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SUSCEPTIBILITY OF SPIN GLASSES
K. Adkins, N. Rivier
To cite this version:
K. Adkins, N. Rivier. SUSCEPTIBILITY OF SPIN GLASSES. Journal de Physique Colloques, 1974,
35 (C4), pp.C4-237-C4-240. �10.1051/jphyscol:1974443�. �jpa-00215634�
SUSCEPTIBILITY OF SPIN GLASSES
K. ADKINS and N. RIVIER Imperial College, London SW7, U. K.
R6sum6.
-La susceptibilite magnetique d'un verre de spin (par ex. Cu avec 1-10 % Mn) a, en champ nul, un pic aigu a la tempkrature d'ordre local. On montre que les theories de champ mole- culaire local peuvent expliquer cet effet. On dkfinit un parametre d'ordre local qui est un effet collectif de tout le solide, dii a la position alkatoire des spins et a la longue portee de leur interaction oscillante (RKKY). La temperature a laquelle I'ordre local disparait est de ce fait unique, comme I'est, par exemple, le rayon d'ecran d'une perturbation dans un mktal, effet collectif de tous les electrons.
Abstract. - The magnetic susceptibility of a spin glass (e.
g.Cu with 1-10 % Mn) has a sharp cusp at the local ordering temperature, rapidly rounded off by an external magnetic field. We show how the local molecular field theories can be generalized to account for this effect. One defines a local order parameter which is a collective feature of the whole solid due to the random position of the spins and their long-range, oscillatory interaction (RKKY). The critical temperature for the disappearance of local order is therefore unique, as, for example, the screening length of a perturba- tion in a metal is a unique, collective contribution of all the electrons.
1. The recent discovery by Cannella and Mydosh of a cusp in the magnetic susceptibility as a function of the temperature of several spin glasses at zero external magnetic field [I], presents an immediate theoretical challenge. The problem is to reconcile the existence of the sharp cusp with the absence of long range magnetic order characteristic of the spin glasses [2]. More specifically, the idea of each spin being in a molecular field which is a random variable with a broad distribution P(H) appears at first sight to be in contradiction with the sharp ordering tem- perature observed by Cannella and Mydosh.
In this paper, we show how a cusp in the magnetic susceptibility can be obtained from a molecular field theory with a distribution P(H) calculated from first principles. The cusp is associated with the disappea- rance of short range order. What makes this short range order a truly collective effect involving all the spins in the alloy is the infinite range of the (Ruder- mann-Kittel-Kasuya-Yosida) interaction J(R) between spins (mean free path effects will be discussed in the conclusion). However, the occurrence of long range order is prevented by the random position of the spins together with the oscillatory nature of their interaction.
In the calculation of P(H), a short range order parameter - the local magnetisation - for which a self-consistent equation can be found, occurs natu- rally. As a function of temperature, this order para- meter goes to zero at some ordering temperature To in the standard fashion of mean-field theories, i. e.
like (To - T ) ~ . To this sudden disappearance of the local magnetisation there corresponds a cusp in the susceptibility.
2. We calculate the distribution of local molecular field P(H) due to N magnetic impurities located at random from first principles, in an Ising, spin 4 model, for simplicity. Generalisation to a Heisenberg model is straightforward, indeed the Ising distribution P(H) is identical to the distribution in the Heisenberg model of molecular field along an arbitrary z direction P(H,)
=2 n 1 dH, H , P ( g ) ('1 ~ 3 1 .
Since H
=C
NH i is the sum of N independent (the
i = 1
magnetic impurities are located at random and their concentration is sufficiently low so that to first appro- ximation there is no excluded volume attached to each of them) random contributions H i
=J(Ri) S , the calculation of P(H) is identical to that of a random walk [3], and one obtains, for N
-,co, with a nor- malisation constant A,
P(H)
=A J dk e-ikH x
x
exp [
-3 1 dT( 2 sin2 x exp (
-(1) Three misprints
should
becorrected in ref.
[3] : a)In eq.
(3), the volume
integral belongs to the exponent, in analogy
with
eq. (1) of the present paper. b) In table I, the first expression for
Ashould be multiplied
by c/a3. c)In the reference list, Klein's paper should face number
8instead of
7,and all subsequent references have their numbers increased
byone.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974443
C4-238 K. ADKINS AND N. RIVIER
The first exponential gives the width of the P(H)
distribution and, for spin 3, it is independent of the nature of the order. I t is equal to exp
-1 k I A , where A
=f
n2c[a/(2kF a)3] (c is the concentration,
athe RKKY coupling constant and a the lattice spacing).
