Quantum Dots – Past, Present and Open Questions

Yigal Meir

Department of Physics &The Ilse Katz Center for Meso- and Nano-scale

Science and Technology

Beer Sheva, ISRAEL

Quantum dot – an artificial device, small enough so that quantization of energy levels and electron charge

are important

vertical quantum dots

Single molecules

Tarucha et al.

Vg

L

R

Transmission resonance when

)0()0(

1)0()0(

1

)0()0(11 )1(

NNggNN

gNgNNN

EEeVeVEEeNVEVNeEEE

00, )2/1()2/1( yxnn nnyx

NNNNg EEeV 1

)(

otherwise

NEEEEVe NNNNNg 0

,...12,6,2)()( 011

)0,0(0

)0,1(),1,0(2 0

)1,1(),0,2(),2,0(3 0

)2,1(),1,2(),0,3(),3,0(4 0

)(2

1),( 22

0 yxmyxV

N

iiNE

1

Example: 2d harmonic oscillator

Coulomb Blockade

eUCeV

CeNCQV

g

g

//

//

UV

eVUNEE

NeVNN

UE

g

gNN

gN

1

2

)1(

charging of a capacitor

0.08

0.06

0.04

0.02

0

g (e

2 /h)

-300 -280 -260 -240 -220Vg (mV)

(a) B = 30 mTT ~ 100 mK

Coulomb blockade peaks

Single electron transistor

Kastner et al.

Now include quantum effects:

• energies

NNg

gNNN

g

N

iiN

UV

eVUNEE

NeVNN

UE

1

11

12

)1(

• wavefunctions

The peak amplitude depends on the wavefunction the electron tunnels into

n=1

n=0

Example - Quantum Hall effect:• All states within a landau level are degenerate, except edge states, En=(n+1/2)hc

• The radii are quantized r2=n0 (n – Landau level index)

McEuen et al.

Spin flips

Kouwenhoven et al.

NNg UV 1

Level statistics and random matrix theory

Artificial molecules

Dynamics

R

L

Probes the excited states

Nonlinear transport

Foxman et al.

Correlation between excited state of N electrons and the ground states of N+1 electrons

Marcus et al.

B

Is transport through a quantum dot coherent ?

Yacoby, Heiblum

CosBABA

eABBABeA ii

2

Re2

22

*222

Checking quantum measurement theory

Aleiner, Wingreen, Meir

Buks et al.

The Kondo effect

Relevant to transport through quantum dots

Ng and Lee

Glazman and Raikh

chemical potential

Conductance (2e2/h)

Goldhaber-Gordon, Kastner (1998)Cronenwett et al. (1998)

Kouwenhoven et al.

arX

iv:c

ond

-mat/

98

072

33

v1

15

Ju

l 1

99

8arX

iv:c

ond

-mat/

98

072

33

v1

15

Ju

l 1

99

8

Vds I

Vg

(b)

-140 -130 -120 -110 -100 -90 -80 -7 00

0.1

0.2

0.3

0.4

0.5

100mK 200mK 300mK 500mK 800mK 1000mK 1500mK 2200mK 3000mK 3800mK

-140 -130 -120 -110 -100 -90 -80 -7 00

0.1

0.2

0.3

0.4

0.5 -4 -2 0 2 4 6 8 10 12

0 2 40

5

10

15

20

25

102

103

104

0

0.1

0.2

0.3

0.4

0.5(a) = -0.91

= -1.26 = -1.60 = -1.95 = -3.67

102

103

104

(b) = -0.74 = -0.48 = -0.22 = 0.12 = 0.47 = 1.50

10-2

10-1

100

101

0

0.2

0.4

0.6

0.8

1

= -0.74, = 280 eV= -0.91= -1.08

= -0.98, = 215 eV

= -1.00, NRG results

= 0.00

h

eG

22

Temperature [K]

Kondo scaling

Goldhaber-Gordon et al.

The Kondo effect out of equilibriumMeir, Wingreen, Lee

The two-impurity Anderson model

Georges & Meir

chang

Kondo vs. RKKY

Marcus et al.

The two-channel Kondo effect

Non- Fermi liquid ground state

Oreg & Goldhaber-Gordon

More open questions

Phase of transmission amplitude

Heiblum

eV=E

Inelastic process ?

Ensslin

Noise measurements and electron bunching

Heiblum

Thomas et al. (1996,1998,2000)

The “0.7 anomalyThe “0.7 anomaly””

Rejec and Meir

conclusions• Quantum dots are controllable miniaturized devices,

which can be instrumental in our understanding of mesoscopic and strongly correlated systems.

• May be the basic ingredient in applications of quantum computing.

• In spite of their apparent simplicity, still many open questions.

P. A. Lee P. Nordlander M. Kastner

N. S. Wingreen M. Pustilnik U. Meirav

J. Kinaret A. Golub P. McEuen

B. L. Altshuler Y. Avishai E. Foxman

X.-G. Wen A. Auerbach D. Goldhaber-Gordon

A.-P. Jauho P. Rojt L. Kouwenhoven

A. L. Aleiner O. Entin-Wohlman R. Ashoori

E. Shopen A. Aharony M. Heiblum

A. Georges T. Aono A. Yacoby

D. C. Langreth Y. Dubi C. Marcus

K. Hirose T. Rejec K. Ensslin

Y. Gefen T. Ihn

Theory: Experiment: