Antiferromagnetic resonance at low frequency

Hardeman, G. E . G . Poulis, N.-J. 1955

Physica X X I 728-736

ANTIFERROMAGNETIC RESONANCE

AT LOW FREQUENCY b y G. E. G. H A R D E M A N a n d N. J. P O U L I S Communication No. 300c from the Kamerlingh Onnes Laboratorium, Leiden, Nederland

Synopsis When interacting with a very small or zero external magnetic field the antiparallel spins in an antiferromagnetic single crystal have fixed orientations along the most preferred axis. This preferred direction is due to the anisotropy energy. In increasing external fields the spins deviate from this preferred direction, the deviation depending on direction and strength of the external field and on temperature. Ar certain field strengths and directions, a transition occurs in which the spins flop over and orient approximately perpendicular to the field, At these so called threshold or critical fields electron resonance absorption can be found at low frequencies. The critical fields were therefore determined by adjusting the external field for maximum absorption. The measurements were done on the orthorombic single crystal of CuC12. 2H20 which is antiferromagnetic below about 4.3°K.

1. Introduction. T h e electron r e s o n a n c e p h e n o m e n a in the a n t i f e r r o m a g n e t i c single c r y s t a l of CuC12.2H20 h a v e b e e n discussed e x t e n s i v e l y by Yosidal), Ubbink ~) a n d N a g a m i y a 3 ) . I n t h e following r e v i e w of t h e i r calculations t h e a n i s o t r o p y of t h e g - f a c t o r is n o t t a k e n into account. T h e e q u a t i o n o f m o t i o n c a n t h e n be e x p r e s s e d as follows:

(l/7)(dI~/dt) = ( I * X H.ff),

(1)

in which Heft is the total effective field acting on the magnetizations I + or I - of the sublattices. As a solution of this equation one obtains the resonance condition which shows the dependence of the electron resonance frequencies on t h e m a g n e t i z a t i o n s a n d t h e e x t e r n a l field. Ubbink's calculations are b a s e d on the t h e o r y of G o r t e r and H a a n t j e s ~), using t h e i r expressions for I ~ a n d Heff. As t h e t h e o r y of G o r t e r a n d H a a n t j e s o n l y applies a t zero t e m p e r a t u r e , U b b i n k ' s resonance condition is also o n l y v a l i d at T = 0. S u b s t i t u t i n g zero for t h e f r e q u e n c y in the r e s o n a n c e condition, one can o b t a i n the locus of t h e e x t e r n a l field v e c t o r for zero f r e q u e n c y r e s o n a n c e (fig. l). According to t h e t h e o r y of G o r t e r and Haantjes, the c u r v e s O~, ~92, O'~ a n d ~9~ in the field s p a c e also show the t r a n s i t i o n f r o m t h e - - 728 - -

G. E. G. HARDEMAN AND N. J. POULIS

729

antiferromagnetic into the paramagnetic state. These curves correspond to absolute values of H of the order of 200000 0. The hyperbola F, also one of U b b i n k's solutions for zero frequency, shows the critical field and corresponds to fieldstrengths > 6500 0. Thus this hyperbola is the only characteristic curve which can be partially realised in the present experiments.

/

t

Z

Dz

AI

/ Fig. 1. T h e zero f r e q u e n c y r e s o n a n c e locii i n H - s p a c e a c c o r d i n g to U b b i n k's reconance theory.

The sPin directions and the resonance conditions for temperatures other than zero have been calculated to a first approximation b y Y o s i d a and N a g a m i y a . Nagamiya calculated the free energy of the crystal as a function of the orientation and strength of the magnetizations and the external field. Those orientations for which the free energy has an absolute minimum at a given temperature and external field are the realised ones. On the critical field, however, two different orientations correspond to the same minimum value of the free energy. Following this line of thought, N a g a m i y a found for this critical field: H~

H~

2AKI

2A(K 2 -- KI)

= -

1

a

(2)

where, K~ =

( N / 2 ) I02 (Cx - - Cy), K 2 =

A -~ 1/Z ± and a = I

( N / 2 ) I02 (C, - - C,)

--Xll/Z±

I 0 is the magnetization of the Cu ions in zero external field. N is the number of the Cu ions of the crystal. Cx, Cy and C, are anisotropy factors. ZIt and Z± are the susceptibilities respectively parallel and perpendicular to the spin directions.

