1 00:00:01,320 --> 00:00:04,560 This is the last part of my guest lectures on electron spin, in which I'll 2 00:00:04,560 --> 00:00:08,178 focus on normally non-magnetic systems which are pushed out of spin degeneracy 3 00:00:08,178 --> 00:00:12,092 equilibrium. By spin injection from an external 4 00:00:12,092 --> 00:00:14,795 thermomagnetic source. I'm going to first tell you why we 5 00:00:14,795 --> 00:00:17,504 want to study these systems, and why this is such a hard problem or at least why 6 00:00:17,504 --> 00:00:22,280 the most straight forward approach to solving it is bound to fail. 7 00:00:22,280 --> 00:00:25,241 Then I'll show you one particular way that I have personally used, to obtain 8 00:00:25,241 --> 00:00:29,399 spin injection in silicon and germanium semiconductor devices. 9 00:00:29,399 --> 00:00:31,580 And a little bit about what can be learned as a result. 10 00:00:34,460 --> 00:00:37,980 To motivate the kinds of information about non-equilibrium spin transport 11 00:00:37,980 --> 00:00:41,225 we're after, I want to appeal to the history of non-equilibrium charge 12 00:00:41,225 --> 00:00:45,085 transport. Namely minority carriers and 13 00:00:45,085 --> 00:00:47,724 semi-conductors. The seminal measurements were done by 14 00:00:47,724 --> 00:00:50,990 Haynes and Shockley in the mid-20th century at Bell Labs. 15 00:00:50,990 --> 00:00:54,752 In these timed domain experiments, a narrow pulse of minority electrons, were 16 00:00:54,752 --> 00:00:59,285 injected into p-type semi-conductors filled with equilibrium holes. 17 00:00:59,285 --> 00:01:02,903 An electric field carried these electrons to a charge detector, where the pulse 18 00:01:02,903 --> 00:01:07,471 could be analyzed. By measuring the time of flight, they 19 00:01:07,471 --> 00:01:11,619 could determine the minority carrier mobility or the proportionality constant 20 00:01:11,619 --> 00:01:15,985 between applied electric field and electron velocity. 21 00:01:15,985 --> 00:01:19,536 By measuring the spreading of the pulse in time, they determined the strength of 22 00:01:19,536 --> 00:01:22,610 random thermal fluctuations from scattering, the minority carrier 23 00:01:22,610 --> 00:01:27,492 diffusion coefficient. And by integrating over the pulse and 24 00:01:27,492 --> 00:01:30,976 determining how many electrons made it without annihilating with a positively 25 00:01:30,976 --> 00:01:36,120 charged hull, the minority carrier lifetime could be determined. 26 00:01:36,120 --> 00:01:39,406 Without these values, none of the solid state devices we use today could be 27 00:01:39,406 --> 00:01:44,085 designed and successfully made. So, if we speculate any use for 28 00:01:44,085 --> 00:01:48,663 spin-polarized electrons out of equilibrium in semi-conductor devices. 29 00:01:48,663 --> 00:01:51,908 Then we at least need to be able to measure the spin analogs of these 30 00:01:51,908 --> 00:01:55,671 parameters. So, let's first look at why it's not so 31 00:01:55,671 --> 00:02:01,264 easy to transfer the substantial spin and balance, spin polarization. 32 00:02:01,264 --> 00:02:05,217 From a ferromagnet, into a non-magnetic electronic material, especially when 33 00:02:05,217 --> 00:02:08,816 using a semiconductor in an effort to make, for example, new kinds of spin 34 00:02:08,816 --> 00:02:13,519 transistors. The most straight forward way one might 35 00:02:13,519 --> 00:02:16,805 naively try, is to make an ohmic or linearly resistive contact between the 36 00:02:16,805 --> 00:02:21,490 ferromagnet and a semiconductor. Let's see how this works for plain old 37 00:02:21,490 --> 00:02:25,030 charge injection. Ohm's law says that the charge current 38 00:02:25,030 --> 00:02:28,330 density flowing, is proportional to the conductivity, and driven by a spacial 39 00:02:28,330 --> 00:02:32,430 gradient of an electro-chemical potential. 40 00:02:32,430 --> 00:02:35,654 This combines the effects of electric field, gradient of electro-static 41 00:02:35,654 --> 00:02:38,774 potential, with flow from high concentration to low via random thermal 42 00:02:38,774 --> 00:02:43,730 fluctuations comprising the random watts of diffusion. 