excitations in heisenberg spin glasses

9
Z. PhysikB-CondensedMatter 38, 243-251 (1980) Condensed Zeitschrift Matter for Physik B by Springer-Verlag 1980 Excitations in Heisenberg Spin Glasses I. The Eu=Sr I _ xS-System U. Krey Fakult~it f~r Physik der Universitgt Regensburg, Regensburg, Fed. Rep. Germany Received January 16, 1980 Within the RPA approach for T = 0, the excitations of the Heisenberg spin glass system EuxSrl_x S are studied by numerical methods, using a continued fraction algorithm. Both the density of states g(E) and also the spectral function S(q, E) are calculated for systems with (16)3 sites, with x=0.4, 0.5, and 0.6 (spin glass phase), and also for x>0.7 (ferromagnetic phase). For q-vectors within the (1, 1, 1) plane, S(q, E) shows magnon peaks even in the spin glass phase, over the whole range of q. However, these peaks are quite broad, and there is considerable intensity at small energies even for large q, leading to a finite intercept of g(E) for E~0. Over a large temperature range, the specific heat is approximately linear in T for x < 0.7. 1. Introduction The EuxSrl_,S-system is presently studied very in- tensively [1-13]. Due to a competition between fer- romagnetic nearest neighbour exchange (J1/kB~-0.21 K, see [14]) and antiferromagnetic next- nearest neighbour exchange (J2/kB ~ -- 0.1 l K, see [14]) this system shows unique properties: Below the percolation concentration %~0.13 for fcc systems with nearest and next-nearest neighbour interaction, it is paramagnetic, however with strong magnetic clustering effects [5, 6]; on the other hand, for x > x: (=0.65) it is ferromagnetic. Finally, and most in- terestingly, at intermediate values of x there is a spin glass phase. For T=0, this phase ranges just from x 1 ~0.13 to x2~0.65; however with increasing tem- perature xl becomes larger and x 2 smaller, until finally at T~2.5K all three phases meet. The cor- responding value of x is x~0.5. A complete phase diagram can be found in [12]. In the present paper the case T = 0 is considered. For x=0.4, 0.5, and 0.6 (spin glass phase), and ad- ditionally for x =0.7, 0.8, 0.9 and 1.0 (ferromagnetic phase), the excitation spectrum of the spin system is considered. Both the density of states g(E) and also the spectral function S(q,E) of the excitations are calculated. From our knowledge of g(E) we have calculated the temperature dependence of the specific heat C~(T), which has recently been measured by Meschede et al. I-7] for x=0.4 and x=0.54. On the other hand, the knowledge of S(q, E) is useful for an interpretation of preliminary inelastic neutron scat- tering results [15]. The present calculations predict for the spin glass phase a proportionality of Cn with T at low temperatures, and also the appearance of broad magnon peaks in S(q, E) in agreement with the experiments [7, 15]. Within the calculation, the conventional linearized semiclassical approach has been used, which is also known as Random Phase approximation (RPA). This means that the excitations are treated as noninteract- ing boson quasiparticles. Otherwise the calculation is essentially exact, corresponding to a computer simu- lation for a very large system of 16 x 16 x 16 lattice sites, which are randomly occupied by the magnetic Eu and nonmagnetic Sr sites, according to the value of x considered [16]. Additionally, periodic boundary conditions have been used. For the spin glass cases, x =0.4, 0.5, and 0.6, the ground state has been pre- pared with the conventional Monte Carlo Method by Kinzel [-17]. Then for the calculation of the exci- tations a continued fraction algorithm has been used. This algorithm, which originally had been devised for surface properties of ideal crystals [18], is extremely 0340-224X/80/0038/0243/$01.80

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Page 1: Excitations in Heisenberg spin glasses

