Magnetic phase transitions and magnetization process in PrCo2Si2

Journal of Magnetism and Magnetic Materials 86 (1990) 225-230 North-Holland

225

MAGNETIC PHASE TRANSITIONS AND MAGNETIZATION PROCESS IN PrCozSi z N. IWATA Faculty of Science, Yamaguchi University, Yamaguchi 753,Japan Received 18 September 1989

The PrCo2Si 2 c o m p o u n d has a high-order commensurate structure with a propagation vector Q3 ffi (0, 0, 0.777) for TN ~ 30 K ~ T > 17 K a n d a structure with Q2 ffi (0, 0, 0.926) for 17 K > T > 9 K, and undergoes a transition between the high-order commensurate structure and a commensurate structure with Qz ffi (0, 0,1) at 9 K. The magnetic m o m e n t s are parallel or antiparallel to the c-axis a n d perpendicular to the ferromagnetic c-planes for all the ordered structures. The magnetization at 4.2 K proceeds by a four-step metamagnetic process. The magnetic transitions a n d the magnetization process have been studied by introducing a wave-like molecular field Hm(i)ffi Eq~(q)( J¢)cos(~qi + % ). Results of the numerical calculations made with the Ising spin chain are presented. The wavenumber-dependent molecular field coefficient ?~(q) h a s a m a x i m u m at q ffi Q3 and is large positive for 1 > q > Q3 and is negative for 0.4 > q > 0. The appearance of ~ e magnetic transitions is shown by calculating the free energy. The magnetization at T ffi 0 K proceeds by three intermediate structures: a structure with Q ffi 1 and m = ~ , a structure with Q = ~ a n d m 2- ; a n d a structure with Q ffi ~ and m ffi ~, whe~'e Q is a wavenumber for which the Fourier component of the total angular m o m e n t u m of the Pr atoms (JQ) is a m a x i m u m a n d m is a reduced magnetization. The temperature dependence of the (Jq~'S is also presented in comparison with the neutron diffraction data.

1. Introduction In a previous paper [1], we have reported magnetic o r d e r - o r d e r transitions observed in PrCo2Si 2. The compound has a body centered tetragonal structure, as shown in fig. 1, and has three different antiferromagnetic phases below T~ (--30 K). The Co atoms bear no magnetic moment, and the moments of the Pr atoms lie parallel or antiparallel to the c-axis and perpendicular to the ferromagnetic c-planes for all the phases below T~. The wavevectors are given as Q= (0, O, Q) (in 2~r/c unit). The values of Q is 0.777 (-- Q3) for T~ >_ T > 17 K and is 0.926(-- Q2) for 17 K > T > 9 K. A change from the high-order commensurate structure with Q2 to a commensurate structure with Q1 ffi I occurs at 9 K. Fig. 2 shows results of neutron diffraction measurements, which were reported by Shigeoka et al. [1]. The measurements were made at the Research Reactor Institute, Kyoto University. As can be seen, there are clear changes in the peak intensity at 9 and 17 K, and for the Q2 structure a higher order superlattice peak is also Observable.

Preliminary results o f magnetization measuremerits have been reported by Shigeoka et al. [2]. The magnetizations have been measured in static fields up to 57 kOe. A two-step metamagnetic

O:Pr •

Co

o

Si

i Fig. 1. The crystal structure of PrCo2Si 2 and arrangement of the Pr m o m e n t s for T ~ 9 K.

0304-8853/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

N. lwata / PrCozSi 2 transitions and magnetization process

226 1

I

l

the molecular field approximation, the Hamiltonian for the i th R-atom is given as follows

20 (100.074)

(100)

¢-

8 %

v

(100.223)

T1

¢'1

°~F(i) =

j 2 ( g , _ 1)2 - L-7.F;" £X~(4)4

gettBV

.~10

01=1

S,.I,

c

o. o.,,,

(4) = E(4)

00

-

~Ix

10

l

20

e-~R',

(2)

where ( ~ ) is the Fourier q component of (./2)'s

o.. r-~

q

e-~K'-

(3)

q

~ _

30

T (K)

Fig. 2. The experimental integral intensities of the (101-q) magnetic superlattice peaks vs. temperature for PrC02Si 2 from ref. [2].

