Single Electron Tunneling Examples

Single Electron Tunneling Examples Danny Porath 2002 (Schönenberger et. al.) It has long been an axiom of mine that the little things are infinitely the most important Sir Arthur Conan Doyle

Books and Internet Sites Electronic Transport in Mesoscopic Systems, S. Datta Single Charge Tunneling, H. Grabert & M.H. Devoret Introduction to Mesoscopic Physics, Y. Imry http://www.iue.tuwien.ac.at/publications/phd%20theses/wasshuber/ http://vortex.tn.tudelft.nl/grkouwen/reviewpub.html http://www.aip.org/web2/aiphome/pt/vol-54/iss-5/archive.html ttp://www.ee.princeton.edu/~chouweb/newproject/research/qdt/singelectunnel.html http://qt.tn.tudelft.nl/publi/papers.html www.unibas.ch/phys-meso....

1.Read the paper: Homework 8 Variation of the Coulomb staircase in a two-junction system by fractional electron charge, by: M.E. Hanna and M. Tinkham, Physical Review B, 44(11), 5919 (1991).

Outline SET Examples: 1. The C 60 work D.P. et. al. a. Background b. Sample preparation and imaging c. SET Theory d. I-V Spectroscopy results and fits e. Summary (of this part) 2. Advanced material from C. Schönenberger a. Introduction b. Law transparency SET c. High transparency Fabry Perot & UCF d. Intermediate transparency Co-tunneling & Kondo e. Conclusions (from this part)

Single Electron Tunneling and Level Spectroscopy of Single C 60 Molecules Danny Porath, Muin Tarabia, Yair Levi and Oded Millo

The General Experimental Scheme I R 1,C 1 V QD C 60 Insulating layer STM tip J-1 J-2 R 2,C 2 gold

C 60 Fullerenes Structure: (icosahedral symmetry) : 1,000,000,000 ~1 m ~10 Å 60 carbon atoms. (20 hexagons, 12 pentagons)

Bacground Before... Significant effort was devoted to studies of the interplay between Single Electron Tunneling (SET) effects and quantum size effects in isolated nanoparticles. The interplay was studied for: E l << E c (Tinkham, Van-Kempen, Sivan, Molenkamp) C 60 spectroscopy was done mainly by optical means, limited by selection rules.

Background What was new here? Extremely small QD (~8 Å) Quantum Size effects even at RT. Interplay between SET and discrete levels in two regimes: E c < E l E c > E l Tunneling spectroscopy of isolated C 60 Electronic level splitting.

Steps: Sample Preparation Rinse C 60 powder in Toluene Evaporate dried powder on a gold substrate covered by carbon and (later) a PMMA layer Gold substrate Prepared sample C 60 Evaporation

100 Å 50 Å STM Images of C 60 Fullerenes 70 Å 40 Å

STM Images of C 60 Fullerenes 3-D

SET Effects in DBTJ I I-V curves CB V Q D R 1,C 1 R 2,C 2 STM tip Tunnel Junction 1 Tunnel Junction 2 I T (V) CS SET effects can be observed If: 1) E C = e 2 /2C > k B T Charging Energy > Thermal Energy 2) R T > R Q = h/e 2 Tunneling Resistance > Quantum Resistance Coulomb Blockade (CB): I T = 0 up to V t e/2c ; V t depends on fractional charge Q 0 Coulomb Staircase (CS): Sequence of steps in I-V. Each step - adding an electron to the island V

Charge Quantization Leads to SET Effects Charge quantization leads to SET effects: V t depends on fractional charge Q 0. Q 0 represents any offset potentials. Q = ne - Q 0 ; Q 0 e/2 E Q 0 = 0-2 -1 1 0 Q(e) 2 E c Q 0 = e/2-1 E 0 1 Q(e) V t = e/2c V t = 0 2 Allowed states with n access electrons indicated on the graphs

Origin of the Fractional Charge Q 0 Difference in contact potentials across the junctions: Q 0 = (C 1 φ 1 -C 2 φ 2 )/e Note: We are interested only in Q 0 mod e. Q 0 can be varied by controlling C 1 through tipsample separation. This is done by changing the STM current and bias settings.

Interplay Between SET Effects and Discrete Energy Levels E L << E C : (Level Spacing << Charging Energy) e No levels E c E L << E C 1 2 (Amman et al.) Additional sub-steps on top of a CS step

Interplay Between SET Effects and Discrete Energy Levels E L ~ E C : (Level Spacing ~ Charging Energy) CS steps are no more equidistantly spaced. zero bias gap is not suppressed for Q 0 = e/2 when the CB is suppressed. Pronounced asymmetry of I-V curves due to different levels on each side.

