1 00:00:01,940 --> 00:00:04,460 So now that we understand some of the experimental and theoretical 2 00:00:04,460 --> 00:00:07,475 underpinnings of electron spin, we're going to be looking at today at effective 3 00:00:07,475 --> 00:00:12,624 interactions between spins. Our goal here is to show how ordered spin 4 00:00:12,624 --> 00:00:16,046 states, like ferromagnets, or spin imbalance, spin polarization and 5 00:00:16,046 --> 00:00:20,015 equilibrium arise, especially in the solid state. 6 00:00:20,015 --> 00:00:22,895 We'll first review the symmetry constraint on the two particle fermion 7 00:00:22,895 --> 00:00:26,303 wave function and see how spin degrees of freedom are coupled through electrostatic 8 00:00:26,303 --> 00:00:30,156 interactions. And then we'll look at a generic 9 00:00:30,156 --> 00:00:33,376 Hamiltonian where this so called exchange interaction is relevant, leading up to a 10 00:00:33,376 --> 00:00:37,510 quasi infinite system of coupled spins as a model for ferromagnets. 11 00:00:38,950 --> 00:00:42,190 As you've heard in professor Galiski's earlier lectures A system of two 12 00:00:42,190 --> 00:00:45,592 indistinguishable particles must yield the same measurement of electron 13 00:00:45,592 --> 00:00:49,048 probability distribution regardless of how the two sets of quantum numbers 14 00:00:49,048 --> 00:00:53,097 apply. To the first, and then second, or 15 00:00:53,097 --> 00:00:56,350 vice-versa. This means that particle exchange implies 16 00:00:56,350 --> 00:01:00,460 the acquisition of a unit amplitude complex phase. 17 00:01:00,460 --> 00:01:03,988 And two distinct phases we see most often in nature are zero and pi, giving an 18 00:01:03,988 --> 00:01:08,250 overall coefficient of plus one or minus one. 19 00:01:08,250 --> 00:01:12,218 These correspond to bosons and fermions with integer or half integer spin 20 00:01:12,218 --> 00:01:17,220 respectively. Electrons are spin one half fermions, and 21 00:01:17,220 --> 00:01:22,000 so, acquire the negative spin upon particle permutation. 22 00:01:22,000 --> 00:01:24,880 For a two electron system, a convenient normalized real space wave function 23 00:01:24,880 --> 00:01:29,354 satisfying this constraint is given here. Switching coordinates r1 and r2 is the 24 00:01:29,354 --> 00:01:34,469 same as multiplying times minus one. However, there is more to the electrons 25 00:01:34,469 --> 00:01:38,420 degrees of freedom than just it's coordinates. 26 00:01:38,420 --> 00:01:42,304 The total wave function is the product. Of this spacial wave length with a spin 27 00:01:42,304 --> 00:01:46,558 wave function constructed from up and down single particle states. 28 00:01:46,558 --> 00:01:51,250 The spatially antisymmetric function must therefore have a spin symmetric wave 29 00:01:51,250 --> 00:01:56,011 function of which there are 3 possible orthogonal states having definite amounts 30 00:01:56,011 --> 00:02:01,500 of spin projections with the spin amount of 0 minus 1. 31 00:02:01,500 --> 00:02:05,190 These three states are therefore called a spin triplet. 32 00:02:05,190 --> 00:02:08,088 If we have a spatially symmetric wave function, then the spin wave function 33 00:02:08,088 --> 00:02:11,813 must be anti-symmetric. There's only one way to do this with 34 00:02:11,813 --> 00:02:17,520 definite total spin along the z-axis. A single state with zero total spin. 35 00:02:17,520 --> 00:02:21,732 This is called the spin singlet. Note that for all these states if one of 36 00:02:21,732 --> 00:02:25,428 the 2 spends is flipped we necessarily change the spin wave function symmetry, 37 00:02:25,428 --> 00:02:30,400 and in turn must change the spacial permutation symmetry as well. 38 00:02:30,400 --> 00:02:34,487 Now the point we're going to make is that these 2 totally antisymmetric states with 39 00:02:34,487 --> 00:02:39,770 different spin and spacial symmetries have different energies. 40 00:02:39,770 --> 00:02:43,960 Let's take an example like the two electron, neutral helium atom. 41 00:02:43,960 --> 00:02:47,080 There's the usual electron nucleus attractive Coulomb interaction for each 42 00:02:47,080 --> 00:02:51,665 of the two electrons independently. But then there's the repulsive Coulomb 43 00:02:51,665 --> 00:02:55,600 between the electrons themselves which involves both coordinates. 44 00:02:56,730 --> 00:03:00,964 Importantly there are no explicit spin spin interactions in the Hamiltonian. 