Phonon-Independent Optical Transitions in Semiconductors

• Phonon-independent optical transitions are direct interband processes in semiconductors where electrons absorb photons and transition vertically in k-space without phonon involvement.
• The NEGF formalism employs Dyson and Keldysh equations to model the quantum-kinetic behavior, capturing coherent and non-equilibrium effects in optical transitions.
• Spatially-resolved NEGF simulations reveal how vertical transitions dictate photocurrent generation and overall performance in ultra-thin, non-homogeneous semiconductor devices.

Phonon-independent optical transitions are fundamental interband processes in semiconductors wherein photons directly induce electronic transitions between valence and conduction bands without the involvement of phonons. Such transitions satisfy momentum conservation intrinsically, occurring “vertically” in k-space and are prominent in materials with direct bandgaps. The quantum-kinetic theory formalism, particularly the non-equilibrium Green’s function (NEGF) approach, provides a rigorous and comprehensive framework for treating these processes, capturing both microscopic non-equilibrium effects and macroscopic observables such as photocurrent, especially in spatially inhomogeneous or nanostructured devices (1012.5462).

1. Nature of Phonon-Independent Optical Transitions

In direct-gap semiconductors, electronic transitions from the valence band (v) to the conduction band (c) under photon absorption do not require a momentum-matching phonon. The physical process involves an electron absorbing a photon and traversing a direct interband pathway, such that the initial and final electronic wave vectors are nearly identical (vertical transition in the Brillouin zone). The absence of phonon involvement differentiates these processes from phonon-assisted (indirect) transitions, which dominate in indirect-gap materials.

Phonon-independent (direct) transitions represent the intrinsic optical response of the semiconductor and define the fundamental absorption edge and radiative recombination mechanisms under equilibrium and non-equilibrium conditions.

2. Quantum-Kinetic Description and NEGF Formalism

The full quantum-kinetic treatment employs the NEGF formalism to describe optical transitions, capturing the coherent and incoherent processes, detailed balance, and non-equilibrium occupation dynamics. The electron states are encoded in lesser ($G^<$) and greater ($G^>$) Green's functions, evolved self-consistently via the Dyson and Keldysh equations.

The electron-photon self-energy, $\Sigma_{e\gamma}$, is incorporated in the equations of motion and is solely responsible for all generation and recombination events in the phonon-independent case:

$$\begin{align*} R_{\text{abs}} &= \frac{2}{V} \int \frac{dE}{2\pi\hbar} \sum_k \sum_{\lambda,q} |M^γ_{cv}(k,\lambda,q)|^2 N^γ_{\lambda,q} \, G^<_v(k; E - \hbar\omega_q) G^>_c(k; E) \ R_{\text{abs,net}}(\lambda, q) &= \frac{2}{V}\int \frac{dE}{2\pi\hbar}\sum_k |M^γ_{cv}(k, \lambda, q)|^2N^γ_{\lambda,q} \left[ G^<_v(k;E-\hbar\omega_q) G^>_c(k;E) - G^>_v(k;E-\hbar\omega_q) G^<_c(k;E) \right] \end{align*}$$

Here, $|M^γ_{cv}(k, \lambda, q)|^2$ is the squared optical matrix element, $N^γ_{\lambda,q}$ is the photon modal occupation, $G^<_v$ and $G^>_c$ are the lesser/greater Green's functions of the valence and conduction bands, respectively. Stimulated emission is captured with the roles of $G^<$ and $G^>$ interchanged. The approach generalizes the Fermi-Golden-Rule to non-equilibrium and spatially dependent regimes; with equilibrium Green’s functions, these expressions collapse to the standard perturbative results based on transition rates.

3. Dyson and Keldysh Equations: Self-Consistent Solution

Retarded (and advanced) Green's functions are governed by the Dyson equation:

$$G^{R/A}(\mathbf{r}_1, \mathbf{r}_2 ; E) = G_0^{R/A}(\mathbf{r}_1, \mathbf{r}_2 ; E) + \int d^3 r_3\, G_0^{R/A}(\mathbf{r}_1, \mathbf{r}_3 ; E)\, \Sigma^{R/A}(\mathbf{r}_3, \mathbf{r}_2 ; E)$$

The correlation (lesser) functions are constructed via the Keldysh equation:

$$G^<(\mathbf{r}_1, \mathbf{r}_2 ; E) = \int d^3 r_3 \int d^3 r_4\, G^R(\mathbf{r}_1, \mathbf{r}_3 ; E) \Sigma^<(\mathbf{r}_3, \mathbf{r}_4 ; E) G^A(\mathbf{r}_4, \mathbf{r}_2; E)$$

When only the electron-photon self-energy is retained, these equations yield a theory that is entirely phonon-independent and only describes direct interband transitions induced by photons. A plausible implication is that no inelastic broadening arises from phonon scattering, which simplifies the spectral features but may neglect effects such as carrier thermalization.

