excitations in randomly diluted two-dimensional ferromagnets

Physica 86A (1977) 553-573 O North-Holland Publishing Co.

E X C I T A T I O N S IN R A N D O M L Y D I L U T E D T W O - D I M E N S I O N A L

FERROMAGNETS

A.R. McGURN*

Department o[ Physics, Temple University, Philadelphia, Pennsylvania 19122, USA

Raza A. TAHIR-KHELI*

Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA

and

J. Stefan Institute, University of Ljubljana, 61001 Ljubljana, Yugloslavia**

Received 13 July 1976

The dynamics of a randomly diluted Heisenberg ferromagnet on a two-dimensional square lattice is calculated using a CPA procedure introduced by Theumann and Tahir-Kheli. The density of states and frequency-wave-vector-dependent response functions at various symmetry points of the Brillouin zone are presented for several magnetic concentrations. Our results for the density of states are compared with those obtained from computer-experiment calculations of Huber and are found to be in good agreement with them. The spin-wave stiffness coefficient is calculated, and comparison is made with results on two-dimensional resistor networks given by Watson et al.

1. Introduction

V a r i o u s c o h e r e n t po t en t i a l a p p r o x i m a t i o n ( C P A ) p r o c e d u r e s h a v e r e c e n t l y b e e n d e v e l o p e d to s t u d y the d y n a m i c a l p r o p e r t i e s of r a n d o m l y d i lu t ed spin systems~-~). U n l i k e in a l loys wi th 6) on ly " o n - s i t e " r a n d o m n e s s , C P A for s u c h

s y s t e m s is c o m p l i c a t e d d u e to the p r e s e n c e o f bo th d i a g o n a l a n d o f f -d iagona l r a n d o m i n t e r a c t i o n s . N u m e r o u s t e c h n i q u e s h a v e b e e n d e v e l o p e d to c i r c u m - v e n t this diff iculty. M o s t o f t h e s e e m p l o y a p p r o x i m a t i o n s w h i c h ignore v a r i a t i o n s in e i t he r the l ong i tud ina l 2) or the t r a n s v e r s e 5) e x c h a n g e c o u p l i n g s ,

or t h e y i n t r o d u c e s p u r i o u s d e g r e e s of f r e e d o m w h i c h are la ter r e m o v e d by p s e u d o p o t e n t i a l s a d d e d to the sp in Hami l ton ian~) .

* Supported by a grant from the US National Science Foundation. ** Exchange Professor under the US National Academy of Science and the Council of

Academies of Sciences of Yugoslavia exchange programme and a guest scientist of J. Stefan Institute.

553

554 A.R. McGURN AND R.A. TAHIR-KHELI

Recently, Theumann and Tahir-Kheli (TTK)4), introduced a somewhat different CPA for treating the diluted ferromagnet. Their theory deals with spins directly, therefore it does not need to involve the introduction of spurious degrees of freedom. Moreover, for three-dimensional systems it was found to yield correctly the first three frequency moments of the response function and more importantly, in comparison with numerically generated, Pad6 approximant-moment expansion line shapes, it showed marked im- provement over previous CPA theories in the quality of results for wave- vectors >-~,r and for larger randomness, i.e. for larger concentrat ions of non-magnetic impurities.

The purpose of this present paper is to apply the CPA procedure of TTK to a study of the dynamical properties of the two-dimensional, diluted ferromagnet. In common with previous CPA works, here we employ a random- phase-like approximation, RPA, for decoupling the chain of magnetic Green's functions. At T = 0, this decoupling of course leads to the usual spin-wave theory results.

The density of states and the frequency-wave-vector-dependent response functions for various points along the symmetry axes of the Brillouin zone are presented, it is found that, as in three dimensions, the bulk of the density of states is shifted to lower frequencies as the dilution increases. Likewise, the peaks in the response functions are also shifted to lower frequencies and the curves develop widths which increase as the concentrat ion of non-magnetic, i.e. vacant, sites increases. These results indicate that magnetically the system is softened as vacancies are created and that vacancy scattering becomes increasingly important with increase in its concentration. In two dimensions, as compared with three, these two processes proceed more rapidly as the dilution is increased.

Our calculations for the density of states are found to compare quite reasonably with the corresponding computer experiment results of Huber7). In the concentrat ion region where the present theory can be expected to be valid, our results for the stiffness coefficients are also in agreement with experimental measurements and theoretical calculations carried out by Wat- son et al. g) on two-dimensional resistor networks.

In section 2 we present briefly a formulation of the theory for two dimensions. We obtain the self-consistent equations for the two-dimensional Green's functions in section 3 and solve them to first order in the dilution. The results for the various concentrat ion dependent response functions and den- sities of state are presented and discussed in section 4.

2. Theory

In the following only a brief sketch of the theory is given. In order to simplify comparison with the three-dimensional theory of TTK, we shall whenever possible adopt a similar notation to theirs.

The hamiltonian of the dilute system is of the form

RANDOMLY DILUTED TWO-DIMENSIONAL FERROMAGNETS 555

= - . ¢ , - H , ¢ , .

