P 8 Y 8 I CA L R E VIE%' 8 VO LUME 11, NUM HER 11

NMR investigation in a single crystal of the two-dimensional ferromagnet K,CnF4

I.e Dang Khoi and P. VeilletInstitut O'Electronique Fondamentale, I.aboratoire assoeie au Centre ¹tional de la Recherche Seientifique, Batirnent 220,

N405-0rsay, France(Received 10 December 1974)

I ocal symmetry around the Cu + ions in K2CuF4 was studied by means of nuclear magnetic

resonance in the ferromagnetic state, The hyperfine interaction tensors of Cu'+ and F were

determined from the resonance spectra in a single crystal of K2CuF4 under a dc magnetic field applied

along the a and e axes. The fluorine octahedron was found to be elongated with tetragonal axis lying

in the e plane, which is contrary to the Knox model. This experimental evidence is consistent with the

recent theoretical predictions of Khomskii and Kugel based on the ferromagnetic interaction within aCu layer. The unpaired spin densities f, and f on the fluorine 2s and 2p orbitals' were evaluated tobe about 0.42 and 6.3%, respectively, for both F in and out of a Cu layer. The field dependence ofthe average value JSz) of Cu +, in the c plane and along the e axis, is also discussed.

I. INTRODUCTION

The compound KBCuF4 has been shown by single-crystal x-ray data to be isostructurejl with KawiF4, 'in particular the fluorine octahedron around theCu ion is compressed Rlong the 0 Rxls. In oux'

previous study on a powdered sample of K2CuF4,we have pointed out that the easy direction of mag-netization is parallel to the local symmetry axis ofthe Cu ' envix'onment. As the magnetization M wasshow'n to lie in the e plane, ' the local symmetryaxis should be in this plane, which is contrary tothe x-x ay data at high temperature. This discrep-ancy has been explained for the similar case of apolycrystaQine sample of RbzCuF4 in terms of arecent model proposed by Khomskii and Kugel, 'that is, the fluorine octahedron is elongated alongan a axis (Fig. l). According to these authors, theexchange interaction within a Cu layer can be fer-romagnetic only in this case, In this paper we re-port our NMR investigation on a oriented singlecrystal of K2CuF4 in a dc magnetic field. The ex-pex imental results enable us to confirm the new

model of Cu ' environment and to determine com-pletely the hyperfine tensors of Cua' and F .

II. HYPERFINE INTERACTION OF Cu2+AND F

The hyperfine field of Cu ' is the sum of thxeetexms: the isotropic Fermi contact texm 8„ theorbital II~b, and the dipolar 0„terms. The two

latter depend on the ground state of the ion, that is,on the crystal field.

The orbital ground state of Cua' in an elongatedoctahedron is described by the wRve function x~

-ya, the z axis being the tetragonal axis. From theexperimental values g, = 2. 30 and g, = 2. 086 of the

g factor, it was straightforward to deduce the prin-cipal values of the g tensor Rs g„=2g, g, =2.52and g, =g, =2.08. From these values one can de-

duce the paxameters 4o and 4~ which are the ener-

gy levels corx esponding to the singlet B3~ Rnd thedoublet 3E~, xespectively, with regard to the ground

state, by using the relations'

g„=2—8&/40, and g, =2 —2&/&g.

With the effective value & of —V10 cm ~, s 40 arid

4~ were evaluated to 10923 and 1V V50cm ', respec-tively.

Taking the effective value of 6. 3 a. u. 8 for (r '),the orbital and dipolar terms were calculated to be+205 and —225 kQe respectively, when the spin~ ls along the symmetry axis Rnd +31.5 Rnd + 122.5

kGe when ~ is perpendicular to it. Assuming thecontact term B, of about —125 kOe, the total hy-perfine field H„=AM/IMI is given by the tensor A

whose px incipal values are A,„=A»=A.,= 19kGe Rnd

145kGe. It should be pointed out that inthe Knox's model, the Cua' orbital ground state isdescribed by the wave function 3za —sa and theprincipal values g„and g, are directly equal to g,Rnd g„x;espectively. In this case the principalvalues of the hyperfine tensor A. are quite differentwith A, = —120kOe and A.„=+132kOe. The calculateddipolar field due to other Cu ' ions in the lattice is—0.46 kOe when M is oriented along the e axis,that is, much smaller than the hyperfine fieM.

