Theory of Kondo Effect in Type II Superconductors

Progress of Theoretical Physics, Sep 1977

We study the equilibrium properties of type II superconductors with Kondo effect. The Kondo effect associated with the impurity spins is taken into account within the interpolation approximation, which was used previously in our calculations of the superconducting transition temperature and the specific heat jump at the transition temperature. Using this approximation, the general Ginzburg-Landau equations are derived for superconductors with Kondo effect and the Ginzburg-Landau parameter κ2(T) is calculated. One finds that the initial slope of the Ginzburg-Landau parameter shows a continuous change with TK / Tc0 and approaches the BCS-value at the end of an infinite Kondo temperature TK / Tc0 = ∞ in contrast to Maki's theory.

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Theory of Kondo Effect in Type II Superconductors

Progress of Theoretical Physics Theory of Kondo Effect in Type II Superconductors Shin'ichi ICHINOSE 0 0 Department of Physics, 1\fagoya University , 1\fagoya 464 Ve study the equilibrium properties of type II superconductors with Kondo effect. The Kondo effect associated with the impurity spins is taken into account within the interpolation approximation, which was used previously in our calculations of the superconducting transition temperature and the specific heat jump at the transition temperature. Using this approximation, the general Ginzburg-Landau equations are derived for superconductors with Kondo effect and the Ginzburg-Landau parameter IC,(T) is calculated. One finds that the initial slope of the Ginzburg-Landau parameter shows a continuous change with TK/T,o and approaches the BCS-value at the end of an infinite Kondo temperature TK/T,,=oo in contrast to Maki's theory. - /Cz (T), which describes the magnetization in this region. Summary is gi\"en in § 4. § 2. Generalized Ginzhurg-Landau equations In the previous papers1l,zl we have assumed that the order parameter is constant 1n space. There are, however, other classes of interesting phenomena, which in­ volve a spatially varing order parameter. To describe the spatial variation of the order parameter it is convenient to start with the generalized Ginzburg-Landau equations. In the following we will consider only the gapless region, where the order parameter .d (r) is small. Then the self-consistent equation is given up to third order in .d(r) by5l J+(r)=[g[T~ sd 3r'Q"'(r,r').:J+(r')+lgiT~ s... sd 3r1 ... d 3rsBw (r, rh .. ·, rs) .:J+ (r1) L1 (rz) .:J+ (r3), (2·1) where Qw(r, r') and Bw(r, r~> ... , r 3) are the two-particle and the four-particle Green's functions, respectively. 1) The first term of Eq. (2·1) Introducing the vertex correction r(?/, w) and the bare two-particle Green's function Clw0 (q) in the presence of magnetic fields, the first term of Eq. (2 ·1) can be written as T ~ w sd 3r'Q"' (r, r') .:J+ (r') = T ~ r (q2, w) (_Lo (7J) .:J+ (r). w Using the renormalized one-particle Green's function in the presence of magnetic field G" (w), 6l the bare two-particle Green's function Clw0 (q) is given as (2·2) (2·3) (2· 4) Q~wo(q ~-) -..::."... G k (W)G-k-q ( - oJ )_- 2rr-N-p-tan-l(v-F--!-7--J-I-) , k vFI7JI 2!&5! where p is the density of states of conduction electrons per atom per spin, 1V is the number of atoms and wis the renormalized frequency. q is defined as q- 2eA or q + 2eA depending on whether it operates on J+ (r) or .d (r) using the external momentum q. Gk (cu) = [icu- ev ·A- ~k -.S(w)] -\ G_k( -cJJ) = [ -iw+ev·A-~k-l:(w)]-\ where OJ= (2n +1) rrT, ~k is the one-electron energy of conduction electrons and 1: (w) is the self-energy correction clue .to both the magnetic and the non-magnetic impurities. A is the vector potential and v the Fermi velocity. The vertex correction r(7J2, w) is given by solving the follmving equation: =1+ 2rrNpT"' r, (w w')~(~q2 cu')tan-1 (vFICll) VF I ~q~ Lw....'J II ' I ' 21~U)'1 ' where T 11 (o), cu') is the irreducible vertex of the effective interaction between elec­ trons due to both the magnetic and the nonmagnetic impurities. Introducing the electronic relaxation time due to nonmagnetic impurities alone r0 and the approxi­ mation used in Ref. 1), vve get an approximate expression for rH (w, w') as r, (u) w') = --_ oro,ro' '1 ' 2rrroTNp +!.:..