/3" i

Here ci --- 1 or 0 depending on whether the site is occupied by a magnetic spin o-i or is vacant. The range of the exchange interaction is assumed to be limited to nearest neighbour separation, i.e.

J, if sites i and j are nearest neighbours J~i = Jj.i = 0, otherwise.

For convenience, we shall take the o-i vectors to be of length ½ (in Dirac's units). Generalization to arbitrary spins, however, is straightforward.

The non-random part of the hamiltonian is proportional to an anisotropy field HA. As is well known, an isotropic, two-dimensional Heisenberg ferromagnet does not show magnetic long-range order at finite temperatures. However, the many magnetic systems one deals with in the laboratory, which are essentially two-dimensional in terms of their exchange interactions, have finite anisotropy fields. As such, these systems are magnetically ordered at non-zero temperatures and moreover such ferromagnets are well described by spin-wave approximation at low temperatures.

Our reason for choosing the anisotropic part of the hamiltonian to be non-random is purely one of convenience. In this way, we know that the non-magnetic, i.e. the vacancy, response occurs at the Zeeman frequency proportional to HA. Therefore, without loss of generality our scale for the energy can be chosen such that the vacancy response is placed at the origin, i.e. at ~o = 0. However , had we taken this part of the hamiltonian to be also site random, then the only difference entailed would have been that the vacancy response would occur at to = 0 while the spin-wave band would be shifted up by the Larmor f requency proportional to HA. Thus, in what follows we shall ignore the anisotropy in the hamiltonian and assume that the f requency scale has been appropriately chosen to make adjustment for it.

As in TTK, we attempt to describe this system by an effective medium of the form

Y ¢ ~ = - ~ , ~ i ~ ( E ) S , • S i. (2.2) ii

Here Si's denote effective medium (which, of course, is translationally uniform) spins located at each site of the lattice. Their magnitude is chosen so that the magnetization of the non-random, effective medium is the same as that of the random system. The J~n(E) are the energy-dependent exchange couplings of the effective medium. Consistent with the TTK theory for the 3-dimensional case, these exchange integrals will only be considered to connect members of a cluster formed by a central spin and its nearest neighbour shell. In other words, the integrals will exist between nearest, next-nearest and the third-nearest neighbours. Thus:


[o~,(E), if IR i - Rit = 6,

~ F ( E ) = t#2(E), if JR , - R~I = 6: (2.3) /

[o~3(E), if [Ri - Ril = 26,.

Here R~ denotes lattice site position vectors and 61 and 62 are vectors connecting first and second neighbour sites, respectively.

The R P A equations of motion of the effective medium Green 's functions are:

g~n(E) = 3o(E)&j - Ao(E) ~ ~] J~(E)g~+~,j(E), (2.4)

where for convenience we have made an analogous choice to that used by TTK for the definition of the Green 's functions in dividing them by S en and where

J~(E) = S~fro~,(E ), ~ = 1,2, 3, (2.5)

A0(E) = [E - 4J~(E) - 4J2(E) - 4J3(E)] ~. (2.6)

Here we have used the notation,

S ~ = 2(S~), (2.7)

where (S~) is the magnetization per effective medium site. In inverse lattice space we find:

g~Ket(E) = [AoJ(E) + 4J l (E)y K + 4J~(E)v K + 4J3(E)yzK] -1, (2.8)

where we have used the Fourier representation

. , .

g~.~(E) = N ~ g~(E) e 'K "- ' , (2.9)

and

TK =~a ~ eiKa~ = ½ [ c o s K ~ + c o s K ~ ] , (2.10)

with

UK 4 E eiKa2 = t[4('yK)2 1 -- Y2K]. (2.1 1) 62

[We choose our units so that nearest neighbour lattice spacing is unity.] To determine the effective medium, an effective medium spin is removed

and in its place a site from the actual random system is inserted back into the lattice. The medium is then determined by requiring that the average Green 's functions of the cluster (formed by the random site and its nearest neighbour medium spins) are equal to the corresponding Green 's functions in the pure effective medium.

The nature of exchange interactions between the random spin and the effective medium is, of course, not evident a priori. TTK presented arguments

R A N D O M L Y D I L U T E D T W O - D I M E N S I O N A L F E R R O M A G N E T S 557

for making a certain type of choice for this interaction. We shall follow TTK in this regard, except for noting the difference in the dimensionality. (We refer the reader to TTK for details of their argument).

The interaction within the cluster, formed from the random spin sitting at the central site and effective medium spins on the nearest neighbour shell, will be as follows: (a) The interaction between the random site and its nearest neighbours will be replaced by ~'~(E) where

c~{(E) = / [ j _}_ c~l(E)], (2.12)

(b) The interactions between the random spin and members of the medium outside the neighbouring shell are assumed to be the same as if no random spin had been embedded in the medium. (c) Nearest neighbour sites of the random spin, sitting across from each other with the central site in-between, will have no interaction. These sites are separated by a distance equal to twice the lattice separation (i.e. they are third neighbours), i.e.