As previously reported, the quadrupole splitting18 1Blportant (59 MHz for Cll) when M is along thelocal symmetry axis z. Now, when the spin is per-pendi. cular to z, the low-frequency transition cor-responding to a small hyperfine field 0, has an ap-parent gyromagnetic ratio enhanced by a factor upto (I+i), where I is the nuclear spin. The exactdiagonalization of the nuclear Hamiltonian was al-ready tabulated by Parkex for I= 1 and ~.

In the light of the new model, the interpxetationof the transferred hyperfine fieMs at the Quorinesites must also be revised. These fields arise

NMR INVESTIGATION IN A SINGLE CRYSTAL OF. . . 4129

~Fa a

Fig. 1. Cu environment after Khomskii and Kugel'smodel. The fluorine octahedron is elongated along ana axis. F, and F, represent F ions in the c plane and onthe c axis, respectively.

from the Cu-F bond causing a transfer of unpaired2s and 2P electrons of F to 3d orbital of Cu~' ion.The spin of the transferred electron is antiparallelto Cu ' spin S. The net spin at 2s and 2P orbitalsis then parallel to S. As the hyperfine field due tothe Fermi contact term is antiparrallel to the sspin density at the nucleus, the transferred hyper-fine field of F due to 2s shell is antiparallel to 8,that is, parallel to the magnetization. As for the2P electrons, they produce at the nucleus a dipolarfield parallel to the magnetization when the latteris along the bond axis. The Fermi contact termand the dipolar field are represented by the con-stants 4 ~ and A.„respectively. The two components&„and II, of the transferred hyperfine tensor arethen given by

H„= (A q+ 2A, ) (S/yh)

sample is put in a narrow Dewar tail such that therf field II„ is along an a axis and a dc magneticfield can be applied in the &e plane.

A. NMR from Cu + sites

When the magnetization is oriented along the &

axis, the nuclear signal was observed from twosites of Cu '. Indeed, in this case there are twomagnetically inequivalent sites of Cu '. one-halfCu„with spin parallel to the local symmetry axisand the others Cu, with spin perpendicular to it.It should be noted that, in the Knox's model, all theCu ' are equivalent whatever the magnetizationorientation is. So far, only the nuclear resonancespectra col responding to Cu~, were observed"'";it is now quite clear why these lines are equallysplit by quadrupole coupling. The resonance fre-quency from Cu, corresponds to the low-frequencytransition predicted in Sec. II. This transition wasalso observed when the magnetization is along thecrystal e axis, i.e. , perpendicular to the localsymmetry axis. The magnetic field dependence ofthese resonance frequencies (Fig. 2) shows clearlyan apparent gyromagnetic factor enhanced by afactor approaching 2 as expected from a spin I= &.

The variation of the Cu resonance frequency ofthe central line as a function of the external fieldapplied along the & axis is shown in Fig. 3. Aspointed out in our previous paper, the averagevalue (Ss) in a two-dimensional ferromagnet depends

50-

H =(H cos 8+Hisln 8)

tan8'= (H~/H„) tan8 .(l)

(2)

III. EXPERIMENTAL RESULTS AND INTERPRZTATION

The magnetic resonance of Cu, Cu, and Fnuclei was detected by a spin-echo method at liquid-helium temperature, the compound K2CuF4 beingferromagnetic below 6.26K. ' The spin-echo spec-trometer was already described in an earlierpaper. ' The single crystal was cut in a parallel-epiped form of dimensions 5&& 5&&10mm. Thee axisis parallel to one side of the square base and thetwo a axes are parallel to the other sides. The

