Magnetic properties of the mixed-spin Ising ferromagnet with a ferrimagnetic surface

Magnetic properties of the mixed-spin Ising ferromagnet with a ferrimagnetic surface

N. Benayad and A. Dakhama

Laboratoire de Physique The´orique, Universite´ Hassan II, Faculte´ des Sciences Aı¨n Chock, B.P. 5366 Maarif, Casablanca, Morocco

~Received 7 November 1996!

The magnetic properties of the three-dimensional mixed spin-¹₂ and spin-1 Ising model with competing surface and bulk exchange interactions are investigated within the framework of the mean-field theory and the finite cluster approximation. In addition to the phase diagrams, which show qualitatively interesting features~a variety of phase transitions and multicritical points!, we find a number of characteristic behaviors of the surface magnetization both in the absence or in the presence of crystal-field interactions. A compensation point in the surface magnetization is found when the surface crystal field and the ratio of the surface and bulk exchange interactions belong to certain ranges. This phenomenon may have important applications in technology such as thermomagnetic writing and erasing at the compensation point.@S0163-1829~97!15017-2#

I. INTRODUCTION

The nature of magnetic ordering near surfaces and inter- faces of magnetic materials is a topic of high current interest in both the theoretical and experimental fields. The magnetic ordering and critical behavior at surfaces are expected to be different from those of the bulk materials due to the different coordination number and symmetry of the atoms at the sur- face. Indeed, both theoretical predictions^1,2and experimental observations suggest surface magnetic behavior different from the bulk.^3–5 These effects have been modeled, by the spin-¹₂ semi-infinite simple cubic Ising ferromagnet with in- trasurface exchange interaction J_S different from the bulk value J_B. It has been extensively studied using a variety of approximations and mathematical techniques, such as mean- field theory,^6,7 the cluster variation method,⁸ effective-field theory,⁹ the finite-cluster approximation,^10,11 renormal- ization-group methods,^11–15 Monte Carlo techniques,^16,17 and series expansions.¹⁸ It exhibits four different types of phase transitions associated with the surface. If the ratio R 5J_B/J_S is greater than a critical value R_C, the surface or- ders at the bulk transition temperature. This is the ordinary phase transition. If R is less than R_C, the system may order on the surface at a temperature higher than the bulk, fol- lowed by the order of the bulk at the bulk transition tempera- ture. These two successive transitions are called the surface and extraordinary transitions, respectively. If R5R_C, the system orders at the bulk transition temperature, but in this case the critical exponents differ from those of the ordinary transition. This is the special phase transition. Using the finite-cluster approximation¹⁹ and real-space renormal- ization-group methods,²⁰similar transitions have been found for the three-dimensional semi-infinite spin-1 ferromagnetic Ising model with crystal-field interaction. As a function of the ratio R of the bulk and the surface interactions and the ratio D of the bulk and the surface crystal fields, it has been shown that the system exhibits a variety of phase transitions and multicritical points.

Ordering in magnetic materials presenting a free surface can in principle exhibit interesting conflicts, when the sur- face favors a type of ordering which competes with that fa-

vored by the bulk. This can be the case of magnetic systems where the surface coupling constant J_S differs in sign from the bulk coupling constant J_B. This seems to be precisely the case of Cr, which is antiferromagnet with a Ne´el temperature of 312 K. Its ~1,0,0! free surface has been investigated,²¹ using angle-resolved photoelectron spectroscopy, and it pre- sents a ferromagnetic ordering up to 780650 K. Magnetic competition between surface and bulk in the simple-cubic spin-¹₂ Ising ferromagnet with ferromagnetic exchange inter- action (J_B.0) in the bulk and antiferromagnetic exchange interaction (J_S,0) between surface spins has already re- ceived theoretical attention through the mean-field approximation,¹⁸ renormalization group method,²² and Monte Carlo ~MC! simulations.^23,24 Below the bulk critical temperature, the bulk is ferromagnetically ordered for all J_S. For surface exchange greater than some temperature- dependent value of J_S, the surface is also in a ferromagneti- cally ordered state, but for more negative values of J_S the surface is antiferromagnetic instead. As the temperature in- creases, the bulk disorders, but for strongly negative J_S the surface remains ordered up to some higher temperature. The phase boundaries for the bulk and surface transitions cross at a tetracritical point. By studying the magnetization profile of the system,²³it has been found that the magnetization under the surface layer was strongly affected by the antiferromag- netic ordering in the surface and that the surface layer be- haves roughly like a two-dimensional Ising antiferromagnet in a ~temperature-dependent! field. It is worthy to note the well-known (J_B,J_S)→(2J_B,2J_S) symmetry, which

~among others!determines the isomorphism between ferro- magnetic and antiferromagnetic Ising models in square and simple cubic lattices.

