The specific heat of the one-dimensional antiferromagnet TMMC above 1K

Solid State Communications,Vol. 15, pp. 1185-1188, 1974.

Pergamon Press.

Printed in Great Britain

THE SPECIFIC HEAT OF THE ONE-DIMENSIONAL ANTIFERROMAGNET TMMC ABOVE 1K R.E. Dietz, L.R. Walker, F.S.L. Hsu and W.H. Haemmerle Bell Laboratories, Murray Hill, New Jersey, 07974, U.S.A. and B. Vis, C.K. Chau and H. Weinstock Physics Department, Illinois Institute 6f Technology*, Chicago, Illinois, 60616, U.S.A.

(Received 31 May 1974 by A.G. Chynoweth)

Specific heat measurements have been carried out in the short range spin correlated phase of the linear antiferromagnet (CH3)4NMnC13 (TMMC) in order to ascertain effects due to dipolar interactions. The specific heat is linear in T at low temperatures as expected from spin wave theory. The measured coefficient is about one-third the isotropic spin-wave value, but is in better agreement with a prediction from anisotropic spin-wave theory (one-half the isotropic value).

RECENT neutron scattering experiments x indicate that (CH3)4NMnC13(TMMC) is a nearly ideal, isotropic, one-dimensional, Heisenberg antiferromagnet with S = 5/2. There must, however, exist dipolar interactions between the spins and the effect of these has been shown in two ways. One is the anisotropic dipolar line broadening of the Mn EPR;2 the other is the anisotropy of the static susceptibility at low temperatures,a,4 which results from the confinement of the spins to the plane normal to the c-axis as a consequence of dipolar interactions. The neutron scattering experiments cannot be carried to very low wave numbers since the scattering cross section is proportional to the staggered susceptibility which vanishes as the wave number goes to zero. The lowest points which have been measured 5 do not show any departure from the isotropic dispersion relation. In EPR measurements, for which the wave number is essentially zero, there is again no evidence for a The portion of this investigation carried out at Illinois Institute of Technology was supported by the U.S. Atomic Energy Commission under contract AT( 11 - 1)- 1629. 1185

dipolar line shift, but, this is somewhat inconclusive since there is no long range order. In this paper measurements of the specific heat of TMMC are reported and compared with the results of isotropic spin wave theory, with the object of assessing dipolar effects. Specific heat measurements were carried out on a 40 gram polycrystalline mass of TMMC between 10 and 300 K using an exchange-gas technique. These are shown in Fig. 1. Measurements were also made on single crystals of (CH3)4NMnC13 and (CD3)4NMnCI3 between 1.0 and 20 K, using isolated sample techniques and results up to 4 ° K are plotted in Fig. 2 as C/T vs T. There are two interesting features in the high temperature data of (CH3)4NMnCls: the anomaly at 126.4 K, apparently corresponding to the first order phase transition observed in (CDs)4NMnCI3 by neutron scatterings at 128 K, and the broad peak at about 50 K. As expected, no sharp anomalies resulting from the one-dimensional ordering of the spins are visible below 100 K since such ordering is a gradual process. However, the peak at 50 K is reminiscent of the one calculated for spin 1/2 by Bonner and Fisher. 7

1186

ONE-DIMENSIONAL ANTIFERROMAGNET TMMC

. . . . . .

~

C /-- 5 ' ~ 3 . . . . . . .

(CHa)aNMnCI3, a least squares fit gives C(T) = (0.088 -+ 0.001)T + (0.047 + 0.001)T 2, where C is expressed in Joules-mol-Ldeg-1. Inclusion of a T a term yields a small negative T a coefficient with a larger standard error suggesting that the T < 4 K data is better characterized without the third order tenn. For the fully deuterated compound, a similar fit gives C(T) = (0.086 + 0.001)T + (0.055 + 0.001)T 2. The agreement of the linear terms is expected from susceptibility measurements if that term arises solely from spin contributions. On the other hand, the quadratic coefficients do not scale in a way consistent with the linear ones, suggesting possible vibrational contributions to the former. Since the (CH3)4N group has been shown to have important low frequency librational motions, 1 it is not surprising that the usual T s dependence due to lattice vibrations is dominated