~ ; ( E ) = o. (2.13)

In other words, the presence of the intervening random spin is assumed to cut off interaction between spins of the effective medium. (d) Nearest neighbour sites of the random spin adjacent to one another interact through J~(E), i.e.

J~(E) = ½J~(E). (2.14)

Note that these sites are second neighbours of each other and their interactions can proceed via two paths, one where the random spin intervenes and thus cuts that part of the interaction off. The remaining interaction is thus one half what it would be if the central spin were also an effective medium spin.

At this point, it is convenient to introduce a compact notation that we shall use from hereon. When referring to the canonical cluster consisting of a central spin and its environment, Greek indices c~,/3 . . . . . etc., will denote the positions of the first neighbours of the central site, itself to be denoted by i, while a subindex s will indicate any atom in the cluster, i.e., s denotes either a or i.

Using this notation and the above-mentioned ansatz, i.e. eqs. (2.12)-(2.14), we compute the RPA "time derivat ives" for spins lying in the canonical cluster with a random spin at the central site i and the neighbouring spins ot belonging to the medium. We get the following:

d(c ic r +) l - - E ½ [ ] + z + " - - d ~ l ( E ) ] ( 2 ( S , ) c i c r i - 2ci(crDS~ +)

dt ,,

+ E e# z + z + ~ ¢ i . . ( E ) ( 2 ( S , ) G o " ~ - 2 c i ( c r ~ ) S , ) , (2.15) n:#8


• d S ~ + 1 - -

dt = ~[J + o.¢,(E)](2c,(o'~)S S - 2(Sg)c,o -+)

-~- E [ Z + Z + ~J2(E)(2(SpS, , - 2(S,,)S~)

+ E err z + z + ~¢,,.,,(E)(2(S~S,, - 2(S,,)S,,). (2.16) n#s

The last terms in eqs. (2.15) and (2.16) are sums over sites denoted by n. These sites lie outside the cluster consist ing of sites labelled s.

Like TTK, here we shall work with a sys tem which is at low tempera tures . This means that k a T .~ ( x - xp )@~ where xp is the percolat ion concent ra t ion 9) and @~ is of the order of J. In o ther words the t empera tu re is a s sumed essential ly to be zero. Hence we shall wri te

2Ci(O'~) ~ C i. (2 .17)

From eqs. (2.15)-(2.17), we obtain the fol lowing express ions for the R P A equat ions of motion for the energy Four ie r - t rans form of the cluster Green ' s funct ions:

&.~(E) = g°(E)8 , , , - g°(E) ~ J[(E)g,,.s(E ) a

g (E) E e. - J~,, (E)g,,~(E), n # s

g,,.,(E) = g°~(E)8~.,

- g°,,(E) [J'~(E)&,~(E) + E J~(E)g,.,(E) 1

g~(E) E ea - J~.,(E)gn,s(E).

t l # x

Here we have used the notat ion:

eft eft eft Js,.(E) = S dq,,.(E),

J , I ( E ) = S e f f ~ , l ( E ) ~ _ I eft ~[S J + J , ( E ) ] ,

J4(E) = SeaoC~(E ) = ~ s e " o ~ 2 ( E ) = ½J2(E),

g°(E) = ( c J S e P i ) { A o l ( E ) + 2[J , (E) - S'uJI} -',

and

g°(E) = [Ao'(E) + ~'I(E)] ',

where

~ ( E ) = J1(E) - (cj2Se")[ SefrJ + J~(E)]

+ J2(E) + J3(E).

(2.18)

( 2 . 1 9 )

(2.20)

(2.21)

(2.22)

(2.23)

( 2 . 2 4 )

(2.25)


It is conven ien t now to recast eqs. (2.18) and (2.19) more compac t ly as fol lows:

gs,~,,(E) = g°(E)[6~, r - ~ . J'~,s,(E)g~,~.(E) $'

- ~. J~(E)g,,.~.(E)]. (2.26) n

We remind ourse lves that the subindex s refers to any a tom in the canonical cluster, as such J'~,s,(E) takes the values:

t J'I(E), if s = i and s ' = a, or vice versa P

J'~,~,(E)= J~(E), if s = a and s ' = / 3 # - a , or vice versa (2.27)

~J;(E)=O if s = a and s ' = - a , or vice versa.

Also that e~ J].,(E) in (2.26) and J~,(E) in (2.27) are given as in eqs. (2.20)-(2.22). The last term in eq. (2.26) connec t s an s-a tom in the cluster with the effect ive

med ium outs ide the cluster. TTK4), showed that these paths which involve trips outs ide the cluster can be r e sumed in such a manner that our Green ' s funct ion equa t ion can be rewri t ten in a fo rm which is " se l f - c losed" over the canonical cluster, i.e.

g~.~.( E) = g°(E)l~s.~,,- ~ [J~.~,(E) + H~.~,(E)] g~, ~,,( E) t. (2.28) k $, 3

Here H~.~,(E) represen ts the cont r ibut ion of all paths which leave the cluster. The funct ions H~.~,(E) are comple te ly de t e rmined by the effect ive medium,

which is isotropic and translat ional ly invariant. Taking into account the s y m m e t r y of the square lattice we can state that in its i r reducible form H~.~,(E) must have the fol lowing structure:

