Green Function Landauer Buttiker Quantum Transport

# Non-Equilibrium Green's Functions (NEGF) and Quantum Transport in Nanostructures

## 1. Introduction to NEGF Formalism

Non-Equilibrium Green's Functions (NEGF) provide a rigorous quantum mechanical framework for describing electron transport in systems far from equilibrium. Unlike equilibrium Green's function methods used in solid-state physics, NEGF explicitly tracks the time evolution of the many-body density matrix under the influence of applied biases and temperature gradients. The formalism is particularly powerful for nanoscale devices (sub-3nm) where quantum coherence effects dominate classical drift-diffusion behavior.

The central object in NEGF theory is the one-particle contour-ordered Green's function:

$$G(x,t; x',t') = -i\langle T_C[\psi_H(x,t)\psi_H^\dagger(x',t')] angle$$

where $T_C$ denotes time-ordering along a closed Keldysh contour comprising forward and backward time paths. This contour formulation naturally encompasses both retarded and advanced propagation, essential for capturing non-equilibrium dynamics.

## 2. Keldysh Contour and Green's Function Components

The Keldysh contour is a closed path in complex time consisting of three segments:
- Forward branch: from $t_0$ to $t_f$
- Backward branch: from $t_f$ to $t_0$
- Imaginary branch: from $t_0 - i\beta$ to $t_0$ (for statistical averaging)

Contour-ordered functions decompose into four components corresponding to time-orderings on different branches:

$$G^{++}(t, t') = -i\langle T_-[\psi_H(t)\psi_H^\dagger(t')] angle$$ (backward-backward)
$$G^{--}(t, t') = -i\langle T_+[\psi_H(t)\psi_H^\dagger(t')] angle$$ (forward-forward)
$$G^{+-}(t, t') = -i\langle \psi_H(t)\psi_H^\dagger(t') angle$$ (forward-backward)
$$G^{-+}(t, t') = -i\langle \psi_H^\dagger(t')\psi_H(t) angle$$ (backward-forward)

where $T_-$ and $T_+$ denote antichronological and chronological ordering respectively.

These four components are not independent; they satisfy the kinetic equation relating them through spectral and kinetic properties. Only two independent components need to be calculated; the others follow from causality relations.

## 3. Retarded and Advanced Green's Functions

For practical calculations, the retarded ($G^R$) and advanced ($G^A$) Green's functions are most useful:

$$G^R_\alpha(E) = \lim_{\eta o 0^+} [(E + i\eta)\mathbb{I} - H_\alpha - \Sigma^R_\alpha(E)]^{-1}$$

$$G^A_\alpha(E) = [G^R_\alpha(E)]^\dagger$$

where $H_\alpha$ is the single-particle Hamiltonian of subsystem $\alpha$ and $\Sigma^R_\alpha(E)$ is the retarded self-energy describing interactions with the environment (scattering, leads, phonons).

The spectral function:
$$A_\alpha(E) = i[G^R_\alpha(E) - G^A_\alpha(E)]$$

encodes the density of states and linewidth broadening due to interactions.

## 4. Self-Energy from Lead Coupling

For a device region coupled to left (L) and right (R) electrodes at distinct chemical potentials $\mu_L$ and $\mu_R$, the self-energy from each lead is:

$$\Sigma^R_{L,R}(E) = g_{L,R}(E)[G_{L,R}^{00,R}(E)]^{-1}$$

where $g_{L,R}$ is the coupling matrix element and $G_{L,R}^{00,R}$ is the retarded Green's function of the lead surface layer.

For a semi-infinite tight-binding lead with nearest-neighbor coupling $t$ and onsite energy $E_0$:

$$G^{00,R}(E) = \frac{1}{E - E_0 - \frac{t^2}{E - E_0 - \frac{t^2}{\cdots}}} = \frac{E - E_0 \pm \sqrt{(E-E_0)^2 - 4t^2}}{2t^2}$$

The coupling strength is defined as:
$$\Gamma_{L,R}(E) = i[\Sigma^R_{L,R}(E) - \Sigma^A_{L,R}(E)]$$

This is a positive-definite matrix whose eigenvalues represent the energy-dependent broadening from each electrode.

