Physica l13B (1982)380-390 North-Holland Publishing Company

T H E M A G N E T I C P H A S E D I A G R A M O F T H E Q U A S I T W O - D I M E N S I O N A L H E I S E N B E R G

A N T I F E R R O M A G N E T KzMnF4

C . A . M . M U L D E R , H.L . S T I P D O N K , P .H. KES , A.J . van D U Y N E V E L D T and L.J. de J O N G H

Kamerlingh Onnes Laboratorium der Rijksuniversiteit Leiden, Nieuwsteeg 18, 2311 SB Leiden, The Netherlands

Received 15 December 1981

We present measurements of the differential susceptibility of the quasi two-dimensional antiferromagnet K2MnF4 as a function of temperature (3K< T<60K) and in external magnetic fields up to 142kOe. The zero-field ordering temperature is found as Tc(H = 0) = 43.94- 0.2 K. Anomalies are found in the x(H) curves at constant temperature near H = 55 kOe, indicative for spin-flop transitions. This transition field remains equal to within 10% of its low-temperature value up to T - 40 K. The transition line in the H, T phase diagram merges smoothly into a second-order line ending at Tc(H = 0). In the present x(T, H) data no evidence is found for a phase boundary separating the spin-flop from the paramagnetic phase. It is not clear whether this is due to the fact that the associated anomalies in x(T, H) are just too small to be detected, or that the supposed spin-flop phase for H > 55 kOe does not correspond to an ordered phase but is an extension of the paramagnetic regime. The latter possibility arises because K2MnF4 in its spin-flop phase might correspond to the two-dimensional planar model with cubic anisotropy. This cubic anisotropy may explain the fact that x(H) does not diverge at the "spin-flop" transition, since it would entail a splitting-up of the first-order spin-flop line into two second-order boundaries encompassing an intermediate phase and meeting in a tetracritical point. The analogy with the tetracritical behaviour of RbMnF3, as recently reported, is discussed.

1. Introduction

In recen t years much effort has been d e v o t e d

to the s tudy of f i e ld - induced magne t i c t r ans i t ions

in an t i f e r romagne t i c ma te r i a l s [1]. In such stu-

dies in te res t ing p h e n o m e n a to e n c o u n t e r a re t he

field d e p e n d e n c e of the o rde r ing t e m p e r a t u r e Tc

itself, as well as the occu r r ence of mul t ic r i t ica l

po in t s in the H, T phase d iagram. T h e poss ib i l i ty

of s tudying mul t ic r i t ica l b e h a v i o u r in magne t i c

phase d i ag rams has a r o u s e d a lot of e x p e r i m e n t a l

and theo re t i ca l in te res t [see e.g. refs. [2-5], for

reviews] . W e m e n t i o n the bicr i t ica l po in t in a

weak ly an i so t rop ic H e i s e n b e r g a n t i f e r r o m a n g e t

[6, and r e fe rences c i ted there in] , the t r icr i t ical

po in t in a m e t a m a g n e t [7], and the te t racr i t i ca l

po in t occurr ing , e.g. in a l loys of mixed com-

p o u n d s with compe t ing an i so t rop ies [8-12], as

well as in uniaxia l a n t i f e r r o m a g n e t s u n d e r cer-

ta in cond i t ions [13-16].

T h e app l ica t ion of a magne t i c field exceed ing

the so-ca l led spin-f lop field (HsF) t r ans fo rms a

H e i s e n b e r g a n t i f e r r o m a g n e t with w e a k uniax ia l

an i so t ropy into an ea sy -p l ane ( X Y ) ant i fer-

r o m a g n e t [2, 17-21]. F o r two-d imens iona l lat-

t ices, the magne t i c o rde r ing in H e i s e n b e r g , X Y or

Ising sys tems d isp lays r a t h e r di f ferent p r o p e r t i e s

[22-25]. Surpr is ingly , the magne t i c phase

d i a g r a m s of the quasi tw o-d ime ns iona l ant i fer -

r o m a g n e t s have ha rd ly been exp lo red . T o ou r

k n o w l e d g e only a few spin-f lop field de te r -

mina t ions [26, 27] and the phase b o u n d a r y for the

quasi tw o-d ime ns iona l Ising a n t i f e r r o m a g n e t

Cs2CoBr4 [28] have been r e p o r t e d . M o r e

recent ly , r a t h e r ex tens ive s tudies on the mag-

net ic p r o p e r t i e s of a new ser ies of two-d imen-

s ional a n t i f e r r o m a g n e t s with gene ra l fo rmula

M(trz)2(NCS)2 were p e r f o r m e d [29-32].

