RKKY indirect exchange in low-dimensional superconductors

0038-1098/93 $6.00 + .00 Pergamon Press Ltd

Solid State Communications, Vol. 88, No. 3, pp. 257-261, 1993. Printed in Great Britain.

R K K Y INDIRECT E X C H A N G E IN LOW-DIMENSIONAL SUPERCONDUCTORS L.R. Tagirov Kazan University, Physics Department, Kazan, 420008, Russia

(Received 16 April 1993; accepted for publication by B. Lundqvist) The indirect exchange interaction between localized moments (LMs), mediated by one- and two-dimensional superconducting electron gas, is calculated by means of thermodynamic Green functions. The interaction potential is expressed in terms of higher transcendental functions of distance R between LMs. The asymptotic behavior at large R is presented for 1D and 2D cases. It is shown that an additional long range potential appears in the superconducting state as compared with the normal one. 1. I N T R O D U C T I O N THE RECENTLY discovered high-temperature superconductors (HTSC) have very anisotropic physical properties. For example, in YBa2Cu3Oystructure compounds the almost two-dimensional conductivity is accomplished by copper-oxygen planes nearby the Y ions or the rare earth ions, which substitute Y in the structure unit. The observation of Gd antiferromagnetic ordering in GdBa2Cu307 with a N6el temperature TN = 2.24K [1], which is considerably lower than the superconducting transition temperature T~, attracts the attention to investigate magnetic interactions mediated by two-dimensional (in general, by lowdimensional) superconducting electrons. The Hamiltonian of exchange interaction between two localized spins $1 and $2, placed at a distance R between them, can be written as = -J(R)SIS>

(1)

For the R K K Y interaction based on the local exchange between a localized spin Sl and a conduction electron spin cri

Here G~(R) and F,~(R) are the "normal" and "anomalous" Green functions of a superconductor, respectively, Jsf is the coupling constant of conduction electrons and localized spins, w = 7rT(2n + 1) is the Matsubara fermion frequency, n = 0, +1, + 2 , . . . , and N is the number of atoms in a volume. Green functions can be calculated as Fourier-transforms of standard momentum dependent Green functions of a superconductor [3]: G ~ ( R ) = f dP.G~(p)eiPR; J (27r)"

iW + {p G~(p)=

w2+A 2+{2,

(4)

A F,0(p) - ¢° 2 -F- A 2 q- { 2 ' where (p = ep - er is the electron band energy with respect to the Fermi energy eF, A -- A(T) is the gap in the excitation spectrum of a superconductor. Hereafter, we assume the band to be parabolic: ep = p2/2m with an effective electron mass m, and begin specific calculations in the one-dimensional (1D) case.

J,: -- - ~ Sio" i.

(2)

2. 1D-SUPERCONDUCTOR

The interaction potential J(R) in equation (1) can be expressed via thermodynamic Green functions (see, e.g., [2], where the R K K Y exchange in threedimensional superconductor has been calculated):

First of all, we convert the one-dimensional integral (4) to the positive half-axis and change variable from p to ~ by the approximation

J(R) =

where Pr and vz are the Fermi momentum and velocity, respectively. Then closing the integration path on the upper half-plane for the positive-sign exponent and on the lower half-plane for the

2N 2

TE{G,,,(R)G,,,(-R ) + F~o(R)Fu(-R)}. gl

(3)

= (p2 _ p~)/2m ~- vF(p - PF),

257

(5)

RKKY INDIRECT EXCHANGE

258

negative-sign exponent we express the integrals via residues of integrand on its simple poles (with eF ~ oo the intendation contribution vanishes): Gu(R) = ___/~

co

Vvl, /(co2 + A2)

Vol. 88, No. 3

2 J(x)

COS(pFR)

(eJ;Zcos2

1-

qgF x

l+-z---|

exp -

/,

R>>~0, (ll)

+isin(pFR)} e x p ( ~/(co2UF_~A 2) R),

i x/(co2 + A2) exp (

F (R) -

(6) These expressions are valid for pFR >> 1, which is the consequence of approximation (5). Substitution of equation (6) in equation (3) gives the interaction potential in the 1D-case: 1

Jsf

2

where ne is the number of free electrons per atom, is the coherence length, q~~_ 0.577. Note here, that at low temperatures the RKKY range function in equations (8) and (9) is only weakly temperature dependent via the temperature dependence of the order parameter A(T). The temperature-dependent corrections are proportional to at least QrT/A) 3. The first term in equations (8) and (10) is just the RKKY interaction potential in the normal conducting one-dimensional metal first derived by Larsen [4] and rederived recently by Yafet [5] in the context of HTSC. Other terms c( A represent the additional long range potential arising in the superconducting phase.

