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Spin wave in an aperiodic ferromagnet
Journal of Magnetism and Magnetic Materials 117 (1992) 75-78 North-Holland
API"
Spin wave in an aperiodic ferromagnet T i a n - s h i Liu a n d G u o - z h u W e i Department of Physics, North-east University of Technology, Shenyang 110006, China Received 22 April 1992
Spin wave in an aperiodic ferromagnetic superlattice is studied by the rescaling approach. The Green's function in an aperiodic ferromagnet has been derived in terms of an exact decimation transformation. Using the iteration of the transformation we calculated the density of states (DOS) and the magnetization, when Jt + J2 = 2./0 = constant, A= J2 ~Jr The numerical results showed that the DOS are the same for ;t = ;to and for ;t = 1/;t 0 for a fixed ;to, the bandwidths of DOS and the magnetization depend on ;t.
1. Introduction Recently a lot of attention has been paid to the T h u e - M o r s e aperiodic structure [1-12]. The structure factor, spectra of accoustic phonons and tight binding electronic properties of the T h u e Morse chain obtained by using the perturbation method, transfer matrix and a renomalization analysis [3-5]. For understanding the magnetic properties of the T h u e - M o r s e chain, Doria et al. considered the T h u e - M o r s e quantum Ising chain [6]. Axel et al. and Luck proved that the energy spectrum of the T h u e - M o r s e sequence is a Cantor-like set [8] and presented the analysis of high-resolution X-ray-diffraction spectra of finite-size T h u e - M o r s e superlattice heterostructures [8-10]. The first experimental realization of a T h u e - M o r s e superlattice is due to Merlin et al. [11]. The outline of this paper is as follow. In section 2 we introduce a model of the 3-D aperiodic structure and the rescaling method for equations of motion of Green's function. The density of states (DOS) and the magnetization are obtained. In section 3 we given the numerical results of
DOS and magnetization with a discussion. In section 4 a brief summary to the present results is given.
2. The model and rescaling method We consider a simple cubic ferromagnetic layered superlattice with spin S. Each layer is a 2-D square lattice with the lattice constant a. The Hamiltonian can be written as /j ~/z
(1)
where i, j are the index numbers of the layers, v, p. are the index numbers of the sites belonging to the i ( j ) layer. Nearest neighbour interactions are considered only. In each layer the exchange interaction is Jo. In the layers closest to each other, the exchange interaction can have Jl or J2, J1 and J2 are arranged in a T h u e - M o r s e sequence, which occurs at any level in the hierarchy generated by successive applications of the rule: o-(A,,) --+ A,,+tB,,+I , o-(B,,) --+ B,,+aA,,+I.
Correspondence to: Dr. T.S. Liu, Department of Physics, North-east Universityof Technology,Shenyang 110006, China.
(2)
In the x - z plane the structure is shown in fig. 1.
0304-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
76
T.S. Liu, G.Z. Wei / Spin wave in an aperiodic ferromagnet t t t
It tI It
I l l t i t t i t
t t !
I l l I l l I l l
tion and taking any a layer as a reference layer, we obtain
t I !
..o,
(to - e,~)Foo = 1 + t l F _ l o + t l F l o ,
(to - et3)Fto = tlFoo + t2F2o, (to - %)F2o = tzFxo + tlF3o,
(6)
(oJ - Et3)F30 = tlF20 + t2F40,
(to -- es)F40 = I + tzF30 + t2Fso, Fig. 1. The aperiodic layered ~ o m a g n e t i c x - z plane.
structure in the
where For determining the D O S we use Green's function method [13]. Defining G r e e n ' s function G/~'(o~) = ( ( S + ; S~))o '
(3)
F i j = F i j ( to , K ) ,
e~ = 4 J I S + A, e~ = % = 2 ( J 1 + J 2 ) S + A, E~ = 4 J 2 S
(7)
+ A,
which satisfies the equation of motion
A = 4JOS(2 - cos Kxa - cos K y a ) ,
toGas(to)
t t = -2SJx,
+(([S+,H];S~))
.