Z. Physik B-Condensed Matter 38, 243-251 (1980) Condensed Zeitschrift Matter for Physik B

�9 by Springer-Verlag 1980

Excitations in Heisenberg Spin Glasses

I. The Eu=Sr I _ xS-Sys tem

U. Krey

Fakult~it f~r Physik der Universitgt Regensburg, Regensburg, Fed. Rep. Germany

Received January 16, 1980

Within the RPA approach for T = 0, the excitations of the Heisenberg spin glass system EuxSr l_x S are studied by numerical methods, using a continued fraction algorithm. Both the density of states g(E) and also the spectral function S(q, E) are calculated for systems with (16) 3 sites, with x=0.4, 0.5, and 0.6 (spin glass phase), and also for x>0.7 (ferromagnetic phase). For q-vectors within the (1, 1, 1) plane, S(q, E) shows magnon peaks even in the spin glass phase, over the whole range of q. However, these peaks are quite broad, and there is considerable intensity at small energies even for large q, leading to a finite intercept of g(E) for E~0 . Over a large temperature range, the specific heat is approximately linear in T for x < 0.7.

1. Introduction

The EuxSrl_,S-system is presently studied very in- tensively [1-13]. Due to a competition between fer- romagnetic nearest neighbour exchange (J1/kB~-0.21 K, see [14]) and antiferromagnetic next- nearest neighbour exchange (J2/kB ~ -- 0.1 l K, see [14]) this system shows unique properties: Below the percolation concentration %~0.13 for fcc systems with nearest and next-nearest neighbour interaction, it is paramagnetic, however with strong magnetic clustering effects [5, 6]; on the other hand, for x > x: (=0.65) it is ferromagnetic. Finally, and most in- terestingly, at intermediate values of x there is a spin glass phase. For T=0 , this phase ranges just from x 1 ~0.13 to x2~0.65; however with increasing tem- perature xl becomes larger and x 2 smaller, until finally at T ~ 2 . 5 K all three phases meet. The cor- responding value of x is x~0.5. A complete phase diagram can be found in [12]. In the present paper the case T = 0 is considered. For x=0.4, 0.5, and 0.6 (spin glass phase), and ad- ditionally for x =0.7, 0.8, 0.9 and 1.0 (ferromagnetic phase), the excitation spectrum of the spin system is considered. Both the density of states g(E) and also the spectral function S(q,E) of the excitations are calculated. From our knowledge of g(E) we have calculated the temperature dependence of the specific

heat C~(T), which has recently been measured by Meschede et al. I-7] for x=0.4 and x=0.54. On the other hand, the knowledge of S(q, E) is useful for an interpretation of preliminary inelastic neutron scat- tering results [15]. The present calculations predict for the spin glass phase a proportionality of Cn with T at low temperatures, and also the appearance of broad magnon peaks in S(q, E) in agreement with the experiments [7, 15]. Within the calculation, the conventional linearized semiclassical approach has been used, which is also known as Random Phase approximation (RPA). This means that the excitations are treated as noninteract- ing boson quasiparticles. Otherwise the calculation is essentially exact, corresponding to a computer simu- lation for a very large system of 16 x 16 x 16 lattice sites, which are randomly occupied by the magnetic Eu and nonmagnetic Sr sites, according to the value of x considered [16]. Additionally, periodic boundary conditions have been used. For the spin glass cases, x =0.4, 0.5, and 0.6, the ground state has been pre- pared with the conventional Monte Carlo Method by Kinzel [-17]. Then for the calculation of the exci- tations a continued fraction algorithm has been used. This algorithm, which originally had been devised for surface properties of ideal crystals [18], is extremely

0340-224X/80/0038/0243/$01.80

Page 2: Excitations in Heisenberg spin glasses

244 u. Krey: Excitations in Heisenberg Spin Glasses. I.

flexible, and its particular power for strongly disor- dered systems has already been proven in a number of recent papers of the present author [-19-21]. But in the present context, a somewhat generalized algo- rithm has to be used since now the equations of motions lead to nonhermitian matrices, which was not the case in [19-21]. However, although the gen- eralized formalism needs twice as much storage, the calculations could have been performed without diffi- culty for 4 times as many sites, if the corresponding ground states would have been available. For x > 0.7 it has been found in [9] that the ground state deviates only slightly from the completely aligned state. Therefore in this case, the completely aligned state has been used as the ground state for the present calculation. Although this is certainly an approximation, it did not lead to difficulties or incon- sistencies in the results. However, in view of the high nontrivial ground state degeneracy of a spin glass, and due to the presence of metastable states which would call for a nonlinear theory, the validity of the RPA theory remains doubtful at least on a large time scale. In particular, concerning the time scales of an inelastic neutron scattering experiment performed on a triple axis spec- trometer (<10 - n s), the present results may apply, whereas on the larger time scales obtained by the neutron spin echo method (<10-Ss) they may perhaps not, see [22-24]. The arrangement of the paper is as follows: In Chap. 2 the basic equations for the different quantities, Green's functions etc., which are needed in our calcu- lations, are derived. Chapter 3 deals with the numeri- cal methods, while finally in Chap. 4 the results are presented. The paper closes with the conclusions, given in Chap. 5.

