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 Bardeens 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 Bardeens 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
3
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.
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 Bardeens theory is applied
Surface integral on the z0 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 Bardeens
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 tunneling conductance
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 2.18.
f 1.29.
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 Bardeens 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 Bardeens 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