F r o m the a b o v e it may be c lear that these

two-d imens iona l an t i f e r romagne t s , in pa r t i cu la r

those of the weak ly an i so t rop ic H e i s e n b e r g type,

p r e sen t cha l lenging ob jec t s for fu r the r s tudies , the

0378-4363/82/0000-0000/$02.75 O 1982 N o r t h - H o l l a n d

C.A.M. Mulder et al. / Magnetic phase diagram of K2MnF4 381

compound K2MnF4 being an excellent candidate. The observed, rather large Tc (=43 K) is thought to be induced primarily by the small uniaxial anisotropy arising (mainly) from the dipolar in- teractions. In the ordered region the magnetic moments for H = 0 are parallel to the c axis (tetragonal crystal axis), i.e. perpendicular to the antiferromagnetic layers. At T = 4.2 K the spin- flop field is of the order of 55 kOe.

In this paper we present measurements of the differential susceptibility of a K2MnF4 single crystal in magnetic fields up to 140 kOe (14T) and for temperatures up to 200K. These data enable us to trace a boundary separating the antiferromagnetically ordered region from the paramagnetic phase and from what appears to be the flopped phase. We have not been able, however, to detect a boundary separating the spin-flopped from the paramagnetic phase. Pos- sible explanations are discussed below.

2. Experimental details

The crystallographic and magnetic structure of K2MnF4 are too well known to be repeated here; for a review of the research on this compound covering the period up to 1974 we refer to ref. [22]. The present experiments were performed on a fairly large (84.0 mg), optically clear, single crystal in the form of a platelet (1.1 mm thick), with the c axis perpendicular to the plane of the platelet.

To generate the strong static magnetic field needed for the experiments a superconducting magnet was used, assembled at our laboratory in cooperation with Intermagnetics General Cor- poration (IGC-31010). The magnet consists of a stack of 26 modules, each containing two coun- terwound discs of NbaSn tape. the modular ap- proach is illustrated in fig. 1 and offers the ad- vantage that the system can be modified to achieve higher field strengths, greater homo- geneity, or a different design (e.g. split coil). Also if one module is inadvertently damaged it can be

Fig. 1. Schematic drawing of two modules of the super- conducting magnet. Each module contains two counterwound "pan-cakes" of Nb3Sn tape (thickness 0.2 mm). The direction of the current through the tape as well as the direction of the corresponding magnetic field are indicated by the arrows. In addition, each module has a shunt assembly (dotted area) which in case of a quench dissipates the magnetic energy in order to prevent the Nb3Sn tape from being damaged.

replaced easily. The magnet has a bore diameter of 52 mm, an outer diameter of 250 mm, while the coil length amounts to 270 mm. By means of a current-controlled power supply (IGC model 180-M) the magnet was designed to generate a magnetic field of about 150 kOe (at 140 A) at a liquid helium bath temperature of 4.2 K. The current output of the power system is controlled by an electronic ramp generator that delivers continuously adjustable rates of current increase or decrease. Safeguards protecting both the powered device and the power supply are pro-

382 C.A.M. Mulder et al. / Magnetic phase diagram of K2MnF4

vided by a circuit breaker , by a reverse voltage turn-on circuit, and by a circuit which protects an excessive tempera ture rise in the power supply. During the measurements a 32-channel transient recorder connected to the individual modules of the magnet is available to detect the instigator of a quench.

Because of the hysteresis of the magnet (Hr 10 kOe) the current is not an accurate measure of the central field of the magnet, so that a calibrated I G C magnetoresist ive probe is attached to the outside of the bore tube at the center of the magnet. The large change in resistance is easily measured as the field is altered. The magnetic field is determined to an absolute accuracy of ap- proximately 0.1%.

