PH YSICAL REVIEW B VOLUME 13, NU MBE 8 5 1 MARCH 1976

Nuclear relaxation of F in the two-dimensional ferromagnet K2CuF4

K. Le Dang and P. VeilletInstitut d'Electronique Fondamentale, Laboratoire associe au C.N. R.S., Batiment 220, 91405 Orsay, France

(Received 21 July 1975)

The spin-lattice and transverse relaxation times T, and T, of "F nuclei at different sites were measured,between 1.65 and 4.3'K, in a single crystal of K,CuF4 with a dc magnetic field Hp applied along one of the

principal axes of the hyperfine-interaction tensor. The field and temperature dependence of T, is close to thatof the three-magnon process calculated for an anisotropic hyperfine interaction, in a two-dimensional

Heisenberg ferromagnet. At 4.3'K, the contributions to the nuclear transverse relaxation are shown to arisechiefly from the two-magnon process for the site with Hp parallel to the bond axis and from the Suhl-

Nakamura interaction for the site with Hp perpendicular to it, as a consequence of the highly anisotropichyperfine interaction.

I. INTRODUCTION

It has been shown that the magnetization of thetwo-dimensional ferromagnet K CuF, decreasesin an unusual way' as the temperature is raised.This variation is supported by a simple model ofmagnons in a two-dimensional lattice, in thepresence of magnetic anisotropy. The field de-pendence of the magnetization previously report-ed' was also quite consistent with this model. Inthis paper we are interested in the dynamics ofthe spin systems in K,CuF~ through the nuclearrelaxation.

Because of the highly anisotropic hyperfine in-teraction of F in KCuF4, the relaxation is ex-pected to depend strongly on the field direction.The calculation of the nuclear spin-lattice andtransverse relaxation rates was carried out inthe following conditions: (a) a dc magnetic field,much higher than the dipolar and anisotropy fields,is applied along one of the principal axes of thelocal symmetry and (b) in the low-temperatureregion where the spin deviation is small. Com-parison between theory and experiments was madeon the field and temperature dependence of therelaxation rates at different inequivalent fluorinesites. It is remembered that the crystal struc-ture of K,CuF, is tetragonal, with lattice constantsa = 4.155 A and c = 12.74 A. The orbital groundstate of Cu" ion is described by the wave function

l.e., each Cu" ion is bonded with fourF ions.

II. THEORY

For simplicity, we shall restrict the calculationto high values of the external field Hp with respectto the dipolar field, such that the dipole-dipoleinteraction may be neglected. Our model is atwo-dimensional Heisenberg ferromagnet with asmall XY-like anisotropy. When the field Hp is

applied in the easy plane the Hamiltonian can bewritten as

gP Hog-S.. .

where 5 are the z nearest neighbors and E is pro-portional to the anisotropy field out of the planeH„defined by

H„= zJSE/P. (2)

In magnon operators, the expression (1}is re-written as

3C =+A~ b~ o~ + 2 B» (b~ b ~ + b~*b*~ ),

A, = g P H, + 2zJS)/2 + 2zJS (1 —g/2) (1 —y, )

and

B~ = $y zJS.In diagonal form the spin-wave energy is given

by

@~a = 8a Ha) (4)

(2HO/H„+ I)» 1.

In the low-temperature region where y, ——1-4k'a', for the square lattice, the dispersion re-lation has the usual quadratic form:

(d~ = (dp+ (d a k

It should be noted that, for high-k values,A&»B&, since («1. This statement is still validfor low-k values in the approximation of a high-field Hp Then the conditions'„»B~, that is, theHamiltonian is diagonal in b, b, , can be written as

1919

1920 K. L E DANG AND P. VK ILL ET

where

(u, = y, (H, (H0+ Hg)]' '

and

(u,„=Jib.(7}

,K= —JQ(S, S;,» —)S, , S. ..»)- gPHDQS, ,j,h

(6)

or

K=+6 (u»b» 5»,

where

ku» = gPH, + 2zJS(1 —y» —$).

