EXCITATIONS OF VARIOUS MAGNONS IN ONE- AND TWO-DIMENSIONAL ANTIFERROMAGNETS

HAL Id: jpa-00214323

https://hal.archives-ouvertes.fr/jpa-00214323

Submitted on 1 Jan 1971

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

EXCITATIONS OF VARIOUS MAGNONS IN ONE- AND TWO-DIMENSIONAL ANTIFERROMAGNETS

M. Date

To cite this version:

M. Date. EXCITATIONS OF VARIOUS MAGNONS IN ONE- AND TWO-DIMENSIONAL ANTIFERROMAGNETS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-837-C1-842.

�10.1051/jphyscol:19711293�. �jpa-00214323�

EXCITATIONS OF VARIOUS MAGNONS

IN ONE- AND TWO-DIMENSIONAL ANTIFERROMAGNET S

M. DATE

Department of Physics, Faculty of Science, Osaka University, Toyonaka, Osaka, Japan

Rburnk. - L'auteur donne une revue de la rksonance du spin klectronique d'un systkme antiferromagnetique A une- et deux-dimension. Dans un systkme de spin #Heisenberg, par exemple dans KCuF3, la rksonance aux temperatures inferieures A TN est attribuk aux RMAF ordinaires sauf pour un comportement particulier du temps de relaxation. Une forte correlation des spins a &te observee dans la rkgion paramagnktique.

D'autre part, I'excitation des spins d'Ising est importante dans les cristaux fortement anisotropes.

Abstract. - A survey of electron spin resonance in one- and two-dimensional antiferromagnets is presented. In the Heisenberg spin system, for example in KCuF3, the resonance below T N can be explained by usual AFMR except a curious behavior of the relaxation time. A strong correlation of spins has been observed in the paramagnetic region. On the other hand, the spin-cluster excitation or Ising spin resonance is dominant in strongly anisotropic crystals.

I. Introduction. - There has been a growing interest in the discovery and study of low dimensional magnets for recent decade. The low dimensional magnets here means one- or two-dimensional ferro- or antiferromagnetic spin systems as are schematically shown in figure 1. Of course, no ideal one- or two-

rn

FIG. 1. - A definition of low dimensional magnets.

dimensional magnets really exist in practical materials so that it may be appropriate to introduce the para- meters J2/J1 and J,/JI for representing a degree of low dimensionality and to discuss various magnetic pro- perties with these parameters.

Now one of the most attractive problem to be expected in these substances may be an anomalously large effect of the short range order which appear in various observations such as the susceptibility, specific heat, neutron diffraction, optical property and magne- tic resonances. In usual ferro- or antiferromagnets, the short range order effect concentrates near the critical temperature i. e. the Curie (Tc) or N6el (TN) tempera- ture, but in low dimensional magnets it is not necessary so. This is because of the fact that Tc or TN of these substances go down to very low temperatures. In a perfect linear chain spin system, for example, no second order phase transition can be expected above 0 OK. On

the other hand, the third law of thermodynamics requests that the spin entropy should be zero near OOK. Accordingly, the spins should be ruled by the short range order especially below the temperature corresponding to zJ,/k, being z to be the number of nearest neighbor spins. In some practical low dimen- sional magnets, zJl/k has been determined to be very large compared with Tc or TN so that one can expect a large short range order effect in a wide temperature region. In other words, one can see a more << amplified ⁾⁾ short range order effect than in usual ferro- or anti- ferromagnets.

When the magnetic properties of the low dimensional magnets are discussed, the anisotropy of the spin system should be considered at the same time because character of the anisotropy strongly affects on the phase transition itself, As a typical example, it may be enough by saying the fact that there is no second order phase transition in the two dimensional Heisenberg spin plane whereas a sharp phase change is expected in the two-dimensional Ising spin network [I]. For simplicity, an uniaxial anisotropic exchange Hamiltonian defined by

H i j = - 2 JII S: Sf - 2 JL(S; SJ

+

4,

Considering both the low dimensionality and aniso- tropy energy, various magnetic substances are classified and ranked as are shown in figure 2.

