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Quantum Dots II
1. Open Dot Experiments
2. Kondo effect
3. Few Electron Dots
4. Double Quantum Dots
Huibers, Ph.D. Thesis (1999)Huibers et al., PRL83, 5090 (1999)
Open Dot Regime
Open Dot
•Vgate set to allow ≥ 2e2/h conductance through each point contact
•Dot is well-connected to reservoirs
•Transport measurements exhibit CF and Weak Localization
point contacts
many open dot slides: A. Huibers and J. Folk
1.41.21.00.8g
(e2 /h
)
-200 -100 0 100 200VGATE (mV)
Open Dot Regime: Conductance Fluctuations
1.21.00.80.6
g (e
2 /h)
50403020100
BZ (mT)
Repeatable randomintereference fluctuationsas function of dot parameters
V(mV)
B T
V
1 µm
B (mT) T
I
g (e
2 /h)
g (e
2 /h)
NL=NR=1
Ψ(x,y)
Goal: use quantum dot as a probe of quantum phase coherence
2D Cavity with Chaotictrajectories:
Simulation by R. Akis, PRL 79, 123 (1997)
Two-Dimensional Quantum Dot
T
1
R
1
0
T
Any parameter that changes path accumulated phase
no dephasing
dephasingof longer paths
Interferometers
Two-arm:
Regular/Integrable:
Chaotic:1. Mostly chaotic/ergodic
2. Interesting physics& complete description
sometimes: reflections or
small signal
Problem: partially chaotic
Quantum Interference in Open Dots
Interference between all possible trajectories gives rise to repeatable random intereference fluctuationsas function of dot parameters
1.21.00.80.6
g (e
2 /h)
20100
B (mT) T
g (e
2 /h)
1.41.21.00.8g
(e2 /h
)
-200 -100 0 100 200VGATE (mV) V(mV)
g (e
2 /h) B T
V
1 µmI
100
80
60
40
20
0
V (m
V)
151050-5B (mT)
86420 g0.5 g (e2/h) 1.6
λF = Fermi wavelength = 50 nm
vF = Fermi velocity = 200 µm/ns
EF = Fermi energy = 7 meV
2D conductor:area = 2.0 µm2
charge density = 2 1011 e/cm2
bulk mean free path le ~ 2- 10 µm
I
1 µm
Typical Quantum Dot
Dwell time in dot: 200 ps
Crossing time: 7 ps
2.0 µm2
30 bounces
G1
G2 G3
G4
G5G6
Weak Localization
At B=0, phase-coherent backscatteringresults in “weak localization”
Conductance dip at B=0
1.00
0.95
0.90
0.85
aver
age
g (e
2 /h)
-4 -2 0 2 4B (mT)
400 mK 1 K
δg
1.00
0.95
0.90
0.85
g (e
2 /h)
2520151050B (mT)
Single trace for each shape Average of ~30 traces
Huibers, 1998
constructive interference of coherently backscattered, time reversed trajectories decreases conductivity
⎟⎟⎠
⎞⎜⎜⎝
⎛ττ
+−∝σ
δσ ϕ1lnk1F
locl
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ττ
+−−∝σ
δσ−
ϕϕ2/1
F
loc 11k1
WL
l
2D (Lϕ<<W)
1D (Lϕ>>W)
(assuming spinless electrons)
*
* *
*
**
magnetic field: AB-flux, cut off trajectories of area A>φ0Bmagnetoconductance
Quantum Correction: Weak Localization
in a given magnetic field B, trajectories enclosing flux acquire additional Aharonov-Bohm phase:
h
r
h
eBS2Sd)A(e2=∫ ⋅×∇=φ
when summing over all trajectories, this φ will effectivelyeliminate trajectories of area A>>φ0/B. (φ0=h/e)
*
*
**
*
*flux
Weak Localization in Magnetic Fields
8 µm2
δg ~ 0.038 e2/h
3 µm2
δg ~ 0.09 e2/h
1 µm
3.0 µm2
1 µm
8.0 µm2
Weak Localization: Measure of Dephasing
φγδ
++=
121
Ng τφ
−1 =∆hγ φ
random matrix theory
T=300mK
T=300mK
DMZ, 2002
Weak Localization vs T
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
δg (e
2/h
)
0.012 3 4 5 6 7 8 9
0.12 3 4 5 6 7 8 9
12 3 4 5
T (K)
8 µm2 N=1(C88/r67)
8 µm2 N=1 (C15/r315)
3 µm2 N=1 (C32/r347)
1.9 µm2 N=1 (I77/r157)
0.5 µm2 N=1 (C14/r23)
0.4 µm2 N=1 (I75/r164)
0.5 µm2
8.0 µm2
1/3
Huibers, 1998
Huibers et al., PRL83, 5090 (1999)
Low Temperature Saturation?
