Bogdan R. Bułka - Magnetismmagnetism.eu/esm/2007-cluj/slides/bulka-slides.pdf · 2017. 6. 20. · Kondo resonance 2. Quantum interference in nanostructures • Fano resonance ...

Bogdan R. BułkaInstitute of Molecular Physics, Polish Academy of Sciences,ul. M. Smoluchowskiego 17, 60-179 Poznań, Poland

European School on MagnetismNew Magnetic Materials and their FunctionsSeptember 9-18th 2007, Cluj-Napoca, Romania

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

Kondo model (s-d model)

∑∑ ++ ⋅+= '' '''σσ σσσσσ σσ σε kk kkk kkkKondo ccSJccH rrFrom the Boltzmann theory of the electrical resistivity

τρ 12nem=

For low temperatures )]ln()(41[2)1(3 22 WTkEJEe SSmJ BFFspin ρπρ −+= h

J

Loca

lden

sity

ofst

ates

Energy

charge fluctuations

spin fluctuations

ε0 EF ε0+U

Abrikosov-Suhl peak

∑∑∑

++

↓+↓↑

+↑

++

++

++=

σσσσσ

σσσ

σσσ εεk kkkk kkkAnderson ccccV ccccUccccH )( 00 0000000

2' 0 0| | | | ( | |)kk k UJ V Uε ε≈ − −

Single impurity Anderson modelU

neglecting charge fluctuations

s-d model with the effective exchange interaction

Kastner, Windsor 2007

Increase of conductance for T→0

)]()([)(2 EfEfETdEheJ RL −= ∫Landauer approachcurrentconductance (for VSD→0)where T(E) is a transmission )())((2 ETEEfdEhe ∫ ∂∂−=�

Coulomb blockade and Kondo effect

U

VgU

NNNN N N N N ---- 1111conductanceVg

N=odd N-1=even

TTK

cond

ucta

nce

Vg

↓+↓↑

+↑

+

=

+++

++

++=

∑∑∑ 00000000 ,,, 00 )( ccccUcc cccctccH RLk kkk kkk

σσσ

σασασασσα

σσσ

ε

ε

Anderson model

d Uε +dε ωdεd Uε +µ

( )ρ ω

Transformation of the density of states with the gate voltage

d Uε +dε ωdεd Uε +µ

( )ρ ω

Transformation of the density of states with the gate voltage

d Uε +dε ωdεd Uε +µ

( )ρ ω

Transformation of the density of states with the gate voltage

d Uε +dε ωdεd Uε +µ

( )ρ ω

Transformation of the density of states with the gate voltage

d Uε +dε ωdεd Uε +µ

( )ρ ω

Transformation of the density of states with the gate voltage

d Uε +dε ωdεd Uε +µ

( )ρ ω

Transformation of the density of states with the gate voltage

d Uε +dε ωdεd Uε +µ

( )ρ ω

Transformation of the density of states with the gate voltage

d Uε +dε ωdεd Uε +µ

( )ρ ω

Transformation of the density of states with the gate voltage

d Uε +dε ωdεd Uε +µ

( )ρ ω

Transformation of the density of states with the gate voltage

d Uε +dε ωdεd Uε +µ

( )ρ ω

Transformation of the density of states with the gate voltage

d Uε +dε ωdεd Uε +µ

( )ρ ω ( )12

d d UUKT Ueπε ε +Γ= Γ

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

tunnel couplingGray scale map of the differential conductance vs. the source-drain and the gate voltageA zero bias peak is a signature of the Kondo effectV

G

Zer

o bi

as p

eak

high conductance

1e2/h

0

Summary on Kondo resonance in quantum dot

D. Goldhaber-Gordon, H.Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav and M. A. Kastner, Nature 391 (1998) 156

N=odd N=oddN=evensource

drain

gate

Increase of the conductance for odd number of electrons

Zero bias peak pinned to theFermi energy

Lateral structuresVertical quantum dots

Carbon nanotubes

GrainsMolecules

Iron atoms on copper surface (Don Eigler, IBM).

Quantum miragein 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

Matrix elements

for discrete state

coupling

for continuum

States for the coupled system

continuum discrete state

|ϕ>ϕ>ϕ>ϕ> |ψψψψE>>>>VE|i>>>>

)'"('|| || || '" '' EEEH VH EH EE EE −=>< =>< =>< δψψ ϕψ ϕϕ ϕ∫+=Ψ ''' EEE bdEa ψϕ ∫ −+=Φ ''P '' EEVdE EE ψϕ

Ionization in He

2s2p

1sEp

1s2

1s2pdirectionization

auto-ionizationFano 1961

Modification of the autoionization absorption line(transition to the continuum) 2 222 1 )(|||| |||| εεψ ++=>< >Ψ< qiT iTEE

Γ−= 21/)( rEEεq is a parameter, which measures the strength of interference and is given by the ratio of direct ionization to autoionization22221 |||| |||| >< >Φ

The Fano resonance is a quantum phenomenon, which was observed in systems of various states and the nature of coupling between them

Physical systems• photoionization of rare gases• bulk GaAs in magnetic field• superlattice in electric field• impurity ions in semiconductors• electron-phonon coupling• and many more …

Observation techniques• optical absorption• Raman spectrosopy• luminescence• STM• conductance characteristics

Ene

rgy

Density of states

In transport through nanostructures• Strongly coupled Quantum Dot• Side attached Quantum Dot• 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 temperaturestep of 50 mK. Lower: Kondo temperatures TK obtained from the temperature dependence. (b) Examples of the fitting to obtain TK. The gate voltages adopted here are indicated byarrows in (a).

How to explain the experiment ?

