Muon spin relaxation measurements of the fluctuation modes in spin-glass Ag Mn

1363

Journal of Magnensm and Magnenc Materials 31 34 (1983) 1363-1364 MUON

SPIN RELAXATION

SPIN-GLASS

MEASUREMENTS

OF THE FLUCTUATION

MODES

IN

AgMn

R.H. HEFFNER,

M. L E O N , M . E . S C H I L L A C I

Los Alamos Nattonal Laboratory, Los Alamo,, N M 87545, USA

D.E

MacLAUGHLIN

Umversltv of Cahforma, RlversMe, CA 92521, USA

a n d S.A. D O D D S Rl~e UmverslO', Houston, T X 77251, USA

Recently reported zero-field ,ttSR measurements below the spin-glass transmon temperature m AgMn (1 6 at%) show a temperature dependent mhomogenous width We discuss these data m terms of a model m which the local field undergoes hmlted-amphtude fluctuations We fred that both very slow ( -~ 0 3 #s i) and rapid ( ~ 3000 ,as i ) fluctuaUons are reqmred

In recently reported zero-field /.tSR experiments on the spin-glass A gMn (1 6 at%), the positive muon (p.+) lnhomogeneous line width, a(T), was found to increase monotonically as the temperature was lowered below the glass temperature Tg [1] This effect was interpreted as evidence for rapid, small-amplitude fluctuahons of the local fields, with the decreasing a m p h t u d e of these fluctuations producing the increasing a ( T ) as the temperature is lowered. The purpose of the present paper is to discuss these data in the context of a spin-glass model based on just such limited-amplitude fluctuations of the local fields We will see that the data provide evidence for the existence of both slow- and fast-fluctuation modes. Models for the depolarization of positive muons in spin glasses which assume complete reorlentation of the local fields during fluctuations have been presented by U e m u r a et al [2] and by Leon [3], these models cannot plausibly produce a temperature-dependent a(T), however A two-state model, one state of which has a fixed local field dlrecnon, has been discussed by Uemura [4] In the present " q u a s i m a g n o n model" we assume that the local dipolar field (corresponding to a particular muon site) has both a static c o m p o n e n t H 0 which is Lorentzian-distrIbuted with width a e, and a fluctuating c o m p o n e n t which is Gausslan-dlStrlbuted with width ,A(Ho). This model is slmdar to one presented by Uemura and Yamazaki [5] For this fluctuating component the " s t r o n g collision" assumption is made [6], i e., there is no correlation before and after the j u m p Finding the evolution of the p.+ polarization then amounts to solving the " K u b o - T o y a b e " problem [7] for an arbitrarily oriented static external field H o Fortunately, this can be done in terms of the polarization functions G,,(t), G z ( t ) for the parallel and perpendicular H o 0304-8853/83/0000-0000/$03

configurations, using the relation

a ( t ) = cos2OoG,, ( t ) + sinZ00G± ( t ),

( 1)

00 being the angle between H 0 and 2 (the lmtlal /L+ polarization direction) Each G, (1 = tl , _L) evolves independently of the other, and is determined from an integral equatmn revolving the corresponding static (no-fluctuation) polarization function G ° ( t ) [6,7] These integral e q u a n o n s are solved numerically, for given H o, A and fluctuaUon rate u, to give the G,(t)=G,(t, H o, A, p) We take A = aHo, so that a represents the characteristic angle of fluctuation of the total field Lastly, an lntegranon over H 0 (1 e., over the different m u o n sites) is performed to give the net polarization function G(t, v) Since we want the rms value of the local field at each site to be i n d e p e n d e n t of the amplitude of the fluctuations, we impose the constraint

aL = . ° / ~ / l + ~,~2

(2)

where a ° = 19 /~s 1 is the frozen spin value [8] In general, of course, the lnhomogeneous width, a, will d e p e n d both on a e and a Finally, since one can show that for very fast fluctuations uo of angular amplitude a o.

