Spin Dependent Thermoelectric Currents of Tunnel Junctions, and Other Nanostructures: Onsager Response-Theory

1. G.E. Bauer, A.M. MacDonald, S. Maekawa, Spin electronics. Solid State Commun. 150, 459 (2010)

   Article  CAS  Google Scholar 

2. K.H. Bennemann, Magnetic Nanostructures. J. Phys. Condens. Matter 22, 243201 (2010)

   Google Scholar 

3. K.H. Bennemann, Photoinduced phase transitions. J. Phys Condens. Matter 23, 073202 (2011)

   Google Scholar 

4. K. Uchida, S. Takahashi et al., Observation of the spin seebeck effect. Nature 455, 778 (2008); F.L. Bakker, A. Slachter, J.P. Adam, B.J. van Wees, Phys. Rev. Lett. (2010); on spin caloritronics: M. Johnson, R.H. Silsbee, Phys. Rev. B 35, 4959 (1987), M. Johnson, Sol. State Commun. 150, 543 (2010); L. Gravier, S. Serrano-Giusan, F. Reuse, and J.P. Ansermet, Phys. Rev. B 73, 024419 (2006)

   Google Scholar 

5. A. Slachter et al., Thermally driven injection from a ferromagnet into a non–magnetic metal. Nat. Phys. 6, 879 (2010)

   Google Scholar 

6. F. Nogueira, K.H. Bennemann, Europhys. Lett. 67, 620 (2004). Note, the gradient of both the magnitude and phase of the magnetization (assuming to be canonical conjugate variables and for which one may use then the classical Hamilton equations) yield for magnetic tunnel junctions with hamiltonian \(H=-E_J S^{2} \cos (\theta _L- \theta _R) + \mu _B^{2}/2C_s (S^{z_L}-S^{z_R})^{2}\) the spin current \(J = 2\frac{E_{J}}{S^{2}}\mu _B\sin \Delta \theta \), and \(\Delta \theta \dot{=}2\mu _B V_S+(...)h\). Here, h refers to an external field and \(V_s= \frac{\mu _B}{C_S}(S_L^z-S_R^z) \) denotes the spin capacitance, see Nogueira, Bennemann. Note , \(J^{J}\) induces electric fields and \(\nabla XE_i = 4\pi J_i^{J}\)

   Google Scholar 

7. B. Kastening, D. Morr, D. Manske, K.H. Bennemann, Phys. Rev. Lett. 96, 047009 (2006); Phys. Rev. 79, 144508 (2009)

   Google Scholar 

8. F. Bloch, Phys. Rev. B 2, 109 (1970)

   Article  Google Scholar 

9. R. Kubo et al., Statistical Mechanics (North–Holland Publishing Co., Amsterdam, 1965); L.D. Landau, E.M. Lifshitz, Statistical Mechanics (revised by E.M. Lifshitz and L.P. Pitaevski), Pergamon Press 1980; S.R. de Groot, Thermodynamik irreversibler Prozesse, Hochschultaschenbücher, Bd. 18, 1960, and Thermodynamics of Irreversible Processes (North–Holland, 1952)

   Google Scholar 

10. S. Takahashi, H. Imamura, S. Maekawa, Phys. Rev. Lett. 82, 3911 (1999)

    Article  CAS  Google Scholar 

11. Note, taking into account magnitude and phase of the spin magnetization one gets generally \(j_M \propto \Delta |M| + b_1 |M|\nabla \exp i\psi \) and similarly for \(j_\sigma \) Josephson like currents. Of course, spin flip scattering plays a role and should be considered

    Google Scholar 

12. P.M. Brydon, B. Kastening, D. Morr, D. Manske, Interplay of ferromagnetism and triplet superconductivity in a Josephson junction, arXiv 0709.2918v1, 19 Sept 2007; K.H. Bennemann, J. Ketterson, Superconductivity, vol. 2 (Springer, 2008)

    Google Scholar 

13. K.H. Bennemann, Ultrafast dynamics in solids. Ann. Phys. (Berlin) 18, 480 (2009)

    Google Scholar 

14. K.H. Bennemann, F. Nogueira, see Lecture–Notes, FU-Berlin. Of course, expressing the Onsager coefficients \(L_{ij}\) by current–current correlation functions and calculating these within Lagrangian theory and electronic Hamiltonian, it is in principle straightforward to get instead of phenomenological Onsager theory results which refer to an electronic Hamiltonian, for example Hubbard hamiltonian. This might be particularly useful for understanding many–body effects, see F. Nogueira, Introduction to the field theory of classical and quantum phase transitions (Lecture Notes, FU–Berlin, September 2010); see also L.D. Landau and E.M. Lifshitz, Kinetics, vol. 10 (Pergamon Press)

