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So now that we understand some of the 
experimental and theoretical 

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underpinnings of electron spin, we're 
going to be looking at today at effective 

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interactions between spins. 
Our goal here is to show how ordered spin 

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states, like ferromagnets, or spin 
imbalance, spin polarization and 

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equilibrium arise, especially in the 
solid state. 

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We'll first review the symmetry 
constraint on the two particle fermion 

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wave function and see how spin degrees of 
freedom are coupled through electrostatic 

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interactions. 
And then we'll look at a generic 

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Hamiltonian where this so called exchange 
interaction is relevant, leading up to a 

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quasi infinite system of coupled spins as 
a model for ferromagnets. 

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As you've heard in professor Galiski's 
earlier lectures A system of two 

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indistinguishable particles must yield 
the same measurement of electron 

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probability distribution regardless of 
how the two sets of quantum numbers 

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apply. 
To the first, and then second, or 

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vice-versa. 
This means that particle exchange implies 

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the acquisition of a unit amplitude 
complex phase. 

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And two distinct phases we see most often 
in nature are zero and pi, giving an 

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overall coefficient of plus one or minus 
one. 

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These correspond to bosons and fermions 
with integer or half integer spin 

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respectively. 
Electrons are spin one half fermions, and 

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so, acquire the negative spin upon 
particle permutation. 

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For a two electron system, a convenient 
normalized real space wave function 

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satisfying this constraint is given here. 
Switching coordinates r1 and r2 is the 

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same as multiplying times minus one. 
However, there is more to the electrons 

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degrees of freedom than just it's 
coordinates. 

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The total wave function is the product. 
Of this spacial wave length with a spin 

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wave function constructed from up and 
down single particle states. 

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The spatially antisymmetric function must 
therefore have a spin symmetric wave 

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function of which there are 3 possible 
orthogonal states having definite amounts 

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of spin projections with the spin amount 
of 0 minus 1. 

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These three states are therefore called a 
spin triplet. 

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If we have a spatially symmetric wave 
function, then the spin wave function 

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must be anti-symmetric. 
There's only one way to do this with 

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definite total spin along the z-axis. 
A single state with zero total spin. 

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This is called the spin singlet. 
Note that for all these states if one of 

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the 2 spends is flipped we necessarily 
change the spin wave function symmetry, 

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and in turn must change the spacial 
permutation symmetry as well. 

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Now the point we're going to make is that 
these 2 totally antisymmetric states with 

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different spin and spacial symmetries 
have different energies. 

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Let's take an example like the two 
electron, neutral helium atom. 

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There's the usual electron nucleus 
attractive Coulomb interaction for each 

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of the two electrons independently. 
But then there's the repulsive Coulomb 

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between the electrons themselves which 
involves both coordinates. 

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Importantly there are no explicit spin 
spin interactions in the Hamiltonian. 

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So let's evaluate the energy associated 
with these wave functions of different 

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symmetry. 
Note that the only difference here is the 

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sign between the product states. 
Calculating the expectation value of the 

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energy, the diagonal matrix elements of 
the hamiltonian gives us many terms. 

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For instance, there are 4 terms related 
to intergrals over the part of the 

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[UNKNOWN] covering electron nucleus 
[UNKNOWN] interaction of the first 

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electrion h1. 
4 for the second electrion h2. 

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And then the electron, electron repulsion 
term h12. 

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Since H one and H two terms are dependent 
on variables R one and R two exclusively, 

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the cross-terms in distributing 
multiplication in the integrals vanish. 

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For instance, in this one. 
Since H one depends only on R one, the R 

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two integral can be done separatley. 
But the way functions are orthogonal so 

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the R two integrals vanish. 
The same happens for the cross terms in 

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the h2 integrals. 
However, h12 depends on both coordinates 

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r1 and r2. 
So this procedure cannot be carried out, 

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in the cross terms remain along with the 
symmetric and anti-symmetric, plus and 

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minus sign. 
These are known as exchange integrals, 

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and we can write the energy of the wave 
function as the following. 

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The plus is for the spacially symmetric 
and minus if for the spacially 

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anti-symmetric wave function. 
This exchange integral J splits the 

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energies of two particle states that are 
distinct only by the spin configuration 

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By an effective interaction, due to the 
spacial and spin wave function total anti 

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symmetry constraint. 
Now you might ask why did we neglect 

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direct spin spin interaction terms in our 
Hamiltonian. 

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The underlying electrostatic route of 
exchange energy between spins is far 

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greater than any direct spin spin 
interaction. 

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This magnetostatic energy is due to one 
magnetic moment's zamon energy in the 

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field of another, which we know from 
classical E&M, falls off as 1 over 

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distance cubed. 
Using the characteristic values of Bohr 

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magneton and Bohr radius, we find the 
energy precisely the same as the 

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interaction scale between the intrinsic 
electron spin, and the orbital moment of 

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an atomically bound electron. 
Evaluating, we once again we find the 

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fine structure scale. 
And despite a res-mass energy of mc 

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squared of 500,000 electron volts. 
Alpha, 1 over 137 to the 4th power is 

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extremely small. 
And the net interaction of millielectron 

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volts is far smaller than the gross 
electronic energy scale. 

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Of electron volts or we can expect our 
exchange cost j to be. 

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Even though the underlying mechanism 
relies more on spatial wave function 

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symmetry we can capture the net effect of 
spin exchange with a [UNKNOWN]. 

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Again, this is because flipping the 
relevant spin orientation of electrons 

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coupled electrostatically through their 
spacial charge distribution, changes the 

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spacial symmetry and induces enormous 
changes in energy due to exchange. 

