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INTERACTION EFFECTS ON THE MAGNETIC PROPERTIES OF TRANSITION IMPURITIES IN
NOBLE METALS
J. Souletie, R. Tournier
To cite this version:
J. Souletie, R. Tournier. INTERACTION EFFECTS ON THE MAGNETIC PROPERTIES OF TRANSITION IMPURITIES IN NOBLE METALS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-172-C1-178. �10.1051/jphyscol:1971155�. �jpa-00214482�
INTERACTION EFFECTS ON THE MAGNETIC PROPERTIES OF TRANSITION IMPURITIES IN NOBLE METALS
J. SOULETIE and R. TOURNIER
Centre de Recherches sur les Tr&s Basses Temp6ratures, Cedex 166, 38, Grenoble-Gare, France
Rksum6. - Les variations de densite d'etat dues a la prksence d'autres impuretQ dans un certain voisinage, semblent btre la cause de modifications locales de la temperature de Kondo et meme de l'apparition d'un moment sur des sites donnes d'un alliage. Nous illustrons ce fait avec des resultats sur AuCo, AuV, CuFe et AuFe. De tels effets entrainent - - -
des deviations importantes au comportement ideal a une impurete pour des temperatures deordre de TK ; par exemple, ils expliquent les apparentes divergences observees dans la susceptibilite du CuFe aux basses temperatures. -
Abstract. - Modifications of the density of states due to the presence of neighbouring atoms in a certain vicinity, seem to be effective in modifying locally the Kondo temperature and even in making a moment appear on given sites of a dilute alloy (illustrations are given with data on AuCo, AuV. CuFe and AuFe). Such effects would introduce simi- - . - -
ficant deviations to the ideal one impurity behaviour at%mperzuresoforder TK ; for example, they would allow forappa- rent divergences which are observed in the low temperature susceptibility of CuFe alloys. -
Since a well known paper by Kondo [I], most of the experimental effort in the field of dilute alloys has been focused on one impurity effects. Very extensive data are now available on quite a number of systems and no doubt remains about the reality of those effects which were successful in inexplaining the resistivity minimum as well as many other properties which appear to scale as functions of TIT, or H/HK where TK of HK are parameters characteristic of the alloy and independent of the concentrations. In our group, in Grenoble, we have been interested in interactions between magnetic moment through Rudermann, Kittel, Kasuya, Yosida (R.K.K.Y.) interactions and evolved towards the study of one impurity effects following a very general trend. Most of our more recent work, however, convinces us that we can use some of our former experience in interactions when trying to understand so called one impurity effects : of course, most experimentalists are conscious of the necessity to make measurements at very low concen- trations ; but the estimation of what a low concentra- tion is, is very often made with reference to the diffi- culty of the measurement and not with regard of the extent of the spoiling which results on the studied effect. We feel this is the reason for the apparent inaptitude of the experimentalists to provide the theo- reticians with sufficiently credible and general laws which may give at least a hint of what is the real thermal evolution of such fundamental properties as for example the susceptibility. Since, it seems to us that interactions may play a very critical role in inducing a moment to appear or vanish on a given impurity, and since a magnetic moment is very effective in spoi- ling a non magnetic effect, it appears worthwhile to recall some very simple facts about magnetic interac- tions between moments in alloys [2]. This will provide us with useful criteria in order not to confuse them with one impurity effects.
