- Email: [email protected]
Landau-Ginzburg equation for a superconductor containing magnetically ordered impurities
Physica 84B (1976) 90-101 © North-Holland Publishing Company
L A N D A U - G I N Z B U R G EQUATION F O R A SUPERCONDUCTOR CONTAINING MAGNETICALLY ORDERED IMPURITIES M. K A U F M A N and O. ENTIN-WOHLMA~ Department of Physics and Astronomy, TeI-A viv University, TeI-A viv, Israel Received 5 January 1976 Revised 9 March 1976
A superconducting alloy containing magnetic impurities, which may become magnetically ordered, is considered. It is found that an antiferromagnetic order is favoured. However, when the superconducting and magnetic transition temperatures are close to each other a large concentration of non-magnetic impurities causes the spontaneous magnetization to be uniform in the vicinity of both transition temperatures. The Landau Ginzburg free energy functional for this case is calculated and the phase-diagram is discussed. We find that for sufficiently strong exchange and spin-orbit scatterings there appears a mixed phase of the two order parameters.
1. Introduction A magnetic order may appear in an alloy containing localized magnetic impurities due to the indirect exchange interaction among the impurities via the conduction electrons [1 ]. This situation may occur also in a superconducting alloy, as discussed by Gor'kov and Rusinov [2]. They found that the superconducting transition temperature T c is reduced when a magnetic order appears. This reduction is caused by the depairing effect o f the spontaneous magnetization, which splits the Fermi levels o f electrons with opposite spins. On the other hand, the superconducting order parameter reduces the magnetic transition temperature T k compared to its value in the normal state. This happens since the opposite spins o f Cooper pairs oppose tile polarization o f the impurities spins. G o r ' k o v and Rusinov [2] assumed that the spontaneous magnetization which may appear in a superconductor containing magnetic impurities is uniform, i.e. they assumed a ferromagnetic order. However, we find that this is not generally the case, and superconductivity favours an antiferromagnetic order. This result can be explained as follows: The spontaneous magnetization o f the impurities depends on the degree of alignment o f the conduction electrons spins, that is, on the electronic spin susceptibility x(q). Consequently, at temperatures close to Tk, the important contribution to the spontaneous magnetization is from wavevectors q close to the one for which x(q) attains its maximum. But because of the polarization o f Cooper pairs [3], x(q) increases with q for small q, q "~ PF. Thus, X does not reach a maximum for q = 0 and the spontaneous magnetization is not uniform. However, we show in section 2 that when T k ~ T c in the vicinity o f T c and in the presence of a large concentration o f non-magnetic impurities (short mean free path), X attains its maximum for q ~ 0 and the assumption o f ferromagnetic order holds. We calculate in section 2 the magnetic transition temperature T k and the temperature dependence o f the spontaneous magnetization close to T k. In section 3 we calculate the superconducting transition temperature T c and derive the L a n d a u - G i n z b u r g equation for a superconducting alloy containing magnetic impurities. In section 4, the free energy functional is constructed. It is shown that it contains a biquadratic coupling term between the two order parameters. Such a functional was already discussed in the literature [ 4 - 6 ] . We discuss the possible phase diagrams which it implies, and in particular the effect o f the exchange interaction strength on them. We find that when the exchange interaction is weak, a first order transition line separates the magnetic and the 90
M. Kaufman and O. Entin-Wohlman/Landau-Ginzburg equation and magnetically ordered impurities
91
superconducting phases. When the exchange interaction is strong, a mixed phase of superconductivity and magnetism may appear. This possibility is analyzed, and a necessary condition for it is derived. We also examine the effect of a strong spin-orbit scattering, in connection with the experiments of Mathias et al. [7] on Y doped with GdOs. Section 5 contains our concluding remarks and a discussion of some limitations of our theory.
2. The magnetic transition temperature We consider a superconducting alloy containing magnetic and non-magnetic impurities and include their potential, spin-orbit and exchange scatterings of the conduction electrons. The magnetic impurities may become magnetically ordered [1, 2] due to the exchange interaction
U(r) = ~ u ( r - R]) o" S/, /
(2.1)
where S! is the spin of the magnetic impurity localized at R] and o is the electronic spin. As a result of this interaction, the impurities magnetic momenta are coupled via the conduction electrons, i.e. through the RKKY interaction. Assuming that the Fourier transform of u (r) is approximately constant, u (r) = u (0)8(r), the RKKY interaction is
u2(o) JfRKKY -
~. X(Ri -- R] )Si" Sj,
.2
(2.2)
t,] where "B is the Bohr magneton and X is the electronic spin susceptibility. The effective magnetic field acting on the impurity spin S i is u2(O)
.BHeff(Ri) -
.~
~. x(R i - Ri)(S]). 1
(2.3)
Assuming the spontaneous magnetization to be in the z-direction, we get
(S)
SB ~"Bneff(Ri)S~
i =
s\
T
]'
(2.4)
where B s is the Brillouin function (we use units4f = k B = c = 1). In the vicinity of the magnetic transition temperature Tk the spontaneous magnetization is small and: .aHeff(Ri)S/T "~ 1. Then (2.4) yields:
s(s + 1) u2(0) (Si) =
3
.2BTk ~ ' x ( R i - RI) (S/). /
(2.5)
Its Fourier transform is:
s(s + 1) u2(o) .2T k nx(q) (Sq),
(Sq)- T
(2.6)
where n is the concentration of the magnetic impurities. The transition temperature T k is defined as the highest
M. Kaufrnan and O. Entin-Btohlman/Landau-Ginzburg equation and magnetically ordered impurities
92
temperature for which a non-zero Fourier component (Sq) exists [8]
Tk -
S(S + 1) u2(O) 3 la2 n x(Q)
(2.7)
and Q is the wave-vector for which X is maximum. We have already found [3] that x(q) increases with q for q "~PF" This happens because the Cooper pairs polarization increases when the wavelength q-1 is sfiorter than the coherence length ~. In the region of very small q (compared to PF), the contribution of the "normal" electrons to × may be approximated by the Pauli susceptibility, independent of q. As q increases, the contribution of the "normal" electrons decreases with q, as in a gas of "normal" electrons. Therefore, ×(q) increases with q for q "~ PF and then decreases, as q becomes larger, thus attaining a maximum at some finite (small) Q. However, in the vicinity of T c, a large concentration of nonmagnetic impurities "pushes" Q towards zero. This results because the mean free path, and consequently the coherence length, becomes shorter, thus weakening the effect of the Cooper pairs polarization. In addition, close to T c the number o f "superconducting" electrons becomes smaller. We now estimate the range of temperatures in the vicinity of T c in which Q ~ 0. From ref. 3, eq. (2.18), we have that at T c - T ' ~ T c the contribution of the Cooper pairs to X, denoted Xs, is Xs(q) 2/a2N(0)
_ 1 - rrTc
~
A2
~
(Icol + 1/rs)2(Icol + 1/3r s + 2/3rso)
A2
+ q2p2 E (icol + l/rs)2(Icoi + 1/2r+)(lcol + l / 3 r s + 2/3rso) 2 ' q/PF ~ 1" 12m2 rrTc n =_oo
(2.8)
Here co = nTc(2n + 1), 1/r s, 1/rSO, 1/r 1 are the exchange, spin-orbit and potential scattering rates, respectively, l/r+ ==-1/r 1 + 1/rso + l/r s and N(0) is the density of states per unit energy at the Fermi surface. (For A = 0, this relation yields the Pauli susceptibility, since it was derived for q ~ PF)" The normal susceptibility Xn dependence on the mean free path, for small q, is negligible (see fig. 2a in ref. 9), and Xn is q Xn(q)=Ia2N(O){I+PF(1-q~F)lnl q + 2 p F I } ~ 2 / j 2 N ( 0 ) ( 1
q2 ,~,
12p2 ]
q/PF `¢ 1.
