GdFe2SI 2
3. Influence of physical properties on magnetism in RT2X ~ compounds
Neutron diffraction data indicate that the rare earth sublattices exhibit a large variety of ordering schemes (see fig. 23):
1. collinear ferromagnetism (F) was found in TbMn2X 2 and ErMn2X ( X = Si, Ge),
2. four types of collinear antiferromagnetic structures:
- t y p e AFI with the wave vector k = (0,0,1) was observed in RCo2X 2 (R = Pr-Tm), RRhzSi 2 and RRh2Ge 2 (R = Nd-Tm), TbIr2Si2, PrCu2X 2 and CeAu2Si 2,
- type AFII with k = (0, 0, }) was found in NdFe2Si2, NdF%Ge2, PrF%G%
and ErFe2Si2,
+
A F [
, i \ ' r
\l \ ,x¢
,\. ~ ,
AF IV
AFIT
[ 2 " x
LSWI f - f ? l LSWII
SPIRAL AXIS
i I
T I a I
',~ ,,~ ,,~
',~ ',~ t~
I\
,~i~- ~,
I i I I l° I° I °
LSW lg LSW IV
Fig. 23. The magnetic ordering of the R sublattice in RT2X 2 compounds.
186 A. SZYTULA and J. LECIEJEWICZ
- type A F I I I described by k = ( I , 1 , 0 ) is adopted by TbNi2Si2, CeRh2Si 2 and CePdaSi 2,
- type A F I V with k = ( ~, 0, ½ ) is exhibited by R C u 2 X 2 ( R = T b - E r ) . 3. a n u m b e r of different non-collinear structures was also discovered:
-cosinusoidally
m o d u l a t e d longitudinal spin wave L S W I propagating along the c-axis was found in PrCozGe2, H o N i z G e 2 and PrNizSi2,- c o s i n u s o i d a l l y m o d u l a t e d longitudinal spin wave LSW II propagating along the a-axis operates in RRuzSi 2 (R = T b - E r ) , T b R u 2 G e 2 , ROszSi 2 ( R = Tb, H o , Er) and CeAgzSi 2. In the case of TbRu2Si 2 and T b R u z G e 2 at low temperatures this type of magnetic structure transforms to the square modulated phase.
-NdRuzSi
2 and N d R u 2 G e 2 have complex magnetic structures. Below T = 10 K they are ferromagnetic with the moments parallel to c-axis. A b o v e T = 10 K NdRuzSi 2 has a cosinusoidally m o d u l a t e d longitudinal spin wave, LSW IIl, propagating along the [110] direction. A squaring of the magnetic structure is observed below T = 15 K. N d R u 2 G e 2 exhibits two types of sine wave modulated magnetic structures, having wave vectors k = (0.19, 0.05, 0.125) and k = (0.12, 0.12, 0) with the amplitude of the m o m e n t along the c-axis,- L S W IV is an incommensurate structure with two c o m p o n e n t wave vectors k = ( k x , 0 , k~) occurs in TbFe2Si2, HoFe2Si2, TbPd2Si2, HoPd2Si 2 and TbPd2Ge 2.
T h e Hamiltonian which describes the magnetic properties of an R ion is usually presented in the form:
Htot --~ Hcoul ~- Hexch ~- Hcf Av gins -[- Hext,
where the term: Hcoul represents the Coulomb interactions, Hexch describes the exchange interactions which in the case of rare-earth systems are usually de- scribed in terms of the R K K Y theory, Hcf is the crystal electric field term, Hms is bound to the magnetostriction effect, H~x t takes into account the interaction within an external magnetic field.
T h e available experimental data clearly indicate that the terms H~xch and Hcf are dominant in the description of the magnetic properties of the R ion in a crystal, so we shall discuss t h e m in detail. But first we shall briefly summarize the information related to the magnetic ordering schemes which have been observed up to now in the ternary RT2X 2 systems. T h e y will be essential in further discussions.
3.1. Exchange interactions
Since the interatomic distances between two R atoms are always about 0.4 nm in the basal plane and a b o u t 0 . 6 n m along the c-axis, the assumption of the Heisenberg type interaction is not justified. T h e models involving indirect cou- pling are usually adopted to explain the stability of particular magnetic structures.
TERNARY INTERMETALLIC COMPOUNDS OF THE RT~X 2 TYPE 187 Since the 4f electrons are strongly localized and the R T 2 X 2 systems show a metallic type of electric conductivity, the exchange interactions of 4f electrons can be described in terms of the well-known R K K Y t h e o r y , which postulates the coupling b e t w e e n the m o m e n t s via conduction electrons.
