Current through a very small conductor nano HUB .org online - PowerPoint PPT Presentation

CQT Lecture #4 nano HUB .org online simulations and more Unified Model for U: Self-consistent Quantum Transport CQT, Lecture#4: Field (SCF) Coulomb blockade and Fock space Far from Equilibrium Objective: Σ s To illustrate the limitations of the model described in Lectures 2,3 and introduce a μ 1 μ 2 completely different approach based on H the concept of Fock space. Σ 1 Σ 2 I believe this will be a key concept in the next stage of development of transport physics. Approach based on “QTAT” (1)Beenakker, Phys.Rev.B44,1646 (1991), (2) Datta, Quantum Transport: Averin & Likharev, J.LowTemp.Phys. 62, Atom to Transistor, 345 (1986) Cambridge (2005) Reference: QTAT, Chapter 3.4. Supriyo Datta 1 Network for Computational Nanotechnology

Current through a very small conductor nano HUB .org online simulations and more γ 2 / � γ � q γ 1 / 1 ⇒ 1 Normalized 2 � 0.9 Current 0.8 F μ 0.7 1 μ 2 0.6 0.5 0.4 ⎡ ⎤ 0.3 γ 1 f 1 + γ 2 f 2 F = 0.2 − qV D ⇒ ⎢ ⎥ γ 1 + γ 2 ⎣ ⎦ 0.1 γ 2 / � 0 γ � -0.2 0 0.2 0.4 0.6 / f γ 1 γ 2 1 1 q [ ] = f 1 − f 2 I γ 1 + γ 2 � γ 2 / � γ � / 1 μ 2 μ 1 γ 1 γ 2 μ 2 q γ 2 / � γ � = / max I 1 γ 1 + γ 2 � μ 1 q γ 1 ⇒ if γ 2 = γ 1 μ 2 2 � μ 1 Supriyo Datta 2 Network for Computational Nanotechnology

Conductance of a very small conductor nano HUB .org online simulations and more γ 2 / � γ � / Assume γ 2 = γ 1 1 2 γ + 4 k B T 1 μ q γ 1 1 ⇒ μ 2 1 2 � Conduc tan ce = ∂ I Normalized 0.8 Current 0.6 ∂ V D 0.4 q γ 1 /2 � 0.2 ~ − qV D ⇒ 0 2 γ 1 + 4 k B T -0.2 -0.2 0 0.2 0.4 0.6 0.8 q 2 /4 � γ 2 / � γ 1 >> k B T γ � γ 2 / � ~ if γ / � / 1 1 μ 2 Conduc tan ce quantum μ 1 μ 2 q 2 /2 π � μ 1 ~ Supriyo Datta 3 Network for Computational Nanotechnology

Effect of “U” on conductance nano HUB .org online simulations and more Assume γ 2 = γ 1 U 0 = 0.5 eV S D U 0 /2 CHANNEL q γ 1 ⇒ V G 1 V D 2 � 0.9 Normalized I 0.8 Current 0.7 0.6 0.5 0.4 U 0 : Increase in 0.3 0.2 0.1 potential due to 0 − qV D ⇒ -0.2 0 0.2 0.4 0.6 SINGLE electron γ 2 / � γ 2 / � γ γ � � / / 1 1 Assume Level floats up U 0 >> k B T , γ 1 , γ 2 μ 2 γ 2 by U 0 γ 1 + γ 2 μ 2 μ 1 μ 1 Supriyo Datta 4 Network for Computational Nanotechnology

SCF with self-interaction correction nano HUB .org online simulations and more Assume γ 2 = γ 1 U 0 = 0.5 eV S D U 0 /2 CHANNEL q γ 1 ⇒ V G 1 V D 2 � 0.9 Normalized I 0.8 Current 0.7 0.6 0.5 0.4 0.3 0.2 0.1 = U 0 ( N − 0 − qV D ⇒ U i N i ) -0.2 0 0.2 0.4 0.6 Self-interaction γ 2 / � γ 2 / � γ γ � � / / 1 1 Level does Correction NOT float up μ 2 μ 1 μ 1 Supriyo Datta 5 Network for Computational Nanotechnology

