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Spin waves at a disordered interface between two ferromagnetic materials
PhysicsLettersA 170 (1992) 159—164 North-Holland
PHYSICS LETTERS A
Spin waves at a disordered interface between two ferromagnetic materials Qiang Hong Department ofPhysics, 239Fronczak Hall, State University ofNew York at Buffalo, Amherst, NY 14260, USA Received 29 June 1992; revised manuscript received 11 August 1992; accepted for publication 19 August 1992 Communicated by J. Flouquet
The spectrum ofmagnetic excitations at a disordered interface between two ferromagnetic materials was studied by using the Bethe lattice model and the Green function method. The effect of the interface composition and chemical short-range order on the local density of states (LDOS) is examined.
1. Introduction The study of magnetic excitations at the interfaces between two crystalline magnetic materials has received much attention in the past few years [1—3].Yaniv [1] has studied the exchange-type spin waves at the interface of a biferroinagnetic system with simple cubic structure and found that up to two branches of bound interface states may exist, depending on the coupling strength between spins in the interface region. In a later work, Xu et al. [2] investigated the effects of crystal structures on the interface spin waves. The conditions for the occurrence ofinterface states were analyzed for a number of different crystal structures ofthe two materials forming the interface system. In addition, Wang and Lin [3] studied the interface excitation modes by using the Bethe lattice model. Interface states were also found. In comparison with pure bulk materials, the crystalline interface system loses the translation symmetry in the direction perpendicular to the interface, while still retaining the symmetry on the planes parallel to the interface. Having considered this feature, most ofthe work cited above employed the mixed Bloch and Wannier representation in a sense that the system was treated in two-dimensional k space for the planes parallel to the interface whilst in real space in the direction perpendicular to the interface. Obviously, these works cannot allow for any deviations from the perfect crystallinity of the interface, which we know exists in almost all real interfaces due to the interdiffusion, dangling bonds, mismatching, etc. The resulting disorders in the interface region will destroy the periodicity of the system in the planes parallel to the interfaceand make the use oftwodimensional momentum representation impossible. One simple model to include the imperfectness of an interface is to consider the alloying effect of the two materials on the interface, i.e. the interfacepossesses two kinds of atoms instead ofone as in a perfect interface. In this paper, we will consider the effect of alloying disorder on the spin wave properties of a biferromagnetic interface system in the context of the Bethe lattice approximation [4,5]. To our knowledge, there is no investigation on such problems in the current literature. The phonon problem of a similar structure has been considered by Kechrakos [6] in the CPA approximation.
0375.960l/92/$ 05.00 @ 1992 Elsevier Science Publishers B.V. All rights reserved.
159
Volume 170, number 2
PHYSICS LEflTERS A
26 October 1992
2. Models The system considered below is composed of two semi-infinite ferromagnetic materials A and B, matching at the interface plane I. The two materials may have different structures and spins. For definition, we assume that the interface plane belongs to A material and some of its atoms were replaced by B atoms. The existence of such alloying disorder destroys the translational symmetry on the interface plane, making the problem hard to solve and approximation methods must be consulted. Considering this, we simulate the system by a mixed Bethe lattice model with the A atoms occupying the upper half space and the B atoms the lower half. We can subdivide the system into planes parallel to the interface and denote them with integers n = 0, ±1, ±2 where positive signs refer to A atom planes and minus signs refer to B atom planes. The interface is the n = 0 plane. We now consider the situation in which the short range interactions between the atomic spins is represented by the Heisenberg Hamiltonian (1)
H=— ~
where (if> implies that only the nearest neighbor interactions between atoms are considered. S~ is the magnetic moment at site i whose value can take SA or SB depending on whether this site is occupied by an A or B atom, respectively. J,, is the exchange integral between the ith and fth atoms. Generally, we may have different values of J~in the interfaceregion and in the two bulk materials. However, to minimize the number of our parameters, we assume that they are only dependent on the bond type between the atoms. Thus, we have only three J values: JAA, JBB and JAB (=JBA), corresponding to the bonds A—A, B—B and A—B (or B—A), respectively. As was mentioned in the first paragraph, we take the interface plane to be composed of both A and B atoms and denote the concentration of A atoms on this plane by x (0 ~ x’~1). Besides, we introduce a chemical short-range parameters a to describe the short-range correlations between atoms. Following Cowley [7], the probability P~, of finding an atom j as nearest neighbor of atom i can be expressed in terms of x and a, as: PAA=x+(l—x)a,
PBA=x(l—a),
PAB=(l—x)(l—a),
PBB=l—X+Xa.
