SLIDE 1
Driven Systems
Non interacting “spinless” electrons in a ring. H(Q, P; Φ(t)) − ˙ Φ = electro motive force (RMS) G ˙ Φ2 = rate of energy absorption
- Flux
Collaborations: Tsampikos Kottos (Wesleyan) Holger Schanz - - PDF document
Collaborations: Tsampikos Kottos (Wesleyan) Holger Schanz - - PDF document
The conductance of small mesoscopic disordered rings Doron Cohen, Ben-Gurion University Collaborations: Tsampikos Kottos (Wesleyan) Holger Schanz (Gottingen) Swarnali Bandopadhyay (BGU) Yoav Etzioni (BGU) Tal Peer (BGU) Rangga Budoyo
SLIDE 2
SLIDE 3
Linear Response Theory (LRT)
H = {En} − Φ(t){Inm} G = π¯ h
- n,m
|Imn|2 δT(En − EF) δΓ(Em − En) G = π¯ h(̺(EF))2 |Imn|2 applies if EMF driven transitions ≪ relaxation
- therwise
connected sequences of transitions are essential. leading to Semi Linear Response Theory (SLRT)
SLIDE 4
Semi Linear Response Theory (SLRT)
H = {En} − Φ(t){Inm} dpn dt = −
- m
wnm(pn − pm) wnm = const × gnm × EMF2 Scaled transition rates: gnm = 2̺−3
F
|Inm|2 (En−Em)2 δΓ(En−Em)
SLIDE 5
Semi Linear Response Theory (cont.)
+J −J
E
J En
gnm = 2̺−3
F
|Inm|2 (En−Em)2 δΓ(En−Em) The SLRT analog of the Kubo formula: G = π¯ h(̺(EF))2 |Imn|2 where |Imn|2 ≡ inverse resistivity of the network |Imn|2harmonic ≪ |Imn|2 ≪ |Imn|2algebraic
SLIDE 6
Conductance of mesoscopic rings
(e) (b) (a) (d) (c)
S S
Naive expectation (assuming Γ > ∆): G = e2 2π¯ hM ℓ L + O
∆
Γ
- L = perimeter of the ring
ℓ = mean free path ∝ W 2 ℓ∞ = localization length ≈ Mℓ Ballistic regime: L ≪ ℓ Diffusive regime: ℓ ≪ L ≪ ℓ∞ Anderson regime: ℓ∞ ≪ L
SLIDE 7
Conductance versus disorder
1
G Drude G Diffusive regime regime Ballistic regime Anderson−Mott ∼(∆/Γ) disorder strength (1/l)
10
- 3
10
- 2
10
- 1
10 10
1
w
10
- 8
10
- 4
10 10
4
G
Drude Kubo Meso
SLIDE 8
The RMT modeling
{|vnm|2} ≡ {X} Characterization of the perturbation matrix:
- bandwidth (b)
- sparsity (p)
- texture
10 10
2
10
4
L / l
10
- 6
10
- 4
10
- 2
10
GMeso /GKubo
numerical results for the tight binding model corresponding RMT results VRH approximation for the RMT results
Comparison between:
- Actual results based on “real” matrices
- RMT results based on “artificial” matrices
- Semi-analytical VRH estimate
SLIDE 9
Ergodicity of the eigenstates
- Weak disorder (ballistic rings):
Wavefunctions are localized in mode space.
- Strong disorder (Anderson localization):
Wavefunctions are localized in real space.
10
−2
10
−1
10 5 10 15 20 25 30
1−gT PR
10
- 2
10 10
2
w
10 10
2
PR
mode space position space mode-pos. space
0.1 0.2 0.3 0.4 0.5 0.6
p 10
- 4
10
- 2
10 10
2
X/<X>
0.2 0.4 0.6 0.8 1
F(X)
W=0.05 W=0.35 W=7.50
p
SLIDE 10
Modeling of sparsity
X ≡ |vnm|2 ∼ 1 M2v2
F exp
- − x
l∞
- log(X)
p <X> = p X log(<X>) X X0
1 1
X X X0 X1 BiModal distribution LogBox distribution
X ∈ LogBox[X0, X1] ˜ p ≡ (ln(X1/X0))−1 p ≡ Prob
- X > X
- ≈
−˜ p ln ˜ p X ≈ ˜ pX1 ∼ pX1
SLIDE 11
The VRH estimate
G = π¯ h
e
L
2
n,m
|vmn|2 δT(En−EF) δΓ(Em−En) G = 1 2
e
L
2
̺F
- ˜
Cqm(ω) δΓ(ω) dω ˜ Cqm-LRT(ω) ≡ 2π̺F X ˜ Cqm-SLRT(ω) ≡ 2π̺F X where by definition:
ω
∆
- Prob
- X > X
- ∼ 1
For strong disorder we get: X ≈ v2
F exp
- −∆ℓ
ω
- G
∝
- exp
- −∆ℓ
|ω|
- exp
- −|ω|
ωc
- dω
SLIDE 12
LRT, SLRT and beyond
− ˙ Φ = electro motive force (RMS) G ˙ Φ2 = rate of energy absorption
Semi linear response theory [1]
- D. Cohen, T. Kottos and H. Schanz,
“Rate of energy absorption by a closed ballistic ring”, (JPA 2006) [2]
- S. Bandopadhyay, Y. Etzioni and D. Cohen,
The conductance of a multi-mode ballistic ring, (EPL 2006) [3]
- M. Wilkinson, B. Mehlig, D. Cohen,
The absorption of metallic grains, (EPL 2006) [4]
- D. Cohen,
“From the Kubo formula to variable range hopping”, (PRB 2007) [5]
- T. Peer, R. Budoyo, A. Stotland, T. Kottos and D. Cohen,
The conductance of disordered rings, (arXiv 2007) Beyond (semi) linear response theory [6]
- D. Cohen and T. Kottos,
“Non-perturbative response of Driven Chaotic Mesoscopic Systems”, (PRL 2000) [7]
- A. Stotland and D. Cohen,
”Diffractive energy spreading and its semiclassical limit”, (JPA 2006) [8]
- A. Silva and V.E. Kravtsov,
Beyond FGR, (PRB 2007) [9] D.M. Basko, M.A. Skvortsov and V.E. Kravtsov, Dynamical localization, (PRL 2003)