Spin-magnon relaxation of Yb 3+-ions in antiferromagnetic cuprate y 1−xyb Xba 2Cu 3o 6+y Текст научной статьи по специальности «Физика»

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Volume 14, 2012 No. 1, 12101 - 9 pages

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Editors-in-Chief Jean Jeener (Universite Libre de Bruxelles, Brussels) Boris Kochelaev (KFU, Kazan) Raymond Orbach (University of California, Riverside)

Executive Editor

Yurii Proshin (KFU, Kazan) Editor@ ksu. ru

Editors

Vadim Atsarkin (Institute of Radio Engineering and Electronics, Moscow) Detlef Brinkmann (University of Zurich, Zurich) Yurij Bunkov (CNRS, Grenoble) John Drumheller (Montana State University, Bozeman) Mikhail Eremin (KFU, Kazan) Yoshio Kitaoka (Osaka University,

Osaka)

Boris Malkin (KFU, Kazan) Haruhiko Suzuki (Kanazawa University, Kanazava) Murat Tagirov (KFU, Kazan)

*

In Kazan University the Electron Paramagnetic Resonance (EPR) was discovered by Zavoisky E.K. in 1944.

Spin-magnon relaxation of Yb3+-ions in antiferromagnetic cuprate

Yi-xYbxBa2Cu3O6+y

A.A. Vishina A. Maisuradze2

1 Kazan Federal University, Kremlevskaya 18, Kazan 420008, Russia

2 Physik-Institut der Universitat Zurich, Winterhurerstrasse 190, ZH-8057, Switzerland

*E-mail: alenavishina@gmail. com

(Received October 26, 2012; accepted November 16, 2012)

The spin relaxation of Yb3+-ions due to their coupling to antiferromagnetic spin waves existing in CuO2 planes in YBa2Cu3O6+y compound is reported. It is shown that it results in a strong temperature dependence of electron paramagnetic resonance (EPR) linewidth. The temperature dependence of EPR g-factor was also obtained and shows a good agreement with experimental data.

PACS: 76.30.-v, 71.27.+a, 74.72.-h

Keywords: YBCO, electron paramagnetic resonance, rare-earth ions

1. Introduction

The quasi-two-dimensional cuprates such as yttrium barium copper oxide YBa2Cu3O6+y (YBCO) whose properties are determined by electrons moving within weakly coupled copper-oxide (CuO2) layers attract much attention due to their transformation into high-temperature superconductors at high oxygen doping. The YBCO compound is characterized by large superexchange integral between nearest-neighbor Cu sites within the CuO2 layers (J « 2000 K [1], J «1700 K [2], J «1400 K [3]), a rather strong coupling between the layers within a bilayer J1 (5 = J1 / J > 4 • 10-2 [1], 5 > 7 • 10-2 [3]), and an extremely small coupling to the next-nearest-layers J2 / J « 2 • 10-5 -3 • 10-4 [1, 3].

Among the other techniques electron paramagnetic resonance (EPR) proves to be an effective tool for studying the properties and spin relaxation in cuprate superconductors [4]. The rare-earth ions are used as an EPR-probe due to the fact that the substitution of yttrium by isovalent rare-earth ions having local magnetic /-moments does not change the critical temperature Tc or other magnetic properties of the material considerably [5]. We analyze the EPR spectra of the YBCO compound doped by Yb3+ ions - Y1-xYbxBa2Cu3O6+y - at low oxygen doping (y < 0.15 ), while the CuO2 planes are not yet doped with the oxygen p-holes.

The explicit expressions for the coupling of ytterbium ions with the antiferromagnetic (AF) spin waves in CuO2 planes and the indirect spin-spin interaction between the Yb-ions due to this coupling were obtained in [6]. The contribution of these interaction to EPR linewidth and g-factors was also studied. In this article we report the Yb3+ ions relaxation due to their coupling with the AF spin waves and the temperature dependence of EPR g-factor.

It was shown in [6] that AF spin waves modes (due to the dipole-dipole interaction between copper ions) have the large energy gaps A ~ 40 - 90 K (at the Brillouin zone center) that are much larger than Zeeman energy. That forbids the one-magnon processes to contribute the Yb3+ spin relaxation due to the energy conservation law. Hence studying the spin relaxation we should consider the two-magnon processes only.

We start this paper with the necessary brief description of the work [6] in Sec. 2. In the third section we derive the two-magnon part of Yb-Cu interaction Hamiltonian. The Sec. 4 describes the Yb-ions relaxation due to their coupling to AF spin waves and its contribution to the EPR linewidth. The last section gives the temperature dependence of EPR g-factor.

