Insights into optical absorption and dark currents of the 6.1Å Type-II superlattice absorbers for MWIR and SWIR applications

The 8-band $\bf k.p$ perturbation approach can precisely determine bandgap energies near the Brillouin zone while offering the correct interface treatment for the non-common atom type-II superlatticesLi et al. (2010); Galeriu (2005); Singh et al. (2022); Szmulowicz et al. (2006). For the type-II superlattice where there is a reduced Brillouin zone, the 8-band $\bf k.p$ method has been extensively utilized to determine the oscillator strength required to calculate the absorption coefficients and spontaneous emission ratesQiao et al. (2012). In this research, we apply the 8-band coupled Hamiltonian—which encompasses the interaction of the conduction, heavy hole, light-hole, and split-off bands, as well as their double spins to calculate the band structure of the InAs/GaSb and InAs/GaSb/AlSb/GaSb superlatticesQiao et al. (2012). The 8-band Lutinger-Kohn (LK) Hamiltonian is first decoupled into two 4x4 upper and lower Hamiltonians Qiao et al. (2012) which are represented as

$$H_{8x8}=\begin{bmatrix}H_{4x4}^{U}(k_{t})&0\\ 0&H_{4x4}^{L}(k_{t})\\ \end{bmatrix},$$

(1)

where, $k_{t}$ is the transverse wavenumber, $k_{t}=\sqrt{(k_{x}^{2})+(k_{y}^{2})}$. The band energies and wavefunction are typically dependent on the magnitude of the $k_{t}$ and the azimuthal angle ($\phi$=$tan^{-1}(k_{y}/k_{x})$), the axial approximation is taken into the accountQiao et al. (2012). The axial approximation simplifies the work that is required to perform the integration over the $k_{t}$ plane in the optical absorption calculations as it makes the energy subbands isotropic in the $k_{t}$ planeQiao et al. (2012). After the axial and basis transformation, the 4x4 upper and lower Hamiltonian can be represented as

$$H_{4x4}^{U}=\\$$

(2)

$\begin{bmatrix}E_{c}+A&-\sqrt{3}V_{p}&-V_{p}+i\sqrt{2}U&-\sqrt{2}V_{p}-iU\\ -\sqrt{3}V_{p}&Ev-P-Q&R_{p}+iS_{p}&\sqrt{2}R_{p}-i(S_{p})/\sqrt{2}\\ -V_{p}-i\sqrt{U}&R_{p}-iS_{p}&E_{v}-P+Q&-\sqrt{2}Q-i(S_{p})\sqrt{3/2}\\ -\sqrt{2}V_{p}+iU&\sqrt{2}R_{p}+i(S_{p})/\sqrt{2}&\sqrt{2}Q+i(S_{p})\sqrt{3/2}&E_{v}-P-\Delta\end{bmatrix},$

$$H_{4x4}^{L}=\\$$

(3)

$\begin{bmatrix}E_{c}+A&-\sqrt{3}V_{p}&-V_{p}+i\sqrt{2}U&-\sqrt{2}V_{p}-iU\\ -\sqrt{3}V_{p}&E_{v}-P-Q&R_{p}+iS_{p}&\sqrt{2}R_{p}-i(S_{p})/\sqrt{2}\\ -V_{p}-i\sqrt{U}&R_{p}-iS_{p}&E_{v}-P+Q&-\sqrt{2}Q-i(S_{p})\sqrt{3/2}\\ -\sqrt{2}V_{p}+iU&\sqrt{2}R_{p}+i(S_{p})/\sqrt{2}&\sqrt{2}Q+i(S_{p})\sqrt{3/2}&E_{v}-P-\Delta\\ \end{bmatrix},$

where, $E_{c}$ and $E_{v}$ are the conduction and valence band edges, respectively. The interband mixing parameter or Kane’s parameter is represented as $P_{cv}$. The $\gamma_{1}$, $\gamma_{2}$, and $\gamma_{3}$ are the modified Luttinger parameters and $m_{c}^{\prime}$ is the corrected effective mass. The $k_{z}$ is replaced by the $-i\partial/\partial z$ operator in the above equations to construct the Hamiltonian. The other parameters are given by

