Phase diagrams of transverse Ising film

Physica A 193 (1993) 133-140 North-Holland

~

A

Phase diagrams of transverse Ising film Xuan-Zhang

Wang

China Center of Advanced Science and Technology (Worm Laboratory), P.O. Box 8730, Beijing 100080, China and Department of Physics, Harbin Normal University, Harbin 150080, China ~

Yan Zhao Department of Physics, Northeast Forestry University, Harbin 150040, China

Received 22 April 1992 Revised manuscript received 11 August 1992 Applying the effective-fieldtheory, we study the phase diagrams of a transverse Ising film. The phase diagrams show that, for Js/J < R c (R c is a critical value of Js/J), the critical temperature Tc of the film is smaller than the bulk critical temperature Tbc of the corresponding semi-infinite system; however, for Js/J>Rc, T~ is larger than the surface critical temperature T~cof the semi-infinite system and as Js/J is increased further, Tc (except Tc of the film with two layers) rapidly approaches Ts~. We also calculate the critical transverse field h c and Tc as a function of the film thickness.

I. Introduction T h e d e v e l o p m e n t of the technique of molecular-beam epitaxy has resulted in the appearance of quality ultrathin films and superlattices, which has produced an increased interest in the experimental [1-4] and theoretical [5-7] investigations of magnetic films, A fascinating problem involves the phase transition of magnetic films. The phase transitions of various transverse Ising systems have attracted the attention of theoretical physicists [8-12]; in particular, the corresponding semi-infinite Ising systems as a model for studying surface magnetism have been investigated with different methods [13-16]. Consider a transverse Ising film composed of a few atomic layers parallel to the surfaces. Because the film is very thin, it generally differs from the semi-infinite transverse Ising system in its magnetic properties, at least quantitatively. As the n u m b e r of atomic layers is increased, the magnetism of the Mailing address. 0378-4371/93/$06.00 I~) 1993- Elsevier Science Publishers B.V. All rights reserved

X . - Z . Wang, Y. Zhao / Phase diagrams o f transverse lsing film

134

film will approach gradually that of the corresponding semi-infinite system. Recently, the critical temperature and magnetisation of an ordinary Ising film were discussed in refs. [7,8] within the framework of the mean-field and the effective-field methods, respectively. Ref. [10] gave the critical behaviour of the simple transverse Ising film where the surface and internal coupling constants are the same. For a more ordinary case, the surface and internal coupling constants can also be unequal, and the surface coupling constant can be larger or smaller than that of the internal one. We have discussed the magnetisational behaviour of this kind of transverse Ising films [11]. We devote this paper to a theoretical study to understand basically the phase transition properties of the transverse Ising film. Using the effective-field theory which was originally applied to investigate the bulk transverse Ising system [17] and semi-infinite transverse Ising system [18], we deal with the phase transition of the transverse Ising film, and for the applied transverse field h = 0, compare the results obtained in this paper with the results in refs. [7,8].

2. Formalism We consider a transverse Ising film with spin s = ½and a simple cubic lattice with two (001) free surfaces. The number of atomic layers in the film, parallel to the surfaces, is represented by n. In transverse fields, the Hamiltonian of the system can be written as

fl =

E g, s;s ij

- h, E s7 - hs E 1

s

(1)

,

where the nearest neighbor coupling constant J~j is equal to Js (Js ~> 0) if sites i and j are on the surfaces, and to J (J > 0) otherwise, h I and h s are transverse fields on the inside and on the surface layers, respectively. F,1 and E s express the sums over the sites on the inside and on the surface layers; S z and S x are the components of a spin-1 operator. Applying ( S ~) to represent the thermal average value of the operator S ~ for any temperature and letting (a ~) = 2 ( S ~ ) , within the effective-field theory [19], then (a ~) is written as

(a~)=((~Ji/a~/Hi)tanh(JflHi))

.

(2)

1

H e r e ( . . . ) indicates the thermal average, /3 =

1/KBT and

X.-Z. Wang, Y. Zhao / Phase diagrams of transverse lsing film

Hi =

2hi) e +

ja zr

.

135 (3)

Using the expression [19] of Honmura and Kaneyoshi, D = O/Ox, we have ~ [cosh(DJii+~) 4- ai+8 z sinh(DJ ii + ~ )])f~(X)lx= 0 ' (a~) = /I~I

(4)

1 2i + x tanh[z/3(4h f/(x) = [(2hi) 2 + X2]I/2

(5)

with x2),,2]

Let us use the approximation as follows: ( a ; a ~ . . . a~) = ( a ~ > ( a ~ ) . . . ( a ~ ) ,

(6)

to decouple the multi-spin correlation functions and note e x p ( a D ) f ( x ) = f ( x + o~) ,

