So now that we understand some of the experimental and theoretical
underpinnings of electron spin, we're going to be looking at today at effective
interactions between spins. Our goal here is to show how ordered spin
states, like ferromagnets, or spin imbalance, spin polarization and
equilibrium arise, especially in the solid state.
We'll first review the symmetry constraint on the two particle fermion
wave function and see how spin degrees of freedom are coupled through electrostatic
interactions. And then we'll look at a generic
Hamiltonian where this so called exchange interaction is relevant, leading up to a
quasi infinite system of coupled spins as a model for ferromagnets.
As you've heard in professor Galiski's earlier lectures A system of two
indistinguishable particles must yield the same measurement of electron
probability distribution regardless of how the two sets of quantum numbers
apply. To the first, and then second, or
vice-versa. This means that particle exchange implies
the acquisition of a unit amplitude complex phase.
And two distinct phases we see most often in nature are zero and pi, giving an
overall coefficient of plus one or minus one.
These correspond to bosons and fermions with integer or half integer spin
respectively. Electrons are spin one half fermions, and
so, acquire the negative spin upon particle permutation.
For a two electron system, a convenient normalized real space wave function
satisfying this constraint is given here. Switching coordinates r1 and r2 is the
same as multiplying times minus one. However, there is more to the electrons
degrees of freedom than just it's coordinates.
The total wave function is the product. Of this spacial wave length with a spin
wave function constructed from up and down single particle states.
The spatially antisymmetric function must therefore have a spin symmetric wave
function of which there are 3 possible orthogonal states having definite amounts
of spin projections with the spin amount of 0 minus 1.
These three states are therefore called a spin triplet.
If we have a spatially symmetric wave function, then the spin wave function
must be anti-symmetric. There's only one way to do this with
definite total spin along the z-axis. A single state with zero total spin.
This is called the spin singlet. Note that for all these states if one of
the 2 spends is flipped we necessarily change the spin wave function symmetry,
and in turn must change the spacial permutation symmetry as well.
Now the point we're going to make is that these 2 totally antisymmetric states with
different spin and spacial symmetries have different energies.
Let's take an example like the two electron, neutral helium atom.
There's the usual electron nucleus attractive Coulomb interaction for each
of the two electrons independently. But then there's the repulsive Coulomb
between the electrons themselves which involves both coordinates.
Importantly there are no explicit spin spin interactions in the Hamiltonian.
So let's evaluate the energy associated with these wave functions of different
symmetry. Note that the only difference here is the
sign between the product states. Calculating the expectation value of the
energy, the diagonal matrix elements of the hamiltonian gives us many terms.
For instance, there are 4 terms related to intergrals over the part of the
[UNKNOWN] covering electron nucleus [UNKNOWN] interaction of the first
electrion h1. 4 for the second electrion h2.
And then the electron, electron repulsion term h12.
Since H one and H two terms are dependent on variables R one and R two exclusively,
the cross-terms in distributing multiplication in the integrals vanish.
For instance, in this one. Since H one depends only on R one, the R
two integral can be done separatley. But the way functions are orthogonal so
the R two integrals vanish. The same happens for the cross terms in
the h2 integrals. However, h12 depends on both coordinates
r1 and r2. So this procedure cannot be carried out,
in the cross terms remain along with the symmetric and anti-symmetric, plus and
minus sign. These are known as exchange integrals,
and we can write the energy of the wave function as the following.
The plus is for the spacially symmetric and minus if for the spacially
anti-symmetric wave function. This exchange integral J splits the
energies of two particle states that are distinct only by the spin configuration
By an effective interaction, due to the spacial and spin wave function total anti
symmetry constraint. Now you might ask why did we neglect
direct spin spin interaction terms in our Hamiltonian.
The underlying electrostatic route of exchange energy between spins is far
greater than any direct spin spin interaction.
This magnetostatic energy is due to one magnetic moment's zamon energy in the
field of another, which we know from classical E&M, falls off as 1 over
distance cubed. Using the characteristic values of Bohr
magneton and Bohr radius, we find the energy precisely the same as the
interaction scale between the intrinsic electron spin, and the orbital moment of
an atomically bound electron. Evaluating, we once again we find the
fine structure scale. And despite a res-mass energy of mc
squared of 500,000 electron volts. Alpha, 1 over 137 to the 4th power is
extremely small. And the net interaction of millielectron
volts is far smaller than the gross electronic energy scale.
Of electron volts or we can expect our exchange cost j to be.
Even though the underlying mechanism relies more on spatial wave function
symmetry we can capture the net effect of spin exchange with a [UNKNOWN].
Again, this is because flipping the relevant spin orientation of electrons
coupled electrostatically through their spacial charge distribution, changes the
spacial symmetry and induces enormous changes in energy due to exchange.