The purpose of this paper is to show a general survey of electron spin resonances in typical low dimensional antiferromagnets both above and below TN with special interests to their low dimensionality and short range order. As typical examples of the one- dimensional Heisenberg spin system, KCuF, and CU(C,H,COO)~ 3 H,O were investigated in detail and Cu(HCOO), 4 H,O was studied as a good model of

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711293

C 1 - 838 M. DATE

ONE DIMENSIONAL MAGNET TWO D I M E N S I O N A L MAGNET

FIG. 2. - Dimensional classification of various magnets with a parameter of anisotropy.

the two-dimensional Heisenberg antiferromagnet.

Generally speaking, the resonances below TN could be explained by usual antiferromagnetic resonance, (AFMR) modes except a curious relaxation pheno- menon. The resonances above TN revealed various kinds of short range order effects as will be shown in the following sections. In the Ising spin system, on the other hand, the main resonance below TN was describ- ed by the spin-cluster model as has been first pointed out in one-dimensional CoCl, 2 H 2 0 with effective spin of S =

4

11. One-Dimensional Antiferromagnets KCuF, and Cu(C,H,COO), 3 H 2 0 . - KCuF, has been believed to be one of the most ideal one-dimensional anti- ferromagnet known at present. The low dimen- sionality of this compound was pointed out by Hira- kawa and his collaborators [3, 4, 51 who found that there is a strong intrachain exchange interaction J, along the c-axis of slightly deformed perovskite struc- ture. The magnitude of the interaction was estimated to be J,/k = - 190OK whereas the interchain exchange interaction was infered to be smaller than 1 OK.

Recently, two modified forms of the crystal structure in KCuF, were discovered by Okazaki [6] and T, of these two crystals were determined to be 38 and 22 OK for type (a) and ( 4 crystals, respectively by using the neutron diffraction technique [7]. Hutchings et al. [7]

also found that below TN the magnetic moment at each spin sites was about 50 % compared with the standard value of Cu2* spins.

We did the magnetic resonance study of this com- pound using microwaves of 10 -- 70 GHz regions in the temperature range from 1.5 to 400 OK [S]. Roughly speaking, the resonance observed below TN can be explained by an antiferromagnetic resonance of ortho- rhombic symmetry and small anisotropy fields of 2.4

+

+

An interesting deviation from usual resonance phenomena was found in the relaxation time both above and below TN.

Concerning with the temperature dependence of

paramagnetic line width in antiferromagnets, Mori et al. [9, 101 calculated it based on a microscopic theory of the critical slowing down of the spin system and they succeeded to explain the temperature dependence of the line width in MnF,. Their results show that the width AH at temperature Tis expressed by a formula given by

where AHw means the width at T = CQ. C and

x

represent the Curie constant and paramagnetic suscep- tibility, respectively. A parameter 5 is written by a variable T* = TITN as

being, a, P and y to be positive constants. It is noticed that 5 is always larger than unity. Using experimental values of AH,, C, and X, an empirical value of 5 is plotted in figure 3 which shows a large deviation from

TEMPERATURE ^{( O K )}

FIG. 3. - &value in KCuF3. Dotted line shows the theoretical curve.

the theoretical curve. As is not shown here, 5-value in all other antiferromagnets show similar tendence.

Accordingly, such a deviation may be due to a short range order effect characteristic in low dimensional antiferromagnets. A satisfactory theory of the line width is desirable.

In some low dimensional antiferromagnets, a strong angular dependence of the paramagnetic resonance line width is seen at low temperatures and as an exam- ple, that of Cu(C,H,COO), 3 H 2 0 is shown in figure 4.

