Recently, InAlAs digital alloys have been shown to exhibit unique electronic dispersion properties, which can be used to make low-noise avalanche photodiodes. In this paper, the strain effect is analyzed for its impact on the band structure of the InAlAs digital alloy. Simulation using a tight binding model that includes the strain effect yields bandgap energies that are consistent with experimental results. The bandgap would be larger without strain. In addition, a positive relationship has been found between minigaps of the InAlAs digital alloy and the band offset between bulk InAs and AlAs at the same position in k-space.

Periodic growth of binary materials instead of random growth has been studied in order to avoid material degradation caused by the miscibility gap effect.1–5 Recently, the emergence of digital alloy material growth techniques has spurred the potential of AlInAsSb for applications in the mid-infrared (IR) and near-IR.2 For example, an AlInAsSb-based digital alloy avalanche photodiode (APD) has exhibited low dark current and very low excess noise.6–12 Recently, we have shown that minigaps in the band structure play an important role in the excess noise of digital alloy InAlAs APDs.1,13 Due to the strong interactions between electron wave function of adjacent layers, minigaps occur at the boundaries of Brillouin zones, which blocks the acceleration of carriers and bring down carrier ionization coefficient. Due to the difference in dispersion properties between electrons and holes in the digital alloy system, the ionization coefficient for one type of carrier is much more deduced compared with another. In our recent work, the ionization coefficient ratio k has been found to be decreased when the temperature decreases,13 indicating that minigaps are preferred to hold value larger than the phonon energy and the larger the better. Strain exists between the layers and plays an important role in the band structure. In this paper, the strain effect is analyzed in terms of its impact on the band structure InAlAs digital alloys, especially on the minigaps. First principles calculation based on the tight binding method is used to calculate the band structure.

The most commonly used composition of InxAl(1-x)As has an In to Al ratio, x, of approximately 0.5 in order to achieve lattice matching to an InP substrate.14,15 We define the n monolayer (ML) structure as a digital alloy composing n/2 monolayers of InAs and n/2 monolayers of AlAs. Figure 1(a) shows the lattice structure of a 6 monolayer (ML) InAlAs digital alloy, which is composed of periodically stacked InAs and AlAs binary materials with three monolayers each. Figure 1(b) shows the lattice structure of an InAlAs random alloy, wherein the In and Al atoms are randomly located.

FIG. 1.

Lattice structures for InAlAs (a) digital and (b) random alloys.

FIG. 1.

Lattice structures for InAlAs (a) digital and (b) random alloys.

Close modal

The tight binding method was used to calculate the band structure. The model is environment (neighboring atoms, bond angles, and bond lengths) dependent. Strain and interface-induced changes play important roles in the calculation results. The parameters were adjusted iteratively in comparison with hybrid functional band structures and functions. This tight binding model has been used in the fast calculation of the band structure for different material structures.16,17 In this work, we used the environment dependent sp3d5s* tight binding model introduced in Ref. 17, wherein there are 20 orbits including spin–orbit coupling. The tight binding parameters were adjusted iteratively to match the hybrid functional (HSE06) band structure and wave functions in order to achieve excellent transferability. Those parameters have been published in the literature.16 Supercells are commonly used in calculating material structures such as superlattices, quantum dots, and defects. Figure 2 illustrates the supercells of (a) the digital alloy and (b) the random alloy and their difference in comparison with (c) a primitive cell. Solid axes (ex,ey,ez) and dashed axes (x,y,z) present the original Cartesian coordinates and lattice vector, respectively.

FIG. 2.

The supercell of (a) a digital alloy and (b) a random alloy is compared with (c) the primitive cell. Unit cells are embedded in the Zinc-blend lattice structure. All atoms in the supercells are marked by different symbols. Solid and dashed axes present the original Cartesian coordinates and lattice vector, respectively.

FIG. 2.

The supercell of (a) a digital alloy and (b) a random alloy is compared with (c) the primitive cell. Unit cells are embedded in the Zinc-blend lattice structure. All atoms in the supercells are marked by different symbols. Solid and dashed axes present the original Cartesian coordinates and lattice vector, respectively.

Close modal

The lattice vectors (x,y,z) of the digital alloy, random alloy, and primitive alloy can be expressed as (1)–(3), respectively,

x=a2(exey),y=a2(ex+ey),z=az2×n×ez,
(1)
x=aex,y=aey,z=aez,
(2)
x=a2(ex+ez),y=a2(ex+ey),z=a2(ez+ey),
(3)

where ex,ey,ez are Cartesian basis vectors, a and az are transverse and vertical lattice constants of the digital alloy, respectively, a is the lattice constant of the random alloy and primitive alloy, and n is the number of monolayers in the digital alloy.