The second exponential gives the shift of the P(H) distribution and depends on the nature of the order through the distribution p,(R, H ) giving the condi- tional probability that a spinvat distance I j from the origin (where the molecular field is calculated) has orientation a when the resulting field at the origin is directed along H. This last restriction is necessary because the short range order region has a given - though arbitrary
-orientation which is that of the resulting field of the origin. Thus,
p,(_R, H ) is a two-site correlation function, and can be obtained from the joint probability distribution P[H(&), H1(0)] which is calculated exactly like P(H) 131 but involves a correlation function of higher order. Alternatively, we can break the infinite chain of higher correlation functions by noticing that [p+ - p - ] (4 H ) is the magnetization at R in a region of short range order of dejinite orienta?on i. e.
a local order parameter which vanishes beyond a correlation length 5 to be determined independently, e. g. by calculating
Assuming that this local magnetization is nearly uniform within the correlation radius, we write, therefore (spin 3, Hex,
=0)
[p+
-p-1 ( R H )
=sig H 2 m(T) R < <
=
0 R > 5 (2)
where m(T) is the local magnetisation averaged within the correlation radius and represents the order parameter in the spin glass. The assumption of uni- formity supposes a single energy determining the orientation of a spin independently of its position, within this region of local magnetic order - cluster for short. The custer holds together magnetically as a single entity.
This assumption is supported by a preliminary calculation of P[H(R), H1(0)]
-P(H) P(H') which exhibits 5 as the onlylength of the problem. 5 has approximately the value found by Klein and Brout [4]
by a different method
:it goes as c-'I3 and is such that the correlation sphere contains 2.4 impurities.
In a finite magnetic field Hex,, eq. (2) is readily generalised as
where < Sz > is the bulk average magnetization
per impurity. t is taken to be field-independent to a first approximation. Eq. (1) and (3) yield the double peaked, Lorentzian-like distribution [5]
P(H)
=A-' d i n ( [H
-mHo sig (H + He,J -
where
< 1
H
- ---2 dR sin
-kJ(R)
N -dRJ(R)
0 -
J
-2
(:3)1:
-and
The normalisation constant A is a function of He,t and T, implicitly through m and < Sz >.
3. We now estimate the bulk average magnetization
< S z > and the local magnetization m
:<
Sz>
=2 '1 dHP(H) tanh P(H + He,,) p, (5a) m
=1 2 / dHP,.,(H) tanh /?(H + HexJ pB (5b) where 9 tanh PHpB is the Brillouin function for spin 3 ( g
=2). P(H) is given by eq. (4). For vanishing external field < S z >
=0. By contrast m is the magnitude of the magnetization when the direction of the cluster is already well defined (cf. eq. (2)), and thus P,,,(H) will be constructed in the same way as the averaged P(H), but without the sig function of (2) and (3). It still involves all the impurities in the alloy ; the order, though short range, is a collective effect. Thus,
a simple peaked, shifted Lorentzian which is obviously normalised. (5b) is then the self-consistent equation for m(T).
For Hex,
=0, (5a) yields < Sz >
=0 and (5b) with (6) gives, after expanding the tanh in an infinite series and integrating term by term, the self-consistent equation
d En T(z)
with $(z)
=- . This equation admits always dz
m
=0 as a solution and, below some temperature To,
a finite order parameter m(T) as well. This ordering
temperature can be found by expanding eq. (7) in
powers of m(T), and is given by the implicit equation
The same expansion implies that m ( T ) oc (T - T ) ~ near the ordering temperature like any mean field theory. Finally, at T
=0, eq. (7) becomes
m ( T ) is plotted schematically in figure 1.
FIG. 1.
- The short range order parameter-
the local magnetization (eq. 7) - as a function of the temperature :- Hext = 0,
....
Hext # 0. (Schematic).The magnetic susceptibility x is estimated from the average magnetisation per impurity <
Sz> given by (5a) from
Since P(H) depends itself on < SZ > one obtains
with A Xo
=nA(Hext=O) +
m(H
-mHo sig H ) tanh ppB H
X J
dH-
- m
[(H
-mHo sig H)' +A2]' . (11) The spins outside the correlation sphere form therefore a polarizable medium. Above the ordering tempera- ture, where m
=0, x has the familiar modified Curie- Weiss form
X- ' ( T > To)
=The Curie temperature
k0
=pB(+ a
-2 A n p 3
)$"($) I)
NpB(+ a - A )
can be negative or positive. x is plotted in figure 2.
