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COLLECTIVE ELECTRON THEORY OF THE VERWEY TRANSITION IN MAGNETITE
J. Cullen, Earl Callen
To cite this version:
J. Cullen, Earl Callen. COLLECTIVE ELECTRON THEORY OF THE VERWEY TRANSI- TION IN MAGNETITE. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-1110-C1-1111.
�10.1051/jphyscol:19711395�. �jpa-00214435�
JOURNAL DE PHYSIQUE Colloque C 1, supplment au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1
-
11 10COLLECTIVE ELECTRON THEORY
OF THE VERWEY TRANSITION IN MAGNETITE
J. R. CULLEN, EARL CALLEN (*) U. S. Naval Ordnance Laboratory, Silver Spring, Md,
American University, Washington, D. C.
R6sumb. - On dkcrit un modble d'dectrons collectifs pour rendre compte du comportement de la magnktite ti basse tempkrature. On prksente un calcul par la methode (( tight-binding )) dans lequel on introduit des paramMres d'ordre abaissant la symetrie afin de decrire la structure kktronique ti basse tempkrature. On explique ainsi directement les spec- tres Mossbauer observes rkcemment dans la magnktite. On montre aussi comment la thCorie peut &re gknbraliske pour stre appliquke a la diffraction des electrons et des neutrons.
Abstract. - A collective electron approach to the low temperature behaviour of magnetite is described. We present the results of a tight-binding band calculation in which symmetry-breaking order parameters are introduced to describe the low-temperature electronic structure. These orderings are used to explain in a direct way the extra spectra observed in recent Mossbauer experiments on magnetite. We also indicate how the theory can be generalized to account for the elec- tron and neutron diffraction measurements.
At the IEEE Magnetism Conference last year [I] we described a one-dimensional band model of the Verwey transition. The model was characterized by an order parameter m, the expectation value of the difference in the number of electrons condensing on alternate sites. m(T) displayed a second order phase transition.
Because the metallic band is half filled, charge modu- lation due to non-zero m(T) doubled the unit cell in the ordered state, reduced the Brillouin zone in half, and produced an energy gap at the halved zone boun- dary. This gap was proportional to m(T), yielding a semiconductor in the ordered state and a metal above the transition temperature T,. For strong coupling, m(0) = 1 (ionic), but when the coupling constant is much less than the band width, 0 < m(0) < 1, and there is only fractional electron occupancy at all T < Tv. The theory is then BCS-like in structure.
The one-dimensional treatment was really too easy.
The metallic band is half filled, and a self-consistent m(T) guarantees a gap and a semi-conductor. In three dimensions the Fermi level is not so easily determined, and the gap must encompass it in all directions in k space. We have now performed enough of a 3 dimen- sional tight-binding calculation to perceive its implica- tions. There is not merely one order parameter, but there can be almost as many as the number of B ions in the cubic unit cell (sixteen). In fact many more, because experiment indicates a symmetry breaking operation, doubling the unit cell along the c axis.
Choosing a small number of all these possible order parameters, we show that a gap can in fact be created, and that we can describe many otherwise-puzzling experimental results.
Figure 1 shows the band structure in the metallic phase along r-X. (The B-site ions form a F. C . C.
Bravais lattice, with four ions in the unit cell. The four B ions are tetrahedrally arranged.) The metallic conduction above Tv results from overlap of two bands, rather than from one half-full band, as in the one dimensional model.
(*) Supported under NASA Grant 09-003-014.
E vs. k
r1001 NO ORDER
FIG. 1. - E vs k in the metallic phase. Note the two overlaping bands each partially full (there are two electrons per primitive
cell).
The four B sites per primitive cell are equivalent above T,. Below Tv, we define three cc order para- meters )>, one of which describes the Verwey order.
We will see that the other two orderings are essential to adequate interpretation of experiments. A non- zero value of
corresponds to ordering in alternate planes along the orthorhombic axis. The n's are averages of the number operators for the B sites. It is important to note that these averages are zero or one in the Verwey scheme, but the actual values, which must be calculated self- consistently, can vary between zero and one, and as a function of temperature.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711395
COLLECTIVE ELECTRON THEORY OF THE VERWEY TRANSITION IN MAGNETITE C 1 - 11 11
With only m, present, the band structure is modified from the high temperature phase mainly by the appea- rance of gaps at
r
and X [OOl]. Figure 2 shows the case - umi = .4, where U/h is the interaction strengthh
to band width ratio. Note that this type of symmetry breaking produces no gaps at [OlO] or [loo], and the resulting structure is still semi-metallic. In fact, no matter how large the interaction strength, the Verwey ordering by itself cannot produce an insulator.
[loo1 SAME AS k 11 [010]
IF m2 =O
FIG. 3. - E vs k, along (001). There are now gaps in all direc- tions. The valence band maximum is at one of the L points.
The gap goes to zero if Urns 5 75.
FIG. 2. - E vs k along (100) for c axis order alone. The band width is taken to be equal to one. Two bands still overlap ;
this is true no matter how large we make Uml.
We now allow two other order parameters
and calculate the band structure as a function of m2 and m3. Non-zero values of m 2 and m, corresponds to further lowering of the symmetry e. g., from tetra- hedral to orthorhombic. Each of the four sites has a different charge. The calculation indicates that the structure finally becomes insulating only when m,, m,, m3 are larger than some critical values. Figure 3 shows the important part of the structure along [OOl]
in the insulating phase. We note that the three
x mi) values must be rather large to make the
(x
structure insulating. We conjecture that further sym- metry lowering (doubling the unit cell along the c axis, for example) will make the passage to insulating behavior easier by allowing a smaller U/h value than what is needed with only three order parameters.
With only a three order parameter theory however, we explain the Mossbauer spectra. With Verwey ordering, one obtains at most three sets of lines (one A, two B spectra) when in fact five sets are observed [2]
(one A, four B spectra). This follows naturally from having four different n values (3 order parameters) on the four B sites in the tetrahedron. Electron [3] and neutron [4] diffraction results indicate even more orderings must be present below Tv including a doubl- ing of the cubic unit cell. These extra orderings can be incorporated in the present theory by introducing more symmetry breaking order parameters. Hall effect [5] and the thermopower measurements also indicate band-type behavior in magnetite. Our results point to an indirect band gap which is temperature dependent, and perhaps to maxima and minima in the valence and conduction bands whose location in the Brillouin zone may shift with temperature right below Tv,
References
[I] CULLEN (J. R.) and CALLEN (Earl), J. Appl. Phys., 1970, [4] SAMUELSON (E. G.), BLEEKER (E. G.), DOBRYZINSKI (L.)
41, 879. and RWTE (T.), J. Appl. Phys., 1968, 39,1114.
121 HARGROVE (R. J.) and KUNDIG (W.), S. S. Comm., [5] SIEMONS (W. J.), Symposium on Magnetic Semicon-
1969, 7, 223. ductors, Yorktown Hts. N. Y., Nov. 13-14 1969 ;
[31 YAMADA (T.), SUZUKI (K.) and CHIKAZUMI 6.). ADDZ. IBM Journal Res. and Dev.. 1970. 14. 245.
. .
.
,,- -
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Phys. G t t e r s , 1968, -13, 172.