(2)
It is easy to see that a= 0 corresponds to a random alloy and a negative value of a represents an attraction between unlike atoms. Following the usual procedures [8], we define the following double-time Green function, G,~(t—t’)=<>
(3)
.
In the one-spin-wave approximation [9], the equations of motion for the Green functions can be written as (w—E~)G,~=~+ ~ V17,G~, where E,= ~ and Vu, = i can be easily obtained:
—
(4) 2J.
From this equation, the diagonal Green function for a bulk material
Gb0= w—z1V1~+z~J~~y1(w)’
(5)
where i=A or B, and z1 is the coordination number of the material. V~=2J11S1. y~(w)is the transfer matrix for the system, which is given by E
160
—
.,.J
w ± (E1 cv) 2_ 4 (z1 —1) V~, 2(z1—l)V,, —
(6)
Volume 170, number 2
PHYSICS LETI’ERS A
26 October 1992
where E. = 2z1Vu. The local density of states of spin waves can be evaluated through the imaginary part of the Green function: D(w)=— -~ImGb0(w).
(7)
In order to study the spin waves at the interface, we must write down the equations of motion of the Green functions for the atoms in the interface region, i.e. [w—E~(A)]G~(A)= ZAI
1
...ZAO[PAA
V~Goi,oc(Ai_A)+P~ V~Go1,oo(B4-A)]
V~~~G10,00(A~A) ..ZBI V~Gio,m(B~—A),
(8)
[w_Ej(A4—A)]Gojoc(Al-A) =
—
V~Goji,oo(A-I)—(ZAO
ZAI
—
l)[P~ V~G0~+1,00(A+-A)+P~ V~G0~+ i,oo(BiA)1
V~Gljoo(A-A)zBl V~G....ijoc(Bi-A),
(9)
[cv—E3(A+-B)]G0~,00(Ai-B) =
—
VABGOJ_!,00(B4-I)
—
(ZAO
—
l)[P~ V~Goj÷i,oc(A4-A) +P~VABGoJ+I,oo(B~-A)]
ZAIV~GlJ,oc(A4A)ZBIV~G_IJ,~(B4A), [w—E_1(B)]G1J,~(B4--A)=
—
(10)
VaxGoj,m(Ai-4)_(Za
—
l)VBBG_2J.oo(B4—B),
(11)
and the equations for A4-B and n~-~n, where ZAO is the number of intraplane nearest neighbors of a site on the interface, z~1is the number of nearest neighbors between planes, z1= z0+ 2z11 is the coordination number of the bulk material i. The meaning of the subscripts of the Green functions are the same as those in ref. [10]. In writing the above equations, we also defined the following expressions, —
VAB=VBA=2JAB~/~~, Eoc(A)=zAo(P~V~+PABYAB)+zAIV~+zBlYAB,
E3(A4—A)= VAA +(zAo
l)(P~.V~+PAB YAB)+zAt ~
—
+ZBI
YAB, 1TAB,
=
VAB + (zAo —1 )(P~V~+PAB VAB) +ZAI V~+ZBI
E_j(B)=YBA+(zB—l)VBB,
YAB=2JABSB,
and the expressions for k-B and nl—. functions on the interface plane, -
Goj,oo(A4—A)
—
Goj,oc(B*-A) Goji,oo(Ai-I)
2(co)= ,
n. These equations can be solved by defining the following transfer
Go~,oo(A4—B)
a
G01_j,00(Ai-I)
—
(12)
a~(w)=
, G0~1,00(B~-I) Goj,oc(Bs—B)
(13)
,
and the transfer functions between the interface plane and its nearest planes, G_ij,oo(B4-A) G1~,00(A*-B) fl~(w)_ , fl2(cv)= , GoJ,~(A4—I) G0~,00(B~—I) —
(14)
where I=A or B. Substitutingeqs. (13) and (14) into eqs. (8)— (11), we obtain the following coupled equations for the transfer functions, 161
Volume 170, number 2
PHYSICS LETTERS A
26 October 1992
1)VAAYA], 2VAB[C0E1(A)+(ZA al=—VAA[w—Ef(A~--A)+zAIVAAyA+zBIVABflI+(zAO—l)(P~VAAaI+PABVABa3)]’, I3I=—VBA[cv—EI(B)+(zB—l)VBByB]~
fl
a2=—VAB[co—EJ(A1--B)+zAIVAAYA+zBI
,
VABfl1+(zAO—l)(PAAVAAal+PABVABa3)]~,
a3=—VBA[w—EJ(B4---A)+zB1VBBYB+zAIVBJ2+(zAO—l)(PBBVBBa4+PBAVBAcX2)1, a4=—VBB[w—EJ(B+-B)+zBIVBBYB+zAIVBJ 2+(zAO—l)(PBBVBBct4+PBAVBAa2)],
(15)
which can be solved numerically. The diagonal Green functions for the atoms on the interface are given by Goo,00(A)[COEoc(A)+zA,VAAYA+zB1VABPI+zAo(PAAVAAaI+PABVABa3)]’, GOO,OO(B)=[w—EOO(B)+zRlVBByB+zAIVBAfl2+zAO(PBBVBBa4+PBAVBAa2)]~
(16) .
(17)
The LDOS at each atom can be obtained through eq. (7).
