the coulomb blockade in quantum boxes avraham schiller racah institute of physics eran lebanon...
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![Page 1: The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University)](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d375503460f94a0fc1d/html5/thumbnails/1.jpg)
The Coulomb Blockade
in Quantum Boxes
Avraham Schiller
Racah Institute of Physics
Eran Lebanon (Hebrew University)
Frithjof B. Anders (Bremen University)
* Funded by the ISF Centers of Excellence Program
![Page 2: The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University)](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d375503460f94a0fc1d/html5/thumbnails/2.jpg)
Charging energy in Quantum boxes
Quantum box: Small metallic grain or large semiconductor
quantum dot with sizeable Charging energy
EC but dense single-particle levels
Charging energy:
QVC
QQE B
0
2
2)(
0
2
2C
eEC
Energy for charging box with one electron
![Page 3: The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University)](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d375503460f94a0fc1d/html5/thumbnails/3.jpg)
T = 0 , t = 0T > 0 and/or t > 0
QVC
QQE B
0
2
2)(g
![Page 4: The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University)](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d375503460f94a0fc1d/html5/thumbnails/4.jpg)
Weak single-mode tunneling to the lead
High temperature: Perturbation theory about the classical limit
Low temperature: PT about well-defined charge configuration
PT breaks down near degeneracy points!
Effective low-energy model: two-channel Kondo model
(Matveev 1991)
T << EC
![Page 5: The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University)](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d375503460f94a0fc1d/html5/thumbnails/5.jpg)
Near perfect transmission (single-mode tunneling)
Treated within an effective model where the QB and the lead are coupled
adiabatically by a 1D geometry with weak backscattering
There are Coulomb-blockade oscillations also near perfect transmission
(Matveev 1995)
![Page 6: The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University)](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d375503460f94a0fc1d/html5/thumbnails/6.jpg)
No unified treatment of all regimes!
Different models and treatments were tailored for the different regimes
Certain treatments are based on effective models involving high-energy
cutoffs which are not fully determined
The crossover behavior between the different regimes is not accessible
Some regimes, e.g. strong tunneling amplitudes, remain unexplored
A unified nonperturbative treatment of all physical regimes is clearly needed!
![Page 7: The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University)](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d375503460f94a0fc1d/html5/thumbnails/7.jpg)
Point-contact tunneling model
g
BBB
kkBkLk
BL kkkk QV
C
QcctccH ˆ
2
ˆH.c.
0
2
,',, ,
,
)(ˆk
BkBkBkB cceQ
Excess charge inside box
![Page 8: The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University)](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d375503460f94a0fc1d/html5/thumbnails/8.jpg)
Our approach: Use wilson’s numerical renormalization group (NRG)
Problem: The NRG is designed to treat noninteracting conducton
electrons. In this case the box electrons are interacting!
Solution: Introduce collective charge operators:
n
nnneQ̂
n
nnQ 1ˆ
Map Hamiltonian onto:
QVC
QccQtccH B
kkBkLk
BL kkkk
ˆ2
ˆH.c.ˆ
0
2
,',, ,
The constraint BQQ ˆˆ can be relaxed for 0 !!!
![Page 9: The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University)](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d375503460f94a0fc1d/html5/thumbnails/9.jpg)
Temperature evolution of the Coulomb-blockade staircase
Coulomb staircase fully develops only well below EC
Capacitance
C(T) =-d<Q>/dVB
![Page 10: The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University)](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d375503460f94a0fc1d/html5/thumbnails/10.jpg)
T = 0: Comparison with 2nd order perturbation theory
CEDd /
Excellent agreement with PT at weak coupling at charge plateaus
NRG and mapping work!
![Page 11: The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University)](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d375503460f94a0fc1d/html5/thumbnails/11.jpg)
Increasing the tunneling amplitude: breakdown of perturbation theory
Reentrance of Coulomb-blockade staircase for t
tT = 0, d = 100
![Page 12: The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University)](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d375503460f94a0fc1d/html5/thumbnails/12.jpg)
Origin of rapid breakdown of perturbation theory and reentrance of CB
The relevant physical parameter is the single-particle transmission coefficient
In the noninteracting case, the latter is given by 22
2
)(1
)(4
t
tT
![Page 13: The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University)](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d375503460f94a0fc1d/html5/thumbnails/13.jpg)
Near perfect transmission
)2sin()(cosln2 2
BBB NNeRR
Nn
Prediction of 1D model :
![Page 14: The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University)](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d375503460f94a0fc1d/html5/thumbnails/14.jpg)
Near perfect transmission
)2sin()(cosln2 2
BBB NNeRR
Nn
Euler’s constant
Reflectance
Prediction of 1D model :
![Page 15: The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University)](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d375503460f94a0fc1d/html5/thumbnails/15.jpg)
Near perfect transmission
)2sin()(cosln2 2
BBB NNeRR
Nn
Single fitting parameter R
Extracted R versus noninteracting 1 - T
Prediction of 1D model :
![Page 16: The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University)](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d375503460f94a0fc1d/html5/thumbnails/16.jpg)
Two-channel Kondo effect at degeneracy points
Two-channel Kondo effect expected to develop at degeneracy points
Characterized by log(T) divergence of the junction capacitance:
)/ln(20
)(2ˆ
TTTk
eTC K
KBBdV
Qd
Kondo temperature
Log(T) divergence for all values of t
![Page 17: The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University)](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d375503460f94a0fc1d/html5/thumbnails/17.jpg)
Conclusions
An NRG approach was devised for solving the charging of a quantum
box connected to a lead by single-mode tunneling, applicable to all
temperatures, gate voltages and tunneling amplitudes.
Rapid breakdown of perturbation theory is found, followed by reentrance
of the Coulomb-blockade staircase for tunneling amplitudes exceeding
perfect transmission.
Two-channel Kondo effect is found at the degeneracy points for all
tunneling amplitudes, directly from the Coulomb-blockade Hamiltonian.
The tunneling Hamiltonian is capable of describing all regimes of the
Coulomb blockade, including the vicinity of perfect transmission.
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Two-channel Kondo effect in charge sector
(Matveev ‘91)
Focus on EC>>kBT and on
vicinity of a degeneracy point
Introduce the charge isospin
NNNNz 112
eVcccctccH zqk
kLqBqBkLBL k
kkk
,,, ,
Lowering and raising isospin operatorsChannel index
NN 1
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Two-channel Kondo effect
2,12,1 ,
)0(
sSJccΗ impk
kkk
Impurity spin is overscreened by two identical channels
rT 0
A non-Fermi-liquid fixed point is approached for T<<TK