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

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
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 atoms with spin
- Corrugation amplitude and decay constant: STM vs. AFM
- Reduction to Tersoff-Hamann and 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)
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:
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:
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)
The most general transformation:
Experimental configuration
Spinor in electrode A:
Spinor in electrode B, different z:
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-insulatorferromagnetic tunneling: Spindependent tunneling and large
magnetoresistance in trilayer
junctions, J. Appl. Phys., 79 47244729, (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
Local density of states at energy E,
counting two spins,
n-th energy eigenvalue
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)
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…
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)
o
Typical feature size: 5A,
o
o
q = p /5A = 0.628 A-1
Effects of non-s states:
SP-STM: k =
f=
o
1 A-1
= 1.29.
Correction factors:
SP-AFM: k = 0.5 A-1
o
f=
= 2.18.
Correction factors:
s-d
d-d
s-d
d-d
1.66
2.77
4.76
22.67
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 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
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 atoms with spin
- Corrugation amplitude and decay constant: STM vs. AFM
- Reduction to Tersoff-Hamann and atom-charge superposition