T h e o t h e r factors are the interaction of the f-electron shells with the crystal field C E F and the interactions with valence electrons of the anionic lattice via the hybridization orbitals.
T h e c o m p e t i t i o n of these factors causes a large variety of magnetic structures in R T 2 X 2 systems to occur. T o e s t i m a t e which of t h e m plays the m o r e i m p o r t a n t role in the particular c o m p o u n d , a plot of the o b s e r v e d N6el points against the de G e n n e s factor
(g-1)2j(J+l)
was constructed (see fig. 24). T h e following conclusions can be drawn f r o m it:- t h e de G e n n e s function is not o b e y e d in RT2X 2 c o m p o u n d s containing light r a r e - e a r t h ions. In m a n y cases the c o m p o u n d s r e m a i n p a r a m a g n e t i c a b o v e 1 . 8 K ,
- in the case of h e a v y r a r e - e a r t h ions, the N6el points follows in principle the de G e n n e s function, however, large discrepancies are o b s e r v e d when T = Cu, R u , Os.
T h e s e observations suggest that different m e c h a n i s m s of interaction are decisive
'mll I
8C
6C
l
\ \I
I
Ix,,
I201-- ×
LQCe PrNd PmsmEU6d TbDyHOgr TmymLU
• RERh2Si 2 A RERh 20e2 o RERu2 Si2 x RERu 2 Oe2
• REOs 2 Si2
o R/Pd2 Si 2
Fig. 24. The N~el or the Curie temperatures observed for RT2X 2 compounds and their relation to the de Gennes function.
188 A. S Z Y T U g A and J. L E C I E J E W I C Z
in the magnetic structure of the R T 2 X 2 systems with light and heavy rare-earth ions.
Attempts to connect the N6el points with the interatomic distances along the a- and c-axes do not provide any satisfactory correlations (Szytula and Leciejewicz 1987), however, in the TbTzX 2 series the oscillatory character of the N6el or Curie temperature is caused by the increase in d-electron number (see fig. 25).
Similar effect is observed in magnetically isostructural ErT2Si 2 (Leciejewicz et al.
1983), EuTzGe 2 and GdTzGe 2 (Felner and Nowik 1978) compounds suggesting that while the magnetic interactions may be discussed in terms of the R K K Y model, the number of free electrons donated to the conduction band depends on the number of d-electrons. In the RF%Si 2 systems where the absence of a local moment on the Fe atom has been found, the observed magnetic fields on the 57Fe site are in reasonable agreement with conduction electron polarization by rare earth moments (Noakes et al. 1983).
In an isotropic RKKY model with a spherical Fermi surface, the Fermi vector is strongly dependent on the c/a ratio and on the number of free electrons Z per magnetic ion:
2kv = 2~r 6f~ a
a c
The analysis of a/c values determined for a large number of RT2X 2 systems containing heavy rare-earth ions ( R - - T b - T m ) shows, that when a/c <0.415, a simple collinear antiferromagnetic ordering is observed, while the compounds
10~
TN(K) 50
0 180 170 V~O-ln m] :~
160 I 150 I Mn
3d 4d 5d
• l o A x TbT2Si 2 ' A \ ~ - • TbT2Ge2
x
o//
I I I i
Fe Co Ni Cu
Ru Rh Pd A 9
Os Ir Pt Au
3d 4d 5d
Fig. 25. The dependence the N6el tempera- tures and the atomic volumes for TbT2X 2 compounds ( T = 3d, 4d and 5d transition metal, X = Si or Ge) in the function of the atomic numbers of T elements.
TERNARY INTERMETALLIC COMPOUNDS OF THE RT2X 2 TYPE 189 with a/c > 0.415 exhibit oscillatory m a g n e t i c structures (Leciejewicz and Szytula 1987).
Following the R K K Y t h e o r y , the energy E of a spin system with an oscillatory character of ordering is given by:
e = - J q , ) ,
where N is the total n u m b e r of m a g n e t i c ions in the system, (/x ) is the a v e r a g e value of t h e magnetic m o m e n t , J(k) represents the Fourier t r a n s f o r m of the exchange integral J(R i - R j) b e t w e e n i and j magnetic ions, positions of which are given by the vectors Ri and R/, respectively. A s s u m i n g that the interaction integral J(k) remains constant, the energy E is directly p r o p o r t i o n a l to a function expressed as
- F ( k ) - [ 1 ( k ) - J ( O ) ] .