I. Magnetic ordering through R. K. K. Y. inter- actions. - The problem of Nc impurities interacting through R.K.K.Y. interactions may be conveniently
approached in a molecular field model [3]. The solu- tion for one impurity pi in the effective field
cos (2 k, r j i + cp)
H , = a C pj
j+i ri 3 j
due to all the other impurities is weighted by a distri- bution function P(H) = dN/dH which gives the statis- tical weight for each value of the field H, and the average is taken for all the values of H. All the geo- metrical data about the relative disposition of the impurities are contained in P(H). We now consider a given alloy of concentration c and retain only the impurities. All we have to do to formally construct an alloy of different concentration c', is to fill the holes between the impurities with bigger or smaller atoms of the host. Obviously cr3 = c' r f 3 (r is any distance in units of the host lattice) because we conserved the same number of impurities. This basic idea was already stated by Blandin [3]. Since cr3 is seen to be concen- tration independent so are the solutions in
H . cos (2 k, r i j f q) I = a C p j
c j+i c r i j 3
(Modifications of the period of the oscillation are neglected, assuming the cos function plays an ave- raging role and is sufficiently defined by a mean value and higher order moments. This is correct for small enough values of the concentration.) If the system of equations in Hi/c for the molecular field is valid for any concentration so is consequently the function
(Incidentally the proportionalities of the width A of P(H) to c and of P (H = 0) to llc are thus demonstra- ted.) If a finite temperature T and an external field h are introduced, the value of ,uj in the expression of Hi/c .will be modified through some function
pj(Hj + h) Bs [-- kT 1.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971155
INTERACTION EFFECTS ON THE MAGNETIC PROPERTIES C 1 - 173 This may be written
For a given T/c and h/c our former argument is still valid since H/c is preserved as the variable and cP(H, T, h) = f (Hlc, TIC, hlc). This dimensional argument is of extended generality : it shows that if the interaction is decreasing like r - 3 in a disordered alloy, the properties of interest to us should scale as func- tions of TIC and hlc, or TITo and h/Ho (where To and Ho are some characteristic order temperature or field proportional to the concentration). Take for example the magnetization.
We give (Fig. 1) the residual magnetization of CuMn
Concentration (at. % ~ n ) - 0.1
.05
0
Concentrahon (at % ~ n ) 1 89
0 0.970
A 0.196 A 0.0992
S 0.0440
17 0.0186
FIG. 1. -Temperature dependence of the residual magneti- zation of CuMn alloys in conventional (oR/c)(T) and reduced
(oR/c)(T/c) diagrams.
alloys 0 ,(h = 0, T) and the same results plotted in a reduced diagram (a,/c) (TIC). A good superposition is obtained when c is varied by a factor 100. Consider an alloy of concentration c much below its Kondo temperature, where, for some reason, a number of impurities in concentration cm c have retained their magnetic moment. We expect the ordering properties to scale as functions of TITo, h/Ho with To and Ho
being proportional to cm in this case (if the one impu- rity effect is the effect studied, it is important that such contributions should be recognized and separated ; for the specific heat and the susceptibility they yield below To - c, concentration independent terms which may for small enough c, dominate the contributions proportional to c due to the Kondo effect). For the AuCo system, superposition of the residual magneti- zation curves is obtained if a,/c3 is plotted against T / C ~
(Fig. 2). In this system, a single parameter is not
I-"'" Remanent Magnefization
FIG. 2. - Temperature dependence of the residual magneti- zation of Au Co alloys (ref. 5) in a reduced diagram U R / C ~ ( T [ C ~ ) .
sufficient to characterize the magnetic state of all impurities but some atoms in number proportional to c3 (groups of 3 atoms probably) have retained a magnetic moment while most do not contribute to magnetic order. In the following we will ask from the experiment a few more indications which may help in tracking the conditions in which the magnetic charac- ters of one impurity may be enhanced or quenched from one to the other site of a given alloy.
11. The effect of interactions on the magnetism of impurities..- We will consider magnetic, all impuri- ties which give rise to magnetic order or which yield a Curie law for the susceptibility in the paramagnetic state over To. If a Curie Wejss like law is obtained with a Curie temperature TK large compared to the temperature range of the experiment, the impurity will be considered non magnetic in this range. This experi- mentalist's criterion of course depends of the range of the measurement and we will eventually speak of more or less magnetic impurities according to the magnitude of TK in our range of interest (usually 0.04 OK to 10 OK). We will start our study with the
&Co system. Hyperfine specific heat results will be
presented in this issue where it is shown that a magne- tic moment is attached to groups of three neighbours [4]. Magnetization measurements (Miss Lecoanet) show exactly the same thing [5]. We will confine ourselves to the analysis of the initial susceptibility which (over the ordering temperature) can be deve- lopped as :
Here pl and TI, p2 and T2 are the respective effective moments and Kondo temperatures for an isolated atom and an atom of a pair and N , and N, the asso- ciated numbers of Co atoms (N1 = c(1 - c)I2, N2 = 12 c2(1 - c)lS, N3 = c - (N1 + N2)). This ex- pression involves five parameters and some explana- tions are needed about their determination. First, in the region much below TI and T2 where the non magnetic contribution may be estimated constant
- xo, a plot of xi T = C + X , T yields straight lines allowing a determination of both C and x0 (Fig. 3).
