Full band Monte Carlo simulation of AlInAsSb digital alloys

Avalanche photodiodes fabricated from AlInAsSb grown as a digital alloy exhibit low excess noise. In this paper, we investigate the band structure-related mechanisms that influence impact ionization. Band-structures calculated using an empirical tight-binding method and Monte Carlo simulations reveal that the mini-gaps in the conduction band do not inhibit electron impact ionization. Good agreement between the full band Monte Carlo simulations and measured noise characteristics is demonstrated.


I. INTRODUCTION
For optical communications and data transmission applications, high-sensitivity receivers can reduce energy consumption and system cost. The internal gain of avalanche photodiodes (APDs) can provide a sensitivity advantage relative to p-i-n photodiodes. The excess noise factor of an APD, which arises from the random nature of impact ionization, is a key factor in the receiver sensitivity. The excess noise power density can be expressed as = 2 ( ) 2 ( ), 1 where q, I, and R represent electron charge, current, and device impedance, respectively. In the local-field model, ( ) = + (1 − )(1 − 1⁄ ), where k is the ratio of the hole ionization coefficient, β, to that of the electron, α. 1 The excess noise increases with gain but increases more slowly for low k values. The highest APD receiver sensitivities in the fiber optic 1310 nm and 1550 nm transmission windows have been achieved by InGaAs/InAlAs and Ge/Si separated absorption, charge, and Random alloys of AlxIn1-xAsySb1-y with a high Al content exhibit a wide miscibility gap A recent set of experiments and calculations on APDs have uncovered some highly promising design possibilities hitherto unexplored in the field. This involves the ability by the Banks group at UT to grow certain digital alloys, such as AlxIn1-xAsySb1-y with a high Al content, which in random alloys exhibit a wide miscibility gap. 2,3 Recently it has been reported that high-Al-content AlxIn1-xAsySb1-y can be grown within the miscibility gap by MBE as a digital alloy of the component binaries, AlAs, AlSb, InAs, and InSb. 4 In this work, we consider 70% Al composition. As shown in Figure 1(A), each period consists of 3 monolayers (ML) InAs, 3 ML AlSb, 1 ML AlAs and 3 ML AlSb. The 3-D bandstructure has been calculated using an empirical tight-binding model. [5][6][7] The model is environmentally dependent, which means the strain and the interface issue have been considered in the calculation.
The empirical parameter was adjusted iteratively to fit empirical tight-binding results with a hybrid functional band structure.
In this work, we used the environment-dependent sp3d5s* tight binding-model introduced in Ref 5. The bandstructure from the Γ point along the [001] direction is shown in Figure 1(B). The bandgap is 1.19 eV. A large minigap is observed in the conduction band. Intuitively, a minigap in the conduction band would prevent electrons from achieving sufficient energy to impact ionize. However, previous work has shown that electron impact ionization occurs at a much higher rate than that of holes in the AlInAsSb digital alloy. 8 In this paper, full band structure based Monte Carlo simulation is used to analyze carrier transport. 9-12 ′ , ( , Ω ± ) = |Δ ( ′ , , , )| 2 | ( , ′ ; , ± )| 2 ′ ( ′ , Ω ′ ) ( where is the lattice density, q is the phonon wave vector of mode , and Δ ( ′ , , , ) is the deformation potential, which is taken to be 0.73 eV/Å in the conduction band and 0.21 eV/Å in the valence band for this calculation. 13,14 It should be noticed that the deformation potential value was taken from the value of InAlAs digital alloy as an approximation.  Table I. In experiment, the epitaxial structure of the digital alloy APD from top to bottom is 100 nm p-GaSb (1×10 19 cm -3 ), 100 nm p-Al0.7InAsSb (2×10 18 cm -3 ), 1000 nm un-doped Al0.7InAsSb, 200 nm n-Al0.7InAsSb (2×10 18 cm -3 ). In the simulation, we take a simplification that the electric field distributes uniformly in the 1000 nm un-doped region and the depletion width is ignored.
The simulated and measured gain curves are plotted in Figure 2(A) as dashed and solid curves, respectively. Figure 2 In our previous study, minigap in InAlAs digital alloy has been found to play an important role in influencing the ionization coefficient of holes. 13,14,19 The accelerating path of holes is blocked by minigap and hole energy is not enough to trigger ionizations. Thereafter, an extremely low k value has been realized in InAlAs digital alloy. As shown in Figure 1(B), a large mingap has been found in the conduction band of AlInAsSb digital alloy, while, there is no significant minigap in the valence band. If the minigap in conduction band could block electrons like the case in the valence band of InAlAs digital alloy, we can infer that the ionization coefficient of electrons is lower than that of holes. However, we found in the experiment that the electron ionization coefficient is much larger than holes. 8 Figure 3. The electric field is taken considering the total bias is 43.9 V (avalanche gain is 10), avalanche region thickness is 1 μm and the electric field is uniformly distributed. There is little difference in the high-energy portion of the energy distributions, which indicates that the minigap does not prevent electrons from accumulating the energy required to impact ionize.
(A) (B) As was explained in our previous work 13 , although the energy is discontinuous in the [001] direction, there are in-plane energy states that have sufficient energy to go across the minigap. The electrons that are scattered to these in-plane states can get over the minigap under the driving force of the electric field. In the simulation, the energy of electron 2 did not drop below the minigap because the 439 kV/cm electric field is high enough to enable electrons to accelerate to higher energy states. Also there are fewer energy states in the minigap energy level for electrons to relax to. The energies of holes are also shown in Figure 3. It can be seen that holes tend to occupy low energy states. The hole energies are smaller than the bandgap and do not trigger ionizations. It follows that the ionization coefficient of holes is much smaller than that of electrons.

FIGURE 3
Energy distributions for carriers in AlInAsSb, wherein electrons with two types of initial states are compared.
In this work, 3-D band structure is used to deploy Monte Carlo simulation. To have an intuitive physical analysis of the energy distribution of carriers, some band structure parameters along different crystal directions are calculated and put in Table II. Important energy parameters are provided in Table III.  Given the disparity between the electron mass and that of the hole in the heavy hole and light hole bands, it can be concluded that the split-off (SO) band plays a critical role in hole acceleration. Large separation (0.48 eV) between the heavy-light hole bands and the split-off band impedes impact ionization. The energy step to the first mini-gap is 0.66 eV, which is much smaller than bandgap (1.19 eV). Since the first mini-gap has an energy of 0.2 eV, which is much larger than phonon energy, the holes could not jump across the minigap by the help of phonon scatterings. Since the hole energy distribution is much more limited than that of the electron, we can infer that it is not easy for holes to find a path (in 3-D Brillouin zone) to get over minigap like electrons.

IV. CONCLUSION
In this paper, we present calculations of the band structure of digital alloy AlInAsSb. Avalanche photodiodes fabricated from AlInAsSb exhibit the lowest noise of any III-V compound material. By analyzing the band structure calculated by the empirical tight-binding method, we have shown that there are mini-gaps in both the conduction band and valence band. Full band structure based Monte Carlo simulation shows that minigap in the conduction band doesn't block the electron accelerating process. The hole accelerating process is limited, wherein large effective mass and minigap might play an important role. Good agreement between the full band Monte Carlo simulations and measured noise characteristics is demonstrated.