T h e stability of a particular m a g n e t i c structure m e a n s that the functions F(k) exhibits a m a x i m u m for n o n - z e r o values of k (the wave vector of the m a g n e t i c structure) (Yosida and W a t a b e 1962).
T h e c o m p u t a t i o n of F(k) function against k for particular values of k v (or Z ) permits o n e to select those values of k for which F(k) exhibits a m a x i m u m . Figure 26 shows the range of k = (k, 0, 0)a* for which F(k) attains m a x i m a , p l o t t e d against the n u m b e r of free electrons p e r T b 3+ ion in TbRu2Si 2 (Slaski et al. 1984).
For these c o m p o u n d s the o b s e r v e d value of the magnetic structure vector k is 0.233a*. It corresponds to Z = 3.13 free electrons p e r T b 3+ ion. T h e Z and k values o b t a i n e d f o r a n u m b e r of R T 2 X 2 c o m p o u n d s are listed in table 14. In R T 2 X 2 c o m p o u n d s the o b s e r v e d m a g n e t i c ordering scheme requires 3 free electrons p e r R 3+ ion. It could indicate that 6s and 5d electrons of R 3+ ion are d o n a t e d to the conduction band. Valence electrons of the o t h e r a t o m s contribute to the chemical bonding. On the o t h e r h a n d the chemical shift m e a s u r e m e n t s p e r f o r m e d by the X - r a y a b s o r p t i o n spectroscop3~ m e t h o d ( X A S ) indicate ,that electrons of all constituents i.e. of R, T and X atoms contribute to the conduction
TABLE 14
The values of k and Z obtained for RT2X 2 compounds.
Compound k Z Ref. Compound k Z Ref.
TbRu2Si 2 0.233a* 3.13 a RC%X 2 1.0c* 2.75-3.75 d DyRu2Si z 0.22a* 3.136 a RRh2Si z 1.0c* 2.77-3.39 d (Ho, Er)Ru2Si 2 0.2a* 3.16 a TbRh2G % 1.0c* 2.84-3.2 e TbRuzG % 0.235a* 3.0 b NdRu~Si 2 0.0c* 2.67-3.4 d
TbOs2Si 2 0.312a* 3.0 c NdFe2Si 2 0.5c* 1.4 d
(Ho, Er)Os2Si 2 0.295a* 3.02 c References:
(a) Slaski et al. (1984), (b) Yakinthos (1986a), (c) Kolenda et al. (1985), (d) Leciejewicz and Szytula (1987), (e) Szytula et al. (1987).
190 A. S Z Y T U , L A and J. L E C I E J E W l C Z
0 - , 0.2
I
• --, 0.1
• ~ 0.0
LL
- 0.I
- 0 . 2
- 0 . 3
- 0.4
- 0 . 5
-0.6
- 0 . 7 - 0 . 8
2kF=1.39
\ \ x \ \ x~xx
\ x
0.1 0.2 0.3
2k_=1.38
2;=137
...~Y2 kF = 1.36
~ , 2 kF= 1.35
;13J
kF=1.32
?kF=1.31
2kF=1.30 :1.29
", \
\x \
\\
\k \
\
%\
\ \
\
2kF=1.28
\ \ \~ \\
\\. 138
\"\ \ ', %1.39 . 2 kF=1.41 \ 2 k v = 1.42 \ 1.40
,, \
0.4 0.5 0.6
w a v e v e c t o r
Fig. 26. A plot of the F(k) function against the wave vector k for different values of 2k F (Fermi vector) in TbRu2Si 2 (Slaski et al. 1984).
band (Darshan et al. 1984). The Fourier map of the electron density obtained recently for DyCo2Si 2 (Szytula et al. 1988) shows S i - C o - S i layers that form an independent sandwich with saturated electric charge.
The comparison of the effective hyperfine magnetic fields Her f in D y T 2 X 2 compounds and of the respective magnitudes of the ordered magnetic m o m e n t u derived from neutron diffraction experiments gives the relation H ~ f f ( k O e ) - 600/x (/XB). This result may be considered as evidence for the localization of 4f electrons (Pinto et al. 1983).