0 2 4 6 T ( K l
FIG. 3. -The initial susceptibility xi of Au Co alloys below 10 OK in a diagram xi T(T) allowing the separation of amagnetic
term C/T and a non magnetic term x o (ref. [51).
For different concentrations C remains proportional to N3 showing that the groups of 3 may be considered magnetic in our temperature range (over 0.04 OK) since a Curie law is observed for them : ,u3 - 3.6 pB per
atom (S - 1.35). Then x,/N, is plotted as a function of N2/N3 and a straight line is observed (Fig. 4). This allows an accurate evaluation of both quantities pf/Tl and &IT2. The thermal dependence of xi(T) - (CIT) up to values of the temperature of the order of TI allows then to estimate p2 - 4 pB (S - 1.5),
T, - 25 OK and also p1 - 4 p, and TI - 225 OK
although with lower accuracy. In our view the pairs are more magnetic than the isolated atoms and two near neighbours make a moment appear on a Co atom.
Very similar is the case of AuV - alloys. TK is about
the same magnitude but here one next neighbour pro- duces the reverse effect, an increase of TK as evidenced from NMR measurements by Narath and Gossard [6].
The Knight shift goes from negative to positive values showing that the impurity becomes less magnetic when
Xd N,
em.u /mole of isolated tmpur~ties
FIG. 4. - The non magnetic term xo of figgure 3 is seen to contain a contribution due to the isolated atoms N1 and a
contribution due to the pairs N2.
c increases ; if we tentatively admit TK is very large for the pairs and groups of higher number, the low temperature initial susceptibility per mole of alloy should be mainly proportional to c(l - c)12 the num- ber of isolated atoms. If we plot (Fig. 5) xo(c) using the data of Creveling and Luo [7] and of Mr. Tissier of our laboratory, we see that the model fits the obser-
v Creveling and Luo
o Present experimenl
FIG. 5. - The low temperature susceptibility of Au V alloys vs.
concentration (the data are from ref. [7] and from Mr. Tissier).
The solid line is the function
c(1 -c)lZ ,&/3 k ( ~ + Ti) + [c-c(l -c)12] $213 k(T + Tz)
yielding indicatively TI = 225 OK and T2 = 1 120 OK for the Kondo temperatures of the isolated and non isolated impurities respectively, if we assume for the effective moments
INTERACTION EFFECTS ON THE MAGNETIC PROPERTIES C 1 - 175 ved dependence quite well in agreement with the
main conclusions of ref. 161.
We see then that in two systems with the same magnitude of TK the presence of a neighbour results in serious modifications (in either direction) of the magnetic character of the impurity. For higher values of TK stronger interactions are required and up to eight neighbours are necessary to make a moment appear in - CuNi [8]. (It is interesting to note that the local concentrations thus determined, 2 neighbours upon 12 in AuCo, - 8/12 in - CuNi, correspond roughly to the values of the mean bulk concentration where long range magnetic effects dominate.) For alloys of lower TK on the contrary, the effect of interactions is sensi- tive to much lower values of the local concentration and pairs at several interatomic distances will even- tually see their properties strongly modified. We will illustrate this point with magnetization results on the CuFe system obtained by Mr. Tholence [9].
-
The magnetization curves at 1.30K are shown (Fig. 6 ) for concentrations between 10 and 600 ppm Fe.
t MAGNETIZATION
FIG. 6. - Magnetization of CuFe alloys (ref. 191) at 1.3 OK.
The differential susceptibility in 60 kOe (AMIAH),, ,,,, is proportional to c, showing that the variations of the high field magnetization are determined by a one impurity effect. An extrapolation to zero field of this slope in 60 kOe, yields a term o, which varies mainly with c2. The initial susceptibility varies like c + c2. This is a hint that interaction effects (mainly pair effects) are decisive in determining the variations of the magnetization below 40 kOe in this concentration range. This is confirmed by the fact that o, (and not (AM/AH),,,,,) is very sensitive to the thermal treatment.