Alternatively, the effective Hamiltonian for the ith atom in terms of the molecular field Hm(i) is given as , ~ ( i ) = - gj~B~ Hm( i ). By equating this and eq. (2), we obtain

° f 2 ( g J - 1) 2 _-7"-Tg_2.'T- E X ~ ( 4 ) e-iqR'. gJgePB V q

Hm(i )

process has been observed at 4.2 K. But the value of the magnetization at the highest field is 0.62#B per formula unit (Lu.), which is much smaller than the value of Pr 3+ moment (3.2/~a). Recently, high-field magnetization measurements have been made up to 300 kOe at the High Magnetic Field Laboratory, Osaka University. The magnetization proceeds by a four-step metamagnetic process and reaches a saturation value of 3.2#a/f.u. [3]. The value at the magnetization for H < 57 kOe is in agreement with that previously obtained in the static fields. This paper reports the results of numerical calculations based o n the molecular field theory in an attempt to understand better the complicated magnetic transitions and unusual magnetization process in the PrCo2Si z compound.

(4)

This wave-like field is determined by a consistency condition, the self-consistent field. As stated before, the magnetic moments and the wavevectors Q~'s of PrC%Si 2 are parallel to the c-axis for all the structures. Then we can give the thermal average ( ~ ) of an atom on the i th c-plane as follows ( Ji) = E ( Jq) cos(,rtqi + el}q). q

(5)

The wavenumber q is defined by this equation hereafter. The molecular field acting on an atom on the i th c-plane is rewritten by introducing a molecular field coefficient h ( q ) as Hm(i )

=

Eh(q)(Jq)

cos(~rqi + %).

(6)

q

2. Wave-like molecular field

The s - d interaction may be an important interaction in metallic rare earth compounds. The R K K Y interaction Hamiltonian is given as follows j2

*j =

-

=, E x ,

g=~Br

(1)

q

Since the total angular momentum ./ is a good quantum number, we let S = ( g j - 1 ) . / . Within

The compound exhibits huge anisotropy, and the crystal-field parameter B ° has been estimated to be - 8 K [3]. The spfitting of the energy between the lowest and the next lowest levels in the ground multiplet is found to be 170 K. Then, we can set J= = + J , and we have ~Ji) -- J tanh{ g j l ~ a J H m ( i ) / k T }.

(7)

We therefore have the problem of the Ising spin chain. The last three equations are to be solved self-consistently.

N. Iwata / PrCozSi 2 transitions and magnetization process

The free energy of an atom on the ith plane is given as

f~ = - k T In [exp( e , / k T ) + e x p ( - e , / k T ) ] + ½gjpn(J,.)Hm(i),

(8)

with the notation

E, = -g.d~mlHm( i ).

(9)

Therefore, the free energy per atom is given as follows

F = ( 1 / 8 ) ~_,f~,

(10)

i

where N is the number of the ferromagnetic planes in one wavelength.

3. Numerical results and discussion

3.1. Molecular field coefficient The compound has been observed to have high-order commensurate structures with Q3-0.777 for T s > T > 17 K and with Q2 -- 0.926 for 17 K > T > 9 K, and below 9 K it is collinear antiferromagnetism with Q1 = 1. In the present calculation, we shall assume that Q3 = 0.777 ffi 7 and Q2 = 0.926 = ~ . There is agreement between the experimental Q and the fractional numbers quoted here to within about 0.1% for Q3 and to within about 0.3% for Q2- For the Q3 structure, the wavelength, therefore, is 9 times as long as the c-parameter, and then there are 18 Pr layers in one wavelength. Whereas for the Q2 structure, there are 28 Pr layers in one wavelength. In this paper, the structure is refered to simply as Q-structure, where Q is the wavenumber for which the magnitude of the (Jq) is a maximum. For the Q3 structure, mathematical considerations predict that ten (Jq)'S with q = 0, 19, { . . . . . ~9 are relevant in the presence of a field and in the absence of a field five (Jq)'S appear with q - - x, 9 { . . . . . 9. On the other hand, for the Q2 structure fifteen (Jq)'S are relevant for H ~ 0 and at H = 0 seven (Jq)'S are relevant. Preliminary calculations predict that the numerical results are sensitive to the values of

227

X(q) with q = {, ~ , ~ , 1 and 0. For temperatures just below the T~, the magnetic structure m a y be purely sinusoidal with q = Q3 and (Jq)'S other than q = Q3 m a y be negligibly small. Then the value of h(7) is determined by the following equation, T~ =

k

'

