Journal of Magnetism and Magnetic Materials 86 (1990) 269-279 269 North-Holland
M O N T E C A R L O S I M U L A T I O N O F H E I S E N B E R G S P I N G L A S S O N F C C L A T I ' I C E I N E X T E R N A L M A G N E T I C F I E L D
J. M L O D Z K I a, F . R . W U E N S C H b a n d R . R . G A L / ~ Z K A a
° Institute of Physics, Polish Academy of Science, AI. Lotnik6w 32/46, 02-668 Warsaw, Poland b Department of Physics, University of Regensburg, D 8400 Regensbur~ Fed. Rep. Germany
Received 11 July 1989; in revised form 16 October 1989
The simple but realistic model described by a Heisenberg Hamiltonian with nearest neighbOurs and next nearest neighbours interactions in an external magnetic field was investigated by use of the Monte Carlo method, Three-dimensional vector spins of length ~ were distributed randomly on a fcc lattice. Different concentrations of spins, x ffi 0.05, 0.10 ... 0.90, were studied.
For low concentrations, simulated samples contained about 1000 spins. For higher values o f x the size of system was about 8000. All the computations were done for high external magnetic fields of around 3 T. During simulation, physical quantities such as magnetization, energy, specific heat and magnetic susceptibility were determined. The results for magnetization differ for the zero-field-cooled (ZFC) and field-cooled (FC) cases for the whole range of concentrations. This difference, also typical for experimental data, seems to vanish after longer simulation. From critical temperatures for'~ computer simulated magnetiza- tion, the magnetic phase diagram was obtained and compared to experimental data foi" Cdl_=MnxTe. Concentration dependent results for magnetization, specific heat and magnetic susceptibility allowed on6 to distinguish three different regions for the simulated system: x < 0.20, 0.30 < x < 0.60, 0.70 < x < 0.90.
1. I n t r o d u c t i o n
S e m i m a g n e t i c s e m i c o n d u c t o r s ( S M S C ) o r di- l u t e d m a g n e t i c s e m i c o n d u c t o r s ( D M S ) a r e m i x e d c r y s t a l s o r s o l i d s o l u t i o n s o f s e m i c o n d u c t o r c o m - p o u n d s , s u c h as eg. C d T e , H g S e a n d PbS, w i t h a p p r o p r i a t e m a g n e t i c c o m p o u n d s l i k e M n T e , M n S e o r M n S [1].
C d l _ x M n x T e , a m e m b e r o f this g r o u p o f m a t e r i a l s , crystalli7~es in z i n c - b l e n d e s t r u c t u r e o v e r the w h o l e x < 0 . 7 0 r a n g e [2]. T h e C d 2+ a n d M n 2+
i o n s r a n d o m l y p o p u l a t e a fcc s u b l a t t i c e . M a n - g a n e s e i o n s a r e m a i n l y c o u p l e d b y the n e a r e s t n e i g h b o u r ( N N ) a n t i f e r r o m a g n e t i c ( A F ) s u p e r e x - c h a n g e i n t e r a c t i o n . T h e n e x t n e a r e s t n e i g h b o u r ( N N N ) a n d m o r e d i s t a n t i n t e r a c t i o n s a r e w e a k e r a n d d e c r e a s e v e r y r a p i d l y w i t h d i s t a n c e b e t w e e n t h e M n i o n s [3]. T h e v a l u e o f J~NW is n o t well k n o w n a n d is b e l i e v e d t o b e in r a n g e 0 . 1 - 0 . 2 5 o f the N N e x c h a n g e c o n s t a n t [ 4 - 6 ] .
T h e fcc l a t t i c e w i t h A F c o u p l i n g s d o e s n o t a l l o w to s i m u l t a n e o u s m i n i m i z a t i o n o f e n e r g i e s o f all e x c h a n g e b o n d s a n d t h u s l e a d s to f r u s t r a t i o n . R a n d o m n e s s a n d f r u s t r a t i o n a r e t w o e s s e n t i a l fac- tors t h o u g h t t o b e r e s p g n s i b l e for the s p i n - g l a s s b e h a v i o u r o b s e r v e d e x p e r i m e n t a l l y in S M S C [7].