2. Basic Equations

In the Eu~Sh_~S systems the Eu and Sr ions occupy a face centered cubic lattice; Sr 2+ is nonmagnetic, whereas the Eu 2+ ions have vanishing orbital mo- mentum and spin quantum number s = 7/2. The mag- netic properties of the systems in the spin-glass and ferromagnetic regimes can be approximately de- scribed by the Heisenberg Hamiltonian

H=-Z4ms,.s~. (1) l,m

Here the l, m denumerate the Eu sites only; Jz,, van- ishes, unless the pair of sites corresponds to the nearest neighbour or next nearest neighbour distance on the fcc lattice. In the nearest neighbour case one has J~m=J1, with Ja/kB~-O.2K, whereas for next

nearest neighbours Jzm=J2, with J2/kB~--OAK (k~ =Boltzmann's constant), see [14]. Here a possible dependence of J1 and J2 on the concentration x is neglected, although such a dependence has been ob- served, see [9, 10]. Furthermore, also the magnetic dipolar interactions are neglected, which is certainly not justified for spin waves in the ferromagnetic phase, in view of the low values of J1 and "/2 and the large value of s. However in the spin glass phase around x~0.5, the dipole interaction is probably much less important, since it does no longer "pile up" as in the ferromagnet, but instead should be largely compensated, as within the antiferromagnet. In fact, there is excellent agreement of recent Monte Carlo calculations [9], based on assumptions identi- cal to those of the present paper, with the experimen- tal phase diagram of [12]. From Eq. (1), in the Heisenberg picture, one obtains the following equations of motions for the spin oper- ators

h dS, 2 ~ 4,.S, x S~. (2) t iT -= l,m

In the RPA approximation this equation is linearized in the following way: One starts from the classical equilibrium configuration as determined e.g. in the Monte Carlo calculation [9]. Then one introduces an arbitrary set of local Cartesean coordinate frames s Y~, ~z, where however the unit vectors Sz must cor- respond to the classical equilibrium spin directions at the site 1. The equations (2) are then linearized with respect to the two transverse components ST, where = 1 refers to 2 z and c~=2 to ~gz. This leads to equa- tions of motion of the form

(ha, a --d S =O. ~ dt tm] (3)

Here 6~m=l if l=m, 6~m=0 otherwise, and double indexes are summed over./t~'~ is a nonhermitian matrix with the imaginary eigenvalues +_iE~, where the E v are the excitation energies of the system. The ele- merits o f / t are

H,,~-x' _- 2 s {Su. ' h , - J,,. 39,. 39m},

/t~'2 = -- 2S. {6,,.. h,-- Jz= x, ' < } .

H,,." x~ = - 2sd~m " {(y,'* x,.),*

H,~ ~*y - - 2 sd,,.09=. 2,),

(4a)

(4b)

(4c)

(4d)

where

P (5)

Page 3: Excitations in Heisenberg spin glasses

U. Krey: Excitations in Heisenberg Spin Glasses. I. 245

In the following, without loss of generality, only the case s = 1/2 is considered. The resolvent operator corresponding to (3) is

d~m(Z) = { ( i z - IZI)- 1~,l,," (6a)

It can be calculated from the equations

(iz 6tm -- ~p ~~' ~ -- 5~ H~,,)G,~,(~)-5~,5~. (6b)

Here z = E + ie, where e is a positive infinitesimal. The operator G is the basic quantity for our calculation. One could also use different resolvents in the present context e.g. that of Bhargava and Kumar 1-25], which

