Atom to Transistor, 2 1 Reference: QTAT, Chapter 1. Cambridge - PowerPoint PPT Presentation

CQT Lecture #2 nano HUB .org online simulations and more Unified Model for Quantum Transport Far from Equilibrium CQT, Lecture#2: Σ s Electrical Resistance: A Simple Model μ 1 μ 2 H Σ 1 Σ 2 Objective: To introduce a simple quantitative model for describing current flow in nanoscale structures and relate it to Simple version Σ s well-known large scale properties like Ohm’s Law. μ 1 μ 2 D(E) D(E) Model based on “QTAT” Datta, Nanotechnology,15, S433 (2004). Datta, Quantum Transport: γ γ Atom to Transistor, 2 1 Reference: QTAT, Chapter 1. Cambridge (2005) 1 Network for Computational Nanotechnology

Equilibrium Energy Level Diagram nano HUB .org online simulations and more V G < 0 <-- L --> n-type S D CHANNEL Vacuum V G > 0 V G Level I V = 0 EMPTY 0.25 0.2 f (E) 0.15 0.1 E No states 0.05 0 -0.05 -0.1 FILLED -0.15 -0.2 -0.25 -0.2 0 0.2 0.4 0.6 0.8 1 Fermi function p-type S Channel D 2 Network for Computational Nanotechnology

What makes electrons flow? nano HUB .org online simulations and more S D CHANNEL S D CHANNEL V G > 0 V G V D V D I I µ1 µ1 µ2 µ2 3 Network for Computational Nanotechnology

Escape rate nano HUB .org online simulations and more γ / � : Escape Rate γ has dimensions of energy γ γ � � 2 / / 1 S Channel D µ1 µ2 γ 1 Small 4 Network for Computational Nanotechnology

Current through a very small conductor nano HUB .org online simulations and more S Channel D Normalized 1 Current 0.8 0.6 I V 0.4 0.2 0 V ⇒ -0.2 -0.2 0 0.2 0.4 0.6 γ γ � � 1 / / µ 2 1 µ 1 μ 2 μ 1 V 5 Network for Computational Nanotechnology

What is Conductance ? nano HUB .org online simulations and more q γ 1 0.1 0.1 γ 1 / � γ � 0.08 0.08 empty / Normalized 1 1 0.06 0.06 2 � Current 0.04 0.04 μ 2 0.8 0.02 0.02 0 0 0.6 -0.02 -0.02 full -0.04 -0.04 0.4 μ 1 -0.06 -0.06 0.2 -0.08 -0.08 -0.1 -0.1 0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 0 V ⇒ -0.2 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 Normalized Conductance 0.8 γ � d I q / 2 0.6 1 ~ 0.4 d V 4 k T / q 0.2 0 -0.2 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 6 Network for Computational Nanotechnology

Conductance quantum nano HUB .org online simulations and more q γ 1 γ � γ � 1 / / Normalized 1 1 2 � Current μ 2 0.8 0.6 0.4 μ 1 0.2 0 V ⇒ -0.2 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 γ � q / 2 d I 1 ~ γ + ( 2 4 ) / d V k T q 1 1 Normalized Conductance 0.8 q 2 /4 � 0.6 γ 1 >> k T ~ if 0.4 0.2 Conductance quantum 0 π Ω � -0.2 2 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ~ q / 2 ~ 1/ 25.8 K 7 Network for Computational Nanotechnology

Conductance: The bottom line nano HUB .org online simulations and more γ γ � � / / q γ µ 1 ~ I D qV � � � 2 � � D Number qV Current of states µ 2 per state 2 q = π γ I D D: Density � � � � � π V � 2 � of states Transmission Conductance Quantum 8 Network for Computational Nanotechnology

Bottom to Top: A Short-cut nano HUB .org online simulations and more Cross-section A ; Length L γ γ � � / / γ γ � � / / µ1 D(E) µ2 <---- L ----> = 2 D ( E ) N ( E ) AL � q � � � � � � = π γ 0 I D 3 π V � nm / eV − 2 3 / eV nm Will show that D � � 2 v γ → γ γ → γ ~ D ~ A / L ~ D ~ A 2 L L 9 Network for Computational Nanotechnology

Ballistic vs. Diffusive channels nano HUB .org online simulations and more Ballistic Diffusive γ γ γ γ <-------- L --------> <-------- L --------> n L ∂ n L n ~ ⇒ = D L ⇒ = − Flux n v Flux L ∂ x γ Flux γ = Flux = � Stored electrons � Stored electrons ν n v ~ ~ = = L D D n / L 2 = = L n L L L 2 n L / 2 L L = 1 / Transit time = 1 / Transit time 10 Network for Computational Nanotechnology

Drift-diffusion equations nano HUB .org online simulations and more Diffusive transport 2 ~ q D � = γ I 2 D γ = V � 2 2 L = D N 0 AL conductivity <---- L ----> S D O q 2 N 0 ˜ D R = IV U ( A / L ) A CHANNEL R I C N E Non-degenerate V I n D → q 2 ˜ ( A / L ) k B T ~ Einstein D k B T = = q n μ Relation : ( A / L ) μ q 11 Network for Computational Nanotechnology

Transmission nano HUB .org online simulations and more One broadened γ γ � � / / 0.2 level : 0.15 0.1 µ1 0.05 0 γ / π -0.05 = D ( E ) ( E − ε ) 2 + γ 2 qV -0.1 -0.15 -0.2 0.5 1 1.5 2 2.5 3 3.5 µ2 D ( E = ε ) = 1/ π γ π D γ = 1 2 q = π γ I D � � � � � π V � 2 � � �� � Transmission Conductance quantum 12 Network for Computational Nanotechnology