I n o u r p a p e r w e p r e s e n t a s i m p l e c o m p u t e r m o d e l o f s u c h m a t e r i a l s ~nd c o m p a r e M o n t e C a r l o s i m u l a t i o n results w i ~ a v a i l a b l e e x p e r i m e n t a l d a t a . S u c h a n a p p r o a c h w a s u s e d b y K e t t , G e b h a r d t a n d K r e y [8] t o i n v e s t i g a t e m a g n e t i z a - t i o n b e h a v i o u r o f Cd0.4iMn0.55Te m i x e d c r y s t a l s in h i g h e x t e r n a l m a g n e t i c fields. W e e x t e n d e d use o f this m o d e l f o r t h e w h o l e r a n g e o f c o n c e n t r a - t i o n s o f M n i o n s ( i n c l u d i n g t h e p h y s i c a l l y i n a c c e s - sible v a l u e s o f x - 0.80, 0.90) a n d d e t e r m i n e d also
i
t h e energy, m a g n e t i c s ~ s c e p t i b i l i t y a n d specific heat.
I n s e c t i o n 2 the d e t a i l s o f the m o d e l , d e f i n i t i o n s o f o b s e r v e d p h y s i c a l quantities~ a n d p r o c e d u r e s o f 0304-8853/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
270 J. Mtodzki et al. / Monte Carlo simulation of Heisenberg spin glass
the Monte Carlo simulations are described. Sec- tion 3 contains our results and comparison with experiment. In section 4 we present final remarks and conclusions.
2. Physical model and conditions of the Monte Carlo simulation
2.1. Model for simulation
In the present model spins are distributed ran- domly over an fcc lattice with imposed cyclic boundary conditions. The number of fcc sites is N s and the number of spins, N s, and these are related to the concentrations of spins, x, by the formula
x -- NJNg. (1)
Each spin is coupled by exchange interactions with its nearest ncighbours (up to 12) and with its next nearest ncighbours (up to 6). Spins also inter- act with the external magnetic field.
The total interaction is given by the following Heisenberg Hamiltonian
H = - J ~ n E S , . S y - Jnn. E S , . s j - g#B
- u B" ES,.
(2)
The first sum is over the pairs of the nearest neighbours, the second over the pairs of next nearest neighbours and the last one over all spins.
J,n and J ~ . are the respective exchange integrals expressed in kelvin. The external magnetic field B in tesla is applied along the z-direction. #B is the Bohr magneton, k the Boltzmann constant and g is the Land6 g-factor.
Here we are concerned only with the classical limit S - , o0, where the quantum nature of the spins is ignored and the Hamiltonian operator reduces to the internal energy E. Spins S i are regarded as three-dimensional vectors
S i = (S:, S y, S:) with fixed length S.
To make computations easier we introduce di- mensionless quantities and unit vector spin. It
enables us to evaluate the energy by the following formula:
e = - - A i E s ~ . s j - - A 2 Y ~ s , ' s j - - b . Y ' . s ~, (3) where
A a = J ~ . / I J . . I , A2---Jn.n/lJ.nl, b = - ~ B / ( S I J n n ] ) ,
e=E/(s2ls.°l), t= r / ( : l s . ° l ) .
and b, e and t are the magnetic field, energy and temperature in dimensionless units. The magni- tude of the vector s~ is equal to one.
To compare our numerical results with experi- mental data for Cd~_~,MnxTe we have chosen:
Sm.
~2, Snn = - 1 0
K and Sn,m- 1.25 K [81. The concentration of spins in the computer simulation corresponds to the concentration of Mn ions in the experiment. In general, the same model with different J n n and J~n can be used to simulate other materials containing Mn ions of spin momentum S = 2 ~ (Hgl_xMnxTe, Znl_xM_nxTe, etc.).2.2. Procedures of Monte Carlo
Our Monte Carlo calculations follow the proce- dure described by Binder [9]. The basic idea is to generate a representative ensemble of states to be used for the determination of thermodynamical quantities. At the beginning the directions of spins are chosen randomly with a uniform distribution of the spins on the unit sphere. The representative ensemble of states is obtained by rotating the spins one at a time. Each Monte Carlo step (MCS) is defined as a selection of one new configuration of the system by attempting to rotate N, spins in a random way and with a random selection of sites.
Each single rotation is performed following the
r u l e *
s7
= R ( o , 1 ) -0.5, (4)
* All vectors with lengths greater t h a n u n i t y before normaliza- tion, are discarded in order to obtain h o m o g e n e o u s distribu- tion of spin vectors in a sphere, n o t in a cube [9].
J. Mtodzki et al. / Monte Carlo simulation of Heisenberg spin glass 271
with a subsequent rescaling of the spin length.
R(O, 1) is a random number taken uniformly be- tween 0 and 1. The new direction of the spin is accepted if it leads to a lowering of the total energy of the system. If the energy increases, the factor e x p ( - A e / t ) is compared against a number chosen randomly and uniformly in the interval from O to 1. If the factor in question is greater than the random number then we accept the new direction of the spin, otherwise it is rejected and the previous one kept. Ae is the change in the energy produced by the attempted rotation.