2. Spin excitations and electron paramagnetic resonance in two-layer antiferromagnetic system Yi-xYbxBa2Cu3O6+y

It was shown in [1, 2] that four modes of spin waves are formed within the CuO2 planes at low oxygen doping while the Cu ions are in AF state: two acoustic and two optical ones. In [6] we studied the way dipole-dipole interaction between the copper ions and an external magnetic field (which is always applied in EPR experiments) contributes the energy gaps of the waves. The energy eigenvalues for acoustic modes are

(E ^ 2

aq

V2 j j

1 -Yq +2bCu l Yq +~ | + A

( J] =[1 -Yq +axyYq + AP] V 2

1 + y +---------------a y

/ q 2 xy< q

1 + Yq +--2bCu l Yq +~

(1)

Aa = 2d 3, A n = —— +—— + 2d 3 +——.

a 3 P 2 4 3 2

The energy eigenvalues for optical modes are given by following expressions

(E Y

v 2 J j

:[ 1 + y —a y + A ]

|_A 1 f q ^xy/q 1 n J

1 — Yq + ~ + 2bCU l Yq — -

( EKq ' 2

V 2 J j

1 + Yq — 2bCu | Yq ~ l + A

An = —1 + -^ + 2d3 + -^, n 2 4 3 2

1 — y +------------+a y

q xy q

Ak= 2d3.

(2)

Here the bn„ = -

B„

= — sin^, where y is an angle by which Cu spins are rotated by the external

J (4 + £)

magnetic field; Yq = 1/2 [cos (qxa) + cos (qya)] , Yyq = cos(qya); ay = AJ / J « 2 -10-4 — 7 -10-4 is a

week planar anisotropy [1, 2], d1 = 3g2 / Ja3, d2 = 3g2 / Jc3, d3 = 3g2fi2B / 4yflJa3,

d4 = 3g2fi2Ba2 / JR5 with R = Va2 + c2 are the terms due to dipole-dipole interaction between Cu-ions; a = 3.85 A is the distance between the nearest Cu-ions within the CuO2 planes and c = 3.38 A is the distance between the bilayers.

The eigenoperators for the modes have the form

a = — q 2

2 {Uaq (a1q + a2q + b1q + b2q )— % (q + a+2-q + Kq + b+-q )} ,

Pq = 2 {UPq (a1q + a2q — b1q — b2q ) — V Pq (a1+-q + a2+-q — Kq — K-q )} ,

nq = 2 {Unn (aq1 — aq2 + bq1 — bq2 )— Vqn (^1 — ^+q2 + b-q1 — b+q2 )^

2 {UqK (aq1 — aq2 — bq1 + b2q )— VqK(a++q1 — a++q2 — b-q1 + b+q2 )} ,

Kq 2

with the coefficients

ulp=-

vl? =

J

a,p

J

u2 =

n,K

a,P

J

2,k

S _ axyYq - — +—— 4 2

± bCu l Yq +-

S axyYq ± b2

□I Ur*

1 + - +

4+ 2

1 + -

4 + 2

± bCu l Yq —T| +

a,P

2 J

a,P

2 J

2 J

v2 =

2,k

J

S a xyY q ± b2

□I ur

1+-+

4 + 2

Yq

2 J

These spin waves modes all have the energy gaps, which are much larger than Zeeman energy. The smallest of the gaps belongs to the a-mode at the zone center q = (0,0) and K-mode at the zone-boundary q = (n / a, n / a) and has the value Ea(0,0) = EK(n / a, n/ a) « 44 K. One can see that Ea0 ~ 4 J^jd^ » gCu^BH0. It means that one-magnon processes are not involved in direct exchange

between Cu and Yb-ions and studying the Yb-ions relaxation to AF spin waves we should take into account the two-magnon processes only.

Actually there are other interactions contributing the energy gaps. But the main goal of the gaps calculation here was to understand whether they allow or forbid the one-magnon processes to contribute the relaxation. Even dipole-dipole interaction itself gives the gaps large enough to prevent the one magnon processes, so calculation of the other interactions contribution is unnecessary.

Yb3+ ions placed between the CuO2 layers interact with the AF spin waves described above. The Hamiltonian of the interaction for the magnetic field oriented along the y-axis has the following form [6]:

r(2)

TT _ L7(°) i Z!7(1) i Z!7( n YbCu _ n YbCu n YbCu n YbCu ?