2 levels: Unrestricted SCF nano HUB .org online simulations and more γ 2 / � γ � γ 2 / � / γ � / 1 1 γ 2 / � γ � / 1 μ 2 μ 2 μ 1 μ 1 μ 1 μ 2 1 0.9 0.8 Restricted SCF 0.7 Unrestricted SCF 0.6 = U i U 0 N 0.5 0.4 = U 0 ( N − 0.3 U i N i ) 0.2 0.1 0 Self-interaction -0.2 0 0.2 0.4 0.6 0.8 Correction Supriyo Datta 6 Network for Computational Nanotechnology

2 levels: SCF versus exact nano HUB .org online simulations and more γ 2 / � γ 2 / � γ 2 / � γ γ γ � � � / / / 1 1 1 One-electron μ 2 energy levels μ 2 μ 1 μ 1 μ 1 μ 2 1 Exact 0.9 Needs picture 0.8 in “Fock” space 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.2 0 0.2 0.4 0.6 0.8 Supriyo Datta 7 Network for Computational Nanotechnology

1-level: the view from “Fock space” nano HUB .org online simulations and more One-electron picture “Fock space” 1 E = ε 1 E = ε 1 E = 0 0 Most of our thinking is based on this picture Supriyo Datta 8 Network for Computational Nanotechnology

2 levels: the view from “Fock space” nano HUB .org online simulations and more 2^2 many-electron levels 2 one-electron levels 11 E = ε 1 + ε 2 + U 0 E = ε 2 E = ε 2 10 E = ε 1 E = ε 1 01 E = 0 00 Supriyo Datta 9 Network for Computational Nanotechnology

1-level: Current flow in “Fock space” nano HUB .org online simulations and more γ 1 γ 2 q γ 2 / � = γ � I / γ 1 + γ 2 E = ε 1 � 1 1 1 = 0 P 1 0.9 γ 2 / � γ � / 0.8 1 0.7 ε 1 μ 2 0.6 μ 1 0 P 0 = 1 0.5 E = 0 0.4 0.3 0.2 0.1 0 -0.2 0 0.2 0.4 0.6 γ 2 / � γ � / E = ε 1 1 = γ 2 /( γ 1 + γ 2 ) P 1 1 = ( q / � ) γ 1 P μ 2 I γ � γ 2 / � 1 / 1 = ( q / � ) γ 2 P 0 ε 1 μ 1 0 E = 0 P 0 = γ 1 /( γ 1 + γ 2 ) Supriyo Datta 10 Network for Computational Nanotechnology

2 levels: Current flow in “Fock space” nano HUB .org online simulations and more 1 11 11 0.9 0.8 0.7 01 10 0.6 01 10 0.5 0.4 00 0.3 00 0.2 0.1 γ 2 / � 11 γ � / 0 γ 2 / � γ -0.2 0 0.2 0.4 0.6 0.8 � 1 / 1 μ 2 γ 2 / � γ � 01 10 / μ 1 μ 2 1 μ 1 μ 2 00 μ 1 Supriyo Datta 11 Network for Computational Nanotechnology

2 levels: Current flow in “Fock space” nano HUB .org online simulations and more 2 γ 1 γ 2 1 I = q 0.9 γ 1 + γ 2 � 0.8 → ( q γ 1 / � ) 2 γ 1 γ 2 I = q 0.7 γ 1 + 2 γ 2 � 0.6 0.5 → (2/3) ( q γ 1 / � ) 0.4 0.3 0.2 = ( q / � ) γ 1 P 0.1 I 1 0 = ( q / � ) 2 γ 2 P 0 -0.2 0 0.2 0.4 0.6 0.8 γ 2 / � γ � / 01 10 1 = 2 γ 2 /( γ 1 + 2 γ 2 ) 1 P μ 2 P 0 = γ 1 /( γ 1 + 2 γ 2 ) μ 1 00 Supriyo Datta 12 Network for Computational Nanotechnology

Coulomb blockade and strong correlation nano HUB .org online simulations and more and γ 2 / � γ f f 00 = (1 − f ↑ )*(1 − f ↓ ) � ↑ ↓ / P 1 10 = f ↑ *(1 − f ↓ ) P “UNCORRELATED” 01 = (1 − f ↑ )* f ↓ μ 2 P μ 11 = f ↑ * f ↓ 1 P 1 11 0.9 STRONGLY 0.8 0.7 CORRELATED 0.6 01 10 0.5 11 = 0 P 0.4 0.3 10 = P 01 = γ 2 /( γ 1 + 2 γ 2 ) P 0.2 00 0.1 P 00 = γ 1 /( γ 1 + 2 γ 2 ) 0 -0.2 0 0.2 0.4 0.6 0.8 Supriyo Datta 13 Network for Computational Nanotechnology