~o "~ Vr/ZcA(T = 0)

co2

z XVF1V/

"~"w ~t- A 2

x e x p ( - 2-~R + . Ux/(co2 F

A2)),

(7) 3.2D-SUPERCONDUCTOR

where the odd term in co cancelled upon summation. Together with definition (1) the expression (7) is the principal result for the ID-superconductor. The remaining summation in equation (7) can be done analytically in two limiting cases: low temperatures (T << Tc) and temperatures close to superconducting transition (Tc - T < < Tc), and give the following results (see Appendix for details):

7r

T<< Tc, x = pF R :

Tcn~J~

x

j(x) --

f l

ln~-~eF+-y-

1 [ e:tZizsin ~d~ = Jo(z)q= iEo(z) 0

7rA

× exp(--~eA),

, p~l<
R >> ~0;

(12)

for the Anger J0(z) and Weber E0(z) functions ([6], Vol. 2, Ch. 7) and their relations with Bessel function of the first kind Jo(z) = J0(z), and Struve function H0(z)---E0(z) ([7], Ch. 12) we obtain the Green functions of a 2D-superconductor:

7rn2j~cos2x~(Trer~( 9eF'~ 4e F 2x \ 2 x A ] 1 - 8xAJ (9)

G~(R) = -

x/(wf-+ A2)

× [ReJ0(p~R) - Im n0(p,oR)]

T c - T << Tc : S(x) ~- ~

In the two-dimensional case we evaluate the integral (4) using polar coordinates: dp = p d p d 0 . First we change the variable by equation (5) and integrate over ~ closing the integration path on the upper half-plane for -rr/2 < ~b_< rr/2 [positive value of cos 0 in the exponent of equation (4)] and on the lower half-plane for Ir/2 < 4~-< 3~r/2 (negative value of cos ~b).Then using a standard integral representation

f l

cos 2 x / 2 x

+ i[ImJo(p~g) + ReHo(p~R)]},

(13)

4eF

mA

X [2

F~,(R) = 2x/(w2 + A2 ) [ReJo(p~,R ) - ImH0(p~R)],

7rTx(2-1nTrTx'~]} "2--~eF]J

pF l << R << ~0;

(14) (10)

p~ -=pF+ i(co2 d- m2)l/2/'u F.

(15)

Vol. 88, No. 3

RKKY INDIRECT EXCHANGE

Substitution of equations (13) and (14) in equation (3) gives the interaction potential j2 m2

J(g)

( w2 _ A2

= ~8N T Z , i w2 ~ ~-~ [Re J°(P~'R)

259

[~12= 1+o32/4e~,

~5= ~/(w2+A2);

(21)

PI = P' - P" - QI - Q" = 1 + ~

(')

1-

+ o ~-~ ,

[ImJo(p~oR )

- Im Ho(p~.R)] 2 -

P2 = P' + P" + Q' - Q" + Re H0(p=R)]2 }.

(16)

Due to the condition pFR >> 1 one can use the asymptotic expressions for the transcendental function Jo(z) and H0(z) ([7], Chs. 9 and 12, respectively):

Jo(z) =~/ ( 2 ) { P(z) cos(z - Tr/4) -

a(z) sin(z - 7r/4)},

(17)

Ho(z) = w/ ( 2 ) { P(z) sin(z - Tr/4) + Q(z) cos(z - 7r/4)}

+1~ = r(l~---k)(z/2) 2k+l' r(k + 1/2)

(18)

9 2 + o (~4) , P(z) = 1 2!(8z) 19.25

a(z)=-~+3!(8z)----- 3

o

(1)

,

(19)

I'(n) is the Euler's gamma-function and z = p,,,R is of complex nature due to equation (15). After the extraction of real and imaginary parts of equations (17) and (18) and substitution of results into equation (16) we arrive at the cumbersome expression, but the ieading contribution in inverse powers of x = pFR >> 1 is

