1
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
http://www.ee.princeton.edu/~chouweb/newproject/research/QDT/SingElecTunnel.html
http://qt.tn.tudelft.nl/publi/papers.html
www.unibas.ch/phys-meso
....
Homework 81. Read the paper:
“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 C60 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. Schönenbergera. 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 C60 Molecules
Danny Porath, Muin Tarabia, Yair Levi and Oded Millo
The General Experimental Scheme
I
V QD
R1,C1
R2,C2
J-1
J-2
STMtip
gold
C60Insulating
layer
2
C60 Fullerenes
~1 m ~10 Å: 1,000,000,000
Structure: (icosahedral symmetry)
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 nano-particles.
The interplay was studied for:∆El << Ec
(Tinkham, Van-Kempen, Sivan, Molenkamp)
C60 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:
Ec < ∆El
Ec > ∆El
Tunneling spectroscopy of isolated C60Electronic level splitting.
Sample Preparation Steps:Rinse C60 powder in Toluene
Evaporate dried powder on a gold substrate covered by carbon and (later) a PMMA layer
C60Evaporation
Goldsubstrate
Prepared sample
STM Images of C60 Fullerenes
70 Å 40 Å
100 Å 50 Å
STM Images of C60 Fullerenes 3-D
3
SET Effects in DBTJ
STMtip
TunnelJunction 1
TunnelJunction 2
I
QD
V
R1,C1
R2,C2
CB
CS
V
I T(V
)
I-V curves
SET effects can be observed If:1) EC = e2/2C > kBT Charging Energy > Thermal Energy2) RT > RQ = h/e2 Tunneling Resistance > Quantum Resistance
Coulomb Blockade (CB):IT = 0 up to |Vt| ≤ e/2C ; Vt depends on “fractional charge” Q0
Coulomb Staircase (CS):Sequence of steps in I-V. Each step - adding an electron to the island
Charge Quantization Leads to SET EffectsCharge quantization leads to SET effects:
Vt depends on “fractional charge” Q0.Q0 represents any offset potentials.Q = ne - Q0 ; |Q0| ≤ e/2
E
Q(e)
Q0 = e/2-1
0 1
2E
Q(e)
-1
Q0 = 0
0
-2
1
2
Ec
Vt = e/2C Vt = 0
Allowed states with n access electrons indicated on the graphs
Origin of the Fractional Charge Q0
Difference in contact potentials across the junctions:Q0 = (C1∆φ1 - C2∆φ2)/e
Note: We are interested only in Q0 mod e.
Q0 can be varied by controlling C1 through tip-sample separation. This is done by changing the STM current and bias settings.
Interplay Between SET Effects and Discrete Energy Levels
∆EL << EC: (Level Spacing << Charging Energy)
e
Ec
1 2
∆EL << EC
(Amman et al.)
No levels
Additional sub-steps on top of a CS step
Interplay Between SET Effects and Discrete Energy Levels
∆EL ~ EC: (Level Spacing ~ Charging Energy)
CS steps are no more equidistantly spaced.
zero bias gap is not suppressed for Q0 = e/2
when the CB is suppressed.
Pronounced asymmetry of I-V curves due to
different levels on each side.
I-V Measurements
STMtip 1
2
I
QV
R1,C1
R2,C2
(a)
(c)-0.6 -0.3 0.0 0.3
-2
-1
0
1
2Single moleculeT = 300 K
Tunn
elin
g C
urre
nt [n
A]
Tip-Bias [V]
(b)
-0.6 0.0 0.6-1.6
0.0
1.6
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
0
2
4
6
8
10 Single moleculeT = 4.2 K
Tunn
elin
g C
urre
nt [n
A]
Tip-Bias [V]
Tunn
elin
g C
urre
nt [n
A]
(d)-1.0 -0.5 0.0 0.5 1.0
-4
-2
0
2
4 Two molecules
Tip Bias [V]
T = 300 K
T = 4.2 K
0.0 0.4 0.80.3
0.6
I-V measurements are done while disconnecting the feed-back loop.Note:
1. Non-vanishing gap.2. Pronounced asymmetry and different steps.3. Negative differential resistance (NDR).
4
I-V Measurements
-1.0 -0.5 0.0 0.5 1.0
-0.8
-0.4
0.0
0.4
I (nA
)
Tip Voltage (V)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
(c)
(b)
(a)
dI/d
V (a
.u.)
0.0
0.2
0.4
0.6
0.8
1.0
T = 4.2 K
T = 4.2 K
dI/d
V (a
.u.)
0.0 0.3 0.6 0.9 1.20.0
0.3
0.6
0.9
dI/d
V
Tip Voltage (V)
-1.0 -0.5 0.0 0.5 1.0
0.0
0.2
0.4
0.6
0.8
1.0T = 300 K
dI/d
V (a
.u.)
Tip Voltage (V)
HOMO+1 (?)