motion in real space
motion in spin space
spin precession affects phase interference(2π in spin space gives -1 to phase)
electrons move with the Fermi velocity, electric fields in material appear as magnetic fields in the rest frame of the electron
spin-precessions
• depend on magnitude of electron velocity (density dependence)• couple to the electron spin via Zeeman coupling
these magnetic fields
• heterointerface (Rashba)• crystalline anisotropy in III-V zincblende crystal (Dresselhaus)
electric fields due to:
Spin-Orbit Coupling
presence of electric fields Ve1E ∇−=rr
electrons are moving in these electric fields
rest frame of electrons: effective magnetic field
EcvBso
rrr×−=
magnetic moment of electron couples to soBr
mcSer
r =µ
soso BHrr ⋅µ−=
electrons precess around BsoBso depends on the electron momentum
spin rotation symmetry is broken, time reversal symmetry is NOT broken
Spin-Orbit Coupling
III-V Semiconductor
Zinkblende crystall structure:two interpenetrating fcc latticeswith only Ga atoms on one lattice,only As on the other
absence of inversion symmetry
.)cycl)kk(k(H 2z
2yxxso +−σγ=
symmetry considerations:G. Dresselhaus, Phys. Rev. 100, 580 (1955)
after size quantization (2D): 0kz =
)kkkk(H 2yxx
2xyy
)3(D σ−σγ=
)kk(H yyxx)1(
D σ−σα=
cubic Dresselhaus term
linear Dresselhaus term
2zkγ=α
Spin-Orbit Coupling due to Crystal Anisotropy
electric field at heterointerfaceperpendicular to 2D plane
)kk(H xyyxR σ−σβ=
k)0,Ek,Ek(B xysorr
⊥−∝E
Rashba term (linear)
coupling strength parameters β and γ can be determinedfrom Band structure, for example in k.p approximation
AlGaAs
GaAs
Spin-Orbit Coupling due to Heterointerface
assuming strong spin-orbit coupling,summing over all trajectories is equivalent to averaging R2 over sphere *
*
**
*
*
R1
R2
R3
R4
R5
R6
initial state: i
final (forward): iRiRRRf 12Nf == K
iRiRRRf 11N
12
11b
−−−− == Kfinal (backward):
Ri: spin rotations 1RR† = †1 RR =−
interference term iRiff 2fb =
12N RRRR K=
(TRS)
21ff bf −= destructive interference
opposite sign for Magnetoconductance
Weak Antilocalization
T=300mK
dots are on different wafersdots are on different wafers
high densitystronger SO couplingantilocalization (AL)
high densitystronger SO couplingantilocalization (AL)
WL+AL
4µm
low densityweaker SO couplingweak localization (WL)
low densityweaker SO couplingweak localization (WL)
4µm
T=300mK
WL
1. Open Dot Experiments
2. Kondo effect
3. Few Electron Dots
4. Double Quantum Dots
Goldhaber-Gordon et al., Nature 391, 156 (1998)Cronenwett et al., Science 281, 540 (1998)S. Cronenwett, Ph. D. Thesis (2001)
Kondo Effect in Metals
1930s experiments:
lattice phonons
1960s: (exp) related to magnetic impurities
theoretical explanation by Jun Kondospin-fip scattering on mag. impurities
Kondo Effect in Metals: Model
Anderson Hamiltonian
free electrons localizedelectrons
on site charging
coupling between localized and free ele.
new energy scale: Kondo temperature TKformation of spin-singlet screening cloud
cloud: more effective scattererincrease in resistance
Kondo Effect in Metals: spin flip scattering
each scattering eventengangles impurity withconduction electron
singlet cloud formation
temperature scale TK
Kondo Effect in Quantum Dots
spin-flip cotunneling (elastic)
unpaired spin
for T<TK : DOS at µS,D enhanced zero bias conductance!!
for T >> TKDOS peaksuppressed
dots: parameters tunableSINGLE impurity
Kondo Effect in Quantum Dots: Experiment
Goldhaber-Gordon et al., Nature 391, 156 (1998)
Kondo Effect in Quantum Dots: Experiments
Goldhaber-Gordon et al., Nature 391, 156 (1998)
Cronenwett et al., Science 281, 540 (1998)
Kondo Effect in Quantum Dots: Experiments
Kondo Effect in Quantum Dots: Experiments
gate voltages into odd valley
Cronenwett et al., Science 281, 540 (1998)
Cronenwett et al., Science 281, 540 (1998)
Kondo Effect in Quantum Dots: Experiments
Kondo Effect in Quantum Dots: Experiments
1. Open Dot Experiments
2. Kondo effect
3. Few Electron Dots
4. Double Quantum Dots
Kouwenhoven, Austing and Tarucha, RPP 64, 701 (2002)Tarucha et al., PRL77, 3613 (1996)Kouwenhoven et al., Science 278, 1788 (1997)
Few Electron Quantum Dots: Vertical
Kouwenhoven, Austing and Tarucha, RPP 64, 701 (2001)
Few Electron Quantum Dots: Lateral
Ciorga et al., PRB61, R16315 (2000)
Elzerman et al., PRB67, R161308 (2003)Petta et al., PRL93, 186802 (2004)
Chan et al., Nanotech. 15, 609 (2004)Zumbuhl et al., PRL93, 256801 (2004)
Rotation Symmetry and Angular Momentum
Ciorga et al., PRB61, R16315 (2000)Tarucha et al., PRL77, 3613 (1996)
circular symmetry: 2D shell filling circular symmetry broken
2D Periodic Table of Elements
Kouwenhoven, Austing and Tarucha, RPP 64, 701 (2001)
Excitation Spectra of Circular, Few Electron Dots
Kouwenhoven et al., Science 278, 1788 (1997)
radial
angular momentum
Fock-Darwin Energies
Fock-Darwin States: Single Particle Levels
Kouwenhoven, Austing and Tarucha, RPP 64, 701 (2001)
Magnetic Field Transitions
“atomic physics” like experiments not accessible in real atoms!!
exact calculation experiment
Kouwenhoven et al., Science 278, 1788 (1997)
Zero to One Electron Transition
Kouwenhoven et al., Science 278, 1788 (1997)
Higher Transitions
Kouwenhoven et al., Science 278, 1788 (1997)
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