In the dispersion relation (6) e, is now given by

(uo —y, (Ho —H„) . (10)

When the field H, is applied along the hard di-rection, i.e. , the c axis, the Hamiltonian becomes

There is no direct process since wp is severalorders of magnitude greater than v„. We there-fore neglect the nuclear energy 0(d„ in the three-magnon process concerning the energy conserva-tion. The nuclear relaxation rate due to this proc-cess is then given by

1

T——2' =z

1

~ (( & +1)(s ) (~ -)4p

X $(h (l1» —5 (Al»» —5 (gp» e )

(13)where (n)»is the thermal average value of theoccupation number for the magnon state k.

The integration in the two-dimensional latticewas carried out by using the dispersion relation(6). Then the integral can be put in the form

ma» ma& e e» e dz dy(e» e* e' —1) (e~ e* —1) (e e' —1)

In the following calculation the external fieldHp is assumed to b e appl ied along the z -direction,one of the principal axes of the hyperfine-inter-action tensor. The expressions of the nuclear re-laxation rate for the x or y direction are deducedby cyclic permutation.

where

A (d A(dn = ' x= " a'k',

k~T' k~T

(14)

A. Nuclear spin-lattice relaxation

Following Oguchi and Keffer, ' we write the non-isotropic hyperfine interaction between the nuc-lear spin I and the electron spin S oriented alongthe z axis as

(d Q»t

y=k, T'

The limits x „.and y „can be extended to in-finity because of the rapid convergence of the inte-gral. The final result is

and

S, = (1 —a*a)a,S = a*(1 —a*a},

S, = S-a*a,(12)

a=A' ~e '" 'bk t

k

el 2~ e k' r b»}»k

JC;., = —.[(A„-A„)S, + (A„„+A „)S ]I,+ —,

' [(A„„+A„)S,+ (A -A„, )S ]I

+A„ I, S, ,

where A„, , A„, and A„are the principal valuesof the hyperfine-interaction tensor A. In the caseof a spin S= —,', such as Cu", the Holstein-Prima-koff transformation4 is simply defined by

T, 16(2w)'ll'(u„h (u, „

1 e2a 1 e2aln

( , + e ln

B. Nuclear transverse relaxation

The transverse relaxation rate due to the longi-tudinal fluctuations of the spin S oriented along thez direction is given by

A' + 0()

T2 28(6S.(0) 6S,(t)) dt.

These fluctuations can be expressed in terms ofmagnon operators, yielding the two-magnon pro-cess. Using the same approximations as before,the relaxation rate in a two-dimensional latticeis given by

It should be pointed out that the relations (12) arephysically admissible only for a*a = 0, 1, i.e.,when the fractional spin reversal is small.

1 A'„k,T 1

T, 167th'(d k(d e ' ~ —1(17)

NUCLEAR RELAXATION OF ' F IN THE TWO-DIMENSIONAL. . . 1921

At low temperatures, the relaxation rate de-creases more swiftly than T2 and the indirect nuc-lear spin-spin interactions via virtual excitation ofmagnons —Suhl-Nakamura (SN) interaction' —maybe dominant.

The SN interaction is written for the nonisotropichyperfine interaction with the spin S along the zdirection as

Xs„— Ui, I; I, ,ij

with

(18)

where r,.&

= r,. —r& is the position vector of the isite relative to the j site. The second moment ofthe homogeneous line is

I(I+ 1) S'(A„'„+ A'„)' P e'"'& e'" ''&

3h' 4' io kIk k + k

S'(A„'„+A.'„)' I(I + 1) ~ 1

Again, using the spin-wave dispersion relation(6) we obtained the result for a two-dimensionallattice:

(A„', + A'„)2S'I(I+1)(20)

The homogeneous linewidth («u )' 3 is indepen-dent of temperature.