This compound has a strongly coupled antiferroma- gnetic linear chain along the c-axis but the interchain exchange interaction is very weak as has been pointed out by Date et al. [ l l ] who could not found the long range order even at 1.4 OK. From a crystallographic

FIG. 4. -Angular dependence of the ESR line width in the ac-plane of Cu(CsH5C00)2 3 Hz0 at 35 GHz region. Dotted

line shows the theoretical results.

consideration, J, % J2 % J3 has been infered so that it is ranked between 1 D and 2 D in figure 2. At 300 OK, the line width can be explained by conside- ring a strong exchange narrowing in addition to the broadening factors of the dipolar part and the aniso- tropic exchange interaction of the order of (AgIg)' J, [12]. However, the line width below the temperature corresponding to J,/k shows a strong angular depen- dence which can not explained by the present theory.

The width becomes sharp near the a-axis but it becomes so large for other axes that the widths along these axes could not be determined at 1.4 OK. Perhaps, it may be said that if one has a satisfactory theory to understand such an angular dependence, one might be informed about more detailed knowledge of the spin correlation in low dimensional antiferro- magnets.

Next, a curious property of the relaxation pheno- menon observed in AFMR in KCuF, is described.

Simply speaking, this is a quenching phenomenon of the resonance at low frequency regions. The AFMR line can easily be observed according a frequency- field relation of the resonance mode above 40 GHz but the resonance line becomes broad below 40 GHz and cannot be observed below 30 GHz region as is shown in figure 5. Such a phenomenon is not expected in usual antiferromagnets and the observed frequency dependence cannot be understood by simple wz relation. So, we tried to explain it by introducing a field dependent relaxation time T I which may come from the field dependent spin fluctuations in one- dimensional antiferromagnet. Removing the inhomo- geneous part of the line width (1.2 kOe) coming from inhomogenuities of the crystal, the relaxation time is estimated and the result is shown in figure 6 . The strong field dependence of T I suggests that the spin system may be stabilized when an applied magne- tic field increases. To check this suggestion, neutron

FREQUENCY

FIG. 5.

-

MAONETIC F I E L D

FIG. 6. - Field dependence of the relaxation time in KCuF3.

C 1 - 840 M. DATE diffraction study under a magnetic field may be effec- tive.

Finally, temperature dependence of the sublattice magnetization estimated from the AFMR is discussed.

Figure 7 shows the magnetization change as a func-

FIG. 8. - Temperature dependence of cr paramagnetic ^)> reso- nance in Cu(HC00)z 4 Hz0 along the magnetic principal

axes at 37 GHz.

FIG. 7. - Temperature dependence of the sublattice magneti- zation.

tion of TITN in type (d) crystal. The magnetization curve does not coincide with Brillouin function and has a tail near TN. It is suggested that the presence of such a tail may offer a reasonable explanation to the fact that there is no change in susceptibility and specific heat near TN [4].

111. Two Dimensional Heisenberg Antiferromagnet Cu(HCOO), 4 H 2 0 . - Several compounds have been believed to show the two-dimensionality in their spin systems. Of these compounds, CuF2 2 H 2 0 has been investigated in detail [13] but it is not discussed here. Instead, a new discovery in Cu(HC00), 4 H 2 0 is presented. The antiferromagnetism and two-dimen- sionality of this compound have been studied by Kobayashi et al. [14] by the susceptibility measure- ment. A broad maximum of the susceptibility lies near 600K and TN = 17 OK. They also suggested that a canting moment may play an important role on the magnetic property of this compound. The electron spin resonance has been investigated by Caster et al. 115, 161 both in paramagnetic and antiferroma- gnetic regions. They found that the line broadening of paramagnetic resonance was observed at low tem- peratures and the resonance below TN could be inter- plated in terms of antiferromagnetic resonance with a Dzyaloshinsky-Moriya vector in the ac-plane.

To investigate the short range order effect more precisely, We did the resonance workusing high sensi- tive ESR spectrometers and found an anomalously large shift of resonance in addition to Castner's result.

Although the resonance signal became very weak below about 20 ^OK, the ⁽⁽ paramagnetic ⁾⁾ resonance survived even below TN and was observed till down to 12 OK. Figure 8 shows the experimental result.