The magnet is placed in the outer helium dewar of a double dewar cryostat. The inner helium dewar fits the clear bore of the magnet and contains the mutual induction coil system. This system consists of two identical but oppositely wound secondary coils and a primary coil extending over the length of the two secon- daries. The advantage of such a system is that a bridge is not needed as the susceptibility is directly obtained from the change in the output voltage over the secondary coils after moving the sample from the center of one secondary coil to the other, while the pr imary coil is fed by an a.c. current stabilized power supply. Both the in- phase component X', and the out-of-phase com- ponent X" of the complex differential suscep- tibility X = x ' - i x " can be detected simul- taneously from the output voltage of a two-phase lock-in amplifier (PAR model 124, 127). The frequency of the a.c. current can be varied in between 1 Hz and 10kHz. Since the present experiment was aimed at mapping the phase boundaries, a systematic study of the frequency dependence of the susceptibility was not per- formed. The data reported below refer to a frequency of 117 Hz, at which the signal-to-noise ratio was found to be opt imum.

At liquid-helium temperatures the empty coil system is properly orientated with respect to the

magnet by feeding an a.c. current through the magnet and then minimizing the output voltage over the secondary coils. In such a way the actual value of the magnetic field at the position of the sample in one of the secondary coils needs only to be corrected for the inhomogeneity of the magnetic field. Within approximately 2 5 m m from the position at which the maximum field is attained the variation of the magnetic field is parabolic while over this distance the reduction of the magnetic field even at the strongest fields is less than 1%. In addition, the distance be- tween the centers of the two secondary coils is 30 mm, so that for a sample of e.g. 2 m m thick- ness the inhomogeneity of a field of 100 kOe over the crystal is less than 100 Oe.

Inside the mutual inductance coil system two concentric glass tubes are placed by which the sample can be isolated from the (inner) 4He bath. This insert is similar to the one described by Groenendi jk et al. [33] and offers the advantage that the coil system remains at the same (liquid 4He) temperature. The tempera ture of the sam- ple may be stabilized in the range from 1.2 K to 300 K by means of regulating the pressure of the helium exchange gas between the tubes and monitoring a current through a heater. In zero applied magnetic field the tempera ture is measured by means of two calibrated carbon resistors (Allen and Bradley R(300 K ) = 5601)). Below 30 K the tempera ture could be kept con- stant to within 0.1%, increasing to 0.5% at 100 K. The absolute accuracy of determining the tempera ture is bet ter than 1%. In the presence of strong magnetic fields the effect of mag- netoresistance strongly influences the values of the carbon resistors, and a field-independent capacity the rmomete r (Lakeshore CS 400 GR) is used to stabilize the temperature. Before and after the field is applied the capacity ther- momete r is calibrated by comparing it with the carbon resistors. Above 150 K the sensitivity of the carbon resistors becomes too small, and we determined the tempera ture in this range by means of a calibrated platinum resistance ther-

C.A.M. Mulder et al. I. Magnetic phase diagram of KzMnF4 383

mometer, with an absolute accuracy better than 1%.

3. Results and discussion

We first report on our measurements of the susceptibility as a function of temperature in zero static external magnetic field. The diamagnetic contributions of the sample holder assembly ( - 1 . 4 x 10 -7emu) and the crystal ( - 0 . 3 × 10 -7 emu) have been subtracted from the data. A correction for demagnetizing effects was not con- sidered necessary. The results in the temperature region between 3 K and 200 K are presented in fig. 2 for the parallel 0(It) as well as for the per- pendicular susceptibility (X_J, i.e. with the oscillat- ing field applied parallel or perpendicular to the preferential axis (c axis) of the crystal, respectively. It is seen that XII approaches zero for T - > 0 K , while X.(0) tends to 2.010x

10-2emu/mol. The absolute accuracy of the measured susceptibilities is better than 0.5× 10-7emu which corresponds with 0.012x 10 -2 emu/(mol K2MnF4). The measurements are in nice qualitative and quantitative agreement with those performed by Breed [26].

