INTROD UCTIO N Diuiiig tho last decade, much attention has bpon paid to magnetic fiu stiated s>s-tems with imilti- sublattice structure [1-8 In theoretical papers [6-8], the Shpningtơii-
Trang 1VNU J O U R N A L OF SCIENCE, Nat Sci., t XV, n^2 - 1999
M A G N E T IC SU SC E P T IB IL IT Y OF T H E T W O -SU B L A T T IC E
FR U ST R A T E D SY STE M S IN T H E ERG O DIC P H A S E
Hoang Zung
Fiwiilty o f Physics College of Natural Science - HCM U
Abstract T h e m a g n e tic susceptM H ty o f I h t t.wo-.sublatt.ice f rn s lr a te d s y s t e m s IV the
ergodtc p hase is derived i n the f r a m e w o r k o f the TTi.ean fie ld theory 'Ịhe in flu e n c e o f the
intrasublatt.rce f r u s t r a t i o n the behavior o f m a g n e t ic susceptihửrtỵ IS studied I t IS sh o w n
th a t o u r m o d e l can qualitatively explain the e x p e r im e n ta l data o f F c r M T i - r T iO ^ i
I INTROD UCTIO N Diuiiig tho last decade, much attention has bpon paid to magnetic fiu stiated s>s-tems with imilti- sublattice structure [1-8 In theoretical papers [6-8], the Shpningtơii- Kirkpatrick model with infinite- ranged iiiteiactioii[9] has been pxtpiided to coustnict a theory for frustrated antifpiTornagnets and fcnimagnots Ono of the most interf'stuig pr<’- dictions of this theory is the possibility of increasing the ii reversibility tem peiatuK ' 1 ],
(at which system undeigoes tiansitioii into the spin glass phase) by tliP extPiaal magiK'tic field k This behavior of Tg{h) has been observed experimentally in many materials [1-3
iế ir
c->
lA
JT
•
z 05 • ư c
i«â7S
■ 4b
T (H)
CO
ư>
-
-w
A
1A
• 7 —# - ^
t-a is
• i rc
ijt
ijo
at
•
• : f c
• i r c
»
«.QAO
10
Fuj I Tlie teụipcraturp (leppiulpiire on the magnetic susceptibility
X { T ) of F e r M n i ^ r T i O i for r = 0.75(fl)'' = 0.65(6),.r =: ().6()(r), [3 Rocently Ito, Aruga and cowoikeis rallied out a detailrd stu(l>’ of solid solutions Fe:,.A/7i,i_ ,T /ơ3[2, 3] The tem poratuie dppendence oil the magnetic s\isc('ptibility \ {T) ot this oompouncl was measured for different F e concentration r and was piPsi'iitod in Fip, 1 Just below Tg the field cooled susceptibility X f c and zero field coolod susceptibility \ z / r start, to differ from each other In the tem perature region bptween Tg and the Nell point
T n (the ergodic phase), the behavior of x(T ) is quite unusual: it has a nmnmurn whose
phase could be explained by the two-sublattice Ising model proposed in [7-8’
10
Trang 2II MODEL
111 oi(J(’i to model fi u stiated antiferrornagiiPtic and ferrimagnetic systems, we con- sulei tho following Hamiltonian [7-8]
vvhci(> we considoi the simplost two-sublattice situation The subscript p = 1,2 numbers tlu> spin subsystems, Sj„ are Ising spin in nature, h is Mie applied magnetic field, and
Iiitei actions J,j and supposed to be Gaussian distributed with the average values and dispersion givpii by
< .7,, > = Jo, < (J,, - J o f J,
Not(' tliat tho paraniotPis J and /p serve as measmes of inter-and intrasublattice
fiustratious
Lsiiii* tli(‘ l i'plica uK'thod, in [7,8] we derived the self-consistent system of sta te equa-tioiis for OUI inod(>l Th(' equations for the siiblattice magnotizatioiis 777 ] 2 and
Edwarcls-Andf'ison p a ia n u “t('is (/I ■> ill tho Pigodic phase arc
= < tanh E,,(2) >^ , fjj, =< >^ , (3)
ÌÌ
< A { z ) > , = ~ [ e - ^ " / ' ^ A { z ) d z
1 - 0 0
(6)
The iiif'vcrsihilitv tcm polatm o Tg is clefinod from tlio do Alineida-Thouless (A-T)
line s (TỊiiatioii
-< c o s l r ' E , (2) > , < c o s l r - ' i Í2( ^ ) > c < ^ o s h - ^ E , { z ) > c - 1 ( 7 )
p
It is well known [10] that tliP loplica syinmetiv equations (3)-(6) are correct, only
in the K'f^ion above tho A-T line (ergodic phase) Since we shall restrict ourselves to stiul\ 111” x ( T ) ill thf' oigodic phase, these equations aie adequate If we are interested in the b('ha\'ioi of the systPin below Tg, we have to use much more complicated equations