(4)
We can perform 2-D Fourier transformation for the G r e e n ' s function, because it is translation invariant in the x - y plane,
G~(to)
=-
2S
EFij(to, K) exp(iK.(R~-R.)), Nil x
(5) where Nil is the number of sites in the x - y plane, K = ( K x , K y ) and R = ( R x , R y ) are the wave vector and coordinate vector in the x - y plane, respectively. If we consider the difference between A on the left and B on the right and opposite case, each of the atoms must be in any one of four different nearest neighbour surroundings in T h u e - M o r s e sequence. In addition, the site energy E~ can have one of four values depending on the local environmental of site i. We shall adopt the notation whereby ~ , i = a, /3, T, 6, refer to fig. 2. Using random phase approxima-
t 2 = -2SJ
2.
In order to solve eqs. (6), we use an exact decimation transformation [14-16]. The decimation is achieved by removing appropriate sites following the rule: AB ~ A', BA ~ B'. If i is one of the sites removed in the decimation having site energy ~ / = G0, the renormalized bound joining site i - 1 to site i + 1 is t ' and the renormalized energy ~' of site i + 1 include contribution ei+l and t Z + l / ( t o - e l ) . The transformation decimation associated with a length rescaling factor b = 2 are given by t
.~-
tl
_
tit2 _
t2
Fig. 2. The T h u e - M o r s e sequence, where o is a site in the a layer, zx is a site in the g layer, [] is a site in the y layer, • is a site in the 8 layer.
w - - ~,/ '
t2+t 2 !
E a = E3, + - w -- e ~
t2 to
--
t2 El3
t2 ¢
- ~,...O...~,--qD-~zx---O---C3~-zx---~zx---O---~ ,',. . . .
tit2
t
to -- El3 '
~3,=Ea+--+ to--E/3 !
w -- e ~
to
(8) --
~T
t2 - , to--E~
T.S. Liu, G.Z. Wei / Spin wave in an aperiodic ferromagnet
Finally, the average density of states p(oJ) is related to the imaginary part of the site-averaged Green's function G(oJ) through the relation 1 p(oJ) = - - - I m "rr
G(oJ),
77
where n i (i = a,/3, y, ~) denote the fractions of sites of that type, and Gii(m) are the corresponding Green's functions, 2S
(9)
Gii(m)
=
-
E
-
Nil K¢BZ
Fii(oJ, I~).
where Im denotes the imaginary part and G(~o) = lira
Gii(m)
Nz "* oo - N z
1
(10)
Performing the sum in (10) and taking the limit one can obtain o, ) = ,
Goo( ,,, ) +
For the Thue-Morse sequence n~ = n , = 1/6, na = nv = 1/3. The magnetization can be obtained by M = M o [ 1 - [o~ p(oJ) d ] l J0 e ~ - - 1 " aJ],
(12)
o, )
+ n,G~,,(~o) + nsG88(oJ),
(11)
where 13 = 1/kBT and M 0 is the magnetization at T = 0 K .
.
.
.
.
.
.
i
C
O0 C)
09 CO £:1
2
E/45Je
-"
."
t
4
6
O
E/4SJe
I
I
!
I
I
o
l
I
I
b
I
d
03 C3 C)
([) C3 C]
me
"
2
"
4
-
E/45J.
S
-
e
e
2
4
6
E/4SJm
Fig. 3. The density of states (DOS). (a) A = 3 (1/3); (b) ~ = 9 (1/9); (c) A = 19 (1/19); (d) A = 99 (1/99).
78
T.S. Liu, G.Z. Wei / Spin wave in an aperiodic ferromagnet
in the T h u e - M o r s e aperiodic crystal for a fixed A
1'
4. Summary
w
TI--T
0
g
o.5
KoT/4SJs Fig. 4. Reduced magnetization M / M o vs. reduced temperature k B T / 4 S J o. The curves a, b, c, d correspond to A = 3 (1/3), 9 (1/9), 19 (1/19), 99 (1/99), respectively.