1 (Jsfm'~ 2T_T_~7.,e -x~/'e { J(x) ~

\ 2N J 7rx/--~'~ lel 2 _

2P~eEsin2x

(p• _ p2) cos 2x a3 [(p1z _ p~) sin 2x + 2PI P2 cos 2x]

w2+A2

,=,

with P' and P" (Q' and Q') are the real and imaginary parts of P(z) [Q(z)] (19), respectively. It can be easily seen that convergence of the frequency summation in equation (20) is provided by exp (--XCO/eF) at w ,.~ £ r / X <<~ f-F, SO the terms with extra ~b/eF-multiplier will have extra x-l<< 1 dependence after summation at short distances R << f0, and extra factor Ale F << 1 at large distances R >> f0. We set 1~[ = 1 in equations (20)-(22) by the same reason. After these simplifications the summation in equation (2) can be done analytically by the same way as in the Appendix. The next distinct contribution in J(R) from equations (16)-(19) has a prefactor x -3/2 and after summation the leading contribution is proportional to X-5/2 at distances R << ~0 [compare with the first term in equation (23)], and to x-2(A/eF) -1/2 at distances R >> ~0 [compare with the first term in equation (24)], that is why we neglect this contribution. Finally, the last distinct contribution comes from the inverse powers of z term in asymptotics of H0(z) (19). Due to the lack of exponential multiplier, the factor [~,[-4 is of crucial importance in providing the convergence of summation. The resulting contribution ~ x-Z(A/£F) 2 or x-2(TrT/eF)2 and is small with respect to corresponding contribution kept in equations (23)-(26). Taking into account the above discussion we obtain the RKKY interaction potential in 2Dsuperconductor (x = pFR):

J(x)

"~n2eg~~sinZx _TrA 1 + sinZx -

lel2(e? +

eF

_1_ + sin 2x (2Pl P2 - ~

+o (,) ,

8x112 1+

T<
£F

,,2

=1

[,(2x)

2

8x

e,~

[ ,+ (inCa

(P~ -

/j

x (2 + sin2x)] },

pF I

~R~0

,

(23)

260 J(x)

RKKY INDIRECT EXCHANGE

"n232f ( rceF~,/2( -- eF \ 2 x A / x

1+

2x J

sin2x (2x) 2 exp

A

4xe F xA

4. 5. 6.

R>>~0, (24)

7. T c - T << T~ :

8.

neJsf sin 2x

7rT 2eF

a(x) ---g/L

x [82

1 + sin 2x X

7rTx(1-1nzrTx' 2-~CF,/J~]c--r

1 /rrT\21"A\2/ ~rTx 2 t~-F ) t ~) tin 2---~e-

J(x)

--

£F

\ £F ,] [

The aim of this Appendix is to perform the frequency summation in expression

A

2x

x exp(-

Tx),

\

eF /

21 + sin2x 2x

R >> ¢0

~v2

s0= (25)

~neJ~ ( ~r---Tx}~sin2x

Statistical Physics, Chapter 7, Section 34. Prentice-Hall, Englewood Cliffs, New York (1964). U. Larsen, Phys. Lett. A85, 471 (1981). Y. Yafet, Phys. Rev. B36, 3948 (1987). H. Bateman & A. Erdelyi, Higher Transcendental Functions, Vols. 1 and 2. McGraw-Hill, New York (1953). Handbook of Mathematical Functions (Edited by M. Abramovitz & I. Stegun), NBS (1964). B. Fischer & M.W. Klein, Phys. Rev. B l l , 2025 (1975).

4. A P P E N D I X

1\ ) )~'

pv I << R << ~0;

Vol. 88, No. 3

/26/

The first term in equations (23) and (25) is the interaction potential in the normal metal [4, 8], but other terms represent the additional interaction inherent to the superconducting phase. In particular interest, the term oc A in equation (23), and the difference of equation (24) and the normal state interaction potential [4, 8], have antiferromagnetic sign and favor or stabilize the N6el state in superconducting two-dimensional antiferromagnet [1]. Let us emphasize here the specific feature of HTSC compounds - because of much higher Tc (and A) and rather small eg the range of R K K Y interaction is considerably shortened by the exponential damping [see equations (24) and (26)] as compared with conventional superconductors. Of course, in HTSC systems like YBaCuO there exists the superexchange interaction between rare earth ions through the system of localized Cu spins, but this problem is beyond the scope of the present paper.

