Phase diagrams of pseudo one-dimensional Heisenberg systems

Phase diagrams of pseudo one-dimensional Heisenberg

systems

Citation for published version (APA):

Hijmans, J. P. A. M., Kopinga, K., Boersma, F., & De Jonge, W. J. M. (1978). Phase diagrams of pseudo one-dimensional Heisenberg systems. Physical Review Letters, 40(16), 1108-1111.

https://doi.org/10.1103/PhysRevLett.40.1108

DOI:

10.1103/PhysRevLett.40.1108

Document status and date: Published: 01/01/1978

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VOLUME 40,NUMBER 16

PHYSICAL

REVIEW

LETTERS

17APRIL 1/78 soya,

J.

Low Temp. Phys. 27, 245 (1977).

'_{S.D. Bader, Q.S, Knapp,}

_S.

_K.Sinha,

_P.

_SchweiP,

and

B.

Renker, Phys. Rev. Lett. 37, 344 (1976);S.D.

Bader, S.K.Sinha, and R.N. Shelton, in Supercondu,

c-tivity in d a-nd

f

Ba-ndMetals, edited by D.H. Doug-lass (Plenum, New York, 1976),p.209.

'5D._{Rainer, to be published.}

'6Q. Bergmann and D. Rainer, Z.Phys. 263, 59 (1973).

Phase Diagrams

of

Pseudo One-Dimensional

Heisenberg

Systems

J.

P.

A. M. Hijmans, K.Kopinga,

F.

Boersma, and W.

J.

M. de Jonge Department ofPhysics, Zindhoven University ofTechnology, Bindhoven, The Netherlands

(Received 13January 1978)

We present the first theoretical results on the anomalous field dependence ofthe Neel

temperature for asystem of loosely coupled classical Heisenberg chains with orthorhom- i

bic anisotropy for different directions ofthe applied magnetic field. The results compare favorably with the experimental phase diagrams of a series of selected S=

& compounds

with varying degrees ofone dimensionality and anisotropy.

The ordering temperature &N(H) ofa pseudo

one-dimensional Heisenberg antiferromagnet may

increase drastically when an external magnetic field

is

applied. This initially surprising effect

has been documented recently by a number of

ex-perimental

results.

'

'

It seems that the theoreti-cal approach suggested by Villain and Loveluck,

'

which

is

based on the behavior of the correlation

length within the individual chains in the

classical

spin model,

'

"

gives at

least

the right order of magnitude. However,

as

we will show, the

dras-tic

influence of some anisotropy resulting in

es-sentially different phase boundaries with the field applied along different directions (including the introduction ofa, spin-flop pha.se) ca.nnot be

re-produced by this isotropic theory. Therefore a,

conclusion about the validity of the description of

this effect

is

precluded so

far.

In this Letter we will present the

first

results

ofa

transfer-matrix

approach for the

classical

Heisenberg chain with small orthorhombic an-isotropy. We will show that this approach

is

cap-able of reproducing the field dependence ofthe

Noel temperature in

a series

of

real

pseudo

one-dimensional Heisenberg S=_{2 systems with the} field along each ofthe three principal

axes.

In

Figs.

1-3

our data on the phase diagrams

are

presented for a selected, representative

series

ofpseudo one-dimensional systems. Some

characteristic

parameter values ofthis

series

are

tabulated in Table

I.

The data were obtained from a continuous heating method, thus

identify-ing the transitions by the maxima in the specific heat. From inspection ofthese results it

is

ob-vious that in a general sense the

rise

in TN(H)

J/4=-0.70K

—

—

isotroprc theory anisotroprc theory a-0.04 P =1.5a 2— (I) Q) CA 0 easy ~ inter 0.6 0.7 0.8 0.9 TN(H)/ TN(0) 1.0

FIG.

1.