45 00:03:02,140 --> 00:03:06,160 So let's evaluate the energy associated with these wave functions of different 46 00:03:06,160 --> 00:03:09,128 symmetry. Note that the only difference here is the 47 00:03:09,128 --> 00:03:14,220 sign between the product states. Calculating the expectation value of the 48 00:03:14,220 --> 00:03:19,340 energy, the diagonal matrix elements of the hamiltonian gives us many terms. 49 00:03:19,340 --> 00:03:22,972 For instance, there are 4 terms related to intergrals over the part of the 50 00:03:22,972 --> 00:03:26,597 [UNKNOWN] covering electron nucleus [UNKNOWN] interaction of the first 51 00:03:26,597 --> 00:03:32,610 electrion h1. 4 for the second electrion h2. 52 00:03:32,610 --> 00:03:35,820 And then the electron, electron repulsion term h12. 53 00:03:36,900 --> 00:03:40,853 Since H one and H two terms are dependent on variables R one and R two exclusively, 54 00:03:40,853 --> 00:03:46,100 the cross-terms in distributing multiplication in the integrals vanish. 55 00:03:46,100 --> 00:03:50,254 For instance, in this one. Since H one depends only on R one, the R 56 00:03:50,254 --> 00:03:55,324 two integral can be done separatley. But the way functions are orthogonal so 57 00:03:55,324 --> 00:04:01,594 the R two integrals vanish. The same happens for the cross terms in 58 00:04:01,594 --> 00:04:06,687 the h2 integrals. However, h12 depends on both coordinates 59 00:04:06,687 --> 00:04:10,032 r1 and r2. So this procedure cannot be carried out, 60 00:04:10,032 --> 00:04:13,583 in the cross terms remain along with the symmetric and anti-symmetric, plus and 61 00:04:13,583 --> 00:04:18,625 minus sign. These are known as exchange integrals, 62 00:04:18,625 --> 00:04:24,020 and we can write the energy of the wave function as the following. 63 00:04:24,020 --> 00:04:26,862 The plus is for the spacially symmetric and minus if for the spacially 64 00:04:26,862 --> 00:04:30,900 anti-symmetric wave function. This exchange integral J splits the 65 00:04:30,900 --> 00:04:35,116 energies of two particle states that are distinct only by the spin configuration 66 00:04:35,116 --> 00:04:39,332 By an effective interaction, due to the spacial and spin wave function total anti 67 00:04:39,332 --> 00:04:45,260 symmetry constraint. Now you might ask why did we neglect 68 00:04:45,260 --> 00:04:50,950 direct spin spin interaction terms in our Hamiltonian. 69 00:04:50,950 --> 00:04:54,214 The underlying electrostatic route of exchange energy between spins is far 70 00:04:54,214 --> 00:04:57,189 greater than any direct spin spin interaction. 71 00:04:58,370 --> 00:05:02,010 This magnetostatic energy is due to one magnetic moment's zamon energy in the 72 00:05:02,010 --> 00:05:05,258 field of another, which we know from classical E&M, falls off as 1 over 73 00:05:05,258 --> 00:05:10,308 distance cubed. Using the characteristic values of Bohr 74 00:05:10,308 --> 00:05:13,324 magneton and Bohr radius, we find the energy precisely the same as the 75 00:05:13,324 --> 00:05:16,912 interaction scale between the intrinsic electron spin, and the orbital moment of 76 00:05:16,912 --> 00:05:22,437 an atomically bound electron. Evaluating, we once again we find the 77 00:05:22,437 --> 00:05:25,675 fine structure scale. And despite a res-mass energy of mc 78 00:05:25,675 --> 00:05:32,434 squared of 500,000 electron volts. Alpha, 1 over 137 to the 4th power is 79 00:05:32,434 --> 00:05:37,610 extremely small. And the net interaction of millielectron 80 00:05:37,610 --> 00:05:41,250 volts is far smaller than the gross electronic energy scale. 81 00:05:41,250 --> 00:05:47,380 Of electron volts or we can expect our exchange cost j to be. 82 00:05:47,380 --> 00:05:50,530 Even though the underlying mechanism relies more on spatial wave function 83 00:05:50,530 --> 00:05:54,810 symmetry we can capture the net effect of spin exchange with a [UNKNOWN]. 84 00:05:56,990 --> 00:06:00,126 Again, this is because flipping the relevant spin orientation of electrons 85 00:06:00,126 --> 00:06:03,654 coupled electrostatically through their spacial charge distribution, changes the 86 00:06:03,654 --> 00:06:08,270 spacial symmetry and induces enormous changes in energy due to exchange. 