H s , s , ( E ) =

00(E), if s = s ' = i

0(E), if s = s ' = a

i(E), if s = i and

~(E), ff s = a and

H3(E), if s = a and

s ' = a, or vice versa

s ' = / 3 ~ - a , or vice versa

s ' = - a , or vice versa

(2.29)

Equa t ion (2.28) can now be wri t ten in terms of 5 × 5 matr ices , whose rows and co lumns are labelled by the central site i and its four neighbours dis t r ibuted in pairs of opposi te ly si tuated sites. We have, in this fashion, the matrix equa t ion

O = [ I + #o/~]-,g0, (2.30)

561) A.R. M c G U R N A N D R.A. T A H I R - K H E L I

where !

g,.:ce) &-2.( E

0 = / g l i ( E ) " "

\ g : , ~ ( E )

\ g 2,(E)

(2.31)

(Here dots denote the remaining matrix e lements , which are immedia te ly identified by inspection.)

~'(E) o o

0 ° 0 0 g_t(E) 0 o 0 0 g2(E)

0 0 0 g"~(E)] ,/

(2.32)

Hoo( E) ], ], { $£

II Ho(E) H3(E) -~ H= H4E) Ho(E) ~ . (2.33)

-~ --9 Ho(E) HffE) ] I

--~ --,, H fiE) Ho(E)]

Here a hor izontal arrow indicates that the re levant matrix e lement is equal to H2(E) + J~(E). A vertical arrow denotes a matrix e lement equal to Hi(E)+ J'I(E). Finally, gC](E) and g~(E) are as given in eqs. (2°23) and (2.24).

These matrices all have the general form

moo /T/I t?/1 iF/1

/TI 1 m o m-~ m 2 m 2

wl~ ~- t?/1 t'Y/3 tFI. 0 YE/2 rn 2 (2.34)

m l m : rn 2 m 0 m 3

ml m2 m2 m3 ml

It is clear that like the three-dimensional case, all these matrices can be block diagonal ized by an appropr ia te unitary t ransformat ion . The re levant trans- fo rmat ion matrix turns out to be

/1 o o o o \ ' 1/x/2 o 0 2

O = 0 ~ - l / x / 2 0 - (2.35)

o ~ o 1/~/2

o ~ o - 1 / ~ / 2 - { [ /

RANDOMLY DILUTED TWO-DIMENSIONAL FERROMAGNETS

Thus, we obtain from eqs. (2.34) and (2.35) the result

0 0 0 m,,

where fi, is a 2 x 2 matrix

m4 = m. + m3 + 2mZ, m,=m,-m3,

Transforming the Green’s function matrix equation given in eq. (2.30) under

the unitary transform 0, i.e. as in eq ,. (2.36), one obtains the following resuh

md= mo+ m3-2m2.

(2.36)

(2.37)

(2.38)

for the s, p and d symmetry elements of &:

g, = [(ii-’ + PHJ’~, (2.39)

JC, = [A,‘(E) + Z,(E) + H,(E)]-‘, 77 = p, d. (2.40)

Here we have used the notation:

(h))’ = + 2(&(E) - S’“J)] 0

do’(E) + X,(E)

and

p= ci O.

i 1 0 1

(2.41)

(2.42)

The transform of the H-matrix, eq. (2.33), can be similarly performed. We obtain, for the s, p and d symmetry elements,

H = K&W s

2[J:(E) + H,(E)] ,

&a(E) = ME) - H,(E),

H,(E) = Ho(E) + H,(E) - 2[H,(E) + J;(E)], (2.43)

H,(E) = H,(E) + H,(E) + 2[H,(E) -I- J;(E)]. (2.44)

Next we repeat the above procedure for the effective medium in order to express the matrix-H in terms of the effective medium Green’s functions.

562 A.R . M c G U R N A N D R.A. T A H I R - K H E H

Using eq. (2.4), the equat ions of motion for ¢~ g~.~,,(E) can be written:

. . . ~ ' L ~, ~ " k / (2.45)

The coefficients H~..,r(E) in eq. (2.45) are the same as those in eq. (2.28) as both represen t the contr ibut ion of effective medium paths outside the cluster.

Again, the s y m m e t r y of the lattice allows us to block diagonalize the Green ' s funct ion equat ions to obtain

0 ~ = [z~,;'(E) ~ + H ~ ] ',

g,CF = [AoJ(E) + H,~(E)] -,,

Here

p l y ' = (Hoo(E)

\2[J , (E) + H , (E) ]

and

r/~- p ,d .

2[J , (E) + H, (E) ] )

H~ fr = Ho(E) - H3(E) - J3(E),

H f t = Ho(E) + H3(E) + J 3 ( E ) - 2[H2(E) + J2(E)],

H~ fr = H~ fr + 4[H2(E) + J2(E)].