For each configuration of spins, xj = (Sl, s 2 ... SN,), obtained after one Monte Carlo step, we determine the magnetization for the aver- age spin
1 N,
i--1
and the internal energy per spin:
S 2
E ( x j ) = ~ s e ( x j ) , (6)
where e(xj) is given by eq. (3). The Monte Carlo algorithm described above allows us to reduce ensemble averages to simple arithmetic averages
1 ~ M ( x j ) ,
( M ) = --~
j-1
(E)
= m(7)
( 8 ) j - - 1
where m is the number of configurations which is taken into account. Fluctuations of the energy and magnetization vector enable us to determine the specific heat, C and the tensor of magnetic sus- ceptibility, X
N$ m
C= ~'-TT2 y~" ( E ( x j ) - ( E ) ) 2"
. . . . j - - 1
X ~ - - --m--T
(9)
j - - 1
x - ( M # > ) , 0 0 )
where a, fl denote Cartesian components.
During simulation, the scatter of specific heat and susceptibility values was greater than for the
magnetization and energy. Because of this we also establish C as an energy derivative
C = OE[~T.
(11)
For each temperature some number of the Monte Carlo steps shoukl be rejected as not being characteristic of true equilibrium. This number depends on temperature and was automatically chosen by the program in the range 200-1400 MCS. At each temperature, 1000 configurations were taken into account when calculating physical quantities. Then we changed temperature quasi- continuously during 500 MCS. To get temperature dependence of the previously mentioned physical quantities we repeated this procedure for different temperatures.
To reproduce experimental measurements of the zero field-cooled system and the field-cooled one we started from a random configuration in the high temperature region (100 K) and quasicon- tinuously, during 4000' MCS, cooled down the system (usually to 1 K) without the magnetic field.
Then, during 500 cycleS we increased slowly the value of the field to 2.8 T. After having the field switched-on, we rejecte d 6000 configurations and then we did the first "measurement". We call this procedure the ZFC landing, because it establishes the starting point for the ZFC curves.
ZFC simulation continued for the next temper- atures by heating the system a little, rejecting some configurations and taking 1000 steps before
"' measurement". When Lhe system was heated to a high temperature we s'~ulated FC behaviour by cooling the syste~ doivn slowly and repeating rejection and measurement for each temperature.
2.3. Various concentrations, number of spins and temperature ranges
In our numerical experiments we studied differ- ent numbers of concentrations of spins, x ranging from 0.05 to 0.90. For small concentrations up to 0.30, it was sufficient to take into account rela- tively small systems Containing approximately 1000 spins. With high¢~ r concentrations it was rather difficult to get good results - smooth curves for magnetization, spedfic heat, etc. - even for
272 J. Mtodzki et al. / Monte Carlo simulation of Heisenberg spin glass systems of about 8000 spins. We used the follow-
ing sizes of sample for concentration: 0.40, 0.70, 0.80, 0.90. For 0.50 and 0.60 we did simulation with a rather small system of about 1000 spins.
Scatter of the data in these cases is relatively high.
We found from our computations that the critical temperature for the z-component of magnetization (corresponding to the maximum of Z F C curve) depends on concentration. To investigate this be- haviour it was necessary to do experiments in the proper temperature range with respect to the value of the critical temperature for a given concentra- tion. We experimentally found these ranges and the physical results are presented in section 3.
coincidence of numerical and experimental data is good - which shows that our Monte Carlo magne- tization can reproduce the experimental magnetic phase diagram.
The Tm~ values derived from numerically calculated Mz curves are scaled by Jr~ according to the formula
T = tS2l Jr~l. (12)
In our calculation we have taken J n n = - - 10 K [8], but the recently
reported
value of J~ = - 1 3 . 8 [15] gives a better fit to the experimental data, especially for concentration of spins, x, greater than 0.20.3. Results 3.2. Temperature-dependent energy and specific heat
3.1. Temperature-dependent magnetization
In this section we present results for the Z F C and FC magnetization as a function of tempera- ture and concentration. In fig. la, b, c the z-com- ponents of magnetization are plotted for both cases. Each pair of curves represents a different concentration, x ranging from 0.05 to 0.80. Mz is measured relative to its maximum M 0 = 1. Tem- perature is in kelvin and the value of the magnetic field along the z-axis is 2.8 T. All our results are consistent with the observation that the FC and Z F C magnetization differs at all x, the latter one always being lower. The Z F C magnetization has a maximum which shifts to higher temperature with increasing concentration. Its value also goes down and curves for both Z F C and FC become flatter for higher x. Finding the position of the maxi- mum, marked as an arrow in the plot, is somewhat arbitrary because of the data scatter, but this does not affect the major features. In the experiment Tmx is referred to as the freezing (critical) temper- ature, TF, and is suggested to characterize the transition point between the paramagnetic and spin-glass states [10-14].
