Nanoelectronics: A New Perspective on TransportWeek 13 · The NEGF method
WEEK 13 · THE NEGF METHOD
Nanoelectronics:
A New Perspective on Transport
Contacts for the Schrödinger equation: broadening, Green's functions, current, dephasing
Prof. Mohammad RazaghiUniversity of Kurdistan · Graduate course in Electronics Engineering
Based on S. Datta, Lessons from Nanoelectronics, Part B: Quantum Transport, Chapter 18
Based on S. Datta, Lessons from Nanoelectronics, Part B: Quantum Transport, Chapter 18
Fixed lecture only; circular session C7 follows next week. The goal is to get the four NEGF equations from one modified Schrödinger equation, with the one-level resistor as the warm-up.
RecapWhere we are
Last week
- The channel: a Hamiltonian matrix H (N = n × b)
- Tight binding: ε, t from E(k); U on the diagonal; phases for B
- But H describes an isolated channel: no current
Today
Add the contacts. The only new ingredients are Σ (outflow and inflow, one per contact) and the rule that sources from different contacts add incoherently.
Fig. 18.1: H, Σ1, Σ2, and Σ0 for interactions inside the channel (the dephasing part comes at the end).
Week 13Today's goals
- Solve the one-level resistor semiclassically and quantum mechanically.
- Understand broadening: γ, the Lorentzian and the dwell time.
- Explain why sources add incoherently and what that does to ψ.
- Write and use Eqs. 18.1–18.4 for an N-level channel; G(E) = (q²/h) Trace[Γ₁GRΓ₂GA].
- Add elastic dephasing: Σ₀ = D × GR, Models A and B.
Datta Ch. 18The whole method on one slide
| Equation | Name | Eq. |
|---|---|---|
| GR = [E I − H − Σ]−1, GA = (GR)† | Green's function | 18.1 |
| Gn = GR Σin GA, Σin = Γ₁f₁ + Γ₂f₂ | electron density (×2π) | 18.2 |
| A = GR Γ GA = i[GR − GA], Γ = i[Σ − Σ†] | spectral function (DOS ×2π) | 18.3 |
| Ĩm = (q/h) Trace[Σinm A − Γm Gn] | current at contact m | 18.4 |
Today: justify these from E ψ = [H + Σ₁ + Σ₂] ψ + s₁, starting with N = 1 where every matrix is a number.
Keep this slide handy: it returns in Weeks 14, 15 and 16. Table 18.1 maps Datta's symbols to the usual −iG<, +iG>, −iΣ<, +iΣ>.
Physics refresherRefresher: a leaky bucket and an exponential
- Rate equation: dN/dt = −νN + S relaxes to N = S/ν with time constant 1/ν.
- A complex energy ε − iγ/2 makes a wave decay: ψ̃ ∝ e−iεt/ħe−γt/2ħ.
- Then |ψ̃|² ∝ e−γt/ħ: the same exponential with ν = γ/ħ.
- ∫ dE γ/[(E−ε)² + (γ/2)²] = 2π: a Lorentzian of width γ has area 2π.
Datta Sec. 18.1.1One level, semiclassical
dN/dt = −(ν₁+ν₂)N + S₁ + S₂Eq. 18.5 · Sm = νmfm (18.7)
N = (ν₁f₁ + ν₂f₂)/(ν₁ + ν₂)Eq. 18.10a
I = q ν₁ν₂/(ν₁+ν₂) · (f₁ − f₂)Eq. 18.10b
- Current = the first bracket in (S₁ − ν₁N) + (S₂ − ν₂N) = dN/dt, evaluated in steady state.
- Slowest contact is the bottleneck: ν₁ν₂/(ν₁+ν₂) ≤ min(ν₁, ν₂).
The sign: S1 − ν1 N is the net flow from the source; S2 − ν2 N is the (negative) net flow from the drain, so I = q(S1 − ν1 N) = q(ν2 N − S2).
Datta Sec. 18.1.2One level, quantum: contacts in the Schrödinger equation
E ψ = (ε − iγ/2) ψ + s₁Eq. 18.15 · γ = γ₁ + γ₂, γm = ħνm
ψ = s₁ / (E − ε + iγ/2)Eq. 18.16
- −iγ/2 is the outflow term (Σ), s₁ the inflow.
- |ψ|² peaks at E = ε but is not small for |E − ε| < γ: broadening.
Why a single source s1: the subtlety is explained at the end of Sec. 18.1; drain electrons are accounted for by the f2 term in the final bilinear equations.