It should be noted that the SN interaction involvesthe transverse components of the spin operator,whereas the two-magnon process involves the lon-gitudinal component. In other words, the two-mag-non relaxation rate is proportional to the squareof the nuclear resonance frequency while the SNsecond moment depends on the transverse com-ponents of the hyperfine field.

the spin S is parallel or perpendicula, r to the bondaxis. Then, the inequivalent fluorine sites are(a) F& ~~& F& && and F«& &

when the field H, is parallelto the a axis and (b) F„,» and F&,~& when Ho is par-allel to the c axis. The symmetry of the F hyper-fine interaction is nearly axial with the z axis de-fined as the bond axis. The x axis is taken asparallel to the tetragonal axis of the correspondingCu" ion. The principal values A„„, A„, and A.„of the hyperfine-interaction tensor are 29 @10 ',32.6x10 ', and 129&10 ' cm ', respectively, forF(,&and31&&10 ', 35x10, and 125&&10 cm ',respectively, for F(,&.

~

A. Temperature dependence of the nuclear relaxation

The nuclear spin-lattice and transverse relaxa-tion times T, and T, were measured for F(,~~~and

F~, &, ina field Hp of 11 kOe parallel to the a axisas a function of temperature. For K,CuF„H„was measured to be about 2.8 kOe' so that thecondition (6) is amply fulfilled. The calculatedrelaxation rates due to the three-magnon process,using h u,„jk = 11.2 K, are in good agreement withthe experimental ones [Fig. 2(a)]. It should benoted that the ratio of the relaxation rates forF[,~~&

and F[,„~ is also consistentwith that expectedfrom the principal values of the hyperfine-inter-action tensor. A similar temperature dependenceof T, was observed for F&„,&

and F&, &[Fig. 2(b)]

when the field H, is parallel to the c axis.The temperature dependence of the transverse

relaxation rates is quite different for F(,)~&and

F(, ~as shown in Fig. 3. For comparison, thecalculated two-magnon relaxation rates as a func-tion of temperature are also given on the figure.From expression (20) the Suhl-Nakamura homo-geneous linewidth (n, &d') &~' was evaluated as

III. EXPERIMENTAL

The "F relaxation was studied at different sitesin a single crystal of the two-dimensional ferro-magnet K,CuF„by a spin-echo method. The spin-echo spectrometers were already described inprevious papers. ' The single crystal is a prolateellipsoid whose major axis is parallel to the aaxis. The two axes are 5 and 10 mm long, re-spectively. The sample is put in a narrow Dewartail such that a dc magnetic field Hp can be ap-plied in the ac plane of the crystal.

We use the symbols F(, &and F„& to indicate the

fluorine sites in the c plane and on the c axis(Fig. 1), and the indexes

~~and & to denote whether

FIG. 1. Different fluorine sites in K2CuF4. F~,~and F~ ~

represent F ions in the c plane and on the c axis, re-spectively.

1922 K. LE DANG AND P. VEILLET

(s)Tq

(sec ")

102

10

theor etical

—10'

4xi0' and 37x10' sec ' for F&,~~) and F(,~), respec-tively. It can be concluded that the transverserelaxation is mainly due to the SN interaction forF(„)whereas the two-magnon process is dominantfor F(,(~), inthehigh-temperature region. The SNcontributions to the relaxation may be overestima-ted because a microscopic inhomogeneous broade-ning would reduce this interaction. ' The fact thatthe spin-echo decays are nearly exponential seemsto support this eventuality. ' From the exp ri-mental and theoretical results for F(,~~) at 4.3 'K itis noted that the calculated value of the two-mag-non relaxation rate is about one half of the effec-tive value. This discrepancy may be due to thapproximations used in the present calculation.

I

3T ('K)

10

B. Field dependence of the nuclear relaxation

The nuclear relaxation time T, was next mea-sured as a function of the external field H„ forthesites F(,~~) and F(,~~), whenH, is parallel to thea and c axes, respectively. The field dep ndenceof the relaxation rate agrees remarkably with the

(b)

(sec ")

102 —10P F(a).)