3. The results and discussion When taking A = J2/J1, fixed J1 + J2 = 2J0, we calculate DOS using (9) from (6) by the above decimation procedure. The results are shown in figs. 3a, b, c and d corresponding to A = 3, 9, 19, 99, respectively. From fig. 3 we can see that the deviations of DOS in the aperiodic system from the one in the 3-D periodic system (A = 1) depend on various A values. The bandwidth is a function of A, it is represented as 16SJ 0 (1 + 2A)/(1 + A) for A > 1. The bandwidth in the 3-D periodic system in which the exchange energy is J0, is always 24SJ 0. From fig. 3 we can also see that the DOS shows mainly the character of double layers for a not too large A and the character of three layers for a large A. The results of reduced magnetization are shown in fig. 4, the curves a, b, c, d correspond to A = 3, 9, 19, 99, respectively. From fig. 4 it is easy to see that when A > 1 the magnetization decreases with increasing A for a given temperature. We also calculate DOS and magnetization for A < 1. The band width is represented as 16SJ0(A + 2)/(1 + A). We found that the DOS for A = 1/3, 1/9, 1/19, 1 / 9 9 are the same with the DOS for A = 3, 9, 19, 99, respectively. This case differs from the Fibonacci quasicrystal [17]. In the Fibonacci quasicrystal the bandwidth is always 24SJ 0 for A > 1, it is the same as the one in the periodic system, but the bandwidth is the same as the one
We considered a simple cubic ferromagnetic layered superlattice with spin S. In each layer the exchange interaction is J0. The exchange interactions between the nearest neighbour layers can have J1 or J2, which are arranged according to the T h u e - M o r s e sequence. Magnons are studied in terms of an exact decimation transformation. Using iteration of the transformation we obtained numerical results for the DOS and the magnetization, when J1 + J2 = 2Jo, A = J2/Jl. The results show that the DOS at A = Ao and A = 1/A 0 are the same for a fixed A0. At a given temperature, for A > 1 (A < 1) the magnetization decreases (increases) with increasing A. The value of A has an important effect on magnons in the aperiodic ferromagnet.
References [1] F. Alex, J.P. Allouche, M. Kleman, M. Mend6s et al., J. de Phys. 47 (1986) C3-181. [2] Z. Cheng, R. Savit and R. Merlin, Phys. Rev. B 37 (1988) 4375. [3] P. Rildund, M. Severin and Y. Liu, J. Mod. Phys. B 1 (1987) 121. [4] S. Tamura and F. Nori, Phys. Rev. B 40 (1989) 9790. [5] M. Qin, H. Ma and C. Tsai, J. Phys.: Condens. Matter 2 (1990) 1059. [6] M.M. Doria, F. Nori and I.I. Satija, Phys. Rev. B 39 (1989) 6802. [7] M. Kolar and F. Nori, Phys. Rev. B 42 (1990) 1062. [8] F. Axel and J. Peyriere, J. Stat. Phys. 57 (1989) 1013. [9] F. Axel and H. Terauchi, Phys. Rev. Lett. 66 (1991) 2223. [10] J.M. Luck, Phys. Rev. B 39 (1989) 5834. [11] R. Merlin, K. Bajema, J. Nagle and K. Ploog, J. de Phys. 48 (1987) C5-503. [12] M. Kolar, M.K. Ali and F. Nori, Phys. Rev. B 43 (1991) 1034. [13] D.N. Zubarev, Usp. Fiz. Nauk 71 (1960) 71. [14] B.W. Southern, A.A. Kumar, P.D. Loly and A.-M.S. Trembelay, Phys. Rev. B 27 (1983) 1405. [15] J.A. Ashraff and R.B. Stinchombe, Phys. Rev. B 37 (1988) 5723. [16] B.W. Southern, T.S. Liu and D.A. Lavis, Phys. Rev. B 39 (1989) 12160. [17] T.S. Liu and G.Z. Wei, to be published.