H YbCu = 4AbCUY0yN1/2 — Ab^N''2 ^a + v]p

22 -V + V

q2 qK

}•

(5)

HYbCu = 2AYj {Fa [YqZ (Uqa + Vqa ) (aq + a++q ) + ^Cu^ (Uqa — Vqa ) (aq — ^q )J

q

+F7[ri(Ur + Vr)(K, +K)+ibCuYq(v - v)(s ~K'+,)J +Y (I - bu f [ F7 (Up-V,p)(P,-P+,) + F7 («„-V„)(n,-2+,)]}.

Here Y*’y’z is the Fourier-transform of the Yb spin operator; and are form factors for the acoustical and the optical modes:

vac qxa qya

F„ = cos—x— cos—y—,

rop ■ qxa ■ qya

F°p = sin —— sin—-—.

The term HYbCu quadratic in the boson operators, which is responsible for two-magnon processes, will be described below.

Using this Hamiltonian the renormalized Zeeman energy can be written in the form

BYb = MbH0

0 A (1 1 2

gYb gCu j I1 4 Vtot

V2t = N ^a

" V]p + V^n + V]k

}* 0.614.

(6)

The renormalized g-factors of the ytterbium ions due to their coupling with the AF spin waves were estimated in the following way: for the initial values of the Yb3+ g-factors the experimental values of the paramagnetic state gYb = g0 = 3 13 (for the parallel orientation) and gYb = g 1 = 3 49 (for the perpendicular orientation) [7] were used. The compound Y098Yb002Ba2Cu3O61 is in paramagnetic state at oxygen doping y = 0.4 (where the AF state is suppressed, while the symmetry remains tetragonal). In formula (6) we use the typical g-factors of the copper ion in the tetragonal field with the orbital ground state of the type (x2 - y2) which are gCu = 2.4 and g^ «2.1 for the parallel and

perpendicular orientations of the external magnetic field, respectively [8]. The exchange coupling between the Cu and Yb ions was found to be A = -120 K [6]; we also use J = 1700 K [2]. Then the theoretical values of g-factors are g* = 3.27 and g* = 3.61, which are quite close to the experimental values g = 3.23 and g1 = 3.54.

3. The two-magnon Yb-Cu interaction Hamiltonian

The exchange coupling between the Yb-ions and AF spin waves within the CuO2-planes has initially the following form:

H ic = AY Z(s;p + + s; ,pl + s;.,,), (7)

p

here p corresponds to the positions of the Cu-ions nearest neighboring the j-th Yb3+-ion, Yj is the spin of that ion.

Following [6], we assume the field rotates the magnetizations of both sub-lattices slightly toward the y direction by the angle p. It is convenient then to turn the axes so that the new x-axis is directed along the corresponding magnetizations of the sub-lattices:

S; = S'x cos p - S'y sin p,

S“ = S'x sin p + S' cos p,

y x r y r (8)

Sb = S'x cosp + S'y sinp,

Sby = -S'x sin p + S'x cos p.

Using the standard Holstein-Primakoff formalism we make the following transformation for the layer:

Sx = 2 -;>j1, Sj1 = (1 - a>j1 )1/2 ;j1, S- = ;+1 (1 - ;+1aj1 )1/2,

21 (9)

^=-1S+.1 = b+.1 (1 - % )1/2, S;1=(1 - % )1/2 bjl,

with S± = Sz + iSx. For the second layer the transformation is the same. Here a+, a, b+, b are the

boson creation and annihilation operators for the two sub-lattices. Then we perform the Fourier

transformation to the reciprocal lattice:

aq = N~1/2 ^aj expOqr,.), bq = N"1/2 ^bj exp(zqrj), (10)

j j where N is the number of unit cells in the bilayer.

After these transformations the Hamiltonian has three parts. Two of them, the one not having the boson operators and the one linear in boson operators were given in (5). The term H Y(2b)Cu quadratic in the boson operators which is responsible for two-magnon processes has the following form

HYbCu = AbCu 'ZYqy-q' 1cosl "x "x a Icos

~a |(aq+1aq'1 + a+2aq2 + bq+1bq'1 + bq2bq'2 )

. Aqxa . kqya

(11)

-sm-

-sin-

(a+1aq'1 - a+2aq'2 - bq+1bq'1 + b++2bq'2 ) |

After the Bogoliubov transformation to the new creation and annihilation boson operators a+, P+, , K+, aq, Pq, Vq, Kq by using (3) the quadratic term takes the form

HYbCu = AbCu Z Yq-q' 1cosaIcos

qq

g=a(3rjK

[u u , g+g , + v v , c+ ,c + 5 ,v v , ] L (12)

_ qg q g bqbq qg q g b-qb-q qq qg q g Jf V >

4. Contribution of the two-magnon processes into EPR linewidth

The contribution of two-magnon processes into the Yb-ions relaxation rate is estimated by the double-time Green functions method described in [9]. The double-time retarded Green function for two operators Y+ and Y- (Y ± = Yz + iYx), which is used here, has the form

0(t -1 x) =

((Y + (t)|Y-(t'))) = -i{{Y + (t),Y-(t')})d(t -1');

[1, t > t',

[0, t < t';

where Y + (t) and Y- (t') are Heisenberg representations of the operators Y + and Y- and {y + (t), Y- (t')} is the anti-commutator of these operators.