0.0 0.3 0.6 0.9 1.2
0.0
0.3
0.6
0.9
dI/d
V
Tip Voltage (V)
12
HOMO
LUMO
LUMO+1
EF
12
Tunneling through LUMO+1
12
Tunneling through LUMOn=2 on the dot
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 C60
+ hu reduces by ~ 0.6 eV
Theoretical Fits (Orthodox theory-Averin & Likharev)
Γi±(n) - Tunneling rate to/from the dot from/to the electrode.
f - Fermi function.
Di(E), Ei - DOS/Fermi energy at electrode i.
Dd(E), Ed - DOS/Fermi energy at the dot.
Ti(E) - Tunneling matrix element.
Schematic of the circuit:
∫ −−−−−=± )]dEEf(E)[1E(E)DE)f(EE(ED(E)T2π(n)Γ dddiii2
ii h
I
QD
V
R1,C1 R2,C2
The tunneling rate on the dot:
5
Theoretical Fits (Orthodox theory-Averin & Likharev)
P(n) – The probability to find n electrons on the dot
We choose the density of states Dd(E):
(n)]Γ(n)P(n)[Γe(n)]Γ(n)P(n)[ΓeI(V) 1122+−−+ ∑∑ −=−=
The tunneling current is:
Energy Level Calculations
Experimental Results – Asymmetrical Curves
-1.6 -0.8 0.0 0.8 1.6-12
-9
-6
-3
0
3
6
9 (a)
Fits
Exp.
Tunn
elin
g C
urre
nt [n
A]
Bias [V]-1.6 -0.8 0.0 0.8 1.6
-9
-6
-3
0
3
6
9(b)
Fits
Exp.
dI/d
V[a
.u.]
Bias [V]
C1 ~ C2 ~ 10-19, EHOMO - ELUMO ~ 0.7 eV, ∆EL ~ 0.05 eV
Source of AsymmetryC1<C2 ==> V1>V2 ==> Onset of tunneling at junction 1
(for the level configuration at hand)
12
12
Positive bias
12
Negative bias
Experimental Results – Symmetrical Curves
C1 ~ C2 ~ 10-19, EHOMO - ELUMO ~ 0.7 eV, ∆EL ~ 0.05 eV
-1.6 -0.8 0.0 0.8 1.6-5
-4
-3
-2
-1
0
1
2
3
4(a)
Fits
Exp.
Tunn
elin
g C
urre
nt [n
A]
Bias [V]-1.6 -0.8 0.0 0.8 1.6
-4
-3
-2
-1
0
1
2
3
(b)
Fits
Exp.
dI/d
V [a
.u.]
Bias [V]
Symmetrical Barriers (C1~C2)12
Positive bias
2 1
Negative bias
Tunneling through the set of levels closer to EF - in
our case the 3 LUMO levels.
6
Checking The Model - Experimentally
-4
-2
0
2
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
-8
-4
0
4
8
B
A
C1 < C2
Vs = 0.75 VIs = 1.6 nA
Tunn
elin
g C
urre
nt (n
A)
dI/d
V (a
.u.)
dI/d
V (a
.u.)
Vs = 0.75 VIs = 6 nA
C1 > C2
Tunn
elin
g C
urre
nt (n
A)
Tip Voltage (V)
The Triple Barrier Tunnel Junction (TBTJ)
STMtip
PMMA
J-1J-2
goldC
C60
J-3
I
QDV
R1,C1
R2,C2
R3,C3
QD
Tunn
elin
g C
urre
nt [n
A]
-1.0 -0.5 0.0 0.5 1.0
-4
-2
0
2
4 Two molecules
Tip Bias [V]
T = 300 K
T = 4.2 K0.0 0.4 0.8
0.3
0.6
Note:Non Vanishing gap.Negative differential resistance (NDR).Non-ohmic curves everywhere on the sample.
Summary - C60 STM SpectroscopyA C60 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
nenbergeröC. Sch
Quantum dotsplanar dot
planar double-dot vertical dot
(„conventional“) Quantum dots
7
quantum dot ?
2 µm
Different energies (what is 0d ?)
δE=h/τround-trip
in addition: eV and kT
example: 0d with respect to quantum-size effects,provided: kT, eV, and Γ < δE
1. low transparencylow transparency single-electron tunnelingdetermined by single-electron chargingeffects, e.g. Coulomb blockade (0d limit)
2. intermediate transparencyintermediate transparency charging effectsbut co-tunneling, e.g. Kondo effect
3. high transparencyhigh transparency quantum interference ofnon-interacting electrons, e.g. Fabry-Perot resonancesand UCF (disordered limit)
Contacts matter...!1. low transparencylow transparency single-electron tunneling
determined by single-electron chargingeffects, e.g. Coulomb blockade (0d limit)
Contacts matter...!