87 00:06:09,530 --> 00:06:13,202 This is the quantam mechanical version of dipole to dipole interaction, where we 88 00:06:13,202 --> 00:06:19,418 have replaced the usual vector dipoles. With new ones having poly-matrix 89 00:06:19,418 --> 00:06:23,080 coordinates. Clearly, from a classical viewpoint, if j 90 00:06:23,080 --> 00:06:26,698 is less than zero, this system favors ferromagnetism, since it's lowest energy 91 00:06:26,698 --> 00:06:31,750 ground state occurs when the spins a line and the dot product is positive. 92 00:06:32,800 --> 00:06:36,673 Now how do we evaluate this Hamiltonian? There are spinup and spindown for each 93 00:06:36,673 --> 00:06:39,610 electron. So we expect exactly 2 times 2, 4 94 00:06:39,610 --> 00:06:43,032 eigenstates. This means our finite-sized Hilbert space 95 00:06:43,032 --> 00:06:47,640 much, must grow from the one-electron two by two Hamiltonian to a four by four 96 00:06:47,640 --> 00:06:52,790 Hamiltonian. We do this with a matrix direct product. 97 00:06:52,790 --> 00:06:55,868 Which is carried out simply by multiplying the elements of the first 98 00:06:55,868 --> 00:07:00,560 matrix times the entire second matrix to form a block in the final result. 99 00:07:00,560 --> 00:07:04,493 The dot product between single particle 2 by 2 spin operators then takes this form, 100 00:07:04,493 --> 00:07:08,597 and upon substitution, we see the need to perform straightforward multiplication of 101 00:07:08,597 --> 00:07:15,180 each term, then sum. This result is non-diagonal. 102 00:07:15,180 --> 00:07:18,390 But we know how to diagonalize it. This can even be done by hand. 103 00:07:18,390 --> 00:07:23,683 Here's the eigen values and associated igon vectors obtained by diagonalizing 104 00:07:23,683 --> 00:07:28,167 the hamiltonion. One with eigen value minus 3 j. 105 00:07:28,167 --> 00:07:32,340 And 3 degenerate vectors with eigen value plus j. 106 00:07:32,340 --> 00:07:35,450 What are these states? Let's remember the correspondance between 107 00:07:35,450 --> 00:07:39,315 our 2 component single particle spin vectors and spin up and spin down. 108 00:07:39,315 --> 00:07:43,284 Assigning the first element to spin up and the lower element to spin down, we 109 00:07:43,284 --> 00:07:47,442 can carry out the direct product and see that the components of a 4 element state 110 00:07:47,442 --> 00:07:54,680 vector define a basis of up up. Up down, down up, and down down. 111 00:07:54,680 --> 00:07:58,772 Then it's clear to see that we have simply calculated the familiar singlet 112 00:07:58,772 --> 00:08:02,770 and triplet spin wave functions all over again. 113 00:08:06,790 --> 00:08:10,560 Now we may ask, what's the net effect of exchange in a system with a large number 114 00:08:10,560 --> 00:08:14,780 spins, on a lattice. This may be a model for a crystalline 115 00:08:14,780 --> 00:08:18,820 solid, for example. In general, our Hamiltonian sums over all 116 00:08:18,820 --> 00:08:24,060 possible spin, spin interactions. Which is incredibly difficult to solve. 117 00:08:24,060 --> 00:08:27,293 To simplify this task, let's consider only short range nearest neighbor 118 00:08:27,293 --> 00:08:31,111 interactions. For a given spin at the i-th lattice 119 00:08:31,111 --> 00:08:34,276 site. We're only going to include interaction 120 00:08:34,276 --> 00:08:38,210 terms with the spins one lattice constant around the crystal axis. 121 00:08:38,210 --> 00:08:43,802 So the exchange integrals are all equal. For a square lattice in two dimensions 122 00:08:43,802 --> 00:08:48,405 that's just four neighbors. And in three dimensions, it's six. 123 00:08:48,405 --> 00:08:53,209 Now, even this is a hard problem. But the form of the Hamiltonian suggest 124 00:08:53,209 --> 00:08:56,751 we make a further approximation. That all the nearest neighbors are 125 00:08:56,751 --> 00:08:59,456 identical. Then this Hamiltonian has the same form 126 00:08:59,456 --> 00:09:05,400 as a magnetic moment interacting with a fictitious exchange meal field. 127 00:09:05,400 --> 00:09:08,520 This mean field is what orients a spins parallel in a ferromagnetic insulator 128 00:09:08,520 --> 00:09:11,304 when the thermal energy is low enough for the system To be below a phase 129 00:09:11,304 --> 00:09:16,818 transition. But what about more common ferromagnets 130 00:09:16,818 --> 00:09:19,580 like iron? A metal. 131 00:09:19,580 --> 00:09:24,002 There, the spins are not localized, but more like free electrons distributed in 132 00:09:24,002 --> 00:09:29,544 space. Before describing ferromagnetic metals, 133 00:09:29,544 --> 00:09:32,676 I'd like to review plain old metals This can be thought of as a gas of non 134 00:09:32,676 --> 00:09:37,550 interacting electrons with plane wave, wave functions. 