43 00:02:43,730 --> 00:02:47,415 If we incorporate the device geometry, we can use this expression to recover the 44 00:02:47,415 --> 00:02:50,590 more familiar V equals IR form of Ohms Law. 45 00:02:50,590 --> 00:02:55,134 In metal-semiconductor ohmic contacts, current is conserved across the interface 46 00:02:55,134 --> 00:02:58,640 but the connectivity in the metal is large. 47 00:02:58,640 --> 00:03:02,800 So, the potential energy provided by voltage q times v drops mostly across the 48 00:03:02,800 --> 00:03:07,424 lower connectivity semiconductor. Now, if we want spin injection to 49 00:03:07,424 --> 00:03:11,243 accompany this charge injection, then, the currents for spin up and down must be 50 00:03:11,243 --> 00:03:15,438 different. Since the conductivities for up and down 51 00:03:15,438 --> 00:03:20,266 are the same in a semi-conductor, the re, the respective electrochemical gradients 52 00:03:20,266 --> 00:03:24,060 must be different. This, is what we need to happen on the 53 00:03:24,060 --> 00:03:27,238 semi-conductor side. Spin up and down electrochemical 54 00:03:27,238 --> 00:03:30,878 potentials have different gradients to drive asymmetric current densities 55 00:03:30,878 --> 00:03:35,059 comprising a spin current. The unavoidable consequence of this 56 00:03:35,059 --> 00:03:39,276 asymmetry, is an electrochemical splitting at the ohmic interface. 57 00:03:39,276 --> 00:03:42,604 Using Ohm's law, we can obtain a relationship between current polarization 58 00:03:42,604 --> 00:03:45,598 and this electrochemical potential splitting. 59 00:03:45,598 --> 00:03:49,054 The first tern, in parenthesis, is due to the average potential drop over the 60 00:03:49,054 --> 00:03:54,566 transport length scale, L. Note that, although, J up and J down are 61 00:03:54,566 --> 00:04:01,230 not equal, their sum does equal the total charge current, as expected. 62 00:04:01,230 --> 00:04:05,199 Now, we have to derive equivalent expressions, on the ferromagnet side, or 63 00:04:05,199 --> 00:04:10,230 we will see the deleterious effect of the splitting on spin eject. 64 00:04:10,230 --> 00:04:12,966 On the ferromagnet side, the electrochemical potential splitting 65 00:04:12,966 --> 00:04:17,065 relaxes to zero in equilibrium, due to spin flips away from the interface. 66 00:04:17,065 --> 00:04:22,190 The spin relaxation length scale is a so called spin diffusion length, lambda. 67 00:04:23,460 --> 00:04:27,117 By once again applying Ohm's Law, we get the following expressions for the current 68 00:04:27,117 --> 00:04:31,020 densities of spin up and, and down on the ferromagnet side. 69 00:04:32,180 --> 00:04:35,430 Note that there are two important differences between these expressions for 70 00:04:35,430 --> 00:04:39,890 the ferromagnet and the ones above describing transport in a semi-conductor. 71 00:04:39,890 --> 00:04:43,547 First, the spin dependent connectivities are not equal to the ferromagnet, due to 72 00:04:43,547 --> 00:04:47,130 their dependence on the asymmetric carrier densitites. 73 00:04:47,130 --> 00:04:51,035 Second, and as a result, the deviation of the interface electrochemical potentials 74 00:04:51,035 --> 00:04:55,141 from equilibrium are not symmetric. In other words, c-up is not equal to 75 00:04:55,141 --> 00:04:59,448 c-down in this figure. However, because the ideal interface 76 00:04:59,448 --> 00:05:03,078 preserves spin, the electrochemical potentials are continuous. 77 00:05:03,078 --> 00:05:06,795 So, we do have the sum rule, giving us a total splitting, which we need to 78 00:05:06,795 --> 00:05:10,512 determine the spin polarization, flowing across the interface in the 79 00:05:10,512 --> 00:05:14,540 semiconductor. Using the definitions of both the 80 00:05:14,540 --> 00:05:17,940 injected current polarization and the bulk ferromagnetic polarization, we can 81 00:05:17,940 --> 00:05:22,306 derive a simple expression for the splitting on the ferromagnet side. 82 00:05:22,306 --> 00:05:25,222 In intimate ohmic contact, the electrochemical potentials are 83 00:05:25,222 --> 00:05:30,410 continuous, so this is the same as the splitting of the semiconductor side. 84 00:05:30,410 --> 