In fig. 2 we plotted the magnetic phase diagram for Tm~ versus concentration. Our numerical re- sults are compared with experimental data from magnetiT.qtion and susceptibility measurements for Cdl_xMnxTe without applied field [10-13], The
In fig. 3 we present curves for the Z F C energy per spin as a function of temperature for the whole range of concentration, x = 0.05, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90. Samples with higher concentrations have lower energy, but the shapes of the curves are similiar.
In fig. 4 energies are plotted for Z F C and FC systems in the low-temperature range. The sep- aration between the curves is independent of con- centration but the point they join together is shifted to higher temperature for higher con- centrations. These results are worth stressing, be- cause we calculated the specific heat as a deriva- tive of the energy with respect to the temperature, and the behaviour of the energy curve is reflected in the behaviour of its derivative.
In fig. 5a, b, e the results for the specific heat per spin, C, as a function of temperature for different concentrations, x = 0.10, 0.40, 0.70 are plotted. The Z F C curve is always below the FC for low temperature and both curves meet at the temperature where the Z F C curve has a maxi- mum. The position of the maximum and its width change with concentration. For low values of x it is rather narrow and becomes wider with higher concentrations, but at x = 0.80 starts to be sharp again. Also, a shift of the maximum to higher temperature with increasing x is a general tend- ency. When we compare the results for specific heat and magnetization, we found that below 70%
J. Mtodzki et al. / Monte Carlo simulation of Heisenberg spin glass 273
0 . 5
0 . 4
0 . 3
0.2
0.1
I~ I I I I I I I I I I I
l ; a
ii g
ll
• • I J! •
T [] • B • x=O, 05
n ~ B
• D ~ × = 0 . H ]
, ~ " ~ i ' e ' " z . . m e ~=o.2o
ooaooo a o o a o x=0.30
~ - • • . = g l i l l i g g g g
i I I I I I I I t I I I
2 4 6 8 10
T [K]
0.035
0.030
::~ 0,025
O. 020
0o015
I I
q:~
cilq" 0 e
" , , f , o _
& ",,
I I I
. g l a
D I iI
o g a a B B a x = 0 . 4 0
• uci ~ •
• • • BO1 I
I
b
• • T • ~. A ., x=O. 50
• .,,. &
o _O..o.
_ oo
l I L l l l
10 20 30
T [K]
I I I I I I I I I
0.015 C
DDQO a
Q a o
0.014 ,,%,, =
IUl • •i~l
till a oat
i o x=O. 70
[ -,'.-
=w ~ h a
0 a am
& &
0.012 '~A
,,%
A I l l ° •& •
• . ; ~ t ~ AAA.~ * x=0.80 .
A • / % ~ a
0.011 . * . A t . . •
• A /1 A
A 8 i .
0.010
I t I I I I I I l
20 40 GO 80 1 O0
T [K]
Fig. 1. Temperature dependence of field-cooled (FC) and zero-field-cooled (ZFC) magnetizations for various concentrations of spins, x. At low temperatures, for each concentration, the ZFC curve (closed points) is always below the FC One (open points). (a) x - 0.05, 0.10, 0.20, 0.30; (b) x =, 0.40, 0.50, 0.60; (c) x = 0.70, 0.80. Arrows indicate the critical teml~atures. Nottce the different
tempecature ranges and values of magnetization for (a), Co), (c). The dashed line in Co) is a 8Oide for the eye only.
274 J. Mtodzki et al. / Monte Carlo simulation of Heisenberg spin glass
40 35 30 25
I--....- 20 15 10 5
I I I
[] Galazka
• Novak - /S OseroFf
• E~corne
I I
A
A a O [] 0
ZX •
o O
~ , ~ m l ~ ~ I L i
I I I I
©
[] O
O
1 I I I I I
0.0" 0.2 0.4 "0.6 0.8 .0 X
Fig. 2. Magnetic phase diagram of temperature versus con-
centrafion of spins, x. A comparison is given between the exI~'imental data for Cdl_xMnxTe [10-13] and our numerical results, m _ ref. [10], • - ref. [11], • - ref. [12], ~ - ref. [13], o
- our data.