H, = (A ~ -A, ) (S/yS'),

where y is the nuclear gyromagnetic factor of ' FWhen the magnetization M makes an angle 8 with

the bond axis the magnitude and the direction, de-fined by an angle 8', of the hyperfine field are givenby:

NxX'LJ

O2: &0-LLI

D0LLj

CLLL

LLI

O'Z ]& 30-]X0LLI

K

20-I ~ ~ I

0 5 10APPLIED MAGNETIC FIELD C kOe)

FIG. 2. Field dependence of the copper NMR frequencyfrom sites with spin perpendicular to the local symmetryaxis (low-frequency transition).

4130 LE DANG KHOI AND P. VEILLET

I el

~ ~

I~

.5

1.05-

0-L,

LLJ

-5

.95

1.05

~++ CU Q =162.5 MHz +x ~o

+++~ i)

I a I E I a I I I I lal)l l I. . . . I I I

1 2 5 10 20APPLIED MAGNETIC FIELD Ho//a C k Oe )

I . I S I

2 F(~.) 1 0

I I I

(b)

Vo =188.2 M Hz

~ 5YH/2P

1.5

FIG. 4. Energy levels of a nuclear spin I= 2 in a dcmagnetic field H normal to the electric-field-gradientaxis as a function of the relative magnitude of the Zeemanenergy with regard to the quadrupole coupling constantP. In the present case, 6P=59MHz for 63Cu.

on the external field Ho. This effect was reportedrecently from the NMR shift of F in K2CuF4 underan external field. In the present case we canseparate out, the sign of H„being negative, the .

NMR shift of ~Cu due only to the change in (Ss).This field dependence is given approximatively bythe expression

(Sg) kT 1S 2ws JS 1 —exp(- h&uo/kT) '

95~

I a I I I Il I I. . . . ) I I I

5 6 7 10 15 20 00APPLIED MAGNETIC FIELD Ho//c C k Oe )

I I I I

F(~) 0

FIG. 3. Resonance frequencies v of 9F line and 6 Cucentral line as a function of the dc magnetic field H0 ap-plied along (a) the a axis and (b) the c axis. The deducedfield dependence of &Sz& /S is compared to the theoreticalcurve from expression (3), (Sz& /S= 1-0.03057 F(cop),F(cop) =ln [1-exp ( Scop/O'T)] '.

where J' is the exchange integral within a Cu layer,s is the number of the spin's nearest neighbors,and a&0=y, [HO(HO+H„, „,)]' ' for the spine lying inthe c plane and &uo=y, (HO —H„,) for the spine ori-ented along the c axis. H„, is the anisotropy fieldout of the easy plane. The anisotropy field in thec plane is negligible, being a few oersted. '

The theoretical curve using the values J/k = 11.2K, H„,= 2. 8kOe ~ = 4, S = 2 for K,CuF4 is closeto the experimental one. The hyperfine field at thecopper nuclei at saturated magnetization is —144kOe, in good agreement with the calculated one of—145 kOe.

The nuclear energy levels of a spin I= 2 in amagnetic field H normal to the electric-field-gra-dient axis as a function of yH/2P, where P is thequadrupole coupling constant, are shown in Fig. 4.From the resonance frequency we deduced the totalfield H, hence the positive field H, . The two values

NMR INVESTIGATION IN A SINGLE CRYSTAL OF. . . 4131

90-

NZ& Zo-

T= 2.15 K

Ho = 11 40~

200-

T= 2.15 K

0ZUJ

0LLI 50-KLI

LLI0Z

ZO 30-

K

e.