In recent years, some interest has developed in the study of the semi-infinite simple cubic consisting of two sublattices of spin-¹₂and spin-1 Ising alloy with a free~1,0,0!surface. It has been studied using effective-field theory with correlations^25,26 and the real-space renormalization-group method.²⁷ Attention has been devoted to the case where the two exchange interactions, J_S on the surface and J_B in the bulk, are both positive ~or negative!. In this case, the only possible states of the surface and bulk are paramagnetic and

55

0163-1829/97/55~18!/12276~14!/$10.00 12 276 © 1997 The American Physical Society

ferromagnetic ~or ferrimagnetic! ordering. However, no at- tention has been devoted to the case of the Ising ferromagnet (J_B.0) with a ferrimagnetic surface exchange interaction (J_S,0) and a surface anisotropy perpendicular to the sur- face. Such systems may describe a variety of magnetic ma- terials, such as thin films of rare-earth–transition-metal al- loys which are important from the technological point of view, because of the large magnetic anisotropy perpendicular to the film plane and their magneto-optic applications.

In contrast to ferromagnets and antiferromagnets, there is in ferrimagnets an important possibility of the existence, un- der certain conditions, of a compensation temperature T_comp, at which the resultant magnetization vanishes below the transition temperature T_c. The appearance of a compen- sation point is due to the fact that the magnetic moments of the sublattices compensate each other completely at T 5T_comp. The existence of such a phenomenon may be very useful in many technological applications, such as thermo- magnetic writing and erasing at the compensation point, be- cause of the high coercivity around it. Because of potential device applications, many ferrimagnets have been exten- sively investigated^28–32and some of them possess a compen- sation point temperature which may vary in the vicinity of room temperature by the proper choice of composition.

The purpose of this work is to investigate the thermal

behaviors of the surface magnetizations in the semi-infinite simple cubic consisting of two interpenetrating sublattices, of a spin-¹₂ and spin-1 alloy, with ferrimagnetic surface cou- pling and ferromagnetic bulk coupling. We also investigate the influence of bulk and surface single-ion crystal-field in- teractions on the surface magnetizations. For this purpose, we use the mean-field theory and the finite-cluster approxi- mation. We note that the latter and effective-field theory with correlations give the same results³³ by two different math- ematical techniques.

Our presentation is as follows. In Sec. II we give a de- scription of the model and the theoretical framework. In Sec.

III we investigate the phase diagrams. In Sec. IV we examine the thermal behaviors of the surface magnetizations, and we present our concluding remarks in Sec. V.

II. THEORETICAL FRAMEWORK AND FOUR-LAYER APPROXIMATION

Let us consider a semi-infinite simple cubic consisting of two sublattices of spin-¹₂ and spin-1 Ising alloy. The ferro- magnetic exchange interaction between all nearest neighbors is J_B, and except for spins on the~1,0,0!surface, they inter- act with one another with a ferrimagnetic exchange param- eter 2J_S (J_S.0). The Hamiltonian of our system takes the form

H5J_S^

(

(

(

(

~1! where S takes the values 61 and 0, and s ^{can be} 1¹2 or 2¹2. The first and second summations are carried out over nearest-neighbor sites and single sites located on the free surface, respectively. The third and fourth summations run over all pairs of remaining nearest-neighbor sites and single sites, respectively. D_Sand D_B, denote the surface and bulk single-ion crystal-field interactions, respectively.

In the framework of the finite-cluster approximation

~FCA!,^34–38we consider a particular spins0 (S₀) and denote

by^s0&C (^^S0&C) the mean value ofs0 (S₀), while all other

spins si and S_i (iÞ0) are kept fixed. Applying this for a spin s0S (S_0S) located on the surface and an other one s0n (S_0n) located in the nth layer (n>1) of the bulk, we obtain

^s0S&C5¹2tanh

F

⁽

G

^^S0S&^C5

2 sinh

F

⁽

G

2 cosh

F

⁽

G

~3!