Number of modes nano HUB .org online simulations and more Ballistic transport Electrons with effective mass ‘m’ γ = � v x / L = L ( m / π � v ) E 1 D : D π D γ = 1 D(E) 2 q = π γ I = LW m /2 π � 2 D 2 D : D � � � π V E � � 2 � � Transmissi on π D γ = W m ν x /2 � Conduc tan ce quantum W ≈ D(E) λ /2 <---- L ----> S D O R U A CHANNEL LAm 2 v /3 π 2 � 3 R I = E C 3 D : D N E V A π D γ ≈ ( λ /2) 2 I D(E) 13 Network for Computational Nanotechnology

Current through one level nano HUB .org online simulations and more γ 2 / � γ � / γ = γ 1 Set : E E 2 1 0.25 0.2 1 - 0 0.25 µ1 0.15 0.2 0.1 0.15 γ 0.05 q 0.1 f − 0 0.05 1 I ~ [ f f ] -0.05 0 -0.05 -0.1 � 1 2 2 -0.1 µ2 -0.15 -0.15 -0.2 -0.2 -0.25 -0.2 0 0.2 0.4 0.6 0.8 1 -0.25 -0.2 0 0.2 0.4 0.6 0.8 1 f 2 ( E ) f 1 ( E ) γ γ [ ] [ ] γ 1 = − = 2 − 1 q I q f f I q f f 1 1 2 2 � � I ~ � 2 γ 1 γ 2 q [ ] = f 1 − f 2 I γ 1 + γ 2 � 14 Network for Computational Nanotechnology

Current with Broadening nano HUB .org online simulations and more S D CHANNEL ⎡ ⎤ γ + γ f f = ∫ dE 1 1 2 2 V G ⎢ ⎥ n D ( E ) V D γ + γ ⎣ ⎦ I 1 2 γ 1 γ 2 γ 2 / � q γ [ ] ∫ dE � = f 1 − f 2 / D ( E ) I γ 1 + γ 2 1 � µ1 qV µ2 15 Network for Computational Nanotechnology

Importance of electrostatics nano HUB .org online simulations and more γ 1 γ 2 γ 2 / � q γ [ ] � = f 1 − f 2 / γ 2 / � γ I � 1 / γ 1 + γ 2 � 1 μ 2 γ 2 / � γ � / 1 μ 1 μ 1 μ 2 γ 2 / � γ � / 1 μ 1 μ 2 μ 2 μ 1 1 1 1 0.8 0.8 0.8 Normalized 0.6 0.6 0.6 0.4 Current 0.4 0.4 0.2 0 0.2 0.2 0 0 0 -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 -0.6 -0.6 -0.6 -0.8 -0.8 -0.8 -1 -1 -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 ⇒ -1 -0.8 -0.6 -0.4 -0.2 0 0 0.2 0.4 0.6 0.8 1 ⇒ ⇒ qV qV qV D D D 16 Network for Computational Nanotechnology

Why does the current in a transistor saturate ? nano HUB .org online simulations and more S D CHANNEL 6x 10 -4 V G Drain V D I current 5 Current (A) ---> 4 γ γ � � / / 3 E 2 µ1 1 qV 0 0 0.2 0.4 0.6 Voltage (V) ---> Drain voltage µ2 Band Edge D(E) 17 Network for Computational Nanotechnology

Self-consistent potential nano HUB .org online simulations and more ⎡ ⎤ γ + γ f f = − 1 1 2 2 U n D ( E U ) ⎢ ⎥ γ + γ S D ⎣ ⎦ CHANNEL 1 2 γ γ V G [ ] q V D = − − 1 2 I D ( E U ) f f I 1 2 γ + γ � 1 2 Well-designed gate D(E) No gate Potential lowered due to decrease in electrons U = + − ∇ U 2 = U ( n n ) U 0 0 0 L ∇ 2 U ~ Charge Density 18 Network for Computational Nanotechnology

Self-consistent field method nano HUB .org online simulations and more Simplified treatment γ 1 γ 2 of a very complicated problem “Poisson” D(E) n --> U Self- Consistent Solution U --> n = + − U U U ( n n ) L 0 0 “Schrodinger” ⎡ ⎤ γ + γ U --> I f f = − 1 1 2 2 ( ) ⎢ ⎥ n D E U γ γ + ⎣ ⎦ 1 2 Nanowires / γ γ [ ] q = − − 1 2 Nanotubes / Molecules I D ( E U ) f f 1 2 γ + γ � 1 2 19 Network for Computational Nanotechnology

Modeling “demons” nano HUB .org online simulations and more γ 1 γ 2 γ s D(E) μ 1 μ 2 D(E) γ γ 1 2 = + − U U U ( n n ) γ L 0 0 [ ] = − 1 I q D ( E ) f f 1 1 � ⎡ ⎤ γ + γ f f = − γ 1 1 2 2 ( ) ⎢ ⎥ n D E U [ ] γ γ = − + 2 ⎣ ⎦ I q D ( E ) f f 1 2 2 2 � γ γ − 2 = [ ] q I I I = − − 1 2 1 I D ( E U ) f f s 1 2 γ + γ � 1 2 20 Network for Computational Nanotechnology