Department of Physics, Kyoto University Kyoto

Transcription

1 29 Progress of Theoretical Physics, Vol. :12, No. 1, July 1964 Pauli Para.mag:netism and Superconducting State. II Kazumi MAKI*) Department of Physics, Kyoto University Kyoto (Received March 11, 1964) The effect of the Pauli param8gnetism on 8 superconducting sm8ll specimen is investigated by the use of the general Gor'kov equation. It is shown that the effect is negligible in a usual specimen but appreciable in an extremely small specimen (thickness d._, 10A). The change of the gap in the energy spectrum and the specific heat at lower temperatures is calculated. I. Introduction In the preceding work 1 ) (which we should refer to as M. T.) we have investigated the effect of the Pauli paramagnetism on superconductors extending the discussions given by Clogston 2 ) and by Chanclrasekhar 3 ) to arbitrary temperatures. We have found also an interesting modification of the Abrikosov's mixed state using the generalized Ginzburg-Landau equation including the Pauli terms. The present paper is devoted to the study of the magnetic properties of superconducting thin films, we take account of the effects of the spin paramagnetism as well as of the diamagnetic current. As is well known, the latter plays an essential role in the phase transition of an usual small specimen. Incorporating the Pauli terms in the general Gor'kov equation, 4 ) we solve the equation on the assumptions: 1) L1 is constant all over the specimen and 2) -r Ll~l -r is the collision time of an electron. The effect of the Pauli terms on the critical field at arbitrary temperatures are obtained. We also calculate the gap in the energy spectrum and the specific heat at lower temperatures. Explicit calculation shows that the effect of Pauli term is appreciable only in an extremely small specimen (thickness d..._, loa). () 2. Formulation According to M. T. we start with the following simplified Hamiltonian: H=- ~ (iv +ea)'f'+(iv -ea)'fpdx+ g~ ('f'+'t) ('f'+1r)dx+ p.~'f'+ah'frdx, (1) p.. is the Bohr magneton and a the Pauli spin matrix. In the following we assume that the magnetic field H is uniform in the specimen. The generalized *) Present Address: Research Institute for Mathematical Sciences, Kyoto University, Kyoto.

2 30 K. Maki Gor'kov equation IS given as (i(j)~- ~) G± (p) + iijft (p) = L (i(j)'t+~)ft(p) -iijg±(p) =0, (2) i(j)r=i(j)- (e/nz)p A+ tth, ~= (1/2m) (p 2 -p 0 2 ), and G± (P) and F ± (p) are the Fourier transfgrms of the Green's functions, G±(x, x') = (-i(7~ 1 (x)y't(x'))) -i( 7~} (x)y! (x')), F±(x, x') = ((T'fr' 1 (x)y~ (x'))) ( Tt~ (x)y 1 (x')). The ordering parameter ij is determined by the following equation: The equation (2) is formally solved as G±(P) =- (i(j)"t:+~)/~(p), F±(P) = -iil/.e(p), In the above derivation we made use of the fact that i1 is constant. In a usual small specimen the electronic mean free path is very short due to the boundary scattering. We can most easily include this effect in the theory considering the scattering due to the random centers. We assume that the scattering amplitude of an electron by a center is written as f(p, p') =a (p, p') + ib (p, p') Po 2 ( [p, p'] a), (8} (3) (4) (5) (6) (7) p and p' are the momenta of the electron before and after the collision, a the Pauli spin matrix and Po the fermi momentum. The standard treatment of impurity scattering in metals leads us to the following expressions: 4 ) G±(P) =- (iw±+n/i::cp), F±(P) = -il jh(p), ~ """'2 ;;± (p) = m.t + ~ 2 + LJ±. (9) (10) a,± and ij± are determined by

3 Pauli Paramagnetism and Superconducting State. Il 31 The terms with 1/'Z" 1 come from the spin orbit interaction. (11) (12) We are interested here in the case 'Z"LI<l. In this case the important expansion parameter is 'Z"(e/m) p A which is of the order of ('Z"L1) 112 Expanding the integrand in Eq. (11) in the power of (e/m) p A and retaining the first two terms in the expansion we finally obtain The above equation is apparently too complicated to be solved. Therefore we consider the case l/'z"1 = 0 in the following for simplicity. In this case Eq. (13) reduces to OJ+ip.H _ ( 1 _,. 1 ) r'= 2'Z"a U-t <., -=-----==j -./1 +uf,.., ij. (14)

4 32 K. Maki Equation (5) is rewritten in terms of U± as follows: 1=!u!~/l 'I_~( ) b 2 ( 2n) 2 L u~ u~ (15) 3. The order of the phase transition and the critical field In the case J~O we expand U± in the powers of.d and the above equation reduces to and p.h p 2 = 2nT fo (p),!1 (p) and!2 (p) are } { f2(p) =Re ~) ( 1 ) 5-3pl~J ( 1 ) 6, n+-2-+p t' (z) = r' (z) jr (z) the di-gamma function. n+2+p The asymptotic forms of fo (p),!1 (p) and!2 (p) are given in Table I. Table I. (16) (17) (18) (19) jo(p) -- 1 pi-p~ '11:2 In( 4'Y I pi) pl +7( (3) p~ 24 lpl 4 jl(p) pi+6pip~-3p TPT6-- 7( (3) - 60( ( 4) p1-186( (5) pz 2 j2(p) -7 py + 95p1 p~- 85pip~ + 5p~ 20lpll0 31((5) We know from the theory of the second order phase transition that the phase transition is always of the second order as long as!1 (p) is positive. As