^s0n&^C512tanh

F

S

⁽

D G

and FIG. 1. ~a!Nearest neighbors of spins S_0Sands0S, located on the surface.~b!Nearest neighbors of spins S0nands0n, located on the nth layer.~c!Part of a two-dimensional cross section of a semi- infinite mixed Ising lattice with a~1,0,0!free surface. d andj denote S- ands-sublattice sites, respectively.

^^S0n&C5

2 sinh

F

S

⁽

D G

2 cosh

F

S

⁽

D G

where K_B5b^JB, K_S5b^JS, andb5(k_BT)²¹. The$s^iS,s51% ^and$^SiS,S51%⁽ⁱ⁵1,2,3,4) are the nearest neighbors of S_0Sand s0S, respectively@Fig. 1~a!#. The $sin,s5(n11),s6(n21)% ^and$^Sin, S_5(n11), S_6(n21)% are the nearest neighbors of S_0n and s0n, respectively@Fig. 1~b!#. Therefore the surface magnetizations per site,mS5^siS&^{, m}S5^^SiS&, and the magnetizations mn and m_n of the nth layer@Fig. 1~c!#satisfy the exact relations

mS5

K

F

⁽

G L

m_S5

K

@

^@

#

^#

L

mn5

K

F

S

⁽

D G L

m_n5

K

@

~ ^@

^~

!

#

^{! #}

L

where^• • •&denotes the full thermodynamic average. It is a formidable task to average the right-hand side of Eqs. ~6!–

~9! over all configurations. The FCA has been designed to treat all spin self-correlations exactly while still neglecting correlations between different spins.³⁴ For our mixed spin-

1

2and spin-1 Ising system described by Eq.~1!, the appropri- ate distributions are

P_FCA~$s^j%^!5

)

~10!

p_FCA~$^Sj%^!5

)

1¹2~2m1x!d~S_j11!#, ~11!

where x5^^S2&^.

When calculating the average on the right-hand side of Eqs.~6!–~9!, it is, however, easier to observe that any func- tions f (s) and g(S) ofs or S can be written as the linear superpositions

f~s!5f₁1f₂s^, ~12!

g~S!5g₁1g₂S1g₃S², ~13!

with appropriate coefficients f_1,2and g_1,2,3. Applying this to all spinssiand S_j in Eqs.~2!–~5!, their right-hand sides are decomposed as

^s0S&C5_q

(

(

^^S0S&C5_q

(

^s0n&^C5_q

(

(

1I_p,q~K_B!~S_5~n11!S_6~n21!!1J_p,q~K_B!~S₅²_~n11!S_6~n21!1S_5~n11!S₆²_~n21!!1K_p,q~K_B!~S₅²_~n11!S₆²_~n21!!%^{, ~16!}

^^S0n&C5q

(

4

$sⁿ%^q$^{Lq~KB!1Mq~KB!~}s5~n11!1s6~n21!!1N_q~K_B!~s5~n11!s6~n21!!%^{, ~17!}

where $^SS

2,S_S%p,q denotes the superposition of all the terms containing p different factors of S²_{j S} and q different factors of S_j8S, with jÞj8. These factors are selected from the set$^{S1S,S2S,S3S,S4S,S}1S

2 ,S_2S² ,S_3S² ,S_4S² %. For example, if p52 and q 51, then

$^S2,S%^2,15S1S~S2S

2 S_3S² 1S_2S² S_4S² 1S_3S² S_4S² !1S_2S~S_1S² S_3S² 1S_1S² S_4S² 1S_3S² S_4S² !1S_3S~S_1S² S_2S² 1S_1S² S_4S² 1S_2S² S_4S² !1S_4S~S_1S² S_2S²

1S_1S² S_3S² 1S_2S² S_3S² !. ~18!

$s^S%^q denotes the superposition of all the terms containing q different factors of siS, selected from the set

$s^1S,s2S,s3S,s4S%^. $^Sn

2,S_n%^p,q is the superposition of all the terms containing p different factors of S²_{j n} and q different factors of S_j8n, selected from the set $^S1n,S_2n,S_3n,S_4n,S_1n² ,S_2n² ,S_3n² ,S_4n² %^{, with j}Þj8^. $sn%q is the superposition of all the terms containing q different factors of sin, selected from $s1n,s2n,s3n,s4n%. The nonzero coefficients A_p,q, B_p,q, . . . , M_q and N_q are listed in the Appendix.