(7)

then we can obtain numerically (a~) (i = 1, 2, 3 , . . . , n) of the ith layer from eq. (4). But in this paper, our aim is to obtain the critical temperature, therefore after substituting (6) into (4) and, when T--* T c, we can expand (4) and then neglect the nonlinear terms of ( a~ ). Thus eq. (4) is changed into a set of equations as follows: ( a l ) ( 4 K , - 1) + K z ( a ; ) = 0 , (a~) K + ( a 2 ) ( a K - 1 ) +

K(a3) =0,

( a ; _ l ) K + ( a ; ) ( 4 X - 1) + K(aZ+l) = 0, (a~_2) K + ( a ~ _ 1 ) ' ( a K - 1 ) +

(8)

K ( a ~ ) = O,

(a~_ l ) K 2 + (a~)(nK, - 1) = 0 , where K 1 = cosh3(DJs) cosh(DJ) sinh(DJs)fs(X)lx= o , Kz = cosh4(DJs) sinh(DJ)fs(X)lx=o,

(9) (10)

136

x . - z . Wang, Y. Zhao / Phase diagrams of transverse lsing film

K = coshS(DJ) sinh(DJ)f~(x)lx= 0 .

(11)

K, K~ and K 2 are functions of temperature as shown in eq. (5), therefore the critical temperature is determined by 4K l - 1

K 0

K2 4K- 1

K

0 K

4K-1

"'" "'"

0 0

0 0

0 0

"'"

0

0

0 =0.

6 0

(1 0

6 0

"" ""

0

0

0

""

4K'-I K 0

k 4K-1 K2

6 K 4K 1 - 1

(12)

3. Results and discussion

From (12), we can obtain numerically the phase diagram of the film. The results show that there can be two phases, a film ferromagnetic phase (F) which means that the longitudinal magnetisation (M = E~_ 1 (S~)) in the film is not equal to zero, and a film paramagnetic phase (P) which corresponds to M = 0. In addition, if the number of layers in the film, n, is very large (or n ~ ~), the film can be considered practically as a semi-infinite Ising system. As is well-known, there are two kinds of transitions in the semi-infinite Ising system: the bulk transition and the surface transition, and the critical temperatures related to them are called the bulk critical temperature (The) and surface critical temperature (Tsc), respectively. First we calculate the T - J s / J phase diagrams for different transverse fields and numbers of layers; typical results can be seen in fig. 1. For comparison we also present the bulk and surface transition temperature lines (dashed lines) of the corresponding semi-infinite Ising system [19]. From fig. la for h = h s = h~ = 0, one can see that the phase boundary curves intersect at the same point for n/> 2. The value of J s / J corresponding to this point is called the critical value R c of Js/J. We find Rc = 1.322, which is slightly larger than 1.3068 presented by G. Wiatrowski et al. [8]. For J s / J < R~, the critical temperature (T¢) of the film is smaller than the bulk critical temperature Tbc of the corresponding semi-infinite Ising system, and the smaller n is, the lower T c is. When J s / J = R e, T~ is independent of n (n ~ 1) and equal to Tbc. These conclusions are qualitatively similar to those obtained within the framework o f the mean-field theory [7]. For J s / J > R c, Tc is larger than T~ of the corresponding semi-infinite Ising system and T c of the bilayer film (n = 2) is

X.-Z. Wang, Y. Zhao / Phase diagrams of transverse lsing film (a)

137

(b)

a 2.0

..?..

ed

2.0

f

R .........

e f

_6_8

1.0

...........

1-192

1.O

' R~=13221 1.0

2.0 Js"

'

3,0

0 1.0

2.0

Js/J

Fig. 1. T-Js/J p h a s e d i a g r a m s for (a) h = h~ = h s = 0 a n d (b) h = h I = h s = J. T h e p h a s e b o u n d a r y c u r v e s are a: n = 1 (of the t w o - d i m e n s i o n a l lattice); b: n = 2; c: n = 3; d: n = 4; e: n = 8 a n d f: n = 12, r e s p e c t i v e l y . W h e r e n r e p r e s e n t s the n u m b e r of the l a y e r s in the film a n d R c is the critical v a l u e of Js/J. The d a s h e d lines s h o w the b u l k or surface critical t e m p e r a t u r e of the c o r r e s p o n d i n g s e m i - i n f i n i t e Ising system.