3.2. Crystalline electric fields
The interaction of crystalline electric fields (CEF) with the multiple moments of R atom electrons in a site of a crystal lattice of 4 / m m m point symmetry is given by the Hamiltonian
Hcf
= B 2 0 2 4 - 0 0 B404 %- 0 0 B 4 0 4 4 - 4 4 B606 %- 0 0 B 6 0 6 , 4 4T E R N A R Y I N T E R M E T A L L I C C O M P O U N D S O F T H E R T 2 X 2 T Y P E 191
where O m is the Stevens o p e r a t o r , B m is a C E F p a r a m e t e r defined by Hutchings (1964). T h e c-axis of a tetragonal unit cell was chosen as a quantization axis.
T h e values of the B," coefficient can be derived from experimental data obtained from the magnetic susceptibility, the Schottky specific heat, the hy- perfine structure, the magnetic form factor and the neutron inelastic scattering measurements. The B m values are shown in table 15. It can be seen that the B2 ° coefficient is dominant since the others are smaller by an order of magnitude.
T h e Orientation of the magnetic m o m e n t in respect to the tetragonal axis can be connected with the sign of the B~ coefficient. It was shown ( G r e e d a n and R a o 1973) that when the magnetic m o m e n t is normal to the tetragonal axis or makes an angle q~ with it, B~ is positive. For a negative value of B~ coefficient the magnetic m o m e n t is parallel to the c-axis. Table 16 contains the values of B ° coefficients determined for a n u m b e r of R T z X 2 compounds. These data indicate that the correlation between the signs of the B~ coefficient and the orientation of magnetic moments agrees with that d e d u c e d from n e u t r o n diffraction experi- ments. H o w e v e r , it has been shown, that the sign of B~ depends also on the n u m b e r of 4f and nd electrons, but the lack of data does not permit us to plot a detailed diagram.
T h e influence of a crystal field on the direction and magnitude of magnetic m o m e n t s in R T z X 2 systems where R = Pr, Tb, E r and T m will be discussed further.
T h e determined values of B~ coefficients for PrNizSi 2 (Barandiaran et al. 1986) indicate that the ground state is a non-magnetic singlet. A n o t h e r non-magnetic singlet is the first excited state separated by 2 K in the energy scale. A doublet at 73 K is the second excited state. Such level scheme is consistent with the m o d u l a t e d magnetic structure observed below 5.5 K.
T h e C E F splittings in YbYzX 2 indicate that the ground state of Tb 3+ is f o r m e d by two very close singlets 1 { 1 6 ) + 1 - 6 ) } and ½{[6} + I - 6 ) } . In this case the magnetic moments are q u e n c h e d along the c-axis giving rise to the Ising-like behaviour. The first excited state is at 100 K in TbFe2Si 2 (Noakes et al. 1983) and 50 K in T b R u 2 G e 2 (Yakinthos 1986). In all TbT2X 2 compounds the magnitude of the magnetic m o m e n t localized on Tb 3÷ ion equals the magnetic m o m e n t value of the T b 3+ free ion g J = 9 pc B.
In all ErT2X 2 systems the magnetic m o m e n t on E r 3÷ is smaller than the free ion value of 9/x B. The magnetic m o m e n t is perpendicular to the tetragonal axis in all E r T 2 X 2 compounds except E r C u 2 G e 2. T h e case of E r C u 2 G e 2 may be explained by considering the influence of CEF. T h e level scheme for the E r 3+ ion was deduced assuming that the charge is localized on the E r and is zero on Cu. T h e magnetic m o m e n t is given by:
~ = ~ c ~ g ( a + l L l a ) P~z = ~ B g ( a + l L I a )
where la+ ) and l a } are two wave functions of the fundamental doublet, [ a ) is the time conjugate of la+), g is the L a n d e factor, Jx and Jz are x- and z - c o m p o n e n t of the rare-earth total angular m o m e n t u m , / x and tx z as the
! 9 2 A . S Z Y T U - L A a n d J . L E C I E J E W I C Z
y.
o
>
I q- + + I
c q
o I + I
I r +
¢3
+ I + +
f-.,
Z" c q
r.f) "--~ t'¢5 ' ~ t ~ t ~
G) I I I
"¢3
,.--2
O
&
,.-z
v
D
,--2
,,...t v
. N
T E R N A R Y I N T E R M E T A L L I C C O M P O U N D S O F T H E R T 2 X 2 T Y P E 193
TABLE 16
The values of B2 ° coefficients and direction of magnetic moment for RT2Si2 compounds.