Using the probability distributions~10!and~11!to treat Eqs.~6!–~9!or averaging Eqs.~14!–~17!over all spin configura- tions and neglecting correlations between different spins, we obtain

mS5_q

(

(

m_S5_q

(

mn5_q

(

(

3~x_n11m_n211m_n11x_n21!1K_p,q~K_B!x_n11x_n21%^, ~21!

m_n5_q

(

withm05mS, m₀5m_S, x₀5x_S, andC_b^a5a!/b!(a2b)! is a combinatorial factor. The parameters x_S and x_n are defined by x_S5^^SiS

2&^{and x}n5^^Sin

2&, respectively. They have to be evaluated in the spirit of the FCA. Similar to Eqs.~7!and~9!, we have

the exact relations

x_S5

K

4 siS1K_Bs51#1exp~b^DS!

L

x_n5

K

L

Expanding the right-hand side of Eqs. ~23!and~24!by using Eq. ~12!and neglecting correlations between different spins or using the probability distribution ~10!, we obtain

x_S5_q

(

x_n5_q

(

The coefficients O_q, P_q, . . . , S_q are given in the Appendix. We note that the bulk magnetizationsmB and m_B, and the bulk parameter x_B are determined by settingmn215mn5mn115mB, m_n215m_n5m_n115m_B, and x_n215x_n5x_n115x_B in Eqs.

~21!,~22!, and~26!. ThusmB, m_B, and x_B are solutions of the equations mB5q

(

4 p

(

C_q⁴C_p^42qx_B^pm_B^q$^{Fp,q~KB!12Gp,q~KB!mB12Hp,q~KB!xB1Ip,q~KB!m}B 2

12J_p,q~K_B!x_Bm_B1K_p,q~K_B!x_B²%^, ~27! m_B5q

(

4

C_q⁴mB

q$^Lq~K_B!12 M_q~K_B!mB1N_q~K_B!mB

2%^{, ~28!}

x_B5q

(

C_q⁴mB

q$^{Qq~KB!12Rq~KB!}mB1S_q~K_B!mB

2%^{. ~29!}

The total magnetizations of the surface and the nth layer (n>1) are then defined by

M_S5N mS1m_S

2 , M_n5N mn1m_n

2 , ~30!

where N is the total number of magnetic atoms in each layer.

In order to investigate the thermal behaviors of the sur- face and bulk magnetizations, we have to solve the coupled equations ~19!–~22!, ~25!, and ~26! obtained in the frame- work of the FCA. However, we are unable to solve them analytically. Even if we use a numerical method, they must be terminated at a certain layer. Note that as n goes to infin- ity, the magnetizations mn and m_n and the parameter x_n should approach the bulk valuesmB, m_B, and x_B. For this purpose, let us assume that the magnetizations remain unal- tered after the third layer, i.e.,

m35m45•••5mB, m₃5m₄5•••5m_B,

x₃5x₄5•••5x_B, ~31! which may be called the four-layer approximation. Analyz- ing Eqs.~19!–~22!and~25!–~29!, the surface and/or the bulk may exhibit an order-disorder transition which can be of first order when D_B/J_B and D_S/J_S belong to narrow ranges. We limit this study to the case of second-order transitions, since we are more interested in the region of the phase diagram where the bulk is already ordered.

Let us end this section by writing down the mean-field coupled equations where all spin correlations are neglected including self-correlations. Within the four-layer approxima- tion, we obtain

mS5¹2tanh@¹2~24K_Sm_S1K_Bm₁!#, ~32!

m_S5 2 sinh@24K_SmS1K_Bm1#

2 cosh@24K_SmS1K_Bm1#1exp~b^DS!, ~33!

mn5¹2tanh

F

G

m_n5 2 sinh@K_B~4mn1m_~n11!1m_~n21!!#

2 cosh@K_B~4mn1m~n11!1m~n21!!#1exp~b^DB!,

~35! for n51,2; with m05mS, m₀5m_S, and mn5mB, m_n 5m_B for n.2. The bulk magnetizations mB and m_B are determined by setting in Eqs. ~34! and ~35! mn215mn

5mn115mB, and m_n215m_n5m_n115m_B. Then mB and m_B are given by

mB5¹2tanh@3K_Bm_B#, ~36!

m_B5 2 sinh@6K_BmB#

2 cosh@6K_BmB#1exp~b^DB!. ~37!