I-V Measurements V I Q R 1,C 1 R 2,C 2 STM tip 1 2 Tunneling Current [na] 10 1.6 Single molecule T = 4.2 K 8 0.0 6 4 2-1.6-0.6 0.0 0.6 (a) 0 (b) -1.5-1.0-0.5 0.0 0.5 1.0 1.5 Tip-Bias [V] Tunneling Current [na] 2 1 0-1 -2 Single molecule T = 300 K (c) -0.6-0.3 0.0 0.3 Tip-Bias [V] Tunneling Current [na] 4 2 0-2 -4 Two molecules T = 300 K (d) 0.6 0.3 T = 4.2 K 0.0 0.4 0.8-1.0-0.5 0.0 0.5 1.0 Tip Bias [V] I-V measurements are done while disconnecting the feed-back loop. Note: 1. Non-vanishing gap. 2. Pronounced asymmetry and different steps.

I-V Measurements di/dv (a.u.) di/dv (a.u.) di/dv (a.u.) 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.8 0.6 0.4 0.2 0.0 I (na) 0.4 0.0-0.4-0.8 0.0-1.0-0.5 0.0 0.5 1.0 0.0 0.3 0.6 0.9 1.2 Tip Voltage (V) Tip Voltage (V) T = 4.2 K (a) (b) (c) T = 4.2 K T = 300 K di/dv 0.9 0.6 0.3-1.0-0.5 0.0 0.5 1.0 Tip Voltage (V)

HOMO+1 (?) di/dv 0.9 0.6 0.3 2 1 LUMO+1 LUMO E F 0.0 0.0 0.3 0.6 0.9 1.2 HOMO Tip Voltage (V) 2 1 2 1 Tunneling through LUMO+1 Tunneling through LUMO

Local Fields (Sarid et. al.) Different electric fields (carbon field, STM tip) effect the molecular orbitals differently (Stark effect).

Molecule Orientation (Guntherodt et. al.) Tunneling channel (depending on the molecule orientation) effects the level splitting.

Jahn - Teller effect (Bendale et. al.) A splitting or a shift of the energy levels due to structural distortion. The C 60+ h u reduces by ~ 0.6 ev

Theoretical Fits (Orthodox theory-averin & Likharev) Schematic of the circuit: R 1,C 1 R QD 2,C 2 I V The tunneling rate on the dot: ± i (n) = 2π h T(E) i 2 D (E i E )f(e i E )D i d (E E d )[1 f(e E d )]de Γ i± (n) - Tunneling rate to/from the dot from/to the electrode. f - Fermi function. D i (E), E i - DOS/Fermi energy at electrode i. D d (E), E d - DOS/Fermi energy at the dot. T i (E) - Tunneling matrix element.

Theoretical Fits (Orthodox theory-averin & Likharev) We choose the density of states D d (E): The tunneling current is: + I(V) = e P(n)[Γ2 (n) Γ2 (n)] = e P(n)[Γ1 (n) Γ1 + (n)] P(n) The probability to find n electrons on the dot

Energy Level Calculations

xperimental Results Asymmetrical Curves 9 (a) Exp. 9 (b) Exp. Tunneling Current [na] 6 3 0-3 -6-9 Fits di/dv [a.u.] 6 3 0-3 -6 Fits -12-1.6-0.8 0.0 0.8 1.6 Bias [V] -9-1.6-0.8 0.0 0.8 Bias [V] 1.6 C 1 ~ C 2 ~ 10-19, E HOMO -E LUMO ~ 0.7 ev, E L ~ 0.05 ev

Source of Asymmetry C 1 <C 2 ==> V 1 >V 2 ==> Onset of tunneling at junction 1 (for the level configuration at hand) 2 1 2 1 2 1 Positive bias Negative bias

Experimental Results Symmetrical Curves 4 (a) 3 Exp. 3 2 (b) Exp. Tunneling Current [na] 2 1 0-1 -2-3 Fits di/dv [a.u.] 1 0-1 -2 Fits -4-3 -5-1.6-0.8 0.0 0.8 1.6 Bias [V] -4-1.6-0.8 0.0 0.8 1.6 Bias [V] C 1 ~ C 2 ~ 10-19, E HOMO -E LUMO ~ 0.7 ev, E L ~ 0.05 ev

Symmetrical Barriers (C 1 ~C 2 ) 2 1 2 1 Positive bias Negative bias Tunneling through the set of levels closer to E F -in our case the 3 LUMO levels.

Checking The Model - Experimentally Tunneling Current (na) Tunneling Current (na) 2 0-2 -4 8 4 0-4 -8 C 1 < C 2 I s = 1.6 na V s = 0.75 V A C 1 > C 2 I s = 6 na V s = 0.75 V B di/dv (a.u.) di/dv (a.u.) -1.5-1.0-0.5 0.0 0.5 1.0 1.5 Tip Voltage (V)

The Triple Barrier Tunnel Junction (TBTJ) V I QD QD R 1,C 1 R 2,C 2 R 3,C 3 Note: PMMA STM tip C 60 J-1 J-2 C J-3 Non Vanishing gap. gold Tunneling Current [na] 4 2 0-2 -4 Two molecules T = 300 K Negative differential resistance (NDR). 0.6 0.3 T = 4.2 K 0.0 0.4 0.8-1.0-0.5 0.0 0.5 1.0 Tip Bias [V] Non-ohmic curves everywhere on the sample.

