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)