In the insert of fig. 2 the susceptibility just around the antiferromagnetic ordering tem- perature Tc is shown on an enlarged scale. From the definition of T¢ as the temperature at which aXII/OT reaches its maximum value, we deduce the ordering (or N6el) temperature to be 43.9-+ 0.2K. Note that at this temperature a small anomaly in X± is observable also. There exists some controversy in the literature about the value of Tc of K2MnF4. The neutron diffraction studies of Ikeda and Hirakawa [34] and Bir- geneau et al. [35] give Tc- -42 .37K and Tc = 42.14K, respectively. On the other hand Breed and co-workers [26, 36] (cf. also ref. [22]) deduce from susceptibility measurements T~ =

o E

E

%

I

. 0

{3.

u

i i i ] r , I I I I ~ I r I l I I I I

e x p o n s l o n ^ o o /

I I I l ]

i . . . . i l i~ ~ .o

0

0

1.7 ÷

4 0 5 0

C I I I I 1 I I I I L I I I I 0 5 0 I O 0 1 5 0 2 0 0

t e m p e P o t u e e ( K )

Fig. 2. The differential susceptibility of 'a single crystal of K2MnF4 as a function of tempera ture in zero static external magnet ic field. The oscillating field ( - 2 . 5 0 e , 117 Hz) is applied either parallel (open circles) or perpendicular (shaded circles) to the preferential axis (c axis) of the crystal. The full curves represent best fits to renormalized spin-wave theory or to the high- temperature series expansion.

384 C.A,M. Mulder et al. / Magnetic phase diagram of K2MnF4

45.0 ___ 1.0 K, which is in good agreement with our present result of Tc = 43.9---0.2 K. In order to check on this discrepancy, we carefully remeasured the zero-field susceptibility in a set- up designed for weak-field measurements [33], where use has been made of independently cali- brated thermometers . We obtained essentially the same result for Tc to within the experimental absolute accuracy of 0.2 K. We point out that in our susceptibility set-ups the difference in tem- perature between the sample and thermometers is determined to be less than 2 0 i n k at 40K, which is much less than the absolute accuracy of our tempera ture determination (0.2K). Un- fortunately, we cannot find any comments in refs. [34, 35] as to the absolute accuracy of the tempera ture determination in those experiments.

In the tempera ture region above 50 K the sus- ceptibility is found to be fairly isotropic and we have analysed our data using the high-tem- perature series expansion [22]. In calculating the antiferromagnetic susceptibility of the S = 5/2, quadratic Heisenberg lattice we used for the exchange constant IJI/k -- 4.20 K [22, 26, 37, 38]. The series prediction (see fig. 2) is in good agreement with the measurements for tem- peratures above 60 K and nicely covers the tem- perature regime in which the broad maximum in

the susceptibility occurs. In the low-temperature region, the parallel

susceptibility XII as well as the perpendicular sus- ceptibility X± have been calculated with a two- sublattice spin-wave formalism [39-42] for the two-dimensional Heisenberg antiferromagnet. Effects of renormalization [43, 44] in terms of the so-called Oguchi corrections [45] were incor- porated in the calculations in a pert urbative way up to second order in 1/2S. With IJI/k = 4.20 K and a temperature- independent , uniaxial aniso- tropy parameter aA = HA~HE = 3.9 X 10 -3 [26, 46] we obtain an excellent agreement (see fig. 2) with the measurements for temperatures up to ap- proximately 20 K, i.e. an overall fit by the spin- wave approach to the experimental susceptibility up to - 0 . 4 T~. This agreement between the pre-

diction by spin-wave theory and eXperiment is somewhat bet ter than obtained previously [36, 47], cf. also [22].

