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.