The reciprocal lattice vectors are thus

b1=4πa(exey),b2=4πa(ex+ey),b3=4πaz×n×ez,
(4)
b1=2πaex,b2=2πaey,b3=2πa×ez,
(5)
b1=4πa(ezey+ex),b2=4πa(ey+exez),b3=4πa(ey+ezex).
(6)

From (1) and (2), we can see that the first Brillouin zone in reciprocal space for the digital and random alloys is represented by rectangular parallelepiped profiles, wherein the E-k relationship is then calculated.

For the digital alloy, since there is a lattice mismatch between binary materials, the strain effect plays an important role in modulating the lattice constant. The lateral and vertical lattice constants (a and ai) and the strain εi of the digital alloy under the strain effect can be calculated by the following formula:19 

a=a1G1h1+a2G2h2G1h1+G2h2,
(7)
ai=ai[1Di001εi],
(8)
Di001=2c12c11,
(9)
εi=a/ai1,
(10)

where i represents the material (1) or (2), ai denotes the equilibrium lattice constants, and h1 and h2 are the respective thicknesses of the unstrained layers for InAs and AlAs. G2 and G2 are the shear moduli along the (001) direction. Di001 is a constant depending on the elastic constants c11 and c12.

In this paper, the stacking directions of the digital alloy are all along the (001) direction. From (10), strain can be calculated in the InAs and AlAs layers. The parameters used in this work and the calculated strain value are provided in Table I. While for the random alloy, the lattice constant of the random alloy is taken to be the average of InAs and AlAs.

TABLE I.

Lattice constant a (in Å), elastic constants c11 and c12, and shear moduli G for InAs and AlAs used in this work. Also given is the calculated strain value εi in 6 ML and 8 ML digital alloy.

ac11c12Gεi in 6 MLεi in 8 ML
InAs 6.08 0.833 0.453 1.587 −0.043 −0.043 
AlAs 5.65 1.25 0.534 2.656 0.0298 0.0298 
ac11c12Gεi in 6 MLεi in 8 ML
InAs 6.08 0.833 0.453 1.587 −0.043 −0.043 
AlAs 5.65 1.25 0.534 2.656 0.0298 0.0298 

After calculating the E-k relationship in the supercell based on the first Brillouin zone, an unfolding process is deployed to replot the E-k relationship in the primary-cell-based reciprocal space figure, which is commonly used to analyze the band structure based on complex supercell.18,20–24 The supercell eigenvector |Km can be expressed as a linear combination of primitive cell eigenvectors |kin (extracting E versus k effective band structure from supercell calculations on alloys and impurities).

From an atom orbital point of view, the eigenstate of energy Ep with wave vector k can be written as a linear combination of atomic-orbital-based wave functions. The wave function |ψmKSC for electron states with wave vector K at the mth band in the supercell can be expressed as a linear combination of wave function basis |ψnkiPC for electron states with the wave vector ki at the nth band in the primitive cell (the same eigen-energy with |ψmKSC) as shown in

|ψmKSC=nki{k~i},a(ki,n;K,m)|ψnkiPC,
(11)

where K and k denote the reciprocal vector in supercell based and primitive cell, which hold the projection relationship via the folding vector GkK, which is expressed as

K=kGkK.
(12)

The projection of |ψmKSC into |ψnkiPC based characters can be expressed as

PmKn|ψmKSC|ψnkiPC|2.
(13)

Figure 3 shows an example of the one-dimensional E-k relationship of a strained 6 ML InAlAs digital alloy and an InAlAs random alloy. Comparing to the random alloy, the digital alloy exhibits mini-gaps, which can affect the ionization coefficient ratio.1,13 In the digital alloy, electron and hole accelerating along the electric field direction might be blocked by the minigaps, which would not happen in the random alloy. Due to the symmetry difference between the conduction band and the valence band, it is easier for electrons than holes to get rid of the restriction of minigap. Thus, a very low k is found in InAlAs digital alloy APD.

FIG. 3.

One-dimensional E-k relationship of (a) a 6 ML InAlAs digital alloy and (b) an InAlAs random alloy.

FIG. 3.

One-dimensional E-k relationship of (a) a 6 ML InAlAs digital alloy and (b) an InAlAs random alloy.

Close modal

Figure 4 shows the external quantum efficiency versus wavelength for random and digital alloy materials with 6 ML and 8 ML. The long-wavelength cutoff of the digital alloys is red shifted compared with the In0.5Al0.5As random alloy. The cutoff wavelengths of the random alloy and the 6 ML and 8 ML digital alloys are 885 nm, 975 nm, and 1070 nm, respectively, corresponding to the bandgaps of 1.4 eV, 1.27 eV, and 1.16 eV, respectively.