7,8]- Howpvor, both thooretical and experimental investigations show th a t even in the ergodic pliase, properties of disordered magnetic systems quite differ from those of the ordeied ones
Trang 314 Hoang Zung
Th(> total susc('ptil)ility of our systoiii can 1)(' f’xpK'Ksrd in the following form
y(T) = / M \ i( ^ ) + "2,\2(T), (8)
wlu'H' X,, is th r susc('ptil)ility of the p - th suhlatricc: \,, = dtii,,/dh Diffcientmting th.' ('xprossioii (3) of IĨ),, ami q,, witli msppct to the f'xtenial Hpld h and introtluciug th(' not at ions Ap = Oqp/dh and
III
we obtain th(> equations for \ p and Ap
\ I + - 7 l)\2 + , -^01 , ,
1 - ^ ( 1 - <i\
0 1 ^ ( 1 - r/2)\i +
T
1 - ^ ( 1 - ^ / 2 )
-^1 + « 2 ^ A., = 1 - <i\ T
1 - <72
T>T r
X2 + ^»1 T p — A i + ~ Y
p \ \
A, - a , : ^ À , = 2 ơ , / T ,
X2 - f i 1 A, + 1 - r 2 Ằ 2 = 2 U 2 / T
(lU)
Together with (3)-(9), pquations (10) make it possible to calculate the susrpptibilit\-
x{T) in th(> eigodic phase.
- ^ 3 : 1 »0.8
I: I «0.6
r
Fi(j.2: The rpiiipeiatuie dependence of the magnetic susceptibility
X { T ) with 7 = 0, Jo = /0 == 1 and different values of frustration I:
(1) I = 0.6, (2) / = 0.7,(3) I = 0.8 and (4) I = 0.9.
To explain the tem perature dependence of x { T ) for F c r M n i ^ ^ T i O z we shall con
sider the case when two sublattices are equivalent; 7Ỉ.1 = 7Ỉ.2 = 0.5, Qi = Q2 = 1, /01 =
/02 = /0, I\ - h - Ỉ ■ In addition, we are especially interested in the situation when
Trang 4fiustiations of tho intrasublattice interactions are much stronger than those of the inter sublattice interactions: J / / ~ 0 To determine the behavior of x (T ) it is necessary to solve equations (3) - (10) The results of our calculations for soưie values of the intrasub"
lattice frustration I with Jo 1 ,/o = 1 , = 0 are presented in Fig.2 One can see that
for tho chosen parametors of the model, x { T ) has a minimum which smear out at / = 0, 6.
IV COM PARISON W ITH E X PE R IM E N T
According to experiincnts [11,12], FeT^Mn\_j.TiO:ị is a typical Ising antiferromag- net with easy-axis anisotropy along the hexagonal c-axis In F e T iO ^, spins within a
c-layer are fenoiriagnetically coupled, and the ferromagnetic sheets stack up a n tife rro
niagiiPtically along the c-axis (see Fig.3) In M v T i O ^ , on the other hand, the intralayer
a n d the^ interlayer spin couplings ar e both a n t i f e i T o m a g n e t i c Therefore, in m i x e d com
pound Fe_j KIn\-.rTiO:^ the competition between ferromagnetic and antiferromagnetic in
teractions creates frustrations among spins In addition, the interlayer antiferromagnetic coupling is much weaker than the intralayer one 12 so we can expect th a t the frustrations
of the intralayei iiiteractions must be dominant This is the reason for F e: rM u i^ j Ti Oz
being described by our model with chosen parameters in Section 3
Í c-axis
FeTiQ
.V; Scliniiatic magnotic stiu ctu re of FcTĩO:>, (a) and MĩìTiO:^ (b) [2'
Lot us coinpaii’ th(‘ results obtained ill Section 3 with the experimental d a ta for
F (\rMvị ,TiO:^ [3] The agmniiont between our calculation (Fig 2) and experiment
is qualitativoly obvious: (1) At the same t.ernperaturo the stronger is the frustra
tion / (or the highen is Mil- concentration) the higher is the value of x; and (2) In the tf'inpeiatuK' region Tg < T < Tv function \ ( T ) has a Iiiiniinurn Its depth increases with
tlio (io»K'(* o f i l i e i n t r a s u b l a t t i c o f r u st r a t i o n
V CONCLUSIONS The foregoing analysis and the agreement between theory [6,8] and experiment [1-3 show that the t w o snblatticp mõcipl with infinito-ranged interaction quite satisfactorily
describes the static properties of real frustrated magnetic systems like F c r M n i - x T i O z ,
at least in the (‘Igodic phase The limit of this inodrl is th a t it can describe these systems only qualitatively but not quantitatively This is the general feature of the models based
on the mean field theory of spin glasses [10
Trang 5R EFER EN C ES 1.] p z Wong, S von Molnar, T T Palstra, J A Mydosh, H Yoshizawa, s M
Shapiro, A Ito Phys Rev Lett 55(1985), p 2043.