## 5. Landauer-Büttiker Transmission Coefficient

The transmission coefficient from left to right electrode at energy $E$ is given by the Caroli formula:

$$T(E) = ext{Tr}[\Gamma_L(E) G^R(E) \Gamma_R(E) G^A(E)]$$

This quantity has remarkable properties:
- $0 \leq T(E) \leq N$ where $N$ is the number of active transport channels
- For single-channel systems: $T(E) \in [0, 1]$ (resonant tunneling reaches $T=1$)
- Dimensionless; independent of device length (ballistic transport)
- Encodes all quantum interference and resonant tunneling effects

The transmission can be reformulated as:
$$T(E) = ext{Tr}[\Gamma_L^{1/2} G^R \Gamma_R G^A \Gamma_L^{1/2}]$$

emphasizing that transmission arises from coupling strengths weighted by Green's function propagation.

## 6. Lesser and Greater Green's Functions

The lesser Green's function $G^<(E)$ carries information about occupied electronic states:

$$G^<(E) = G^R(E) \Sigma^<(E) G^A(E)$$

where the lesser self-energy from leads is:
$$\Sigma^<_L(E) = 2i f_L(E) \Gamma_L(E)$$
$$\Sigma^<_R(E) = 2i f_R(E) \Gamma_R(E)$$

with $f_L(E) = 1/[1 + \exp((E-\mu_L)/k_B T)]$ being the Fermi-Dirac distribution in the left electrode.

The greater Green's function $G^>(E)$ represents unoccupied states:
$$G^>(E) = G^R(E) \Sigma^>(E) G^A(E)$$

with $\Sigma^>(E) = -2i[1 - f(E)] \Gamma(E)$.

The local density of states (LDOS) at site $n$:
$$ ho_n(E) = -\frac{1}{\pi} ext{Im}[G^R_{nn}(E)]$$

## 7. Current and Power Dissipation

The current from lead $\alpha$ is determined by the Landauer formula:

$$I_\alpha = \frac{q}{h} \int dE \, T(E) [f_\alpha(E) - f_\alpha'(E)]$$

For a simple two-terminal device:
$$I = \frac{q}{h} \int dE \, T(E) [f_L(E) - f_R(E)]$$

This elegant result shows that current is proportional to transmission weighted by the difference in lead populations—a direct manifestation of quantum ballistic transport.

The power dissipation in the device:
$$P = \int dE \, (E - E_F) \frac{dI}{dE} = V \cdot I$$

where $V$ is the applied bias. Energy is dissipated through inelastic scattering processes represented by imaginary parts of the self-energy.

## 8. Inelastic Scattering: Electron-Phonon Interactions

Beyond elastic transport, electron-phonon coupling introduces the inelastic self-energy:

$$\Sigma^R_{ep}(E) = \sum_ u \int \frac{d\omega}{2\pi} \frac{|M_ u|^2}{E - \hbar\omega_ u - \omega \pm i\eta}$$

where $M_
u$ is the electron-phonon matrix element for mode $
u$ and $\omega$ is the phonon frequency.

This contributes to:
1. Linewidth broadening: Elastic collisions reduce coherence length
2. Hot carrier relaxation: Energy dissipation to phonon bath
3. Phonon-assisted tunneling: Resonances shifted by $\hbar\omega_
u$

The Born approximation is valid when $|M_
u|^2 \ll \hbar\omega_
u$, which holds for most semiconductors but breaks down in strong-coupling regimes.

## 9. Steady-State Transport Equation

Under steady bias $V_{DS}$, the current reaches a stationary value determined by:

$$I = \frac{q}{h} \int dE \, T(E) [f_L(E) - f_R(E)]$$

with $\mu_R = \mu_L - qV_{DS}$. The transmission $T(E)$ contains the full quantum mechanical structure: resonances, tunneling barriers, and band structure effects.

For a 1D quantum channel with length $L$ and confining potential $U(x)$:

$$H_0 = -\frac{\hbar^2}{2m^*} \frac{d^2}{dx^2} + U(x)$$

The retarded Green's function in the channel interior satisfies:
$$(E + i\eta - H_0)G^R(x,x',E) = \delta(x-x')$$

For open boundary conditions with incoming waves from the left, the Green's function has the form:
$$G^R(x,x'<x) = \frac{e^{ik(x-x')}}{i\hbar v_g}$$

where $v_g$ is the group velocity.

## 10. Resonant Tunneling in Double Barrier Structures

Consider a double-barrier resonant tunneling diode (RTD) with two barriers separated by a quantum well of length $d$. The resonances occur when quasi-bound states align with lead Fermi levels.

The resonance condition:
$$k d = n\pi, \quad n = 1, 2, 3, \ldots$$

gives discrete energy levels in the well. Near resonance:
$$T(E) \approx \frac{4\Gamma_L \Gamma_R}{(\Delta E)^2 + (\Gamma_L + \Gamma_R)^2/4}$$

where $\Delta E = E - E_n$ is the detuning from resonance and $\Gamma_L, \Gamma_R$ are lead couplings. This Lorentzian lineshape shows perfect transmission ($T=1$) at exact resonance when $\Gamma_L = \Gamma_R$.

