Theory of spin-polarized STM and AFM: A tutorial presentation

1. Theory of spin-polarized STM and AFM:A tutorial presentation C. Julian Chen December 12, 2006 Institut für Angewandte Physik und Zentrum für Mikrostrukturforschung Universität Hamburg Jungiusstrasse 11, Hamburg

2. Outline The original paper of Tersoff and Hamann - The original derivation from Bardeen’s theory - atom-charge superposition Spin-valve effect: in the light of Bardeen The Landauer formalism of tunneling problem - Concept and an elementary derivation - Relation with Bardeen’s tunneling theory Pair-wise treatment of SP-STM and AFM - Tunneling conductance between two atomswith spin - Corrugation amplitude and decay constant: STM vs. AFM - Reduction to Tersoff-Hamann and atom-charge superposition

3. References and Acknowledgements • D. Wortmann et al, Resolving complex atomic-scale spin structures by spin-polarized scanning tunneling microscopy, Phys. Rev. Lett. 86, 4132 (2001). • S. Heinze, Simulation of spin-polarized scanning tunneling microscopy images of nanoscale non-collinear magnetic structures, Appl. Phys. A, (2006). • H. J. Reittu, Analysis of spin-dependent tunneling of electrons in solid state structures using the transfer-Hamiltonian method, J. Phys. Condens. Matter, 9, 10651 (1997). The Author sincerely acknowledge numerous discussions with Stefan Heinze, Mattias Bode, and Oswald Pietzsch. The presentation contains no new physics. It is a pedagogic presentation of the known results.

4. The original paper of Tersoff and Hamann (1) Sample wavefunction is expended into a two dimensional Fourier transform Tip wavefunction is also expended… The original Bardeen’s theory is applied: Surface integral on the z=0 plane:

5. The original paper of Tersoff and Hamann (2) Tunneling matrix element is proportional to the sample wavefunction at tip center: The charge density of the sample at the tip center can be estimated using atom charge superposition: wavefunction: charge density:

6. Spin-valve effect: in the light of Bardeen (1) General formalism: Using spinors instead of spatial wavefunctions

7. Spin-valve effect: in the light of Bardeen (2) In a coordinate system the z-spin of electrode A is diagonized, Starting with a spin-up state, Starting with a spin-down state, Following the procedure of deriving Bardeen’s theory…

8. Spin-valve effect: in the light of Bardeen (3) The most general transformation: through the Euler angles. Experimental configuration Spinor in electrode A: Spinor in electrode B, different z:

9. Spin-valve effect: in the light of Bardeen (4) In the coordinate system of spin polarization of electrode A… The total tunneling conductance is… It can be simplified by introducing…

10. Spin-valve effect: in the light of Bardeen (5) Finally, a familiar result of Slonczewski… Further, by defining We obtain For SP-STM, the above results can be further simplified by using the Landauer formalism.

11. Spin-valve effect: experimental verifications J. S. Moodera and L. K. Kinder , Ferromagnetic-insulator-ferromagnetic tunneling: Spin-dependent tunneling and large magnetoresistance in trilayer junctions, J. Appl. Phys., 79 4724-4729, (1996).

12. The Landauer formalism of tunneling problem (1) The tunneling conductance has an exponential dependence on z. What is the absolute value?

13. The Landauer formalism of tunneling problem (2) n-th wavefunction n-th energy eigenvalue Local density of states at energy E, counting two spins, Classical velocity

14. The Landauer formalism of tunneling problem (3) Bias and Fermi levels Tunneling conductance Impinging current Finally…

15. The Landauer formalism of tunneling problem (4) Supriyo Datta made a connection between the Bardeen tunneling theory and the Landauer formalism (pp. 161-163 of Electronic Transport in Mesoscopic Systems ): The tunneling conductance according to Landauer… The tunneling conductance according to Bardeen… Consequently, The spin-polarized tunneling conductance between two atoms is…

16. Pair-wise Model of SP-STM and SP-AFM (1) For each atom on the sample surface… The total tunnelingconductance…

17. Pair-wise Model of SP-STM and SP-AFM (2) For periodic surfaces, the sum can be evaluated using a mathematical identity, And the corrugation amplitudes can be predicted:

18. Pair-wise Model of SP-STM and SP-AFM (3) o Typical feature size: 5A, q = p /5A = 0.628 A-1 o o Effects of non-s states: o o SP-AFM: k = 0.5 A-1 SP-STM: k = 1 A-1 f = = 1.29. f = = 2.18. Correction factors: s-d d-d 1.66 2.77 Correction factors: s-d d-d 4.76 22.67

19. Pair-wise Model of SP-STM and SP-AFM (4) If either the tip or the sample is not spin-polarized, Tersoff-Hamann model with atom-charge superposition!

20. The Logic Time-dependent perturbation theory Schrödinger equation Pauli equation Bardeen theory without spin Bardeen theory with spin Spherical tip model Spin-valve effect Tersoff-Hamann basic Landauer-Datta Atom-charge superposition Individual orbital model no spin Tersoff-Hamann full Heinze model

21. Summary The original paper of Tersoff and Hamann - The original derivation from Bardeen’s theory - atom-charge superposition Spin-valve effect: in the light of Bardeen The Landauer formalism of tunneling problem - Concept and an elementary derivation - Relation with Bardeen’s tunneling theory Pair-wise treatment of SP-STM and AFM - Tunneling conductance between two atomswith spin - Corrugation amplitude and decay constant: STM vs. AFM - Reduction to Tersoff-Hamann and atom-charge superposition