The unambiguous determination of a c2 depen- dence in the initial susceptibility necessitated a careful study over a large concentration range. NMR measu- rements by Golibersuch and Heeger [lo] yield the equivalent of a magnetization curve where similar cur- vatures are observed. Those results were interpreted as experimental evidence for the existence around the impurity of a (< quasi particle D. It would be of interest to determine unequivocally whether those effects are proportional to c, as claimed by the authors, or if they are the same effects that are observed here and for which a very different interpretation is proposed.
The actual magnetization is analysed as M(H) = cM1(H) + c2 M2(H).
Separation of Ml(H) and M,(H) is made at each temperature for different field values by plotting
The one impurity part M,(H) is given for different temperatures in figure 7 where we see that the main curvatures have been removed. The related suscepti- bility x,(T) follows a Curie Weiss law of the form p&/3 k(T + TK) yielding TK = 29 OK and peff = 3.4 p, in good agreement with Hurd's high temperature mea- surements [l l]. On the same graph plots of the actual bulk susceptibility per unit concentration are given where low temperature divergences are observed. In our scheme they are due to the pairs which we will now consider : the general trend of M2(H) (Fig. 7) suggests magnetism ; x2(T) is seen to follow a Curie
C u - F e
-
TEMPERATURE
20 40 60 kCe
..* TEMPERATURE
0 l b 20 30 C
FIG. 7. - The magnetisation results on CuFe of ref. 191 analysed as cMl(H) + cz M2(H) yield Ml(H) and M2(H) as given above.
The corresponding susceptibilities XI(T) and x2(T) follow res- pectively a Curie Weiss and a Curie law within experimental error. Together with the one impurity result l/xl(T) is given (broken lines) for different concentrations, the thermal evolu- tion of the bulk susceptibility c/xi(T) per impurity when no
separation of the effects of the pairs has been done.
law above 1.3 OK within experimental error : assuming we observe the paramagnetism of Nc2 moments of spin S, both N and S can be tentatively determined from the two equations :
saturation = gNcZ ,uB S ,
Curie constant = Nc 2 gZ S(S + 1) 3 k '
The value obtained for S is 2.6 p, i. e. two times the value which may be deduced for S from the p,, of the isolated atoms, in agreement with the pair model.
The value obtained for N is large and shows that the pairs considered here are not restricted to first neigh- bours but that the interaction is of much larger extent ; allowing for an equal number of ferromagnetic and antiferromagnetic pairs one finally finds that there are approximately N = 560 given critical sites around where a given Fe atom may find a partner to build a pair. In other terms, two neighbours within 10 A
give a magnetic (Curie like) susceptibility above 1.3 OK.
Whether all those pairs are really magnetic (TK = 0) or only nearly magnetic (TK lowered) is another ques- tion. Comparison of our results with the results at lower temperature by Daybell and Steyert [12] and Golibersuch and Heeger [lo] suggests that quite a number of those pairs will lose their moment below 1 OK. This, we think, is along one's natural conception that we deal with a complete distribution of Kondo temperatures between a maximum value and zero according to the extent of the interaction which is present on a given impurity site. Our approach is an over simplification analogous to the one which consists in attributing the bulk effect to the isolated atoms alone ; but we have taken the analysis a step further and although the description of the pair effect is very crude, such mean values as the critical distance 10 A
or the critical local concentration 11560 (to be compa- red to the values 2/12 and 8/12 obtained for AuCo or CuNi) are interesting parameters to test thestrength -
of the interaction which is necessary to induce magne- tism on a given impurity.
This analysis has been kept much on the experi- mental side and although much evidence has been given for the existence of interactions, very little has been said about what those interactions are.
In 1959 Blandin and Friedel [3] already remarked that the Hartree Fock criterion for the occurence of magnetism (1 - Up(EF) < 0) could be sensitive to modifications of the density of states p(EF) due to the presence of other impurities. A mechanism for such modifications has been demonstrated by Caroli [13].
In both the Kondo and the localised spin fluctuation (LSF) theories, the characteristic temperature TK depends on p, the local density of states. If we think the conditions in a disordered alloy are better approa- ched by allowing p to vary from site to site according to the local modifications of the ideal matrix due to the presence of other impurities, we are led to the idea that TK also may have different values TKloo depending on the site. A parameter
< T~ >av = T~lo. P(T~103 d T ~ b .
the properties of alloys with non zero concentration, at least at temperatures large compared with the width of P(TKIoc) where we may assume this distribu- tion is sufficiently defined by a mean value < TK >,, ;
this parameter would be shifted from the value TK towards lower (or higher) values when c increases.