After this recapitulation of the susceptibility behaviour in zero field, undertaken mainly to check on the magnetic purity of our K2MnF4 crystal and on the reliability of our new experi- mental equipment, we now turn to the field- dependent measurements. Before presenting our data we briefly summarize the behaviour expec- ted for a weakly anisotropic, uniaxial Heisenberg antiferromagnet in a field applied parallel to the easy axis [17, 18, 22, 48-51]. The field-depen- dence of the magnetization and the differential susceptibility at T = 0 K in the simplest case is as sketched in fig. 3a. As soon as the fieid-depen-

1 2 • dent magnetic energy, ~(X~-XJO H , exceeds the anisotropy energy K the magnetic moments will become directed perpendicular to H. This tran- sition is thought to be of first order, entailing a discontinuity in the magnetization and a diver- gence in X- The corresponding spin-flop field is to a good approximation given by

= ~ 2K ~,/2 ~ 2HAHE ~,/2 HsF tX---~-XIIJ = l 1 - XII/X±J '

where the last equality follows by introducing the exchange field HE = 2zlJIS/gtxa and an aniso- tropy field by H A = K/Ms, Ms being the sublattice magnetization. If the magnetic field increases from USE to the saturation field He, the moments are gradually rotated back towards the easy axis, the Susceptibility being approximately given by X=X±~-Ng21z2/4z[J [. The saturated paramag-

netic phase is then attained via a second-order transition at Hc = 2HE-- HA = 4z[JiS/gttB. Both these critical fields HSF and H¢ are dependent on temperature , and for low-dimensional systems a phase diagram may be obtained as sketched in fig. 3b, where also abbreviations for the phase boundaries are introduced. Current ideas agree in that the first-order AS-line and the second- order SP and AP-lines join (tangentially) in a bicritical point, indicated by (/-/2, T2). Usually

C.A.M. Mulder et al. / Magneac phase diagram of K2MnF4 385

M V- HSF HC

O

M

t / - / / i

"'t---H H1 H2 Hc

C

HSF :H

,H

~H

H

HC ~ p a n o

spm-flap )

~,--'~"~ (H2,T 2 ) ontifenpo

I • T Tc (0)

b H

H C ~ p a n a

spin - flop

,/ H, Infenm. ~ (H4.,T 4) H , ~ -T

T c (01 d

Fig. 3. The expected behaviour at low temperature of the magnetization M and the differential susceptibility X = (eM/aH)r as a function of field for weakly anisotropic anti- ferromagnets (H is applied parallel to the easy axis). Fig. 3a displays the usual situation of a uniaxial anisotropic antifer- romagnet, the corresponding phase diagram (fig. 3b) exhibits a bicritical point (/-/2, 7"2). Fig. 3c and 3d refer to a situation where the first-order spin-flop boundary is split up into two second-order ones encompassing an intermediate region, which give rise to a tetracritical point (//4, T4).

HsF does not change much with temperature, which implies the anisotropy energy K and g i - XII to have similar temperature dependences. For a number of compounds the temperature varia- tion of K has been studied by antiferromagnetic resonance, the result being K ~ Ms" with n be- tween 2 and 3. Indeed a similar temperature dependence is found [36, 43] for -XII, whereas Xl is not much dependent on T.

So-called tetracritical behaviour [17, 18, 49-51] may occur, e.g. if a cubic anisotropy is added to the above described uniaxial one. In that case the magnetization and the susceptibility behave as sketched in fig. 3c and, correspondingly, the

phase diagram looks like fig. 3d. The first-order spin-flop boundary is now split up into two second-order ones, which encompass an inter- mediate region in which the alignment of the magnetic moments is along a direction inter- mediate between the easy axis and the so-called basal plane. Two second-order transition fields /-/1 and /-/2 are now found at T = 0 K, and the corresponding second-order lines meet the AP and SP-boundaries in a tetracritical point (/-/4, T4). In between H1 and /-/2 the susceptibility attains a (fairly) constant value, /~12, which, as well as the difference (HE--H1), will be deter- mined by the ratio of the cubic to the uniaxial anisotropy.

The above discussion around fig. 3 refers to the situation of internal fields and internal sus- ceptibilities. Demagnetizing effects prevent the susceptibility from diverging at HsF (Xm~x = l /D) , and allow the spin-flop transition to occur over a range of the applied field (from HSF to HSF +

Hdem) [1]. In a favourable case one has/-/2 - H1 ~> Hdem and thus X~2 ~ l /D, so that the occurrence of the intermediate phase may be distinguished from the normal demagnetizing effects. To our knowledge an intermediate phase has thus far been observed in two compounds [15, 16].

