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
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