Ising model of phase transitions in ultrathin films

0038-1098/9053.00+.00 Pergamon Press plc

Solid State Communications, Vol. 74, No. 3, pp. 155-158, 1990. Printed in Great B r i t a i n .

ISING MODEL OF PHASE TRANSITIONS IN ULTRATHIN FILMS F. Aguilera-Granja and J. L. Mor~n-LSpez Instituto de Flsica, "Manuel Sandoval Vallarta ~ Universidad AutSnoma de San Luis Potosf 78000 San Luis Potosi, S.L.P., MEXICO (Received on 11 December 1989 by A.A. Maradudin) The phase transitions in ultrathin Ising films are studied within the mean field approximation. Analytic expressions for the thickness dependence of the Curie temperature are given. The model is applied to estimate the magnetic interactions in Fe and Gd ultrathin films. The experimental results in the case of Fe films on Au substrates can be very well reproduced.

The study of the phase transitions plays a central role in modern condensed matter physics. Although much is known about phase transitions in 3d-systems, many aspects remain to be understood in systems of low dimenslonality. Very often one finds unexpected and interesting properties at surfaces, thin films, etc. For example, recent experimental studies (1--4] on the magnetic properties of surfaces of Gd, Cr, and Tb have shown that a surface ordered magnetically can coexist with a magnetically disordered bulk phase.

Here, we study some of these new experimental results by modelling the ferromagnetic thin film within the framework of the spin-~ Ising Hamiltonian. This model, solved in the mean field approximation, allows us to study the thickness dependence of the magnetization. As a result, we obtain an analytic expression of the thickness dependence of the Curie temperature Tc. Our model is applied to the cases of Fe on Au [11] and of Gd on W [10]. Despite the simplicity of the model, the first set of experiments can be very well reproduced. The model used to describe the magnetic properties of the thin film is shown in Fig. 1. The crystal is subdivided into planes parallel to the surface and we assume that the two surfaces are equivalent. The coordination numbers on the parallel planes and i n t e r - p l a n e s are denoted by Zo and ZI, respectively (Z = Z0 +2Z~). To follow the magnetic transitions, we define a magnetic long-range-order parameter ~j. at each plane, i as the difference of the spin-up and spin-down probabilities at the j - t h plane. Within the mean field approximation and taking only first nearest neighbor interactions into account, the free energy of the system can be written in a very simple way [8]. For a temperature close and below temperature Tc, it is possible to perform a linearization on the free energy of the system. In that range of temperatures, it is easy to show [8] that the free energy reduces to the following set of homogeneous linear equations:

The most studied systems are those with magnetic phase transitions and much effort has been devoted to their understanding. From the theoretical point of view, one of the models more widely used to study the magnetic properties of surfaces, is the Ising model. Within that model one can take into account in a straightforward manner the presence of the surface. The environmental effects produced by the surface can be simulated by assuming a location dependent coupling constant J,,~; m and n denote the position of the magnetic atoms in the lattice. The simplest case corresponds to a situation in which only the surface coupling constant Js is assumed to be different from the rest. In that context, the coexitence of a paramagnetic bulk with a magnetic surface can be obtained if Js exceeds a critical value Js c . The estimation of Js c has been the subject of various publications [5 9]. In general, it is found that it depends sensitively on the surface geometry, bulk structure and range of interactions.

,i0 = 0,

On the other hand, with the developement of modern vacuum science it is now possible to study experimentally the magnetic properties of low dimensional systems. In particular, by depositing magntic atoms on the top of non-magnetic substrates, the thickness dependence of the Curie temperature of ultrathin films of G d on W ( U 0 ) [10] and of Fe on Au(100) [11], has been measured.

(1)

where q is a vector with the ~. components, and A is a square n x n tridlagonal matrix with elements:

A..

=

(~rc-ZoJ...)6...-z, J..(6.+,,. +~.,.+I). (2)

The existence of the two symmetric surfaces in 155

156

Vol. 74, No. 3

PHASE TRANSITIONS IN ULTRATHIN FILMS

The equation (5) was obtained by the using a cofactor expansion and the identity:

~0 /

4,

J /

/

•

/

if

•

1

•

2

J /

1 --y .

-y 1 :

... ... ...

0

0

...