the largest for a given value of Js/J. As n is increased, T c decreases and rapidly approaches T~c in the figure and for n = 12 we are not able to distinguish between T~ and Tsc. An exceptional case is that T~, for n = 1, is even smaller than Ts~. We think that the film with n = 1 is just a complete two-dimensional Ising lattice, so that this is due to the difference between the n u m b e r of neighbors for atoms in the two-dimensional lattice and the n u m b e r of neighbors for atoms at the edges in the film (n > 1). On the other hand, as Js/J is enlarged in the region of Js/J > R~ in the figure, Tc of the films with n > 2, including Tsc , tend towards Tc of the two-dimensional lattice, and in our numerical calculation we find that T~ of the film with n = 2 approaches very slowly T~ of the two-dimensional lattice. This shows that as Js/J is enlarged, the relative importance of the surfaces of the films increases, but one cannot obtain this conclusion in fig. 3 of ref. [7]. In fig. lb, we present the T vs. Js/J phase diagram for h = h s = h~ = J. We find R e = 1.308 for this transverse field. Comparing fig. l a with lb, one can see that the transverse field obviously does not change the value of Rc, which is a result of us taking h~ = h s. For the film with n = 2, when Js/J is smaller than some value, there cannot be a ferromagnetic phase. O f course, this situation can also exist for a film with n > 2 in a field which is transverse enough. T h e three T-h phase diagrams of the film with n = 2, 4 and 12, respectively, are. given in fig. 2, and they also show the relations between the critical t e m p e r a t u r e and transverse field for different parameters (Js/J). The T-h curve for a given value of Js/J and the h-axis intersect at some point, and the

138

X.-Z. Wang, Y. Zhao / Phase diagrams of transverse lsing film

-rn"

(a)

I,v

lI

1.0 I

1,0

2.0

3.0

-2

(b)

t-

1.0

0 .-,L

I

1.0

2.0

3.0

(c)

I

I

1.0

2.0

I

II

h/a

3.0

F i g . 2. T-h p h a s e d i a g r a m s f o r (a) n = 2, ( b ) n = 4 a n d (c) n = 12. I: Js/J = 0 . 5 ; II: Js/J = 1.0; III: Js/J = 2 . 0 .

value of h corresponding to this point is usually called the critical field (he). When h > he, at any temperature, there cannot be a ferromagnetic phase. In order to understand the critical temperature and critical field as a function of the thickness of the film, we present figs. 3 and 4. Fig. 3 shows the relation between T c and n for different values of Js/J and transverse fields. The two short-dashed and the two long-dashed lines in the figure represent the bulk critical temperatures and the surface critical temperatures of the corresponding semi-infinite Ising system, respectively. One can see that T c changes monotonously with n for a given value of Js/J and as n is increased, the critical temperatures for Js/J > R~ and Js/J < R~ approach the surface and bulk critical temperatures of the semi-infinite Ising system, respectively, which are different from those in fig. 5 of ref. [7] where they approach the same value T¢(Tbc ). Fig. 4 gives the critical field as a function of n for Js/J = 0.5, 1.0 and 2.0. The short-dashed and the long-dashed line correspond to the bulk and the surface critical fields, respectively.

X.-Z. Wang, Y. Zhao / Phase diagrams of transverse Ising film

139

2.0

~_u

KTsc/d

...=.%_~_/."!_........... 1,0

b'

C C

0

¢,*

I

I

I

I

2

4

6

8

!

n

.

10

Fig. 3. Dependence of the critical temperature on the number of layers comprised by the film. a:

Js/J=2.0 and h =0.0; a': Js/J=2.0 and h = 1.0J; b: Js/J= 1.0 and h =0.0; b': Js/J= 1.0 and h = 1.0J; c: Js/J = 0.5 and h = 0,0; c': Js/J = 0.5 and h = 1.0J The horizontal short-dashed and long-dashed lines represent the bulk and surface critical temperatures of the corresponding semi-infinite Ising system, respectively.

a

3.0 .hsc/J ,._"_~.~/_J. ...........................

t-o

2.0

1.0 0

C

I

I

I

2

4

6

I

n

8

Fig. 4. Dependence of the critical transverse field on the number of layers contained by the film. a:

Js/J = 2.0; b: Js/J = 1.0; c: Js/J = 0.5. The short-dashed and the long-dashed line correspond to the bulk and surface critical fields of the semi-infinite Ising system, respectively. In summary, transverse

we have calculated and discussed the phase diagrams

Ising films within the framework

have obtained

some typical temperature

of the

of the effective-field theory.

- Js/J a n d t e m p e r a t u r e

field phase diagrams, and given the critical temperature

We

vs. t r a n s v e r s e

a n d f i e l d as a f u n c t i o n

o f t h e film o f t h e film t h i c k n e s s f o r d i f f e r e n t v a l u e s o f Js/J. F o r t h e t r a n s v e r s e

140

x . - z . Wang, Y. Zhao / Phase diagrams of transverse Ising film

field h = 0, t h e film b e c o m e s a u s u a l Ising film. H o w e v e r , s o m e o f o u r r e s u l t s a r e q u a l i t a t i v e l y d i f f e r e n t f r o m t h e c o r r e s p o n d i n g r e s u l t s in t h e w o r k o n t h e u s u a l I s i n g film w i t h i n t h e f r a m e w o r k of t h e m e a n - f i e l d t h e o r y [7].

Acknowledgement T h i s w o r k was f i n a n c i a l l y s u p p o r t e d b y t h e S c i e n c e F o u n d a t i o n o f H e i l o n gjiang Province.

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