New Magnetic Materials and their Functions September 9-18 th 2007, - PowerPoint PPT Presentation

European School on Magnetism New Magnetic Materials and their Functions September 9-18 th 2007, Cluj-Napoca, Romania Bogdan R. Bułka Institute of Molecular Physics, Polish Academy of Sciences, ul. M. Smoluchowskiego 17, 60-179 Poznań, Poland

Outline 1. Kondo resonance 2. Quantum interference in nanostructures • Fano resonance • Aharonov-Bohm effect 3. Many body effects in double dot systems 4. Summary

Minimum resistance

ρ = 1 From the Boltzmann theory of the electrical resistivity m τ 2 ne Kondo model (s-d model) + + = ε + ⋅ σ r calulation of the r H c σ c σ J S c σ σσ c σ ∑ ∑ relaxation time Kondo k k k k ' k ' ' σ σσ k kk ' ' π + 2 ρ = − ρ 3 mJ S ( S 1 ) k T For low temperatures B [ 1 4 J ( E ) ln( )] h spin F 2 2 e E W F J<0 Kondo state cloud of spins of conducting electrons screens the localized spin

Single impurity Anderson model U + + + + = ε + ε + σ σ σ σ ∑ ∑ ↑ ↑ ↓ ↓ H c c c c U c c c c Anderson k k k 0 0 0 0 0 0 0 σ σ k + + + + σ σ σ σ ∑ V ( c c c c ) k 0 k k 0 σ Abrikosov-Suhl peak k Local density of states neglecting charge fluctuations spin fluctuations s-d model charge fluctuations with the effective exchange interaction ≈ − U 2 ε − ε J | V | | kk ' k | ( U | |) 0 0 ε 0 ε 0 +U Energy E F

7 0 0 2 r o s d n i W , r e n t s a K

Increase of conductance for T → 0 Landauer approach current = − 2 e ∫ J dE T ( E ) [ f ( E ) f ( E )] L R h conductance (for V SD → 0) ∂ e ∫ = − 2 f ( E ) � dE ( ) T ( E ) ∂ h E where T(E ) is a transmission

Coulomb blockade and Kondo effect conductance T<<T K N=odd N-1=even U T>>T K N N N N N N N N - - - 1 - 1 1 1 V g U V g Anderson model + + + = ε + + σ σ α σ ασ ασ σ ∑ ∑ conductance H c c t ( c c c c ) k k k 0 k k 0 σ α = σ k k , L , R , + ε + + + + σ σ ∑ ↑ ↑ ↓ ↓ c c U c c c c 0 0 0 0 0 0 0 0 σ V g

Transformation of the density of states with the gate voltage ε + U d µ ε d ρ ω ( ) ω ε ε + U d d

Transformation of the density of states with the gate voltage ε + U d µ ε d ρ ω ( ) ω ε ε + U d d

Transformation of the density of states with the gate voltage ε + U d µ ε d ρ ω ( ) ω ε ε + U d d

Transformation of the density of states with the gate voltage ε + U d µ ε d ρ ω ( ) ω ε ε + U d d

Transformation of the density of states with the gate voltage ε + U d µ ε d ρ ω ( ) ω ε ε + U d d

Transformation of the density of states with the gate voltage ε + U d µ ε d ρ ω ( ) ω ε ε + U d d

Transformation of the density of states with the gate voltage ε + U d µ ε d ρ ω ( ) ω ε ε + U d d

Transformation of the density of states with the gate voltage ε + U d µ ε d ρ ω ( ) ω ε ε + U d d

Transformation of the density of states with the gate voltage ε + U d µ ε d ρ ω ( ) ω ε ε + U d d

Transformation of the density of states with the gate voltage ε + U d µ ε d ρ ω ( ) ω ε ε + U d d

ε + U d µ ε d ρ ω πε ε + ( ) ( d U ) = Γ 1 d Γ T Ue U K 2 ω ε ε + U d d

Conclusion: The Abrikosov-Suhl peak in the local density of states is pinned to the Fermi energy electrons in the electrodes, even when the local state is shifted by the gate potential The conductance is large, when the local state is shifted by the gate potential

Gray scale map of the differential conductance vs. the source-drain and the gate voltage A zero bias peak is a signature of the Kondo effect tunnel coupling 1e 2 /h Zero bias peak V G high 0 conductance

Summary on Kondo resonance in quantum dot N=even Increase of the conductance for odd number of electrons N=odd N=odd Zero bias peak pinned to the Fermi energy source gate drain D. Goldhaber-Gordon, H.Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav and M. A. Kastner, Nature 391 (1998) 156

Lateral structures Vertical quantum dots Carbon nanotubes Grains Molecules

Iron atoms on copper surface (Don Eigler, IBM). Quantum mirage in the ellipse of 36 cobalt atoms (Monoharan et al., Nature 2000)