-2
-4 z n u'l - 6 cIc LO n
>.. - 8 C-~
CIC L.l.l
Z w -ID
-12
- 1 4
I I I I I ! I I I
a a
a a a a
a a a a a a a
a a D a
o o a a a a
a a a
a D a a a a o a n a
o a a a a a m ~
o oO @Ooo × = 0.05
OoOS? o
o = o 0.10
- a o o ° a ° a o Q ° ° o ° m o a °
a ~ a a a a
a a a a a ~ a
• o °°o °°oan n QD on on a a •
o n a ° a ° m u a
~
a Q a a a a ~~ a a a
I m I I I
2O
0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
I I l I m l
40 60 80 1 O0
T [K]
Fig. 3. Z F C eaergy per spin as a function of tempzrature for various concentrations of spins, x = 0 . 0 5 , 0.10 . . . 0.90 (from
top to bottom).
- 8 . 0 ' ' '
a FC
• ZFC
6,-
- 8 . 8 • o
- - m _ m °
- 9 . 0 ~ i i
0 2
z - 8 . 2 o._ u ' ) r,.- - 8 . 4 I,.J.J Q _
L.~
Z l u_l
I I I I I
B B
I
@
@
m
l l . a
x=O. 40
-d
i i i i I I i
4 6 8 10
T {K]
Fig. 4. Energies per spin for Z F C and F C cases in the low temperature range for x = 0.40.
spin concentration, the m a x i m u m for C is at a higher t e m p e r a t u r e and b r o a d e r than the respec- tive one for M,. At 0.70 a n d 0.80 the position and width of M, a n d C e x t r e m a axe the same.
It m u s t b e mentioned that all our results are c o m p u t e d for high external magnetic fields and some difficulties m a y arise when we try to com- pare numerical results with experimental measure- ments done without fields. This is a p r o b l e m espe- cially for specific heat of low concentration sam- pies where a strong dependence on field is ob- served [16]. W e tested b y M o n t e Carlo simulation the systems with x = 0.30 for different values of magnetic field. I n fig. 6 three curves are plotted for B = 2.8, 5.6, 8.4 T, Because of litde depen- dence on field for this case, we feel justified to c o m p a r e our results with experimental data for concentrations starting f r o m 0.30.
It was reported that, in the vicinity of the magnetization cusp, the specific heat behaves lin- early and shows no a n o m a l y [14,16]. The same tendency is reproduced b y c o m p u t e r simulations for concentrations lower than 0.70.
3.3. Temperature-dependent magnetic susceptibility A p a r t f r o m the experimentally m e a s u r e d specific heat a n d magnetization, we also de- termlned the diagonal c o m p o n e n t s of the mag- netic susceptibility tensor, X, for samples with
J. Mtodzki et al. / Monte Carlo simulation of Heisenberg spin glass 275
I - -
< [ : IJ.I - r "
(._) [ J - ( . 3 LLJ E L rJ')
iJ.i -1- (,.3 i t _ (_.) LLI {:Z- U')
0 . 1 6
0 . 1 3 ]
0,10 °0
o °
0,07 1 ,"
0 . 0 4 ~ , "
0.011
F
.~ i 00.1G
0.13
0,10!
-,,
0.07
0.04
0.01 I
I I I
a FC
• ZFC
i l i l w_ g =
o
x=O.lO
I I I I
2 4
T [K]
I
0 . 1 5
0.12i
L.cJ
~
0,0~u_
~
O,OGa
I g
I I I I I I I I I [ I J
~ ~ o ~ . .
[]
-O
A2.BT
o 5 , 6 T O 8 . 4 T
- o x=0.30
0.00 - ~
0
1'o ' ~ ' ~ ' ~ ' & ' =T [K]
Fig. 6. Specific heat versus temperature for different external magnetic fields B = 2.8, 5.6, 8.4 T. Spin concentration, x = 0.30.
I I I I I
I o FC
• ZFC
d 3 " ml~° I : 1 " ] I M ~ " [3 --
~ i ~ ' = . e 0 . , oe = _
x=0.40
I I I |
10 20
T [K]
bi
30
different concentrations. For systems below 0.30 we found that xx, y y components of X almost do not depend on temperature, while X , goes down with temperature. We present these results in fig.
7. The low values of X,, results from the fact that t h e s u s c e p t i b i l i t y t e n s o r w a s d e t e r m i n e d f o r h i g h e x t e r n a l m a g n e t i c f i e l d i n t h e z - d i r e c t i o n . H o w - ever, f o r h i g h e r concentr~ttions, this b e h a v i o u r was n o t o b s e r v e d a n d t h e r e w a s n o d i s t i n c t i o n b e t w e e n d i a g o n a l c o m p o n e n t s o f X- S u c h a n i s o t r o p y o f m a g n e t i c s u s c e p t i b i l i t y , i f o r s a m p l e s w i t h s m a l l v a l u e s o f x , h a s n o t b e e n o b s e r v e d e x p e r i m e n t a l l y . 0 . 1 6
0.13
0.10 (..3
U._
0.070.04
0.01 0
I I / I I ! I I I I
/ ~
ta
FC• ZFC
x=O. 70 • •
i I f •
C
I I I I I I I I I
20 40 60 60 100
T [K]
Fig. 5. Specific heat as a function of temperature for various concentrations of spins, x. (a) x - 0.10, Co) x = 0.40, (c) x - 0.70. The arrows indicate critical temperatures obtained from
the magnetization for correslxmding x.