-

--

2Yi+* --sin(i+l)0,

+x+

(7) I

,J

.-2

/

/j /

/

,f

/~

with tan 8 = V / ~ 1. The equation (5) is valid for values of the parameters such that Tc _< TcB- In the particular case of Joo and J0, # J , the critical surface in the parameter space of the J ' s that separates the two behaviors for n _> 3 is given by the relation:

n-1

Jo, / ¢ ,0

Fig. 1 Illustration of the model used for the thin film. (2-ac)2-2(2-a the system impose the additional symmetries on A: Akh = A . + , - k , . + , - h for 1 < k < n, and A~-,,I = A . - 1 + 2 . . - i ÷ l for 2 < l < . . Here, n is the thickness of the film. The same properties are obviously satisfied by J . , . . For semi-infinite systems, it is well known [59] that depending on the J m . , the surface Curie temperature Tcs is equal or higher that the bulk Curie temperature Tcs = ZJ/k. The set of Jm. that makes Tcs = Toy defines the critical surface f(J.~.,c) and is obtained from the equation:

det~l =

0.

(3)

In principle, one can take any number of Jm, different from the bulk coupling constant J. Here, just for simplicity, we take only Joo and Jox # J. In a thin film there is strictly only one transition temperature, the one that occurs at the surface of the thin film. We call that temperature To. In the monolayer and two-layer films (n=0,1) kTo = ZoJoo and kTv = ZoJoo + Z1Jo,, respectively. The transition temperature for n = 2 is given by:

c 2"n-2" c 4 r~-3 n-2 0 +)o (VC~_3)+ o (~--C~_2)(~:7_~) = • (S)

Here, the subindice c denotes the critical value of the quantities defined in Eq. (6). For situations where the Curie temperature is higher the one in the infinite solid, the equation that holds for n _> 3 is:

z 2 (z - a)2L(n)-4z(z

- a)c2L(n-1)+4c'L(n-2)

= O, (9)

where

L(n) = (1 + X/[4z -2 - 1[)" - (1 - ~/[4= -2 - 1[)".

(1o) Now, we apply our simple model to the recent experimental results on the thickness dependence of To. The magnetic properties of Fe films

| .0"

t Va~ + 8c~), ~= ~(a+

(4) 0.8 '

and in general, for n > 3: t~ ¢3

(z - - a)2sinnO--2(z

--

< ,_o

a)c2sln (n -- 1)e

+c4sin(n--2)8 =0,

0.6.

0.4.

(5) 0.2.

where

0.0 0 Thlckneaz

:r,--

(kTcs - ZoJ) Z, J '

c = (-~-),

a ~

Z0(Jo0 - J) Z, J

tano = 4v/G~-~ - 1.

(6)

Fig. 2 Dependence of (To ~Toy ) on sample thickness in Fe thin films deposited on An. The circles correspond to the experimental results [ill.

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PHASE TRANSITIONS

deposited on Au(100) were investigated by means of spin-polarized secondary electron emission spectroscopy (SPSEE) and spin-polarized low-energy electron diffraction (SPLEED) [11]. The thicknem dependence of the Curie temperature is shown by open circles in Fig. 2. Based on the results of spin-polarized Auger spectroscopy, which show that no induced magnetic moment occurs in the Au subsurface, one can infer that the Fe films are isolated magnetically. One can then assume that the interfaces Fe-vacuum and the F e - A u are magnetically similar. The parameters that define the geometry of the bcc Fe-thin film are Z0 = 0 and Zs = 4. Furthermore, the two parameters, J0s and J , are estimated from the experimental values of the Curie temperatures of the monolayer and the bulk. We obtain for ¢ -- J o s / J the value 0.558. The results for the thickness dependence of the Curie temperature are shown in the Fig. 2. It is remarkable that with such a small number of parameters one can reproduce reasonably well the experimental results. We show also the results for c = 1, which correspond to a situation in which the coupling constants are the same in all the system. In the case of Gd, it has been observed by means of SPLEED and electron capture spectroscopy (ECS) that the surface Curie temperature of macroscopic samples is higher that the one in the bulk [1,2] by about 22 K. This means that the coupling constant at the surface is larger than Js c . More recently, electron--spin resonance experiments in ultrathin films, show that the Curie temperature is a monotonic function of the thickness (see experimental points in Fig. 3). Even for sampies as thick as 80 ~ ( ~ 27 layers) the Tc is 5K smaller than Tc ~. To model this system, we take a fcc thin film with (111) surfaces. The coordination numbers are Z0 = 6, Z~ = 3 and the coupling constants are estimated from the bulk Curie temperature and the surface Curie temperature of the macroscopic sample. In order to reduce the number of parameters we assume the following three different relations: i) J00 = a J and J01 = J ,

