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
mL
mR
Transmission resonance when
m EN 1  EN  EN( 0)1  e( N  1)Vg  EN( 0)  eNVg 
(0)
(0)
(0)
( 0)
 EN 1  EN  eVg  eVg  EN 1  EN  m
1
2
2
V
(
x
,
y
)

m

(
x

y
)
Example: 2d harmonic oscillator
0
2
 nx ,ny  (nx  1 / 2)0  (ny  1 / 2)0
N
40
(0,3), (3,0), (2,1), (1,2)
30
(0,2), (2,0), (1,1)
20
(0,1), (1,0)
0
(0,0)
EN    i
i 1
eVg( N )  EN 1  EN   N
0
eVgN  ( EN 1  EN )  ( EN  EN 1 )  
0

N  2,6,12,...
otherwise
Coulomb Blockade
Vg  Q / C  eN / C
Vg  e / C  U / e
N ( N  1)
EN  U
 NeVg
2
E N 1  E N  UN  eVg
 Vg  U
charging of a capacitor
Coulomb blockade peaks
0.08
g (e2/h)
0.06
(a)
B = 30 mT
T ~ 100 mK
0.04
0.02
0
-300
-280
-260
Vg (mV)
-240
-220
Single electron transistor
Kastner et al.
Now include quantum effects:
• energies
N ( N  1) N
EN  U
   i  NeVg
2
i 1
E N 1  E N  UN   N 1  eVg
 Vg  U   N 1   N
• wavefunctions
The peak amplitude depends on the wavefunction the
electron tunnels into
Example - Quantum Hall effect:
• All states within a landau level
are degenerate, except edge states,
En=(n+1/2)hc
• The radii are quantized pr2=nf0
(n – Landau level index)
n=1
n=0
McEuen et al.
Spin flips
Kouwenhoven et al.
Level statistics and random matrix theory
Vg  U   N 1   N
Artificial molecules
Dynamics
Nonlinear transport
mL
mR
Probes the excited states
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
A  Be
if 2

2
2
A  B  2 A B Cosf   
2
2

 A  B  2 Re AB*e if 
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
Conductance (2e2/h)
chemical potential
Goldhaber-Gordon, Kastner (1998)
Cronenwett et al. (1998)
Kouwenhoven et al.
Kondo scaling
 2e 2 
G

 h 
Temperature [K]
Goldhaber-Gordon et al.
The Kondo effect out of equilibrium
Meir, 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
Ensslin
eV=E
Inelastic process ?
Noise measurements and electron bunching
Heiblum
The “0.7 anomaly”
Thomas et al.
(1996,1998,2000)
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.
Theory:
Experiment:
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