(2.9)
The temperature dependence was also neglected, since T21~2c ~ F ~ 1 . The q2 term in Xs [last term on the right-hand side of (2.8)], is approximately q2(A2/T2) (/1120/~0), where ll,/SO are the mean free paths due to potential and spin-orbit scatterings, respectively, and ll,/SO "~ G0 ("dirty" limit). The maximum of × is attained at Q ~ 0 if
(A)
2 11120
1
(2.1o)
Using (A/Tc) 2 ~ 1 - T/Tc, we obtain
1---
- -'~0
Tc - p2 ll it O
which gives the temperature range in the vicinity of T c in which x(Q) = x(O).
(2.11)
M. Kau fman and O. Entin-Wohlman/Landau-Ginzburg equation and magnetically ordered impurities
92
We see that the shorter are l 1 and lso, the broader is the range of temperatures for which x(Q) = x(0). Finally if the magnetic impurities concentration n is such that T k ~ Tc, at temperatures close enough to the transition temperatures, a uniform spontaneous magnetization may appear and Tk is obtained by replacing x(Q) with X(0) [3] in eq. (2.7)
Zk
:
'
Tk
[
'
1 -- 7rTl~
(Icol + 1/rs)2(Icoi + 1/r a)
]
+ (p ( A 4 ) .
(2.121
Here T~ ~S(S + 1)77(0)U2(0)n is the magnetic transition temperature o f the normal state (i.e. non-super conducting), lit a = 2 / 3 r s o + 1/3rs, co = (2n + l)7rT~ and the sum runs over all the integers. Thus T k decreases compared to its value T~ in the normal state. Using (2.3) and (2.4), the uniform magnetization is =
l (o) hx(O)S
h = nu(O)SB s ~
~
],
(2.13)
where h = nu(O) (S>. At T close to Tk, (2.13) becomes h
[Tk-
T
(S + 1) 2 + S 2 (2U(0)N(0)) 2 30
T2
h2
]
]
=0.
(2.14)
Note that since T k depends on A, eq. (2.14) contains both order parameters, h and A. This equation will be used in section 4 to construct the free energy functional. The uniform magnetization h and the exchange relaxation time r s are both related to the exchange interaction (2.1). However, the exchange relaxation time contains the localized spins correlation function, which does not vanish above the magnetic transition temperature, contrary tc the uniform magnetization. The exchange time r s is in second order in u [10], whereas h is proportional to u(0). Thus the calculation is valid for T close to Tk, where the spontaneous magnetization is small. On the other hand, r s, being a scattering time, is calculated in the Born approximation [2], where the lowest order is in second order in u.
3. The superconducting transition temperature In order to calculate the superconducting transition ter~perature Tc, we retain in eq. (A.19) of the appendix only linear terms in A 1
A*(r) g TcnN(O) S tcol
A*(r).
h2
(3.1)
+ ! + _ _ Ts
Icol + 1 Ta
The sum over co diverges logarithmically because for large n it behaves like ~n 1In. This divergence results from th~ fact that in the calculation o f Qto (see the appendix), the integration was performed over all momenta, without taking into account the cutoff corresponding to the energy cutoff coD . To eliminate this divergence we subtract and add the contribution o f the pure superconductor [ 11 ]. Eq. (3.1) thus becomes
M. Kaufman and O. Entin-Wohlman/Landau-Ginzburg equation and magnetically ordered impurities
94
in
7rTc
)
1 h+2
+ 1 +
[el + rs
(3.2)
h2
Icoi
where TO is the transition temperature of the "pure" superconductor. From (3.2) it is seen that the spontaneous magnetization decreases the superconducting transition temperature and that the spin-orbit scattering, which appears in 1 counteracts the effect of the spontaneous magnetization on T c. In the absence of spin-orbit scatterings, 1/f a --- 1/3rs, and (3.2) becomes:
Ira,
lnTO:TrTc~.( 1 Tc to Icol
tcol+ 1/3r s ) (Icol+ 1/rs)(Icol + 1/3r s ) + h 2 "
(3.3)
By neglecting l/r s compared to 1/rso , 1/r a = 2/3rSO , and since h 2 is small, (3.2) yields:
ln TO: TrTch2~ rc
1
(3.4)
to 6°2(16°1 + 2/3rSO)"
Equations (3.3) and (3.4) were derived by Gor'kov and Rusinov [2]. Using eq. (3.2), eq. (A.19) becomes
1
Tc
2
(Icoi + 1/rs)4 tA(r)12
12 nTc
(3.5)
(tcol + 1/rs)2(Icol + 1/2r+) b-~ A*(r) = 0.
The characteristic length for spatial variation of A(r) (the coherence length) is
T -~[TrTc~
OF( ,~(T) :-~- 1 - ~ee)
~
1
(Icoi + 1/rs)2(ico[ + 1/2r+)
]~
(3.6)
"
We can now check the Ansatz made in the appendix that A(r) varies more slowly in space than the normal Green's function (~n(r)). The range of (&n(r)) is found from (see the appendix) m
(&~ (r)) = - - 2nr exp (irp F sgn c o -
Icol~?O/VF)
X( exp (-irh sgnco/oF O .\ 0
OF/(2nT+
exp (-irh
) sgn co/vF
(3.7)
T/Tc)-},
to be VF/21eolr/~< l/r+). By comparing it to ~(T) ~(1 it is seen that for temperatures close enough to Tc, A(r) changes over distances which are much larger than the range of (~n(r)).
M. Kaufman and O. Entin-Wohlman/Landau-Ginzburg equation and magnetically ordered impurities
95
4. The free energy functional Here we construct the free energy functional which yields the equations for T k and T c derived above, and discuss the possible phase diagrams. In the Landau theory of second order phase transitions, the free energy functional of a system characterized by an order parameter ff is
F_Fo=aff2+_~ff4,
with/3>0.
(4.1)
Our system is characterized by two order parameters `5 and h. Consider the free energy functional FSM
FN = 0t`52 + ~Z `54 + ,),A2h2 4- ah 2 4 -Z~ h 4
(4.2)
with/3, 7, b > 0. Here FSMrefers to the free energy of a system with superconducting and magnetic order parameters, and F N - to the normal state. A and h have been assumed to be constant in space. The variation of FSM with respect to `5 and h gives (a +/3,5 2 + 7h 2) `5 = O,
(4.3)
(a + bh 2 + 7A2)h = 0.
(4.4)
To find the coefficients a,/3 and 7 we insert the expression for T c [eq. (3.2)] into (3.5) (the term proportional to a2A*(r)/~r2does not appear because A is constant in space), and the expression for T k [eq. (2.12)] into (2.4). Then, for small h and `5, i.e. in the vicinity of T k ~ T c, we obtain t
Tc
t
'
T - Tt~ a =- r~
~
b -
~' = 7rT
(16o[ + 1/7"s)4 '
(16ol + 1/rs)2(l~l + 1/'ra) '
(S + 1) 2 + S 2 [2u(0)N(0)] 2
'
30
rl~ 2
(4.5) '
where T c is the superconducting transition temperature for h = 0 and TI~ is the magnetic transition temperature fm A=0. The stationary points (`5, h) of the fuctional (4.2) are obtained from the solutions of eqs. (4.3) and (4.4). They are (a) h = `5 = 0; __
(b)`52=1~1 /3' (c)`52=0,
(d) A 2 -
FSM -- F N = 0,
normal phase;
h 2=0;
N-
h 2 = -b lal ;
FsM-F
FSM _ F N -
I~1 1 - 7lal/bl~l , /3 1 - 72 /b/3
mixed phase.
h2 .
ial 2
2/3' lal 2b2 '
superconducting phase;
magnetic phase;
lai 1 - ~'l~i//31al . . b 1 - ~[2/b/3 '
.
FSM - FN
1~12/2/3 + lal2/2b - I~llal~///3b 1 -- 72 /b/3 (4.6)
M. Kaufman and O. Entin-Wohlman/Landau-Ginzburg equation and magnetically ordered impurities
96
The next step is to determine the conditions for which the stationary points give the minimum free energy. We find (a) A superconducting phase exists for 71c~1//7ial > 1. (b) A magnetic phase exists for 71al/bl~l > 1. If both ")'lc~l/Nal and 71al/bl~l are larger than 1, the superconducting phase exists if its free energy is smaller than that of the magnetic phase, i.e. 1c~12//7> lal2/b while the magnetic phase exists if lal21b > la12//3. For lal21b = Ic~12//3, the free energies of the two phases are equal, and this equation gives a line of first order transitions. (c) A mixed phase exists if 7lc~l//Tlal < 1 and Tlal/bloel < 1. It follows that a necessary condition for the existence of a mixed phase is ,i2/b~ < 1. Tile transition between the mixed phase and the magnetic or the superconducting phases is o f the second order. At ylc~l//7tal = ylal/bl~l = 1 the transition is changed from a first order to a second order, i.e. a tricritical point. We now examine the effect of the exchange interaction strength u on the phase diagram in the neighbourhood of the intersection point of T c and T k curves in the (T, n) plane (n is the concentration of magnetic impurities). The explicit calculations were done with the assumption that only linear terms in 1/rsT and I/raT are retained. These assumptions are made for simplicity but it is expected that the main features of the phase diagram, caused by u, are not altered. The necessary condition for the existence of a mixed phase is 126 1+(1 + l/S) ~
~--2
1
-1.92 -+
0.01 ~ <
rrTra
~rs. I
(4.7)
1,
where ~"= 2rrsu(O)N(O) is the Born parameter [2]. Hence ~"must be larger than ~8 in order to ensure the existence of the mixed phase. Tile boundary lines of this phase are
( m - 1)TcT ~ T * - mT~,- T c p
wherem=~_2 I + ( I + I / S ) 2 63
,
[1+o9' 2
]
7rT \ r s
and ' ' T* - (n - 1)TcTI~
nT~
{ 0.93 where n = 2 1 +
Tc
rrTr s
0.96)
(4.8)
rrTr a
Note than when ~"increases (stronger exchange interaction) the mixed phase expands (fig. 1). Using (4.6d)we find that in the mixed phase A2 ~ T~ c T and h 2 ~ T T*. Since T* < T < Tc*,h 2 increases while A 2 tends to zero when T approaches Tc* (fig. 2). For a weaker exchange interaction, for which inequality (4.7) is reversed, the mixed phase disappears. In this case, the superconducting and the magnetic phases are separated by a first order transition line determined by the equation lal2/b = 1~12//3 (fig. 3). The first order transition line is
TI =
(1 - ~ / ~ ) T~T c , , , rc - V~ri
(4.9)
where
mn=~2 1+(1+1/S)2 31.5
1 + 3.85 ) . rrTr s
It is seen that the spin-orbit scattering has no influence whatsoever on the first order phase transition line.
M. Kaufman and O. Entin.Wohlman/Landau-Ginzburg equation and magnetically ordered impurities
97
&,h I
T~:
N
T~:
-
T T¢ T~
T
Fig. 2. Fig. 1. T
T~:
N
'Tk
n
Fig. 3. Fig. 1. Phase diagram in (T, n) plane, in the case where a mixed phase exists. Second order transitions lines separates the mixed phase from the superconducting and magnetic phases. Fig. 2. The temperature dependence of the order parameters (for a given concentration n) in the case where a mixed phase exists. Fig. 3. Phase diagram in (T, n) plane, in the case where a mixed phase does not exist. A first order transition line separates the magnetic and superconducting phases.
Let us now analyze the necessary condition for the existence of a mixed phase, 72/b/3 < 1, in the vicinity of the intersection point of the T k and T c curves in the (T, n) plane. Inserting the expressions for 7,/3 and b from (4.5) we get 607r3T 3 ~--2
1 + (1 + l/S) 2
{~, [([c°[ + 1/Ts)2 ([w[ + 1/~'a)]-1 } 2 Icol
<1,
T~T~T
c.
(4.10)
w~ (Icol + 1/rs)4 From (4.10) we see that a strong exchange interaction (large ~') and a strong spin-orbit scattering (large 1/ra) contribute to the appearance of the mixed phase by lowering 72/b/3. The role played by the spin-orbit scattering can be understood as follows: The spin-orbit scattering does not conserve the electronic spin. Thus the sliding o f the Fermi surface, produced by the spontaneous magnetization, is decreased. It follows that the opposing effect ot magnetism on superconductivity is weakened, and a mixed phase may appear. Strong spin-orbit scattering has been assumed [2] to explain the phase diagram obtained by Matthias et al. [7] in experiments on Y doped with GdOs. We consider this case in order to estimate the spin-orbit interaction
M. Kaufman and O. Entin-Wohlman/I.andau-Ginzburg equation and magnetically ordered impurities
98
strength necessary to ensure the appearance of a mixed phase. For this end we approximate (4.10) as follows: (a) the magnetic impurities concentration in the neighbourhood of the intersection point is such that it is sufficient to consider only linear terms in 1/rsT. Such a situation occurred in the experiment o f Watson et al. [12] and of Matthias et al. [7], where T c decreases approximately linearly with n till the intersection point (in agreement with the Abrikosov and Gor'kov [11] approximated expression T c = T~c- n/4rs). (b) The spin-orbit mean free path/SO is much shorter than the coherence length G0. We then obtain
3]
390.5 r_27r2T2~2 [ 1+(1+1/S)2 ~ 1,SO 1+-"(1. nT7 s
(4.11)
Note that at least in this approximation, the spin-spin correlation function which appears in 1/r s opposes the occurrence of the mixed phase. From eq. (4.11) we obtain nTTso < ~'{390.5 _x_ [1 + (1 + l/S)] )~1 ~ ~/14. Some values o f f and ~'/14 are listed in table I. In real metals, where ~"~ 0.2 [2], nT~-so must be smaller than 0.014 in order to ensure the occurrence of a mixed phase. This value corresponds to a very small spin-orbit mean free path,/SO ~ 10-2 G0.
5. Discussion We have studied a superconducting alloy, containing magnetic impurities which may become ordered. It was shown that for a concentration of magnetic impurities for which the two transition temperatures are close, at temperatures close enough to them, and in the presence of a large amount of non-magnetic impurities, the spontaneous magnetization is uniform. Assuming that this is the case, w'e derived the coefficients of the L a n d a u Ginzburg free energy functional of a system with superconducting and magnetic order parameters. The fluctuations o f the order parameters were neglected. As is well known, this approximation breaks down at temperatures close to the transition temperature, where the range of fluctuations becomes very large. In a pure superconductor, due to the long coherence length (G0 ~ 10-5 cm), the effect of the fluctuations becomes important only when 1 - TIT c is very small (e.g. 10 14 for tin [13] ). For an alloy, the size of a Cooper pair is smaller than G0, thus the temperature range on which the fluctuation effect can be neglected is reduced. In our case, to ensure the uniformity of the spontaneous magnetization, the non-magnetic impurities concentration is large (i.e. small "dirty" coherence length) and temperatures close to T c are considered. Moreover, for the magnetic order, the range of temperatures on which the fluctuations may be neglected is much smaller than that o f superconductivity [13]. These are strong limitations on the theory proposed in section 4. Examining the effect of the exchange interaction strength, it was found that for a weak interaction, a line of first order transitions separates the superconducting and magnetic phases, while for a strong interaction a mixed phase may appear in the vicinity of the intersection point of T c and T k curves [in the (T, n) plane]. The transition from the mixed phase to the superconducting or magnetic phases is of the second order. The spin-orbit interaction, which weakens the opposing effect of magnetism on superconductivity, favours the occurrence of a mixed phase. However, we find that a very short spin-orbit free path (/SO ~ 10-2~0) is needed to ensure the occurrence ,,f thie nh~o of this phase.
Table I ~"
0.1
0.2
1
2
4
~'/14
0.007
0.014
0.07
0.14
0.28
M. Kaufman and O. Entin-Wohlman/Landau-Ginzburg equation and magnetically ordered impurities
99
Acknowledgements We acknowledge useful conversations with Guy Deutscher and Shlomo Alexander.
Appendix Here we calculate the coefficients of the Landau-Ginzburg equation for a superconducting alloy in which ther~ is a uniform spontaneous magnetization. The spin-orbit, exchange and potential interactions are included. We assume that the magnetic (Tk) and the superconducting (Tc) transitions temperatures are close to each other. Since temperatures close to the transition temperatures are considered, A is small and it is convenient to work with the integral form of Gor'kov's equations which allows a "perturbation" type calculation [11, 14]. The differ. ential Gor'kov equations are written in a compact form by using a 4 X 4 matrix notations. [iw/33 + V2/2m +/a -- A(x) -- v(x, x')] G w ( x , x ' ) = 8(x - x').
(A.1)
Here
(i i)
(A.2)
where i is the 2 × 2 unit matrix,
A=
(o °
where ~ = (_o
(A.3)
o1) and the scattering potential is
(?x> o )
o(x, x') =
(A.4)
f~t(x', x)
where 17includes potential, spin-orbit and exchange scattering [10]. The scattering potential is non-local because of the spin-orbit interaction so that uG in (A.1) denotes an integral [3]. The 4 × 4 Green's function matrix is [2] X p)
O,~(x,x') =[~¢'~(x' [
\~+(~, x')
x')]\ , ~¢~(x, x) ]
t
,
where ffw(x, x') =if_to(x, x).
(A.5)
The description of the superconductor is completed by writing the self-consistency relation
zX*(x )R = g T ~.~'~ (x, x).
Here w = (2n + 1)TrT and g > 0 is the BCS coupling constant.
(A.6)
M. Kaufman and O. Entin-Wohlman/Landau-Ginzburg equation and magnetically ordered impurities
100
The normal Green's function satisfies (icofi 3 + V2/2m + ta - v(x, x'))Gn(x, x') = 6(x - x'),
(A.7)
where
~ ( ~ ~')
aco(~,x)= n
t
C \0
@n(x,x')) 0
(A.8)
and d.~ nco(x, x ' ) =.~ n_ co (x , , x). From (A.1) and (A.7) we obtain
G c o ( x , x, )_- G cmo ( x , x )' + f d3x '' Gn(x, x") A(x")Gco(x " , x )' .
(A.9)
Since A is small, an iteration of (A.9) in terms of IAI is done up to the third p o w e r . ~ -+ (x, x) is obtained from this approximation and (A.6). Making the Ansatz that A(r) varies in space much slower than the normal Green's function [11, 14] we find A*(r) = gT~. Q (q = 0, co) A*(r) -- ggT ~ 82Q(q' co) 1 co co 8q2
q=0
82A*(r) Or2
gr~_BwlA(r)12 A*(r),
(A.10)
O3
where
Q(q, co) = ½ t r f d 3 r' e-iq(r'-r) < ~ - 1 (#n_t(r'r)~fyn(r'r)>,
(A.11)
Bco =.[ d3r' d3r" d3r ''' (~-lfgn_t (r'r)~f~n (r . .r. .)g . . lcent.~_wtr . . . . .r. .)gNw(rcan . . . . . .r); .
(A.12)
and <) denotes an average over the impurities positions and their spin orientations. Q (q, co) was calculated before [10]. F o r q = 0 it is
Q(q : 0, co) = 7rN(0)
h2
icol+ ! + rs
-
(A.13)
-
Icol + 1 Ta
Since A(r) varies with r much slower than the normal Green's function, the term proportional to 82A*(r)/Or 2 is much smaller than the term proportional to A* [in the right-hand side of (A.10)]. Hence, we neglect in this term (~ 0 2 A , /Or2 ) the spontaneous magnetization contribution which is by itself Small because T k - T ~ T k ~ T, Then
82Q(q=O,w) 8q 2
-
7rN(0) ~-
1
(A.14)
02 (]coi + 1/rs)2([col + l / 2 r + ) "
The calculation of Bco is done as follows. Since IAI3 is much smaller than IAI, we neglect the spontaneous magnetization's contribution. We then use the averaging technique of Abrikosov and Gor'kov [11 ], and find B w in terms of <~n (p))
M. Kaufman and O. Entin-Wohlman/Landau-Ginzburg equation and magnetically ordered impurities 1
( ~ n (p)) = .
1CdT/ - -
101 (A.15)
~p
with r~ = 1 + 1/21wlr+, ~p = p2/2m - li and V~, the vertex correction Q (q = o, w)
Vw
(A.16)
OO(q:O,w)
Here
QO(q = 0, co) : f d3r ' ~-1 (.~n ( r' r))g(~n(r' r)).
(A.17)
A straightforward calculation yields
Bt.o
~--
nN(0)
Icol
2
(icoL + 1/~'s) 4"
(A. 18)
Finally, substituting expressions (A. 13), (A. 14) and (A. 18) into (A. 10), we obtain 1
A *(r) = gT 7rN(O) ~,
h2
A*(r) + 02 g T T r N ( O ) ~
|
I~o1 + ± + -I ~ 1- + 7" s
1
w
1 (
Ico[+
~s)2(
~)2A*(r) [col+
l
) 8 r ~
_
Ta
- ½gTTrN(O)~ .
IA(r)l 2 A*(r).
References [1] A. A. Abrikosov and L. P. Gor'kov, Soviet Phys. JETP 16 (1963) 1575. [2] L. P. Gor'kov and A. I. Rusinov, Soviet Phys. JETP 19 (1964) 922. [3] M. Kaufman and O. Entin-Wohlman, Physica 84B (1976) 77 (preceding paper). [4] Y. Imry, D. J. Scalapino and L. Gunther, Phys. Rev. B10 (1974) 2900. [5] K. Levin, D. L. Mills and S. L. Cunningham, Phys. Rev. B10 (1974) 3821. [6] O. Entin-Wohlman, G. Deutscher and S. Alexander, Phys. Rev. B12 (1975) 4854. [7] H. Suhl, B. T. Mathias and E. Corenzwit, J. Phys. Chem. Solids 11 (1959) 346. [8] Quantum Theory of Magnetism, R. M. White (McGraw Hill, New York, 1970). [9] P. G. de Gennes, Le Journal de Physique et le Radium. 23 (1962) 630. [10] O. Entin-Wohlman, Phys. Rev. B12 (1975) 4860. [11] A. A. Abrikosov and L. P. Gor'kov, Soviet Phys. JETP 12 (1961) 1243. [12] H. L. Watson, W. J. Keeler, B. J. Beaudry and D. K. Finnemore, J. Low Temp. Physics 12 (1973) 171. [13] L. Kadanoff et al., Rev. of Mod. Physics 39 (1967) 395. [14] N. R. Werthamer in Superconductivity, vol. 1, R. D. Parks, ed. (M. Dekker, New York, 1969).
(A.19)
L A N D A U - G I N Z B U R G EQUATION F O R A SUPERCONDUCTOR CONTAINING MAGNETICALLY ORDERED IMPURITIES M. K A U F M A N and O. ENTIN-WOHLMA~ Department of Physics and Astronomy, TeI-A viv University, TeI-A viv, Israel Received 5 January 1976 Revised 9 March 1976
A superconducting alloy containing magnetic impurities, which may become magnetically ordered, is considered. It is found that an antiferromagnetic order is favoured. However, when the superconducting and magnetic transition temperatures are close to each other a large concentration of non-magnetic impurities causes the spontaneous magnetization to be uniform in the vicinity of both transition temperatures. The Landau Ginzburg free energy functional for this case is calculated and the phase-diagram is discussed. We find that for sufficiently strong exchange and spin-orbit scatterings there appears a mixed phase of the two order parameters.
1. Introduction A magnetic order may appear in an alloy containing localized magnetic impurities due to the indirect exchange interaction among the impurities via the conduction electrons [1 ]. This situation may occur also in a superconducting alloy, as discussed by Gor'kov and Rusinov [2]. They found that the superconducting transition temperature T c is reduced when a magnetic order appears. This reduction is caused by the depairing effect o f the spontaneous magnetization, which splits the Fermi levels o f electrons with opposite spins. On the other hand, the superconducting order parameter reduces the magnetic transition temperature T k compared to its value in the normal state. This happens since the opposite spins o f Cooper pairs oppose tile polarization o f the impurities spins. G o r ' k o v and Rusinov [2] assumed that the spontaneous magnetization which may appear in a superconductor containing magnetic impurities is uniform, i.e. they assumed a ferromagnetic order. However, we find that this is not generally the case, and superconductivity favours an antiferromagnetic order. This result can be explained as follows: The spontaneous magnetization o f the impurities depends on the degree of alignment o f the conduction electrons spins, that is, on the electronic spin susceptibility x(q). Consequently, at temperatures close to Tk, the important contribution to the spontaneous magnetization is from wavevectors q close to the one for which x(q) attains its maximum. But because of the polarization o f Cooper pairs [3], x(q) increases with q for small q, q "~ PF. Thus, X does not reach a maximum for q = 0 and the spontaneous magnetization is not uniform. However, we show in section 2 that when T k ~ T c in the vicinity o f T c and in the presence of a large concentration o f non-magnetic impurities (short mean free path), X attains its maximum for q ~ 0 and the assumption o f ferromagnetic order holds. We calculate in section 2 the magnetic transition temperature T k and the temperature dependence o f the spontaneous magnetization close to T k. In section 3 we calculate the superconducting transition temperature T c and derive the L a n d a u - G i n z b u r g equation for a superconducting alloy containing magnetic impurities. In section 4, the free energy functional is constructed. It is shown that it contains a biquadratic coupling term between the two order parameters. Such a functional was already discussed in the literature [ 4 - 6 ] . We discuss the possible phase diagrams which it implies, and in particular the effect o f the exchange interaction strength on them. We find that when the exchange interaction is weak, a first order transition line separates the magnetic and the 90
M. Kaufman and O. Entin-Wohlman/Landau-Ginzburg equation and magnetically ordered impurities
91
superconducting phases. When the exchange interaction is strong, a mixed phase of superconductivity and magnetism may appear. This possibility is analyzed, and a necessary condition for it is derived. We also examine the effect of a strong spin-orbit scattering, in connection with the experiments of Mathias et al. [7] on Y doped with GdOs. Section 5 contains our concluding remarks and a discussion of some limitations of our theory.
2. The magnetic transition temperature We consider a superconducting alloy containing magnetic and non-magnetic impurities and include their potential, spin-orbit and exchange scatterings of the conduction electrons. The magnetic impurities may become magnetically ordered [1, 2] due to the exchange interaction
U(r) = ~ u ( r - R]) o" S/, /
(2.1)
where S! is the spin of the magnetic impurity localized at R] and o is the electronic spin. As a result of this interaction, the impurities magnetic momenta are coupled via the conduction electrons, i.e. through the RKKY interaction. Assuming that the Fourier transform of u (r) is approximately constant, u (r) = u (0)8(r), the RKKY interaction is
u2(o) JfRKKY -
~. X(Ri -- R] )Si" Sj,
.2
(2.2)
t,] where "B is the Bohr magneton and X is the electronic spin susceptibility. The effective magnetic field acting on the impurity spin S i is u2(O)
.BHeff(Ri) -
.~
~. x(R i - Ri)(S]). 1
(2.3)
Assuming the spontaneous magnetization to be in the z-direction, we get
(S)
SB ~"Bneff(Ri)S~
i =
s\
T
]'
(2.4)
where B s is the Brillouin function (we use units4f = k B = c = 1). In the vicinity of the magnetic transition temperature Tk the spontaneous magnetization is small and: .aHeff(Ri)S/T "~ 1. Then (2.4) yields:
s(s + 1) u2(0) (Si) =
3
.2BTk ~ ' x ( R i - RI) (S/). /
(2.5)
Its Fourier transform is:
s(s + 1) u2(o) .2T k nx(q) (Sq),
(Sq)- T
(2.6)
where n is the concentration of the magnetic impurities. The transition temperature T k is defined as the highest
M. Kaufrnan and O. Entin-Btohlman/Landau-Ginzburg equation and magnetically ordered impurities
92
temperature for which a non-zero Fourier component (Sq) exists [8]
Tk -
S(S + 1) u2(O) 3 la2 n x(Q)
(2.7)
and Q is the wave-vector for which X is maximum. We have already found [3] that x(q) increases with q for q "~PF" This happens because the Cooper pairs polarization increases when the wavelength q-1 is sfiorter than the coherence length ~. In the region of very small q (compared to PF), the contribution of the "normal" electrons to × may be approximated by the Pauli susceptibility, independent of q. As q increases, the contribution of the "normal" electrons decreases with q, as in a gas of "normal" electrons. Therefore, ×(q) increases with q for q "~ PF and then decreases, as q becomes larger, thus attaining a maximum at some finite (small) Q. However, in the vicinity of T c, a large concentration of nonmagnetic impurities "pushes" Q towards zero. This results because the mean free path, and consequently the coherence length, becomes shorter, thus weakening the effect of the Cooper pairs polarization. In addition, close to T c the number o f "superconducting" electrons becomes smaller. We now estimate the range of temperatures in the vicinity of T c in which Q ~ 0. From ref. 3, eq. (2.18), we have that at T c - T ' ~ T c the contribution of the Cooper pairs to X, denoted Xs, is Xs(q) 2/a2N(0)
_ 1 - rrTc
~
A2
~
(Icol + 1/rs)2(Icol + 1/3r s + 2/3rso)
A2
+ q2p2 E (icol + l/rs)2(Icoi + 1/2r+)(lcol + l / 3 r s + 2/3rso) 2 ' q/PF ~ 1" 12m2 rrTc n =_oo
(2.8)
Here co = nTc(2n + 1), 1/r s, 1/rSO, 1/r 1 are the exchange, spin-orbit and potential scattering rates, respectively, l/r+ ==-1/r 1 + 1/rso + l/r s and N(0) is the density of states per unit energy at the Fermi surface. (For A = 0, this relation yields the Pauli susceptibility, since it was derived for q ~ PF)" The normal susceptibility Xn dependence on the mean free path, for small q, is negligible (see fig. 2a in ref. 9), and Xn is q Xn(q)=Ia2N(O){I+PF(1-q~F)lnl q + 2 p F I } ~ 2 / j 2 N ( 0 ) ( 1
q2 ,~,
12p2 ]
q/PF `¢ 1.
(2.9)
The temperature dependence was also neglected, since T21~2c ~ F ~ 1 . The q2 term in Xs [last term on the right-hand side of (2.8)], is approximately q2(A2/T2) (/1120/~0), where ll,/SO are the mean free paths due to potential and spin-orbit scatterings, respectively, and ll,/SO "~ G0 ("dirty" limit). The maximum of × is attained at Q ~ 0 if
(A)
2 11120
1
(2.1o)
Using (A/Tc) 2 ~ 1 - T/Tc, we obtain
1---
- -'~0
Tc - p2 ll it O
which gives the temperature range in the vicinity of T c in which x(Q) = x(O).
(2.11)
M. Kau fman and O. Entin-Wohlman/Landau-Ginzburg equation and magnetically ordered impurities
92
We see that the shorter are l 1 and lso, the broader is the range of temperatures for which x(Q) = x(0). Finally if the magnetic impurities concentration n is such that T k ~ Tc, at temperatures close enough to the transition temperatures, a uniform spontaneous magnetization may appear and Tk is obtained by replacing x(Q) with X(0) [3] in eq. (2.7)
Zk
:
'
Tk
[
'
1 -- 7rTl~
(Icol + 1/rs)2(Icoi + 1/r a)
]
+ (p ( A 4 ) .
(2.121
Here T~ ~S(S + 1)77(0)U2(0)n is the magnetic transition temperature o f the normal state (i.e. non-super conducting), lit a = 2 / 3 r s o + 1/3rs, co = (2n + l)7rT~ and the sum runs over all the integers. Thus T k decreases compared to its value T~ in the normal state. Using (2.3) and (2.4), the uniform magnetization is =
l (o) hx(O)S
h = nu(O)SB s ~
~
],
(2.13)
where h = nu(O) (S>. At T close to Tk, (2.13) becomes h
[Tk-
T
(S + 1) 2 + S 2 (2U(0)N(0)) 2 30
T2
h2
]
]
=0.
(2.14)
Note that since T k depends on A, eq. (2.14) contains both order parameters, h and A. This equation will be used in section 4 to construct the free energy functional. The uniform magnetization h and the exchange relaxation time r s are both related to the exchange interaction (2.1). However, the exchange relaxation time contains the localized spins correlation function, which does not vanish above the magnetic transition temperature, contrary tc the uniform magnetization. The exchange time r s is in second order in u [10], whereas h is proportional to u(0). Thus the calculation is valid for T close to Tk, where the spontaneous magnetization is small. On the other hand, r s, being a scattering time, is calculated in the Born approximation [2], where the lowest order is in second order in u.
3. The superconducting transition temperature In order to calculate the superconducting transition ter~perature Tc, we retain in eq. (A.19) of the appendix only linear terms in A 1
A*(r) g TcnN(O) S tcol
A*(r).
h2
(3.1)
+ ! + _ _ Ts
Icol + 1 Ta
The sum over co diverges logarithmically because for large n it behaves like ~n 1In. This divergence results from th~ fact that in the calculation o f Qto (see the appendix), the integration was performed over all momenta, without taking into account the cutoff corresponding to the energy cutoff coD . To eliminate this divergence we subtract and add the contribution o f the pure superconductor [ 11 ]. Eq. (3.1) thus becomes
M. Kaufman and O. Entin-Wohlman/Landau-Ginzburg equation and magnetically ordered impurities
94
in
7rTc
)
1 h+2
+ 1 +
[el + rs
(3.2)
h2
Icoi
where TO is the transition temperature of the "pure" superconductor. From (3.2) it is seen that the spontaneous magnetization decreases the superconducting transition temperature and that the spin-orbit scattering, which appears in 1 counteracts the effect of the spontaneous magnetization on T c. In the absence of spin-orbit scatterings, 1/f a --- 1/3rs, and (3.2) becomes:
Ira,
lnTO:TrTc~.( 1 Tc to Icol
tcol+ 1/3r s ) (Icol+ 1/rs)(Icol + 1/3r s ) + h 2 "
(3.3)
By neglecting l/r s compared to 1/rso , 1/r a = 2/3rSO , and since h 2 is small, (3.2) yields:
ln TO: TrTch2~ rc
1
(3.4)
to 6°2(16°1 + 2/3rSO)"
Equations (3.3) and (3.4) were derived by Gor'kov and Rusinov [2]. Using eq. (3.2), eq. (A.19) becomes
1
Tc
2
(Icoi + 1/rs)4 tA(r)12
12 nTc
(3.5)
(tcol + 1/rs)2(Icol + 1/2r+) b-~ A*(r) = 0.
The characteristic length for spatial variation of A(r) (the coherence length) is
T -~[TrTc~
OF( ,~(T) :-~- 1 - ~ee)
~
1
(Icoi + 1/rs)2(ico[ + 1/2r+)
]~
(3.6)
"
We can now check the Ansatz made in the appendix that A(r) varies more slowly in space than the normal Green's function (~n(r)). The range of (&n(r)) is found from (see the appendix) m
(&~ (r)) = - - 2nr exp (irp F sgn c o -
Icol~?O/VF)
X( exp (-irh sgnco/oF O .\ 0
OF/(2nT+
exp (-irh
) sgn co/vF
(3.7)
T/Tc)-},
to be VF/21eolr/~< l/r+). By comparing it to ~(T) ~(1 it is seen that for temperatures close enough to Tc, A(r) changes over distances which are much larger than the range of (~n(r)).
M. Kaufman and O. Entin-Wohlman/Landau-Ginzburg equation and magnetically ordered impurities
95
4. The free energy functional Here we construct the free energy functional which yields the equations for T k and T c derived above, and discuss the possible phase diagrams. In the Landau theory of second order phase transitions, the free energy functional of a system characterized by an order parameter ff is
F_Fo=aff2+_~ff4,
with/3>0.
(4.1)
Our system is characterized by two order parameters `5 and h. Consider the free energy functional FSM
FN = 0t`52 + ~Z `54 + ,),A2h2 4- ah 2 4 -Z~ h 4
(4.2)
with/3, 7, b > 0. Here FSMrefers to the free energy of a system with superconducting and magnetic order parameters, and F N - to the normal state. A and h have been assumed to be constant in space. The variation of FSM with respect to `5 and h gives (a +/3,5 2 + 7h 2) `5 = O,
(4.3)
(a + bh 2 + 7A2)h = 0.
(4.4)
To find the coefficients a,/3 and 7 we insert the expression for T c [eq. (3.2)] into (3.5) (the term proportional to a2A*(r)/~r2does not appear because A is constant in space), and the expression for T k [eq. (2.12)] into (2.4). Then, for small h and `5, i.e. in the vicinity of T k ~ T c, we obtain t
Tc
t
'
T - Tt~ a =- r~
~
b -
~' = 7rT
(16o[ + 1/7"s)4 '
(16ol + 1/rs)2(l~l + 1/'ra) '
(S + 1) 2 + S 2 [2u(0)N(0)] 2
'
30
rl~ 2
(4.5) '
where T c is the superconducting transition temperature for h = 0 and TI~ is the magnetic transition temperature fm A=0. The stationary points (`5, h) of the fuctional (4.2) are obtained from the solutions of eqs. (4.3) and (4.4). They are (a) h = `5 = 0; __
(b)`52=1~1 /3' (c)`52=0,
(d) A 2 -
FSM -- F N = 0,
normal phase;
h 2=0;
N-
h 2 = -b lal ;
FsM-F
FSM _ F N -
I~1 1 - 7lal/bl~l , /3 1 - 72 /b/3
mixed phase.
h2 .
ial 2
2/3' lal 2b2 '
superconducting phase;
magnetic phase;
lai 1 - ~'l~i//31al . . b 1 - ~[2/b/3 '
.