2R

2

Low temperatures: T<< TC. The condition ~rT/A << 1 provides the validity of Euler-Maclaurin summation formula ~_F(a+n)~_ n=0

F(x)dx+

F(a)

a

1 OF(a) (A2) 12 Oa Denoting y = 2RA/vF and using equation (A2) with a = 1/2 we obtain for equation (A1) SO-~ 5 ( 1 0 - I T ) + - ~ [ 1 + 26(1 + y)]e -yV(I+261, (A3) where 6 = (1/2)(rrT/A) 2 << 1, contribution I0 = /01 + 102 =

zero-temperature

t2dt / ~--7~,2exp [-yx/(1 + t2)] Jl+t

(A4)

0

and finite-temperature correction Ir =

~T/A I t2dt

l---~-~exp[-yv/(1 + t2)].

(A5)

0

REFERENCES 1. 2. 3.

J.O. Willis et al., J. Magn. Magn. Mater. 67, L139 (1987). B.I. Kochelaev, L.R. Tagirov & M.G. Khusainov, Soy. Phys. - JETP, 49, 291 (1979); 51,826 (1980). A.A. Abrikosov, L.P. Gor'kov & I.E. Dzyaloshinski, Methods of Quantum Field Theory in

Splitting equation (A4) on two integrals we get for the first one I01 = i d t e x p [ - Y x / ( 1 +fl)] = Kl(y);

(A6)

0

hereafter K~(y) is the McDonald function [7]. The

RKKY INDIRECT EXCHANGE

Vol. 88, N o . 3

remaining integral obeys the differential equation OIo2(Y)_

(A7)

Ko(y)

Oy

with boundary condition I02(Y= 0 ) = 7i"/2. The exact integration of equation (A7) ([7], Ch. 11) gives: 7f

261

where now y = 27rTR/vF, w = (2n + 1). Realizing that for R << ~0, 27rTR/ve >> 1, only n = 0, 4-1 terms in $2 and $3 relevant, and S1 is geometric progression, we get So ,-~ 2e-Y l -

\Tr// \ 1 +

.

(All)

Io2(y) = ~ - yro(y) Try[K0(y)L 1(y) + Kl (y)L0(y)], (A8) 2 where Lu (y) is the modified Struve function. The finite-temperature correction (A5) can be expressed via probability integral eft(z) ([7], Ch.7) Ir ~-

/ A \ 2 oo e-Y(2n+l )

erf [x/( y6)] - 6 l/2e-y6

e -y

=

Y

x(1

For p~l<< R << (0, the generating sum in the set (A10) is $3, the sums $2 and Sl can be obtained differentiating $3 with respect to y.

--

~y)

+

3x21/263/2 4y e-Y(l+6)"

(A9)

Thus, substitution of the formulae (A6) and (A8) in equation (A4) and then equations (A4) and (A9) in equation (A3) solves the problem of frequency summation in equation (A1) at low temperatures. Expansions in equation (A3) with conditions y << l, y6 << 1 correspond to p~l << R << ~0; with condition y >> l, but y6 << 1 correspond to R >> ~0. High temperatures: Tc - T << Tc. At T ~ Tc we expand So in powers of (A/TrT)2:

[Li2(e-Y) - Li2(-e-Y)],

(A12)

where Li2(z) is the Euler's dilogarithm. Using the relation between Li2(z) and transcendental function ~(z, s, v), and expanding the latter one in the logarithmic series ([6], Vol. 1, §1.11) we obtain S3~ (~----T)2[-~-y(l-lnY)].

(A13)

Differentiating equation (A12) with respect to y we get $2-

ROS3

,oF ~

RA2 l n [ l + e - Y ] -- 71-,Or--~ [1~-"~-~J'

(A14)

RA 2

So- $1- $2- $3-- ~--~e -yI~I. - ~rTl~,l-----VF e-yl~l A2 - X-~--e

/--, ( ~ - T ~ )

-yI~I ~

,

(A10)

1 02S3

1

(A15)

$1 = A 2 0 y 2 -- sinhy"

Thus the formulae (A11)-(A15) solve the problem of frequency summation at T "~ Tc.