3. Results and discussions Using the method outlined in section 2, we calculated the LDOS of spin waves for a set of different values ofthe interface concentration x and short-range order a. The results are contained in figs. 1—3. In all the graphs, the full lines are the LDOS for A atoms and the dotted lines those for B atoms. We choose the parameters ZAO = 4, ~ = 1, ZBO = 4 and ZBI = 2 to simulate the situation that the (100) plane of a simple cubic crystal A matches with the (111) plane of a body-centered cubic lattice B. The parameters J and S are chosen so that there is no overlap between the spectra of the two pure systems (see fig. la). In fig. ib, we display the results for a situation when x approaches 1, i.e. a vanishing small number of B atoms on the interface. The LDOS of B atoms in this case reduces to the single-impurity states of B atoms in an otherwise perfect interface system.
~ 0.8 ~
06
(b)
~0.4
(a)
0.2 0.1 Fig.
0.0 0.0
162
-
10.0
(i)/JS 20.0
30.0
‘
40.0
1. (a) LDOSforthebulkatoms. (b) LDOSfortheatomsat 707, the interface when x-. 1. (c) LDOS for the atoms at the interface SA=O.5andSB=I.5. when x—.0. The parameters are J,~=0.5,JBB= 1, JAB=°.
)
Volume 170, number 2
26 October 1992
PHYSICS LETTERS A
Figure 1 c gives the results when x—. 0. Generally, we can specify the spectra of an interface atom in different energy regions: those within the energy range of that of bulk A atoms can propagate into the A material without damping but decay exponentially when going into the B material, and vice versa. Those states with an energy lying inside the energy gap of the two materials or outside the spectra of both materials can only exist in the interface region and decay when propagating into A or B materials. We call these states interface states. One may see from fig. lb that when the interface has a predominant number ofA atoms, the spectra of both A and B atoms are mainly located at those ofbulk A atoms. Those states within the energy range ofB atoms contribute for a small part to the interface spectra. Opposite behaviors were observed in fig. ic when x-+0. Besides, there is a big contribution of interface states in both cases. In fig. 2, we present the results for the calculation of LDOS for different values of x for a random interface (a=O). Two features were observed: (1) The LDOS for an interface atom is a mixture of those of the two bulk materials. From fig. 2a to fig. 2c, as more and more B atoms reside on the interface, the spectra as a whole shift toward the high energy region. This behavior can be easily understood in considering that more energy should be required to flip a spin at the interface as there are more B atoms around it, because we have chosen a bigger spin value and stronger interactions for B atoms. (2) The energy gap between the spectra of the two materials narrows and vanishes in fig. 2a. This means interface states always exist in the system. We also studied the influence of the chemical short-range order on the LDOS. The results are shown in fig. 3. In the calculation, we have chosen x=0.5. One noticeable property is that when a= —1, i.e. for a completely ordered interface, the interface states are located isolatedly in the gap region of the two bulk materials (see fig. 3a). As I al becomes smaller from fig. 3a to fig. 3c, the energy gap of the interface spectra becomes small and eventually the interface states are connected to the bulk states. 0.4
0.4
~
.4
(c
(b)
03
A
02
1V~
(a)
0.6
(a)
0.4 0.2/(,
0.0
0.0
~ ‘~~:i----10.0 20.0
30.0
40.0
(~)/JS Fig. 2. (a)—(c) LDOS for the interface atoms when x=0.8, 0.5 and 0.2, respectively,
0.0
0.0
~
10.0
20.0
30.0
40.0
CO/JS Fig. 3. (a)—(c) LDOS for the interface atoms when a= —1, —0.6 and —0.2, respectively. The concentration xis 0.5.