The experimental susceptibility measured for a number of constant temperatures with the field parallel to the c axis of K2MnF4 is depicted in fig. 4. This figure shows the real part X' of the complex susceptibility only, the imaginary com- ponent X" being always smaller than 1% of X' with no apparent anomaly in X" at the transition field.

The behaviour of x ' ( H ) resembles that for a spin-flop transition, substantial anomalies are observed in the x - H curves, and in the field range above the peaks the susceptibility ap- proaches X_L. However, the maximum value of the susceptibility does by no means reach the reciprocal of the demagnetizing factor, i.e. 1/D = 6 emu/mol, while Xma, -- 0.15 emu/mol. The temperature variation of gm~x is illustrated in fig. 5. The behaviour of X around the maximum is

386 C.A.M. MuMer et al. I Magnetic phase diagram of K2MnF4

0 o - o - - o - . - o . o . O ~ I I I

, D

O ~ 1 T A

0 50 100 150 mognetic field (kOe)

Fig. 4. T h e differential susceptibility of K2MnF4 as a function of the external magnetic field applied parallel to the c axis for various temperatures. The drawn lines are a guide to the eye only.

given on an expanded scale in fig. 6. If evidence for the rounding of the susceptibility peak in fig. 6 is to be found from the demagnetizing effects, we expect for our platelet a peak width of 100 Oe, whereas our data are constant within 1% over a field range as large as 1 kOe. Furthermore, the inhomogenei ty of the applied field over the sample at 55 k O e is about 25 Oe, excluding this as a possible source of the excessive broadening.

1

0

T I I - -

\ ~'o \

°" I '

f \ 5 b. 4

J -- I I

10 2 0 3 0 4 0

t e m p e P a t u p e (K )

Fig. 5. T h e m a x i m u m value of the differential susceptibility of K2MnF4 as obtained from external magnetic field sweeps (nllc axis) at various temperatures. The dashed line is for visual aid only. The circles are obtained from fig. 3, whereas the squares refer to a direct measurement along the spin-flop transition line.

Misalignment of the field with the easy axis is thought to be less than 0.5 ° so that this will also not explain the discrepancy.

A similar problem was recently by Basten et al. [15], in a study of transition in COC12.61-/20. These gested the existence of a possible phase as described above. This behaviour was attributed to the

encountered the spin-flop authors sug- intermediate

tetracritical presence of

]51--

E

% _

-1-,

x5

S m ~3

10

I I I

T = 4 , 3 9 K

/ ?

I I

I

5 4

o" t\ I o

\

\o \

\

HSF \ \

L I §\ 5 5 5 6 5 7

m a g n e t i c f i e l d ( k O e )

Fig. 6. The differential susceptibility of K2MnF4 as a function of the magnetic field (/-/[Ic axis) around the spin-flop tran- sition field Hsv at T = 4.39 K. The dashed line is for visual aid only.

C.A.M. Mulder et al. / Magnetic phase diagram of K2MnF4 3 8 7

magneto-elastic coupling terms. In addition,

recent experiments in RbMnF3 [16] present evi- dence for an intermediate phase in this cubic antiferromagnet. The tetracritical behaviour stems in this case [51] from the sign of the cubic anisotropy constant a, which is such as to favour

alignment of the magnetic moments along the diagonal directions (a > 0). For cubic lattices the dipolar anisotropy is extremely small, and thus

the main source of anisotropy in RbMnF3 is thought to arise from the cubic crystal field split- ting term [52, 53]

a{S 4 + Sy4 + S~4 _ ½ S ( S + 1)(3S 2 + 3S - 1)}.