4 levels: USCF versus exact nano HUB .org online simulations and more Unrestricted SCF γ 2 / � γ = U 0 ( N − � / U i N i ) 1 1 0.9 μ 2 USCF plateaus have μ 1 0.8 wrong width: U 0 /2 0.7 wrong height: 1/N 0.6 0.5 Exact 0.4 Fock space 0.3 Restricted SCF approach = 0.2 U i U N 0 0.1 0 -0.5 0 0.5 1 1.5 2 Supriyo Datta 14 Network for Computational Nanotechnology

Equilibrium is different … nano HUB .org online simulations and more SCF: Restricted γ 2 / � γ � / Exact 1 and unrestricted μ 2 4 μ 1 3.5 3 of electrons Non-equilibrium Number 2.5 2 μ 1 = μ 2 ⇒ 1.5 1 Normalized 0.9 1 0.8 Current 0.7 0.5 0.6 0 0.5 -0.5 0 0.5 1 1.5 2 0.4 0.3 Equilibrium 0.2 0.1 0 -0.5 0 0.5 1 1.5 2 − qV D ⇒ Supriyo Datta 15 Network for Computational Nanotechnology

Being “close” to equilibrium helps too nano HUB .org online simulations and more SCF: Restricted γ 2 / � γ � / Exact 1 and unrestricted μ 2 μ 1 ---> Closer ---> to equilibrium ---> γ 2 = γ 1 γ 2 = 100 γ 1 γ 2 = 10 γ 1 1 1 1 Normalized 0.9 0.8 0.8 0.8 Current 0.7 0.6 0.6 0.6 0.5 0.4 0.4 0.4 0.3 0.2 0.2 0.2 0.1 0 0 -0.5 0 0.5 1 1.5 2 -0.5 0 0.5 1 1.5 2 0 -0.5 0 0.5 1 1.5 2 − qV D ⇒ Supriyo Datta 16 Network for Computational Nanotechnology

General “Fock space” approach nano HUB .org online simulations and more 2 one-electron 2^2 many-electron levels levels γ 2 / � γ E = 2 ε + U 0 � 11 / 1 ε + U 0 γ 1 (1 − f 1 ( ε + U 0 )) γ 1 f 1 ( ε + U 0 ) ε μ 1 μ 2 E = ε 01 10 γ 1 f 1 ( ε ) γ 1 (1 − f 1 ( ε )) E = 0 00 Supriyo Datta 17 Network for Computational Nanotechnology

Equilibrium in “Fock space” nano HUB .org online simulations and more 11 E = 2 ε + U 0 γ 1 f 1 ( ε + U 0 ) γ 1 (1 − f 1 ( ε + U 0 )) γ 1 f 1 ( ε ) γ 1 (1 − f 1 ( ε )) 01 10 E = ε E = 0 ⇑ f 1 ( Δ E ) 00 P N + 1 e − ( Δ E − μ ) / k B T = = = ⇓ 1 − f 1 ( Δ E ) P N Law of γ 2 / � γ � / 1 Z e − ( E i − μ N i ) / k B T Equilibrium i = 1 P ε + U 0 ε No general solution μ 1 μ 2 out-of-equilibrium with µ 2 ≠ µ 1 Supriyo Datta 18 Network for Computational Nanotechnology

Rate equations in “Fock space” nano HUB .org online simulations and more 11 γ 1 f 1 ( ε + U 0 ) E = 2 ε + U 0 γ 1 f 1 ( ε + U 0 ) + γ 2 f 2 ( ε + U 0 ) + γ 2 f 2 ( ε + U 0 ) 01 10 E = ε γ 1 f 1 ( ε ) γ 1 f 1 ( ε ) + γ 2 f 2 ( ε ) E = 0 + γ 2 f 2 ( ε ) 00 ⎧ ⎫ ⎧ ⎫ ⎡ ⎤ ⎧ ⎫ ∗ 0 0 P 00 P 00 ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ⎪ P 01 0 0 P 01 d ⎢ ⎥ ⎨ ⎬ = ⎨ ⎬ = ⎨ ⎬ ⎢ ⎥ ∗ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dt P 10 0 0 P 10 ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ⎩ ⎭ ⎩ ⎭ ⎣ ⎦ ⎩ ⎭ P 11 0 0 P 11 Each column 0 0 0 0 adds to zero Supriyo Datta 19 Network for Computational Nanotechnology