00:05:34,619 We can therefore substitute it into our previously-derived expression to obtain 85 00:05:34,619 --> 00:05:38,956 this result. Note that this is very different from our 86 00:05:38,956 --> 00:05:41,798 naive expectation, since it depends strongly on the magnitude of the 87 00:05:41,798 --> 00:05:46,297 dimensionalist parameter, epsilon. The ratio of conductivities and transport 88 00:05:46,297 --> 00:05:51,060 lengths across the interface. If epsilon is much less than 1, then, 89 00:05:51,060 --> 00:05:55,959 only when the bulk magnetic polarization, beta, is approximately 1 a half-metallic 90 00:05:55,959 --> 00:06:00,580 ferromagnet. Do we recover the desired case, where the 91 00:06:00,580 --> 00:06:04,582 injected current polarization P, is approximately equal to the bulk 92 00:06:04,582 --> 00:06:12,190 ferromagnetic polarization beta. Unfortunately, the bulk polarization of 93 00:06:12,190 --> 00:06:17,930 typical ferromagnets is around 50%, so as this plot shows ohmic injection is doomed 94 00:06:17,930 --> 00:06:23,612 unless epsilon is at least 0.01. However, the relevant materials 95 00:06:23,612 --> 00:06:26,890 properties are not forgiving in this respect. 96 00:06:26,890 --> 00:06:30,494 The ratio of conductivities between a semi-conductor and a ferromagnetic metal 97 00:06:30,494 --> 00:06:34,500 is significantly below unity. Even for highly-doped semi-conductors, 98 00:06:34,500 --> 00:06:37,625 and highly disordered, amorphous ferromagnetic metals. 99 00:06:37,625 --> 00:06:42,114 Likewise, the ratio of length scales is small, due to the fast spin relaxation in 100 00:06:42,114 --> 00:06:46,469 the ferromagnet, leading to a spin diffusion length Lambda of approximately 101 00:06:46,469 --> 00:06:51,591 ten nanometers. Whereas in semiconductors with low spin 102 00:06:51,591 --> 00:06:55,879 orbit interaction, such as silicon, transport lengths can be ten microns or 103 00:06:55,879 --> 00:07:00,220 longer even at elevated ambient temperatures. 104 00:07:00,220 --> 00:07:04,508 Therefore, even in the best scenario, epsilon is approximately ten to the minus 105 00:07:04,508 --> 00:07:07,788 four. Leading to the negligible polarization 106 00:07:07,788 --> 00:07:11,725 shown in the figure. Over the range of expected values for 107 00:07:11,725 --> 00:07:16,210 epsilon, one needs bulk polarization of at least 95% for injected polarization of 108 00:07:16,210 --> 00:07:22,422 greater than 10%, or so. So, elemental ferromagnets, Iron, Cobalt 109 00:07:22,422 --> 00:07:29,070 and Nickel, are you useless for spin injection in the ohmic regime. 110 00:07:29,070 --> 00:07:32,220 In fact, the problem is evident even graphically. 111 00:07:32,220 --> 00:07:35,496 The splitting delta mu, which is necessary for a non-zero injected current 112 00:07:35,496 --> 00:07:38,824 polarization, also tends to reverse the spin up electrochemical potential 113 00:07:38,824 --> 00:07:43,160 gradient at the interface on the ferromagnet side. 114 00:07:43,160 --> 00:07:46,640 Inhibiting injection of the very spin specise we want to inject into the 115 00:07:46,640 --> 00:07:50,024 semi-conductor. Therefore, in order to maintain the 116 00:07:50,024 --> 00:07:53,632 constraint of current conservation across the interface. 117 00:07:53,632 --> 00:07:57,280 The steady state inter-facial splitting is small, and the injected current 118 00:07:57,280 --> 00:08:01,260 polarization, p, is negligible. Modern techniques to overcome this 119 00:08:01,260 --> 00:08:03,822 problem include quantum mechanical tunneling. 120 00:08:03,822 --> 00:08:07,122 And, in my lab Ballistic hot electron injection, which circumvents the issues 121 00:08:07,122 --> 00:08:11,296 relevant for ohmic injection, here. I'm not going to describe the details of 122 00:08:11,296 --> 00:08:15,048 spin injection and detection, that's a whole other course in device physics and 123 00:08:15,048 --> 00:08:19,000 magnetism. But rather, what we can learn from 124 00:08:19,000 --> 00:08:22,877 measurements of spin transport and manipulation, by any means. 125 00:08:22,877 --> 00:08:25,937 The key to extracting the most information from these measurements, is 126 00:08:25,937 --> 00:08:30,890 exploiting a topic I mentioned several segments ago, spin procession. 