163
Volume 170, number 2
PHYSICS LETTERS A
References [1] A. Yaniv, Phys. Rev. B 28 ( 1983) 402. [2]B.X. Xu, M. MostollerandA.K. Rajagopal, Phys. Rev. B 31(1985) 7413. [3] X.F. Wang and T.H. Lin, Chinese Phys. Lett. 4 (1987) 29. [4] R.C. Kittler and L.M. Falicov, J. Phys. C 9 (1976) 4259. [5] R.C. Kittler and L.M. Falicov, Phys. Rev. B 18 (1978) 2506. [6] D. Kechrakos, J. Phys. Cond. Matter 2 (1990) 2637. [7] J.M. Cowley, Phys. Rev. 77 (1950) 669. [8] D.N. Zubarev, Soy. Phys. Usp. 3 (1960) 320. [9]A. Theumann, J. Phys. C 6(1973)2822. [10] J.L. Morán-Lopèz and L.M. Falicov, Phys. Rev. B 20 (1979) 3900.
164
26 October 1992
PHYSICS LETTERS A
Spin waves at a disordered interface between two ferromagnetic materials Qiang Hong Department ofPhysics, 239Fronczak Hall, State University ofNew York at Buffalo, Amherst, NY 14260, USA Received 29 June 1992; revised manuscript received 11 August 1992; accepted for publication 19 August 1992 Communicated by J. Flouquet
The spectrum ofmagnetic excitations at a disordered interface between two ferromagnetic materials was studied by using the Bethe lattice model and the Green function method. The effect of the interface composition and chemical short-range order on the local density of states (LDOS) is examined.
1. Introduction The study of magnetic excitations at the interfaces between two crystalline magnetic materials has received much attention in the past few years [1—3].Yaniv [1] has studied the exchange-type spin waves at the interface of a biferroinagnetic system with simple cubic structure and found that up to two branches of bound interface states may exist, depending on the coupling strength between spins in the interface region. In a later work, Xu et al. [2] investigated the effects of crystal structures on the interface spin waves. The conditions for the occurrence ofinterface states were analyzed for a number of different crystal structures ofthe two materials forming the interface system. In addition, Wang and Lin [3] studied the interface excitation modes by using the Bethe lattice model. Interface states were also found. In comparison with pure bulk materials, the crystalline interface system loses the translation symmetry in the direction perpendicular to the interface, while still retaining the symmetry on the planes parallel to the interface. Having considered this feature, most ofthe work cited above employed the mixed Bloch and Wannier representation in a sense that the system was treated in two-dimensional k space for the planes parallel to the interface whilst in real space in the direction perpendicular to the interface. Obviously, these works cannot allow for any deviations from the perfect crystallinity of the interface, which we know exists in almost all real interfaces due to the interdiffusion, dangling bonds, mismatching, etc. The resulting disorders in the interface region will destroy the periodicity of the system in the planes parallel to the interfaceand make the use oftwodimensional momentum representation impossible. One simple model to include the imperfectness of an interface is to consider the alloying effect of the two materials on the interface, i.e. the interfacepossesses two kinds of atoms instead ofone as in a perfect interface. In this paper, we will consider the effect of alloying disorder on the spin wave properties of a biferromagnetic interface system in the context of the Bethe lattice approximation [4,5]. To our knowledge, there is no investigation on such problems in the current literature. The phonon problem of a similar structure has been considered by Kechrakos [6] in the CPA approximation.