## 11. Python Implementation: Transmission in 1D Channel

import numpy as np
import matplotlib.pyplot as plt
from scipy import special
from scipy.linalg import eigh

def transmission_1d_wellbarrier(E_array, L_well=5e-9, d_barrier=2e-9, 
                                 V_barrier=0.3, m_eff=0.067):
    """
    Compute transmission spectrum for double-barrier resonant tunneling diode.
    
    Parameters:
    -----------
    E_array : array (eV)
        Energy range for transmission calculation
    L_well : float (m)
        Width of quantum well
    d_barrier : float (m)
        Barrier thickness
    V_barrier : float (eV)
        Barrier height
    m_eff : float (in units of m_e)
        Effective mass
    
    Returns:
    --------
    T_array : array
        Transmission spectrum T(E)
    E_array : array (eV)
        Energy array
    """
    
    # Physical constants
    hbar = 1.055e-34  # J·s
    m_e = 9.109e-31   # kg
    q_e = 1.602e-19   # C (elementary charge)
    
    m_star = m_eff * m_e
    
    # Dimensionless parameters
    E_scaled = E_array / V_barrier
    
    # Tunneling decay in barrier: κ = sqrt(2m(V-E))/hbar
    kappa_array = np.sqrt(2 * m_star * np.maximum(V_barrier*q_e - E_array*q_e, 1e-40)) / hbar
    
    # Wave vector in well: k = sqrt(2mE)/hbar
    k_array = np.sqrt(2 * m_star * np.maximum(E_array*q_e, 1e-40)) / hbar
    
    # Resonance condition: phase in well = nπ
    phase_well = k_array * L_well
    
    # Single-barrier transmission (WKB approximation)
    T_barrier = np.exp(-2 * kappa_array * d_barrier)
    
    # Resonance enhancement factor
    # When phase_well ≈ nπ, resonance condition satisfied
    resonance_factor = np.zeros_like(phase_well)
    
    for n in range(1, 6):
        phase_diff = np.abs(phase_well - n*np.pi)
        resonance_factor += 1.0 / (1.0 + (phase_diff / (0.1*np.pi))**2)
    
    # Total transmission: includes resonance enhancement
    T_array = T_barrier**2 * (1.0 + 10.0 * resonance_factor)
    T_array = np.clip(T_array, 0, 1.0)  # Physical bounds
    
    return T_array, E_array

def lead_coupling_strength(E_array, t_coupling=0.1):
    """
    Compute coupling matrix Γ(E) from lead-device interaction.
    
    For semi-infinite tight-binding lead:
    Γ(E) ∝ |t_coupling|^2 × ImG_lead(E)
    
    Parameters:
    -----------
    E_array : array (eV)
        Energy values
    t_coupling : float (eV)
        Lead-device coupling parameter
    
    Returns:
    --------
    gamma_array : array (eV)
        Coupling strength vs energy
    """
    
    # Semi-infinite lead density of states (approximate)
    # For tight-binding lead: ρ(E) = 2/π × 1/sqrt(4t^2 - E^2)
    bandwidth = 4 * np.abs(t_coupling)
    
    gamma_array = np.zeros_like(E_array)
    inside_band = np.abs(E_array) < bandwidth
    
    gamma_array[inside_band] = 2 * t_coupling**2 / np.sqrt(
        bandwidth**2 - E_array[inside_band]**2 + 1e-10
    )
    
    return np.abs(gamma_array)

def landauer_current(T_array, E_array, mu_L, mu_R, T_phonon=300):
    """
    Compute current via Landauer formula.
    
    I = (q/h) ∫ T(E)[f_L(E) - f_R(E)] dE
    
    Parameters:
    -----------
    T_array : array
        Transmission spectrum
    E_array : array (eV)
        Energy values
    mu_L, mu_R : float (eV)
        Left and right chemical potentials
    T_phonon : float (K)
        Temperature
    
    Returns:
    --------
    I : float (A)
        Net current
    """
    
    k_B = 8.617e-5  # eV/K
    h = 4.136e-15   # eV·s
    q_e = 1.602e-19  # C
    
    # Fermi-Dirac distributions
    f_L = 1.0 / (1.0 + np.exp((E_array - mu_L)/(k_B*T_phonon)))
    f_R = 1.0 / (1.0 + np.exp((E_array - mu_R)/(k_B*T_phonon)))
    
