Coulomb Blockade and Kondo Effect in Nanostructures Marcin M. Wysokioski 1,2 1 Institute of Physics Albert-Ludwigs-Universität Freiburg 2 Institute of Physics Jagiellonian University, Cracow, Poland 2.VI.2010 Seminar Quantum dynamics in mesoscopic systems 1
Outline Charging energy and charge quantization Single electron transistor (SET) Coulomb staircase Cotunneling Kondo effect Summary 2
Charging energy and charge quantization number of electrons on the island must be integer charge Q produces electric field, which accumulates energy: Increase in energy by adding Nth electron to the island e e - (N-1) e - 3
How big does the island can be if we still want to observe discrete energy levels? consider cubic island of size L - number of atoms on island, where a is interatomic distance - mean level spacing, where is Fermi energy - charging energy Charging energy and charge quantization Large, what means that electron spectrum is approximately continuous. 4
Gate Charging energy and charge quantization n electrons in a box: Voltage drop on the two capacitances Box Electrostatic energy Box Where a charge induced in the island by the gate is and -1 0 1 5
Outline Charging energy and charge quantization Single electron transistor (SET) Coulomb staircase Cotunneling Kondo effect Summary 6
Single electron transistor (SET) Source Drain Source dot gate Drain Drain Gate Electrostatic energy: Source Gate Where,, and 7
Available electron tunneling processes and associated energy differences: From source Single electron transistor (SET) To source FS TD From drain Source dot Drain To drain TS FD 8
Single electron transistor (SET) Positive energy difference means that electron transport is forbidden Transport regime: ; and all other energy differences for other processes at N and N+1 must be positive In particular in order to get one-by-one electron tunneling: 9
Single electron transistor (SET) Coulomb diamonds in a SET (for and ) - Coulomb blockade - One-by-one transport 1->2 2->1 0 1 2 Experimental results Kouwenhoven et al. Rep. Prog. Phys. 64 (2001) 10
Outline Charging energy and charge quantization Single electron transistor (SET) Coulomb staircase Cotunneling Kondo effect Summary 11
Coulomb staircase Explanation (roughly) I(V) Increase of the voltage V The current increase every time when additional electron is able to tunnel through the dot. Number of level on the dot between Fermi s Energies on the source and drain (picture) is tunned by the Voltage beetwen leads. Because of discrete energy level on 12 the dot we obtain staircase-like current-voltage dependence.
j Coulomb staircase Transition rate for an electron in the initial state using Fermi s golden rule: to a final state For a steady state averge charge at the island is constant. Current from source to dot: Rate of electrons entering the island occupied by n electrons, equals the rate of electrons leaving the island when occupied by (n+1) electrons: Normalization condition 13
Coulomb staircase I(V) The visibility of the staircase strongly depends on the sample parameters. V The steps become most pronounced if resistence and capacitance of one junction are large compared to those of the second junction 14
Outline Charging energy and charge quantization Single electron transistor (SET) Coulomb staircase Cotunneling Kondo effect Summary 15
u Cotunneling Higher order tunneling process, significant when sequential tunneling (first order tunneling process) is suppresed Tunneling through the intermediate virtual state which has larger energy than initial state Is allowed for a very short time by Heisenberg's uncertainty principle Inelastic cotunneling Elastic cotunneling 16
Outline Charging energy and charge quantization Single electron transistor (SET) Coulomb staircase Cotunneling Kondo effect Summary 17
Resistance Kondo effect Kondo effect in metals (left picture): resistance temperature dependence; (red) metal with magnetic impurities (cobalt in copper system), (blue) metal, (green) superconductor Temperature presence of magnetic impurities in metal below particular temperature (called Kondo temperature ) increase electron scattering Cloud of spins shielding impurity spin increase scattering 18
Conductance u Kondo effect Kondo effect in quantum dots (picture): (red) conductance curve for odd number of electron on the quantum dot; (blue)-conductance curve for even number on the quantum dot Temperature (picure): quantum dot with non zero total spin embedded between leads can be treated in simmilar way as a magnetic impurities in metals. 19
Kondo effect Spin flips Dh a) Adding electron to dot is prohibited by the Coulomb energy U, while removing electron from the dot would cost at least εo. However Heisenberg's uncertainty principle enable electron to tunnel through the impurity (virtual state) with spin flip. b) Many such events combine to produce Kondo effect leads to appearance of an extra energy resonance at Fermi energy. What at low temperature can significantly increase conductance through the quantum dot (new tunneling channel ) 20
Outline Charging energy and charge quantization Single electron transistor (SET) Coulomb staircase Cotunneling Kondo effect Summary 21
Summary To get discrete energy spectrum on an island, it has to be made out of few atoms Electrostatic energy of an island consisting N electrons. Positive energy difference in the tunneling process in SET means that electron transport is forbidden Coulomb diamonds 1->2 2->1 0 1 2 -Coulomb blockade -One-by-one transport 22
j Summary Increase of the voltage enable more and more electrons to tunnel what increase the current. Because charge is quantized we get staircase-like graph I(V) V cotunneling Kondo effect 23
Coulomb Blockade and Kondo Effect in Nanostructures Thank you for your attention! 24