API"
Spin wave in an aperiodic ferromagnet T i a n - s h i Liu a n d G u o - z h u W e i Department of Physics, North-east University of Technology, Shenyang 110006, China Received 22 April 1992
Spin wave in an aperiodic ferromagnetic superlattice is studied by the rescaling approach. The Green's function in an aperiodic ferromagnet has been derived in terms of an exact decimation transformation. Using the iteration of the transformation we calculated the density of states (DOS) and the magnetization, when Jt + J2 = 2./0 = constant, A= J2 ~Jr The numerical results showed that the DOS are the same for ;t = ;to and for ;t = 1/;t 0 for a fixed ;to, the bandwidths of DOS and the magnetization depend on ;t.
1. Introduction Recently a lot of attention has been paid to the T h u e - M o r s e aperiodic structure [1-12]. The structure factor, spectra of accoustic phonons and tight binding electronic properties of the T h u e Morse chain obtained by using the perturbation method, transfer matrix and a renomalization analysis [3-5]. For understanding the magnetic properties of the T h u e - M o r s e chain, Doria et al. considered the T h u e - M o r s e quantum Ising chain [6]. Axel et al. and Luck proved that the energy spectrum of the T h u e - M o r s e sequence is a Cantor-like set [8] and presented the analysis of high-resolution X-ray-diffraction spectra of finite-size T h u e - M o r s e superlattice heterostructures [8-10]. The first experimental realization of a T h u e - M o r s e superlattice is due to Merlin et al. [11]. The outline of this paper is as follow. In section 2 we introduce a model of the 3-D aperiodic structure and the rescaling method for equations of motion of Green's function. The density of states (DOS) and the magnetization are obtained. In section 3 we given the numerical results of
DOS and magnetization with a discussion. In section 4 a brief summary to the present results is given.
2. The model and rescaling method We consider a simple cubic ferromagnetic layered superlattice with spin S. Each layer is a 2-D square lattice with the lattice constant a. The Hamiltonian can be written as /j ~/z
(1)
where i, j are the index numbers of the layers, v, p. are the index numbers of the sites belonging to the i ( j ) layer. Nearest neighbour interactions are considered only. In each layer the exchange interaction is Jo. In the layers closest to each other, the exchange interaction can have Jl or J2, J1 and J2 are arranged in a T h u e - M o r s e sequence, which occurs at any level in the hierarchy generated by successive applications of the rule: o-(A,,) --+ A,,+tB,,+I , o-(B,,) --+ B,,+aA,,+I.
Correspondence to: Dr. T.S. Liu, Department of Physics, North-east Universityof Technology,Shenyang 110006, China.
(2)
In the x - z plane the structure is shown in fig. 1.
0304-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
76
T.S. Liu, G.Z. Wei / Spin wave in an aperiodic ferromagnet t t t
It tI It
I l l t i t t i t
t t !
I l l I l l I l l
tion and taking any a layer as a reference layer, we obtain
t I !
..o,
(to - e,~)Foo = 1 + t l F _ l o + t l F l o ,
(to - et3)Fto = tlFoo + t2F2o, (to - %)F2o = tzFxo + tlF3o,
(6)
(oJ - Et3)F30 = tlF20 + t2F40,
(to -- es)F40 = I + tzF30 + t2Fso, Fig. 1. The aperiodic layered ~ o m a g n e t i c x - z plane.
structure in the
where For determining the D O S we use Green's function method [13]. Defining G r e e n ' s function G/~'(o~) = ( ( S + ; S~))o '
(3)
F i j = F i j ( to , K ) ,
e~ = 4 J I S + A, e~ = % = 2 ( J 1 + J 2 ) S + A, E~ = 4 J 2 S
(7)
+ A,
which satisfies the equation of motion
A = 4JOS(2 - cos Kxa - cos K y a ) ,
toGas(to)
t t = -2SJx,
+(([S+,H];S~))
.
