MAGNETISM ELECTRON TRANSPORT MAGNETORESISTIVE LANTHANUM CALCIUM MANGANITE

120 Chapter 6
resistivity ρ ∝ M 2 [113, 155, 156] (which is isotropic) with limited experimental
verification.
6. 2. 1 General Model
The difference above and below T C as well as the relationship to the
magnetization M can be shown more generally by only considering the
symmetry of the resistivity tensor - with no assumptions about or even a
reference to microscopic transport mechanisms. Following Landau and
Lifshitz [157], the relation between the electric field E and the current density J
is given by the resistivity tensor ρ: E i = ρ ik J k where the components of ρ are
functions of H and M. Since H and M are vectors which are antisymmetric
under time reversal, ρ must have the following symmetry: ρ ik (H,M) =
ρ ki (-H,M) = ρ ki (H,-M) = ρ ik (-H,-M). The symmetric, s ik , and antisymmetric, a ik ,
parts of ρ ik = s ik + a ik then have the following properties: s ik (H,M) = s ik (-H,M) =
s ik (H,-M) and a ik (H,M) = -a ik (-H,M) = -a ik (H,-M). Thus the components of s ik are
even functions of H and M while those of a ik are odd functions.
The measured resistivity ρ, the diagonal component of the resistivity
tensor, contains no contribution from a ik and therefore must be even in H and
M. Assuming the resistivity is an analytic function with respect to H and M
(which themselves may not be analytic in T), the first terms in the expansion
of ρ in powers of H and M is then ρ = ρ 0 + αM 2 + βH 2 + γH•M. At this point,
one can already expect that for T > T C (M=0) the low field magnetoresistance
should be proportional to H 2 while for T < T C (M≠0) the term linear in H may
be expected to dominate. Clearly a negative magnetoresistance can not obey
this relation for arbitrarily large H, for the resistivity would eventually
become negative; higher order terms or inverse powers of H are then
required. The above argument is as valid for the conductivity tensor σ, where
J i = σ ik E k , as it is for ρ. In the case of the manganites, the magnetoconductivity