M. Sato, et al., PRL2006

VgConductance Conductance for the Kondo resonance

Conductance for the Fano resonance

Modeling of transport: quantum dot + wireMany-body effects treated within the Interpolative Perturbative Scheme

P. Stefański, Solid St. Commun. 128, 29 (2003)-0,0020 -0,0015 -0,0010 -0,0005 0,0000 0,0005 0,00100,00,2

0,4

0,6

0,8

1,0

Strong couplingWeak coupling

U=0

Γ1,max=0.025 meVT=50 mKT= 100 mKT=150 mK

Γ1,max=0.28 meVT=1000 mK

T=100 mKT=500 mK

T=0

G [

2e2 /

h]

εεεεd [eV]

Σ(2)(ω)=

Second order term for self-energy

ExperimentC. Fuhner, et al., PRB 66, 161305 (2002)cond-mat/0307590Fano resonance in semi-open large quantum dot

More in: P. Stefanski, A. Tagliacozzo, B.R.B, Phys. Rev. Lett. 93, 186805 (2004)

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

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 Φ0 = e/h.

∫ ⋅= L de sAhϕ

ΦΦΦΦmagnetic field flux

electronic trajectory

]exp[)(exp),( tirAekitr ωψ

⋅+∝ rrh

rr

wave function of an electron in a magnetic field

Magnteic field potential

ABrrr

×∇=

R.A. Webb, et al., PRL 54, 2696 (1985)

Conductance oscillationswith the period ΦΦΦΦ0 = h/e

Conductance of 1D ring vs. magnetic flux for various geometry of attached wires

Multiple reflections were taken into account

Φ

Aharonov-Bohm effect in a metallic ring with a multi-level quantum dot

FIG. 1. (a) Schematic representation of the experimental setup. (b) Scanning electronmicrograph of the correspondent device fabricated by wet etching the 2DEG at anAlGaAs_GaAs heterostructure. The white regions indicate the Au_Ti metallic gates. Thethree gates (VL , VR and Vg ) at the lower arm are used for controlling the QD, and thegate at the upper arm is for VC .

K. Kobayashi, et al, Phys. Rev. Lett. 88, 256806 (2002)

FIG. 4 (color). (a) Conductance of twoFano peaks at 30 mK at the selectedmagnetic fields. The direction of theasymmetric tail changes between B =0.9140 and 0.9164 T and the symmetricshape appears in between. K. Kobayashi, et al. Phys. Rev. Lett. 88, 256806 (2002)Figure: Conductance through the metallicring with the two-level quantum dotcalculated within the bridge model. Thecoupling of the QD to the electrodes issymmetric tLi = tRj and the bridge channelwas described by tLR = |tLR| exp[iΦ]. Theparameters were taken as tLi = 0.008, |tLR| = 0.133, the separation of the energy levels∆ε = 0.14, temperature T = 0.0032 (in units the half-band width D=1). The blue, the green and the red curve corresponds to the phase shift in presence of the magneticflux Φ = 0, π/2 and π, respectively.Bulka, et al., 2003bridge model

experiment

Φ=0Φ=π/2Φ=π

Differential conductance vs the source-drain voltage for Φ = 0.5 hc/e (black curve), 0.25 hc/e (blue curve) , 0.125 hc/e (green curve), and 0 (magneta curve) at T = 2 x10-6, the level position ∆ε = 0.05.

Change of the profile of the zero-biasanomaly due to the Aharonov-Bohm effect

Φ

Φ= 0.5 hc/eΦ= 0.25 hc/eΦ= 0.125 hc/eΦ= 0

Brandes, et al, PRL(2001)van der Wiel, et al., RMP(2003) Ono, at al. Science (2002)

Experiments on Double Quantum DotsRogge at al., APL (2003)

Motivation for studies of DQD• Construction of multi-dot electronic devices• Construction of qubits

drainsource

Competition:Kondo coupling vs. Antiferromagnetic coupling

JK JKDouble-Kondo

Strong dot-electrode coupling

JAFAntiferromagnetic

Strong inter-dot coupling

competitionDouble-Kondo system

K. Ono, D. G. Austing, Y. Tokura, S. Tarucha, Science 297, 1313(2002)

Spin-blockade in double dot system

A.W. Holleitner, C.R. Decker, H. Qin, K. Eberl, and R. H. BlickPhys. Rev. Lett. 87, 256802 (2001)

Double-Quantum Dot connected in parallel: Kondo coupling vs. Antiferromagnetic coupling

Strong dot-electrode coupling

JAF AntiferromagneticStrong inter-dot couplingDouble-Kondo

competitionJK JKRecent experiment: Chen, Chang and Melloch, PRL (2004)

P. Jarillo-Herrero, et al., Nature 434, 484 (2005); E. Minot et al. Nature 428, 536 (2004); Zaric et al., Science (2004); Coskun et al., ibid

µorb ≈ 0.8 meV/T (>> µB = 0.06 meV/T)Orbital magnetic moment

Orbital Kondo effect in carbon nanotubes

1. Spin of a single electron can be seen in quantum dots

2. In multi-dot systems local spins can be coupled and form multi-electron statesCan the current switch between various configurations?

3. Quantum interference should be taken into account in construction of nanodevices

Φ

Side-attachedquantum dot

Spin Correlations in Y structures

GaAs/GaAlAs hybrid structure

Stern-Gerlach experimenton electrons

J. Wrobel, T. Dietl, A. Łusakowski, G. Grabecki, K. Fronc, R. Hey, K. H. Ploog, and H. Shtrikman, PRL 93, 246601 (2004)

R. M. Potok, I. G. Rau, Hadas Shtrikman4, Yuval Oreg4& D. Goldhaber-Gordon, Nature 446, 167 (2007)