III. ORDER-DISORDER TRANSITIONS AND OTHER SURFACE TRANSITIONS

We are first concerned with the evaluations of the order- disorder transition temperatures for the bulk and surface or- derings based on the four-layer approximation. In the frame- work of the FCA, the bulk reduced critical temperature (K_B^C)²¹ is determined from the coupled equations~27!–~29! as

15

H

⁽

14_p

(

1K_p,1Q₀²!

J

Thus K_B^Cis a function of D_B. For D_B50, the value of K_B^Cis 0.4736, to be compared with the mean-field result 0.4082.

One should note that for D_B52` the spin configurations in the bulk are completely dominated by S561 on each site of the S sublattice. Therefore the bulk corresponds to the three- dimensional spin-¹₂ Ising model. Comparing with Monte Carlo critical value K_B^C50.4433,³⁹ the FCA result 0.3942 improves the mean-field approximation~MFA!value 0.3333.

In order to obtain the surface order-disorder critical tem- perature, we have to linearize Eqs.~19!–~22!and~25!–~26!. Within the four-layer approximation, we obtain

mS5p

(

C_p⁴B_p,0O₀^pm₁

14p

(

C_p³O₀^p~A_p,11C_p,1Q₀!m_S, ~39!

m_S54D₁mS1E₀m1, ~40!

m15_p

(

1~G_p,01J_p,0O₀!m₂%

14_p

(

1K_p,1O₀Q₀!%^{m1, ~41!} m₁5M₀~mS1m2!14L₁m1, ~42!

m25p

(

C_p⁴Q₀^p~G_p,01J_p,0Q₀!~m₁1m_B!

14_p

(

m₂5M₀~m11mB!14L₁m2. ~44!

Therefore the critical ordering frontier~for K_B<K_B^C!is ana- lytically obtained through a determinantal equation. In Figs.

2, 3, and 4, which correspond, respectively, to (D_B/J_B,D_S/J_S)5(0,0), ~0.5,21!, and ~0.5,11!, we repre- sent the critical line of ferrimagnetic-paramagnetic surface transitions (S). As is seen from these figures, if the ratio R 5J_B/J_S is less than a critical value R_C, the surface may ferrimagnetically order at a temperature (K_S^C)²¹ higher than the bulk. K_S^Cand R_Cdepend on the values of the surface and bulk anisotropies D_S and D_B. In particular, for D_S5D_B 50, the FCA critical value of R_Cis 0.6125, to be compared with the mean-field result 0.6781, which is obtained from Eqs. ~32!–~37!.

The steps described before are not sufficient to obtain the remaining part of the phase diagram. In fact, any two nearest neighbors on the surface interact via a ferrimagnetic cou- pling. At the ground state of the Hamiltonian~1!, the system

~surface!makes a first-order transition from a ferrimagneti- cally ordered state for R,4 to a ferromagnetically ordered state for R.4. The critical value of 4 does not depend on the anisotropies. In order to obtain the rest of phases and transi- tions in the framework of the FCA, we must solve numeri- cally the coupled equations~19!–~22!,~25!, and~26!within the four-layer approximation scheme ~31!. The analysis of the above equations leads to very interesting surface phe- nomena. The surface behaviors and their dependences on the anisotropies are shown in Figs. 2, 3, and 4, corresponding to the values of (D_B/J_B,D_S/J_S)5(0,0), ~0.5, 21!, and ~0.5, 11!, respectively. As is shown in Fig. 2, five physically different phases are identified. These phases are indicated on the phase diagrams by the following symbols: SP, BP, sur- face and bulk paramagnetic; SFi, BP, surface ferrimagnetic and bulk paramagnetic; SF, BF, surface and bulk ferromag- netic; SFi¹², BF, surface ferrimagnetic and bulk ferromag- netic with mS.0 and m_S,0; and SFi²¹, BF, surface ferri- magnetic and bulk ferromagnetic with mS,0 and m_S.0.