Summary - C 60 STM Spectroscopy A C 60 molecule is an ideal quantum dot for studies of single electron tunneling and quantum size effects. The interplay between SET effects and molecular levels is clearly observed at 4.2 K as well as at RT for the double barrier tunnel junction configuration. The degeneracy of the HOMO, LUMO and LUMO+1 levels was fully resolved, and the degree of splitting is consistent with the J-T effect.

Quantum Dot Physics C. Schönenberger

Quantum dots

( conventional ) Quantum dots planar dot planar double-dot vertical dot

quantum dot? 2 µm

Different energies (what is 0d?) δe=h/τ round-trip in addition: ev and kt example: 0d with respect to quantum-size effects, provided: kt, ev, and Γ < δe

Contacts matter...!. low transparency single-electron tunneling determined by single-electron charging effects, e.g. Coulomb blockade (0d limit). intermediate transparency charging effects but co-tunneling, e.g. Kondo effect. high transparency quantum interference of non-interacting electrons, e.g. Fabry-Perot resonances and UCF (disordered limit)

Contacts matter...!. low transparency single-electron tunneling determined by single-electron charging effects, e.g. Coulomb blockade (0d limit)

Single-electron tunneling greyscale plot of di/dv (V gate,v bias ) here: black = low differential conductance Coulomb blockade

tutorial on di/dv plots V sd (mv) E add add addition energy, i.e. sum of: single-electron charging energy U C V g level-spacing δe Change V sd Change V g

even even even odd odd odd filling of states according to S = 1/2 0 1/2... odd number of electrons: Ε add = UC even number of electrons: Ε add = UC + δe

single-electron-tunneling if tunneling probability p of each junction is small : current determined by accessible levels in dot i.e. by level spacing and Coulomb charging energy uncorrelated sequential tunneling dominates. Current I p

T~1 limit (interaction-free wire) bias voltage 5 mv -5 mv gate voltage -2... +2 V

T~1 limit (wire) T ( E) = 2 1+ R C + TC 2R cos( Θ + C 2δ R ) Θ = L( k + k ) round-trip phase depends on E I V = e 2 { T ( ev / 2) + T ( ev / 2) }

Fabry-Perot (interference) 5 mv -5 mv their data gate voltage -2... +2 V simple model

co-tunneling if tunneling probability p of each junction is large : if tunneling probability p of each junction is small : uncorrelated sequential tunneling dominates. Current I p coherent 2nd (and higher) order processes add substantially p 2 we call this co-tunneling

Remainder Coulomb blockade even even even odd odd odd I jump back to low transparent tunneling contacts for reference black regions = very low conductance G G is suppressed due to Coulomb blockade (CB)

V sd (mv) δe ~ 0.55 mev U C ~ 0.45 mev V g Elastic co-tunneling Inelastic co-tunneling

V sd (mv) When the number of electrons on the quantum dot is odd, spin-flip processes (which screen the spin on the dot) lead to the formation of a narrow resonance in the density-of-states at the Fermi energy of the leads. V g This is called the Kondo effect

Kondo effect resistance R(T) of a piece of metal log( T / T K ) conductance G(T) through a single magnetic impurity (e.g. an spin ½ quantum dot) unitary limit 2e 2 /h T K ideal metal, e.g. Au wire superconductor, e.g. Pb magnetic impurities in ideal metal e.g. (ppm)fe:au Kondo system T K

S=1/2 Kondo in Q-dots

at 50 mk

why interesting? because it is complicated it s many-body physics (all orders are relevant) more precisely: it s many-electron physics in Condensed Matter Physics: Superconductivity Superfluidity Luttinger liquid (non-fermi liquids) Kondo physics

Kondo physics + superconductivity Kondo effect Superconductivity Al Kondo effect and superconductivity are many-electron effects can Kondo and superconductivity coexist or do they exclude each other?

spin 1/2 Kondo + S-leads normal case superconducting case U E F 1. a gap opens in the leads 2. Cooper pairs have S=0 Kondo effect is the screening of the spin-degree of the dot spin by exchange with electrons from Fermi-reservoirs (the leads) Hence: Kondo effect suppressed, but...

Kondo effect Superconductivity A cross-over expected at k T ~ Cooper pair Energy scale : ~ k b T K S = 0 Cooper pair S = 0 Energy scale : ~

Conclusions nanotubes may be one part of the toolbox of nanotechnology nanotubes can serve as a model system to study exciting physics Kondo physics (co-tunneling) Interplay between Kondo physics & superconductivit www.unibas.ch/phys-meso group Web-page www.nccr-nano.org nano.org NCCR on Nanoscience