FIG. 4.

External quantum efficiency versus wavelength for random alloy and digital alloy materials with 6 ML and 8 ML.

FIG. 4.

External quantum efficiency versus wavelength for random alloy and digital alloy materials with 6 ML and 8 ML.

Close modal

The calculated bandgaps of the random alloy and the 6 ML and 8 ML strained digital alloys are compared with experimental results in Table II in second and third columns. The calculated bandgap of different structures is very close to the experimental results. If strain effects are not considered, the calculated bandgaps for the digital alloy are much bigger than the experimental results, which are provided in the fourth column of Table II. We conclude that the strain reduces the bandgap of digital alloys.

TABLE II.

Experimental and theoretical bandgap for InAlAs 6 ML, 8 ML digital alloy and random alloy.

StructureExperimental bandgap (eV)TB (strained) bandgap (eV)TB (no strain) bandgap (eV)
6 ML digital alloy 1.27 1.24 1.33 
8 ML digital alloy 1.16 1.17 1.30 
Random 1.4 1.38  
StructureExperimental bandgap (eV)TB (strained) bandgap (eV)TB (no strain) bandgap (eV)
6 ML digital alloy 1.27 1.24 1.33 
8 ML digital alloy 1.16 1.17 1.30 
Random 1.4 1.38  

Since the thickness of InAs and AlAs in the digital alloy is on the order of a few monolayers, the electron wavefunctions strongly overlap between the adjacent binary materials. The E-k relationship at the boundary of the Brillouin zone may have a large distortion due to the strong perturbation caused by the periodic stacking. In order to illustrate the distortion, especially its relationship with the band structure of the primitive-cell-based InAs and AlAs binary materials, we use an unfolding algorithm to project the calculated band structure based on a rectangular parallelepiped Brillouin zone [Eq. (4)] into a primitive-cell-based parallelepiped diagram [Eq. (6)].

Figures 5(a) and 5(b) show the band structure of AlAs and InAs along the [001] direction starting from the Γ point with (solid line) and without strain (circle). The lateral lattice constant under the strain condition is regarded as the same as the InP substrate. For AlAs at the Γ point, with strain, the bottom of the conduction band decreases and the top of the valence band goes up, resulting in a reduced bandgap. For InAs, both the bottom of the conduction band and the top of the valence band at the Γ valley slightly decrease with the strain, resulting in a nearly unchanged bandgap.

FIG. 5.

(a) Strain effect on the band structure of AlAs and (b) InAs along the [001] direction starting from the Γ point with (solid line) and without strain (circle). The bandoffset between InAs and AlAs (c) without strain and (d) under strain.

FIG. 5.

(a) Strain effect on the band structure of AlAs and (b) InAs along the [001] direction starting from the Γ point with (solid line) and without strain (circle). The bandoffset between InAs and AlAs (c) without strain and (d) under strain.

Close modal

Figures 5(c) and 5(d) display the band structure of AlAs and InAs together. From the two figures, the band offsets between AlAs and InAs are sensitive to strain. In Fig. 5(c), the band structures of both binary materials are calculated without strain. The conduction band-offset is as large as 1.7 eV, while the valence band-offset is 0.5 eV. If the strain is included, there is a significant change in the band profiles, as shown in Fig. 5(d). The conduction band-offset is reduced to 1.25 eV, while the valence band-offset is unchanged.

The band structure of digital alloys with 6 ML, 8 ML, and 10 ML has been calculated and unfolded. They are compared with InAs and AlAs binary material in Fig. 6. From Eq. (4), the super-cell-based reciprocal vector along the growth direction is 4πaz×n, which is inversely related to the monolayer number of each period. In Figs. 5(a)5(c), the primary-cell-based k-scales of the digital alloys with 6 ML, 8 ML, and 10 ML are divided by dashed lines indicating the positions of related boundaries in the super-cell-based k-scale, of which the interval width is 4πaz×n. From the plot, it can be seen that the minigap exists at the positions of each dashed line for the digital alloy. The minigap values labeled in the figures are listed in the second and fourth columns of Table III. The band structures of InAs and AlAs under the strain are plotted adjacent to the band structures of the digital alloys with the same interval scale as dashed lines. From Figs. 6(d)6(f), there are bandoffsets between InAs and AlAs at the dashed lines. The bandoffset values labeled in the figures are listed in the third and fifth columns of Table III. It can be seen that for different monolayer numbers in the digital alloys, the minigap exhibits a positive relationship with the band offset of binary materials. The larger the minigap is, the better the blocking performance exhibits.

FIG. 6.

