1.7 Spin-Dependent Band Gap in Ferromagnetic Materials
15
constructing a new effective Hamiltonian which aligns the spin parallel or antiparallel depending on the sign of the exchange term. For example, in iron (Fe, Z =
26), cobalt (Co, Z = 28) and nickel (Ni, Z = 29) atoms, the exchange term has a
negative sign resulting in a lower total energy that favour the alignment of the spin
to be parallel in the 3d shell. However, in the period table, the subsequent element
copper (Cu) with Z = 29 does not exhibit any signature of ferromagnetism. This is
because of completely filled 3d shell of Cu, in which there is no flexibility of the
electronic arrangement that is requisite to show ferromagnetism. The Stoner criteria
determine if a 3d transition metal is stable against the formation of a ferromagnetic
or paramagnetic state. This criterion is schematically illustrated by the spin-resolved
3d energy band structure, where the up-spin and down-spin 3d energy bands are
represented at the two sides of the energy axis (Fig. 1.2). According to Stoner model,
there are formation of energy bands where the 3d-spin sub-bands, i.e., 3d up-spin
and down-spin bands are shifted with respect to each other along the vertical energy
axis owing to the presence of exchange interaction. As a result of this shift between
the 3d-spin sub-bands, if the increase in exchange energy supersedes the increase in
the kinetic energy, ferromagnetic ordering occurs. Thanks to their spin-dependent
states, the flow of current inside a ferromagnet is affected by the direction of the
electron spin, leading to two transport channels, spin-up channel and spin-down
channel; thus ferromagnetic material can be considered as two different materials
simultaneously occupying the same space. Those ‘two materials’ have different elecFig. 1.2 Schematic representation of Stoner model of ferromagnetic transition metals illustrated
for 3d shell. Spin states with the largest number of electrons are called the majority spins, and those
with the smallest number of electrons are called the minority spins. Centre of the majority and
minority spin bands are separated by the exchange splitting E ex
15
constructing a new effective Hamiltonian which aligns the spin parallel or antiparallel depending on the sign of the exchange term. For example, in iron (Fe, Z =
26), cobalt (Co, Z = 28) and nickel (Ni, Z = 29) atoms, the exchange term has a
negative sign resulting in a lower total energy that favour the alignment of the spin
to be parallel in the 3d shell. However, in the period table, the subsequent element
copper (Cu) with Z = 29 does not exhibit any signature of ferromagnetism. This is
because of completely filled 3d shell of Cu, in which there is no flexibility of the
electronic arrangement that is requisite to show ferromagnetism. The Stoner criteria
determine if a 3d transition metal is stable against the formation of a ferromagnetic
or paramagnetic state. This criterion is schematically illustrated by the spin-resolved
3d energy band structure, where the up-spin and down-spin 3d energy bands are
represented at the two sides of the energy axis (Fig. 1.2). According to Stoner model,
there are formation of energy bands where the 3d-spin sub-bands, i.e., 3d up-spin
and down-spin bands are shifted with respect to each other along the vertical energy
axis owing to the presence of exchange interaction. As a result of this shift between
the 3d-spin sub-bands, if the increase in exchange energy supersedes the increase in
the kinetic energy, ferromagnetic ordering occurs. Thanks to their spin-dependent
states, the flow of current inside a ferromagnet is affected by the direction of the
electron spin, leading to two transport channels, spin-up channel and spin-down
channel; thus ferromagnetic material can be considered as two different materials
simultaneously occupying the same space. Those ‘two materials’ have different elecFig. 1.2 Schematic representation of Stoner model of ferromagnetic transition metals illustrated
for 3d shell. Spin states with the largest number of electrons are called the majority spins, and those
with the smallest number of electrons are called the minority spins. Centre of the majority and
minority spin bands are separated by the exchange splitting E ex