4. Spatially-Resolved NEGF Implementation in Ultra-Thin Devices

For ultra-thin crystalline silicon p-i-n junctions lacking complete translational symmetry in the growth ($z$) direction, a spatially resolved NEGF is required. The field operators are expanded in an envelope function basis distinguishing in-plane and growth directions:

$$\Psi_b(\mathbf{r}, t) = \sum_{\mathbf{k}_\parallel, i} \psi_{ib,\mathbf{k}_\parallel}(\mathbf{r})\, c_{ib,\mathbf{k}_\parallel}(t),\qquad \psi_{ib,\mathbf{k}_\parallel}(\mathbf{r}) = \phi_{i,\mathbf{k}_\parallel}(\mathbf{r})\, u_{n,k_0}(\mathbf{r})$$

A finite-difference (or tight-binding-like) approach is used for $\phi_{i,\mathbf{k}_\parallel}(r)$ along the $z$ direction, enabling layer-resolved matrix representation for Green’s functions and self-energies. The local current conservation law across a layer of thickness $\Delta$ is expressed as:

$$\frac{J_i - J_{i-1}}{\Delta} = -\frac{2e}{\hbar A \Delta} \sum_{\mathbf{k}_\parallel} \int \frac{dE}{2\pi} \left\{ \left(\Sigma^R G^< - G^R \Sigma^< + \Sigma^< G^A - G^< \Sigma^A\right)_{i,i} \right\}$$

where $A$ is the cross-sectional area. Integration over all layers yields the device photocurrent as the difference in interband currents at the contacts.

5. Procedural Steps and Photocurrent Computation

The phonon-independent simulation workflow consists of:

• Constructing the device Hamiltonian in the selected envelope basis (including conduction and valence effective mass models).
• Introducing only the electron–photon self-energy for direct optical transitions, with electron–phonon interactions deliberately neglected.
• Self-consistently solving the coupled Dyson and Keldysh equations for the Green's functions to obtain energy-resolved spectral functions and state occupations.
• Calculating the local optical generation rate from the scattering term:

$$R_{rad} = \frac{2}{\hbar A} \sum_{\mathbf{k}_\parallel} \int \frac{dE}{2\pi}\, \operatorname{Tr}\left\{ \Sigma^<_{e\gamma}(\mathbf{k}_\parallel; E) G^>_e(\mathbf{k}_\parallel; E) - \Sigma^>_{e\gamma}(\mathbf{k}_\parallel; E) G^<_e(\mathbf{k}_\parallel; E) \right\}$$

With only photon self-energies present, all generated carriers and the resultant photocurrent originate exclusively from vertical transitions. This simulation paradigm captures non-equilibrium physics and spatial inhomogeneities and does not account for inelastic relaxation via phonons.

6. Advantages, Limitations, and Physical Interpretation

The phonon-independent NEGF methodology provides several key advantages:

• Non-equilibrium occupations and detailed spatial inhomogeneity are inherently included.
• Energy-resolved mapping of generation and transport is facilitated.
• Self-consistency ensures conservation laws (e.g., continuity equation) are satisfied when coupled to electrostatics (e.g., Poisson’s equation).

A significant limitation is the neglect of phonon-induced processes, meaning the description is incomplete where phonon scattering is essential (e.g., indirect gap materials or situations dominated by relaxation effects). For direct-gap materials or cases where optical processes are dominant, this approach yields fundamental insights into the interplay of photogeneration and carrier collection.

A summary of the central mathematical relationships is provided in the table below:

Rate/Quantity	Expression	Physical Meaning
Net absorption rate	$$R_{\text{abs,net}} = \frac{2}{V} \int \frac{dE}{2\pi\hbar}\ldots$$	Net photon-induced interband transitions
Layer-resolved current	$(J_i-J_{i-1})/\Delta = -\ldots$	Current conservation per layer
Optical generation	$R_{rad} = \frac{2}{\hbar A} \sum \ldots$	Local carrier generation rate

7. Relation to Macroscopic Device Performance

By linking the microscopic quantum-kinetic description to layer-resolved observables, the framework bridges the gap between quantum processes at the interband level and macroscopic device characteristics, such as external quantum efficiency and current-voltage response. In ultra-thin, non-homogeneous architectures, this approach rigorously captures the essential physics of direct optical transitions and their impact on photocurrent, establishing the upper bounds set by purely photon-induced carrier generation and collection processes. This methodology specifically addresses device questions where phonon processes are secondary and vertical transitions dictate device operation (1012.5462).