0375.960l/92/$ 05.00 @ 1992 Elsevier Science Publishers B.V. All rights reserved.
159
Volume 170, number 2
PHYSICS LEflTERS A
26 October 1992
2. Models The system considered below is composed of two semi-infinite ferromagnetic materials A and B, matching at the interface plane I. The two materials may have different structures and spins. For definition, we assume that the interface plane belongs to A material and some of its atoms were replaced by B atoms. The existence of such alloying disorder destroys the translational symmetry on the interface plane, making the problem hard to solve and approximation methods must be consulted. Considering this, we simulate the system by a mixed Bethe lattice model with the A atoms occupying the upper half space and the B atoms the lower half. We can subdivide the system into planes parallel to the interface and denote them with integers n = 0, ±1, ±2 where positive signs refer to A atom planes and minus signs refer to B atom planes. The interface is the n = 0 plane. We now consider the situation in which the short range interactions between the atomic spins is represented by the Heisenberg Hamiltonian (1)
H=— ~
where (if> implies that only the nearest neighbor interactions between atoms are considered. S~ is the magnetic moment at site i whose value can take SA or SB depending on whether this site is occupied by an A or B atom, respectively. J,, is the exchange integral between the ith and fth atoms. Generally, we may have different values of J~in the interfaceregion and in the two bulk materials. However, to minimize the number of our parameters, we assume that they are only dependent on the bond type between the atoms. Thus, we have only three J values: JAA, JBB and JAB (=JBA), corresponding to the bonds A—A, B—B and A—B (or B—A), respectively. As was mentioned in the first paragraph, we take the interface plane to be composed of both A and B atoms and denote the concentration of A atoms on this plane by x (0 ~ x’~1). Besides, we introduce a chemical short-range parameters a to describe the short-range correlations between atoms. Following Cowley [7], the probability P~, of finding an atom j as nearest neighbor of atom i can be expressed in terms of x and a, as: PAA=x+(l—x)a,
PBA=x(l—a),
PAB=(l—x)(l—a),
PBB=l—X+Xa.
(2)
It is easy to see that a= 0 corresponds to a random alloy and a negative value of a represents an attraction between unlike atoms. Following the usual procedures [8], we define the following double-time Green function, G,~(t—t’)=<
(3)
.
In the one-spin-wave approximation [9], the equations of motion for the Green functions can be written as (w—E~)G,~=~+ ~ V17,G~, where E,= ~ and Vu, = i can be easily obtained:
—
(4) 2J.
From this equation, the diagonal Green function for a bulk material
Gb0= w—z1V1~+z~J~~y1(w)’
(5)
where i=A or B, and z1 is the coordination number of the material. V~=2J11S1. y~(w)is the transfer matrix for the system, which is given by E
160
—
.,.J
w ± (E1 cv) 2_ 4 (z1 —1) V~, 2(z1—l)V,, —
(6)
Volume 170, number 2
PHYSICS LETI’ERS A
26 October 1992
where E. = 2z1Vu. The local density of states of spin waves can be evaluated through the imaginary part of the Green function: D(w)=— -~ImGb0(w).