10—

H, = 'll gP(F(all)

I

3T( K)

105 (2 magnon pr ocess)

t

-F(ai)

& T('K)

FIG. 2. Nuclear spin-l. attice relaxation rates for Fin K2CuF4 as a function of temperature. The solidcurves are theoretical estimates of the three-magnonrelaxation rate: (a) the external field H p is parallel tothe a axis; (b) the external fiel, d Hp is parallel to thec axis.

FIG. 3. Nuclear transverse relaxation rates for 'PF

in K2CuF4 as a function of temperature. The externalfield Hp is parall. el to the a axis. The dashed curvesgive the calculated 1/T2 from the two-magnon processat the bvo magnetically inequivalent sites.

NUCLEAR RELAXATION OF ' F IN THE TWO-DIM ENSIONAL. .. 1923

theoretical curve, especially for F~, ~~~, showing thatthe approximation A,'»B~ is still good for thefield H, as low as 2 kOe (Fig. 4). It should benoted that the high-field approximation for F(,~~~,

that is, the effective field H, «- = H, -H„muchgreater than the dipolar field, of the order of4 @M= 1 kOe, is not very good for H, = 5 kOe.This explains the rather poor agreement betweentheory and experiments in the low-field region.

The nuclear transverse relaxation time T, forthe sites F&, ~~&

was measured, at 1.74 K, as afunction of the external field (Fig. 5). For com-parison, the field dependence of the calculatedtwo-magnon relaxation rate is also indicated on

the figure. But, it must be reminded that theactual rate may be larger by a factor of about 2.It is apparent from these curves that the two-magnon process is not important in the high-fieldregion as already expected from the thermal vari-ation of T, at low temperatures. It is noted thatthe SN homogeneous linewidth (b, e') ' ' varies as

'/' while the two-magnon relaxation rate variesmore swiftly than (d, '. As a consequence, thetwo-magnon process becomes dominant in the low-field region where the field dependence of the ob-served relaxation rate agrees practically with thisprocess.

IV. CONCLUSION

The nuclear relaxation of ' F resulting from theanisotropic hyperfine interaction was investigatedin a single crystal of the two-dimensional ferro-magnet K,cuF, . The calculation of the nuclearrelaxation rates was carried out, on the basis ofa two-dimensional Heisenberg ferromagnet witha small XY-like anisotropy. The approximationto the high values of the external field H, with re-gard to the dipolar and anisotropy fields was alsoused. This simplifying approximation is shownto be justified by the actual experimental condi-tions. The temperature and field dependence ofthe nuclear spin-lattice relaxation at differentinequivalent sites agrees remarkably with thethree-magnon process. The relaxation rate isalso consistent with the calculated value. Thecontributions to the transverse relaxation 1/T,can be attributed predominantly, for a given site,to the Suhl-Nakamura interaction or to the two-magnon process. The field dependence of 1/T,is also coherent with the theory. It should bepointed out that these processes depend morestrongly on the spin-wave gap 6 (d, for a two-dim-ensional ferromagnet than for a three-dimensionalone. For the latter, (n &u') ' ~ and (1/T, ) two-magnon are proportional to ao ' ~ and ln (e, '), re-spectively, at a moderate external field (h a, «kT)

(sec )

10

Ho/I a

I

5 1P

H. (ka )

FIG. 4. Theoretical and experimental nuclear spin-lattice relaxation rates for F in K2CuF4 as a functionof external field 8 p. The field Hp is parallel to the a orc axis .

10

ca I cul a t ed

two-magnon process&

I I

10 20

Ho Ik Oe)FIG. 5. Nuclear transverse relaxation rates for ~pF

in K&CuF4 as a function of external field Hp. The fieldHp is parallel to the a axis. The dashed curve repre-sents the calculated values from the two-magnon process.

K. LE DANG AND P. VEILLET

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4T. Holstein and H. Prixnakoff, Phys. Rev. 58, 1098(1940).

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