It is noticed that the resonance line does not connect to the antiferromagnetic resonance branch. As is shown in figure 8, the observed shifts for various direc-

tions always point toward the lower field side com- pared with the high temperature values. Such large shifts have not been reported in any ferro- or antifer- romagnets.

This large shift may be explained by introducing a model that the resonance near TN comes from the canted short range order spin-clusters in two dimen- sional layers. The cluster, different from spin-clusters in the Ising spin systems [2], consists of many spins in a layer and acts as a two-dimensionally ordered spin plane with a limited area though the size and lifetime of the cluster are not so easy to determine.

At high temperatures, of course, the lifetime becomes very short and the cluster itself becomes meaningless so that the resonance will gradually alter to usual paramagnetic resonance when temperature increases.

Assuming an external magnetic field and the effective exchange field coming from averaged spins out of the cluster, the resonance condition can be obtained by the two sublattice model under an orthorhombic anisotropy and the Dzyaloshinsky-Moriya interactions.

It is emphasized that the resonance field shifts largely from the paramagnetic position to the lower field side mainly due to the Dzyaloshinsky- Moriya effect. The general feature of the shift can well be explained by this model but the quantative estimation is difficult.

Such a cluster model seems to be effective for explain- ing the survived ⁽⁽ paramagnetic ⁾⁾ mode below TN.

Presumably, such a cluster may be excited even below 17 OK and the resonance as a cluster may become possible. It is also found that the line width below TN becomes sharp as temperature decreases.

IV. One-Dimensional Ising Antiferromagnet, CoCl, 2 H 2 0 . -As has been first pointed out by Date et al. [2], an important character of the Ising spin excita- tions is their localizability in the spin network as is seen in antiferromagnetic CoC1, 2 H 2 0 . It should be noticed that such a localized spin-cluster state cannot be the eigen state if the system consists of Heisenberg spins because terms such as J S + S ~ in the Heisenberg spin Hamiltonian act so as to mix the localized spin-

cluster state with other states. In the Ising spin system, however, the state of the localized spin-cluster is the exact eigen state because there is no transverse compo- nent of spins. Schematical medols of the spin-cluster states and their transitions in a ferromagnetic linear chain of S = 112 are shown in figure 9. These transi-

FIG. 9. - Spin-cluster excitations in ferromagnetic Ising chain.

tions are practically observed in CoCl, 2 H 2 0 . In addition to intrachain exchange field and external magnetic field, interchain exchange field must be considered for determining the resonance condition of the clusters. As is easily seen from figure 9, the transitions like (b) do not change the intrachain exchange energy so that the resonance frequencies are l'ow and come into the microwave region. These tran- sitions have been called the spincluster resonance [2, 17, 181. Direct Ising spin excitations such as figure 9a can be observed in the far-infrared region and beha- viors of these excitations are summalized by Torrance Jr. et al. 119, 201. They pointed out that small devia- tions of the resonance frequencies from the simple cluster model can be removed by introducing an Ising based Bloch type function.

V. Two-Dimensional Ising Antiferromagnet, Ni(CN)2 NH,C6H6. - The first observation of the spin- cluster excitation in two-dimensional spin system has been done in hexagonal FeC1, [18]. However, the observed line width was very broad. This may be due to the fact that the transverse components of spins are not negligible in this compound. Recently, we found that a new compound Ni(CN2)NH,C,H6 shows a typical character as an Ising-like spin system of S = 1 and succeeded to observe a new type excitation.

The static magnetic properties of this compound were investigated by Takayanagi et al. [21]. They found that this compound is antiferromagnetic below 2.36 OK

with the spin easy axis along the c-direction. Although the chemical formula seems to be complex, the anti- ferromagnetic configuration is rather simple. We found that the uniaxial crystalline anisotropy constant D is about 4.5 cm-I which is larger than the exchange energy. It is also confirmed that the spin system shows a metamagnetic steep jump of magnetization near 30 kOe when an external magnetic field is applied

[ l ] ONSAGER (L.), Phys. Rev., 1944, 65, 117.