ii) Joo = aaJ and Jo~ = ctJ, iii) Joo = Jo~ = " J . In all the cases the parameter a is choosen to fit the Tc s of a macroscopic Gd sample [2]. Those values are 1.845, 1.282 and 1.485, respectively. The results of our calculation for the three different relations are shown in Fig. 3. The solid line corresponds to case i), in which one assumes that only the Jo0 differs from the rest. The dashed line shows the results in which one assumes a smoother variation of the J ' s (case ii)). Finally, assuming that the coupling constants Joo and Jol are affected to the same extent (case iii)), is shown by the dotted line.

157

IN ULTRATHIN FILMS 1.4-

1.2m 1.0 "

/: 0.8

"

0.6

~lokneu

Fig. 3 Dependence of the (To ~To 8 ) on sample thickness in Gd thin films deposited on W. The continuos, the dashed and the dotted line, correspond to the three models discussed in the text. The circles correspond to the experimental results [10]. For the three cases considered, Tc for the monolayer is smaller than T c s , and model i) agrees better with the experimental value. In the two--layer film, as a consequence of the larger coordination number, Tc > T c s . A common characteristic of the three models, is that Tc reaches a maximum, higher than TcB, then it decreases as a function of thickness and goes to the surface Curie temperature of the semi-infinite system. This behavior is not in accordance with the experimental findings [10]. Other fact that our model can not reproduce is that Tc < Tc B in the 80 ~ film. To get that kind of behavior it may be necessary to assume surface perturbations over very large distances. This problem has been studied within a more accurate method [12] and by assuming anisotropic interactions. The improvements on the model do not remove the disagreement between theory and experiment. In conclusion, by using a very simple [sing film with two free surfaces, we can obtain analytic expressions for the thickness dependence of the Curie temeperature. This allows us to estimate the magnetic interactions in these low dimensional systems. We can reproduce fairly well the recent experimental results of Fe on Au. In contrast, our model fails to explain the experimental findings of Gd on W, where interactions of larger range may be important.

Acknowle~dgeme,ts. This work was partially supported by Direcci6n General de Investigaci6n Cientifica y Superacidn Acad~mica de la Secretarfa de Educaci6n Pdblica through Grant C89-08-0081 and by Programa Regional de Desarroilo Cientffico y Tecnol6gico de la Organizaci6n de Estados Americanos.

REFERENCES [1] C. Rau and S. Eichner, in Nuclear Methods in Materials Research, edited by K. Bethege, H. Banman, H. Hex, and F. Rauch (Vieweg, Braunschweig, 1980), p.354.

[2.] D. Weller, S.F. Alvarado, W. Gudat, K. Schroder, and M. Capagna, Phys. Rev. Left. 64, 1555 (1985). [3] C. Rau and S. Eichner, Phys. Rev. Lett. 47, 939

(19Sl).

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PHASE TRANSITIONS IN ULTRATHIN FILMS

[4] C. Rau, C. Jin, and M. Robert, J. Appl. Phys. 63, 3667 (1988). [5] D.L. Mills, Phys. Rev. B 3, 3887 (1971). [6] K. Binder and D.P. Landau Phys. Ray. Lett. 52, 318 (1984). [7] E.F. Sarmento and C. Tsallis, J. Phys. C 18, 2777 (1985). [8] F. Aguilera-Granja and J. L. Mor(m-L6pez, Phys. Rev. B 31, 7146 (1985). [9] J.M. Sanchez and J.L. Mor~n-L6pez, in Magnetic Properties of Low Dimensional Systems, edited by L.M. Falicov and J.L. Morgn-L6pez, Springer Proceedings in Physics, Vol. 14 (Springer Verlag, Berlin, 1986).

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[I0] M. Farle and K. Baberschke, Phys. Rev. Lett. 58, 511 (1987). [11] W. Dfirr, M. Taborelli, O. Paul, R. Germar, W. Gudat, D. Pescia, and M. Landot, Phys. Rev. Lett. 62,206 (1989). [12] J.M. Sanchez and J.L. Morgn-L6pez, in Magnetic Properties of Low Dimensional Systems 1I, edited by L.M. Falicov, F. Mej|a-Lira and J.L. MorgnL6pez, (Springer Verlag, Berlin, in press).