$$A=\frac{\hbar^{2}}{2m_{c}^{\prime}}(k_{t}^{2}+k_{z}^{2}),$$

(4)

$$P=\gamma_{1}\frac{\hbar^{2}}{2m_{0}}(k_{t}^{2}+k_{z}^{2}),$$

(5)

$$Q=\gamma_{2}\frac{\hbar^{2}}{2m_{0}}(k_{t}^{2}-2k_{z}^{2}),$$

(6)

$$R_{p}=-\frac{\hbar^{2}}{2m_{0}}\sqrt{3}\frac{(\gamma_{1}+\gamma_{3})}{2}(k_{t}^{2}),$$

(7)

$$S_{p}=-\frac{\hbar^{2}}{2m_{0}}2\sqrt{3}\gamma_{3}(k_{t}*k_{z}),$$

(8)

$$V_{p}=-\frac{1}{\sqrt{6}}P_{cv}k_{t},$$

(9)

$$U=-\frac{1}{\sqrt{3}}P_{cv}k_{z}.$$

(10)

In these superlattices, as the additional potential due to the band, alignment is superimposed over the underlying atomic potentials, we implement the 8-band $\bf k.p$ method with the envelope function approximation (EFA). To obtain the energies and their corresponding wavefunctions, the finite difference method (FDM) with periodic boundary conditions are employed to solve this HamiltonianGaleriu (2005); Mukherjee et al. (2021); Seyedein Ardebili et al. (2023). Additionally, we treat the interface as an InSb layer with 3Åwidth in one superlattice period to account for the lattice mismatch in these non-common atom configurationsSzmulowicz et al. (2006); Delmas et al. (2019). The interband momentum matrix parameter is calculated by the weighted average of InAs, GaSb, InSb, and AlSb layers. The parameters for InAs, GaSb, InSb, and AlSb materials are provided in Table 1. Figure. 1 shows the band alignment in T2SL and MSL. The following equations are implemented to calculate the density of states of these superlattices utilizing the 8-band $\bf k.p$ method Benchtaber et al. (2020); Benaadad et al. (2021),

$$\left\{\begin{aligned} DOS^{i}(E)\ &=\frac{m^{*}}{\pi^{2}\hbar^{2}}k_{z}(E)\ E_{min}^{i}\leq E\leq E_{max}^{i},\\ &=0\ otherwise,\end{aligned}\right.$$

(11)

where, i is the index that represents the band, $E_{min}$ and $E_{max}$ are the energies at the $k_{z}=0$, and $k_{z}=\pi/L$, respectively, $m^{*}$ is the effective mass, and $k_{z}$ is the wavevector along the growth direction. The miniband width of the superlattice is defined by the term $Emax^{i}-Emin^{i}$.

The optical absorption coefficient represents the value that how far the light travels in the material before being absorbed. It gives an idea of how readily the light is absorbed in the material and contributes to the generation of electron-hole pairs, which will later contribute to the photocurrent in the device. It is a product of the joint densities of states and the Fermi function, which can be calculated by Fermi’s Golden Rule Qiao et al. (2012) and represented as

$$\begin{split}\alpha(\hbar w)=\frac{\pi e^{2}}{(n_{r}c\epsilon m_{0}^{2}w)}\sum_{\sigma_{1},\sigma_{2}}^{U,L}\sum_{n,m}\frac{1}{L_{T}}\int_{0}^{2\pi}\frac{d\phi}{2\pi}\int_{0}^{\infty}\frac{d\phi}{2\pi}\frac{k_{t}dk_{t}}{2\pi}\\ |<\psi_{c}^{\sigma_{1},n}|e.p|\psi_{v}^{\sigma_{2},m}>|^{2}(f_{v}^{\sigma_{2,m}}(k_{t})-f_{c}^{\sigma_{1,n}}(k_{t})L(k_{t},\hbar*w),\end{split}$$