1 x is related to equations for the amplitudes a ~ : = - ~ (S~ v -

• In this case one would define a resolvent, where i. z is replaced by ( - 1) ~- z, while the redefined /4 becomes hermitian. However o u r / 4 has only real matrix elements, whereas for the redefined operator they would be complex. Therefore, to avoid unnec- essary computer storage problems, it is advantageous to use (6). In the following the relation between G and the interesting physical quantities is shortly described. At first, the local density of states

g~(E) = ~ I(L, ~)~)I ~ 6 ( E - ~ ) (7) v,c~

is considered. Here v denumerates the N different eigenstates Iv) of (3) with positive energies (those with negative energies must be excluded, as is known from antiferromagnets); N is the number of magnetic sites. Furthermore, c~ i s = x o r=y , that is (l, ctlv) corresponds to the S~-amplitude of the eigenstate, g~ can be calculated from G by means of the following equation:

g~(E)= 1 ~ [Re{(l,c~[(~(E+iE)[1,~)}l. (8) 7"C c ~ _ 1

Here Re means the real part, and the matrix element ~aft Glm has been written as (1,c~]GIm, fi). An equation

similar to (7) is obtained for the spectral function Si~(q, E), which is defined by

S"(q, E) = ~ I(q, ilv)t 2 5 ( E - E~). (9) v

Here ]q, i) means a state, which corresponds in global coordinates (not in the set of local frames !) to a plane wave polarized in the/-direction ( i=x , y, z). The local amplitudes S~ corresponding to such a plane wave state are easily calculated since the orientation of the local unit vectors 2z, Yz and 2~ with respect to the global frame is known. Actually, for the spin glass,

the result should not depend on i, therefore in the following we write simply S(q, E) instead of S~(q, E). (For the ferromagnetic case, instead of the present formalism we apply the simpler one as described in [19, 20], where the index i does not appear from the beginning.) The importance of the function S(q, E) is the follow- ing: For given q it is directly proportional to the inelastic scattering cross section for neutrons with wave number transfer q and energy loss E. The relation between S(q, E) and d is similar to (6) and (7), namely

S(q, E) = l~lRe {(q, il d(E + i8)[q, i)}l. (10)

In a way similar to (9) also the averaged density of states (g(E)): = N- 1 ~ gl(E) can be calculated. Name-

l

ly instead of averaging a large number of gz(E) functions, it suffices to perform one single calculation if one uses for {g(E)) a representation similar to (9), where however the plane wave state [q, i) is replaced by a random state IqS~), i.e. a state where the local

tS 1 $2~ vectors t t, z~ are chosen as N-1/2(cos(p~,sin(p~), where the phases q~ are chosen at random between 0 and 2~. For ferromagnets, this method has already been applied in [19].

3. Numerical Algorithm

3a) ResoIvent Operator and Continued Fraction

According to Eqs. (7), (9), and (11), for the calculation of g~(E), S(q, E) and (g(E)) it is necessary to calculate matrix elements {~bldt0) of the resolvent G. Within the algorithm described below this is no problem as long as the states ]q~) and 10) are not orthogonal. The essential point of the numerical algorithm is to find a biorthogonal basis set of pairs of states lq~,), 10,) (n=1,2,3, ...), where ]@1>=]~>, and 10~>=10>, and which at the same time tridiagonalizes ITI. Here bi-orthogonality means that

<05,,] ~,,,) =5,, m (11)

and the "basis" property means completeness:

10,)" (q~,[ = 1. (12) n

Finally, tridiagonalization means a representation of /~ as follows

Ft= ~ I0.) (r (q).l, (13) n,m

Page 4: Excitations in Heisenberg spin glasses

246 U. Krey: Excitations in Heisenberg Spin Glasses. L

where the matrix (~b,]/~[O,.) has the special tri- diagonal form

0 , . . . !i) (~b,i/~l~,m) = [Ibll , a2, b2, 0, ~ 0 : Ib2l, a3, b3, "

, " .

