SPIN RELAXATION

We reviewed mechanisms of spin relaxation in electronic systems (metals and semiconductors) in  J. Fabian and S. Das Sarma, J. Vac. Sc. Technol. B 17, 1708 (1999).

The spin-hot-spot model was introduced in J. Fabian and S. Das Sarma, Phys. Rev. Lett. 81,5624 (1998). The first realistic calculation of spin relaxation time (T1) in a metal (aluminum) was reported in J. Fabian and S. Das Sarma, Phys. Rev. Lett. 83, 1211 (1999). A semi-popular account of how band-structure affects  spin relaxation is in J. Fabian and S. Das Sarma, J. Appl. Phys. 85, 5075 (1999).

My own spintronics research focuses on theoretical understanding of the ways electron spins decay in metals and semiconductors (slide1, slide2). As was shown in the 50's and 60's by Overhauser, Elliott, and Yafet, there are two ways for spins to decay, and both include spin-orbit coupling of some kind. First, impurities can induce a spin-orbit interaction that can flip an electron spin. Second, a spin-orbit interaction can be induced by host-lattice ions. The second mechanism is important at high temperatures where electrons scatter off phonons, but also at low temperatures, if the impurities are light—meaning they induce small spin-orbit coupling. The second mechanism is somewhat tricky. One has to realize (as Elliott did first) that in the presence of spin-orbit coupling, spin up and spin down states are no longer good quantum numbers. Instead, new Bloch eigenstates are mixtures of spin up and down states, although usually one state dominates and electrons can still be called "up" and "down."  As a result, even scalar (spin-independent) interactions due to impurities or phonons can cause spins to flip (slide). First CESR (conduction electron spin resonance) experiments seemed to be consistent with the above--now called Elliott-Yafet--picture, although no real calculation based on this mechanism had been done. One interesting observation of Yafet was that spin relaxation rate should be proportional to resistivity (equivalently, to momentum relaxation rate). The coefficient of proportionality is b^2, which is the probability of finding a Bloch state with spin down, if the large amplitude is spin up (that is, the Bloch state is in state "up").  Monod and Beuneu set out to check this relation and thus the Elliott-Yafet theory itself. They took b^2 values from atomic physics, collected then known data of spin relaxation rates of different metals, and tried to reproduce the Gruneisen behavior, as explained in the following Figures. The result was surprising:

Vertically the above figure plots spin relaxation rate (Gamma s) divided by spin-orbit-coupling strength (b^2) and resistivity at Debye temperature Td. Horizontal scale is given by the reduced temperature, T/Td.  Monod and Beuneu expected that data for all metals will fall onto a single, Gruneisen-like curve. Only alkali and noble metals followed the expected behavior.

The above is what Monod and Beuneu expected to find: a nice Gruneisen scaling valid for all (simple) metals. Reduced resistivity R/Rd (Rd is the resistivity at Td) is plotted as a function of reduced temperature T/Td and is found to be a single, universal function of T/Td, as first noticed by Gruneisen.

Together with S. Das Sarma, I explained the  Monod-Beuneu surprising result by pointing out that the two groups of metals (one that follows the scaling and one with spin relaxation rates off by orders of magnitude) have different valence: monovalent metals (Na, Cu, ...) follow the Gruneisen behavior, while polyvalent metals (Al, Pd, Be, and Mg) do not. What is so peculiar about polyvalent metals? Band structure. Because of the complicated character of Bloch bands in polyvalent metals one cannot use b^2 from atomic physics. Instead, b^2 is band-renormalized by the presence of band-structure anomalies--spin hot spots. Spin hot spots are points on the Fermi surface where the surface cuts through a Brillouin zone boundary, special symmetry point, or a line of accidental degeneracy. If an electron jumps from (or, into) a spin hot spot, the electron’s spin flips with much larger probability than usual. Since the resulting spin-flip probability is an average over the whole Fermi surface, on has to know how large spin hot spots are. We showed that they are large enough to completely monopolize spin relaxation: to calculate the average it suffices to count contributions from spin hot spots only. 

Above: The spin-hot-spot model was introduced to explain the Monod-Beuneu scaling. For alkali and noble metals, which are monovalent, an electron performing a random walk on the Fermi surface has a small chance of flipping its spin everywhere on the surface. All Fermi states are equivalent. By contrast, polyvalent metals (like Al, Be, Mg, and Pd from the Monod-Beuneu graph) have so called spin hot spots (red) where a chance of a spin flip is much greater, by several orders of magnitude.

The following Figures illustrate the occurrence of spin hot spots on the Fermi surface of Aluminum:

Stereograph of the almost spherical Fermi surface of aluminum. The center of the stereograph is the north pole, the circumference is the equator. The Fermi surface cuts through the Brillouin zone boundaries at white areas (here the Bragg scattering makes some spherical angles inaccessible for electrons). The blue part is the second band, the red is the third band of Al (the first band is completely occupied--aluminum has 3 valence electrons).

Above: A quarter of the stereograph with only spin hot spots (points with the largest spin-flip probabilities) shown. Violet points are at the Brillouin zone boundaries, blue, green, and yellow are around accidental degeneracy points which are red.

The spin-hot-spot model is a general concept which explains the details of spin relaxation in metals within the framework of the Elliott-Yafet mechanism. A real calculation that would clearly show, without any fitting or adjusting, that the mechanism works, was still lacking. We therefore decided to do such a calculation and chose aluminum, for it is both simple to calculate and complicated enough (polyvalent) to exhibit the spin-hot-spot model attributes. We used realistic pseudopotentials to model the electron-ion interaction (which included the spin-orbit interaction) and electronic band structure, realistic empirical force constants between aluminum ions to get a good phonon structure, some techniques to calculate Fermi surface averages (the tetrahedron summation with an adaptive mesh), and got a wonderful curve which is in excellent agreement with experiment. In addition, our curve predicts what spin relaxation time T1 should be at room temperature, since no experiments there exist. Below I also show our calculated spin-flip Eliashberg function, which we used to get the final curve for T1. 

Above I plot spin relaxation time T1 of aluminum in nanoseconds (on a logarithmic scale) versus temperature in kelvins. The solid curve is our first-principles calculation. Symbols come from two measurements: spin injection (Johnson and Silsbee) and CESR (Lubzens and Schultz). The agreement between theory and experiment is very good, showing, for the first time, that the Elliott-Yafet
mechanism and the spin-hot-spot model works.

Above: Calculated spin-flip Eliashberg function. The function measures how effective phonons with a given frequency (Omega--horizontal scale) are in scattering electrons in such a way that the electrons' spins flip. Here I plot it for aluminum (solid line). The long-dashed curve is the phonon density of states (F) and the short-dashed curve is the ordinary (non-spin-flip) Eliashberg function which is important for superconductivity.