I I I I I
= -'- cmpmm~
N ' ' . N •
1 5 " I ~ b l • a I I I
N lO .--',•" " "• • , / " - ' n u t - , m
5 ~,0.~
I I I I I
0 1 0 20 3O
T [K]
Fig. 7. Magnetic susceptibility ~ a function of temperature for low concentration of spins, xi,= 03.0. The diffecenoe between yy component (open squares) and zz (dosed squares) is seen.
276 J. Mtodzki et al. / Monte Carlo simulation of Heisenberg spin glass
O. 156
O. 150
:LV
0.144
I I I I o I I o ~ o o ~ .
o
o ° o ° ° o o ° o ° o - o o o o O o O O o ° • oo • i o o nw o ° ° o o 0 • m i l l I i i I l n n l l l l l I
n l l l l l • l •
• o FC
ell
• ZFC
T=IK, x=0.20
0.138 0
I I I I I I
0 10 20 30 40
m m
I
MCS/1000
: £
0.038
0.036
O. 034
O. 032
I I & [] I I I I I
- o o o o _
o ° ° ° ° ° ooo o o o o o o o
[] o o o o o o ooOO ° []
- oOO o I~.
i n m i m i l l m m mmmi=
limB I
m% o FC
= % = • ZFC
m i l l l l •
= i T = I K , x = 0 . 4 O
l i b b
I I I I I i I
10 20 30 40
MCS/1000
Fig. 8. M a g n e t i z a t i o n as a f u n c t i o n o f M o n t e C a r l o steps ( M C S ) . ( a ) N u m b e r o f spins, N s = 2195, c o n c e n t r a t i o n o f spins, x = 0.20;
(b) N, = 7861, x = 0.40.
3.4. Dependence of magnetization and energy on number of Monte Carlo steps
For typical simulations which allowed us to get the temperature dependence of thecgfiodynamical quantifies, we used about 2000 MCS for each temperature point. Additionally, we did longer computations of 40000 MCS for several tempera- tures and concentrations of spins.
From these long runs we wanted to observe the large "time"-scale evolution of magnetization and
energy. Another point was to determine the num- ber of MCS's necessary for the system to reach thermodynamical equilibrium and the correspond- ing number of MC steps needed to average physi- cal quantifies. We found that for the same temper- ature and concentration of spins, the ZFC and FC systems behaved in different ways below the criti- cal temperature.
In general, rejecting several hundreds of MCS's and taking 1000 steps for the average FC values was enough, but in the ZFC case the magnetiza-
z t-t Lr) t - ~ 1.1.1 (3-
>.- i , i Z LI..J
-5.79
-5.80
-5.91
-5.82
I I I I I I I
a
me T=1K, x=0.20
m I
• [] FC
== • ZFC
@i I l i l m
mmeimm i
o mmmilwi • l m
o o o D me me
O pOD Q l a
° ° o ° ° ° ° ° O o o Ooo oo o o o ml o o oo •
I I I i I I I
10 20 30 40
MCSI1000
Z - - -8.90
o _ u ' ) E E W ( 2 .
> - -8.93
LLI Z LIJ
-8.96
I I I I I I I
T=IK, x=0.40
b
l l l l i i i l l l B i I 0 FC
• ZFC
I I I I I I I I I I I i i i i i i i i i i i i i i i
°°ooooOooooooooOOOoOOOoOOoooOooooOoOOOO I
I I I I I I I
0 10 20 30 40
MCS/1000
Fig. 9. E n e r g y p e r spin as a f u n c t i o n o f M o n t e C a r l o steps ( M C S ) . ( a ) N u m b e r o f spins, N s = 2195, c o n c e n t r a t i o n o f spins, x ~ 0.20;
(b) N s = 7861, x = 0.40.
J. Mtodzki et aL / Monte Carlo simulation of Heisenberg spin glass 277
d o n i n e r ~ (energy decreases) steadily over tens of thousands of MCS's. Using the FC "time"-seale for Z F C system produced the results presented in sections 3.1 and 3.2. The examples of long " t i m e "
evolution of magnetization and energy are given in figs. 8 and 9.