(11)

here we let Jz = + J , as stated before. The values of ~ ( q ) with q - - 1 1~ 4 , _u 14, and with q = 1, were roughly determined by an attempt to fit the calculated transition temperatures to the data. The magnetization reaches saturation at a critical f e l d of 122 kO¢. Since the value of h(0) is very sensitive to the calculated critical field, we can roughly estimate the value from the data. Values of the other ?~(q)'s were estimated b y assuming that X(q) varies monotonically with q except in the region around q -- Q3- Numerical calculations were made for a number of combinations of X(q)'s. The appropriate fit parameters obtained in this study are plotted as a function of the wavenumber infig. 3. ]

30

I

i

i

]

i

i

I

t

i

i

~32 0.7

035

q

0.8

"~20 C)

~O"1 0 0

-10

-20 I

I

I

I

I

i

i

Fig. 3. Molecular field coefficmnt ~,(q) vs. wavenumber q. The

inset shows a magnification df the region in which the ~(q) takes a maximum.

N. lwata / PrCozSi2 transitionsand magnetizationprocess

228

10

r"

i

i

T (K) 15 20 I

l

25

30

'i

i

" ' ~ 0

2

O"

V

5

-1 2

Q'=t

A

0

511 "

q=9114

~

,

-,6 .

0 7114

-2

519

111 I/*

0

5'

10'

'

2'0 15 T (K)

2'5

30t

,," -20

03~ T N

Fig. 4. The calculated Fourier components of the total angular momentum vs. temperature.

I

I

|

I

i

i

Fig. 5. The calculated free energies for the Q,, Q2 and Q3 structures vs. temperature.

3.2. Magnetic phase transitions and magnetic structures The behavior of the calculated (Jq>'S is illustrated in fig. 4. For T N > T > 17 K, (Jq> with q --- Q3 = ~ is the main component throughout the region and vanishes at T N. The magnitude of the (Jq) with q = ~9 is the next largest component, which develops with decreasing temperature. However, the high-order superlattice peak which corresponds to the q -- ~ component has not been observed yet. The contribution to the free energy F from this component is positive: it increases the value of F, since A(q) is negative at q = ~. For the Q2 structure, the (Jq> with q = Q2 -- {~4 is the largest component, and magnitudes of (Jq>'S with q--- ~11, q = ~ and q = ~ are relatively large as shown in fig. 4. The superlattice peaks corresponding to the (J~>'s with q = Q 2 and q = ~ have been observed, as is shown in fig. 2. And also the superlattiee peak corresponding to the (Jq> with q - - - ~ has been observed from detailed experiments at T = 12 K [3]. The contribution to the free energy from these components are negative. The growing tendency of these components is to give rise to a crossover in the free energy curves for the Q~ and Q2 structures, as shown in fig. 5. Under these circumstance, there is a magnetic transition between the Q3 and the Q2 structures.

Below 9 K, the value of the free energy for the Q1 = 1 structure is lowest, and the structure becomes stable. An important and suggestive fact follows. Though the A(q) takes a maximum at q = Q3, the magnetic structure with q = Q2 or q = Q, can be stable over a wide temperature range. This is entirely expected for a compound which has a huge uniaxial anisotropy, as predicted by Bak and

Bochm[4].

The moment arrangements resulting from this work are illustrated in fig. 6. Since we take 4,q = 0 for all q, a paramagnetic layer appeares every 14 layers in the Q2 structure. We can choose the

(a) T=26K.0='//9(=03). I.,

{

I

~"

~

I

I

"°

I

I

,.

° . °

,{

(b) T=12K. O=13114(=Q;r)"

IITI_ lIT[ 1].IT]... lltlllllll Ill (c) T=4.2K.O=I(=Qt). ok,,)

Fig. 6. Arrangementsof the 1~"moments at 25, 12 and 4.2 K.

N. lwata / PrCozSi2 transitionsand magnetizationprocess ep13/14 among the value of 0 < ¢13/]4 < ~ / 2 8 which covers all independent values of e&3/14. A n d the others •q'S are determined as predicted by Mashiyama [5]. F o r the ~13/~4--~/28 case, the atoms on the 8th layer are ferromagnetic and the moments point upward, whereas the ferromagnetic moments on the 22rid layer point downward. However, for T = 12 K, the difference in the free energies for the two cases remains less than 1 0 - 3 ~ . The contribution to the free energy from the itinerant electron energy has been ignored in the present calculation. The spin density wave that arises in the itinerant electron system may favour some angle %. The values of ¢q'S will be those which make the total energy a minimum. However, it is likely that the important aspects of the magnetic transitions of the PrCo2Si 2 compound will be understandable within the scope of this study.