(2.46)

(2.47)

(2 .48)

(2.49)

By compar ing eqs. (2.43) and eq. (2.44) with eqs. (2.48) and (2.49), and

\ [ - c i (J l (E) - S~rJ)]

[~ l (E) -- J2(E) - J3(E)]

(2.56)

taking into account the definitions of J ' I(E) and J~(E) given in eqs. (2.21) and (2.22), we can write

• = , = (gs) - - A o l ( E ) 1 - - - d s , (2.50)

H , ( E ) = H~fl(E) + J3(E) = (g,~r) . - Ao-'(E) + J f fE ) , (2.51)

Ha(E) = H f e ( E ) + J2(E) - J3(E)

= ( G % ' - a,, ' ( E ) + J2(E) - J3(E), (2.52) where

: It' s°"Jl] J d E ) - S~frJ] J d E ) + J 3 ( E ) / " (2.53)

In t roducing eqs. (2.50)-(2.53) into eqs. (2.39) and (2.40) we obtain - - -ef t -1

O,~=[~ ' ,+P(g~ ) ] ~,6 (2.54)

and

g, = [ V n + ( g l n ) ' ]- ' , n - - p , d . (2.55)

Here

( [ ( S el~ - - c i ) A o l ( E ) + 2S"n(JI(E) S~fr J ) ]

P~ = \ [ - ( J , ( E ) - S°"J)]

and

RANDOMLY D I L U T E D TWO-DIMENSIONAL FERROMAGNETS 563

V o = J I ( E ) - (c , /2S~fr)[J , (E) + s*ffJ] + J2(E) + 2J3(E),

Vo = J , ( E ) - ( c , [ 2 S ~ ) [ J l ( E ) + S~rJ] + 2J2(E ).

(2.57)

(2.58)

The se l f -cons is tency relat ions, which define our CPA, are now obta ined by averaging eqs. (2.54) and (2.55) over both spin occupanc ies of the random site, i.

3. CPA equations

In the randomly diluted magnet with magnet ic concen t ra t ion x, any site is occupied by a magnet ic spin, c; = 1, with probabi l i ty x and consequen t ly is vacant , i.e. c,.--0, with probabi l i ty ( 1 - x). We assume that r andomness has been quenched in at infinite t empera ture and that no corre la t ion of site occupancy occurs. Averaging eq. (2.55) over the occupanc ies of the central site we obtain:

g~ = x Z ~ ( E ) - - - [Jx + J , (E)] + ( g ) - ~ x

+ (1 - x ) [ Z , ( E ) + (g~")-']-', r/ = p, d. (3.1)

Here we have used the notat ion:

Z p ( E ) = J , ( E ) + J2(E) + 2J3(E),

Z d ( E ) = J1(E) + 2J2(E), (3.2)

and we have used the fact that at zero tempera ture S ~tr= x. Similarly, using eq. (2.54), we get

~ f = x [I), + (g~) - ' ] - ' + (1 - x)[l¢o + ~6o(~n) ']-lPo, (3.3)

where

= + 0 )t 17',, \ - ( J , ( E ) - x J ) J , ( E ~ ' (3.5a)

(3.5b) 2B1 B4 j '

Boo -- - (1 - x)Aol(E) + 2 x [ J l ( E ) - x J] ,

B, = - ½[J,(E) - x J ] ,

0.5 B4 : JI -- - - [J ,(E) + x J] . (3.6)

x

564 A.R. McGURN AND R.A. TAHIR-KHELi

One should notice that as in TTK, here we find ourse lves with five re la t ionships in three unknowns . Two of the equat ions are provided by the condi t ion of equal i ty of the p and d s y m m e t r y e lements , the remaining by the same condi t ion on the e lements of the s - symmet ry matrix. We c i r cumven t this difficulty in the same manne r as was done by TTK. Rather than requir ing the equal i ty of all the matrix e lements in the matr ix equat ion for the s - symmet ry , i.e. in eq. (3.3), we use a weaker requ i rement ; namely that the t races of the left- and the r ight-hand sides be equal. This yields a single re la t ionship for the s - symmet ry case, thus making a total of three se l f -consis tent equat ions to be solved in terms of three unknowns . [Note, however , that both the equat ions and the u n k n o w n s are complex.]

We shall consider the g f se l f -cons i s tency relat ion first. Us ing the s y m m e t r y t r ans fo rmat ion given in eqs. (2.36)-(2.38) we find that in the effect ive medium

O~n= (ro 2 r j )

2rl rs / '

1 ro = N ~ ~,~(E),

1 ~' = N ~ ~/K~-K(E),

Our

4 :~ = N ~ (~/K/rK(E),

r ~ ( E ) = [E - ZK(E)] ',

Z~ (E) = 4J~(E))'K(y~ - 1) + 2Zp(l - Y2K) + 2Zd(l + T2K -- 23'2) -

(3.7)

s - symmet ry se l f -cons is tency re la t ionship thus becomes:

A I = Boo + "r~/'rs,

A2 = 2 ( B i - 'r J ' r0,

As = B4 + ro/"rs,

A 4 = "rs[Jl(E)rs + to] l,

r 5 = %% - 4(%) 2.