(7)
In order to study the spin waves at the interface, we must write down the equations of motion of the Green functions for the atoms in the interface region, i.e. [w—E~(A)]G~(A)= ZAI
1
...ZAO[PAA
V~Goi,oc(Ai_A)+P~ V~Go1,oo(B4-A)]
V~~~G10,00(A~A) ..ZBI V~Gio,m(B~—A),
(8)
[w_Ej(A4—A)]Gojoc(Al-A) =
—
V~Goji,oo(A-I)—(ZAO
ZAI
—
l)[P~ V~G0~+1,00(A+-A)+P~ V~G0~+ i,oo(BiA)1
V~Gljoo(A-A)zBl V~G....ijoc(Bi-A),
(9)
[cv—E3(A+-B)]G0~,00(Ai-B) =
—
VABGOJ_!,00(B4-I)
—
(ZAO
—
l)[P~ V~Goj÷i,oc(A4-A) +P~VABGoJ+I,oo(B~-A)]
ZAIV~GlJ,oc(A4A)ZBIV~G_IJ,~(B4A), [w—E_1(B)]G1J,~(B4--A)=
—
(10)
VaxGoj,m(Ai-4)_(Za
—
l)VBBG_2J.oo(B4—B),
(11)
and the equations for A4-B and n~-~n, where ZAO is the number of intraplane nearest neighbors of a site on the interface, z~1is the number of nearest neighbors between planes, z1= z0+ 2z11 is the coordination number of the bulk material i. The meaning of the subscripts of the Green functions are the same as those in ref. [10]. In writing the above equations, we also defined the following expressions, —
VAB=VBA=2JAB~/~~, Eoc(A)=zAo(P~V~+PABYAB)+zAIV~+zBlYAB,
E3(A4—A)= VAA +(zAo
l)(P~.V~+PAB YAB)+zAt ~
—
+ZBI
YAB, 1TAB,
=
VAB + (zAo —1 )(P~V~+PAB VAB) +ZAI V~+ZBI
E_j(B)=YBA+(zB—l)VBB,
YAB=2JABSB,
and the expressions for k-B and nl—. functions on the interface plane, -
Goj,oo(A4—A)
—
Goj,oc(B*-A) Goji,oo(Ai-I)
2(co)= ,
n. These equations can be solved by defining the following transfer
Go~,oo(A4—B)
a
G01_j,00(Ai-I)
—
(12)
a~(w)=
, G0~1,00(B~-I) Goj,oc(Bs—B)
(13)
,
and the transfer functions between the interface plane and its nearest planes, G_ij,oo(B4-A) G1~,00(A*-B) fl~(w)_ , fl2(cv)= , GoJ,~(A4—I) G0~,00(B~—I) —
(14)
where I=A or B. Substitutingeqs. (13) and (14) into eqs. (8)— (11), we obtain the following coupled equations for the transfer functions, 161
Volume 170, number 2
PHYSICS LETTERS A
26 October 1992
1)VAAYA], 2VAB[C0E1(A)+(ZA al=—VAA[w—Ef(A~--A)+zAIVAAyA+zBIVABflI+(zAO—l)(P~VAAaI+PABVABa3)]’, I3I=—VBA[cv—EI(B)+(zB—l)VBByB]~
fl
a2=—VAB[co—EJ(A1--B)+zAIVAAYA+zBI
,
VABfl1+(zAO—l)(PAAVAAal+PABVABa3)]~,
a3=—VBA[w—EJ(B4---A)+zB1VBBYB+zAIVBJ2+(zAO—l)(PBBVBBa4+PBAVBAcX2)1, a4=—VBB[w—EJ(B+-B)+zBIVBBYB+zAIVBJ 2+(zAO—l)(PBBVBBct4+PBAVBAa2)],
(15)
which can be solved numerically. The diagonal Green functions for the atoms on the interface are given by Goo,00(A)[COEoc(A)+zA,VAAYA+zB1VABPI+zAo(PAAVAAaI+PABVABa3)]’, GOO,OO(B)=[w—EOO(B)+zRlVBByB+zAIVBAfl2+zAO(PBBVBBa4+PBAVBAa2)]~
(16) .
(17)
The LDOS at each atom can be obtained through eq. (7).
3. Results and discussions Using the method outlined in section 2, we calculated the LDOS of spin waves for a set of different values ofthe interface concentration x and short-range order a. The results are contained in figs. 1—3. In all the graphs, the full lines are the LDOS for A atoms and the dotted lines those for B atoms. We choose the parameters ZAO = 4, ~ = 1, ZBO = 4 and ZBI = 2 to simulate the situation that the (100) plane of a simple cubic crystal A matches with the (111) plane of a body-centered cubic lattice B. The parameters J and S are chosen so that there is no overlap between the spectra of the two pure systems (see fig. la). In fig. ib, we display the results for a situation when x approaches 1, i.e. a vanishing small number of B atoms on the interface. The LDOS of B atoms in this case reduces to the single-impurity states of B atoms in an otherwise perfect interface system.