R / T M n Fe Co Ni Cu R h
Ce -3.0 ~ ±c x
Pr - 0 . 7 ", ][c a ]lc k'j - 3 . 9 9 q, lie q lie ~
N d - 0 . 6 3 a, Iic e Ilc k lie A
Zb t[C b --3,0 d, tl cf lie j*k'l'nl []C 1 +0.8 t, ± c U 'n' - R A w, Nc z'w D y - 1 . 3 5 e 1.8 ~a, - I . 0 " , []c i'~ +0.17 c + 0 . 5 7 ; ± c " q n
H o _0.61 a, pg tic k ... +0.175', ± c .... - 0 . 2 2 y, q~
E r c h +0.67 a, ~_c h L c k, @P --0.2 t J~C A
T m +2.54 d ± c ~ - 0 . 7 9 ' , + 0 . 1 2 ~ ± c c
Y b +7.65 a
References:
(a) Iwata et al. (1986), (b) Shigeoka et al. (1986), (c) G6rlich et al. (1982), (d) Noakes et al. (1983), (e) Pinto and Shaked, (1973), (f) Szytula et al. (1987b), (g) Leciejewicz and Szytula (1985), (h) Leciejewicz et al. (1984), (i) Leciejewicz and Szytula (1983), (j) Yakinthos et al. (1984), (k) Leciejewicz et al. (1983), (1) Nguyen et al. (1983), (m) Pinto et al. (1985), (n) Pinto et al. (1983), (o) Schobinger-Papamantellos et al. (1983), (p) Yakinthos et al. (1983), (q) Barandiaran et al. (1986), (r) Horn et al. (1981), (s) Szytula et al. (1983), (t) Budkowski et al. (1987), (u) Leciejewicz et al. (1986), (v) Stewart and Zukrowski (1982), (w) Chevalier et al. (1985), (z) Slaski et al. (1983), (x) Quezel et al.
(1984), (y) Takano et al. (1987), (A) Szytuta et al. (1984), (B) Melamud et al. (1984), (C) Yakinthos (1986a).
m a g n e t i c m o m e n t in x a n d z directions (x is n o r m a l to z). T h e w a v e f u n c t i o n s w e r e c a l c u l a t e d to be:
la+ ) = 0.1561_+7/2 ) + 0.9841-+1/2 ) + 0 . 0 8 1 1 + 9 / 2 )
h e n c e /x, = 4 . 8 / x B and /x z = 0 . 5 / x B. H o w e v e r , the m o d e l d o e s n o t agree with e x p e r i m e n t a l data. If o n e a s s u m e s t h a t t h e c h a r g e o n C u ion is n o n - z e r o , t h e splitting o f the I-+15/2) m u l t i p l e t is r e v e r s e d a n d t h e f u n d a m e n t a l d o u b l e t has 1+ 1 5 / 2 ) as its strongest c o m p o n e n t . T h e calculated w a v e functions f o r qcu = 0.6+
are:
I b + ) = 0.9201-+15/2) + 0 . 2 4 9 1 + 9 . 2 ) + 0 . 2 9 7 1 + 1 / 2 )
and t h e n t h e m o m e n t s are /x x = 0 . 4 / % a n d /x z = 7 . 9 / x B. T h e d i r e c t i o n o f t h e m a g n e t i c m o m e n t is along the c-axis of the crystal. T h e s e results are in a g r e e m e n t with t h e n e u t r o n diffraction e x p e r i m e n t ( Y a k i n t h o s 1985).
I n t h e case of T m R h 2 S i 2, T m F e 2 S i 2 a n d TmCu2Si2, t h e g r o u n d state o f T m is d e d u c e d to be 10) and t h e first excited state is I-+1} at 4 9 K f o r T m R h 2 S i a ( Y a k i n t h o s 1986a), 8 K f o r T m C u 2 S i 2 ( S t e w a r t a n d Z u k r o w s k i 1982) and 8 K f o r T m F e 2 S i 2 ( U m a r j i et al. 1984), S u c h splitting leads to the r e d u c t i o n o f t h e m a g n e t i c m o m e n t value o n T m 3+ i o n yielding 4 . 2 / x B in T m R h 2 S i 2 ( Y a k i n t h o s 1986a) and 3 . 2 / x B in T m C u 2 S i 2 ( K o z l o w s k i 1986). B o t h / x values are smaller t h a n the f r e e T m 3+ ion value o f 7 / z B.