Obviously the susceptibility results on CuFe (Fig. 7) may be fitted in the high temperature region by a formula
C
Xi = T + < T K
with < TK >, as given in figure 8. The knowledge of the whole distribution P(TKloF) would be desirable to
CONCENTRATION
1 -
0 200 400 600 at. ppm
FIG. 8 . - < TK VS. concentration in CuFe. < TK >av is determined by fitting a Curie Weiss law
C/(T + < TK to the high temperature part of the (c/xi)(T) data of figure 7.
account for the low temperature divergences also ; but, this first approximation with only a mean value shows that a distribution may account for the same results than the probably too schematic criterion that we used before (pairs at 10 magnetic).
The next and final illustration we propose is from resistivity data on the - AuFe system. Loram and al. [14]
have shown that over TK, for low enough concentra- tions (< 25 ppm) the results are well fitted with Hamman's law :
R(u) = Ro(l - u(u2 + I)-') with
taking S = 0.77 and TK = 0.25 OK. Changing TK by
we find
could then be a better parameter than TK to describe
INTERACTION EFFECTS ON THE MAGNETIC PROPERTIES C 1 - 177
where
p(u) = (u2 + I)-% (1 + u(u2 + I)-%) is a function (Fig. 9) which may be considered constant (-- 1) in the region where R undergoes it's main variations. Then the resistivity a t non zero concen-
trations should still approximately follow Hamman's law but should not be proportional to c but rather to
Figure 10 shows resistance measurements by Miss La- pierre of our laboratory together with results of four other groups of experimentalists [I41 1151 in a range of concentrations between 10 and 2000 ppm. The results are plotted against Hamman's law with the values of S and TK taken from Loram and al. low concentration results (the intercept at infinite tempe- rature was subtracted in all cases so that we consider here only the Kondo anomaly AR = R(T) - R(co)).
We see that linearity is correctly observed between the low temperature maximum which signals the occu- rence of ordering effects [16] and the high temperature anomaly to Mathiessen's law [17]. We do not observe a proportionality of AR to c (this fact was already noticed by Brewer and al. [14] who considered the slope of the logarithmic increase) ; even for concentrations of the order 800 ppm there is a maximum of the FIG. 9. - The function ~ ( u ) = (uz + 1)-112 (1 + u(u2 + 1)-112)
is seen to be nearly constant when the resistivity R undergoes its major variations.
FIG. 10. - The resis- tance of AuFe alloys vs.
Hamman's law F(T) = 1 -
13.5 -112
(' + ~ 2 )
The data are from ref.
[14] and [I 51 from Miss Lapierre. The linear re- gion is tentatively extra- polated to F ( T = 0 ) i. e.
infinite temperature and the extrapolated part is subtracted so as to give an indicative plot of AR, the Kondo term of the resistance alone, for dif- ferent concentrations. AR is not proportional to c even in the linear re- gion where Hamman's law seems almost obeyed.
The insert shows that ARlc at T = 8 O K
(F(T) = 0.31) decreases with c.
AR(T) deviation. The plot of (AR/c)(c) a t T = 8 OK shows that a decrease of < TK > ,, with c is observed in this system as in CuFe.
I n conclusion, it seems that in many alloys a critical value of the local concentration c,,, may sufficiently modify the density of states o n an impurity to make a moment appear on the site considered. When the mean concentration c of the alloy is much below c,, this situation is obtained on the sites where local fluc- tuations of the concentration reach c,,. The variation with c of the ordering temperature To then reflects the statistical weight attached to such fluctuations. Fur-
thermore it seems that a unique value of TK is not suffi- cient to completely determine the magnetic state of all other impurities. Rather a complete distribution of TK would account better for the experimental data ; a t temperatures much over the range of variations of TK, this distribution would be sufficiently defined by a mean parameter < TK >,, depending on c.
When the mean concentration c becomes higher than c,, most impurities become magnetic through inter- actions and magnetic cooperative effects dominate with ordering temperatures To proportional to c.
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