FSM - FN
1~12/2/3 + lal2/2b - I~llal~///3b 1 -- 72 /b/3 (4.6)
M. Kaufman and O. Entin-Wohlman/Landau-Ginzburg equation and magnetically ordered impurities
96
The next step is to determine the conditions for which the stationary points give the minimum free energy. We find (a) A superconducting phase exists for 71c~1//7ial > 1. (b) A magnetic phase exists for 71al/bl~l > 1. If both ")'lc~l/Nal and 71al/bl~l are larger than 1, the superconducting phase exists if its free energy is smaller than that of the magnetic phase, i.e. 1c~12//7> lal2/b while the magnetic phase exists if lal21b > la12//3. For lal21b = Ic~12//3, the free energies of the two phases are equal, and this equation gives a line of first order transitions. (c) A mixed phase exists if 7lc~l//Tlal < 1 and Tlal/bloel < 1. It follows that a necessary condition for the existence of a mixed phase is ,i2/b~ < 1. Tile transition between the mixed phase and the magnetic or the superconducting phases is o f the second order. At ylc~l//7tal = ylal/bl~l = 1 the transition is changed from a first order to a second order, i.e. a tricritical point. We now examine the effect of the exchange interaction strength u on the phase diagram in the neighbourhood of the intersection point of T c and T k curves in the (T, n) plane (n is the concentration of magnetic impurities). The explicit calculations were done with the assumption that only linear terms in 1/rsT and I/raT are retained. These assumptions are made for simplicity but it is expected that the main features of the phase diagram, caused by u, are not altered. The necessary condition for the existence of a mixed phase is 126 1+(1 + l/S) ~
~--2
1
-1.92 -+
0.01 ~ <
rrTra
~rs. I
(4.7)
1,
where ~"= 2rrsu(O)N(O) is the Born parameter [2]. Hence ~"must be larger than ~8 in order to ensure the existence of the mixed phase. Tile boundary lines of this phase are
( m - 1)TcT ~ T * - mT~,- T c p
wherem=~_2 I + ( I + I / S ) 2 63
,
[1+o9' 2
]
7rT \ r s
and ' ' T* - (n - 1)TcTI~
nT~
{ 0.93 where n = 2 1 +
Tc
rrTr s
0.96)
(4.8)
rrTr a
Note than when ~"increases (stronger exchange interaction) the mixed phase expands (fig. 1). Using (4.6d)we find that in the mixed phase A2 ~ T~ c T and h 2 ~ T T*. Since T* < T < Tc*,h 2 increases while A 2 tends to zero when T approaches Tc* (fig. 2). For a weaker exchange interaction, for which inequality (4.7) is reversed, the mixed phase disappears. In this case, the superconducting and the magnetic phases are separated by a first order transition line determined by the equation lal2/b = 1~12//3 (fig. 3). The first order transition line is
TI =
(1 - ~ / ~ ) T~T c , , , rc - V~ri
(4.9)
where
mn=~2 1+(1+1/S)2 31.5
1 + 3.85 ) . rrTr s
It is seen that the spin-orbit scattering has no influence whatsoever on the first order phase transition line.
M. Kaufman and O. Entin.Wohlman/Landau-Ginzburg equation and magnetically ordered impurities
97
&,h I
T~:
N
T~:
-
T T¢ T~
T
Fig. 2. Fig. 1. T
T~:
N
'Tk
n
Fig. 3. Fig. 1. Phase diagram in (T, n) plane, in the case where a mixed phase exists. Second order transitions lines separates the mixed phase from the superconducting and magnetic phases. Fig. 2. The temperature dependence of the order parameters (for a given concentration n) in the case where a mixed phase exists. Fig. 3. Phase diagram in (T, n) plane, in the case where a mixed phase does not exist. A first order transition line separates the magnetic and superconducting phases.
Let us now analyze the necessary condition for the existence of a mixed phase, 72/b/3 < 1, in the vicinity of the intersection point of the T k and T c curves in the (T, n) plane. Inserting the expressions for 7,/3 and b from (4.5) we get 607r3T 3 ~--2
1 + (1 + l/S) 2
{~, [([c°[ + 1/Ts)2 ([w[ + 1/~'a)]-1 } 2 Icol
<1,
T~T~T
c.
(4.10)
w~ (Icol + 1/rs)4 From (4.10) we see that a strong exchange interaction (large ~') and a strong spin-orbit scattering (large 1/ra) contribute to the appearance of the mixed phase by lowering 72/b/3. The role played by the spin-orbit scattering can be understood as follows: The spin-orbit scattering does not conserve the electronic spin. Thus the sliding o f the Fermi surface, produced by the spontaneous magnetization, is decreased. It follows that the opposing effect ot magnetism on superconductivity is weakened, and a mixed phase may appear. Strong spin-orbit scattering has been assumed [2] to explain the phase diagram obtained by Matthias et al. [7] in experiments on Y doped with GdOs. We consider this case in order to estimate the spin-orbit interaction
M. Kaufman and O. Entin-Wohlman/I.andau-Ginzburg equation and magnetically ordered impurities
98
strength necessary to ensure the appearance of a mixed phase. For this end we approximate (4.10) as follows: (a) the magnetic impurities concentration in the neighbourhood of the intersection point is such that it is sufficient to consider only linear terms in 1/rsT. Such a situation occurred in the experiment o f Watson et al. [12] and of Matthias et al. [7], where T c decreases approximately linearly with n till the intersection point (in agreement with the Abrikosov and Gor'kov [11] approximated expression T c = T~c- n/4rs). (b) The spin-orbit mean free path/SO is much shorter than the coherence length G0. We then obtain
3]
390.5 r_27r2T2~2 [ 1+(1+1/S)2 ~ 1,SO 1+-"(1. nT7 s
(4.11)
Note that at least in this approximation, the spin-spin correlation function which appears in 1/r s opposes the occurrence of the mixed phase. From eq. (4.11) we obtain nTTso < ~'{390.5 _x_ [1 + (1 + l/S)] )~1 ~ ~/14. Some values o f f and ~'/14 are listed in table I. In real metals, where ~"~ 0.2 [2], nT~-so must be smaller than 0.014 in order to ensure the occurrence of a mixed phase. This value corresponds to a very small spin-orbit mean free path,/SO ~ 10-2 G0.
5. Discussion We have studied a superconducting alloy, containing magnetic impurities which may become ordered. It was shown that for a concentration of magnetic impurities for which the two transition temperatures are close, at temperatures close enough to them, and in the presence of a large amount of non-magnetic impurities, the spontaneous magnetization is uniform. Assuming that this is the case, w'e derived the coefficients of the L a n d a u Ginzburg free energy functional of a system with superconducting and magnetic order parameters. The fluctuations o f the order parameters were neglected. As is well known, this approximation breaks down at temperatures close to the transition temperature, where the range of fluctuations becomes very large. In a pure superconductor, due to the long coherence length (G0 ~ 10-5 cm), the effect of the fluctuations becomes important only when 1 - TIT c is very small (e.g. 10 14 for tin [13] ). For an alloy, the size of a Cooper pair is smaller than G0, thus the temperature range on which the fluctuation effect can be neglected is reduced. In our case, to ensure the uniformity of the spontaneous magnetization, the non-magnetic impurities concentration is large (i.e. small "dirty" coherence length) and temperatures close to T c are considered. Moreover, for the magnetic order, the range of temperatures on which the fluctuations may be neglected is much smaller than that o f superconductivity [13]. These are strong limitations on the theory proposed in section 4. Examining the effect of the exchange interaction strength, it was found that for a weak interaction, a line of first order transitions separates the superconducting and magnetic phases, while for a strong interaction a mixed phase may appear in the vicinity of the intersection point of T c and T k curves [in the (T, n) plane]. The transition from the mixed phase to the superconducting or magnetic phases is of the second order. The spin-orbit interaction, which weakens the opposing effect of magnetism on superconductivity, favours the occurrence of a mixed phase. However, we find that a very short spin-orbit free path (/SO ~ 10-2~0) is needed to ensure the occurrence ,,f thie nh~o of this phase.