127 00:08:30,890 --> 00:08:34,428 Again, this is magnetic analog of a spinning top or gyroscope, with an off 128 00:08:34,428 --> 00:08:38,350 axis gravitational force causing a mechanical torque. 129 00:08:39,600 --> 00:08:42,948 In spin transport devices, we apply a magnetic field perpendicular to the 130 00:08:42,948 --> 00:08:46,827 injective spin direction. But parallel to the transport direction 131 00:08:46,827 --> 00:08:50,599 caused by electric fields and the spin with process in a plane. 132 00:08:52,350 --> 00:08:55,714 The final spin procession angle, is determined by the product of spin 133 00:08:55,714 --> 00:08:59,948 procession frequency determined by magnetic field strength. 134 00:08:59,948 --> 00:09:03,368 About 28 gigahertz per Tesla in a material with weak spin orbit coupling 135 00:09:03,368 --> 00:09:07,073 like silicon, and the transit time inversely proportional to electric field 136 00:09:07,073 --> 00:09:12,701 strength. If we apply a perpendicular magnetic 137 00:09:12,701 --> 00:09:16,796 field with the appropriate strength. Then you cause the spins to process an 138 00:09:16,796 --> 00:09:20,100 average of 180 degrees, fully flipping with respect to their injected 139 00:09:20,100 --> 00:09:24,466 polarization. Your experimental measurement of sigma z, 140 00:09:24,466 --> 00:09:28,550 the spin along the initialization axis, will then vary. 141 00:09:28,550 --> 00:09:31,930 Doubling the magnetic field doubles the procession frequency and therefore 142 00:09:31,930 --> 00:09:35,730 results in an average procession angle of 360 degrees. 143 00:09:35,730 --> 00:09:39,400 A coherent full rotation restoring the expectation value of sigma z. 144 00:09:40,600 --> 00:09:43,165 As you can see from this actual experimental data, it doesn't matter 145 00:09:43,165 --> 00:09:46,660 whether the field polarity is positive or negative. 146 00:09:46,660 --> 00:09:49,476 In other words, it doesn't matter if the spin processes clockwise or counter 147 00:09:49,476 --> 00:09:52,605 clockwise. Now if all electrons had the same transit 148 00:09:52,605 --> 00:09:56,685 time from injector to detector, We would expect this cosine-like oscillation to 149 00:09:56,685 --> 00:10:02,495 continue indefinitely for higher and higher orders of procession rotations. 150 00:10:02,495 --> 00:10:06,925 But that's not what happens. In reality, not all electrons have the 151 00:10:06,925 --> 00:10:11,690 same transit time due to random scattering processes. 152 00:10:11,690 --> 00:10:15,162 Therefore, an uncertainty in transit time gives rise to an uncertainty in 153 00:10:15,162 --> 00:10:19,577 procession angle. When the precession frequency grows in 154 00:10:19,577 --> 00:10:22,713 higher an higher magnetic field, the affects of partial cancellation can be 155 00:10:22,713 --> 00:10:27,940 seen an the oscillations diminish. We can model this measurement with a 156 00:10:27,940 --> 00:10:31,880 transport simulation. Summing up the cosine like contributions 157 00:10:31,880 --> 00:10:35,592 from electrons, or the distribution of arrival times, in order to fit the non 158 00:10:35,592 --> 00:10:40,430 equilibrium spin mobility. And diffusion coefficients we're after. 159 00:10:41,600 --> 00:10:45,110 However, there's a model independent method with far greater utility. 160 00:10:45,110 --> 00:10:49,054 The key is to recognize that this integral summation, is really just a 161 00:10:49,054 --> 00:10:53,705 fourier transform. Therefore, the oscillations we measure, 162 00:10:53,705 --> 00:10:57,173 can be inverted to yield the empirical transport distribution without any model 163 00:10:57,173 --> 00:11:01,948 dependence whatsoever. In this example we can see the effects of 164 00:11:01,948 --> 00:11:05,562 increasing the electric field. Oscillation period increases, and the 165 00:11:05,562 --> 00:11:10,028 number of oscillation themselves grows. But the transformed, clearly shows that 166 00:11:10,028 --> 00:11:15,130 this is the result of smaller from mean and transit time and standard deviation. 