Experimental phase diagram of (NC5H6) MnC13 H20

(PIC),

together with the theoretical

pre-dictions. Data for the hard axis are not shown since

they reveal a more complicated phase diagram which is most likely due to asma11 canting ofthe magnetic moments.

strongly depends on the degree of one

dimension-ality, characterized by the entities in Table

I.

This indicates that basically the understanding of this anomalous behavior must be sought in the

properties ofthe individual chains. Therefore,

we will

treat

the system

as

consisting of loosely coupled chains.

For

such a system TN

is

implicity given in the mean-field approach by'

2zJ''y(T~(H), H)=

_1,

VOLUME 40,NUMBER 16

PHYSICAL

REVIEW LETTERS

17APRIL 1978 0. 8- 04-0.3— 0,4— 0.2- -2.6K .01

4

0. 2-0.1₀ Q4— 0.2— I 0.9 1.0 1.1 Tg (H)/TN(0) 1.2 I 0.8 1.0 1,2 1.4 15 Tq (H)/Tg (0)

FIG.

3.

Experimental phase diagram of (CH3)2NH2

xMnC13 (DMMC), together with the theoretical

pre-dictions.

FIG.2. Phase diagram ofCsMnBr3 2820 {CMB) and CsMnC13 2820 (CMC). The drawn curves denote the theoretical prediction inthe presence of ortho-rhombic anisotropy. The theoretical results for the

isotropic case are not plotted separately, since they

almost coincide with the predicted behavior for the

intermediate axis.

where X

is

the staggered susceptibility ofan

iso™

lated chain and

s

J'

is

interehain interaction.

Hence we will have to evaluate the staggered

sus-ceptibility, X(T, H), ofan isolated chain with

or-thorhombic anisotropy. We will not apply the

us-ual approximation ofX in terms ofthe

correla-tion length.

'-"

As we

are

dealing with high-spin (8=—,_{) systems we will evaluate X in the}

frame-work ofthe

classical

spin model. Moreover,

it

has previously been shown that the

classical

mod-el

gives remarkably good

results

for those

ther-modynamic variables in the paramagnetic state

which greatly depend on long-range

correlation.

"'

We will consider here the

classical

nearest-neighbor Hamiltonian oforthorhombic symmetry,

TABLE

I.

Review ofsome characteristic parameters ofthe compounds studied inthis

Letter. TN(0) isthe zero-field ordering temperature;

J

and

J'

are the intrachain and

interchain exchange coupling, respectively; and

S«,

_,

denotes the entropy ratio S(TN)/

S(~).

The anisotropy parameters + and P are defined in the text.

Compound TN(0) (K) seri& (NC5H6)MnC l3 H20 CsMnBr3~_2H20 CsMnC13~_2H20 (CH3)2NH2MnC13

2.

38 5.75 4.89

3.

60 —

_0.

7 —2.6 —

_3.

₀

—

_5.8 39% 15% 12% 39'

Vx]0

~ 10

8x

10 10

4x10

2

1x10

'

4x10

'

1.

15x10

1.

5 3 4 8 11 12,13 13 14 1109

VOLUME 40,NUMBER 16

PHYSICAL

REVIEW-

LETTERS

17APRIL 1978 given by

K.

~~(&

',

K.

_,

)

Z(2g

HS(S

+S;,

')+2JS(S+l)[5;

5;,

+&(S

S;,

'

—35;

5~,

)+&(S;"S;,

"-S

S;„,

')]),

(2) where

S;

=

_(S;",

S,

S ) =

(cosy;

sin8;, sing;

sin8;,

cos8;).

(5)

and

exp[

-X$„$,

)/kT] =_(2w)

'

Z

Z

_K„,

_(8„8,

_)exp[i(my,

_-kp,

_)1.

The central problem in the

transfer-matrix

approach

is

to solve the integral equation

f

'd8,

fo"

dy, sin8, exp[

-K(5„'5,

)/k T]

((8„y,

)=&

₍₍₈

„y,

) (4)

because all static properties ofthe chain can be formulated in terms of the eigenvalues and eigenfunc-tions ofthis equation.

