Modeling Infrared Superlattice Photodetectors: From Nonequilibrium Green’s Functions to Quantum-Corrected Drift Diffusion

Carrier transport in type-II superlattice photodetectors is investigated by means of a rigorous nonequilibrium Green’s function model based on a physics-based Büttiker-probe formalism. Intraband scattering self-energies (carrier-phonon interactions) are computed in the self-consistent Born approximation, while interband self-energies (Shockley-Read-Hall and optical transitions) are included in terms of semiclassical generation-recombination rates, neglecting interband renormalization effects. Current conservation is achieved with an efficient Newton-Raphson algorithm. While carrier transport in infrared detectors is usually understood in terms of quantities (e.g., mobilities and quasi-Fermi-levels) that are admittedly not germane to nonequilibrium Green’s function theory, the proposed model provides a quantum-kinetic description of tunneling, miniband transport, hopping, and carrier extraction within a drift-diffusion-friendly framework. The connection with semiclassical theories allows exploration of the possibilities offered by Poisson-Schrödinger or localization landscape drift-diffusion approaches.

Figure 1

Sub-band structure of a superlattice consisting in the alternating sequence of a 18/24 $\AA$ $\mathrm{In}\mathrm{As}$/$\mathrm{Ga}\mathrm{Sb}$ T2SL, computed with a multiband $k\cdot p$ model for wave vectors along the growth (left panel) and in-plane (right panel) directions. The in-plane dispersion for $k_z=0$ is shown in black, while the blue curves are for equally spaced values of $k_z$ up to the mini-Brillouin-zone boundary $\pi /d$ ($d$ is the superlattice period). Luttinger parameters are from Ref. [36]. The dashed red lines are the sub-bands computed in the effective mass approximation. The inset in the left panel shows the first heavy-hole (HH1) sub-band dispersion along $k_z$ for different values of the (normalized) transverse wave vector $k_t$; the colored strips mark the corresponding minibands.

Figure 2

Local density of states of the $n\mathrm{B}n$ structure, shown for zero transverse momentum. The reverse bias applied to the top (right) contact is $0.1\mathrm{V}$. The superlattice absorber consists of 40 $\mathrm{In}\mathrm{As}$/$\mathrm{Ga}\mathrm{Sb}$ ($18/24\AA$) periods with $n$-type ${10}^{17}{\mathrm{cm}}^{-3}$ doping; the same superlattice is used also for the top contact region. The doping is increased to $5\times {10}^{18}{\mathrm{cm}}^{-3}$ on both sides of the structure. The barrier is implemented with an undoped $\mathrm{Ga}\mathrm{Sb}$/$\mathrm{Al}\mathrm{Sb}$ ($18/24\AA$) superlattice to avoid the valence-band offset that typically arises when $(\mathrm{Al},\mathrm{Ga})\mathrm{Sb}$ is used as the barrier material [39]. The first conduction miniband of the superlattice absorber extends approximately $0.2\mathrm{eV}$ above the effective conduction-band edge ${\tilde{E}}_C$ (blue solid line), while the first valence miniband is strongly localized in the weakly coupled $\mathrm{Ga}\mathrm{Sb}$ quantum wells, just below ${\tilde{E}}_V$ (red solid line). The band diagram (thin blue and red lines) computed with NEGF is shown for a reverse bias of $0.1\mathrm{V}$; the SPDD band diagram (not shown for clarity) is almost superimposed to the NEGF result, save for small differences arising from the broadening effects, which are not included in the Schrödinger equation. All simulations are performed at 150 K.

Figure 3

Dark-current-voltage characteristics of the $n\mathrm{B}n$ structure computed at $T=150\mathrm{K}$ and $T=200\mathrm{K}$, with SPDD (solid lines) and NEGF (open circles).

Figure 4

Spatially and spectrally resolved current density (color maps) with a light illumination intensity of $P_{\mathrm{opt}}=0.1{\mathrm{W/m}}^2$, and at a reverse bias of $0.1\mathrm{V}$. The Fermi levels of the Büttiker probes (NEGF) and the quasi-Fermi-levels of the carriers (SPDD) are shown in solid and dashed lines, respectively.

Figure 5

Energy-integrated spatial profile of electron and hole currents obtained from NEGF (cyan and magenta solid lines), and the corresponding currents computed with the SPDD model (blue and red dashed lines). As GR processes are suppressed in the barrier (see Fig. 6), the electron and hole currents remain constant in this region. The total NEGF current (solid black line) is perfectly conserved.

Figure 6

Spatial profile of the SRH recombination rate in dark and illumination conditions. For simplicity, the optical generation is assumed constant in the absorbing region. The competition of SRH recombination and optical generation in the superlattice absorber is clearly visible.