167 00:11:15,130 --> 00:11:19,490 This method of obtaining time of flight is called the Larmor clock. 168 00:11:19,490 --> 00:11:22,946 We don't make a explicit measurement of transit time, we measure the angle of 169 00:11:22,946 --> 00:11:26,564 rotation at a known angular velocity, the same way we measure time from an analog 170 00:11:26,564 --> 00:11:30,253 clock. We know the rotation speed of the hand 171 00:11:30,253 --> 00:11:33,698 for the clock, 360 degrees per hour for the minute hand, and infer time from the 172 00:11:33,698 --> 00:11:38,792 instantatous oreintation. We're likewise measuring the spin 173 00:11:38,792 --> 00:11:43,575 orientation and determining how long it processed in a known magnetic field. 174 00:11:43,575 --> 00:11:47,791 For measurements of spin transport, we can correlate the transit time with final 175 00:11:47,791 --> 00:11:52,590 spin polarization and extract the spin lifetime. 176 00:11:52,590 --> 00:11:56,510 In silicon, we can see that, although the non-equilibrium lifetimes of hundreds of 177 00:11:56,510 --> 00:12:00,859 nanoseconds are fairly long. In comparison to the momentum relaxation 178 00:12:00,859 --> 00:12:04,331 time of picoseconds or less, they're strongly dependent on temperature, 179 00:12:04,331 --> 00:12:08,970 increasing dramatically as the sample is cooled. 180 00:12:08,970 --> 00:12:11,940 This demonstrates the importance of relaxation, via a nominally 181 00:12:11,940 --> 00:12:15,342 spin-independent process, electrons scattering off of thermal phonons, 182 00:12:15,342 --> 00:12:20,362 distortions in the crystal lattice. This electron-phonon spin relaxation 183 00:12:20,362 --> 00:12:24,394 process, results from the fact that due to the weak but non-zero spin-orbit 184 00:12:24,394 --> 00:12:28,045 coupling. The electron wave functions are not pure 185 00:12:28,045 --> 00:12:32,670 spin eigenstates up and down. Rather, spin up has a small amount of 186 00:12:32,670 --> 00:12:37,170 spin down and vice versa, but remain fully orthogonal. 187 00:12:37,170 --> 00:12:40,602 We can calculate the transition rate between these states constituting a spin 188 00:12:40,602 --> 00:12:44,034 flip, due to momentum scattering of these free electrons from wave vector k to k 189 00:12:44,034 --> 00:12:48,200 prime. By using the so-called Fermi Golden rule. 190 00:12:49,490 --> 00:12:52,370 This first order expression is proportional to the square of the matrix 191 00:12:52,370 --> 00:12:55,202 element of a scattering potential, coupling the two initial and final 192 00:12:55,202 --> 00:12:58,475 states. And the density of final states row. 193 00:12:58,475 --> 00:13:02,763 Now, even if the scattering potential only couples states of different momenta 194 00:13:02,763 --> 00:13:05,873 k. Due to the spin-orbit mixing of the wave 195 00:13:05,873 --> 00:13:10,117 function, we see that there is a non-zero matrix element. 196 00:13:10,117 --> 00:13:14,537 And this exactly equal to the quantity determining the spin preserving momentum 197 00:13:14,537 --> 00:13:17,958 relaxation rate. The spin relaxation is therefore 198 00:13:17,958 --> 00:13:21,426 proportional to the momentum relaxation, and also proportional to the square of 199 00:13:21,426 --> 00:13:24,550 the typically small spin mixing amplitude. 200 00:13:26,360 --> 00:13:29,576 This being the end of my contribution to this course, I'm obliged to acknowledge 201 00:13:29,576 --> 00:13:33,453 support, not only from my experimental research on spin transport. 202 00:13:33,453 --> 00:13:36,573 But also support for scientific outreach efforts for students and the public 203 00:13:36,573 --> 00:13:40,440 outside my institution, the University of Maryland. 204 00:13:40,440 --> 00:13:43,275 In particular, the National Science Foundation Career Award has made this 205 00:13:43,275 --> 00:13:47,510 work possible. It's been my great pleasure to share this 206 00:13:47,510 --> 00:13:51,531 quick story of electron spin with you. And I invite you to learn more about spin 207 00:13:51,531 --> 00:13:54,381 through own study and perhaps even original research in Physics and 208 00:13:54,381 --> 00:13:57,210 Engineering labs around the world.