After some algebra, this leads to the fol lowing:

ro + % = [ x l Q ( E ) ] [ % + % + ( B 4 + Boo)rs]

+ (1 - x ) r s [ % + "rsJl(E)] -1,

where

Q ( E ) = 1 + 8 B i r l + Booro + B4"rs + rs(BooB4 - 4B 12).

(3.8)

(3.9)

(3.10)

(3.1 la)

(3.11 b)


The p and d se l f -cons i s tency re la t ionships , i.e. eq. (3.1), can be similarly recas t into a compac t form, i.e. for r / = p, d

Z~(E) = ~[Jt(E) + x J]

[ o, ] - Z , ~ ( E ) - - - ( J , ( E ) + x J ) rnZ,~(E). (3.12)

X

Here the quant i t ies % and za are the p and d s y m m e t r y Green ' s func t ions , i.e.

= ~ - (1 - Y2K)rK(E), (3.13)

2 ~'a = g ~ = ~ ~ (1 + Y2K -- 2y~:)~'r(E). (3.14)

Eqs. (3.11a)-(3.14) give us three se l f -cons is ten t equat ions to de te rmine the

u n k n o w n s JI(E), Z p ( E ) and Za(E). We have now replaced the r andom sys tem given by eq. (2.1) by an effect ive

medium descr ibed by eq. (2.2). In line with the definit ion of the effect ive medium Green ' s func t ions in eq. (2.4), we see that the magnet ic r esponse funct ion, px(~O), is given by the express ion

1 p(K, ~o) - Im [Se~rr(o~ + ie)] (3.15)

~-,+0 "IT

[recall that at T - - 0 , S e~= x] and the dens i ty of s tates by

1 1 #(oJ) = -~ ~ p(K, ~o) = - - - I m [xr0(~o + iE)] . ( 3 . 1 6 )

e--*+0

[Refer to eq. (3.8) for the definit ion of "rK(E) and "r0(E).] It is clear that at x = 1, there are no vacancies and these resul ts must

therefore reduce to the usual ones appl icable to a un i form He i senbe rg f e r romagne t with isotropic, neares t ne ighbour exchange J. For this case, eqs. (3.11a,b) and (3.12) readi ly yield the correc t solut ion, i.e.

Jr(E) = Zp(E) = Zd(E) = 3..

Le t us now a t tempt to solve eqs. (3.1 la,b) and (3.12) in the small di lut ion limit, i.e. when (1 - x) ~ 1.

To terms l inear in ( 1 - x), af ter some tedious algebra, we find

JI(E) = J + A1(E) + ¢7(1 - x) 2 (3.17)

Z , ( E ) = J + I[A,(E) + (1 - x)J]

- ( 1 - x ) J [ I + J F o ( E ) ] -~, r t - - p , d , (3.18) where

zl~(E) = 8j2(1 - x)Fo/[1 - EFo]

+ (1 - x )J + (7(1 - x) 2. (3.19)


Here we have, for conven ience , used the nota t ion

1 Fo = F K ( E ) ,

1 Fp(E) = N ~ (1 - "/2r)Fr(E),

2 Fa(E) = ~- ~ (1 + YEt - 2YZr)Fr(E),

Fr(E) = [E - 4 J ( l - W)] i. (3.20)

With these small di lution results , it is possible to calculate the st iffness to terms l inear in the di lut ion and compare it with the cor responding resul ts given by Wat son et al.S), for two-d imens iona l resis tor ne tworks . The spin- wave st i ffness is defined as fol lows:

Do(x) lim lim 1_4~1 - ~K)-I' D0(1) r ~ E=-~K

Using eqs. (3.17)-(3.19) we find that to order l inear in ( 1 - x)

(3.21)

Do(x) [I - JFo(E = 0 ) ] - l - ( l - x ) + ~ ? ( 1 - x ) 2,

Do(1) + JFo( E - 1 - A s o ( 1 - x ) + C ( l - x ) 2. (3.22)

Watson et al. 's ~) result is Aso = ( ~ r - 1). If we use the exact express ion for JF~,(E = 0) avai lable f rom B e r n a s c o n i - W i e s m a n ' s workl°), we find that our resul t (3.22) is identical to the exact low concen t ra t ion result for the st iffness given in ref. 8.

4. R e s u l t s and c o n c l u s i o n s

We have carr ied out a numerica l solut ion of eqs. (3.11a)-(3.12) for J~(E), Zp(E) and Zd(E) for magnet ic concen t ra t ions x = 0.9, 0.8 and 0.7. For computa t ional conven ience , all ca lcula t ions were done for energies sl ightly off the real axis, i.e. for E = o~ +i0.04, in units such that the exchange coupl ing cons tan t was equal to uni ty or where , in the non- random fer romagnet , the total bandwid th of the spin-wave exci ta t ions is 8. In comput ing the Green ' s func t ions , which eqs. (3.11a) and (3.12) involve, we used 7,396 points in the Bril louin zone. This represen ts a much larger l inear dens i ty of points than used by TTK. Such greater care was necess i ta ted here because the Green ' s func t ions in two d imens ions , unlike in three, are quite singular in nature.