I I I ~ I I I I

Qo 300 600 9Q0

Ho//c Ho//

0 R

Z+ 150-DQLLI

KLL

LLI

UZ

Z 100-0W

LLI

K

FIG. 5. Resonance frequencies of two 63Cu lines at thetwo magnetically inequivalent sites as a function of thedc magnetic field orientation in the ac plane. H0 is theangle between the applied field Ho and the c axis.

of H„at saturated magnetization, are then +15and +1kOe, along the a axis and c axis, respective-ly. They are coherent with the prediction of asmall field H, in the case of pure axial symmetry.In fact, the axis of the axial crystal field at a Cu2'

site due to other ions in the lattice is along the caxis, that is, perpendicular to the local symmetryaxis such that the actual crystal field at Cu ' ioncan not be purely axial. The resonance frequen-cies at different sites as a function of the magnet-ization orientation in the ac plane are given in Fig.5. The research for the nuclear quadrupole transi-tions when the spin is perpendicular to the sym-metry axis was unsuccessful. It should be pointedout, in this connection, that the enhancement factor

of the rf field and of the nuclear signal in aferromagnetic compound is much smaller for aquadrupole transition than for a magnetic transi-tion.

B. NMR from F sites

50 I I E I

00 300 600Ho//c

e.900

FIG. 6. Resonance frequencies of SF nuclei at differ-ent sites as a function of the dc magnetic field orientationin the ac plane. 80 is the angle between the applied fieldHo and the c axis.

latter case. Thus a pulse &m was obtained withpulse magnitude of 500 V and pulse width of about0.8 and 3 p.sec for F„and F„, respectively, show-ing that the enhancement factor is nearly four timeshigher for F,~.

For an anisotropic hyperfine field H„, the factorp actually depends not only on the direction of Mbut also on that of the transverse rf field. It canbe written

H„dnn Ho+H

where dn' and dn are the deviations of H„and M,respectively, and HA the anistropy field. The ex-pression of dn'/dn was calculated, using the rela-tion (2) in the case of axial symmetry with, say

The resonance frequencies of F nuclei at dif-ferent sites as a function of the magnetization ori-entation are shown in Fig. 6. The symbols I', and

F, indicate the fluorine sites in the c plane and onthe c axis, respectively. Using the indexes t~ and~ to denote 8 = 0 and 8 = ~m, the inequivalent fluorinesites are one hand F,(, and F„when the magnetiza-tion M is along the c axis, and the other hand F„,F„and F„,when M is along the a axis. It shouldbe pointed out that the enhancement factor g of therf field H„ is very different for F„-and I'„ in the

I li a axis I II e axis

Foll Fal Fcl Fcn Fol

H@& kOe

Hn kOeat O'K

4.13 —l.77

A~=12 ~ 6 Axx= 12 ~ 8A~=14.4

2.47 -2.36

A~ = 44. 5 A~ = 14.6

TABLE I. Calculated dipolar fields at fluorine sitesdue to Cu moments in the lattice and deduced hyper-fine fields at saturated magnetization. A~, A~, andAg~ are the principal values of the hyperfine tensor inthe principal-axis system shown in Fig. 7.

LE DANG KHQI AND P. VEILI.ET

g2 )f2(:c 3

FIG. 7. Principal-axis system of the F hyperfinetensor. The diagrams of fluorine p~ and d~p ~2 orbitalsare shown in solid line and dashed line, respectively.

the z axis as principal axis, to be

dn ' d8' H,P~dc' d8 H(( cos 8+H~ sin 8

for H, in the (M, s) plane and

fine tensor A. whose principal values A.„„, A», andA„are referxed to the principal-axis system givenln Fig. 7.