    Google Scholar 

15. F. Nogueira, K.H. Bennemann, Current Correlation Functions (FU–Berlin, to be Published, 2010)

    Google Scholar 

16. B. Tatievski, P. Stampfli, K.H. Bennemann, Ann. der Physik 4, 202 (1995); Comp. Mat. Sci.2, 459 (1994); B. Tatievski, Diploma– thesis (FU–Berlin, 1993)

    Google Scholar 

17. A.C. Bleszynski-Jayich, W.E. Shanks, B. Peaudecerf, E. Ginossar, F. von Oppen, L. Glazman, J.G. Harris, Science 326, 272 (2009 ); P. Michetti, P. Recher, Bound States and persistent currents in topological insulator rings (2010). arXiv: 1011.5166v1

18. E.I. Rashba, Phys. Rev. B 68, 241315-1 (2003); B.A. Bernevig, Phys. Rev.B 71, 073201 (2005); A.V. Balatsky, B.L. Altshuler, Phys. Rev. Lett. 70, 1678 (1993). Spin–orbit coupling causes in a ring a persistent current due to the induced electron phase \(\phi _{AC}\) (Aharonov–Casher effect), see studies in above references. Using Faraday’s law \(\partial _{t} \phi _{AC} = -c \oint E_{AC}\bullet dl\) it is \(j_{\varphi ,\sigma _z}=-(c/2\pi R)\frac{\partial E}{\partial \phi _{AB}}\), \(\sigma _z=\pm 1/2\), and electron energies \(E_i=E_i(\phi _{AB}+\phi _{AC})\)

    Google Scholar 

19. Assuming particle–hole symmetry it is \(j=-\Sigma _i\frac{dE_i}{d\phi }\tanh (\frac{E_i}{2kT})\). Here, i refers to the electron states with energy \(E_i\) and for example to the polygonal orbits of the ring, see Stampfli et al., \(E_{t,p} \propto \cos (SB)\) assuming that all orbitals t,p enclose same area S. Then approximately \(j=-\int dE N(E) \frac{\partial E}{\partial \phi }\tanh (E/2kT)+...\) and thus at \(T= 0\) it is \(j=-\Sigma _i \frac{dE_i}{d\phi }\)

    Google Scholar 

20. Note, the Landau–Lifshitz equation, \(\frac{d\overrightarrow{M}}{dt}=\gamma \overrightarrow{M}\times \overrightarrow{H_{eff}}+\alpha \overrightarrow{M}\times \frac{d\overrightarrow{M}}{dt} + \beta \overrightarrow{j}\times \partial _t\overrightarrow{M}\), includes damping. A time dependent spin current yields in accordance with the Maxwell equations an electric field

    Google Scholar 

21. A.A. Shanenko, M.D. Critoru, A. Vagov, F.M. Peeters, Phys. Rev. B 78, 024505 (2008), and More Recent Publ. by F.M. Peeters et al

    Google Scholar 

22. M.E. Garcia, Habilitation thesis, Physik (FU Berlin, 1999)

    Google Scholar 

23. To include chemical reactions occurring in (open) magnetic systems with diffusion currents etc. one may extend Onsager theory taking into account spin and magnetization. For simplicity one may first neglect nonlinear behavior. (However, note frequently this may not be valid.). Then, \(j_i=L_{ij}X_j+...\) and chemical reactions are taken into account by the forces \(X_j=A_{j}/T\) with spin dependent chemical affinities \(A_j=-\sum _{k}\nu _{k,j}\mu _k\). Here, the substance k with chemical potential \(\mu _k\) is also characterized by its spin. The stoichometrical coefficients \(\nu _{k,j}\) characterize the chemical reaction which couples to the currents. It is straightforward to work out details of the theory, see de Groot, Landau. Thus one gets spin dependent coupling of diffusion currents and chemical reactions, for example effects due to concentration gradients depending on spin and magnetization, etc.

    Google Scholar 

Download references