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 HamburgJungiusstrasse 11, Hamburg

Outline

The Landauer formalism of tunneling problem - Concept and an elementary derivation - Relation with Bardeen’s tunneling theory

Spin-valve effect: in the light of Bardeen

Pair-wise treatment of SP-STM and AFM - Tunneling conductance between two atoms with spin - Corrugation amplitude and decay constant: STM vs. AFM - Reduction to Tersoff-Hamann and atom-charge superposition

The original paper of Tersoff and Hamann - The original derivation from Bardeen’s theory - atom-charge superposition

References and Acknowledgements

1. D. Wortmann et al, Resolving complex atomic-scale spin structures by spin-polarized scanning tunneling microscopy, Phys. Rev. Lett. 86, 4132 (2001).

2. S. Heinze, Simulation of spin-polarized scanning tunneling microscopy images of nanoscale non-collinear magnetic structures, Appl. Phys. A, (2006).

3. 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.

The original paper of Tersoff and Hamann (1)

Tip wavefunction is also expended…

Sample wavefunction is expended into a two dimensional Fourier transform

The original Bardeen’s theory is applied: Surface integral on the z=0 plane:

The original paper of Tersoff and Hamann (2)

The charge density of the sample at the tip center can be estimated using atom charge superposition:

wavefunction:

charge density:

Tunneling matrix element is proportional to the sample wavefunction at tip center:

Spin-valve effect: in the light of Bardeen (1)

General formalism: Using spinors instead of spatial wavefunctions

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…

Spin-valve effect: in the light of Bardeen (3)

Experimental configuration

Spinor in electrode A:

Spinor in electrode B, different z:

The most general transformation:

through the Euler angles.

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…

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.

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).

The Landauer formalism of tunneling problem (1)

The tunneling conductance has an exponential dependence on z. What is the absolute value?

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

The Landauer formalism of tunneling problem (3)

Bias and Fermi levels

Tunneling conductance

Impinging current

Finally…

The Landauer formalism of tunneling problem (4)

The tunneling conductance according to Landauer…

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 Bardeen…

Consequently,

The spin-polarized tunneling conductance between two atoms is…

Pair-wise Model of SP-STM and SP-AFM (1)

For each atom on the sample surface…

The total tunneling conductance…

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:

Pair-wise Model of SP-STM and SP-AFM (3)

Typical feature size: 5A,

q = /5A = 0.628 A-1

o

o o

o o

f = = 2.18. f = = 1.29.

Correction factors:

s-d d-d

1.66 2.77

SP-AFM: 0.5 A-

1

SP-STM: 1 A-1

Correction factors:

s-d d-d

4.76 22.67

Effects of non-s states:

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!

The LogicTime-dependent perturbation theory

Schrödinger equation

Pauli equation

Bardeen theory without spin

Bardeen theory with spin

Spherical tip model

Tersoff-Hamann basic

Atom-charge superposition

Spin-valve effect

Tersoff-Hamann full

Landauer-Datta

Heinze model

Individual orbital model

no spin

Summary

The Landauer formalism of tunneling problem - Concept and an elementary derivation - Relation with Bardeen’s tunneling theory

Spin-valve effect: in the light of Bardeen

Pair-wise treatment of SP-STM and AFM - Tunneling conductance between two atoms with spin - Corrugation amplitude and decay constant: STM vs. AFM - Reduction to Tersoff-Hamann and atom-charge superposition

The original paper of Tersoff and Hamann - The original derivation from Bardeen’s theory - atom-charge superposition