The cusp is associated with the sudden onset of short range order m below To
(2).Incidentally, in this spin 3,
low concentration theory, the scaling laws of Souletie and Tournier [6] are obeyed throughout. Super- paramagnetism has not been considered. For a dis- cussion of the concentration range of this (Blandin) scaling regime, see ref. [4] and especially [7]. Finally from eq. (8) and (12), one notices that the suscepti- bility at the ordering temperature has the simple expression
:x(T0)
=2 NpB/(Ho - a).
FIG.
2. - The inverse magnetic susceptibility (eq. 11)....
no short range order, ----short range order. (Schematic).The calculation of the susceptibility in a finite magnetic field is slightly more complicated in that the two self consistent equations (5a) and (5b) are now coupled. If the field is sufficiently small so that the system is still in the linear response range then
in (6), and a self-consistent equation for m alone can be obtained
This equation admits non-zero solutions at all tem- peratures, and, if the cluster is aligned along the external magneiic field, which is certainly the case if the system has been cooled down in the presence
( 2 ) The cusp in the susceptibility does not necessarily imply
a discontinuity in the specific heat (which.is not observed) since most of the entropy in the system is due to the interaction between spins and not to the short range order. There is a peak in the specific heat even if m = 0 at all temperatures.
C4-240 K. ADKINS AND
N.RIVIER of a field, m(T) is an infinitely differentiable function
of the temperature. The cusp in the susceptibility disappears into a broad peak, as observed by Cannella and Mydosh and others [7, 81.
4. In conclusion, we have produced a theory of spin glasses where the absence of long range order is compatible with the presence of a sharp cusp in the susceptibility. It is based on the introduction of a short range order parameter which is a collective quantity involving all the impurities in the alloy, and appears naturally in the construction of the distribu- tion P ( H ) . The random position of the spins implies that the local molecular field is the superposition of an infinite number of oscillating contributions (RKKY) with random phases. The resulting wave packet has a well-defined, finite size 5. Hence the absence of long range order. However, the interaction between impu- rities extends beyond the size of this wave packet.
The local order is therefore a truly collective effect.
Indeed, one would expect an EPR line to be strongly narrowed by the fast modulation due to all the other
precessing spins located at random (recent measu- rements in PdGd by Shaltiel (private communication) seem to confirm this fact.) The situation is analogous to the screening of a charge in a free electron gas, where the screening radius is a well-defined, unique property of the whole gas, in that both problems involve all the degrees of freedom in the system to construct an object of finite size.
The theory relies on two assumptions. The first is that of a single order parameter within a cluster, independent of the position, and is supported by the fact that the only relevant length of the problem is 5.
The second is essential
:the size of the interaction (which, in a real alloy, is the electronic mean free path L [9]) is longer than the range of the magnetic short range order.
If L is sufficiently decreased, for instance by adding non magnetic impurities, such that L < 5, the magne- tic cluster is no longer a collective entity, and the cusp of the susceptibility should be washed out into a broad peak.
References [I] CANNELLA, V. and MYDOSH,
J.A., Phys. Rev. B
6(1972)
4220
;ref.
2,p. 195
;Magnetism and Magnetic Mate- rials 1972, AIP. Conference Proceedings
No10,
ed.C. D. Graham, Jr. and
J. J. Rhyne ;Proceedings of the International Conference on Magnetism, Moscow 1973, to
be published.[2]
Amorphous Magnetism, H.
0.Hooper and A. M. de Graaf, ed. (Plenum Press) 1973, article
byANDERSON, P.
W.(p.
1) and COLES, B. R.
(P.169).
[3]
RIVIER,
N.and A ~ m s , K., ref. 2, p. 215.
ADKINS, K., thesis, London 1973.
[4] KLEIN, M.
W.and BROUT, R., Phys. Rev.
132(1963) 2412.
[5] MARSHALL,
W.,Phys. Rev.
118(1960) 1519.
[6] SOULETIE,
J. andTOURNIER, R.,
J.LOW Temp. Phys.
1(1969) 95.
[7]
THOLENCE,
J.L. and TOURNIER, R.,
J.Physique
35(1974) C4-229.
THOLENCE,
J.L., thesis, Grenoble 1973.
[8]
OWEN,
J.,BROWNE, M. E., AKP, V. and KIP, A. F.,
J.Phys.
&