Datta Sec. 18.1.2–18.1.3Fixing the source and getting the current
∫ dE ψψ* = γ₁/(γ₁+γ₂) ⇒ 2π s₁s₁* = γ₁Eqs. 18.17–18.20
I = q ∫ dE (γ₂/ħ) ψψ*
= (q/h) ∫ dE γ₁γ₂/[(E−ε)² + (γ/2)²]Eqs. 18.21–18.22 · f₁ = 1, f₂ = 0 → (q/ħ)γ₁γ₂/γ
= (q/h) ∫ dE γ₁γ₂/[(E−ε)² + (γ/2)²]Eqs. 18.21–18.22 · f₁ = 1, f₂ = 0 → (q/ħ)γ₁γ₂/γ
- Same as the rate equation: the quantum treatment reproduces I.
- But now we also know the energy resolution: D(E) = (γ/2π)/[(E−ε)² + (γ/2)²] (18.23).
- Time–energy: γτ ≥ h; strong coupling means short stay and big broadening.
Datta Sec. 18.1.3Conductance of the one-level resistor
G(E) = (q²/h) γ₁γ₂/[(E−ε)² + (γ/2)²]compare with G(E) of Eq. 3.3
- Equal couplings: peak G = q²/h at E = ε.
- Unequal: peak 4γ₁γ₂/γ² of q²/h, and broader.
- Experiment: a hydrogen molecule between good contacts shows 2q²/h (levels come in spin pairs; Smit et al., 2002).
InteractiveExplore: the one-level resistor
One level, two contacts: quantum broadening (Datta Part B, Ch. 18, Sec. 18.1)
A single level ε coupled to a source and a drain with escape rates γ1/ℏ, γ2/ℏ. Contact turns the sharp level into the Lorentzian D(E) = (γ/2π)/[(E−ε)² + (γ/2)²] (Eq. 18.23), and the current is I = (q/h) ∫ T(E)[f1 − f2] dE with T = γ1γ2/[(E−ε)² + (γ/2)²] (Eqs. 18.22, 18.30). Energies are measured from the equilibrium µ0 = 0.
Level occupation N (< 1)–
Current I = (q/h)∫T(f1−f2)dE–
Rate-equation I at E = ε (18.10b)–
Peak G (kT ≪ γ) = 4γ1γ2/γ² q²/h–
Width γ = γ1 + γ2–
Dwell time τ = ℏ/γ–
Linear-response G at µ0 (q²/h)–
kT–
Try this
- Equal couplings, 100 K, ε = 0: what is the peak of G(µ₀)?
- Make γ₂ = 5γ₁: does the peak drop to 4γ₁γ₂/γ²? Check.
- Raise T until kT exceeds γ: what controls the width of the peak now?
- Put ε in the middle of a bias window much larger than γ: compare I with the rate-equation estimate.
Scroll inside the panel for all controls. Swipe or use the arrow buttons outside it to change slide.
Energies relative to µ0. The rate-equation I assumes f1 and f2 are constant across the level; the quantum integral accounts for the Lorentzian tails.
Datta Sec. 18.1.4Do multiple sources interfere?
ψψ* = [|s₁|² + |s₂|² + s₁s₂* + s₂s₁*]
/ [(E−ε)² + (γ/2)²]if both sources are put in one equation
/ [(E−ε)² + (γ/2)²]if both sources are put in one equation
- Cross terms are never seen: electrons from different contacts have uncorrelated, rapidly changing phases.
- Like two light bulbs: powers add, not fields.
- Rule: one source at a time, form bilinear quantities (ψψ†), add them.
Coherent sources (lasers, superconducting contacts: Josephson) are the exception.
Datta Sec. 18.2.1–18.2.2N levels: Green's function and electron density
E{ψ} = [H + Σ₁ + Σ₂]{ψ} + {s₁}Eq. 18.24 · N × N matrices, Γm = i[Σm − Σm†]
ψ = GRs₁, Gn = 2πψψ† = GR Γ₁ GAEqs. 18.1, 18.2 · 2πs₁s₁† = Γ₁
- Add all contacts incoherently: Σin = Γ₁f₁ + Γ₂f₂.
Datta Sec. 18.2.3–18.2.4Spectral function, and the current equation
- Fill every state (f₁ = f₂ = 1): Gn becomes the density of states, A = GRΓGA.
- Subtract (GA)−1 from (GR)−1: i Γ = (GR)−1 − (GA)−1 ⇒ A = i[GR − GA].
- Current: d(ψ†ψ)/dt from the contact-ed equation; its two parts belong to contacts 1 and 2:
Ĩm = (q/h) Trace[Σinm A − Γm Gn]Eq. 18.4 · in-flow minus out-flow
The steady-state dN/dt = 0 means I1 + I2 = 0: Kirchhoff. The widget shows I1 + I2 at the marker energy: it is zero to machine precision.
