MAGNETITE AS NORMAL SPINEL AND THE SEMI-CONDUCTOR METAL TRANSITION

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MAGNETITE AS NORMAL SPINEL AND THE

SEMI-CONDUCTOR METAL TRANSITION

X. Oudet

To cite this version:

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JOURNAL D E PHYSIQUE Colloque C1, supplkment au no 4, Tome 38, Avril 1977, page C1-223

MAGNETITE AS NORMAL SPINEL AND THE SEMI-CONDUCTOR

METAL TRANSITION

X. OUDET

Laboratoire de Magnitisme, Centre National de la Recherche Scientifique 1, Place Aristide Briand, 92190 Meudon, France

R h m 6 . - L'interprBtation des propriCtt5-s Clectriques de Fe304 est bask sur un model d'6lectron liB, et sur une Ctude statistique de la rkpartition de I'energie thermique reCu par cet electron.

L'expression de la rCsistivitC est

A et a sont deux constantes, Eg est 1'Bnergie de la barrikre de potentiel que doit franchir 1'Clectron pour contribuer au passage du courant, U est 1Vnergie thermique moyenne reGu par l'blectron, et

pc la r6sistivitC qu'aurait le corps si tous les Blectrons participaient au passage du courant.

Abstract. - The electrical properties of FesO4 areexplained considering the model of a bound electron and the statistical distribution of the thermal energy received by this electron.

The resistivity is expressed by

A and a are two constants, Eg is the energy necessary for the electron to contribute to the current flow, U the average thermal energy received by the electron and P C the resistivity of the compound

in the case when all electrons contribute to the current flow.

1. Introduction. - Since the experiment carried out by von Laue in 1912 [I] a great number of crystal structures has been discovered, but much remains to be explained as to why a compound should crystallize in a particular structure. In view of this, we have assumed that : in a given compound if an element occupies two crystallographic sites, then the number of electrons exchanged by the element in the two sites with its neighbours must be dzflerent. We shall call this hypo-

thesis the symmetry hypothesis. The good agreement observed between the calculated and experimental values of the saturation magnetic moment, for gar- nets [2] and spinels [3, 41, as well as the Curie constant for spinels [5], for more than 70 compounds, all seem to prove the validity of this hypothesis.

In the case of MgFezO,, for example, which is an inverse spinel, this hypothesis enables us to predict the presence of ~efi: in one of the two sites (2' for the ionicity and I11 for the valency) and of Fe3+ in the other one, which gives within the framework of the Dirac's theory the value of 0.80 p, (p, = Bohr magne- ton) for saturation moment, 0.82 p, being the experimental value (6).

From the symmetry hypothesis it follows that in Fe30,, Fez+ must occupy the tetrahedral site A and Fe" the octahedral site B. The study of the saturation magnetic moment at 0 K leads us to divide the site B

into two subsites in which the magnetic moment of the ions are antiparallely coupled antiparallel to each other, and to attribute it the contribution of Fez+ [4].

As the present choice of the inverse type follows from the interpretation of the electrical properties [7] and is used also in other interpretations as for instance to understand the Mossbauer experiment [8] it seemed to us interesting to show the possibility of interpreting the electrical properties independently of the type cho- sen.

2. The mechanism of electrical conductivity. - At room temperature in cubic phase Fe,O, is a conductor of metallic type (M) Below 110 K in orthorhombic phase it is a semi-conductor (S. C.) [9, 10, 111. The M-type is presently attributed to the hopping effect occurring between the Fez+ and Fe3 + ions of the site

B [8] and the S. C.-type to an ordered arrangement of the Fez+ and Fe3+ ions in the site B 1121.

Large variations in the electrical conductivity asso- ciated with a phase transformation are also to be found in other compounds such as V z 0 3 in which there is only one type of cations [13, 141. Even if no phase transformation occurs a compound can pass progressively from the S. C.-type into the M-type, as is the case for SmB, [15]. This leads us to think that the large variations in the electrical conductivity of a

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C1-224 X. OUDET

compound are rather realted to a single mechanism than to the particular compound itself, therefore, inde- pendent of normal or inverse type in the case of Fe,O,.

Let us now consider the theoretical results obtained for the magnetic moment and the Curie constant. We have found that only the spin contribution is concern- ed here and that the moment of each electron is gp,

with g = - j j being the quantum number intro-

i

+

112

duced by b i r a c in its electron theory, characteristic of the subshell to which the electron belongs 1161. Thus

in these oxides the electrons have their quantum properties governed by the same quantum numbers as the ones they have in the isolated atom or ion. It follows that in these oxides each electron belongs to one and only one ion.

With a localized electron how can the conductivity be understood ? The conductivity for example in the case of only one type of electrons contributing to the current flow can be explained as follows : to contribute to the current flow the electron must cross a potential barrier or gap, characterized by ail energy E, and, for this, should receive an energy E > E,. This energy is supplied by the thermal agitation. As the compound is at a temperature different from 0 K, each electron has a mean energy U and, on account of the statistical distribution, some of them have an energy E > E,. If the solid is then submitted to an electrical polarization, the electrons with E > E, will leave their ion and move through the solid until they lose a part of this energy so that it becomes lower than E,, and they are trapped by ions having lost this electron. Let P(E,, U) be the probability for an electron to have received the energy E higher than E, and let p, be the resistivity of the solid in the case when all electrons contribute to the current flow, it follows : p = p,/P(E,, U).

