Chapter 8: Coulomb blockade and Kondo physics

Transcription

1 Chater 8: Coulomb blockade and Kondo hysics 1) Chater 15 of Cuevas& Scheer. REFERENCES 2) Charge transort and single-electron effects in nanoscale systems, J.M. Thijssen and H.S.J. Van der Zant, Phys. Stat. Sol. (b) 245, 1455 (2008). 3) Coulomb blockade oscillations in semiconductor nanostructures, H. van Houten, C.W.J. Beenakker, and A.A.M. Staring in Single Charge Tunneling, edited by H. Grabert and M.H. Devoret, NATO ASI Series B294 (Plenum, New York, 1992). 4) The revival of the Kondo effect, L.P. Kouwenhoven and L. Glazman, Physics World, ag. 33, January 2001.

2 8.1 Introduction E = ε 0 E F = injection energy Traversal time: Γ = Γ L + Γ R = level width 2 2 τ = / E +Γ Energy scale of the Coulomb interaction: U In this chater we focus on situations in which and therefore, the transort is dominated by the Coulomb reulsion of the electrons inside the molecule. This situation is realized when the metalmolecule couling is relatively weak.

3 8.2 Charging effects in transort through nanoscale devices How small and how cold should a conductor be so that adding or subtracting a single electron has a measurable effect? 1. The caacitance C of the island (or dot) has to be such that the charging energy (e 2 /C) is larger than the thermal energy (k B T): 2. The barriers have to be sufficiently oaque such that the electrons are located on the dot: In molecular transistors these two requirements can be met.

4 8.2 Charging effects in transort through nanoscale devices To resolve the discrete electronic levels of a quantum dot: The level sacing at the Fermi energy for a box of size L deends on the dimensionality: π N / 4 (1D) 1/ (2D) 2 2 E = 2 π ml 2 1/3 (3 π N) (3D) The level sacing of a 100 nm 2D dot is around 0.03 mev, which is large enough to be observable at dilution refrigerator temeratures (100 mk mev). Using 3D metals to form a dot, one needs to choose a radius of around 5 nm in order to see atom-like roerties. In the case of molecular junctions, the level sacing is essentially the HOMO-LUMO ga and it is tyically several electronvolts. Therefore, the level quantization is easily observable in molecular transistor even at room temerature.

5 8.2 Coulomb blockade: a well-known henomenon in mesoscoic hysics Metallic islands

6 8.2 Coulomb blockade: a well-known henomenon in mesoscoic hysics Nanoarticles

7 8.2 Coulomb blockade: a well-known henomenon in mesoscoic hysics Semiconductor quantum dots

8 8.2 Coulomb blockade henomenology in carbon nanotubes S.J. Tans et al., Nature 386, 474 (1997) (i) Coulomb oscillations (ii) Coulomb staircase (iii) Characteristic temerature deendence (ii) Coulomb staircase

9 8.3 Single-molecule three-terminal devices

10 8.4 Coulomb blockade theory: constant interaction model E ( N) U( N) E Dot N = + = 1 U(N) = (Ne) 2 /2C NeV ext E ( =1,2,...) = single - electron energy levels Γ L ( ), Γ R ( ) tunneling rates C = C S + C D + C G V ext = ( C S V S + C G V G + C D V D )/C (weak couling)

11 8.4 Coulomb blockade theory Periodicity of the oscillations Dot chemical otential: µ 2 1 e ( N) = E ( N) E ( N 1) = N ev + E 2 C Dot Dot Dot ext N Electrons can flow from left to right when: µ L > µ Dot > µ R

12 For small bias voltages, V SD 0: 2 1 e µ Dot ( N) = N eαvg + EN ; ( α = CG / C = gate couling) 2 C Thus, the addition energy is given by: 2 2 e e µ ( N) = µ Dot ( N + 1) µ Dot ( N) = + EN + 1 EN = + E C C In the absence of charging effects, the addition energy is determined by the irregular sacing E of the single-electron levels. The charging energy e 2 /C, in contrast, leads to a regular sacing. When it is much larger than the level sacing (as in metallic islands), it determines the eriodicity of the Coulomb oscillations. From an exerimental oint of view, the Coulomb oscillations are measured as a function of the gate voltage and the eak sacing is given by: G while the condition Coulomb eak. 8.4 Coulomb blockade theory V = µ N eα = e C+ E eα 2 ( )/( ) ( / )/( ) eαv G N = (N 1/2)e 2 /C + E N gives the gate voltage of the N-th

13 Different tunneling rocesses (energy conservation): f, l State in the dot (N electrons) left lead at energy E ( N) : E ( N) = E + U( N) U( N 1) (1 η) ev f, l il, Left lead at energy E ( N) state in the dot (N electrons) : E ( N) = E + U( N + 1) U( N) (1 η) ev il, f, r State in the dot (N electrons) right lead at energy E ( N): E ( N) = E + U( N) U( N 1) + ηev f, r ir, Right lead at energy E ( N) state in the dot (N electrons): E ( N) = E + U( N + 1) U( N) + ηev ir, Stationary current through the left barrier: = 1{ n } ( δ ),0 δ,1 I = e Γ P({ n}) f( E ( N) E ) [1 f( E ( N) E )] i 8.4 Coulomb blockade theory Amlitude and line-shae of the oscillations ( ) il, f, l L i n F n F η: fraction of voltage droing at the right barrier