The relationship between digital alloy's minigap and binary materials' band offset. (a) Band structure for 6 ML InAlAs digital alloy. (b) Band structure for 8 ML InAlAs digital alloy. (c) Band structure for 10 ML InAlAs digital alloy. (d) Band structure for strained InAs and AlAs with dashed lines holding the same interval with 6 ML InAlAs digital alloy. (e) Band structure for strained InAs and AlAs with dashed lines holding the same interval with 8 ML InAlAs digital alloy. (f) Band structure for strained InAs and AlAs with dashed lines holding the same interval with 10 ML InAlAs digital alloy.

FIG. 6.

The relationship between digital alloy's minigap and binary materials' band offset. (a) Band structure for 6 ML InAlAs digital alloy. (b) Band structure for 8 ML InAlAs digital alloy. (c) Band structure for 10 ML InAlAs digital alloy. (d) Band structure for strained InAs and AlAs with dashed lines holding the same interval with 6 ML InAlAs digital alloy. (e) Band structure for strained InAs and AlAs with dashed lines holding the same interval with 8 ML InAlAs digital alloy. (f) Band structure for strained InAs and AlAs with dashed lines holding the same interval with 10 ML InAlAs digital alloy.

Close modal
TABLE III.

The value of minigaps and bandoffsets.

StructureConduction band label (eV)Band offset at conduction band (eV)Valence band label (eV)Band offset at valence band (eV)
6 ML digital alloy 0.48 0.57 0.26 0.55 
8 ML digital alloy 0.60 0.67 0.21 0.25 
10 ML digital alloy 0.63 0.85 0.11 0.11 
StructureConduction band label (eV)Band offset at conduction band (eV)Valence band label (eV)Band offset at valence band (eV)
6 ML digital alloy 0.48 0.57 0.26 0.55 
8 ML digital alloy 0.60 0.67 0.21 0.25 
10 ML digital alloy 0.63 0.85 0.11 0.11 

In avalanche photodiodes (APDs), the ionization coefficient ratio k (0–1) is a key parameter that influences the speed and noise performances of the device. k reflects the difference in ionization coefficient between electron and hole. APDs with low k values exhibit better performance.1,13,25–29 In the previous work,1,13 we have shown that minigaps in the InAlAs digital alloy significantly impact the k value. A minigap larger than the phonon energy can prevent a carrier from attaining sufficient energy to impact ionization. Owing to symmetry differences between the conduction band and the valence band, it is easier for electrons than holes to find a path to get over minigap through in-plane scattering and gain energy. Therefore, electrons in the InAlAs digital alloy exhibit much larger ionization coefficient than holes, which results in a low k value.

In this work, we take a step further to reveal the exact location of the minigaps in conventional zinc-blend materials reciprocal space. From Figs. 6(a)6(c), it can be seen that minigaps in the valence band are present in the light, heavy, and spin split-off bands. We believe that the minigaps in the spin split-off bands are the most influential in the hole ionization coefficient. If there were no minigaps in the spin split-off band, owing to its low effective mass, holes could easily gain energy. However, this is suppressed by minigaps in the spin split-off band, which greatly reduces the hole ionization rate. From the data in the fourth column of Table III, the minigap becomes larger in the spin split-off band as the period of the digital alloy decreases from 10 ML to 6 ML, resulting in a greater suppression of the hole ionization coefficient. From the second column of Table III, minigaps in the conduction band decrease for 10 ML to 6 ML periods, opposite to that of holes. Thus, it can be inferred that k decreases as the digital alloy structure period decreases from 10 ML to 6 ML.

Excess noise factors, F(M), have been measured versus gain, M, on three samples, InAlAs random alloy, 8 ML digital alloy, and 6 ML digital alloy. Material growth and device fabrication details have been provided in Refs. 1 and 13. The measured excess noise factor versus gain is shown in Fig. 7. The dashed lines are plots of F(M) for different k values based on the local-field model, in which F(M) is given by the expression13 

F(M)=kM+(1k)(21/M).
(14)
FIG. 7.

Experimental results of excess noise factor versus multiplication gain for random alloy (♦), 8 ML digital alloy (■), and 6 ML digital alloy (▴).

FIG. 7.

Experimental results of excess noise factor versus multiplication gain for random alloy (♦), 8 ML digital alloy (■), and 6 ML digital alloy (▴).

Close modal

From Fig. 6, it can be seen that digital alloy APD exhibits a lower k value than random alloy APD and the 6 ML digital alloy APD exhibits a lower k value than the 8 ML digital alloy APD, which is consistent with the band structure calculations.

In this paper, the effect of strain on the band structure of InAlAs digital alloys has been analyzed. It is found that strain reduces the bandgaps for the InAlAs digital alloy. In addition, minigaps exert a positive relationship with band offset between related binary materials, which is sensitive to strain.

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