2.] H Yoshizawa, s Mitsuda, H Aruga, and A Ito Phys Rev Lett 59(1987), p
2364; J Phys Soc Jpn 58(1989), p 1416.
3.] A Ito, H Aruga, M Kikuchi, Y Syono and H Takei Solid State Commun
66(1988), p 475; H Aruga, A Ito, H Wakabayashi, T Goto J Phys Soc
Jpn 57(1988), p 2363.
4.] K Gunnarsson, p Svedlindh, J.-O Anderson, p Nordblad, L Lundgren, H Aruga,
and A Ito Phys Rev B 46(1992), p 8227.
5.] Tran Quang Hung, Mai Suan Li, M Cieplak J Magn Mater 138(1994), p 153.
6 I Ya Korenblit, Ya V Feodorov and E.F Shender Zh Eksp Tear Fiz
92(1987), p 710; Ya V Feodorov, I Ya Korenblit and E.F Shender Europhys
Lett 4(1987) p 827 and references therein.
[7.] I Ya Korenblit, Ya V Feodorov and H.Zung Fiz Tverd Tela 32(1990), p
1441
8.] M.S Li, L.Q Nguyen, A v Vedyaev and H.Zung J Magn Magn Mater
96(1991),p 175
9.] D Sherrington and s Kirkpatrick Phys Rev Lett 35(1975), p 1792; G Parisi
J Phys A 13(1980), p 1101.
10.] K Binder and A p Young Rev Mod Phys 58(1986), p 801; I Ya Korenblit and E.F Shender t/sp Fiz Nauka 157(1989),p.267.
11.] H Kato, M Ymada, H Ymauchi, H Hiroyoshi, H Takei and H W antanabe J
Phys Soc Jpn 51(1982), p 1769; H Kato, y Ymauguchi, M Ymada, s
Funahashi, Y Nakagawa, H Tekei J Magn Mater 31-34(1993), p 671.
[ 1 2 ] Y Y a m a g u c l l i , I I K a t o , I I T t i k e i , A I G o l d i i i i x i i , a n d G S h i i a u < - S o l i d S t a t e
Commun 59(1986), p 865; G Shirane, S.J Pickart, and Y Ishikawa ,/ Phys Soc Jpn 14(1959), p 1352.
TAP CHÍ KHOA HOC ĐHQGHN, KHTN, t.x v n°2 - 1999
■
-ĐỘ CẢM T Ừ CỦA HỆ T Ừ MAT T R Ậ T T Ự HAI PHÂN MẠNG TRONG PH A ERGODIC
Hoàng Dũng
Đại học K H Tự Nhiên - DHQG TP HCM
Trong khuôn khổ lý thuyết trư ờng trung bình, đ ả tính độ cảm từ của hệ từ mất
t r ậ t t ự gồm hai phân mạng trong pha ergodic Nghiên cứu ảnh hường của th ăn g giáng
t ư a n g tác trao đổi trong phân m ạng lên độ cảm từ Chứng tỏ rằng mò hình trên có thể
giải thích định tính k ế t quả khảo sát thực nghiệm đối với hợp kim F c x M n x - r T i O s