(

IT= 126.4 K

I

300

o~ 200 la.i .-I o

I00

I

0

4.0

i

i

80

I

I

I

120

I

160

I

i

200

I

I

I

240

I

Vol. 15, No. 7

I

280

T(K)

FIG. 1. The specific heat of polycrystaUine (CHa)4NNnCla. The anomaly at 126.4°K corresponds to a first order phase transition that distorts the hexagonal arrangement of chains, but does not affect the internal structure of the chains.

.30

txl x,"

.26

CD

0

.22

tj') IM ..J

.18

NMnCI

"~''""~/ "

0 I.tj

.14 .10 .O6 0

I

I

I

I

2

3

4

T(K) FIG. 2. The low-temperature specific heat of (CDa)4NMnCIa and (CH3),NMnC13. A least squares fit of the data between 1.9 and 4.2 K to an expansion of C(T) in powers of T yielded 0.086 and 0.088 respectively for the coefficients of the linear terms. These are indicated in the figure as the ordinate intercepts. The coefficients of the second order terms are given by the slopes. Below 1.5 K C(T) turns upward as a consequence of the phase transition to a 3-dimensional long range ordered state at 0.86 K. For this reason we restrict our analysis to the data above 1.9 K. Data shown below 2°K were measured on a different apparatus, and are discussed in the following paper. 9 The data shown for (CHs)4NMnC13 are in good agreement with previously reported data. 1°' n Our main interest is to study the limiting behavior of the spin specific heat for small T, since in this region the specific heat will be most sensitive to those low-lying excitations near k = 0 which are strongly perturbed by the anisotropy. As shown in Fig. 2, C(T) appears to be well described by a linear and a second-order term below 4 K. In fact, for

by a lower order term. It is not possible to say whether there is a spin contribution to the T 2 term: consequently, we will consider only the linear T coefficient in the following discussion, and assume that it is due entirely to the spin entropy.

Vol. 15, No. 7

ONE-DIMENSIONAL ANTIFERROMAGNET TMMC

There are no exact calculations for the excitation spectrum of S = 5/2 chains. One must, therefore, have recourse to standard spin wave theory. The fact that S is large gives one some confidence in the applicability of the latter. The form of the dispersion relation as observed by neutron scattering, z e(k) = 01 fin c k I, where 0 = 70.7 K and c = 3.25 A, does not in itself validate spin-wave theory. One knows that for S = 1/2, the form is as stated above, but the relation between 0 and J is not that of orthodox spin wave theory. As a corollary of the latter we assume the excitations to be bosons. A calculation of the specific heat on this basis yields nl2c C2Nc a f dke(k) _ 2zrNkT rr a T J ~ ! 0

for the two branches are [el,'q ~ = 2SS = [2 - - A]

We now consider the effect of dipolar anisotropy, again in the context of standard theory, starting from a Ndel state of the spins lying in a plane normal to the c-axis. The axis in the plane is arbitrary since the inplane anisotropy is extremely small. We take as our Hamiltonian ~f = 2 J .Z. Si" Sj -- ~.,Dii'Si x S i'x -- .~.,Djj,Sjx S j,x _ 1,1

l, I

]

j

h2 4JA--Dn+D, ] 2J(2 -- A) + Od -- Ds + ~'~ - - - - ~ 6 ~-~2 (8J--D s --Oa) ] where

A = e ikc + C #'c

O s = ~, O i i ' e g~ (ri-ri'), 1l

Da = .E. D i f z]

~ (rj- ri)

D ° = iEi,D,,',

D d0 = ~iiDij

70

60

,•(d) k

50 4O

30 2O IO /

0.0

I

I

I

I

0.5

1.0

1.5

2.0

2.5

kc FIG. 3. The dispersion eka of spin excitations in TMMC as calculated by spin wave theory is shown by the solid line. For comparison, the dashed line gives e(k) = 70.7 [sin ckl, while the points are neutron scattering measurements of Birgeneau, Shirane and Kitchens. s