730

ANTIFERROMAGNETIC RESONANCE AT LOW FREQUENCY

When" the field in the xz-plane passes this critical hyperbola, the magnetizations flop over from the xz-plane to the 4- y-axis. To a first approximation Cx, Cy, C, and A can be considered as independent of temperature. The temperature dependence of this critical field hyperbola is then only determined b y the temperature dependence of Xll and of I0. So it is to be expected that the system of temperature dependent critical field hyperbolae consists of a locus with the asymptotes in common. The resonance theory for finite temperatures was formulated b y Y os i d a 1) who arranged K i t t e l's equation of motion as follows:

1 dI* _ i± x (H-- AI T) + E d-7

(K2 -- Kt)

K, r±i_ ~ 17,

P0 -x

K 11.1±

,, -- P0

"

7

-J (3)

H is the external field, while A I represents the field which corresponds to the exchange interaction. The second term between the brackets represents the torque acting on I ± due to the anisotropy energy. As a result he obtained a complicated general resonance condition. The following special cases can be derived from this general condition. A. The external/ield H along the x-axis, but below threshold value. In this case, the antiparallel spins can be taken as directed along the + or -x-axis and the following resonance condition is obtained:

(~o/7)4 -- (o~/7)2 {/-/~2(1 + a 2) + 2 A K , + 2AK2} +

a2H4x - -

-- aH~ (2AK, + 2AK2) + 2 A K 1. 2 A K 2 -----0.

(4)

o~ is the resonance frequency, 7 is the magnetogyric ratio of the electron. Putting ~ = 0 in order to find the values of H , at which zero frequency resonance occurs we obtain the following solutions: H ° = -4- ~/(2AKt/a) or 4- ~/(2AK2/a )

(5)

The first solution corresponds to the intersection of the critical hyperbola (2) with the x-axis and is therefore equal to the critical value of H along the x-axis. The second solution cannot be realised, being above the critical value. B. H along the x-axis but above critical value. In this case the antiparallel spins can be assumed to be directed along the + or -- y-axis. The resonance condition being: (oJ/7)2 = 2A (K 2 -- Kt),

(6)

and _-

-- 2AK,.

(7)

Those resonance conditions for H along the x-axis which are of interest

G. E. G. HARDEMAN AND N. J. POULIS

731

for the experiment are plotted in fig. 2. The left and righthand part of the drawn curve represent conditions (4) and (7) respectively. The dotted part shows the part of curve B which is not realised, being below critical value, it violates the assumption on which B was founded. At T = 0 however, the zero points of A and B coincide on H ° = ~ / 2 A K 1, the critical value along the x-axis at T = 0 already predicted b y U b b i n k. Only in that case is B completely reahsed. 12000

/

8000

/

/ t

0

0

~

. 4CX30

8000

Fig. 2. The resonance conditions for H along the x-axis at 0 < T < TN. The dotted curve represents the part of formula (7) which is not realised.

Of,2 D1,2

A1, 2 $I, 2

Fig. 3. The experimental a r r a n g e m e n t : Oscillator _~ Frequency meter Detector P Power amplifier Amplifier T Time base amplifier Oscilloscope G Audio oscillator

From the foregoing considerations we see that for 0 < T < TN (where T N represents the N ~ e 1 temperature) and H along the x-axis, the zero frequency resonance field coincides with the critical field. Moreover, at T = 0 zero frequency resonance should be found on the whole critical hyperbola according to U b b i n k's theory. Therefore it seems reasonable to expect that this is also true for 0 < T < TN. Though the analysis of

732

ANTIFERROMAGNETIC

RESONANCE

AT L O W

FREQUENCY

Y o s i d a's formulae for this general case leads to very complicated expressions, there can be no doubt that in principal zero frequency resonance should be found when H has the critical value. In that case there is no orienting force acting on the spins which means that the torque of formula (1) becomes zero, resulting in a zero solution of ~o.

2. Experimental method. The electron resonance absorption measurements on antiferromagnetic copper chloride were performed, at a frequency of 3.5 MHz. This value can be consideredas zero frequency because of the high value of the magnetogyric ratio of the electron and the large line width. The experimental arrangement is shown in the block diagram on fig. 3. The coil containing a single crystal of CuC12. 2H20 was placed vertically in a cryostat. A coaxial cable connected this coil with an oscillator 01 of a type designed b y P o u n d . The external field H was obtained b y a normal Weiss electromagnet, which could be rotated in a horizontal plane in order to adjust the direction of H relative to the crystal axes. This external field was modulated at a frequency of 31 Hz b y means of a 30 W a t t amplifier P driven b y an audio oscillator G. The auxiliary coils used for this purpose, were fixed around the pole pieces of the magnet.

/

8500

8000

7500

/

7000

.J 6500 o

T

1.0

2.0

3.0

4.0

°K

Fig. 4. The temperature dependence of the critical field Hc along the x-axis. The curve shown was determined b y previous proton resonance experiments. The results of the zero frequency electron resonance experiment are shown b y the circles.

The 3,5 MHz output of 01 was detected (D1) in order to obtain the 31 Hz modulation of the R F voltage due to the resonance absorption. The modu-

733

G. E. G. HARDEMAN AND N. J. POULIS

lation field was adjusted to be only a fraction of the line width of about 30 O. Thus the detected signal was proportional to the derivative of the absorption line. This signal was amplified b y a narrow band amplifier A t tuned to 31 Hz. The output voltage was observed on an oscilloscope $1.

12OOO

I1000

I0000

9000

8000

J

7000

6O00 40 °

30 °

20 °

tO°

o( 0

o( 10°

20 °

30 °

aO°

Fig. 5. T h e c r i t i c a l field s t r e n g t h as a f u n c t i o n of t h e a n g l e b e t w e e n t h e e x t e r n a l m a g n e t i c field a n d t h e x-axis.