U. Fano, Nuovo Cimento 12 (1935) 177 (in Italian) cited > 5 000

Energy scheme for Fano resonance Ionization in He V E | ϕ> ϕ> ϕ> ϕ> | ψ ψ ψ E > ψ > > > 2s2p 1sEp direct 1s2p ionization 1s 2 | i > > > > auto-ionization Fano 1961 Matrix elements < ϕ ϕ > = for discrete state ϕ | H | E < ψ ϕ > = coupling | H | V E ' E ' < ψ ψ > = δ − for continuum | H | E ' ( E " E ' ) E " E ' States for the coupled system E ψ Φ = ϕ + Ψ = ϕ + ψ V ' E ' ∫ ∫ − P dE ' a dE ' b E E ' E ' E E ' continuum discrete state

Modification of the autoionization absorption line (transition to the continuum) < Ψ > + ε 2 2 | | | | = T i ( q ) E < ψ > + ε 2 2 | | | | 1 T i E ε = − Γ ( E E ) / 1 r 2 Γ = π 2 – broadening of the 2 | | V E resonant level q is a parameter, which measures the strength of interference and is given by the ratio of direct ionization to autoionization < Φ > 2 π Γ = | | T | i | 2 1 q < ψ > 2 2 | | T | i | E

The Fano resonance is a quantum phenomenon, which was observed in systems of various states and the nature of coupling between them Physical systems Energy • photoionization of rare gases • bulk GaAs in magnetic field • superlattice in electric field • impurity ions in semiconductors • electron-phonon coupling • and many more … Density of states Observation techniques • optical absorption In transport through nanostructures • Raman spectrosopy • luminescence • Strongly coupled Quantum Dot • Side attached Quantum Dot • STM • conductance characteristics • For edge states in nanorings in magnetic field

M. Sato, et al., PRL 2006 (a) Schematic diagram of a stub-resonator. (b) Scanning electron micrograph of the device. The white areas are metallic gates made of Au/Ti. The dot and the wire are indicated by dotted lines. . (a) Upper: Conductance as a function of gate voltage at temperatures from 750 mK to 50 mK with the temperature step of 50 mK. Lower: Kondo temperatures T K obtained from the temperature dependence. (b) Examples of the fitting to obtain T K . The gate voltages adopted here are indicated by arrows in (a).

M. Sato, et al., PRL2006 How to explain the experiment ? Conductance for the Kondo resonance e c Conductance for the Fano resonance n a t c u d n o C V g

Modeling of transport: quantum dot + wire Many-body effects treated within the Interpolative Perturbative Scheme 1,0 0,8 U=0 0,6 G [2e 2 /h] Weak coupling 0,4 Σ (2) ( ω )= Strong coupling Γ =0.025 meV 1,max Γ =0.28 meV 1,max T=50 mK T=1000 mK 0,2 T= 100 mK T=500 mK T=150 mK Second order term for T=100 mK self-energy T=0 0,0 -0,0020 -0,0015 -0,0010 -0,0005 0,0000 0,0005 0,0010 d [eV] ε ε ε ε P. Stefa ń ski, Solid St. Commun. 128, 29 (2003)

Fano resonance in semi-open large quantum dot Experiment C. Fuhner, et al., PRB 66 , 161305 (2002) cond-mat/0307590 More in: P. Stefanski, A. Tagliacozzo, B.R.B, Phys. Rev. Lett. 93, 186805 (2004)

magnetic field flux Φ Φ Φ Φ Magnteic field potential electronic trajectory = ∇ × r r r B A wave function of an electron in a magnetic field r r r r ψ ∝  + ⋅  ω e ( r , t ) exp i ( k A ) r exp[ i t ]   h   Schematic presentation of the Aharonov-Bohm effect in a nanoscopic metallic ring in magnetic field B . The phase shift of the electronic wave traveling through the ring ϕ = ⋅ e ∫ A d s h L depends on the trajectory of in the upper and in the lower arm of the ring and on the magnetic field potential A ( B = rot A ). The traveling waves interfere, which is observed in the oscillations of the conductance with the period Φ 0 = e/h .

Conductance oscillations with the period Φ Φ 0 = h/e Φ Φ R.A. Webb, et al., PRL 54, 2696 (1985)

Conductance of 1D ring vs. magnetic flux for various geometry of attached wires Φ Multiple reflections were taken into account

Aharonov-Bohm effect in a metallic ring with a multi-level quantum dot FIG. 1. (a) Schematic representation of the experimental setup. (b) Scanning electron micrograph of the correspondent device fabricated by wet etching the 2DEG at an AlGaAs_GaAs heterostructure. The white regions indicate the Au_Ti metallic gates. The three gates (V L , V R and V g ) at the lower arm are used for controlling the QD, and the gate at the upper arm is for V C . K. Kobayashi, et al, Phys. Rev. Lett. 88 , 256806 (2002)