J,]

- h Z. (S; + s;j., 1,1

+ v,)- p

[e~a)] = = 2JS=[2 + A]

= 0.246 T Joules mo1-1 K -z

It is known from the neutron work s that the observed excitations are sharp only when the wave number exceeds r, the inverse correlation length of the spin ordering. It can be argued that at any temperature only a certain fraction o f the excitations which contributed to the above calculation of the specific heat are describable as magnons, namely those with Ikl > K(T). From the known dependence o f t upon T, one finds that about 70% of the estimated specific heat at any low temperature comes from good magnons. The magnitude o f the error committed by treating the broad ones as normal is not clear, but in any case this effect alone cannot account for the observed discrepancy.

h= 4JA-Ds-Dd

2a2 + A ) -

o

The observed specific heat is, therefore, only about one-third of that predicted by this analysis.

1187

L]

The D's are dipolar sums, while h is an external field directed along the chain axis. The dispersion relations

It is assumed here that we are working within the Brillouin zone o f the ordered state. If we use the structural (or disordered) zone, this is twice the size o f the former; there is one branch and it is given by e~. One thus predicts one branch for which e~ = 0, and one for which ego = 11.8 K. The complete disper-

5.0

1188

ONE-DIMENSIONAL ANTIFERROMAGNET TMMC

sion relation is plotted in Fig. 3 where it is compared with the relation e(k) = 70.7 Isin ckl, and with the neutron data. ~ Clearly, if there exists a branch with a gap of 11.8 K at k = 0, there will be no appreciable excitation of this branch at low temperatures. The predicted specific heat would then be diminished by a factor of 2. The observed low specific heat (relative to the calculated spin wave theory result) can therefore be taken as an indication that there may be a magnon branch with a gap. The predicted dispersion curve for e a lies above the lowest k experimental point

Vol. 15, No. 7

observed by neutron scattering, with the difference in energy falling just barely outside the probable error. Attempts to observe an optical mode by Raman 12 and IR 13 measurements have not been successful and since, as was the case with EPR, we are dealing here with wavelengths very long compared to the correlation lengths, this is not surprising. It does not seem to be possible at this time to determine directly whether a gap exists for small but finite K.

We are grateful to R. Dingle for the crystals of TMMC.

REFERENCES 1.

HUTCHINGS M.T., SHIRANE G., BIRGENEAU R.J. and HOLT S.L., Phys. Rev. BS, 1999 (1972)

2.

DIETZ R.E., MERRITT F.R., DINGLE R., HONE D., SILBERNAGEL B.G. and RICHARDS P.M., Phys. Rev. Letters 26, 1186 (1971).

3.

DINGLE R., LINES M.E. and HOLT S.L.,Phys. Rev., 187,643 (1969). The value o f J = 6.45°K in this paper is the same as obtained by a high temperature expansion technique in ref. 1..

4.

WALKER L.R., DIETZ R.E., ANDRES K. and DARACK S., Solid State Comm., 11,593 (1972).

5.

BIRGENEAU R.J., SHIRANE G. and KITCHENS T.A., private communication.

6.

Ref. 1 suggests this phase change is associated with the ordering of the (CD4)N+ groups. However, the small difference between the transition temperatures for TMMC prepared with hydrogen and deuterium indicates that interpretation may be questionable.

7.

BONNER J.C. and FISHER M.E., Phys. Rev., 135, A640 (1964).

8.

Actually, the neutron measurements (ref. 1) apply only for the case Jk -- 7r/cl > K in the extended zone.

9.

VIS B., CHAU C.K., WEINSTOCK H. and DIETZ R.E., following paper.

10.

DIETZ R.E., HUS F.S.L and HAMMERLE W.H.,Bull. Am. Phys. Soc., 17,268(1972).

11.

WHITE H.W., MILAN J.M., LEE K.H. and HOLT S.L., preprint.

12.

DIETZ R.E. and MEIZNER A.E., (private communication).

13.

ALLEN S.J., Jr. (private communication).