(4)
We can perform 2-D Fourier transformation for the G r e e n ' s function, because it is translation invariant in the x - y plane,
G~(to)
=-
2S
EFij(to, K) exp(iK.(R~-R.)), Nil x
(5) where Nil is the number of sites in the x - y plane, K = ( K x , K y ) and R = ( R x , R y ) are the wave vector and coordinate vector in the x - y plane, respectively. If we consider the difference between A on the left and B on the right and opposite case, each of the atoms must be in any one of four different nearest neighbour surroundings in T h u e - M o r s e sequence. In addition, the site energy E~ can have one of four values depending on the local environmental of site i. We shall adopt the notation whereby ~ , i = a, /3, T, 6, refer to fig. 2. Using random phase approxima-
t 2 = -2SJ
2.
In order to solve eqs. (6), we use an exact decimation transformation [14-16]. The decimation is achieved by removing appropriate sites following the rule: AB ~ A', BA ~ B'. If i is one of the sites removed in the decimation having site energy ~ / = G0, the renormalized bound joining site i - 1 to site i + 1 is t ' and the renormalized energy ~' of site i + 1 include contribution ei+l and t Z + l / ( t o - e l ) . The transformation decimation associated with a length rescaling factor b = 2 are given by t
.~-
tl
_
tit2 _
t2
Fig. 2. The T h u e - M o r s e sequence, where o is a site in the a layer, zx is a site in the g layer, [] is a site in the y layer, • is a site in the 8 layer.
w - - ~,/ '
t2+t 2 !
E a = E3, + - w -- e ~
t2 to
--
t2 El3
t2 ¢
- ~,...O...~,--qD-~zx---O---C3~-zx---~zx---O---~ ,',. . . .
tit2
t
to -- El3 '
~3,=Ea+--+ to--E/3 !
w -- e ~
to
(8) --
~T
t2 - , to--E~
T.S. Liu, G.Z. Wei / Spin wave in an aperiodic ferromagnet
Finally, the average density of states p(oJ) is related to the imaginary part of the site-averaged Green's function G(oJ) through the relation 1 p(oJ) = - - - I m "rr
G(oJ),
77
where n i (i = a,/3, y, ~) denote the fractions of sites of that type, and Gii(m) are the corresponding Green's functions, 2S
(9)
Gii(m)
=
-
E
-
Nil K¢BZ
Fii(oJ, I~).
where Im denotes the imaginary part and G(~o) = lira
Gii(m)
Nz "* oo - N z
1
(10)
Performing the sum in (10) and taking the limit one can obtain o, ) = ,
Goo( ,,, ) +
For the Thue-Morse sequence n~ = n , = 1/6, na = nv = 1/3. The magnetization can be obtained by M = M o [ 1 - [o~ p(oJ) d ] l J0 e ~ - - 1 " aJ],
(12)
o, )
+ n,G~,,(~o) + nsG88(oJ),
(11)
where 13 = 1/kBT and M 0 is the magnetization at T = 0 K .
.
.
.
.
.
.
i
C
O0 C)
09 CO £:1
2
E/45Je
-"
."
t
4
6
O
E/4SJe
I
I
!
I
I
o
l
I
I
b
I
d
03 C3 C)
([) C3 C]
me
"
2
"
4
-
E/45J.
S
-
e
e
2
4
6
E/4SJm
Fig. 3. The density of states (DOS). (a) A = 3 (1/3); (b) ~ = 9 (1/9); (c) A = 19 (1/19); (d) A = 99 (1/99).
78
T.S. Liu, G.Z. Wei / Spin wave in an aperiodic ferromagnet
in the T h u e - M o r s e aperiodic crystal for a fixed A
1'
4. Summary
w
TI--T
0
g
o.5
KoT/4SJs Fig. 4. Reduced magnetization M / M o vs. reduced temperature k B T / 4 S J o. The curves a, b, c, d correspond to A = 3 (1/3), 9 (1/9), 19 (1/19), 99 (1/99), respectively.
