Quantum Transport — NEGF

Non-Equilibrium Green's Function: transmission coefficient, conductance, and current spectrum

Parameters
Transmission Coefficient T(E)
Conductance G(E) [G₀]
Quantum Transport — Physical Intuition

Classical electronics relies on Ohm's law: R = ρL/A. This holds when the system size greatly exceeds the electron mean free path. At the nanoscale — below ~10 nm — electrons are quantum objects with a wave phase: they interfere coherently, tunnel through barriers, and propagate ballistically. Ohm's law breaks down. Instead, charge transport becomes a quantum-mechanical scattering problem: a left contact (lead) injects electrons, they attempt to reach the right contact, and the Landauer formula computes the probability of this process — the transmission coefficient T(E). T=1 means perfect transparency (ballistic transport); T=0 means complete blockade. In nanodevices T(E) can take any value between 0 and 1.

Historical Context

In 1957, Rolf Landauer introduced a new perspective: electrical resistance is not "energy dissipation" but "electron scattering." In his formula G = (2e²/h)·T, the transmission coefficient T is the central quantity. In 1988, Büttiker extended this to multi-terminal systems (Landauer-Büttiker formalism). In the late 1980s-90s, Kadanoff-Baym quantum statistical mechanics and Keldysh's non-equilibrium formalism were combined, giving birth to NEGF. Today NEGF is the standard for nanodevice simulation, molecular junctions (break junctions), and nanowires. In practice, NEGF+DFT combinations (SIESTA, TranSIESTA) compute I(V) characteristics with atomic accuracy.

Tight-Binding Hamiltonian

1D chain Hamiltonian in matrix form:

$$ H = \begin{pmatrix} \varepsilon_0 & t & 0 & \cdots \\ t & \varepsilon_0 & t & \cdots \\ 0 & t & \varepsilon_0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{pmatrix} $$

ε₀ — on-site energy. t — hopping integral. This model is an extension of Hückel theory to solids. Band structure: E(k) = ε₀ + 2t·cos(k) → bandwidth 4|t|.

$$ E(k) = \varepsilon_0 + 2t\cos(k), \quad \text{Bandwidth} = 4|t| $$
Green's Function in NEGF

The retarded Green's function of the system is defined as:

$$ G^r(E) = \left[(E + i\eta)I - H - \Sigma_L - \Sigma_R\right]^{-1} $$

H — system Hamiltonian (tight-binding matrix). Σ_L, Σ_R — contact self-energies, encoding the influence of leads on the device. η — infinitesimal imaginary part (finite in numerics for broadening). Γ = i(Σ - Σ†) — broadening matrix.

$$ \Gamma_{L,R} = i\left(\Sigma_{L,R} - \Sigma_{L,R}^\dagger\right) $$
Self-Energy

Analytical self-energy for a semi-infinite 1D tight-binding lead:

$$ \Sigma_L(E) = t_c \, g_s(E) \, t_c^\dagger, \qquad g_s(E) = \frac{E - \varepsilon_0}{2t^2} - \frac{i}{2t^2}\sqrt{4t^2 - (E-\varepsilon_0)^2} $$

This formula is exact for a 1D semi-infinite chain. k(E) is determined from E = ε₀ + 2t·cos(k). Γ_L,R = -2·Im(Σ_L,R) — broadening. For 2D/3D leads, self-energies are computed as surface Green's functions.

Transmission Coefficient

T(E) is computed from the broadening matrices and Green's functions:

$$ T(E) = \mathrm{Tr}\left[\Gamma_L \, G^r(E) \, \Gamma_R \, G^a(E)\right], \quad G^a = (G^r)^\dagger $$
Landauer Formula

Quantum conductance is computed from the transmission coefficient:

$$ G = G_0 \int_{-\infty}^{\infty} T(E) \left(-\frac{\partial f}{\partial E}\right) dE, \qquad G_0 = \frac{2e^2}{h} \approx 7.748 \times 10^{-5}\,\mathrm{S} $$

G₀ = 2e²/h ≈ 7.748×10⁻⁵ S — the quantum of conductance. The factor of 2 comes from spin degeneracy. T(E) — transmission probability at energy E. f(E) — Fermi-Dirac distribution. This formula holds for ballistic (lossless) transport.