[2. r (w)o N T 1 ro,ro ,_f(w)f(())_')_] 4TxP2 ' where f(cu) = ( 1 + :~~) -2, and n is the magnetic impurity concentration. If we neglect the effect of an applied magnetic field on impurity spins, rl (w) Is independent of the magnetic field and is given in Ref. 1) as (2· 5) (2· 6) (2·7) (2·8) (2·9) Using the expansiOn the vertex correction r (qZ, w) is finally obtained as r (q2, U)) = ______ lcul±:_l I (u)) I jciJI +na(u)) + (1/2)D1 (o))q2 . [1 n . [Juj) fff1 (ql -], 4TxP 1+ (n/4Txp)(})2(q2) (2·10) and a(cu) is the pair-breaking parameter which appears in Ref. 1). Two kinds of diffusion constants D 1 (rJ)) and D 2 (w) are written by the diffusion constant Do which is determined by nonmagnetic impurities alone, 1.e., D 1 (w) =Do·1+2rrronPT1 ~W), [1 + !'o/rJ] D2 (w) = IJ_o_ 1 + ro/r1 vvhere D 0=r0v//3 and T1 is the electronic relaxation time due to magnetic impurities alone. When the electron mean free path is so short that it is hardly affected by a small amount of magnetic impurities, we can approximate them by D 0 , Finally substituting Eq. ( 2 -10 ) into Eq. (2 · 2), we arrive at T~ sd 3r'Qw(r,r')LJ+(r') =Np[i§o(7/)"' n . {i§ 1 (q2)}~ 4TxP 1+ (nj4TKp){]) 2 (q 2) 2) The second term of Eq. (2 ·1) N mv \ve turn to the analysis on the second term o£ Eq. (2 -1). It is conven­ ient to rewrite it as follows: T ~ S···Sd 3r1· ··d 3rsBw (r, rh · • ·, r 3) J+ (r1) L1 (r,) J+ (r3) = lim T ~ lJw (qi) Ji· (r1) L1 (r,) J+ (rs). ;r!,} ~r w The quantity lJ, (qi) can be calculated by means of the same perturbation method that was used previously by the present author in the calculation of the specific heat jump at the transition temperature,") though in this case the dependence of the external momentum 7j is introduced into the s-electron Green's function Gk (1u) and vertex corrections r(q2, w). Since it is tedious to vvrite down various contri­ butions of diagrams for lJ"' (q_J, here we show only the results. jJ (?f-)= _rrNp .lo)[?Js(w) + (1/8)D2(w) [(q_~-?fsr+C?fz-?f,2_"l w ' 2 [u)[ 4 {ID=117z(?f/, w)} ' ( 2 -17 ) vvhere ~;=l?fi = 0 and ?ji operates only on L1 (r;). Here use has been made of the fact that in the dirty limit, the term associated with the non-commutative nature of the operators ?ji only gives rise to a higher order correction in lj !;0 even at low temperature,6) -vvhere l is the electronic mean free path and !;0 is the BCS coherence length. Thus we arrive at a quasi-local equation even at low temper­ ature. Here [wf-t?s(W) is giVen in Ref. 2) as where the magnitude o£ the impurity spm 1s assumed 1/2. The renormalization (2·14) ]J+(r). factor 1!2(qj, w) is the extension of 1J2 (w), which appears in Ref. 2), into a magnetic field dependent case and is given by , Y(q·) = 7r T 2.:: [cu[·1s(o))+ (1/S)~z(w) [_(?J.J-?J.s) 2 + (q2 ~q4L]_. ' 2 w !cunn~~J1/z(?J./, w)] 3) Ginzburg-Landau current density The second Ginzburg-Landau equation which relates the current to the vector potential is obtained by a similar procedure. Since the reduction involves methods already described, we \vill not take the space to give more than the final result6l \vhere Equations (2 · 21) and (2 · 24) are the basic equations for our discussion. As we will see later, the temperature dependence of the Ginzburg-Landau parameter /C2 (T) is quite different from that of Maki's theory, mainly due to the behavior of ;.:: (cu) at low temperature and at low frequency. However, there remains some ambiguity in the form of a(w) an~ ;.::(cu) in the region w<TK. Fortunately no sharp transi­ tion between two limiting regwns, and so we can believe such interpolation gives semi-quantitatively correct results. § 3. Magnetic properties Using the basic equations in the preceding section, we can discuss quite gener­ ally the magnetic properties of superconductors near the upper critical field. In the following we describe several applications of the above equations. 1) The ujJper critical field Hc2 (T) D If the transition from the superconducting state to the normal state 1n the presence of a magnetic field is of second order, the upper critical field Hc2 (T) is determined from the first term in Eq. (2 · 21) by where q02 1s the lo~west eigenvalue of the following equation: D,(w)7'tL1+(r) =D,(cu)q02Ll+(r), qo2 = 2eHc, (T). If we confine ourselves to the case of alloys where the electronic mean free path is extremely short, then we can obtain the previous resultll using D0 = r 0v//3 instead of D2 ((})). 