low dimensional systems and nanostructuresnano-bio.ehu.es/files/presentación.pdf · kronig-penney...
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Surface StatesLow dimensional systems and
nanostructures
Maider Ormaza Saezmiera
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Motivation
• Understand of heterogeneous catalysis.
• Semiconductor Industry andNanotechnology.
• Study of two dimensional systems.
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Basic Theory
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To desbribe electrons in a crystal (1023 e-) we have to use
•Adiabatic Approximation.
•One electron approximation, where each e- moves under theinfluence of a periodic potential.
Now, to obtain the band structure, we have to solve…
Bloch´s Theorem tolds that the solution…
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Kronig-Penney model
Direct matching procedure
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Electrons at surfaces
Igor Tammshowed first theexistence ofsurface states in1932.
What happens if our crystal is finite?
We could have real and complex k!
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One-dim. Schrödinger equation:
Periodic Potential:
We try this solution:
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We get this secular equation:
The eigenvalues are
The eigenvectors are
where
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Bulk zk real Semi-infinite crystal
zk real
zk complex
The wave function monotonicallydamped in the direction ofvacuum and damped in anoscillatory way in the direction ofthe crystal
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The energy levels that are in the gaps, are known as surface states.
Gap
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Shockley state exists, if:
(1) Vg>0(2) matching conditions are fulfilled for ,
Tamm StatesThis surface states are obtained using the tight-binding model.
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DFTdensity functional theory
Hohenberg-kohn Theorem states that the total energy of a systemis a unique functional of the electron density.
Kinetik energyCoulomb term
Exchange-correlation term
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The energy of the ground state is found by looking for thedensity which minimizes the energy functional.
To obtain this density…Kohn-Sham equation
Effective potential for one e-
The density is given by…
Self-consistent solution!
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Jellium Model + DFT
Density
Positive fixed background
e- spill out!
Friedel oscillations
It can be also written
Volume of an e-
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Experimental Techniques
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Surface science
We need ultra high vacuum
Vacuum pumps Bake out
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Scanning Tunneling microscope(STM)
•Sharp tip very close to the surface
•Measure the current between thetip and the surface.
•Quantum tunnelling
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STM modes
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If we change the applied voltage…
STM image of Ag rows on vicinal Si(111).a)Empty states (Utip=-1V), b)FilledStates(Utip=1.5V)
STM measures the electronic topology of a surface
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Low Energy Electron diffraction(LEED)
The mean free path of low energy e- is small Surface sensitive
d
When
DiffractionQualitative information about thesurface symetries and periodicities.
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Diamond-type (1 1 1) surface
Electrons interact strongly withmatter.
Multiple scattering
Dynamic theoryRods
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ARPES
Based on the photoelectric effect
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We obtain directly the band structure
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References•Solid State Physics, Aschcroft & Mermin•Physics at surfaces, A.Zangwill•Basic Theory of Surface States, Sydney G. Davison & Maria Steslicka.•Lecture Notes on Surface Science, P.Hofmann (2005)•Electronic Structure of low-dimensional systems analized by ARPES},Aitor Mugarza (Doctoral thesis)•Introduction to nanotechnology, Henrik Bruus•www.physinet.uni-hamburg.de•www.nanoed.org•Estructura electrónica de superficies: estados de superficie}, H.Herrera &C.Mora.•Surface Science, K.Oura & Co.•http://newton.ex.ac.uk/research/qsystems/people/jenkins/mbody.html•www.unc.edu/shubin/dft.html
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Thank you for yourattention!!!