This is the last part of my Guest Lectures on Electron Spin, in which I'll focus on normally non-magnetic systems which are pushed out of spin degenerate equilibrium by spin injection from an external thermomagnetic source. I'm going to first tell you why we want to study these systems, and why this is such a hard problem, or at least why the most straightforward approach to solving it is bound to fail. Then I'll show you one particular way that I have personally used to obtain spin injection in silicon and germanium semi-conductor devices. And a little bit about what can be learned as a result. To motivate the kinds of information about non-equilibrium spin transport we're after, I want to appeal to the history of non-equilibrium charge transport. Namely, minority carriers and semiconductors. The seminal measurements were done by Haynes and Shockley in the mid 20th century at Bell Labs. In these time-domain experiments, a narrow pulse of minority electrons were injected into p-type semiconductors filled with equilibrium holes. An electric field carried these electrons to a charge detector, where the pulse could be analyzed. By measuring the time of flight they could determine the minority carrier mobility or the proportionality constant between applied electric field and electron velocity. By measuring the spreading of the pulse in time, they determine the strength of random thermal fluctuations from scattering, the minority carrier diffusion coefficient. And by integrating over the pulse and determining how many electron made it without annihilating with a positively charged hole, the minority carrier lifetime could be determined. Without these values, none of the solid state devices be use today can be designed and successfully made. So, if we speculate any use for a spin polarized electrons out of equilibrium in semi-conductor devices, then we at least need to be able to measure the spin analogs of these parameters. So, let's first look at why it's not so easy to transfer the substantial spin imbalance, spin polarization from a feral magnet into a non-magnetic electronic material. Especially when using a semiconductor in an effort to make, for example, new kinds of spin transistors. The most straightforward way one might naively try is to make an ohmic or linearly resistive contact between the feral magnet and the semiconductor. Let's see how this works for plain old charge injection. Ohm's law says that the charge current density flowing is proportional to the conductivity and driven by spacial gradient of an electrochemical potential. This combines the effects of electric field gradient of electrostatic potential with flow from high concentration to low via random thermal fluctuations comprising the random walks of diffusion. If we incorporate the device geometry, we can use this expression to recover the more familiar V=IR form of Ohm's Law. In metal-semiconductor ohmic contacts, current is conserved across the interface but the conductivity in metal is large so the potential energy provided by voltage qV drops mostly across the lower conductivity semiconductor. Now, if you want spin injection to accompany this charge injection then the currents for spin up and down must be different. Since the connectivities for up and down are the same in the semiconductor, their respective electrochemical gradients must be different. This is what we need to happen on the semiconductor side. Spin up and down electrochemical potentials have different gradients to drive asymmetrical current densities comprising of spin current. The one avoidable consequence of this asymmetry is electrochemical splitting at the ohmic inter-phase. Using Ohm's Law we can obtain a relationship between current polarization and this electrochemical potential splitting. The first term in parenthesis is due to the average potential drop over the transport length scale L. Note that although J up and J down are not equal, their sum does equal the total charge current as expected. Now, we have to derive equivalent expressions on the ferromagnetic side where we will see the deleterious effect of the splitting on spin inject. On the ferromagnetic side, the electrochemical potential splitting relaxes to zero in equilibrium due to spin flips away from the interface. The spin relaxation length scale is the so-called spin diffusion length lambda. By once again applying Ohm's Law we get the following expressions for the current densities of spin up and down on the ferromagnetic side. Note that there are two important differences between these expressions for the ferromagnet and the ones above describing transport in a semiconductor. First, the spin dependent connectivities are not equal in the ferromagnet due to their dependence on the asymmetric carrier densities. Second, and as a result the deviation of the interface lecture chemical potential from equilibrium are not symmetric. In other words, C up is not equal to C down in this figure. However, because the ideal interface preserve spin, the electrochemical potentials are continuous. So, we do have the sum rule giving us a total splitting, which we need to determine the spin polarization flowing across the interface in the semiconductor. Using the definitions of both the injected current polarization and the bulk ferromagnetic polarization, we can derive a simple expression for the splitting on the ferromagnetic side. In intimate omega contact, the electrochemical potentials are continuous. So this is the same as a splitting on the semiconductor side. We can therefore substitute it into our previously derived expression to obtain this result. Note that this is very different from our naive expectation, since it depends strongly on the magnitude of a dimensionalist parameter Epsilon, the ratio of connectivities and transport lengths across the interface. If Epsilon is much less than 1 then only when the bulk magnetic polarization beta is approximately 1, a half metallic ferromagnet, do we recover the desired case where the injected current polarization P is approximately equal to the bulk ferromagnetic polarization beta. Unfortunately, the bulk polarization of a typical ferromagnets is around 50%. So as this plot shows, I'll make an injection is doomed in the last epsilon is at least .01. However, the relevant materials properties are not forgiving in this respect. The ratio of conductivities between a semi-conductor and a ferromagnetic metal is significantly below unity, even from highly dope semi-conductors and how we disordered amorphous ferromagnetic metals. Like wise the ratio of length scales is small due to the fast spin relaxation in the ferromagnetic leading to the spin defusion length lambda of approximately tem nanometers. Where as in semiconductors with low spin orbit interaction such as silicon. Transport lengths can be ten microns or longer, even at elevated ambient temperatures. Therefore, even