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Entropy Driven Inductive Response of Topological Insulators
by Ahmet Mert Bozkurt, Sofie Kölling, Alexander Brinkman, İnanç Adagideli
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Submission summary
| Authors (as registered SciPost users): | Inanc Adagideli · A. Mert Bozkurt · Sofie Kölling |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2403.00714v1 (pdf) |
| Date submitted: | 2024-03-27 08:54 |
| Submitted by: | Bozkurt, A. Mert |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
3D topological insulators are characterized by an insulating bulk and extended surface states exhibiting a helical spin texture. In this work, we investigate the hyperfine interaction between the spin-charge coupled transport of electrons and the nuclear spins in these surface states. Previous work has predicted that in the quantum spin Hall insulator phase, work can be extracted from a bath of polarized nuclear spins as a resource. We employ nonequilibrium Green's function analysis to show that a similar effect exists on the surface of a 3D topological insulator, albeit rescaled by the ratio between electronic mean free path and device length. The induced current due to thermal relaxation of polarized nuclear spins has an inductive nature. We emphasize the inductive response by rewriting the current-voltage relation in harmonic response as a lumped element model containing two parallel resistors and an inductor. In a low-frequency analysis, a universal inductance value emerges that is only dependent on the device's aspect ratio. This scaling offers a means of miniaturizing inductive circuit elements. An efficiency estimate follows from comparing the spin-flip induced current to the Ohmic contribution. The inductive effect is most prominent in topological insulators which have a large number of spinful nuclei per coherent segment, of which the volume is given by the mean free path length, Fermi wavelength and penetration depth of the surface state.
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In this manuscript, the authors investigate theoretically the charge and spin dynamics of the surface states of a disordered three-dimensional topological insulator, including the Fermi contact hyperfine interaction with the nuclear spins of the host material. They employ the Keldysh formalism within a quasiclassical approach, introducing the coupling of the electrons to (nonmagnetic) impurities and the nuclear spins to lowest order. This allows to derive the quantum kinetic equations describing the coupled dynamics of the electronic spin and charge and the nuclear spins. These equations can then be reduced to Boltzmann-like equations for the charge and spin dynamics, from which simple transport equations follow.
The authors find that the spin-momentum locking of the electrons on the surface yields a direct coupling between the nuclear spin dynamics and the surface charge currents. They show that in a transport setup this results in (i) a small correction to the conductance of the surface states, due to scattering off nuclear spins and (ii) a small contribution to the current driven by the entropy of the nuclear spin system. The latter contribution can be seen as a small inductive contribution to the impedance of the surface, the magnitude of which depends directly on the geometry of the device. This could potentially lead to applications in the form of small and controllable inductive elements. Finally, the authors estimate the magnitude of the effect in a few candidate materials, confirming that it will indeed probably be on the small side.
The manuscript is very well written and easy to follow: Sections 2.1-2.5 give a very readable and pedagogic overview of the calculation, making it possible to follow all steps in detail. Although there is a large overlap of ideas between this work and Ref. [1] and the estimated magnitude of the resulting inductive effects is somewhat disappointing, I think that the manuscript is timely, interesting and of value for the community. Especially the very clearly worked out interpretation of the effect of the nuclear spin dynamics in terms of an inductive contribution is interesting and does provide a novel and synergetic link between different concepts, to my taste. Therefore, altogether I recommend publication of this work in SciPost. I do have a few comments which I think the authors could address first:
- The Fermi contact hyperfine interaction as written in Eq. (2) is directly proportional to the weight of the electronic wave functions at the position of the nuclei. For the usual s-type states in the conduction band of most semiconductors this will indeed be by far the dominating contribution to the interaction. For p-type states, this contact interaction will in principle be absent and the interaction will be much smaller and can be qualitatively different. I am not an expert on the band structure of real-life topological insulators and the resulting orbital structure of the surface states, but I imagine that they are not of (pure) s-type. This issue is addressed in Sec. 3.2 by a statement that the interaction will be reduced because of this, with a reference to a PhD thesis I cannot access. It would be good if the authors could address this slightly more thoroughly: Why is the resulting interaction expected to be of the Heisenberg type, and not, e.g., dominated by Ising terms, depending on the detailed orbital angular momentum of the states? Is the behavior of all materials investigated in Sec. 3.2 expected to be the same in this respect? Etc.
- The authors write below Eq. (44) that the entropy-induced current results from the fact that a polarized nuclear spin system prefers to raise its entropy, which "can only be achieved by transferring their spin angular momentum to electron spins via hyperfine interaction." In reality, such entropy gain will also be achieved via nuclear spin diffusion into the bulk (through nuclear spin-spin interactions). Typically, this is a relatively slow process, but since the effect on the charge current found here is also very small it would be good to compare the relevant time scales more quantitatively.
- Since I consider the pedagogic style of the manuscript one of its key values, I would recommend adding one initial step at the beginning of App. B, defining the greater and lesser Green functions and showing how (67,68) follow. This would balance that part with the very detailed introduction provided around Eqs. (3-7).
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