Local Density of States (LDOS)

LDOS(i, E) measures the density of electronic states on atom i at energy E:

$$ \mathrm{LDOS}(i, E) = -\frac{1}{\pi} \mathrm{Im}\left[G^r_{ii}(E)\right] $$

LDOS visualization shows where the electron spends the most time — this directly corresponds to STM experimental images. High LDOS = electron localized; uniform LDOS = delocalized (ballistic) transport.

Current Calculation

Under non-equilibrium conditions (V≠0), electric current is:

$$ I(V) = \frac{2e}{h} \int_{\mu_R}^{\mu_L} T(E)\, dE, \qquad \mu_{L,R} = E_F \pm \frac{eV}{2} $$
$$ \frac{dI}{dV}(V) \propto T\!\left(E_F + \frac{eV}{2}\right) + T\!\left(E_F - \frac{eV}{2}\right) $$

μ_L = E_F + eV/2 and μ_R = E_F - eV/2 — chemical potentials. Integration over the "transport window" [μ_R, μ_L]. Non-linear I(V) — nano-diode or tunnel diode effect. dI/dV(V) — spectroscopic tool, direct reflection of LDOS.

Quantities Reference
Gr(E)Retarded Green's function
T(E)Transmission coefficient [0, 1]
ΣL,RLead self-energy
ΓL,RBroadening matrix = i(Σ − Σ†)
G₀Quantum of conductance = 2e²/h
LDOS(i,E)Local density of states at atom i
μL,RChemical potentials of leads
ηInfinitesimal imaginary broadening
tHopping integral
ε₀On-site energy
Step-by-Step Guide
1
Choosing a model
"1D chain" — simplest model. N atoms in sequence; hopping t only between nearest neighbors. Despite simplicity, captures ballistic transport, band structure, localization. "Resonant level" — single atom between two leads: ideal model for molecular junctions. "Potential barrier" — tunneling simulation.
2
Parameter meaning
t — hopping integral. Typical values: Si ~1 eV, benzene ~2.5 eV, carbon nanotube ~2.7 eV. η — broadening: small η (0.001–0.01 eV) = sharp features; large η = smeared. Γ — coupling strength: Γ ≫ t → broad; Γ ≪ t → narrow, well-defined resonance.
3
Interpreting T(E)
T(E)=1 — perfect transparency (ballistic). T(E)=0 — complete blockade. T peaks = resonant states. T=0 outside the band. Fermi level (E=0) sets the zero-bias conductance. T=1 inside band (clean 1D chain) — signature of ballistic transport.
4
LDOS and localization
LDOS heatmap shows electron distribution along the chain at different energies. Uniform color = ballistic; edge localization = edge states; chaotic = Anderson localization (with disorder).
5
I(V) characteristic
Gradually increasing V shows T(E) integration over the transport window. Near V=0: I=G·V (Ohmic). At higher V: I(V) can become nonlinear — negative differential resistance (NDR) in molecular systems.

Experimental comparison: STM dI/dV spectroscopy measures exactly LDOS(E). Break junction experiments measure G(V). This calculator simulates those experiments at the tight-binding level.

Self-Assessment

Q1. In the Landauer formula G = G₀·T(E_F), G₀ = ?

A 2e²/h ≈ 7.75×10⁻⁵ S
B e²/h ≈ 3.87×10⁻⁵ S
C e/h
D 1 Ω⁻¹

Q2. T(E) = 1 means:

A Electron is fully reflected
B Electron is fully transmitted — ballistic transport
C System is superconducting
D Tunneling occurs

Q3. In NEGF, Σ (Sigma) represents:

A System energy
B Temperature
C Influence of the contact (lead) on the device — self-energy
D Number of electrons

Q4. A peak in LDOS(i, E) at atom i means:

A Atom i is rotating
B An electronic state at energy E is localized at atom i
C Conductance is maximum
D Barrier is zero