Anisotropic Kondo effect Wael Chibani (email: chibani@fhi-berlin.mpg.de) Using the numerical renormalization group (NRG), we study the STM tunneling current through a Co atom embedded on an anisotropic lattice and experiencing a magnetic field in both directions, parallel and perpendicular to the anisotropy, as was measured by Otte et al. [1]. We introduce the Kondo-Anderson hybrid model (KAHM) Hamiltonian, by which we describe the system, where we take the spin of the Co atom as being S=3/2, and present the mapping of the self energy representation [2] onto our model. After discussing the easy-axis and easy-plan anisotropy, we demonstrate, that our problem is best described by an easy-axis anisotropy. Moreover, the experimental spectra show a dependence of the splitting of the Kondo resonance at finite magnetic fields on the direction of the magnetic field with respect to the anisotropy, which we will also discuss. Finally, when comparing our NRG calculated current with the experimentally measured one, we found, that, the Kondo temperature as given in the experiment is too small and thus, we choose an effective temperature to describe the system. [1] Otte A. F., Ternes. M., von Bergmann K., Loth S., Brune H. Lutz C. P., Hirjibehedin C. F. and Heinrich A. J., Nature Physics, Vol 4, November 2008. [2] Bulla R., Hewson A. C. and Pruschke T., J.Phys. : Condens. Matter 10, 8365- 8380 (1998).

1. Analysing the anisotropicAnalysing the anisotropic
Kondo-effect with NRGKondo-effect with NRG
Wael Chibani, Andreas Weichselbaum and Jan von Delft
24.02.2011

2. Outline
• The STM method
• Measurement of magnetic anisotropic Kondo-effect
• The KAHM Hamiltonian
• The structure of the impurity
• Brief introduction to NRG (identification of the
challenge)
• The self-energy representation
• The NRG - calculated differential conductance

3. The Kondo-effet
Leo Kouwenhoven and Leonid Glazman (2001)

4. The STM experiment
•Due to rapidly developing field of nanotechnology
•STM: well established method on imaging individual
atoms
•Tunneling current
Meir Y. and Wingreen N. S. Phys. Rev. Letters 68, 16 (1992)
•Differential conductance is measured quantity
•With

5. Measurement of anisotropic Kondo-
effect
• Measurement of differential
conductance
• Cobalt atom (S=3/2) embedded
on Cupper Nitrate
• Side-shoulders due to anisotropy
Otte A. F. et al., Nature Physics, Vol 4, November 2008.
• Kondo temperature given
by HWHM at T=0.5 K

6. Magnetic anisotropic Kondo-effect
•Side-shoulders shift inward for parallel B-field and outward for perpendicular B-field!!
•Kondo-resonance splitting depends on B-field direction and strength !!
Spectra show structure at high energies!!
• Measurement of differential
conductance
• with anisotropy acting
in z-direction
• Experiencing and
Otte A. F. et al., Nature Physics, Vol 4, November 2008.

7. The KAHM Hamiltonian
For a degenerate impurity orbital
The single impurity Anderson Model
•We define the KAHM Hamiltonian
(i.e. half-filling)
With the width of the localized level
For (i.e. half-filling)

8. The anisotropic KAHM
Taking the anisotropy of the underlying metal into account
The anisotropic KAHM is
where
But which sign for the anisotropy constant ?

9. Easy-axis & Easy-plane
Anisotropy
• Study structure of S=3/2-impurity
• In the limit
• Due to anisotropy, splitting of GS
•In the experiment:
• Thus, at B=0:

10. Ground state Splitting with B-field
•Energy splitting in parallel B-field:
•Energy splitting in perpendicular B-field:
•Since
Diagonalisation for

11. Wilson’s NRG
(a) Logarithmic discretization of conduction band (b) Defining Fourier transform in each interval:
discretization of the coupling
(c) Mapping onto a semi-infinite chain (Wilson Chain)
Using NRG we want to calculate, the spectral density
given in Lehman representation by

12. Problem: NRG is known for its limited resolution at high
energies!!
Wilson’s NRG

13. Wilson’s NRG
Why?
•Raw NRG spectral function is discrete
•And due to log-discretization raw data peaks are log-spaced!
Smoothening at high frequencies,
by convolution with Log-Gaussian of width
Bulla R., et al., J.Phys. : Cond. Matter 10, 8365- 8380 (1998)
Problem: NRG is known for its limited resolution at high
energies!!

14. Wilson’s NRG
Why?
•Raw NRG spectral function is discrete
•And due to log-discretization raw data peaks are log-spaced!
Smoothening at high frequencies,
by convolution with Log-Gaussian of width
Smaller Better resolution
Bulla R., et al., J.Phys. : Cond. Matter 10, 8365- 8380 (1998)
Can be reached by:
(i) z - averaging
(ii) Self-energy representation
Problem: NRG is known for its limited resolution at high
energies!!

15. Self-energy representation
for KAHM
• Our goal, is to calculate
•Using the self-enegy representation we get an improved correlation function
•For our S=3/2 model, we get
Where:

16. Self-energy for KAHM
Now we have mighty tools to improve the resolution

17. The NRG-current
Because of trace in the differential conductance formula
we have to plot
All system parameters of the experiment,
in units of Kondo temperature

18. The NRG-current
We found

19. Conclusion
We have
• Presented the KAHM Hamiltonian
• Studied the structure of the S=3/2 impurity
• Mapped the self-energy representation onto KAHM
• Calculated the NRG differential condutance
• Found that, Kondo temperature of the experiment is too
large, whereas its temperature is too small.
Open question
• Why is the “real” Kondo temperature that small?

20. Thank you for your attention!Thank you for your attention!

21. The NRG-current