The Hamiltonian we use has the form

TT _ ZjYb I ZJ~Cu I tJ(2) •

H H 0 + H 0 + H YbCu;

ЯCu ___ Zj-Cu

0 = H Zeem

(13)

q

g=a(3rjK

H 0Yb = BybY0y.

Equations of motion for the Green function [9] are

E((Y' lY ~))=({Y',Y D+«[Y' •H11Y i E{( _Y * ^H JlY D=({[Y * ^H1 ■Y ~})+({[[Y * ■H J ■H J|Y1

(14)

etc.

Here we obtain the infinite chain of equations that could be solved in different ways: by decoupling [10] or a special perturbation theory [10, 11]. We use the one proposed by Izyumov [11].

Applying the equations of motion (14) to the Green function we obtain the formulas

EG( E) = 1 + (([Y+, H J| Y -)),

E« [Y * ^H ]|Y D={([y * ■H 1|[H ^Y ~J)).

Substituting the second equation into the first we have the expression for the Green function [11]

G( E)=(e4o+(db(< _y • ■H - ]|[H ffiu.Y1» •

On the other hand if we compare it to the Dyson equation [11] (with a yet unknown eigenenergy part E( E)) expansion to the terms of the second order in E(E)

G( E) =

1

1

1

E -

BYh -E(E) E - BYb (E - Byb )

-2( E),

we can see that

Z( E) = -/([Y +, H YC J_ H YbCu, Y -])).

(15)

The imaginary part of this eigenenergy part gives the contribution of Yb-Cu interaction into the EPR linewidth. Calculating the energy part we should omit all the divergent terms of the form

(E - BYb) n since they have already been taken into account by the structure of Dyson equation [11].

After the commutators calculation the imaginary part of (15) takes the form:

Im E( E) =

16 A2k

cos21 ^^a I cos2

[ qy - qy |

x[(u u,„+v v ,„) n (1 + n,„)5( E - (E,„-E ))

I y qa q p qa q p J qa y q p J y q p qa ' /

+ (uq 'auqp + Vq 'aVqP ^ ^ (1 + n q' a)'5 ( E - (E q' a - Eqp) ) + (uqnuqK +VqnVq'Kf n qn (1 + ^k)5^ - (EqK - E qn) )

+ (uq 'nuqK +Vq n qK (1 + ^ n) '5 ( E - (Eq 'n - E qK) )J

(16)

+sin21 ———a I sin2

[% - qy

a |_(uqauq'n+VqaVq'n) nqa (1 + nq'n)5(E - (Eqn- Eqa ))

■( uq 'auqn+Vq' aV ^ n n (1 + nq' a)'5 ( E - (Eq' a - EqJ )

■( uqpuq' K + VqPVq' K ^ n qP (1 + nq k)5( E - (Eq' K - Eqp) )

-(u ,„u + v ,pv ) n (1 + n,„')5( E - (E ,„- E ))] I.

\ q p qK q p qK J qK \ q p) \ ' q p qK ') J |

Here nq is magnon Bose-Einstein distribution ng = [exp (Eqg / kBT) -1J .

Using the formulas (4) and neglecting the dipole-dipole terms we can substitute the terms of the

form (

u u ,» + V V ,

qa q p qa q

\ 2

•p) by

(

u u ,* + V V ,

qa qp qa q

•p)

2 1

1

(2J)2 (

EqaEq' p

1+55'I ~[Y>+5)ir<+55

(17)

In order to obtain the analytical formula for Im E( E) by (16) we use the linear dispersion law for

the eigenenergies Egq « uq, where u = V2Ja. It is reasonable since the EPR experiments were made

at low temperatures where magnon frequencies are small enough to use the linear approximation typical for all antiferromagnets. Using this approximation and neglecting the small E term (which is the frequency of Fourier transformation and is of the order of Zeeman frequency, and for EPR-experiments is much smaller than the magnons frequencies (1)) we can replace the 5-functions