The field strength was measured b y means of proton resonance. For this purpose a separate coil containing water was placed just outside the cryostat where the field strength proved to have the same value as in the coil containing the crystal. This coil formed part of the tank circuit of a second oscillator 02 whose frequency was variable from 25 to 60 MHz. At different orientations of the magnet, the field strength was varied while at the same time the signal proportional to the derivative of the electron resonance line was observed. Exactly on top of the electron resonance line the signal

734

ANTIFERROMAGNETIC RESONANCE AT LOW FREQUENCY

becomes zero. While the external field was kept on top of the resonance line, the oscillator 02 was tuned until the proton resonance line was visible on the centre of the oscilloscope $2. The frequency of 02 was then determined b y a crystal controlled frequency meter F. The time bases of $1 and S 2 were obtained from a combined phase regulator and amplifier T driven b y the generator G. 3. Results and discussion, a. H i n t h e xy-p 1 a n e. The z-axis of the crystal was vertically adjusted using a polarizing microscope. In this case zero frequency absorption was found only if H was directed along the x-axis. At different temperatures ranging from 2°K to TN the zero frequency value of H , was measured. As a result we obtained the curve plotted in fig. 4. Within measuring accuracy this curve coincides with the critical field curve previously determined from the decomposition of the resonance lines of the protons belonging to the crystalwater. b. H i n t h e xz-p 1 a n e. Again the y-axis was adjusted vertically b y means of a polarizing microscope. At different temperatures the zero frequency field strength was measured as a function of the angle of the field with the x-axis (fig. 5). With the help of this diagram we plotted H ,2 = H 2 cos2a against H,2 = H 2 sin 2 a (fig. 6). Apart from non-systematic errors due to the limited measuring accuracy a system of straight, parallel lines was obtained for T < 4°K. This shows that up to 4°K the critical curves are hyperbolae with "identical values of (K 2 -- K1)/K t. Thus the ratio between the anisotropy constants Cy -- C, and C, -- Cy is invariant. Apparently the curves are no longer hyperbolae above 4°K. This must be due to the proximity of the transition to the paramagnetic state. From fig. 6 the common asymptotes were determined. As a result we found: H , / H , = + 0.740.

In fig. 7 four measured hyperbolae are plotted while the dotted line represents the critical hyperbola for T = 0 obtained b y extrapolation of the curve in fig. 4. For this critical hyperbola at T = 0 we found:

H,

42.2

1.

- - =

77.0

(8)

H , and H, are expressed in kilo-oersteds. As to the T = 0 hyperbola derived b y G o r t e r and H a a n t j e s a correction has to be made because of t i e anisotropy of the g-factor, which we had neglected until now:

H, 2AK,

(g~/g~)22A (K 2 -- K,)

=

1,

g,/g.

=

1.03

(9)

G.E.G. HARDEMAN

...........

AND

735

N. J. POULIS

I00

eke)'

51i :

3.

80

~,~~

(93".018

J

7

60

1 40 0

HI

IL

20

40

Fig. 6. H 2 as a f u n c t i o n of

H2~, d e r i v e d

(k¢,)'

f r o m fig. 5.

10000,

I

¢~

8000

6000

4000

\

/

Y

2000

8000

6000

4000

2000. Hz

0

Hz, 2 0 0 0

4000

6000

8000

Fig. 7. T h e four d r a w n curves are critica] hyperbolae m e a s u r e d for different temperatures. T h e dotted line shows the extrapolated curve for T = 0.

736

ANTIFERROMAGNETIC RESONANCE AT LOW FREQUENCY

comparing (9) with (8) we obtain: (K 2 -- KI)IK

1=

(Cy --

c21(cx -

cy)

= 1.94.

This is in good agreement with the results of the microwave experiments of Gerritsen 5) e t a l . Our thanks are due to Prof. C. J. G o r t e r for his suggestions and criticism and to H. J. G e r r i t s e n, phys. drs., for his suggestions, which led to this experiment. We are also indebted to Dr M. B 1 o o m and Mr. W. V a n d e r L u g t, nat. phil. cand., for their valuable assistance during the measurements. This work is part of the research programme of the. "Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)" and was made possible by a financial support from the "Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek (Z.W.O.)". Received 2 I-3-55. REFERENCES I) 2) 3) 4)

Y o s i d a, K., Progr. theor. Phys. 7 (1952) 425. U b b i n k , J., thesis Leiden (1953). Nagamiya, T., Progr. theor. Phys. I I (1954) 309. G o r t e r , C.J. and H a a n t j e s , J., Commun. KamerlinghOnnesLab.,Leiden, Suppl. 104b; Physica IB (1952) 285. 5) G e r r i t s e n , H . J . , O k k e s , R., B S l g e r , B. and G o r t e r , C . J . , Comrnun. No. 300b; Physica 88 (1955) 629.