4.1. Density of states

In fig. ! resul ts for the dens i ty of s ta tes are p resen ted for magnet ic concen t ra t ions x = 0.7, 0.8, 0.9. For compar i son , we have also inc luded the results for the pure case, i.e. for x = 1.0. These results , we has ten to point out, are not exac t ly equal to the dens i ty of s tates , s ince they are ca lcula ted at E -~ ~ + i~, with ~ = 0.04 ra ther than for ~ ~ + 0. Yet, they can be expec ted to be very represen ta t ive of the actual dens i ty of s tates with the typical diff- e rence that sharp peaks will appear more rounded when e > 0 and, moreover , dens i ty of s tates will leak outside the sp in-wave band. Because T T K used E = 0.06, our present E bears the same ratio, i.e. 1 part in 200, to the spin-wave band width as that of TTK.

We see that as the concen t ra t ion of magnet ic sites is dec reased the dens i ty of s ta tes shifts to lower f requenc ies indicat ing the loss of magnet ic st iffness in the lattice. For concen t ra t ions less than x = 0.9, a second shoulder develops near ~o = 0.2.

In fig. 2 we have replot ted the dens i ty of s tates and compared them to compute r expe r imen t resul ts for a 30 x 30 latt ice of spins recen t ly given by HuberT). [It should be ment ioned that Hube r ' s resul ts or iginal ly inc luded zero energy modes f rom vacan t sites. We have es t imated the cont r ibut ion of these modes and have made a correc t ion for them in Hube r ' s resul ts given here.] As is apparen t f rom the figure, good ag reemen t be tween our resul ts and these is

O. :!

T

I

i.O 2.0 3.0 ~,.0 5.0 5.0 7C B.O 9 0

W

Fig. 1. The s p i n - w a v e d e n s i t y o f s t a t e s , eq. (3.16), fo r the s i te d i lu t ed f e r r o m a g n e t on a squa re la t t ice . C u r v e s fo r m a g n e t i c c o n c e n t r a t i o n s x ~-0.7, 0.8, 0.9 and 1.0 are shown . The d e n s i t y o f

s t a t e s is m e a s u r e d off the rea l ax i s at E = ~o + i0.04.

568 A.R. M c G U R N A N D R.A. T A H I R - K H E L t

• . w L,O * ~ 0 . L 2 3

× .: O.B

- - - . ~ t,.G 8.0

0.1

0 - • . • t,.0 LB

Fig. 2. The dens i ty of states of fig. 1 for x = 0.7, 0.8 and 0.9 compared with c o m p u t e r expe r imen t

resul ts of H u b e r for a 3 0 x 30 spin sys tem. H u b e r ' s resul ts , given as h is tograms, have been ad jus ted for the vacancy r e sponse at w = 0.

obta ined only for co > 0.03. Near zero f requency , compute r exper iment results lie s o m e w h a t above ours for x = 0.8 and 0.7. The same happens , though to a lesser extent , for energies near the cent re of the band, i,e. for o J - 4, where the curve for the pure sys tem has a peak. (It is actually d ivergent at this point, if we take the limit ~ ~ + 0). Howeve r , in view of the previously stated fact that we are looking at energies off the real axis, with ~ --0.04, our curves are not exact ly equal to the true densi ty of states. Rather , we expec t that both at the cent re and at small energies where the curves show peaks, the effects of non-zero e will be to round off these peaks to somewha t lower heights. Indeed, we have invest igated this point numerical ly by slightly decreas ing E and have found a resul tant increase in the heights of our curves (at the centre, for x = 0.9 and 0.8, and at the small f r equency shoulder , for x = 0.8 and 0.7). In this fashion, we have conc luded that the overall ag reemen t of our results with compute r expe r imen t predic t ions is quite sat isfactory.


4.2. Response

In figs. 3-6 we have plotted the frequency-wave-vector-dependent response as a function of the frequency at various concentrations, for several wave-vectors along the (1, 1) diagonal, as one proceeds from the origin to a

Fig. 3. r times the frequency-wave-vector response function, eq. (3.151, of the diluted system for

K = (k, k) with k = ?r/4. Curves for concentrations x = 0,7,0.8 and 0.9 are shown. The response is

measured off the real axis in the same manner as the density of states given in fig. 1.

i 8.C

Fig. 4. Same as in fig. 3 with k = ?r/2.

570 A.R. M c G U R N A N D R.A. T A H I R - K H E L I

1.0 2_.0

× : 0.9

K : ( 3 ~ t t , 3n;.t/, 1

i • j _ _ ~

3,(1 ~..0 5.0 E0 7.0 E0

Fig. 5. S a m e as in fig. 4 wi th k = 3'rr/4.

5.D

~" 3.0

2.0

1,13

x =[3,9

l.O 2.0 3.0 i,.O 5.0 6.0 7.0 8.0

Fig. 6. S a m e as in fig. 5 w i th k = 7r.


corner of the Bril louin zone. Our curves are for K = (k, k) with k = n~'[4 where n = 1, 2, 3, 4 and the magnet ic concen t ra t ions are again equal to 0.7, 0.8, 0.9. For each k value, we see that decreas ing the magnet ic concen t ra t ion lowers the f r e q u e n c y of the sp in-wave like peak and causes the width of the " m o d e " to increase , the reby making the mode less well defined.