    # Integrand: T(E)[f_L(E) - f_R(E)]
    dE = E_array[1] - E_array[0]
    integrand = T_array * (f_L - f_R)
    
    # Integrate
    integral = np.trapz(integrand, E_array)
    
    I = (q_e / h) * integral
    
    return I

# Main calculation
E = np.linspace(-0.1, 0.4, 500)  # Energy range in eV
T, E = transmission_1d_wellbarrier(E, L_well=10e-9, d_barrier=3e-9, 
                                    V_barrier=0.3, m_eff=0.067)

# Coupling strength
Gamma = lead_coupling_strength(E, t_coupling=0.05)

# I-V characteristic
V_bias_array = np.linspace(0, 0.3, 30)
I_array = np.zeros_like(V_bias_array)

for i, V_bias in enumerate(V_bias_array):
    mu_L = 0.0  # Reference level
    mu_R = mu_L - V_bias
    I_array[i] = landauer_current(T, E, mu_L, mu_R, T_phonon=300)

# Plotting
fig, axes = plt.subplots(1, 3, figsize=(14, 4))

# Panel A: Transmission
axes[0].plot(E, T, 'b-', linewidth=2)
axes[0].set_xlabel('Energy (eV)', fontsize=11)
axes[0].set_ylabel('Transmission T(E)', fontsize=11)
axes[0].set_title('RTD Transmission Spectrum', fontsize=12, fontweight='bold')
axes[0].grid(True, alpha=0.3)
axes[0].set_ylim([0, 1.05])

# Panel B: Coupling strength
axes[1].semilogy(E, Gamma + 1e-6, 'r-', linewidth=2)
axes[1].set_xlabel('Energy (eV)', fontsize=11)
axes[1].set_ylabel('Γ_L, Γ_R (eV)', fontsize=11)
axes[1].set_title('Lead Coupling Strength', fontsize=12, fontweight='bold')
axes[1].grid(True, which='both', alpha=0.3)

# Panel C: I-V curve
axes[2].plot(V_bias_array*1000, I_array*1e6, 'g-o', linewidth=2, markersize=5)
axes[2].set_xlabel('Bias Voltage (mV)', fontsize=11)
axes[2].set_ylabel('Current (μA)', fontsize=11)
axes[2].set_title('RTD I-V Characteristic', fontsize=12, fontweight='bold')
axes[2].grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('negf_transport.png', dpi=150, bbox_inches='tight')
plt.show()

print("=== NEGF Transport Calculation Complete ===")
print(f"Peak transmission: {np.max(T):.4f}")
print(f"Max current @ {np.argmax(I_array)} mV bias: {np.max(I_array)*1e6:.2f} μA")

## 12. Steady-State vs. Transient NEGF

The treatment above assumes steady-state conditions where time derivatives vanish. For ultrafast phenomena (sub-picosecond), the full time-dependent NEGF formalism is required:

$$i\hbar\frac{\partial G^<(t,t')}{\partial t} = [H, G^<(t,t')] + \int dt_1 [\Sigma^R(t,t_1) G^<(t_1,t') - \Sigma^<(t,t_1) G^A(t_1,t')]$$

This Keldysh equation couples forward and backward propagation, requiring sophisticated numerical integration schemes (Runge-Kutta on Keldysh contour, recursive Green's function, etc.).

## 13. Applications to GAA FinFETs and Gate-All-Around Devices

For 3nm and sub-3nm nodes, gate-all-around (GAA) architectures rely on quantum confinement to suppress short-channel effects. NEGF accurately captures:

1. Subthreshold swing: Limited by $\kappa = k_B T / q$ times transmission turn-on steepness
2. Ballistic performance: In nanowire channels, elastic transmission dominates for lengths < mean-free-path (~20nm in Si)
3. Quantum capacitance: Related to $dN/dE$ where $N$ is charge from Green's function
4. Frequency response: Governed by transit time through channel

## 14. Comparison with Boltzmann and Drift-Diffusion

For sub-5nm devices, NEGF is mandatory for accurate IV, C-V, and noise predictions.

## 15. Extensions and Advanced Topics

Many-body NEGF includes vertex corrections accounting for two-particle correlations—necessary for Coulomb blockade in single-electron transistors and Kondo effects in quantum dots.

Photo-excited NEGF extends the formalism to optical pumping and laser-driven transport, relevant to THz quantum cascade lasers and nanoscale photodetectors.

Spin-NEGF incorporates spin-orbit coupling and magnetic exchange interactions—essential for spintronics devices and topological transport phenomena.

The NEGF framework remains the gold standard for nanoscale device simulation, bridging quantum mechanics, statistical physics, and transport theory.