(14)

Here the quantities a, and b, can be calculated together with the l~b,), l~,) by a simple and fast algorithm, see Nex [27]: One starts with the states I~bl):=14) and I t ) l ) :=10). Introducing the dummy auxiliary quantities bo:=0, [4o).'=0, and 1~o):=0, one then performs the following recursion, starting with n = 1 :

a.: = (qS.I/~10.), (15a)

lu,+ 1): =(/~-a,,)[ 7J.) - b . _ 1' 11//_ 1), (15b)

Iv,+l):=(f t+-a,)14),)-Ib,- l l '[c~, 1), (15c)

b,. Ib,]: =(v ,+ 1 lu,+ ~), (15d)

I~.+ z> ' = lu.+ ~>/{b.t, (15e)

[qS+ 1) := Iv,+ t)/b,. (15t)

After performing (15t), taking the next value of n, one returns to (15a), and so on. For the desired matrix element (4)[ G(z)10) one then obtains the following continued fraction represen- tation:

<4 [ d(z)I0> = tl (z), (16 a)

where tl(z ) is calculated from the recurrence

1 t.(z) - (16b)

i z -a . -b . . Ib . .L . t .+l(z )"

Before discussing the evaluation of (16) let us note an important property of Eq. (3) which simplifies the calculation of (7). Namely, one can prove that simul- taneous interchange of all S~ and S~ components in the states [~) and I~b) leads to a simultaneous change of sign of all G, whereas the b~ remain unchanged. The proof is based on the arbitraryness of the direc- tion of 2 l (apart from the condition 2l-2z=0 ). In particular, in Eq. (3) one can reverse the ground state (2~-~ ' t :=-~) , and interchange the perpendicular vector ( 2 ~ ' t : =Yl; )~l~29't " =21)' If at the same time S~ and S~ z are interchanged, one obtains equations which are identical to (3) apart from a time reversal (H~--* -H~m ). This leads to a change of sign of the a~, whereas the b~ are unchanged. As a consequence, for the evaluation of Eq. (7), it is sufficient to calculate the a, and b~ for the If, 1) state,

since for the 1/,2) state only the a~ have to be replaced by -a, , . Actually, since in the evaluation of gt(E), S(q, E), and (g(E)) only the absolute values of real or imaginary parts are involved, and since in (16) the term iz=i(E+ia) becomes purely imaginary for e-~0, whereas all a~ and b, are real, one gets identical results both with the {a,.} and { - G} sets. This would not be the case, if instead of i z - G there would appear expressions of the form z - G in (16).

3 b) Termination of the Continued Fraction

The main problem with the evaluation of (16) is how to terminate the continued fraction. Namely, because of finite size effects e.g. in the evaluation of gz(E) only the first n pairs of coefficients G, b~, v=l . . .n are correct, where n corresponds to the number of hops necessary to reach the boundary when starting from a central site, see e.g. [18-20]. Therefore, we have performed the calculation only up to no=16 and terminate the calculation analytically in an anal- ogous way as described in [18-20]. Namely, it turned out in the calculation that the a~ are generally quite small, typically one order of magnitude or even less smaller than the b~, and fluctuating around zero. On the other hand, the b~ are negative, and in magnitude comparable to energies near the band edge. Therefore, the pairs (a,,,b,) with n>n o have been replaced by (a~,b~):=(O,b,o). According to (16b) it follows that t,o+~(z ) has the same value for every v = 1,2, 3,..., namely

1 t,,o+t(z)=2b~lbo~] {iz- l / ( iz)Z-4b| (17)

Although in the present case boo is <0, we have written down the general solution applying for both signs; with z = E + ie, one must choose that branch of the square root with positive imaginary part. Finally, since t~o+l is known from (17), one can solve (16) to obtain the desired expression for tt(z), and thus finally the results for g~(E), g(E), and S(q, E).

4. Results

4 a) Density of States and Specific Heat

In Fig. 1 the results for the density of states are presented, for the Eu-concentrations x = 0.4, 0.5, 0.6, ..., 1.0. With the exception of the pure case (x= 1), where a single-site state has been used as starting state of the recursion algorithm, all other results have been obtained by the "random wave" method, as