Table I ~"
0.1
0.2
1
2
4
~'/14
0.007
0.014
0.07
0.14
0.28
M. Kaufman and O. Entin-Wohlman/Landau-Ginzburg equation and magnetically ordered impurities
99
Acknowledgements We acknowledge useful conversations with Guy Deutscher and Shlomo Alexander.
Appendix Here we calculate the coefficients of the Landau-Ginzburg equation for a superconducting alloy in which ther~ is a uniform spontaneous magnetization. The spin-orbit, exchange and potential interactions are included. We assume that the magnetic (Tk) and the superconducting (Tc) transitions temperatures are close to each other. Since temperatures close to the transition temperatures are considered, A is small and it is convenient to work with the integral form of Gor'kov's equations which allows a "perturbation" type calculation [11, 14]. The differ. ential Gor'kov equations are written in a compact form by using a 4 X 4 matrix notations. [iw/33 + V2/2m +/a -- A(x) -- v(x, x')] G w ( x , x ' ) = 8(x - x').
(A.1)
Here
(i i)
(A.2)
where i is the 2 × 2 unit matrix,
A=
(o °
where ~ = (_o
(A.3)
o1) and the scattering potential is
(?x> o )
o(x, x') =
(A.4)
f~t(x', x)
where 17includes potential, spin-orbit and exchange scattering [10]. The scattering potential is non-local because of the spin-orbit interaction so that uG in (A.1) denotes an integral [3]. The 4 × 4 Green's function matrix is [2] X p)
O,~(x,x') =[~¢'~(x' [
\~+(~, x')
x')]\ , ~¢~(x, x) ]
t
,
where ffw(x, x') =if_to(x, x).
(A.5)
The description of the superconductor is completed by writing the self-consistency relation
zX*(x )R = g T ~.~'~ (x, x).
Here w = (2n + 1)TrT and g > 0 is the BCS coupling constant.
(A.6)
M. Kaufman and O. Entin-Wohlman/Landau-Ginzburg equation and magnetically ordered impurities
100
The normal Green's function satisfies (icofi 3 + V2/2m + ta - v(x, x'))Gn(x, x') = 6(x - x'),
(A.7)
where
~ ( ~ ~')
aco(~,x)= n
t
C \0
@n(x,x')) 0
(A.8)
and d.~ nco(x, x ' ) =.~ n_ co (x , , x). From (A.1) and (A.7) we obtain
G c o ( x , x, )_- G cmo ( x , x )' + f d3x '' Gn(x, x") A(x")Gco(x " , x )' .
(A.9)
Since A is small, an iteration of (A.9) in terms of IAI is done up to the third p o w e r . ~ -+ (x, x) is obtained from this approximation and (A.6). Making the Ansatz that A(r) varies in space much slower than the normal Green's function [11, 14] we find A*(r) = gT~. Q (q = 0, co) A*(r) -- ggT ~ 82Q(q' co) 1 co co 8q2
q=0
82A*(r) Or2
gr~_BwlA(r)12 A*(r),
(A.10)
O3
where
Q(q, co) = ½ t r f d 3 r' e-iq(r'-r) < ~ - 1 (#n_t(r'r)~fyn(r'r)>,
(A.11)
Bco =.[ d3r' d3r" d3r ''' (~-lfgn_t (r'r)~f~n (r . .r. .)g . . lcent.~_wtr . . . . .r. .)gNw(rcan . . . . . .r); .
(A.12)
and <) denotes an average over the impurities positions and their spin orientations. Q (q, co) was calculated before [10]. F o r q = 0 it is
Q(q : 0, co) = 7rN(0)
h2
icol+ ! + rs
-
(A.13)
-
Icol + 1 Ta
Since A(r) varies with r much slower than the normal Green's function, the term proportional to 82A*(r)/Or 2 is much smaller than the term proportional to A* [in the right-hand side of (A.10)]. Hence, we neglect in this term (~ 0 2 A , /Or2 ) the spontaneous magnetization contribution which is by itself Small because T k - T ~ T k ~ T, Then
82Q(q=O,w) 8q 2
-
7rN(0) ~-
1
(A.14)
02 (]coi + 1/rs)2([col + l / 2 r + ) "
The calculation of Bco is done as follows. Since IAI3 is much smaller than IAI, we neglect the spontaneous magnetization's contribution. We then use the averaging technique of Abrikosov and Gor'kov [11 ], and find B w in terms of <~n (p))
M. Kaufman and O. Entin-Wohlman/Landau-Ginzburg equation and magnetically ordered impurities 1
( ~ n (p)) = .
1CdT/ - -
101 (A.15)
~p
with r~ = 1 + 1/21wlr+, ~p = p2/2m - li and V~, the vertex correction Q (q = o, w)
Vw
(A.16)
OO(q:O,w)
Here
QO(q = 0, co) : f d3r ' ~-1 (.~n ( r' r))g(~n(r' r)).
(A.17)
A straightforward calculation yields
Bt.o
~--
nN(0)
Icol
2
(icoL + 1/~'s) 4"
(A. 18)
Finally, substituting expressions (A. 13), (A. 14) and (A. 18) into (A. 10), we obtain 1
A *(r) = gT 7rN(O) ~,
h2
A*(r) + 02 g T T r N ( O ) ~
|
I~o1 + ± + -I ~ 1- + 7" s
1
w
1 (
Ico[+
~s)2(
~)2A*(r) [col+
l
) 8 r ~
_
Ta
- ½gTTrN(O)~ .
IA(r)l 2 A*(r).
References [1] A. A. Abrikosov and L. P. Gor'kov, Soviet Phys. JETP 16 (1963) 1575. [2] L. P. Gor'kov and A. I. Rusinov, Soviet Phys. JETP 19 (1964) 922. [3] M. Kaufman and O. Entin-Wohlman, Physica 84B (1976) 77 (preceding paper). [4] Y. Imry, D. J. Scalapino and L. Gunther, Phys. Rev. B10 (1974) 2900. [5] K. Levin, D. L. Mills and S. L. Cunningham, Phys. Rev. B10 (1974) 3821. [6] O. Entin-Wohlman, G. Deutscher and S. Alexander, Phys. Rev. B12 (1975) 4854. [7] H. Suhl, B. T. Mathias and E. Corenzwit, J. Phys. Chem. Solids 11 (1959) 346. [8] Quantum Theory of Magnetism, R. M. White (McGraw Hill, New York, 1970). [9] P. G. de Gennes, Le Journal de Physique et le Radium. 23 (1962) 630. [10] O. Entin-Wohlman, Phys. Rev. B12 (1975) 4860. [11] A. A. Abrikosov and L. P. Gor'kov, Soviet Phys. JETP 12 (1961) 1243. [12] H. L. Watson, W. J. Keeler, B. J. Beaudry and D. K. Finnemore, J. Low Temp. Physics 12 (1973) 171. [13] L. Kadanoff et al., Rev. of Mod. Physics 39 (1967) 395. [14] N. R. Werthamer in Superconductivity, vol. 1, R. D. Parks, ed. (M. Dekker, New York, 1969).
(A.19)