5 Pauli Paramagnetisn~ and Superconducting State. II 33 / 1 (p) being the monotonic increasing function of the temperature, we expect a second order phase transition at arbitrary temperatures if lim(2n:t)- 2 / 1 (p)';>o. In this case the critical field is determind by -ln 7/Tao= /o(p). (20) T->0 Using the asymptotic expression of / 0 we obtain and In( 2 H(~:[i+;l2) = - -lif ( 2 }/?: (:2~~ 2 + ;2)2, (21) for T~Tao, -ln T/ Tao= -~2_ (~~~~;) + 7 t: (3) c;~ifr-r, (22) for Ta- T<Tco, Lloo=n:Tao/r the gap in the energy spectrum at T=0 K and 11=0. Especially at the absolute zero of the temperature we obtain Lloo _/----2 ah 2 = --Cv 1+-z -z), --2 The critical field at intermediate temperatures are numerically evaluated and given in Fig. 1 for various values of z. The critical thickness of a specimen is given from the following equation: air Juu (23) Combining this equation with Eq. (23) we obtain or de= (2-rtr.doo) -l/ 2 v'3 (18-10v'3 ) 114 X (p/ev). (26) Fig. 1. The critical field H is depicted for various values of z = 3 (2-nr L1 oo) -l X (p/evd)2. For the thinner specimen, the order of the phase transition changes from the second order to the first order as the temperature decreases. The critical thickness 0 is. roughly of the order of loa for "rtrlioo""' 1/100, which is extremely small.

6 34 K. Maki 4. The gap in the energy spectrum and the specific heat at lower temperatures In this section we are concerned with the behavior of L1 at lower temperatures. In this case Eq. (15) are written as ~ = dm { ( 1 ln(li/ L1 0 o) = -- Im -./.. 2 2T 2T o L1 1-u 2 ) - 1 -(tanh m- tj._lf_ +tanh_!!)+ pli) - Im-./ 1 - C~ I Lf)2 }. In the limit T~0 K we can carry out further calculations. ln(.d/lioo) = _ ~ r:-r:-213 (1-(213)114/ ~ (- ~-r 12 ~( _ 1)n+ln-312e-ncwo-p.H)IT, = _ n ( [ 2 ( ~~!! - 3 r:-213 (1-(213) 114 (27) for m 0 - pl-i>o, (28) ~- 1 = - arccosh ( - ~ ( C: arcsin ( /1 - (~2) m 0 = L1 ( 1 - ( 213 ) mo_\ (n1_,) 2 -] L1 J 12LI 312 V ph- mo ' for ph-m 0 >0 and (<1, (29) for (> 1, (30) The gap in the energy spectrum IS given as m 0 - ph for m 0 - p1--l>o and vanishes for the field H 1 defined as (31) At T= 0 K we can solve this expression for small value of z as ah12 ~ Llooe-~14 j 1- (ze~/4) 1/3 + ( ~- + ; --) (z:~/4 )_2/a --'!3 (32) ~ 1- _!!_ (ze7rl4)113+}_n (1+ _!!~)Cze7r/4) Comparing this expression with Eq. (23) one sees that the change in H1 due to the Pauli term is rather remarkable. Now we are going to discuss the change of the specific heat at lower temperatures. According to Abrikosov and Gor'kov 5 ) we calculate the thermodynamic potential as follows:

7 Pauli Paramagnetism and Suj;erconducting State. II 35 (33) we find the expression of ofr(y) from Eqs. (28), (29) and (30} respectively; ofr(l1) = - v TC /6- T3/2 ij(-;3/2 (1-,213) 1/4:2:: ( -1) n+l n-312 e-n(wo-mh)it, for w 0 - ph>o, C34) = ;3-iJfj3/2 (1-(213) 1/4_ /S.f!!}~ ' v ph- Wo fj= ((LJ)- 1 = (ah 2 )- 1. = -fj2(1-(-2)-112~~ (rct)2, for w 0-11}{ <O and ( <1, (~35) for (>1, (36) Using the thermodynamic potential thus obtained we find the following expressions for the specific heat; for w 0 - pi-l>o, (37) == 1 rt~_g_,j~j6 (15/ 4) ((5/2; ( ) T312 8s12 LJ(1-(21s)-11\ TC. for (flo- ph~o, (38) for ph-aj 0 >0 and (<1, (39) = m._:()_ (1-(-2) 112 rr: for (> 1. ( 40) Except the case (J)o- ph> O no exponential term appears in the expression of the specific heat, which is a direct consequence of the vanishing gap in the energy spectrum. 5. Concluding remarks As we have seen above, the effect of Pauli paramagnetism is very small in a usual superconductor and appreciable only for an extremely small specimen. The most important effect may be a rather remarkable change of the gap

8 36 K. Maki in the energy spectrum which may be detectable by the measurement of the specific heat at lower temperatures. In conclusion the author wishes to thank Dr. T. Tsuneto for interesting discussions. References 1) K. Maki and T. Tsuneto, Prog. Theor. Phys. 31 (1964), ) A. M. Clogston, Phys. Rev. Letters 9 (1962), ) B. S. Chandrasekhar, Appl. Phys. Letters 1 (1962), 7. 4) K. Maki, Prog. Theor. Phys. 32 (1964), 5) A. A. Abrikosov and L. P. Gor'kov, ZETF 39 (1960), 1781.