2) The Ginzburg-Landau parameter !C2 (T) The general Ginzburg-Landau parameter tC2 (T) introduced by Maki, which may be determined from the slope of the magnetization curve near He,, is defined as follovvs :6l 1 where IC z(T) - 1 . ~Y(qo2) 2 - 4e Z (q02) X' (q02) ' of /C2 (T) at the transition temperature: ,,~ 1 (3 ·9) where a,= na (T,) /2-r.T,. The calculation of JC* can be simplified by the use of the approximations a(T,o):::::: 37r(ln TJ(_)-2, 8p T, 0 dn dt, t,~l ::::::3?pT,a(ln TK)' 37:"2 T, 0 c):::::n:!2_ ( ln T K_) -2• 6 Teo , (3 ·10) The result 1s given by IC* = IC* + AG n4 252( ( 3 ) ( In _T___!£_) -2 T co ' where (J:.G = -0.091. One easily finds that IC* approaches the AG value "XG in the limit Tco/Tx---->;oo. b) On the other hand, if T<f:..Tx, we can put mo (qo2) :::.:::_@1 (qo2) :::.:::_@2 (qo2) :::.:::_~' jgjNp where jgj = lgl-n/4TxNp2• Using the approximations we can simplify the calculations of /C2 (T) as ( !C2 (t)) 2 ~ n4 -( n ) -1J!C2l((1/2)+X) - , . - -56(( 3 ). 1 + 4TxP . [1J!C1l((1/2) +X)J2' where When T = Tc, this result reduces to the simple formula, !wlna(w):::::. (1+-n-) jwj, 4TxP jwj +na(w)::::::. (1+ _77-_) jwj, 4TxP X= eDoHc2 (T) . 2nT 1 1+ (nj4TxP) ('IC2(tc)) 2:::::.1+_n_. " 4TxP Then the Tx/Tc0-dependence of IC* is given by Here use has been made of the approximation -d-n:' dtc 1 tc=l :::::.4TxP ( In___T_!£_)-2• Teo One easily finds that IC* approaches the BCS-value ~Cics = 0 in the limit of high Kondo temperature, T x/Tc0---->;oo, in contrast to Maki's theory. The behavior of /C2 (tc) / /C as a function of Tc/Tco is given by the numerical calculation for the wide region of T x/Tco· Results are given in Fig. 1. Here the parameter used for the numerical calculation is wDjnTco = 1169. Results of numerical calculations for IC* are given in Fig. 2. The initial slope of the Ginzburg-Landau parameter In this paper we have discussed how the Kondo effect affects the magnetic properties of type-Il superconducting alloys based on the interpolation approxi­ mation. Though our results contain the already known limiting cases: the AG- or BCS-like behavior, they are different from those of Maki's theory. In particular, the initial slope JC* of the Ginzburg-Landau parameter tends to the BCS-value at the end of an infinite Kondo temperature. Therefore we can say that the study of the magnetic properties near H,2 in type-Il superconductors with Kondo effect will also provide important information on the Kondo effect in superconduc­ tors. Throughout this paper we have neglected completely the effect of the mag~ *> The definition of the Kondo temperature in the Yamada-Yosida theory is different from that of the most divergent approximation. Since we made an interpolation between them, our calculation contains some ambiguity about TK. Therefore, quantitative comparison of our results with experiments does not seem so much meaningful. S. Iclzinose netic field on magnetic impuntles. Since the magnetic field characteristic of the Kondo effect is given by Hx = kBTx/flB, this approximation is allowed only when Hc2<f::_Hx. In general, the magnetic-field effect on magnetic impurities is expected to appear as (H/ Hx) 2 in the lowest order. Therefore, the Ginzburg-Landau param­ eter /C2 Ctc) at Tc and IC* are not affected by this effect, and our approximation of neglecting it is justified. Experimentally, the Kondo effect in type II superconductors has not been studied extensively. Most of the previous work'J was carried out in alloy systems with Tx/Tco<l. For the most interesting case of TK/Tco>l there exist no reliable experimental results. However, if alloy systems with TK/Tco>l are found in the future, \Ve belieye that a continuous change in the quantity IC* from the negati\-e yalue to the positive ones can be observed experimentally. Acknowledgement Thanks are due to Professor Y. Nagaoka for critical reading of the manuscript. 1) T. Matsuura and Y. Nagaoka , Solid State Comm. 18 ( 1976 ), 1583. T. Matsuura, S. Ichinose and Y. Nagaoka , Frog. Theor. Phys . 57 ( 1977 ), 713 . 2) S. Ichinose , to he published in Frog. Theor. Phys. 3) K. Maki , J. Low Temp . Phys. 6 ( 1972 ), 505 . 4) E. Mi. iller-Hartmann and J. Zittarz , Z. Phys . 54 ( 1970 ). 234 , 5) A. A. Abrikosov and L. P. Gor 'kov , J. Exp. Theor. Phys. (U.S.S.R.) 39 ( 1960 ), 1781 [ Soviet Phys . -JETP , 12 ( 1961 ), 1243 ]. 6) See, e.g., Superconductivity I , edited by R. D. Parks (Marcel Dekkar , New York, 1969 ). 7) K. Winzer , Z. Physik 265 ( 1973 ), 139. T. Aoi and Y. Masuda , Proceedings of the Thirteenth International Conference on Low Temperature Physics ( 1972 ), p. 574 .


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Shin'ichi Ichinose. Theory of Kondo Effect in Type II Superconductors, Progress of Theoretical Physics, 1977, 733-742, DOI: 10.1143/PTP.58.733