For the system under study, we extend the accepted termi- nology used in semi-infinite Ising models with no competing exchange interactions.19,27,40,41

As seen from Fig. 2, the above phases are separated by different transition lines. Among them, we find all critical lines obtained in the semi-infinite simple cubic ferromagnetic Ising model.^11,17,18 They correspond to the surface (S), the extraordinary (E), and the ordinary (O) transitions. When the bulk is ferromagnetically ordered, the surface exhibits, at finite temperature, a second-order transition (L) from the ferromagnetic phase ~SF,BF! to the ferrimagnetic phase

~SFi,BF!. One of the most interesting features of the surface is the existence of two possible ferrimagnetic orderings

~SFi¹² and SFi²¹!, which are separated by a first-order transition line (Q). This surface behavior does not occur in

three-dimensional semi-infinite monoatomic Ising models with competing surface and bulk exchange interactions.^18,22–24 One can note that the surface layer be- haves roughly like a ferrimagnetic mixed spin-¹₂ and spin-1 Ising model in a nonuniform ~temperature-dependent!field.

At T50, this field is equal to J_B and J_B/2 acting ons ^and S sublattices, respectively; then, the configurations SFi²¹ and SFi¹² have the same energy. However, at a finite tem- perature ~K_B,` and R,4!and due to the entropy and the bulk boundary conditions, the stability of each one of them depends on the ratio R and K_B as is shown in Fig. 2. In this FIG. 2. ~a!The phase diagram of the FCA in the K_S-K_B plane for the three-dimensional semi-infinite mixed spin-¹₂ and spin-1 Ising model with competing surface and bulk exchange interactions, (D_B,D_S)5(0,0). ~b! The phase diagram of the FCA in the K_B²¹-R²¹plane for the three-dimensional semi-infinite mixed spin-

1

2and spin-1 Ising model with competing surface and bulk exchange interactions, (D_B,D_S)5(0,0).

work, we choose positive boundary conditions ~i.e., mB.0 and m_B.0!. As can be seen from Eqs.~19!–~22!, changing mBto2mB and m_Bto2m_B is equivalent to changingmS to 2mS and m_Sto2m_S. If we choose negative boundary con- ditions, the phase diagrams are the same as those in Figs. 2, 3, and 4, except that the locations of the two phases SFi¹² and SFi²¹ are interchanged.

In the absence of the bulk and surface anisotropies ~Fig.

2!and within an appropriate range of R (R,R_c50.6125), the system exhibits two successive transitions: The surface orders (SP/SFi²¹) at a temperature higher than the bulk,

followed by the bulk ordering~BP/BF!at the bulk transition temperature (K_B^C)²¹. For 4.R.R_C, the surface and bulk order ferromagnetically at (K_B^C)²¹, and at a lower tempera- ture ~which depends on the value of R!, the surface under- goes a second-order transition from a ferromagnetically or- dered state ~SF! to a ferrimagnetically ordered state (SFi¹²). Furthermore, for certain values of R the surface may undergo a transition from a SFi¹² to SFi²¹ phase, when the temperature is decreased. From the figure, we also point out that for R.4, the bulk promotes its order to the surface in such a way that they are both paramagnetic or ferromagnetic for K_B,K_B^C and K_B.K_B^C, respectively.

FIG. 3. ~a!The phase diagram of the FCA in the K_S-K_Bplane for the three-dimensional semi-infinite mixed spin-¹₂ and spin-1 Ising model with competing surface and bulk exchange interactions, (D_B/J_B,D_S/J_S)5(0.5,21).~b!The phase diagram of the FCA in the K_B²¹-R²¹ plane for the three-dimensional semi-infinite mixed spin-¹₂ and spin-1 Ising model with competing surface and bulk exchange interactions, (D_B/J_B,D_S/J_S)5(0.5,21).

FIG. 4. ~a!The phase diagram of the FCA in the K_S-K_B plane for the three-dimensional semi-infinite mixed spin-¹₂ and spin-1 Ising model with competing surface and bulk exchange interactions, (D_B/J_B,D_S/J_S)5(0.5,11).~b!The phase diagram of the FCA in the K_B²¹-R²¹ plane for the three-dimensional semi-infinite mixed spin-¹₂ and spin-1 Ising model with competing surface and bulk exchange interactions, (D_B/J_B,D_S/J_S)5(0.5,11).