In fig. 8a we plot the magnetization as a func- tion of Monte Carlo steps for T = 1 K, x = 0.20 and N, = 2195. Each point on the curve represents average values of Mz over 1000 configurations.
The Z F C magnetization tends toward the FC which is almost constant with MCS. In fig. 8b the same results are presented for x = 0.40 and N s = 7861. It can be seen from both plots that for lower concentration of spins less MCS's are needed, for the Z F C M, to reach a value corresponding to the F C level. In fig. 9a, b energies are plotted for the same values of T, x and N,. Again the FC curves are almost constant and Z F C values tend toward them.
We also did longer simulations for T = 3, 10 K.
F o r samples with x = 0.20 and Tm~ x ~-2 K we found no differences in behaviour of Z F C and FC quantities. However, systems with x = 0.40 and Tma ~ - - 7 K behaved in the same way only for T = 10 K, but at 3 K the Z F C energy and magne- tization were approaching FC values.
3.5. Other effects
F r o m our numerical work we could produce results which are in good agreement with experi- ment. However, by changing simulation parame- ters we also obtained another picture, which is not experimentally observed.
In fig. 10 we present M, curves for Z F C & FC systems for x = 0.30. The size of the sample was 1200 spins. In this simulation temperature was changed in greater steps than usual. In an experi- ment, these changes correspond to a greater rate of cooling or heating of a real sample. Comparing fig. 10 to fig. l a we can see that the maximum for the Z F C magnetization is shifted to the right and the FC curve is shifted to the left. Such effects were neglected in the experimental works. Some evidence that empirical results depend on rate of cooling was reported in ref. [17].
0.06
0.05
0.04
0.03
!
o o
o
I I I I !
o FC
• ZFC
i •
o o @
o
r
l l l
I0
T
x=O. 30
D m u
° o i i Ig
l I I
20 30
[K]
Fig. 10. Temperature dependence of the magnetization in case of rapid temperature changes, (8 "measurements" in 1-15 K interval), N, ffi 1200, x ffi 0.30. The arrow indicates the critical temperature for magnetizatioin of a typical x = 0.30 run [see
fig. l a for comparison].
Doing several Z F C landings in high concentra- tions ( x = 0.70, 0.80), we found that starting points of Z F C curves for magnetization are distributed randomly over the area where the FC curve goes through. This was the reason that not only one curve for the Z F C system was observed in our experiments but a whole family of curves. It was even possible to get to a situation where the be- haviour of Z F C and FC systems was reversed.
This is probably a numerical artifact coming from high fluctuations of both magnetization curves, which are very flat and close each to other. For lower concentration o f spins, the difference be- tween Z F C and F C magnetizations was greater and Z F C landings were far from FC curves.
4. Final c o n d u s i o n s
4.1. Magnetic phase diagram
Using the M o n t e Carlo simulations described we were able to reproduce numerically the experi- mental magnetic p h a s e diagram, extending it to the physically inaccesible concentrations of M n ions in C d l _ x M n x T e . Since the critical tempera- ture for every concentration is scaled in our calcu- lations b y J ~ , fitting the numerical results for the
278 J. Mlodzki et al. / Monte Carlo simulation of Heisenberg spin glass phase diagram to experimental data allows us to
derive a value of J ~ and to study its possible dependence on concentration. The same method can be used for other SMSC with unknown Jm~.
The common feature of the model system over the whole range 0.05 < x < 0.90 was the difference of the ZFC and FC magnetization, energy and specific heat. However, more detailed studies have allowed us to distinguish three different con- centration-dependent regions.
In the near percolation region with low con- centration, 0.05 < x < 0.20, we observed spin glass behaviour with a cusp in ZFC magnetization and a broad maximum of specific heat shifted to higher temperature. The dominant features were the ani- sotropy of the magnetic susceptibility and smooth curves for the magnetization. The relaxation
"times", measured in MCS's, for ZFC energy and magnetization, were short; the system quickly evolved to FC state.
In the intermediate region, 0.30 < x < 0.60, the system also behaved like a spin glass, but with a longer relaxation of the ZFC quantities. The mag- netization curves showed larger fluctuations and the difference between perpendicular and parallel field components of the magnetic susceptibility vanished. As in the first region, the difference between the cusp in ZFC magnetization and the broad maximum of the specific heat was distinct.
The last interval of concentrations, 0.70 < x <
0.90, is partially not accessible in experiment, since Cdl_xMnxTe crystailiTes in fcc structure only for x < 0.70 [2]. The main feature of this area in our simulation was the common position of magneti- zation and specific heat maxima. This result is typical for an antiferromagnetic phase, which is also apparently evidenced in experiment [10,18].