This is the quantam mechanical version of dipole to dipole interaction, where we
have replaced the usual vector dipoles. With new ones having poly-matrix
coordinates. Clearly, from a classical viewpoint, if j
is less than zero, this system favors ferromagnetism, since it's lowest energy
ground state occurs when the spins a line and the dot product is positive.
Now how do we evaluate this Hamiltonian? There are spinup and spindown for each
electron. So we expect exactly 2 times 2, 4
eigenstates. This means our finite-sized Hilbert space
much, must grow from the one-electron two by two Hamiltonian to a four by four
Hamiltonian. We do this with a matrix direct product.
Which is carried out simply by multiplying the elements of the first
matrix times the entire second matrix to form a block in the final result.
The dot product between single particle 2 by 2 spin operators then takes this form,
and upon substitution, we see the need to perform straightforward multiplication of
each term, then sum. This result is non-diagonal.
But we know how to diagonalize it. This can even be done by hand.
Here's the eigen values and associated igon vectors obtained by diagonalizing
the hamiltonion. One with eigen value minus 3 j.
And 3 degenerate vectors with eigen value plus j.
What are these states? Let's remember the correspondance between
our 2 component single particle spin vectors and spin up and spin down.
Assigning the first element to spin up and the lower element to spin down, we
can carry out the direct product and see that the components of a 4 element state
vector define a basis of up up. Up down, down up, and down down.
Then it's clear to see that we have simply calculated the familiar singlet
and triplet spin wave functions all over again.
Now we may ask, what's the net effect of exchange in a system with a large number
spins, on a lattice. This may be a model for a crystalline
solid, for example. In general, our Hamiltonian sums over all
possible spin, spin interactions. Which is incredibly difficult to solve.
To simplify this task, let's consider only short range nearest neighbor
interactions. For a given spin at the i-th lattice
site. We're only going to include interaction
terms with the spins one lattice constant around the crystal axis.
So the exchange integrals are all equal. For a square lattice in two dimensions
that's just four neighbors. And in three dimensions, it's six.
Now, even this is a hard problem. But the form of the Hamiltonian suggest
we make a further approximation. That all the nearest neighbors are
identical. Then this Hamiltonian has the same form
as a magnetic moment interacting with a fictitious exchange meal field.
This mean field is what orients a spins parallel in a ferromagnetic insulator
when the thermal energy is low enough for the system To be below a phase
transition. But what about more common ferromagnets
like iron? A metal.
There, the spins are not localized, but more like free electrons distributed in
space. Before describing ferromagnetic metals,
I'd like to review plain old metals This can be thought of as a gas of non
interacting electrons with plane wave, wave functions.
The quantum numbers are just the wave vector components kx, ky, and kz.
By applying energy minimization, we account for all electrons in the system
by filling the lowest energy state at the orgin of k space.
And then remembering poly exclusion, fill in states ever farther isotropically
until every electron has filled the state.
This creates a so called Fermi's sphere with the radius given by the Fermi wave
number ksabeth/g. But this picture does not capture the two
fold spin degenerancy of every state with definite kx, ky and kz.
A better picture, in this regard, is to plot states not in three-dimensional K
space, but simply as a function of energy, determined by the square of the
distance from the origin. Then, we can plot states up to the Fermi
energy, which have the maximum Fermi wave vector.
However, as we move farther away from the origin in K space, we construct a
spherical shell It gets bigger and bigger, and accounts for more electron
states. The density of states, then, at a given
energy, is growing. Plotting the density of states as a
function of energy reveals that for a parabolic dispersion relation that we
have here for mass and particles, gives a square root of energy dependence, and it
is of course the same for spin up and spin down.
Now we can easily but only partially spin polarize this system by applying a
magnetic field. Zeeman splitting states spin up and down
moving while relaxation processes keep the fermi energy constant in equilibrium.
But in a system with farromagnetic exchange.
The mean field, which is far stronger than any terrestrial magnetic field, does
this for us even in the absence of a real external magnetic field.
The density of states at the firmia energy is then asymmetric and a spin
polarization in an equilibrium is maintained.
Now, we might ask how we can make full use of these spin polarize electrons in
ferro magnets. For example, can we transfer this spin
asymmetry to otherwise non magnetic systems?
Either other metals or even semiconductors?
And if we can, what are the processes that govern relaxation of that spin
polarization back to equilibrium in the non magnetic material.
If those time scales are long enough. Can we manipulate spin asymmetry before
equilibrium is obtained? All of these questions are relevant to
exploiting the spin degree of freedom in solid state devices, with the goal of
addressing unique applications not effectively dealt with by ordinary
electronics. This is the field of spin electronics.
Which has impacted us all through the development and use of the giant
magnetoresistance effect and hard drive read sensors, the discovery of which was
recognized in 2007 with the Nobel Prize. In the next lecture, I'll tell you a
little bit about spin polarized electron transport, especially the unique
difficulties of spin injection. And what you can learn once that
challenge has been resolved.