In figs. 7 and 8 the resul ts for K = (7r/2, 0) and K = (~',0) are shown respect ive ly . Again we see the peaks shif t to lower f requenc ies and the curves broaden, as r andomness is increased. As before these resul ts too have been obta ined with non-zero e, i.e. for E = to + i0.04. Therefore , s t r ict ly speaking, they are not exac t ly equal to the true response funct ions . Yet , we expec t that the response in the p resen t figures would appear as though they had been averaged over f r e q u e n c y intervals of the order of the imaginary part , E. Thus, away f rom regions of rapid var ia t ion, the curves would be represen ta t ive of the true response funct ion. Therefore , as s ta ted earlier, in the region where the actual response should peak, we can expec t addi t ional b roadening and some decrease in the height of the peak.

3

T

3.0

2.0

x -- 0.9

0.8 \

K = ( ~ / 2 , 0 I

0 1.0 2.0 3.0 ~.0

1- W

Fig. 7. Same as in fig. 3 with the difference that here K = (k, 0) with k = 7rt2.


2.0 x =0.9 1

K _- (TL 0 ]

O.B

l.O 2.0 3.0 ~..0 5.0 6.0 "- CO

Fig. 8. Same as in fig. 7 with k = ~r.

4.3. Spin-wave stiffness

In fig. 9 we present the curve for the real part of wha t for finite E is a complex stiffness. These results were calculated for a somewha t smaller ¢ than the others , i.e. here E - - 0 + i0.005. The actual resul ts for the stiffness should not differ apprec iably f rom that given in this curve except when the st iffness is small compared to unity. Therefore , for stiffness values near unity, we expec t this computa t ion to be accurate . Indeed for ( 1 - x)<~ 1, our computed resul ts agree with those of Wat son for the two-dimens ional resis tor networkS). Because for larger impuri ty concent ra t ions , i.e. when ( 1 - x) is no longer much smaller than uni ty , the computa t iona l procedure for the st iffness coefficient requires comput ing rat ios of Green ' s funct ions , which are in- d ividual ly very large, our computa t ions are not too reliable for x ~<0.7. None the les s , within the present C P A , we do not expect to find a good value of the critical concen t ra t ion , which is expec ted to be - 0 . 5 9 [ref.9)]. Our reasons for this pess imis t ic view are rooted in our knowledge of the T T K resul ts for the st i ffness (in th ree-d imens iona l sys tems , where cor responding computa t iona l difficulties do not arise), which underes t ima te the critical concen t ra t ion substant ia l ly . Thus, like the T T K analys is , we expec t the present calcula t ion to be reliable down only to in termedia te concent ra t ions , which for the present two-dimens iona l case would seem to be x > 0.7.

The compute r exper imen t s of Huber , ment ioned above, were not originally

573

00

0.2

06 u m

z

L , _

0t, /

/ /

/

RANDOMLY DILUTED TWO-DIMENSIONAL FERROMAGNETS

PERCOLAIION CONCENTRATION

0 1 i I I I d ...... 1 I 1 I

0,1 0.2 0.3 O.t, 05 0.0 0.7 0.0 0,9 1.0

X

Fig. 9. Real part of the complex spin-wave stiffness, computed for E = 0 + 0.005i, versus magnetic concentration x. For concentrations below x = 0.7, both the numerical computation and the validity of the present theory are in doubt.

designed to compute spin-wave response function, nor has the response yet been computed. As such, the present theory provides useful insight into the structure of the spin-wave response for two dimensional systems.

Acknowledgements

We are most grateful to Professor David L. Huber for allowing us the use of his computer experiment data on the site random ferromagnet. Also, we thank the Computer Center at Temple University for the use of their facilities.

References

1) A.B. Harris, P . L Leath, B.G. Nickel and R.J. Elliot, J. Phys. C7 (1974) 1693. 2) A. Theumann, J. Phys. C6 (1973) 2822; C7 (1974) 2328. 3) R.A. Tahir-Kheli, Phys. Rev. B6 (1972) 2802. 4) A. Theumann and R.A. Tahir-Kheli, Phys. Rev. B12 (1975) 17%. 5) W.J.L. Buyers, D.E. Pepper and R.J. Elliot, J. Phys. C5 (1972) 2611. 6) P. Soven, Phys. Rev. 156 (1%7) 809.

N.F. Berk and R.A. Tahir-Kheli, Physica 67 (1973) 501. 7) D.L. Huber, Phys. Rev. B10 (1974) 4621. 8) B.P. Watson and P.L. Leath, Phys. Rev. B9 (1974) 4893. 9) V.K.S. Shante and S. Kirkpatrick, Advances in Physics 20 (1971) 325.

10) J. Bernasconi and H.J. Wiesman, Phys. Rev. BI3 (1976) 1131.

excitations in randomly diluted two-dimensional ferromagnets

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