in the best scenario, epsilon is approximately 10 to the minus 4, leading to the negligible polarization showing the figure. Over the range of expected values for epsilon, one needs bulk polarization of at least 95% for injected polarization of greater than 10% or so. So elemental ferro magnets iron, cobalt, and nickel are useless for spin injection in the Ohmic region. In fact, the problem is evident even graphically. The splitting delta mu which is necessary for a non-zero injected current polarization also tends to reverse the spin of electro-chemical potential gradient at the interface on the ferromagnetic side. Inhibiting injection, the very spin species we want to inject into the semiconductor. Therefore in order to maintain the constraint of current conservation across the interface the study state inter-facial splitting is small and the injected current polarization, P is negligible. Modern techniques to over come this problem include quantum mechanical tunneling, and in my lab ballistic hot electron injection, which circumvents the issues relevant for ohmic injections here. I'm not going to describe the details of spin injection and detection, that's a whole other course. In device physics and magnetism, but rather what we can learn from measurements of spin transport and manipulation by any mean. The key to extracting the most information from these measurements is exploiting a topic I mentioned several seconds ago spin precession. Again this is the magnetic analog of a spinning top or gyroscope, with off axis gravitational force, causing a mechanical torque. In spin transit devices, we apply a magnetic field perpendicular to the injected spin direction but parallel to the transport direction and caused by electric fields. And the spin will precess, in a plane. The final spin precession angle is determined by the product of spin procession frequency, determined by magnetic field strength, about 28 gigahertz per Tesla. And a material with weak spin orbit coupling like silicon, and the transit time inversely proportional to electric field strength. If we apply a perpendicular magnetic field with the appropriate strength, then you can cause the spins to precess an average of 180 degrees, fully flipping with respect to their ejected polarization. Your experimental measurement of sigma, the spin along the initialization excess will then vary. Doubling the magnetic field, doubles the procession frequency and therefore results in an average procession angle of 360 degrees. A coherent full rotation restoring the expectation value of sigma Z, as you can see from this actual experimental data, it doesn't matter whether the field polarity is positive or negative. In other words it doesn't matter if the spin processes clockwise or counterclockwise. Now if all the electrons have a same transit time injector to detector. We would expect this cosine like oscillation to continue indefinitely for higher and higher orders of precession rotations, but that's not what happens. In reality, not all electrons have the same transit time due to random scattering processes. Therefore an uncertainty in transit time gives rise to an uncertainty in precession of angle. When the procession frequency grows in higher and higher magnetic field, the effects of partial cancellation can be seen and the oscillations diminish. We can model this measurement with a transport simulation, summing up the cosine-like contributions from electrons where the distribution of arrival times in order to fit the non-equilibrium spin mobility and diffusion coefficient we're after. However, there's a model independent method with far greater utility. The key is to recognize that this integral summation is really just a Fourier transform. Therefore, the oscillations we measure can be inverted to yield the empirical transport distribution without any model dependence whatsoever. In this example we can see the effects of increasing the electric field. Oscillation period increases, and the number of oscillations themselves grows. But the transforms clearly show that this is a result of smaller mean transit time and standard deviation. This method of obtaining time of flight is called the Larmor clock. We don't make an explicit measurement of transit time. We measure the angle of rotation at a known angular velocity. The same way we measure time from an analog clock. We know the rotation speed of the hand for the clock, 360 degrees per hour for the minute hand, and infer time from the instantaneous orientation. We're likewise measuring the spin orientation and determining how long it processed in a known magnetic field. For measurements of spin transport, we can correlate the transit time with final spin polarization and extract the spin lifetime. In silicon, we can see that although the non-equilibrium lifetimes of hundreds of nanoseconds are fairly long In comparison to the momentum relaxation time of pico seconds or less is strongly dependent on temperature, increasing dramatically as the sample is cooled. This demonstrates the importance of relaxation via a nominally spin-independent process, electron scattering off of thermal phonons, distortions in the crystal lattice. This electron phonons spin relaxation process results from the fact that, do to the weak but non zero spin orbit coupling, the electron wave functions are not pure spin eigenstates up and down. Rather spin up has a small amount of spin down and vice versa, but remain fully orthogonal. We can calculate the transition rate between these states constituting a spin flip due to momentum scattering of these free electrons from wave vector k to k prime by using using the so called Fermi golden rule. This first order expression is proportional to the square of the matrix element of the scattering potential, coupling the two initial and final states, and the density of final states row. Now, even if the scattering potential only couples states of different momentacae due to the spin order mixing of the wave function, we see that there's a non-zero matrix element. And this is exactly equal to the quantity determining the spin preserving momentum relaxation rate. The spin-relaxation is therefore proportional to the momentum relaxation and also proportional to the square but typically small spin mixing amplitude. This being the end of my contribution to this course I'm obliged to acknowledge support, not only for my experimental research on spin transport, but also support for scientific outreach efforts to students and the public outside my institution, the University of Maryland. In particular, the National Science Foundation Career Award has made this work possible. It's been my great pleasure to share this quick story of electron spin with you. And I invite you to learn about more about Spin through your own study and perhaps even original research in physics and engineering labs around the world.