InteractiveExplore: the NEGF equations on a small molecule
The NEGF equations on a small molecule (Datta Part B, Ch. 18, Eqs. 18.1–18.4, 18.30)
Give it a Hamiltonian H (hopping t = −1 eV between neighbours) and attach a source and a drain with broadening γ1, γ2. At each energy: GR = [E−H−Σ]−1, A = GRΓGA, Gn = GRΣinGA, T = Trace[Γ1GRΓ2GA]. Click the plots to move the energy marker.
T(E) at the marker–
Ĩ1 + Ĩ2 (Eq. 18.4, q/h units)–
Total current I (T=300 K)–
∫ Trace A dE / 2π (should be N)–
Eigenvalues of the isolated H (eV)–
Try this
- Single level: reproduce the one-level T(E). Then 5 sites: where are the peaks?
- Check I₁ + I₂ at any energy: Kirchhoff in disguise.
- Benzene ring: compare para (1, 4) and meta (1, 3) contacts: which has the larger T near E = 0.5 eV?
- Raise γ₁ and γ₂: what happens to the peaks and to ∫ Trace A dE/2π?
Scroll inside the panel for all controls. Swipe or use the arrow buttons outside it to change slide.
Wide-band contacts: Σ = −iγ/2 on one site each. In Week 14 the contacts become energy-dependent self-energies t·e^(ika).
Datta Sec. 18.3Conductance function for coherent transport
Ĩ(E) = (q/h) Trace[Γ₁GRΓ₂GA] (f₁ − f₂)from Eq. 18.4
G(E) = (q²/h) T̄(E)Eq. 18.30
T̄ = Trace[Γ₁GRΓ₂GA]Eq. 18.30
- Every coherent conductor is an elastic resistor (Week 3): I = (q/h)∫T̄(f₁−f₂)dE.
- Several contacts: Ĩm = (q/h) Σn T̄mn(fm − fn), T̄mn = Trace[ΓmGRΓnGA] (18.31–18.32): the Büttiker equations of Week 8.
- The bottleneck rule of the one-level case generalises: T̄ ≤ number of modes.
Datta Sec. 18.4Elastic dephasing: a third contact
Σ = Σ₁ + Σ₂ + Σ₀, Σin = Γ₁f₁ + Γ₂f₂ + Σ₀inEq. 18.34
Σ₀ = D × GR, Σ₀in = D × Gn, Γ₀ = D × AEq. 18.35 · × = element by element
- Model A: Dij = D₀ — phase only.
- Model B: Dij = D₀δij — phase and momentum.
D_ij = . The reinjection picture: Model A reinjects in the same state (multiplication by a constant), Model B throws away off-diagonal elements.
Datta Sec. 18.4Why Γ₀ = D × A, and how to compute it
Zero current at terminal 0
Terminal 0 is not a physical contact: electrons removed must be reinjected, so Trace[Σ₀inA − Γ₀Gn] = 0. This holds when Σ₀in = D × Gn and Γ₀ = D × A (and D is symmetric).
A Büttiker probe is the same idea with Γ₀ chosen proportional to A.
Self-consistency and cost
- Σ₀ depends on GR, which depends on Σ₀: iterate with mixing.
- Model A needs the full G; Model B only its diagonal: much cheaper.
- Averaging over random realisations of UR in H is not the same as Σ₀.
The notebook implements the self-consistent loop and prints the zero dephasing-contact current.
CheckConcept check
One minute with a neighbour, then click to reveal.
1. γ₁ = 10 meV, γ₂ = 40 meV: peak conductance? (click)
4·10·40/50² = 0.64 q²/h.
2. Why is ψ not enough for several contacts? (click)
Sources are incoherent, so only bilinear quantities (Gn) can be added.
3. What does A = i[GR − GA] reduce to for N = 1? (click)
The Lorentzian γ/[(E−ε)² + γ²/4] = 2πD(E).
4. What does Model A dephasing not destroy? (click)
Momentum: the electron is re-injected into the same state.
Wrap-upSummary and what comes next
Take-home
- E ψ = [H + Σ] ψ + s: outflow + inflow
- GR, Gn, A, Ĩ (18.1–18.4)
- G(E) = (q²/h) Trace[Γ₁GRΓ₂GA]
- Dephasing: Σ₀ = D × GR (A or B)
Week 14
Put the equations to work: 1D wires with scatterers, can two resistors offer less resistance than one? (Ch. 19), and the quantum of conductance in 2D wires, nanoribbons and the quantum Hall effect (Ch. 20). Circular session C7: an NEGF lab. Open week13_negf.ipynb and the Week 13 toolkit.
Prof. Mohammad Razaghi · University of Kurdistan · after S. Datta, Lessons from Nanoelectronics