The problem is thus reduced to the calculation of P(E,, U) i. e. to a statistical problem. To solve it we shall first calculate the probability per unit of energy, D(E, U), for an electron to have the energy E, its mean energy being U . This function has been already used successfully for the calculation of the Curie constant [ 5 ] .

3. The statistical problem. - The method used for determining the distribution D(E, U) is fairly similar to the classical one [17]. It consists in calculating the number n, of particles, electrons in the present case, the thermal energy of which has occupied for a given interval of time a segment i of given energy, rather than in determining a t a given time the number of particles occuping a given segment of energy.

This method offers the advantage of giving directly the mean value of the instantaneous energy distribu- tions which are fluctuating in the course of time. In fact, the longer the lapse of time during we consider the particles to come and occupy a segment of given energy, the more balanced are the deviations from the mean value and the more the real value tends to the mean one.

The method then consists of determining the most probable series of the n, for all the possible values of energy, which implies, to calculate for each ni the number v, of the particles from which the n,s have been drawn. So, we shall proceed first to the determination of the n, and v,.

3.1 D~TERMINATION OF THE ni AND v,. - Let D(E) be the probability per unit of energy for a particle to have at a given time the thermal energy E. Let U be the mean value of this energy, and N the total number of particles of a set. The whole possible values of the energy, form a positive half-axis. Let us divide this half axis into segments i of width AE, (i is a positive whole number above zero) so that any value of the energy corresponds to one and only one segment. For this purpose let us introduce the energies E, defined

by the relations

E~ = el-,

+

AEi and 8, = 0 .

A particle will be regarded as belonging to the segment

i if its energy Ei is such that :

In order, to neglect the variations of D(E) when E,

varies within the segment i, we shall require that any AEi satisfies the relations : AE,

<

U. At time t

the number of particles of the segment i is n(Ei, t ) . This number varies with time. Let n(EJ be its mean value, Since we have AE, 4 U, we may write :

In order to determine n(E,) rahter than n(E,, t ) let us introduce z,, the average time necessary for a particle to leave the segment i and evaluate the n, particles which have left the segment i during the time t , t + A t .

As At increases this number tends to be equal to that of the particles which have occupied the segment i. Moreover, if any n, is dependent upon t , its mean value in the course of time is : n(E,) At/zl.

Let us determine now the number v, of particles from which the n, have been drawn in the interval of time t, t

+

At. At the time t there is no specific property on the v, particles, whence their average energy is U . The fact that the v, particles have not specific properties according to i except the width AE,, shows that their probability per unit of energy is the same whatever be i. In order to determine this probability per unit of energy, let us consider the segment which contain the energy U. Let n be the number n, for this segment and v

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MAGNETITE AS NORMAL SPINEL AND THE SEMI-CONDUCTOR METAL TRANSITION C 1-225

cannot be kept in the case of segment not containing U

because in this case there is a change in the mean value of the energy, and 'v, is necessarily different from 2 n,. In other words this comes from the fact that, for instance in the case of high values of energy which are less probable, the density of vi is higher than that of n, a result which is natural seen from this angle. Furthermore, the mean time necessary for a particle of the segment i to leave its segment is identical to that during which it occupies this segment i. As the particles

vi have no particular property, this time does not depend on the segment i. Thence ti is the mean time z

between two energy exchanges. Therefore :

The determination of D(E) is then a relatively classical problem.

3 . 2 CALCULATION OF D ( E ) . - Let Wi be the number of different combinations to obtain ni particles taken amoung the v, particles. We have :

Let W be the total number of possible combinations giving a series of n,. We have : W =

n,

W,.

The most probable series of n, is the one for which W

is maximum and, therefore, for which d(ln W) = 0. In this differentiation v, must be taken as a constant, because these particles belong to all the segment i so in the set there is n o variation. Using the Stiring's relation, this yields :

vi

-

n,

Ci In - dn, = 0 .

"

i

If this relation is satisfied, it comes from the diffe- rential physical properties of the set : Ci dn, = 0, resulting from

and Ci E, dn, = 0, resulting from

Thus whatsoever be dn,, we must have :

vi

-

ni

Xi(- a

+

PE,) dn, = Ci ln

-

"

i dni (3)

we have introduced -a and

P

rather than a and

/?

as Lagrange's multipliers.

The equality (3) can be satisfied whatever be dn, only if we have :

Introducing the relations (1) and (2) we get :

Now, when allowing the different AE, to tend towards zero, the sequence of the D(E,) tends to the

continuous function of E :

In this expression for D(E) three parameters remain to be calculated :a,

P

and D(U). Three relations are thus

to be found. The first one is obtained by substituting E by U in the expression (5), which gives :

a = f l u . (6) The second one is obtained by writing :

which is the classical standard relations expressing that the probability, of finding a particle with some energy is always 1.