Page 5: Excitations in Heisenberg spin glasses

U. Krey: Excitations in Heisenberg Spin Glasses. I. 247

described above. For x=0.7, 0.8, 0.9 an aligned ground state has been assumed, which is reasonable according to the Monte Carlo results of [9], and an average has been performed over 10 different random wave states as starting states of the recursion (ac- tually, within the present accuracy this averaging turned out to be unnecessary). On the other hand, for x=0.6 , 0.5, and 0.4, in the spin glass regime, the calculation is based on the nonaligned ground states prepared in [9]. Here, for x=0.4, only one random wave has been used, whereas for x=0 .5 and 0.6 the results present an average from two random waves. The continued fractions have been terminated as described above, after n o = 10 (no=6 for x = 1), and the imaginary part e of the energy has been chosen as

= 10-3 do, where do =2s J1 is the natural energy unit (Jr = nearest neighbour interaction). As a general trend one can see from Fig. 1 that low energy states become more and more frequent with decreasing x. However, there is an important differ- ence of the low-energy behaviour of g(E) for the ferromagnetic and the spin glass cases, respectively. Namely, in the ferromagnetic case g(E) behaves o c ] ~ for E--+0, as it should according to the pres- ence of spin waves with energy E(q)ocq 2, whereas in the spin glass regime g(E) apparently extrapolates to a finite value go for E ~ 0 . Furthermore, go increases drastically with decreasing x, and for x=0 .4 one observes a steep rise of g(E) for E--,0. These results are confirmed by a recent calculation of Ching et al. [28], who evaluated g(E) for smaller samples by direct evaluation of the eigenvalues and ei- genstates for x = 0.4 and x = 0.54. The excellent agree- ment of the present results for g(E) and those of [28] lends additional support also to the reliability of our results for the magnetic spectral function S(q,E), see below, which has not been calculated in [-28]. As a further confirmation of our results for g(E) we have calculated g(E) for x=0 .5 and 0.4 also by averaging over the local densities of states for a large number of sites. In Fig. 2 the result for x = 0.4, calcu- lated from 55 randomly chosen single sites, are repre- sented by the dotted curve. Here the termination has been performed with n o = 16, and the imaginary part

has been chosen as 10 aJo. From the spiky be- haviour of the dotted curve one concludes that part of the excitations are strongly resonant at certain sites, that is the local densities gl(E) are very spiky both with respect to the energy and also with respect to the position. Here one sees the advantage of the random wave method, which yields results (the dark curve in Fig. 2), which would only be obtained by the single-site method by averaging over an extremely large number of sites. On the other hand, by the single site method one can directly study the problem

024 g(E)J1 t 0'16 Eux Srl-xS

o.o

, / , , 06

04 L 8 12 16 E/J1

Fig. 1. The averaged density g(E) of spin excitations of the EuxSrl_xS system is presented over the energy for x=0.4, 0.5, 0.6 (spin glass regime), 0.7, 0.8, 0.9, 1.0 (ferromagnetic regime). J , =2SJNN, where s = 7 / 2 is the spin quantum number of Eu 2+ and JNN(=0.2K) the nearest neighbour exchange. Next nearest neigh- bour exchange was assumed to be (--JNN/2). For x < 1 the results are obtained by the random-wave method

02/+ t

016i

A

8

o /, 8 12 E - -~m. -

Fig. 2. For x =0.4 the result for g(E) obtained by the random wave method ( - - ) is compared with results ( . . . . . ) obtained by an average over local densities &(E) for 55 randomly chosen spins. Reduced energies corresponding to those of Fig. 1

of the localization properties of the excitations; this problem will be considered in a subsequent work in a somewhat more general context. From the densities of states obtained by the random wave method one calculates the specific heat per spin, C m according to the formula

CIt = k2~ - r . ~ dx g(x . k B T)x2 e x (18) o ( e x - 1) 2 "

Page 6: Excitations in Heisenberg spin glasses

248 U. Krey: Excitations in Heisenberg Spin Glasses. I.

CH/(N k B)

X • 0.6

X x

o 5 x

0A- x 0.6

0.3- x 7

0.2- x x 0.8 x 0.9

0.1- 1.o

0 0 0:5 i 1:5 T / ( 2 s JNN)

Fig,& The specific heat C u per spin is calculated according to Eq.(18). The crosses (x x x x x) denote experimental results of [7] for x=0.4

1.6-

1.4-

CA =0.80

Richtung = 1.0.0.

1.2.

