## Saturday, August 28, 2021

### What is the spin Seebeck effect?

Thermoelectricity is an old story, and I've also discussed it here.  Take a length of some conductor, and hold one end of that conductor at temperature $T_{\mathrm{hot}}$, and hold the other end of that conductor at temperature $T_{\mathrm{cold}}$.  The charge carriers in the conductor will tend to diffuse from the hot end toward the cold end.  However, if the conductor is electrically isolated, that can't continue, and a voltage will build up between the ends of the conductor, so that in the steady state there is no net flow of charge.  The ratio of the voltage to the temperature difference is given by $S$, the Seebeck coefficient.

It turns out that spin, the angular momentum carried by electrons, can also lead to the generation of voltages in the presence of temperature differences, even when the material is an insulator and the electrons don't move.

Let me describe an experiment for you.  Two parallel platinum wires are patterned next to each other on the surface of an insulator.  An oscillating current at angular frequency $\omega$ is run through wire A,  while wire B is attached to a voltage amplifier feeding into a lock-in amplifier.  From everything we teach in first-year undergrad physics, you might expect some signal on the lock-in at frequency $\omega$ because the two wires are capacitively coupled to each other - the oscillating voltage on wire A leads to the electrons on wire B moving back and forth because they are influenced by the electric field from wire A.  You would not expect any kind of signal on wire B at frequency $2 \omega$, though, at least not if the insulator is ideal.

However, if that insulator is magnetically interesting (e.g., a ferrimagnet, an antiferromagnet, some kinds of paramagnet), it is possible to see a $2 \omega$ signal on wire B.

In the spin Seebeck effect, a temperature gradient leads to a build-up of a net spin density across the magnetic insulator.  This is analogous to the conventional Seebeck effect - in a magnetically ordered system, there is a flow of magnons from the hot side to the cold side, transporting angular momentum along.  This builds up a net spin polarization of the electrons in the magnetic insulator.  Those electrons can undergo exchange processes with the electrons in the platinum wire B, and if the spins are properly oriented, this causes a voltage to build up across wire B due to the inverse spin Hall effect.
So, in the would-be experiment, the ac current in wire A generates a temperature gradient between wire A and wire B that oscillates at frequency $2 \omega$.  An external magnetic field is used to orient the spins in the magnetic insulator, and if the transported angular momentum points the right direction, there is a $2 \omega$ voltage signal on wire B.

I think this is pretty neat - an effect that is purely due to the quantum properties of electrons and would just not exist in the classical electricity and magnetism that we teach in intro undergrad courses.

(On writing this, I realized that I've never written a post defining the spin Hall and related effects. I'll have to work on that....  Sorry for the long delay between postings.  The beginning of the semester has been unusually demanding of my time.) Anon, you are right that there are other ways to get a 2$\omega$ signal beside spin-related physics. For example, if the dielectric response of the insulator is electric field dependent, then the capacitance between the wires will be C(V), and a nonzero dC/dV will lead to a 2$\omega$ effect. To prove that spin is at work, you need to look at the magnetic field dependence of that signal - in particular, how that signal depends on the angle $\alpha$ shown in the figure (if H is large enough to coerce the magnetization of the insulator). As drawn, the spin Seebeck signal should be maximized when $\alpha = \pi/2$ and zero when H (and therefore M) is parallel to the Pt wires.