nonlineary and nonlocality, the effect of impurities in the magnetic penetration depth when both s-wave and p-wave Cooper pairings coexist, has been considered. For superconductor CePt 3 Si, I have shown that such a model with different symmetries describes the data rather well. In this system the low temperature behavior of the magnetic penetration depth is consistence with the presence of line nodes in the energy gap and a quadratic dependence due to nonlocality may accrue below 0.015 nonloc TK   . In a dirty superconductor the quadratic temperature dependence of the magnetic penetration depth may come from either impurity scattering or nonlocality, but the nonlocality and nodal behavior may be hidden by the impurity effects. [...]... Vol 73 , No 7, (13 April 2004) pp 16 57- 1660, ISSN: 13 474 073 Yogi, M; Kitaoka,Y; Hashimoto,S; Yasuda, T; Settai,R; Matsuda, T D; Haga,Y; Onuki, Y; Rogl, P & Bauer,E (2004) Evidence for a Novel State of Superconductivity in 152 Superconductivity – Theory and Applications Noncentrosymmetric CePt3Si: A 195Pt-NMR Study Physical Review Letters,Vol 93, No 2, (8 July 2004), pp.0 270 03 [4 pages], ISSN 1 079 -71 14... Kitaoka, Y (2005) Unconventional superconductivity and magnetism in CePt3Si1-xGex Physica B, Vol 359-361, (30 April 2005 ), Pp 360-3 67, ISSN 0921-4526 Blatsky, A.V; Vekhter,I & Zhu, J.X (2006) Impurity-induced states in conventional and unconventional superconductors Reviews of Modern Physics, Vol 78 , No 2, (9 May2006), pp. 373 –433, ISSN 1539- 075 6 150 Superconductivity – Theory and Applications Bonalde,I; Brämer-Escamilla,... 2010 17: 39:10 STEP=1 SUB =1 TIME=1 MX ELEM=1459 MIN=.353E-10 MAX=.925E-06 353E-10 1 103E-06 206E-06 308E-06 411E-06 514E-06 617E-06 72 0E-06 822E-06 925E-06 VECTOR JUL 14 2010 17: 15:56 STEP=1 SUB =1 TIME=1 JX NODE=1 37 MIN=3.865 MAX=5999 3.865 669.99 1336 2002 2668 3334 4001 46 67 5333 5999 Fig 3 Force, torque and current density distributions per surface element 160 Superconductivity – Theory and Applications. .. superconductor are shown 158 Superconductivity – Theory and Applications Fig 1 Flow-diagram of the algorithm Fig 2 Small permanent magnet (m=0.016 Am2) over superconductor 159 Foundations of Meissner Superconductor Magnet Mechanisms Engineering 1 VECTOR JUL 14 2010 17: 33:09 STEP=1 SUB =1 TIME=1 FX ELEM=1458 MIN=.157E-08 MAX=.141E-03 157E-08 157E-04 314E-04 471 E-04 628E-04 78 5E-04 942E-04 110E-03 126E-03... Function of Noncentrosymmetric CePt3Si Physical Review Letters, Vol 94, No 19, (16 May 2005), pp.1 970 02 [4 pages], ISSN 1 079 -71 14 Kosztin, I & Leggett, A.J (19 97) Nonlocal Effects on the Magnetic Penetration Depth in dWave Superconductors Physical Review Letters, Vol 79 , No 1, (7 July 19 97) , pp.135– 138, ISSN 1 079 -71 14 Effects of Impurities on a Noncentrosymmetric Superconductor - Application to CePt3Si 151... status report Low Temp Phys, Vol 31, No 8, (7 October 2005), pp .74 8 -75 7, Bauer, E; Hilscher, G; Michor, H; Paul, Ch; Scheidt, E W; Gribanov, A; Seropegin, Yu; Noël, H; Sigrist, M & Rogl, P (2004) Heavy Fermion Superconductivity and Magnetic Order in Noncentrosymmetric CePt3Si Physical Review Letters, Vol 92, No 2, (13 January 2004), pp.0 270 03 [4 pages], ISSN 1 079 -71 14 Bauer, E; Hilscher, G; Michor, H; Sieberer,... F; Chandrasekhar, B.S; Andres, K; Ott, H R; Beuers, J; Fisk, Z & Smith, J.L (1986) Magnetic Field Penetration Depth in the HeavyElectron Superconductor UBe13 Physical Review Letters, Vol 56, No 23, (9 June 1986), pp.2513–2516, ISSN 1 079 -71 14 Edelstein, V M (1995) Magnetoelectric Effect in Polar Superconductors Physical Review Letters, Vol 75 , No 4, (4 September 1995), pp 2004–20 07, ISSN 1 079 -71 14 Frigeri,... No 3, (16July 2001), pp.0 370 04 [4 pages], ISSN 1 079 -71 14 Gross, F; Chandrasekhar, B.S; einzel,D; Andres, K; Herschfeld, P.J; Ott, H.R; Beuers, J; Fisk, Z & Smith, J.L (1986) Anomalous temperature dependence of the magnetic field penetration depth in superconducting UBe13 Zeitschrift für Physik B: Condensed Matter, Vol 64, No 2, pp. 175 -188, ISSN 072 2-3 277 Hayashi, N; Wakabayashi, K; Frigeri, P A & Sigrist1,... 29-32, ISSN 1361-648X Anderson, P W (1959) Theory of dirty superconductors Journal of Physics and Chemistry of Solids, Vol 11, No 1-2, (September 1959), pp 26-30, ISSN 0022-36 97 Anderson, P W (1984) Structure of triplet superconducting energy gap Physical Review B, Vol 30, No .7, (1 October 1984), pp 4000-4002, ISSN 1550-235X Bauer, E; Bonalde, I & Sigrist, M (2005) Superconductivity and normal state properties... set of 12 references forces The calibration constant was established by least squares fitting in K = (3. 87 0.14)×10-4 N/με, with a correlation coefficient of R2 = 0.9 97 162 Superconductivity – Theory and Applications Fig 7 Coordinate system of the PM-SC configuration The superconductor is down and the permanent magnet is over it Figure is not scaled to real sizes The measurement for every position . states in conventional and unconventional superconductors. Reviews of Modern Physics, Vol. 78 , No. 2, (9 May2006), pp. 373 –433, ISSN 1539- 075 6 Superconductivity – Theory and Applications 150.  0 22 ReNN        (76 ) For the gap parameter with line nodes from Eq. (76 ) one gets  0 0 2 NN      (77 ) Thus from Eq. (75 ) one has 22 23 0 2 1 0 1 2 JNT T    (78 ) Therefore,. scattering phase shift), a is a constant, and   0N the zero energy   0   quasi-particle density of state is given by Superconductivity – Theory and Applications 148  1 2 2 0 11 1 42 o NN          