In order to reduce the number of integration variables in Eq. (4)we introduce the Fourier expansions

g(8;,

y,

) =(2msin8, )

"'

Z

@' _{(8;)exp(imp;)}

m=-~

Substitution of

Eqs.

(5)and (6)in Eq. (4) yields [because ofthe orthogonality ofthe functions exp(imp, )]

the

set

ofcoupled integral equations

d82(sin8, sin8,

)"'K,

(8

„8,

)4',(82)=X4

„(8,

_),

m =0,

+1,

+_2,

...

.

$=~ao

It will be

clear

that Eq. (V) cannot be handled

nu-merically unless the infinite summation

is

trun-cated in some way. In the

case

of axial symmetry around the field direction'

"

this gives no com-plications because

K,

(8

„8,

)

&„,

&K,

(8„8,

),

and only one term in the summation survives. Ifthe diviations from axial symmetry

are

small, that

is l2JS(S+1)e l«kv', a

meaningful

truncation

is

possible because one finds

a

rapid

decrease

in magnitude ofthe

K,

&(8„8,

) with

in-creasing 1m

-I

I.

Moreover, only a

restricted

number

of

equations has to be retained because

the susceptibilities depend mainly on the

per-turbed eigenstates related to m=O,

+1

in the

axi-ally symmetric

case.

Even after this truncation,

practical

calculations

are

almost impossible

with-out further simplifications. However, exploiting

the

C,„symmetry

ofthe system, Eq. (7) can be

factorized into four smaller subsets. The inte-grals over

~,

occurring in each of these subsets

are

approximated by a

discrete

quadrature

scheme. The choice of

a

set

ofvalues for

~,

iden-tical

to the

abscissas

used in the integration over ~2 results in

a

matrix eigenvalue equation' which can be diagonalized by standard routines. The dimension of these

matrices,

which ultimately

sets

a limit to the applicability ofthe method,

in-creases

progressively when I

J

I/OT'

increases

because both the number of coupled equations and

the number of

abscissas

needed to obtain results

of sufficient accuracy grow rapidly at lower tern

peratures.

Using the method sketched above, we calculated

phase diagrams for (NC,H,)MnC1, H,Ot(PMC)

CsMnBr,'2H, O (CMB), CsMnCl, '2H,O (CMC),

and (CH,

_),

NH, MnC1, (DMMC). In order to

elimi-nate the interchain molecular-field constant

zP',

Eq.

(1)was applied in the form y(&N(H), H

)

=g(T~(0),

0),

where y

is

the appropriate staggered

susceptibility. Basically we did not use adjustable

parameters; the values for

J

were fixed to the

re-ported values. From the zero-field energy

dif-ferences,

per spin, between the

antiferromag-netically aligned states parallel to the easy and

intermediate axis (&E,)and the easy and hard

axis

(&E,) we define the reduced anisotropy en-ergies

o

and P

as

a

=&E,

/2I

J

lS(S+1),

p =&E2/2 I

JIS(S+

I).

(8)

Note that the values of e and & for _given n _{and p}

depend upon the correspondence between the

co-ordinate system

_{x, y,}

~ and the magnetic

axes.

For

CMC and CMB, &and p were taken from the

literature,

"'"

whereas for PMC and DMMC the

anisotropy parameters were obtained in an

VOLUME 40,NUMBER 16

_PHYSICAL

_{REVIE%' LETTERS}

₁₇APRiL 1978

gous way,

as

will be reported elsewhere.

"

Theo-retical

phase diagrams obtained with quadratic

single-ion anisotropy instead of Eq. (2) showed only minor deviations from the results presented

in

Figs.

1-3.

With the field applied along the easy

axis,

we

have to consider two different susceptibilities

corresponding to the directions of the staggered

field in the antiferromagnetic and spin-flop

state.

The phase boundaries found in this way

intersect

at

the

bicritical

field IIb;, which

is

given to

a

very

good approximation by gtLBHb;,=4I

J

IS~2o.