300-

CD

[XT

200-

100-

1 1 1 1 ... - n : ...... 1 1 1 1 umerical calcula alculation by the 1 1 1 1 i i i i tion by the formu analytical formu 1 1 1 1 la (16) -a (18) -

i i —J L—i i i -4-4-4-4““ j L J I j j j j / / / : /: / /

II '-TT"TT^ ■■■■■ 1 1" I 1 —\ [—\ ;;±±;;|;;;±;;; -4--1-4-4-- // . - : ! i i i

L:. -■■■H+f- '"'T" 1 1 ■'■f j. L i J- 1 1 1 1 y s' 1 1 1 1 i \ [ : ■ i i i i""

50

100 T, K

150

200

Figure 1. The contribution of the two-magnon processes into EPR linewidth. Solid line describes the numerical calculation by the formula (16), dashed line is the calculation by the analytical formula (18).

in (16) by S(q - q') / u. We also suppose exp (Eqg / kBT) »1, which allows us to replace nqg by exp (-Eqg / kBT). Now using (16) and (17) we have

Im E( E) =

16 A2 a4

nu

| dqq2

4J2 l1 -Yq )[1 + Yq +J ^

exp

' vq} ' kT

which, after the y expansion and the integral calculation, takes the form

Im E(T )>

16A2 (kBT

nJ I J

(18)

Calculating (18) for T = 40 K (which is the lowest temperature used in EPR experiments for Y1-xYbxBa2Cu3O6+y [7]) we can see that at low temperatures contribution of two-magnon processes into the EPR line is negligible E «1 G. But it increases substantially with temperature (see Fig. 1).

The numerical evaluation of (16) with the explicit form of Egq taken into account shows that the

approximations we made in the analytical formula are good enough to closely describe the more accurate values of Im E(T) (see Fig. 1).

0

0

5. Temperature dependence to EPR g-factor

Using the Hamiltonian H(^Cu given above (12) we obtain its contribution into the EPR g-factors for the ytterbium ions. The part of H YbCu which doesn't contain the magnon operators has already been taken into account in (6). We now consider the other part, quadratic in magnon operators, which defines g-factors' temperature dependence. After we put q = q' the renormalized Zeeman energy due

to HY(2b)Cu can be written in the following form

_i____________i_________i_____________i—

—i___________________i_i_________i—

—i_______________________i_i__________i—

E

Q

OQ

190 25-1 " - experimental points

--------- numerical calculation by the formula (19)

-------- - calculation by the analytical formula (20)

190.20

■

190.15

25

50

75

T, K

100

125

Figure 2. Temperature dependence of the resonance field (for H * c ) in YBCO61 with 2% of Yb: experimental points are marked by squares, dashed line gives the numerical calculation by the formula (19) for gYk = g* = 3.4 and solid line is the calculation by the analytical formula (20).

BYb = ^BH 0

0 - A

g Yb g Cu j

1 1 2 A 4 V“ 4

4 X {K+ vqs)2 nqf}

(19)

J

To calculate the sum in (19) we make the same approximations we did above for Im E(T) (see Sec. 4) but take the energy gaps as the lower limits of the integrals. It gives us the following approximate analytical expression:

BYb =MbH 0 g eff(T ),

0 A J , 1 2 1 kBT

g eff (T) = g Yb - g Cu J j 1 - ^ Vtot - ^ -J

exp

( Ea (0,0) ^ yf2kBT

exp

Ep (0,0) 4lkRT

(20)

We use the values of the energy gaps Ea p (0,0) obtained by (1) (Ea= 44 K, Ep = 109 K ). At T = 40 K the contribution of temperature-dependent part is negligible and it raises slightly with temperature. The calculated value of the g-factor at T = 40 K (g* = 3.61) is a little higher than the experimental one (g*xp = 3.54). To compare the temperature dependence we shift the theoretical points down by choosing gYb = g * = 3.4 in (19) instead of g Yb = g* = 3.49 (which doesn't change the temperature dependence itself), see Fig. 2. Along with the analytically obtained result we also show the more accurate numerical calculation by the formula (19).

As can be seen theoretical temperature dependence shows a good agreement with the experimental data.

6. Results and conclusions

The contribution of the two-magnon processes to Yb-Cu interaction and EPR linewidth was considered. It is shown that these processes result into strong temperature dependence of the EPR linewidth, and being negligible at low temperatures give the considerable input as the temperature rises. The temperature contribution to EPR g-factor was also obtained which fits nicely the experimental data.

Acknowledgments

We thank Prof. B.I. Kochelaev for the valuable discussions. This work was supported by the Swiss National Science Foundation, the SCOPES grant No. IZ73Z0_i2S242, and partially by the Ministry of Education and Science of the Russian Federation.

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