Análisis de los determinantes del acceso a la educación superior

2.4.1 A nderson m odel

This model was introduced [16] with the aim to describe the magnetic properties of a single magnetic impurity (here we will assume /-electron impurity) immersed in a non-magnetic metallic host lattice. It uses the concept of a resonant state, which can be occupied simultaneously by two interacting particles. The model includes the Coulomb repulsive interaction U between localised / states, their hybridisation with conduction-electron states described by m atrix element with dispersion e(^):

'HAnd = Ç + « / I T + U n / ^ n f i + Ç - I - % c ±

ka ^ ka

(2.17) where cr is a spin index (either / or j.), /+ and c+ ( / and c) are the creation (an­ nihilation) operators of the localised and conduction electron states, respectively. Ufff and rij^^ are the corresponding number operators.

The key term is the Coulomb repulsion representing the interaction energy between the two spin states. The evaluation is usually treated in some suitable approximation together with simplifying assumptions. The treatm ent within the Hartree-Fock approximation leads to the conclusion th a t if the Coulomb repulsion param eter U is high enough, double occupancy is not favourable and one state is pushed up above the Fermi energy, and one electron of either spin will fall into

the c?-shell state and form a magnetic 5' = | state. Due to the hybridisation both / states (spin t and 4^) are broadened, enabling partial occupancy, if all states up to Ey are filled. When the hybridisation is weak the occupancies for the occupied and unoccupied states are 1 and 0, respectively. In the strong hybridisation limit, the impurity level becomes broader, intersects the Fermi level and gets partially depopulated. In this case there is an additional contribution to the density of states at the Fermi level and hence also an enhanced specific heat.

Although the Anderson model explains qualitatively the im portant phenom­ ena related to the hybridisation between localised states and conduction electrons, it requires non-trivial extensions to become relevant to actinide materials: It ne­ glects orbital degrees of freedom, the periodicity of magnetic “im purities” and it is limited to a single conduction band. Although it is possible to construct a “periodic Anderson Hamiltonian” , there are reasonable doubts to such extension. First, the number of electrons released by the impurities could be a significant factor in changing the position of the Fermi level. Second, the strength of the hybridisation between the localised and conduction states is treated in the single-impurity model as constant, independent of the geometric details of the surrounding of the localised electron which is an im portant factor in a crystalline solid. Third, the Anderson model cannot predict or even account for cooperative phenomena at low tem peratures which are consequences of lattice periodicity and coherence.

However the Anderson model provides a good starting point for more elabo­ rate approaches and gives a clear idea concerning the infiuence of hybridisation on magnetic properties of solids.

2.4.2

K ondo m odel

In the magnetic regime, when one impurity state is above the Fermi level and the other one below, it was shown [17] th a t the Anderson Hamiltonian is equivalent

Chapter 2. Theory 36

to another im portant type of Hamiltonian, which involves the interaction term given by

J* =

^

[(^/) (£/ + U)]

called the Kondo exchange parameter. The Kondo model involves, apart from the energy of conduction electrons, the interaction term

Hex = - 2 J * s - 5 (2.19)

which couples the localised moment (S) and the conduction electron spin mo­ ments (s] antiferromagnetically through the effective interaction param eter J*. Due to this interaction, in the ground state the impurity spin is completely screened and compensated by conduction electrons. The ground state is a non magnetic singlet. Such a type of ground state is characterised by a logarithmic increase of electrical resistivity below a characteristic tem perature given by

where Tp denotes the Fermi tem perature. There are also other significant con­ sequences of the formation of Kondo state on physical properties. The specific heat for instance becomes linear in tem perature well below T p and exhibits a maximum near it. The magnetic susceptibility shows in the vicinity of Tk also a maximum, but it becomes tem perature independent at lower tem peratures.

The infiuence of coherence effects at low tem peratures is successfully included in the Kondo lattice models [18].

The interaction of localised moments with conduction electrons may lead to an effective interaction between localised moments themselves. This type of in­ teraction is of the RKKY-type.

The final magnetic state of a Kondo lattice depends on the balance between the RKKY interaction which pushes the system towards magnetic ordering and the Kondo interaction which generates a non magnetic ground state. Both inter­ actions depend on J*, in different ways. Tr k k y oc | a n d Tr is shown in equation (2.20). For high values of J*, the Kondo interaction dominates and the system does not order magnetically. On the other hand, if J* is small, the RKKY interaction becomes stronger and the system would order magnetically.