In this region we observed large fluctuations of physical quantities and long relaxations of the ZFC magnetization and energy toward FC values.
The M, curves for the ZFC and FC systems were very flat and close each to other.
4.2. New results
Apart from the results which were compared with experiment we also observed some new ef- fects not previously reported. Anisotropy of the
magnetic susceptibility for low concentration of spins is one such example. Also the different tem- perature-dependence for the ZFC and the FC specific heat has not been observed experimen- tally.
Investigation of the energy and magnetization during long runs of 40000 MCS's gives some in- sight in the time-dependent behaviour of the ZFC states, under the assumption that Monte Carlo steps are related in some way to real time in an experiment. If this condition is fulfilled, we can derive the conclusion that the ZFC system is a metastable state relaxing toward the FC one. The time of relaxation depends on the concentration of spins - the higher value of x, the greater time.
Another effect, which can be attributed to time scales, is the dependence of the magnetization results on the rate of cooling. Quick changes in temperature shift the position of the cusp for the ZFC magnetization. This effect can explain the different critical temperatures reported by various authors for the samples with the same concentra- tions of Mn ions [10-13].
In summary, we conclude that the Monte Carlo method is a useful tool to investigate SMSC sys- tems and produces results in good agreement with experiment as well as predicting some new effects, not observed experimentally at present.
Acknowledgements
We acknowledge that most of the computations presented in that paper were done at the Univer- sity of Regensburg. We would like to express our sincere thanks to Professor Wolfgang Gebhardt and his group at the Department of Physics for the hospitality they extended to one of us (JM) during the stay at the Regensburg University.
References
[1] R.R Ga)~zkA and J. Kossut in: Landolt-Bornstein, New Series 1To, eds. O. Madelung, M. Schultz and M. Weiss (Springer, Berlin, 1982) p. 302.
[2] J. BL~k, U. Dcbska, R.R. Galtgka, G. Jasiolek, E. Mizera and B. Bryza, Suppl. Acta Crystallogr. A 34 (1978) 245.
[3] W.H. Brumage, C.IL Yager and C.C. Lin, Phys. Rev. 133 (1964) A765.
J. Mtodzki et aL / Monte Carlo simulation of Heisenberg spin glass 279 [4] T.M. Giebultowicz, J.J. Rhyne, W.Y. Ching, D.L. Huber
and R.R. G~¢zka; J. Magn. Magn. Mat. 54-57 (1986) 1149.
[5] B.E. Larson, K.C. Hags and L. Aggarwal, Phys. Rev. B 33 (1986) 1789.
[6] A. Lewicki, J. Spalek, J.K. Furdyna and 1L1L Ga/gzka, Phys. Rev. B 37 (1988) 1860.
[7] K. Binder and A.P. Young, Rev. Mod. Phys. 58 (1986) 801.
[8] H. Kett, W. Gebhardt and U. Krey, J. Magn. Magn. Mat.
46 (1984) 5.
[9] Monte Carlo Methods in Statistical Physics, ed. K.
Binder, vol. 7 (Springer, Berlin, 1979).
[10] R.IL Gala Tka; S. Nagata and P.H. Keesom, Phys. Rev. B 22 (1980) 3344.
[11] M.A. Nowak, O.G. Symko, D.J. Zheng and S. Oseroff, J.
Appl. Phys. 57 (1985) 3418.
[12] M. Escome, A. Manger, R. Triboulet and J.L. Tholence, Physica B 107 (1981) 309.
[13] S.B. Oseroff, Phys. Rev. B 25 (1982) 6584.
[14] A. Twardowski, C.J.M. Denissen, W.J.M. de Jonge, A.T.A.M. de Waele, M. Demianiuk and 1~ Tdboulet, Solid State Commun. 59 0986) 199.
[15] J. Spa/ek~ A. Lewicki, Z. Tamawski, J.K Furdyna, R.R.
Gaigzka and Z. Obuszko, Phys. Rev. B 33 (1986) 3407.
[16] S. Nagata, ILR. Ga/gzka, G.D. Khattak, C.D. Amara- sekara, J.K. Furdyna and P.H. Keesom, Physica B 107 (1981) 311.
[17] 1LR. GalLtzka, W.J.M. dei Jonge, A.T.A.M. de Waele and J. Zeegers, Solid State Commun. 68 (1988) 1047.
[18] T.M. Giebultowicz, J.J. Rhyne, W.Y. Ching, D.L. Huber, J.K. Furdyna, B. Lebech and ILR. Ga/~ka, Phys. Rev. B 39 (1989) 6857.