The third relation is obtained by writing : OC

1

ED(E)dE = U

0 (8)

which indicates that the mean values of the energy of a particle is U.

Using the variable substitution x = fl(E

-

U) and the relations ( 9 , (6) and (7) it follows :

5

dx

2 D ( U ) = P with A =

A - a 1

+

ex

Using the same variable substitution and the relations ( 9 , (6) and (8) it follows :

This relation is an equationfhat determines a. It gives

and by developing xll

+

ex into series with x c 0 ;

we get

from the relations

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Cl-226 X. OUDET

it follows

a is very close to 1.5. A more accurate calculation gives a = 1.5049. With this value of a we get :

A = 1.7054 and finally :

and

OD

P(E,, U) = D(E, U) dB = Es

4. Theoretical and experimental results. - We thus obtain the relation :

which is represented in figure 1 as a function of

x = UIE,. We do not discuss in this work the variation

FIG. 1. - Solid curve p = pc Alln [I

+

exp - z(E,/U - l)]

as a function of UIE,. Circles the experimental values for SmB6 after Menth et al. [15], with VIE, = T/21.

of p, with T as for example the Joule effect. Is should be noted first that p is an explicit function of U, and not of T. It is a result which follows from the treat- ment of the statistical problem and which points out that the condition for an electron to contribute to the current flow is not that the compound is at the temperature T, but that at this temperature it has rcceived the energy U which depends on the considered compound.

As the variations of p are determined experimentally as a function of T, the knowledge of U(T) is therefore necessary for determining p(T). We are faced here with a new problem for which there are no experimen-

tal data available. Only U(Fe30,, T) is known by the measurement of the specific heat of Fe304 [18, 191. But, as for the amount in this quantity corresponding to U(T) for the electron responsible for the conduction, we are not able to determine it presently. So, we shall confine ourselves to the qualitative verification of the expression for p(U).

Let us first consider the variation of p in the orthorhombic S. C.-phase. It has been shown that p does not Es follow an exponential law of type : p = Bexp --

KT but that the energy E, decreases with T [ l l ] .

If the term exp - a in the expression of p(U) is small as compared to "nity for instance with

E g > 2, it follows

U

E*

p = p, A e-" exp a -

U

The large variations of p show that we are really in this case. The measurement of the specific heat of Fe,04 [18] shows'that in this temperature zone U(Fe304) varies in the same way as T Y with 4

>

y > 3 and that the specific heat per ion is much lower than 3 K. We can thus assume that U(T) varies as T Y with j ) > 1. AS a

Es result, if we determine E, using a law of type exp -

K T in place of exp a - , we shall find values of E,

U

decreasing with T, as is shown by experiment. Failing an accurate determination of E,, we can nevertheless determine its upper limit. For this purpose we firstly consider for p, the value of p at room temperature at which p is nearly constant and, therefore, P(E,, U) close to I (assuming that the cubic p, is not very diffe-

rent from the orthorhombic p,) and then we assume that all the energy received by the divalent iron ion can be used by the electron to contribute to the current flow and that this energy is the third of that received by Fe304, it follows then : E, < 0.06 eV.

Let us consider now the cubic M-phase between 110 K and 300 K. In this phase the compound passes progressively from the S. C.-type at 110 K into the M-type at 300 K [lo]. This can be explained qualita- tively as follows : for a relatively low value of E,,

when the mean energy U received by the electron is suficiently high as compared to E,, the quasi-totality

of the electrons contributes to the current flow and the resistivity becomes nearly constant, thus the compound behaves like a metal. This is given in fact, by the variations of p for UIE, > 2. The higher limit of E,

gives : E, E,< 0.02 eV.

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MAGNETlTE AS NORMAL SPINEL AND SEMI-CONDUCTOR METAL TRANSITION C1-227

to the S. C.-phase, as seems to be proved for these two phases by the estimations of E,. This leads us to suppose

that the bond state of the electron responsible for the current flow is strongly bound in the orthorhombic phase than in the cubic phase.

5. Conclusion. - We can verify that this interpretation of the electrical properties of Fe,04 is not specific to this compound. For instance, the experi- mental results obtained for SmB, are well described in figure 1 by the curve representing p as a function of

UIE, [20]. Furthermore, it must be noted that this

interpretation does not enable us to choose between

the normal and the inverse type. Also in Mossbauer spectrum with a model of a bound electron one ought to expect three Zeemann patterns. For an inverse spinel, however only two are found whence we could conclude, as from the interpretation of the two iron sites with the symmetry hypothesis, Fe,O, is a normal spinel.

Note added in prooJ

-

Recent experimental results (B. Balko and G . R. Hoy, J. Physique 37 (1976) Colloq.

C6-89) in double Mossbauer spectroscopy fail to find

the relaxation time on site B of Fe,04, a result easy to understand from this work.

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der Buyerischen Akadenlie der Wissenschaften, Math- phys. Klasse, pp. 303-322 (1912) ; die Naturwissens-

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[5] OUDET, X., 18th Ampere Congress Nottingham, Proc. 1

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