.

For

fields higher than Hb;, the spin-flop phase yields the highest, and thus physically realized, value

of T~(H), while for fields smaller than Hb;, the situation

is

reversed.

The agreement between experimental and

theo-retical

results

is

satisfying but it deteriorates

somewhat

at

higher values ofthe reduced field gpsH/2 I

/IS.

Whether this

is

due to a deficiency

in the applicability of Eci.

(l)

at high fields

re-mains to be seen. Possibly

a

self-consistent type

ofapproach would give better

results.

However,

these high-field deviations could also be due to

the suppression ofquantum fluctuations,

as

de-scribed by Imry, Pincus, and Scalapino.

"

There-fore

it

seems worthwhile to investigate the effect ofquantum

corrections

on the present theory.

The results ofthe present analysis may be sum-marized by stating that the,

at

first

sight,

anoma-lous behavior of the phase boundaries of pseudo one-dimensional Heisenberg systems can be

quan-titatively understood from the behavior ofthe

staggered susceptibilities of the isolated chains,

provided that the orthorhombic anisotropy terms

are

not neglected.

The authors wish toacknowledge the

assistance

of

J.

C.

Schouten,

J.

Millenaar, and

J.

A. M.

Boos.

This

research

is

partially supported by

the Stichting Fundamenteel Onderzoek der Mater-ie

'D.

P.

Almond and

J.

A. Rayne, Phys. Lett. 55A, 137

(1975).

2%'._G.Clark,

_L.

J.

_{Azvedo, and}

_E.

_{O. McLean in}

Proceedings ofthe Fourteenth Internationat Conference

on Low TemPexatuye Physics, Otaniemi, Einland, 1975 (North-Holland, Amsterdam, 1975), p. 391.

3C.Dupas and

J.

P.

Renard, Solid State Commun. 2O, 381 (1976).

J.

P.

Goren,

T.

O.Klaassen, and ¹

J.

Poulis, Phys.

Lett. 62A, 453 (1977).

W..

J.

M. deJonge,

J.

P, A. M,Hijmans,

F.

Boers-ma,

J.

.C.Schouten, and K.Kopinga, Phys. Rev. B{to

bepublished)

.

J.

ViOain and

J.

M. Loveluck,

J.

Phys. (Paris),

Lett.39, L77 (1977).

VM.

E.

_{Fisher, Am.}

J.

_{Phys. 32, 343 (1964).}

M.Blume,

P.

HeBer, and ¹ A. Lurie, Phys. Rev.

B

11,

4483 (1975).

J.

M. Loveluck, S.W.Lovesey, and

S.

Aubry,

J.

Phys. C8, 3841 (1975).

'

_S.

_{W. Lovesey and}

_J.

_{M. Loveluck,}

_J.

Phys. C 9,

3639 (1976).

"P.

M.Richards, R.K.Quinn, and

B.

Morosin,

J.

Chem. Phys. 59, 4474 (1973). '

_T.

_{Iwashita and}

N. Uryu,

J.

Phys. Soc.Jpn. 39, 1226

(1975).

' See, for instance, K.Kopinga, Phys. Rev. B 16,

427 (1977),.and references therein.

'4R._D.Caputo and R.D.Vfillett, Phys. Rev. B 13,

3956(1976).

'_{See, for instance, Y.Imry, Phys. Rev. B 13,3018}

(1976),and references therein.

' Y.Imry, -P. Pincus, and D. Scalapino, Phys. Rev.

B12, 197S(1975).

' R.

J.

Birgeneau, R.Dingle, M.

T.

Hutchings, G.

Shir-ane, and

S. L.

Holt, Phys. Rev. Lett. 26, 718 (1971).

S.

A.Scales and H.A.Gersch, Phys. Rev. Lett. 28, 917 (1972).

' K.Takeda,

J.

C.Schouten, K.Kopinga, and W.

J.

M. de Jonge, Phys. Rev. B17, 1285 (1978),