Bogdan R. Bułka - Magnetismmagnetism.eu/esm/2007-cluj/slides/bulka-slides.pdf · 2017. 6. 20. ·...
Transcript of Bogdan R. Bułka - Magnetismmagnetism.eu/esm/2007-cluj/slides/bulka-slides.pdf · 2017. 6. 20. ·...
-
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)