135 00:09:37,550 --> 00:09:40,490 The quantum numbers are just the wave vector components kx, ky, and kz. 136 00:09:40,490 --> 00:09:45,162 By applying energy minimization, we account for all electrons in the system 137 00:09:45,162 --> 00:09:50,820 by filling the lowest energy state at the orgin of k space. 138 00:09:50,820 --> 00:09:54,628 And then remembering poly exclusion, fill in states ever farther isotropically 139 00:09:54,628 --> 00:09:58,350 until every electron has filled the state. 140 00:09:58,350 --> 00:10:01,340 This creates a so called Fermi's sphere with the radius given by the Fermi wave 141 00:10:01,340 --> 00:10:05,178 number ksabeth/g. But this picture does not capture the two 142 00:10:05,178 --> 00:10:11,080 fold spin degenerancy of every state with definite kx, ky and kz. 143 00:10:11,080 --> 00:10:14,135 A better picture, in this regard, is to plot states not in three-dimensional K 144 00:10:14,135 --> 00:10:17,002 space, but simply as a function of energy, determined by the square of the 145 00:10:17,002 --> 00:10:22,340 distance from the origin. Then, we can plot states up to the Fermi 146 00:10:22,340 --> 00:10:26,450 energy, which have the maximum Fermi wave vector. 147 00:10:26,450 --> 00:10:29,622 However, as we move farther away from the origin in K space, we construct a 148 00:10:29,622 --> 00:10:32,846 spherical shell It gets bigger and bigger, and accounts for more electron 149 00:10:32,846 --> 00:10:37,370 states. The density of states, then, at a given 150 00:10:37,370 --> 00:10:41,530 energy, is growing. Plotting the density of states as a 151 00:10:41,530 --> 00:10:44,730 function of energy reveals that for a parabolic dispersion relation that we 152 00:10:44,730 --> 00:10:48,130 have here for mass and particles, gives a square root of energy dependence, and it 153 00:10:48,130 --> 00:10:52,299 is of course the same for spin up and spin down. 154 00:10:54,930 --> 00:10:58,521 Now we can easily but only partially spin polarize this system by applying a 155 00:10:58,521 --> 00:11:02,978 magnetic field. Zeeman splitting states spin up and down 156 00:11:02,978 --> 00:11:08,900 moving while relaxation processes keep the fermi energy constant in equilibrium. 157 00:11:08,900 --> 00:11:11,540 But in a system with farromagnetic exchange. 158 00:11:11,540 --> 00:11:14,957 The mean field, which is far stronger than any terrestrial magnetic field, does 159 00:11:14,957 --> 00:11:19,870 this for us even in the absence of a real external magnetic field. 160 00:11:19,870 --> 00:11:22,997 The density of states at the firmia energy is then asymmetric and a spin 161 00:11:22,997 --> 00:11:26,300 polarization in an equilibrium is maintained. 162 00:11:27,560 --> 00:11:30,773 Now, we might ask how we can make full use of these spin polarize electrons in 163 00:11:30,773 --> 00:11:35,132 ferro magnets. For example, can we transfer this spin 164 00:11:35,132 --> 00:11:38,260 asymmetry to otherwise non magnetic systems? 165 00:11:38,260 --> 00:11:41,550 Either other metals or even semiconductors? 166 00:11:41,550 --> 00:11:44,550 And if we can, what are the processes that govern relaxation of that spin 167 00:11:44,550 --> 00:11:49,400 polarization back to equilibrium in the non magnetic material. 168 00:11:49,400 --> 00:11:53,374 If those time scales are long enough. Can we manipulate spin asymmetry before 169 00:11:53,374 --> 00:11:58,828 equilibrium is obtained? All of these questions are relevant to 170 00:11:58,828 --> 00:12:02,013 exploiting the spin degree of freedom in solid state devices, with the goal of 171 00:12:02,013 --> 00:12:05,002 addressing unique applications not effectively dealt with by ordinary 172 00:12:05,002 --> 00:12:09,970 electronics. This is the field of spin electronics. 173 00:12:09,970 --> 00:12:13,160 Which has impacted us all through the development and use of the giant 174 00:12:13,160 --> 00:12:16,955 magnetoresistance effect and hard drive read sensors, the discovery of which was 175 00:12:16,955 --> 00:12:23,053 recognized in 2007 with the Nobel Prize. In the next lecture, I'll tell you a 176 00:12:23,053 --> 00:12:26,061 little bit about spin polarized electron transport, especially the unique 177 00:12:26,061 --> 00:12:29,950 difficulties of spin injection. And what you can learn once that 178 00:12:29,950 --> 00:12:31,530 challenge has been resolved.