Single-electron tunneling
„greyscale plot“ of dI/dV (Vgate,Vbias)
here: black = low differential conductance
Coulomb blockade
Vg
Vsd
(mV)
tutorial on dI/dV plots
Change Vsd Change Vg
single-electron chargingenergy UC
∆∆E E addadd addition energyaddition energy, i.e. sum of:
level-spacing δE
8
filling of statesaccording to
S = 1/2 0 1/2 ...
even number of electrons:∆Ε add = UC + δE
odd number of electrons:∆Ε add = UC
even even evenodd odd odd
single-electron-tunneling
if tunneling probability pof each junction is „small“:
current determinedby accessible levels in dot
i.e. by level spacing andCoulomb charging energy
„uncorrelated“ sequential tunnelingdominates. Current I ∝ p
1. low transparencylow transparency single-electron tunnelingdetermined by single-electron chargingeffects, e.g. Coulomb blockade (0d limit)
2. intermediate transparencyintermediate transparency charging effectsbut co-tunneling, e.g. Kondo effect
3. high transparencyhigh transparency quantum interference ofnon-interacting electrons, e.g. Fabry-Perot resonances and UCF (disordered limit)
Contacts matter...!
W. Liang et al., Nature 411, p 665 (2001)
5 mV
-5 mV
gate voltage -2 ... +2 V
bias
vol
tage
T~1 limit (interaction-free wire)
T~1 limit (wire)
W. Liang et al., Nature 411, p 665 (2001)
)2cos(21)( 2
RCC
C
RRTET
δ+Θ++=
{ })2/()2/(2
eVTeVTeVI
−+=∂∂
)( −+ −=Θ kkL round-trip phasedepends on E
Fabry-Perot (interference)
simplemodel
5 mV
-5 mV
gate voltage -2 ... +2 V
their data
9
1. low transparencylow transparency single-electron tunnelingdetermined by single-electron chargingeffects, e.g. Coulomb blockade (0d limit)
2. intermediate transparencyintermediate transparency charging effectsbut co-tunneling, e.g. Kondo effect
3. high transparencyhigh transparency quantum interference ofnon-interacting electrons, e.g. Fabry-Perot resonances and UCF (disordered limit)
Contacts matter...!„co-tunneling“
if tunneling probability pof each junction is „small“:
„uncorrelated“ sequential tunnelingdominates. Current I ∝ p
if tunneling probability p of each junction is „large“:
coherent 2nd (and higher) orderprocesses add substantially
∝ p2
we call this „co-tunneling“
even even evenodd odd odd
black regions =very low conductance G
G is suppressed due toCoulomb blockade (CB)
I jump back to low transparent tunneling contacts for reference
Remainder Coulomb blockade
Vg
Vsd
(mV
)
Elastic co-tunneling Inelastic co-tunneling
δE ~ 0.55 meV UC ~ 0.45 meV
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.
Related work: J.Nygard et al, Nature 408, 342 (2000)
Vg
Vsd
(mV
)
This is called the Kondo effect
Kondo effectresistance R(T) of a piece of metal
„ideal“ metal, e.g. Au wire
superconductor, e.g. Pb
magnetic impurities in „ideal“ metale.g. (ppm)Fe:Au Kondo system
)/log( KTT−
TK
TK
conductance G(T) through a singlemagnetic impurity
(e.g. an spin ½ quantum dot)
2e2/hunitary limit
10
S=1/2 Kondo in Q-dots
from L. Kouwenhoven & L. Glazman, Physics World, June 2001
at 50 mK
Mark Buitelaar and Thomas Nussbaumer
why interesting?
it‘s manymany--body physicsbody physics (all orders are relevant)
more precisely: it‘s manymany--electron physicselectron physics
in Condensed Matter Physics:Superconductivity
Luttinger liquid (non-Fermi liquids)Kondo physics
because it is complicated
Superfluidity
Kondo effect Superconductivity
∆∆
Al
Kondo effect and superconductivity are many-electron effects• can Kondo and superconductivity coexist or do they exclude each other ?
Kondo physics + superconductivity
spin 1/2 Kondo + S-leads
UEF
Kondo effect is the screeningof the spin-degree of the dot spinby exchange with electrons fromFermi-reservoirs (the leads)
normal case superconducting case
1. a gap opens in the leads
2. Cooper pairs have S=0
Hence: Kondo effect suppressed,but ....Cooper pair
A cross-over expected at kb TK ~ ∆
S = 0Energy scale : ~ ∆
Cooper pairS = 0
Energy scale : ~ kb TK
Kondo effect Superconductivity
∆∆
11
Conclusions
nanotubes can serve as a model systemmodel systemto study excitingexciting physics
Kondo physics (co-tunneling)Interplay between Kondo physics & superconductivity
www.unibas.ch/physwww.unibas.ch/phys--mesomeso group Web-page
www.nccrwww.nccr--nano.orgnano.org NCCR on Nanoscience
nanotubes may be one part of the toolboxof „nanotechnology“