229

(cl) H
kl.!-i(b) 10kOe~H<37kOe, 0=13114, m=1/14.

l TIllllllll LLL

LLLLLLL

1111

LLL

_._

(c) 37kOe<~H<89kOe, Q=7/9, m=219.

11111I 11111 l tl[lt

t

(d) 89kOe<~H<123kOe, Q=3/4, m=ll/,.

T TT TT I

[

(e) 123kOe<~H, Q=O,m=l.

3.3. Magnetization process along the c-axis The behavior of the calculated magnetization at T = 0 K, as a function of the magnetic field is illustrated in fig. 7 by broken lines. The solid curve represents the magnetization data reported by Shigeoka et al. [3]. The measurements were made at 4.2 K. We confine ourselves to the calculation of the 0 K magnetization, since the temper-

T=4.2K

~m2 :2.

'-I

1

oS o

lOO

i

200 (kOe)

300

H Fig. 7. Experimental magnetization vs. applied field for PrCozSi2 at T = 4.2 K (solid curve from ref. [3]) and calculated magnetization (broken lines).

Fig. 8. Arrangements of the Pr moments under magnetic fields. Notations Q and m represent the wavenumber for which the (JQ) has a maximum and the reduced magnetization, respectively.

ature at which the measurements have been made is sufficiently low in comparison with the value of T~. The magnetization fitted values were to the data under assumption that the intermediate phases involve high-order commensurate structures with N < 28, where N is the number of the Pr layers in one wavelength. For magnetic structures with Q = ~11 and a reduced magnetization m = 3 , with Q = 18/23 and m = ~ and with Q = ~1° and m = ~3, energies are sufficiently low, but none of them are the lowest. The arrangements of Jmoments which appear in the magnetization process are illustrated in fig. 8. The compound reaches the ferromagnetic arrangement via three intermediate structures. As can be seen, the main feature of these arrangements is that two adjacent planes with moments parallel to the field appear in the antiferromagnetic arrangement as - + + - + . . . . These structures raise the level of the exchange energy slightly. It is unlikely that the magnetization proceeds by re-

230

N. lwata / PrCo2Si 2 transitions and magnetization process

versing the antiparallel ferromagnetic planes one after another in the Q = 1 or Q = 13 structures. The reason is as follows. By this process, the intermediate structures will have a sequence -+++-+ ..or - + + + + - + ..., in which the moments on several planes in a row point to the field direction. This will increase the values of the (Jq)'S near q = 0. And then the exchange energy will tend to increase considerably, since h ( q ) is negative for q < 0 . 4 . The calculated magnetization and the structures are somewhat different from those proposed by Shigeoka et al. [3]. Finally, brief comments will be made on the Q3 structure. The experimental value of the Q3 is 0.777. A fractional number ~z n ( = 0.786) is also near to this value. There is small difference of 1.2%, which is not much larger than the limits of experimental error. If we take Q3 = ~11, then the )~(q) will have a maximum at q = ~ . A structure with Q = ~ and m = ~ will appear in the magneti-

zation process. And there is some improvement in the calculated fit to the data for 34 kOe < H < 54 kOe. However, no definite reason has been found to choose between Q3 = 7 and Q3 = ~11. The actual value would have to be determined by results of neutron diffraction intensities of high-order harmonics. Neutron diffraction experiments in the presence of a magnetic field are needed. However, it should be mentioned that the present analyses give some explanations of the main features of the experimental results.

References [1] T. Shigeoka, N. Iwata, Y. Hashimoto, Y. Andoh and H. Fujii, Physica B 156&157 (1989) 741. [2] T. Shigeoka, N. Iwata, H. Fujii, T. Okamoto and Y. Hashimoto, J. Magn. Magn. Mat. 70 (1987) 239. [3] T. Shigeoka, H. Fujii, K. Yonenobu, K. Sugiyama and M. Date, J. Phys. Soc. Japan 58 (1989) 394. [4] P. Bak and J.V. Bochm, Phys. Rev. 21 (1980) 5297. [5] H. Mashiyama, J. Phys. C 16 (1983) 187.