As the tw'o tensors for F, and F, are quasi-iden-tical it can be concluded that the interionic dis-tances Cu-F, and Cu-F, are practicaUy the same,in agreement with the elongated-octahedron model.The tensor A. can be separated into an isotropicand an anisotropic parts corresponding to the Fermicontact term and the dipolar field, respectively.The hyperfine constants A. s and A were than esti-mated to about 64& 10 4 and 27& 10 4em ~. Assum-ing they arise chiefly from 2s and 2P electrons ofF and using the corresponding wave functions givenby Froese, ~4 the fractions f, and f, of unpaired spinon these orbitals were evaluated to 0.42% and6. 3%, respectively. The slight difference in thebvo values A.„, and A» would arise from the factthat the Cu ' orbital is only of binary symmetryaround the bond axis. In order to strengthen theCu-F bond, some d character of the same symme-try, i.e. , za-ya, may be induced on the fluorinebond orbital which produces a dipolar field whosesymmetry axis is x, that is, perpendicular to thebond axis (Fig. 7). It can be shown, in this assump-tion, that the dipolar field is given by f(A„„-A,„),which is negative as expected from this symmetry.

dn sin8 (H'„cos 8+H', sin'8}'~'

for H„perpendicular to the (M, s) planeFor 8 =0 the two expressions are identical as a

consequence of axial symmetry. For 0&0 the fac-tor g is. a tensor whose principal values are givenabove. The factor dn'/dn for F„and F„was ob-tained from the expressio ns (&} and (6) with 8 =~vas H„/H, and I,, respectively. The observed ratioH„/H, of 3.3 is coherent of the NOR pulse condi-

tion.In order to deduce the hyperfine constants &&

and +6, the dlpolar fieMS due to atomic momentsin the lattice were calculated at different fluorinesites. These dipolar fields and the hyperfine fieldsat saturated magnetization (Fig. 3) are given in

Table I. The latter ean be represented by a hyper-

IV. eoNn, USIA

The hyperfine interaction of both Cu * and F ionsresulting from this NMR study clearly supports thetheoretical predictions of Khomskii and Kugel, thatis, the fluorine octahedron is distorted such thatit is elongated with tetragonal axis lying in the eplane. Thus, the NMRappears tobe asuitable meth-od to check the local symmetry; unfortunately itis practically limited to ferromagnetic compounds.For a two-dimensional ferromagnet the field de-pendence of magnetization can be deduced from theresonance-frequency shift corrected for the directeffect of the dc magnetic field. Beyond the unpairedspin densities on the 2+ and 2P fluorine orbltals itwas suggested owing to the asymmetry of the hy-perfine tensor that some d character is also pres-ent on the fluorine bond orbital.

K. Knox, J. Chem. Phys. 30, 991 (1959).Le Dang Khoi, P. Veillet, J-P. Henard, and C. Jaco-boni Phys. Lett. A 43, 39 (1973),

I, Yamada, J. Phys. Soc. Jpn. 33, 979 (1972).L. C. Gupta, R. Vijayaraghavan, S. D. Daple, U. R.K.Rao, Le Dang Khoi, and P. Veillet, in Proceedingsof the Fifth International Symposium on Magnetic Reso-nance, Bombay, India, 1974 (unpublished)

~D. I. Khomskii and K. I. Kugel, Solid State Commun.13, 763 (1973).

6K. Hirakawa, I. Yamada and Y. Kuragi, J. Phys. 32,C1-890 (1971).

~A. Abragam and B. Bleaney, EEectxon Pmamagnetic

Resonance of &mnsition Ions (Clarendon, Oxford,England, 1970).

BB. Bleaney, K. D. Bowers, and M. H. L. Pryce, Proc.R. Soc. A 228, 166 (1955).

'P. M. Parker, J. Chem. phys. 24, 1096 (1956).OLe Dang Khoi, Rev. Phys. Appl. ~3 193 (1968).

~'I. Yamada, H. Kubo and K. Shimohigashi, J. Phys.Soc. Jpn. 30, 896 (1971).H. Kubo, J. Phys. Soc. Jpn. 36, 675 (1974),A. M. Portis and A. C. Qossard, J. Appl. Phys.Suppl. 31, 205S (1960).C. Froese, Proc. Camb. Philos. Soc. 53, 206 (1957).