13- ~O

0.4-

0 L 8 12 E/J 1

-q=0.7 ~ q : 0 . 6

q:0.5 q:O.L

- - q = O . 3 - - q = O , 2

q=0.1

Fig.4a-c. The spectral functions S(q,E) are plotted for x=0.8 (ferromagnetic regime) over the energy E for various values of the q-vector in the (1, 0, 0)-, (1,1, 0)-, and (1, 1, 1)-directions. The edge of the Brillouin zone is at q = 1 ; Yz : see Fig. 1 !

1.6-

1.4-

1.2-

1.0-

o8 l A 0.6 uJ o'-

o3 0.4-

0.2-

0 0

b

1.6-

1.4

1.2-

1.0-

0.8-

o: 0.6- o3

Q4

0.2

0 0

e

CA =0.80

Richtung= 1.1.0,

A q=10 //,

q=0.1

E/J1

CA =0.80

R ichtung = 1.1.1.

8 1'2 1'6

E /J 1

q = 1.0 q=08

q=06 q=0.L

q=02

O n e shou ld note , however , tha t this f o r m u l a is str ict on ly in the l inear ized theory ( R P A a p p r o x i m a t i o n ) ; k R is B o l t z m a n n ' s cons tan t . I n Fig. 3, ou r resul ts for CH(T ) are presen ted , to- gether wi th the recent e x p e r i m e n t a l resul ts of M e-

schede et al. [7] for x = 0.4. F o r this c o n c e n t r a t i o n , the

sp in glass freezing t e m p e r a t u r e is at T = 1.7 K. C o n - c e rn ing the e x p e r i m e n t the m o s t a s t o n i s h i n g p o i n t is the wide r ange over which CH is j u s t p r o p o r t i o n a l to T. Since in ou r t heo ry the e q u a t i o n s of m o t i o n have

Page 7: Excitations in Heisenberg spin glasses

U. Krey: Excitations in Heisenberg Spin Glasses. I. 249

0.56-

0.48

0.40-

0.32

0.2L (D"

O3

0.16

0.08

~ _ ~ --q=0.613

0 o

CA = 0.40 Richt ung = 1.0.0.

q=oq;l~ - - q = 0 8

N ~ q-07 "

O oS: oo

q=0.1

8 12 16 I = / J l ~

Fig.Sa-c. The same as in Fig. 4, however for x=0.4 (spin glass regime)

0,64-

0.55

0.48-

0.40

0.64-

0.32

0.24 E-

016

008

0 - -

CA :0.40 Richtung : 1,1.0.

q = 1.00 q =0.90

q:0.80 q=070

~ q:0.4O

.~._._._L q=0.30

4 8 12 16 E/J 1

0.64-

0.56-

O.L8-

0.40-

O. 32

~ .08[/ 0

CA=0,40 Richtung=l.1,1.

q=0.40

q=020

q =1.00

q=0.80

4 8

e E/J1

been linearized, and the magnetic dipole interaction has been completely neglected, the theoretical results should apply only at temperatures T <~ Tf, if at all. In fact, for x=0 .4 , the asymptotic slope of Cn(T) for T-*0 is only 10% smaller than the slope of the experimental curve. In view of the above mentioned approximations, this agreement is quite satisfactory.

4b) Spectral Functions and Dispersion Curves

In Fig. 4a-c the magnetic spectral function S(q,E) is presented for the case of x = 0 . 8 (ferromagnetic case) for q-vectors in the [1,0,0], [1, 1,0], and [1, 1, 1] directions, respectively. These results are obtained with n 0 ~ 1 0 and ~=10 - 3 . 2 s J 1. The corresponding results for x = 0 . 4 (spin glass case) are shown in Fig. 5 a-c. From these figures one concludes that in both cases the spectral properties are highly anisotropic. For x =0.8 there are well defined magnons for all q-vectors of the BrilIouin zone leading to well defined disper- sion curves with small widths, as shown in Fig. 6. However it is obvious from Fig. 4 and Fig. 6, that the excitations are very soft for q o c [ l l l ] already for the pure system (x--1). In fact, for x=0 .6 , the spectral functions S(q,E) for given q-vector have still well defined "magnon peaks" at finite E, however for x =0.4 only if q lies within an (1, 1, 1)-plane; whereas

for qoc[1, 1, 1] and x = 0 . 4 the maxima of S(q,E) are

Page 8: Excitations in Heisenberg spin glasses

250 U. Krey: Excitations in Heisenberg Spin Glasses. I.

16

14

Eu x Sr1_ x S

tk,O,01 ~ ,

: lll ! 8

6

0 0.5 k.a 2n

12

5

I 0.5 k.a

[ k,k,k ]