For RbMnF3 this leads to a minute anisotropy field HA of =5 Oe [52] which should be com- pared with the dipolar anisotropy field of

-~2.5 kOe in K2MnF4. However, for K2MnF4 the same intermediate phase may occur as soon as

the uniaxial dipolar anisotropy is balanced by the

applied field near HSF. E.g. for K2ZnF4:Mn 2+ Folen finds a = +(5.6___0.3)x 10-4cm -1 [54] ,

quite similar to the result of a = + 4 × 10 -4 c m -1

obtained for RbMnF3 [55]. Finally, we recall that the present data were all

taken at the same frequency of l l 7 H z . It has

been shown previously [56-58] that the time constants r involved in the spin reorientation

processes near field-induced magnetic transitions may become very large, in particular when they have first-order character. Thus the measured a.c. susceptibility near the transition will become

considerably reduced with respect to the iso- thermal susceptibility Xr. However, since r usu-

ally increases on decreasing the temperature, such relaxation effects are unlikely to explain the

observed temperature dependence of the X peaks measured at a single frequency (see fig. 5).

For simplicity, we will call the observed behaviour around 55 kOe in K2MnF4 a spin-flop transition, although the true nature of the tran- sition remains to be determined.

We have collected in fig. 7 the various

experimentally determined transition fields to

1 5 0 ~ I I I i

O

7:3

-+~

(3

E

100

_.eoo_.o__ o . o - - o - - - o - O. ©

I antlfenpo o papa . +

0 20 40 60 tempeputupe IK)

Fig. 7. The magnetic phase diagram of K2MnF4. The closed circles refer to determinations from field sweeps at constant temperatures, whereas the open circles are obtained from temperature sweeps at constant external magnetic fields. The shaded circle corresponds to a measurement in a direction perpendicular to the phase boundary. The second-order phase transition line from the spin-flop to the paramagnetic phase is not observed experimentally (see the text).

construct the antiferromagnetic phase diagram. The closed circles refer to determinations from field sweeps at constant temperatures, as shown in fig. 4. For temperatures above 35 K the peaks

in the x - H curves become hardly discernable and the transition was detected by means of

temperature sweeps at constant external mag-

netic fields. These experiments yielded the open circles shown in fig. 7. For fields up to 30 kOe the

X versus T curves obtained were hardly different from the measurements of a'll ( H = 0) shown in

fig. 2. Thus Tc (H) is fairly constant but the experimental error increases so that To(H)= 43.9 --- 0.6 K for H ~< 30 kOe. For H > 30 kOe the scatter in the data again becomes considerable,

whereas the structure in the X versus T curves is less and less pronounced. In order to match the two experimental regions we scanned the H - T diagram around 40 K, 50 kOe in a direction per- pendicular to the phase boundary.

As was mentioned in the above, the measurements were confined to the situation

388 C.A.M. Mulder et al. / Magnetic phase diagram of K2MnF4

where H was applied parallel to the easy axis of the K2MnF4 crystal. However , we also deter- mined the field dependence of the ordering tempera ture with the magnetic field applied perpendicular to the preferential direction of the magnetic moments , i.e. To(H±). The result for H = 0 has already been given in fig. 2. In fields up to 30 kOe we found To(H±) = 44_+ 2 K, the relatively large error increases with H due to the difficult measurement as to detect a small anomaly in the fairly constant susceptibility, g±. Above 30 kOe we were no longer able to detect

the anomaly in g±. Returning to fig. 7 we note that (as usual) the

"spin-flop" field hardly varies in the low-tem- perature range. Up to T = 40 K the value of Hsr remains equal within 10% to that found at T = 4.39 K (fig. 6: HsF = 55.25 -- 0.04 kOe). The latter value is in good agreement with the one obtained earlier by Breed [26] f rom the magnetization curve at 4.2 K (HsF = 55.1-----1.0 kOe). From the gap temperature , i.e. the spin-wave energy gap extrapolated to zero tempera ture (T~(0) = 7.40 _+ 0.05K), as deduced from antiferromagnetic

resonance data by de Wijn et al. [46], one may derive HsF(T = OK) = 55.1 - 0.04 kOe, again in good accord with the above results.