G,(t) = P(t)a°(t,

H o, A ) ,

(3)

where

we can combine these very fast fluctuations with slower ones in an approximate way by simply substituting these G,(t) for G°(t) in the integral equations In this

O0 '~) 1983 N o r t h - H o l l a n d

1364

R tt tlefjner et a/ // ,tluon spin relaxatmn m A gMn spin glas~

[able I Parameters used to generate lit', to the AgMn (16 atC l d a t a [ l ) 1/ I~

a i : a,

a

.(/is

i)

T~=75

K and a~l'

19 f t`, I ~,(l~

~,

t)

X:/N

(1 30

0 92

/)

0

0 24

~000

(1 3{)

0 86

0 24

03

0 24

3000

2S

{1 80

0 6()

1)

0

0 76

3(100

36

0 8(t

0 47

0 76

03

0 76

3000

[4

o 6 /- -

04-

02-

-0 2

i

1

i

t

i

AgMnCI6at

%]

T =0 8 T g

-

~

~

___L~ 2

I 4

I

I

6

8

I0

t(~s)

Fig 1 Zero-field muon spm relaxauon luncuon for spm-gla,,s AgMn (1 6 %) .it £ = 0 8Tg The dotted t.urxes from bottom to top v, ere calculated from the quasunagnon model usmg a stogie fluctuation rate of 1000, 3000 and 900(I #s i respective b Fhe solid curve v.as generated using rate,, 0 3 and 3000 tzs Fable I gl,,es other relevant parameter,,

60

that these p,nameter,, are certalnl', not u n i q u e In conclusion, we have p r e s e n t e d a model [or inuon s p m relaxation in a s p m - g l a s s system v~hl~.h m c o r p o rates h n n t e d - a m p h t u d e f l u c t u a t i o n s of lhe local held a n d a c c o u n t s for the o b s e r v e d t e m p e r a t u r e d e p e n d e n c e of the i n h o m o g e n e o u s hne width in a physical b plausible way O u r model a s s u m e s the ,,ame tx~o lluctuatlon rates and a n g u l a r a m p h t u d e s at each m u o n szte An equally plausible picture m which different sites ha~c d d f e r e n t (smgle) fluctuation rates and a m p h t u d e s m a y also be consistent wnth o u r data In reality, we expect the s p m glass to p o s s e s s a c o n t m u u m of fluctuatlorL m o d e s , as has been o b s e r v e d with the n e u t r o n s p m - e c h o t e c h n i q u e [9], h o w e v e r w h e t h e r a ~lngle site experiences this c o n t n l u u m or is d o n l l n a t e d by a stogie m o d e ~s not k n o w n Finally, we note that the existence of spln-glas~ m o d e s at frequencies m u c h lower t h a n that f o u n d m numerical s n n u l a t i o n s of harInomt, q u a s l m a g n o n ex~.lt.itlons [10], as prewousl~y r e p o r t e d [1], persists in this new analysis, this p r o v i d e s evidence that small a m p h t u d e b a r n e r m o d e s m a y be u n p o r t a n t in the rclaxmg o[ the nluon spin

case eq (2) IS replaced b} a I = a'l'/~; I + 3a e + 3 ~

(5

T h e data at T - 0 37~ a n d F = 0 8T~ were first fitted a s s u m i n g a stogie, fast fluctuation m o d e of h i l l e d a m p l i t u d e , a p p l y m g the c o n s t r a i n t of eq (2) Excellent fits to the first 0 5 /*s of data were o b t a m e d for the fluctuation rate of 104 ~ts i As can be seen f r o m the d o t t e d curves in fig 1, however, b o t h the long- and s h o r t - t i m e b e h a v i o r of the relaxation function c a n n o t s i m u l t a n e o u s l y be well fitted with this model Next b o t h fast (v 0) a n d slow ( v ) fluctuation m o d e s were included Tht> p r o v i d e s a better fit to the d a t a as s h o w n m fig 1 T h e fit p a r a m e t e r s are s u m m a r i z e d tn table 1, for convenlence we have taken c~ = c% It m u s t be e m p h a s i z e d

References [1] [2] [3] [4] [51 [6] [7] [81 [9] 110]

R it Heffneret a l . J Appl Ph~s 53 (19821 2174 Y J Uemura et al, Ph',s Rev Lett 45 (19801 583 M Leon. Hyperfme Interacnon,,8(1981)781 Y J Uemura, Sohd State Commun 36 (19801 369 Y J Uemura and T Yamazakl. Physlca 109&110B ( 19821 1915 R S H a y a n o e t al, Ph~s Rev B20 (1979) 851) S Dattagupta, H?perftne lnteracnons 11 (1981) 77 R Kubo and T Toyabe, Magnetm Resonance and RclaxaUon, ed R Bhnc (North-Holland. Amsterdam) p 81() A T Flory, Hyperfinelnteractlons8(1981)777 F M e z m a n d A P Muram, J Magn Magn Mat 14(1980) 211 L R Walker and R E Walstedt, Phys Rev B22 (198{)) 3816