0.5

2~

12

~8

,,'2,6

Eux Srl-x S 16

[k,O,O]

H

0 0.5

2re

[k,k,O]

k '~x : 1.0 ~t~x: 0.8

0.5

2~

[k,k,k]

A 0.5

k.a 2rr

Fig.6a and b. Dispersion curves; the dark points denote the position of the maximum of the spectral function for given q, while the bars denote the half-widths

at E=0 . For x=0 .6 and 0.5, which corresponds al- ready to the spin glass regime, the magnon peaks are still relatively sharp, as can be seen from the disper- sion curves of Fig. 6a; however there is an important distinction between the results for these concen- trations and those for the ferromagnetic regime. Namely, in the spin glass regime, similar to the be- haviour of g(E), also the S(q, E) functions extrapolate to a finite value for E--+0. For x =0.4, the calculated intercept S(q, 0), see Fig. 5a-c, is always larger than half the maximum, with the exception of the im- mediate neighbourhood of the Brillouin zone bound- ary (q.a/(2~)>0.8) for q-vectors within (1,1,1)- planes. Furthermore, contrarily to what has been predicted e.g. by Halperin and Saslow [29] and Huber and Ching [30], we do not see a linear dispersion in the

spin glass regime. Instead, the dispersion curves look more like those of a ferromagnet, i.e. E(q)ocq z, al- though the limited accuracy of our results at low q forbids a definite conclusion. However it is interest- ing to note that the author has found the linear dispersion in Mattis spin glasses [31], and in particu- lar not only in Mattis spin glasses with (_+ 1) Heisen- berg interaction, but also in case of a " random" Mattis model, where not only the sign, but also the strength of the interaction was subject to random- ness. For the (+1) Mattis spin glass the magnon peaks are quite sharp over the whole Brillouin zone, whereas for the " random" Mattis model, propagating excitations appeared only for (q.a)/(2~)<0.3, see Fig. 1 in [31]; in both cases the dispersion was linear in contrast to the present case. From these results, and from related calculations of Ching et al. [32] one concludes, that the dynamics of Mattis models with Heisenberg interaction is not representative for the Eu~S h _xS system.

5. Conclusions

In the present paper, the low energy spin excitations of the Heisenberg spin glass system EuxSrl_xS have been studied by a continued fraction algorithm. In particular, not only the density of states g(E) but also the spectral functions S(q, E) have been calculated. In the calculation, magnetic dipolar interactions have been neglected, and for the exchange interactions it has been assumed that the nearest neighbour in- teraction J1 is positive favouring ferromagnetism whereas the next-nearest neighbour interaction J2= -0 .5J1 would favour antiferromagnetism. These as- sumptions have already been successfully applied in [9]. From the calculation it is found that g(E) extra- polates to a finite value for E--*0 in the spin glass regime (0.13 < x < 0.65) leading to a linear behaviour of the specific heat at low temperatures. At the same time S(q, E) exhibits magnon peaks almost over the whole Brillouin zone. Although in the spin glass regime these peaks are quite broad, it is still possible to define a dispersion curve E(q), which looks roughly similar to that observed in the ferromagnetic regime (0 .65<x<1) . Preliminary measurements on a poly- crystal [15] seem to support the present results for S(q, E).

The author has profited from discussions with K. Binder, W. Kinzel, D. Kumar, K. Maletta, and J. Stein. He expresses his particular gratitude to W. Kinzel for the data of the ground states, and to D.L. Huber for communicating his results prior to publi- cation.

Page 9: Excitations in Heisenberg spin glasses

U. Krey: Excitations in Heisenberg Spin Glasses. I. 251

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U. Krey Fakult~it ftir Physik der Universit~it Regensburg Universit~itsstr. 31 D-8400 Regensburg Federal Republic of Germany