We now turn to the discussion of the "missing

curve" in fig. 7, namely the SP phase boundary separating the spin-flop phase from the paramagnet ic one. In fact the diagram shown represents the field and tempera ture ranges scanned in the experiments. Up to the maximum magnetic field of 142 kOe we made several runs as a function of temperature , and up to tem- peratures of 60 K we per formed several magnetic field runs, without detecting any significant anomaly. Typical examples of strong field measurements as a function of tempera ture are shown in fig. 8, where we have included the H = 0 data for x±(T) for comparison. It may be seen that the measured xH(T) curves show no apparent structure, the susceptibility being in fact within the errors hardly dependent on field and tempera ture over the whole region of the

o E

IN

% ,2- 2

25 ;5 o_

' I ~ - - I ' -

oOo~ O°oOoO~ - -%- - - _ _ T °

o6--

° °oO°~o ° _

D

....... __

0 I I L I I 0 2 0 4 0 6 0

t e m p e r a t u r e ( K )

Fig. 8. The differential susceptibility of K2MnF4 as a function of temperature at 79.9 kOe (squares) and at l17 .2kOe (cir- cles) (H[[c axis). The dashed lines refer to the zero-field x l ( T ) measurements (cf. fig. 2).

phase diagram outside the antiferromagnetic phase. We remark that the experimental errors increase substantially for stronger fields. There- fore, if the anomalies in X occurring at the SP transition would be small, it is quite possible that we were just not able to detect them. Evidence that this might indeed be the case has been claimed for M n C I 2 . 4 H 2 0 by Butera et al. [59].

On the other hand we cannot exclude the possibility that what we have identified as the "spin-flop" phase does in fact not correspond to an ordered region, but is merely an extension of the paramagnetic regime. We have ment ioned in the above that K2MnF4 in its flopped phase would be equivalent to a two-dimensional XY magnet with cubic anisotropy and a considerable reduction of the ordering tempera ture could be expected. The fact that the susceptibility in the field-range 60 kOe < H < 142 kOe and tem- perature range 4 K < T < 60 K is nearly a con- stant is not so unusual as it may seem (if this part of the diagram would be an extension of the paramagnet ic regime). From numerical cal-

CA.M. Mulder et al. / Magnetic phase diagram o[ K2MnF 4 389

culations for antiferromagnetic Heisenberg chains and exact results for the S = ½ antifer- romagnetic X Y chain it appears that in those systems the susceptibility is nearly independent of field and temperature in a large part of the paramagnetic region [60-64], namely for tem- peratures below that of the maximum in x ( H = 0) (i.e. for kTmax~S(S+ 1)lJI) and for fields smaller than El-Hc. This independence of X is of course widely different from what one predicts on the basis of effective field theories, but it can be ascribed to the large degree of short-range order in low-dimensional antiferromagnetic sys- tems. Thus the behaviour seen in fig. 8 would not be contradictory to that expected for a low- dimensional antiferromagnet.

Clearly, additional experiments using other techniques are necessary to determine whether or not the strong-field region in K2MnF4 still corresponds to an ordered magnetic phase. At present a further discussion about the possible existence of a bicritical or a tetracritical point in K2MnF4 is of little value. In the case of a bicriti- cal point, experiments on other systems indicate it to occur near the maximum in the boundary enclosing the antiferromagnetic phase in the H - T diagram (Tb = 30 K, Hb = 60 kOe, see fig. 7). If, as we are inclined to believe, in K2MnF4 the SF transition is replaced by an intermediate phase, then the tetracritical point should be located by the vanishing difference H 2 - / - / 1 , see fig. 3d. This is difficult to determine from the present experiments, however we no longer find a traject of constant X in the x (H) curves for temperatures above T = 35 K. We stipulate that as long as no evidence is available for an SP boundary emanating from a multicritical point, such considerations reduce to mere speculation.

Acknowledgements

We wish to thank the Intermagnetics General Corporation for their cooperation in assembling the magnet. Further technical assistance was

provided by Drs. J.J.M. Peters, Dr. J.J. Smit, J.A. van der Zeeuw, W. Klijn and T.P.M. van den Burg.

Dr. A.F.M. Arts from the University of Utrecht is acknowledged for placing a single crystal of K2MnF4 at our disposal.

We thank Prof. W.J. Huiskamp and Prof. H.W. Capel for their interest in the present research.

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