14 In equilibrium the robability distribution P({n i }) is given by the Gibbs distribution in the grand canonical ensemble: P eq ({n i }) = 1 Z ex 1 k B T i=1 E i n i +U(N) NE F ; Z = artition function The non-equilibrium robability distribution P is a stationary solution of the kinetic equation: P({ ni}) = 0 t = P n δ Γ f E N E +Γ f E N E ( ) i, l ( ) i, r ({ i}) n,0[ L ( ( ) F) R ( ( ) F)] P n δ Γ f E N E +Γ f E N E ( ) f, l ( ) f, r ({ i}) n,1[ L (1 ( ( ) F)) R (1 ( ( ) F))] ( ) f, l ( ) f, r + P( n1,, n 1,1, n+ 1, ) δn,0[ Γ (1 ( ( 1) )) (1 ( ( L f E N + EF +ΓR f E N + P n n n δ Γ f E N E +Γ f E N E 8.4 Coulomb blockade theory ( ) i, l ( ) i, r ( 1,, 1, 0, + 1, ) n,1[ L ( ( 1) F) R ( ( 1) F)] + 1) E ))] F

15 Linear resonse theory: The joint robability that the quantum dot contains N electrons and that the level is occuied is: P eq (N,n =1) = P eq ({n i })δ N, δ ni n,1 In terms of this robability the conductance is given by: e ΓΓ G = Peq ( N, n = 1) 1 f( E + U( N) U( N 1) EF ) kt Limit: 2 l r l r B = 1 N= 1 Γ +Γ kt e 2 / C, E B 8.4 Coulomb blockade theory P({n i }) P eq ({n i }) 1+ ev k B T Ψ({n }) i {n i } eα(v G(V G,T) /G max = cosh 2 G V 0 ) 2k B T G max = e2 (N π Γ ) 0 (N L Γ 0 ) R (N h 2k B T Γ ) 0 (N L + Γ 0 ) R V 0 : gate voltage at N 0 th resonance (maximum of the CB oscillation) i

16 8.4 An examle: Coulomb oscillations and staircase E 1 E F = 50 mev; E 2 E F = 80 mev E = 30 mev; e 2 /C =100 mev T = 30 K; Γ L ( ) = Γ R ( ) = 1 mev; η = 0.6

17 8.4 An examle: Stability diagrams and Coulomb diamonds Diamonds inclined because η 0.5

18 8.4 Coulomb blockade henomenology in carbon nanotubes S.J. Tans et al., Nature 386, 474 (1997) (i) Coulomb oscillations (ii) Coulomb staircase (iii) Characteristic temerature deendence (ii) Coulomb staircase

19 8.4 Single-molecule transistors: Observation of Coulomb blockade S. Kubatkin et al., Nature 425, 698 (2003). (OPV5)

20 8.4 Single-molecule transistors: Observation of Coulomb blockade Nanomechanical oscillations in a single-c 60 transistor Park et al., Nature 407, 57 (2000)

21 8.4 Single-molecule transistors: Observation of Coulomb blockade Park et al., Nature 417, 722 (2002)

22 8.5 Elastic and inelastic cotunneling Elastic cotunneling rocess: Non-zero background conductance within diamonds Inelastic cotunneling rocess: Horizontal line in diamonds at ev SD = E

23 8.6 Kondo effect Sin-fli cotunneling rocesses can change the sectrum of the dot leading to the screening of the localized sin and to the aearance of the so-called Kondo resonance. The Kondo resonance lies exactly at the Fermi energy, indeendent of the osition of the original level. For this reason, the Kondo effect leads to an enhancement of the conductance. The only requirement for this effect to occur is that the temerature is below the Kondo temerature (see below). The width of the Kondo resonance is roortional to the characteristic energy scale for Kondo hysics, the so-called Kondo temerature. For a single-level model it reads: k B T K = ΓU 2 ex πε (ε +U) 0 0 ΓU

24 8.6 Kondo effect Transort signatures of the Kondo effect Zero-bias line in the stability diagram for an odd number of electrons in the dot. Peak in the low-bias conductance at low temeratures with a width equal to the Kondo temerature. Kondo ridge Characteristic temerature deendence of the linear conductance.

25 8.6 Single-molecule transistors: Observation of the Kondo effect Park et al., Nature 417, 722 (2002) Molecules: Co-ion comounds; T K = K.

26 13.6 Single-molecule transistors: Observation of the Kondo effect Liang et al., Nature 417, 725 (2002) Molecules: Divanadium comounds.

27 Summary: Chater 8: Coulomb blockade and Kondo hysics Different transort regimes Oen QD regime: Γ» E C = e 2 /C Quantum interference is imortant (classical analogon: Fabry-Perot) Intermediate QD regime: Γ E C Interference, charging effects, higher order rocesses in the couling Closed QD regime: Γ «E C Charging effects dominate (Coulomb blockade for: Γ «k B T «E C ) S. Samaz et al., Phys. Rev. B 71, (2005)

28 Summary Chater 8: Coulomb blockade and Kondo hysics Quantum dot made of a carbon nanotube E(k)=ћv F k ΔE=ћv F Δk=hv F /(2L) with Δk=π/L T. Delattre, Current quantum fluctuations in carbon nanotubes, PhD Thesis, University Paris VI (2009)