121 DATE (M.) and MOTOKAWA (M.), Phys. Rev. Letters, 1666, 16, 1111.

[3] HIRAKAWA (K.) and HASHIMOTO (T.), J. Phys. Soc.

Japan, 1960, 15, 2063.

141 KADOTA (S.), YAMADA (I.), YONEZAWA (S.) and HIRAKAWA (K.), J. Phys. Soc. Japan, 1967, 23. 751.

[5] HIRAKAWA (K.) and KADOTA (S.), J. Phys. Soc.

Japan, 1967, 23, 756.

[6] OKAZAKI (A.), J. Phys. SOC. Japan, 1969, 26, 870.

along the c-axis. Considering the crystal and spin structures, this field can be considered as the inter- sublattice exchange field HE.

The frequency-field diagram of the resonance is shown in figure 10. It was easily found that the results

- EPR(42'K) -+- ONE SPIN PROCESS -.*.- ^(OM-2)

100 TW SPIN PRQCESS

(AM-1)

Magnetic f i l d

FIG. 10. - Ising spin excitations observed in N ~ ( C N ) Z N H ~ C ~ H ~ at 1.4 OK and the corresponding spin flops are schematically

shown by dotted arrows.

cannot be explained by usual antiferromagnetic resonance but rather be interplated by the Ising spin resonance schematically shown in figure 10. Detailed analysis shows that the resonance branch I in figure 10 corresponds to the single spin flopping in one sublattice.

As this transition is Am =

+

The branch I1 shows a new type Ising spin resonance.

Consider first a thermally excited m = 0 state in a sublattice. Interacting with the radiation field, this flops to the opposite direction and at the same time, one spin belonging to another sublattice flops. It is noticed that although such a transition concerns with two spins, change of the resultant magnetic quantum number is f 1, so that it is an allowed transition. Thus it has become clear that the Ising spin excitation in two-dimensional spin system of S = 1 is practically possible.

[7] HUTCHINGS ( M . T.), SAMUELSEN (E. J.), SHIRANE (G.) and HIRAKAWA (K.), Phys. Rev., 1969, 188, 919.

[8] IKEBE (M.) and DATE (M.), to be published in J.

Phys. Soc. Japan.

[9] MOM (H.) and KAWASAKI (K.), Prog. Theor. Phys., 1962, 28, 971.

[ l o ] Mom (H.), Prog. 172eor. Phys., 1963, 30, 578.

[ l l ] DATE (M.), MOTOKAWA (M.) and YAMAZAKI (H.), J. Phys. SOC. Japan, 1963, 18, 911.

C 1-842 M. DATE

[I21 MORIYA 0.) and YOSIDA (K.), Prog. Tkear. Phys., [17] DATE (M.) and MOTOKAWA (M.), J. Phys. Soc. Japan,

1953, 9, 633. 1968, 24, 41.

[13] NAGATA (K.) and DATE (M.), J. Phys. SOC. Japan, [IS] DATE (M.) and MOTOKAWA (M.), J. App. Phys.,

1964, 19, 1823. 1968, 39, 820.

[14] KOBAYASHI (H.) and HASEDA (T.), J. Phys. Soc. Japan, 1191 TORMNCE Jr (J. B.), and TINKHAM (M.), Phys.

1963, 18, 541. Rev., 1969, 187, 587.

[I51 SEEHRA (M. S.) and CASTNER Jr (T. G.), Physik [20] TORRANCE Jr (J. B.) and TINKHAM (M.), Phys. Rev., Kondensierten Materie, 1968, 7 , 185. 1969, 187, 595.

[I61 SEEHRA (M. S.) and CASTNER, Jr (T. G.), Phys. Rev., [21] TAKAYANAGI (S.) and WATANABE (T.), J. Phys. Soc.

1970, B-1, 2289. Japan, 1970, 28, 296.