3. The results and discussion When taking A = J2/J1, fixed J1 + J2 = 2J0, we calculate DOS using (9) from (6) by the above decimation procedure. The results are shown in figs. 3a, b, c and d corresponding to A = 3, 9, 19, 99, respectively. From fig. 3 we can see that the deviations of DOS in the aperiodic system from the one in the 3-D periodic system (A = 1) depend on various A values. The bandwidth is a function of A, it is represented as 16SJ 0 (1 + 2A)/(1 + A) for A > 1. The bandwidth in the 3-D periodic system in which the exchange energy is J0, is always 24SJ 0. From fig. 3 we can also see that the DOS shows mainly the character of double layers for a not too large A and the character of three layers for a large A. The results of reduced magnetization are shown in fig. 4, the curves a, b, c, d correspond to A = 3, 9, 19, 99, respectively. From fig. 4 it is easy to see that when A > 1 the magnetization decreases with increasing A for a given temperature. We also calculate DOS and magnetization for A < 1. The band width is represented as 16SJ0(A + 2)/(1 + A). We found that the DOS for A = 1/3, 1/9, 1/19, 1 / 9 9 are the same with the DOS for A = 3, 9, 19, 99, respectively. This case differs from the Fibonacci quasicrystal [17]. In the Fibonacci quasicrystal the bandwidth is always 24SJ 0 for A > 1, it is the same as the one in the periodic system, but the bandwidth is the same as the one
We considered a simple cubic ferromagnetic layered superlattice with spin S. In each layer the exchange interaction is J0. The exchange interactions between the nearest neighbour layers can have J1 or J2, which are arranged according to the T h u e - M o r s e sequence. Magnons are studied in terms of an exact decimation transformation. Using iteration of the transformation we obtained numerical results for the DOS and the magnetization, when J1 + J2 = 2Jo, A = J2/Jl. The results show that the DOS at A = Ao and A = 1/A 0 are the same for a fixed A0. At a given temperature, for A > 1 (A < 1) the magnetization decreases (increases) with increasing A. The value of A has an important effect on magnons in the aperiodic ferromagnet.
References [1] F. Alex, J.P. Allouche, M. Kleman, M. Mend6s et al., J. de Phys. 47 (1986) C3-181. [2] Z. Cheng, R. Savit and R. Merlin, Phys. Rev. B 37 (1988) 4375. [3] P. Rildund, M. Severin and Y. Liu, J. Mod. Phys. B 1 (1987) 121. [4] S. Tamura and F. Nori, Phys. Rev. B 40 (1989) 9790. [5] M. Qin, H. Ma and C. Tsai, J. Phys.: Condens. Matter 2 (1990) 1059. [6] M.M. Doria, F. Nori and I.I. Satija, Phys. Rev. B 39 (1989) 6802. [7] M. Kolar and F. Nori, Phys. Rev. B 42 (1990) 1062. [8] F. Axel and J. Peyriere, J. Stat. Phys. 57 (1989) 1013. [9] F. Axel and H. Terauchi, Phys. Rev. Lett. 66 (1991) 2223. [10] J.M. Luck, Phys. Rev. B 39 (1989) 5834. [11] R. Merlin, K. Bajema, J. Nagle and K. Ploog, J. de Phys. 48 (1987) C5-503. [12] M. Kolar, M.K. Ali and F. Nori, Phys. Rev. B 43 (1991) 1034. [13] D.N. Zubarev, Usp. Fiz. Nauk 71 (1960) 71. [14] B.W. Southern, A.A. Kumar, P.D. Loly and A.-M.S. Trembelay, Phys. Rev. B 27 (1983) 1405. [15] J.A. Ashraff and R.B. Stinchombe, Phys. Rev. B 37 (1988) 5723. [16] B.W. Southern, T.S. Liu and D.A. Lavis, Phys. Rev. B 39 (1989) 12160. [17] T.S. Liu and G.Z. Wei, to be published.