~ 0.8 ~
06
(b)
~0.4
(a)
0.2 0.1 Fig.
0.0 0.0
162
-
10.0
(i)/JS 20.0
30.0
‘
40.0
1. (a) LDOSforthebulkatoms. (b) LDOSfortheatomsat 707, the interface when x-. 1. (c) LDOS for the atoms at the interface SA=O.5andSB=I.5. when x—.0. The parameters are J,~=0.5,JBB= 1, JAB=°.
)
Volume 170, number 2
26 October 1992
PHYSICS LETTERS A
Figure 1 c gives the results when x—. 0. Generally, we can specify the spectra of an interface atom in different energy regions: those within the energy range of that of bulk A atoms can propagate into the A material without damping but decay exponentially when going into the B material, and vice versa. Those states with an energy lying inside the energy gap of the two materials or outside the spectra of both materials can only exist in the interface region and decay when propagating into A or B materials. We call these states interface states. One may see from fig. lb that when the interface has a predominant number ofA atoms, the spectra of both A and B atoms are mainly located at those ofbulk A atoms. Those states within the energy range ofB atoms contribute for a small part to the interface spectra. Opposite behaviors were observed in fig. ic when x-+0. Besides, there is a big contribution of interface states in both cases. In fig. 2, we present the results for the calculation of LDOS for different values of x for a random interface (a=O). Two features were observed: (1) The LDOS for an interface atom is a mixture of those of the two bulk materials. From fig. 2a to fig. 2c, as more and more B atoms reside on the interface, the spectra as a whole shift toward the high energy region. This behavior can be easily understood in considering that more energy should be required to flip a spin at the interface as there are more B atoms around it, because we have chosen a bigger spin value and stronger interactions for B atoms. (2) The energy gap between the spectra of the two materials narrows and vanishes in fig. 2a. This means interface states always exist in the system. We also studied the influence of the chemical short-range order on the LDOS. The results are shown in fig. 3. In the calculation, we have chosen x=0.5. One noticeable property is that when a= —1, i.e. for a completely ordered interface, the interface states are located isolatedly in the gap region of the two bulk materials (see fig. 3a). As I al becomes smaller from fig. 3a to fig. 3c, the energy gap of the interface spectra becomes small and eventually the interface states are connected to the bulk states. 0.4
0.4
~
.4
(c
(b)
03
A
02
1V~
(a)
0.6
(a)
0.4 0.2/(,
0.0
0.0
~ ‘~~:i----10.0 20.0
30.0
40.0
(~)/JS Fig. 2. (a)—(c) LDOS for the interface atoms when x=0.8, 0.5 and 0.2, respectively,
0.0
0.0
~
10.0
20.0
30.0
40.0
CO/JS Fig. 3. (a)—(c) LDOS for the interface atoms when a= —1, —0.6 and —0.2, respectively. The concentration xis 0.5.
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PHYSICS LETTERS A
References [1] A. Yaniv, Phys. Rev. B 28 ( 1983) 402. [2]B.X. Xu, M. MostollerandA.K. Rajagopal, Phys. Rev. B 31(1985) 7413. [3] X.F. Wang and T.H. Lin, Chinese Phys. Lett. 4 (1987) 29. [4] R.C. Kittler and L.M. Falicov, J. Phys. C 9 (1976) 4259. [5] R.C. Kittler and L.M. Falicov, Phys. Rev. B 18 (1978) 2506. [6] D. Kechrakos, J. Phys. Cond. Matter 2 (1990) 2637. [7] J.M. Cowley, Phys. Rev. 77 (1950) 669. [8] D.N. Zubarev, Soy. Phys. Usp. 3 (1960) 320. [9]A. Theumann, J. Phys. C 6(1973)2822. [10] J.L. Morán-Lopèz and L.M. Falicov, Phys. Rev. B 20 (1979) 3900.
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