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This is the quantam mechanical version of 
dipole to dipole interaction, where we 

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have replaced the usual vector dipoles. 
With new ones having poly-matrix 

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coordinates. 
Clearly, from a classical viewpoint, if j 

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is less than zero, this system favors 
ferromagnetism, since it's lowest energy 

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ground state occurs when the spins a line 
and the dot product is positive. 

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Now how do we evaluate this Hamiltonian? 
There are spinup and spindown for each 

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electron. 
So we expect exactly 2 times 2, 4 

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eigenstates. 
This means our finite-sized Hilbert space 

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much, must grow from the one-electron two 
by two Hamiltonian to a four by four 

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Hamiltonian. 
We do this with a matrix direct product. 

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Which is carried out simply by 
multiplying the elements of the first 

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matrix times the entire second matrix to 
form a block in the final result. 

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The dot product between single particle 2 
by 2 spin operators then takes this form, 

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and upon substitution, we see the need to 
perform straightforward multiplication of 

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each term, then sum. 
This result is non-diagonal. 

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But we know how to diagonalize it. 
This can even be done by hand. 

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Here's the eigen values and associated 
igon vectors obtained by diagonalizing 

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the hamiltonion. 
One with eigen value minus 3 j. 

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And 3 degenerate vectors with eigen value 
plus j. 

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What are these states? 
Let's remember the correspondance between 

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our 2 component single particle spin 
vectors and spin up and spin down. 

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Assigning the first element to spin up 
and the lower element to spin down, we 

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can carry out the direct product and see 
that the components of a 4 element state 

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vector define a basis of up up. 
Up down, down up, and down down. 

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Then it's clear to see that we have 
simply calculated the familiar singlet 

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and triplet spin wave functions all over 
again. 

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Now we may ask, what's the net effect of 
exchange in a system with a large number 

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spins, on a lattice. 
This may be a model for a crystalline 

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solid, for example. 
In general, our Hamiltonian sums over all 

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possible spin, spin interactions. 
Which is incredibly difficult to solve. 

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To simplify this task, let's consider 
only short range nearest neighbor 

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interactions. 
For a given spin at the i-th lattice 

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site. 
We're only going to include interaction 

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terms with the spins one lattice constant 
around the crystal axis. 

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So the exchange integrals are all equal. 
For a square lattice in two dimensions 

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that's just four neighbors. 
And in three dimensions, it's six. 

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Now, even this is a hard problem. 
But the form of the Hamiltonian suggest 

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we make a further approximation. 
That all the nearest neighbors are 

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identical. 
Then this Hamiltonian has the same form 

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as a magnetic moment interacting with a 
fictitious exchange meal field. 

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This mean field is what orients a spins 
parallel in a ferromagnetic insulator 

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when the thermal energy is low enough for 
the system To be below a phase 

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transition. 
But what about more common ferromagnets 

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like iron? 
A metal. 

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There, the spins are not localized, but 
more like free electrons distributed in 

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space. 
Before describing ferromagnetic metals, 

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I'd like to review plain old metals This 
can be thought of as a gas of non 

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interacting electrons with plane wave, 
wave functions. 

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The quantum numbers are just the wave 
vector components kx, ky, and kz. 

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By applying energy minimization, we 
account for all electrons in the system 

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by filling the lowest energy state at the 
orgin of k space. 

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And then remembering poly exclusion, fill 
in states ever farther isotropically 

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until every electron has filled the 
state. 

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This creates a so called Fermi's sphere 
with the radius given by the Fermi wave 

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number ksabeth/g. 
But this picture does not capture the two 

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fold spin degenerancy of every state with 
definite kx, ky and kz. 

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A better picture, in this regard, is to 
plot states not in three-dimensional K 

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space, but simply as a function of 
energy, determined by the square of the 

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distance from the origin. 
Then, we can plot states up to the Fermi 

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energy, which have the maximum Fermi wave 
vector. 

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However, as we move farther away from the 
origin in K space, we construct a 

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spherical shell It gets bigger and 
bigger, and accounts for more electron 

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states. 
The density of states, then, at a given 

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energy, is growing. 
Plotting the density of states as a 

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function of energy reveals that for a 
parabolic dispersion relation that we 

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have here for mass and particles, gives a 
square root of energy dependence, and it 

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is of course the same for spin up and 
spin down. 

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Now we can easily but only partially spin 
polarize this system by applying a 

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magnetic field. 
Zeeman splitting states spin up and down 

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moving while relaxation processes keep 
the fermi energy constant in equilibrium. 

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But in a system with farromagnetic 
exchange. 

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The mean field, which is far stronger 
than any terrestrial magnetic field, does 

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this for us even in the absence of a real 
external magnetic field. 

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The density of states at the firmia 
energy is then asymmetric and a spin 

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polarization in an equilibrium is 
maintained. 

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Now, we might ask how we can make full 
use of these spin polarize electrons in 

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ferro magnets. 
For example, can we transfer this spin 

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asymmetry to otherwise non magnetic 
systems? 

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Either other metals or even 
semiconductors? 

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And if we can, what are the processes 
that govern relaxation of that spin 

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polarization back to equilibrium in the 
non magnetic material. 

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If those time scales are long enough. 
Can we manipulate spin asymmetry before 

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equilibrium is obtained? 
All of these questions are relevant to 

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exploiting the spin degree of freedom in 
solid state devices, with the goal of 

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addressing unique applications not 
effectively dealt with by ordinary 

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electronics. 
This is the field of spin electronics. 

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Which has impacted us all through the 
development and use of the giant 

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magnetoresistance effect and hard drive 
read sensors, the discovery of which was 

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recognized in 2007 with the Nobel Prize. 
In the next lecture, I'll tell you a 

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little bit about spin polarized electron 
transport, especially the unique 

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difficulties of spin injection. 
And what you can learn once that 

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00:12:29,950 --> 00:12:31,530
challenge has been resolved.