(12)

where n represents the refractive index of the superlattice, $\epsilon$ denotes the permittivity of the free space, $E_{c}(k_{t}),E_{v}(k_{t})$, are conduction and valence band energies calculated by the 8 band $\bf k.p$ theory at a particular $k_{t}$, m and n are the valence and conduction band indexes, w is the incident photon frequency, and $f_{c},f_{v}$ are the Fermi functions and $E_{fn},E_{fp}$, are the quasi-Fermi levels calculated at certain doping, and $L$ is the Lorentzian function, given by

$$L(k_{t},\hbar w)=\frac{1}{\gamma\sqrt{2\pi}}exp{\frac{-({E_{c}(k_{t})-E_{v}(k_{t})-\hbar w})^{2}}{2\gamma^{2}}},$$

(13)

where the $\gamma$ parameter is used for the scattering processes, also called as broadening function and the value taken in this work is $9meV$ both for the T2SL and MSL. The Fermi functions are denoted as $f_{c}$ and $f_{v}$ and can be expressed as

$$f_{c}^{\sigma_{1},n}(k_{t})=\frac{1}{1+e^{\frac{{E_{c}(K_{t})-E_{fn}}}{k_{B}T}}},$$

(14)

$$f_{v}^{\sigma_{2},m}(k_{t})=\frac{1}{1+e^{\frac{{E_{v}(K_{t})-E_{fp}}}{k_{B}T}}}$$

(15)

where, $k_{B}$ is the Boltzmann constant, and other parameters are the same as in the functions discussed above.
Further, the term in equation 12, i.e., $|<\psi_{c}^{\sigma_{1},n}|e.p|\psi_{v}^{\sigma_{2},m}>|$ is the optical matrix moment elements which represents the transition probability from the filled valence band to the empty conduction band and is calculated by equations given in Qiao et al. (2012), which mainly depends on the overlap of the envelope wavefunction w.r.t to the space and the transverse wavevector. Both the TE and TM polarization matrix elements which are required for the calculation of the absorption coefficients are calculated by the equation as given in Qiao et al. (2012). Where, the absorption includes the upward transition, the opposite of absorption is spontaneous emission, which considers only the downward transition and requires the initial conduction band state to be occupied, and the final valence band state to be empty. The spontaneous emission rate is given by

$$\begin{split}e_{spon}(\hbar w)=\frac{n_{r}e^{2}w}{c^{3}\pi\hbar\epsilon m_{0}^{2}}\sum_{\sigma_{1},\sigma_{2}}^{U,L}\sum_{n,m}\frac{1}{L_{T}}\int_{0}^{2\pi}\frac{d\phi}{2\pi}\int_{0}^{\infty}\frac{d\phi}{2\pi}\frac{k_{t}dk_{t}}{2\pi}\\ |<\psi_{c}^{\sigma_{1},n}|e.p|\psi_{v}^{\sigma_{2},m}>|^{2}(f_{c}({k_{t}})^{\sigma_{1},n})(1-f_{v}({k_{t}})^{\sigma_{2},m})\\ L(k_{t},\hbar*w)),\end{split}$$

(16)

where all the terms are similar to absorption but according to the occupancy of electrons and holes in conduction and valence bands, there is a difference in the terms representing the Fermi function.

The generated electron-hole pairs in the active region should get transmitted to their respective contacts for the photocurrent. In addition, the diffusion length of the minority carriers are to be higher than the absorption length only then they can participate in the photoconduction, otherwise, they will just lost by the recombination process in the device. Therefore, the minority carrier’s lifetime is a crucial parameter, as it determines both the dark current and maximum operating temperature for the acceptable performance of the photodetectors. The diffusion-limited detector dark current Olson et al. (2015) in the n-type doped material is given by

$$J_{diff}=q\frac{ni^{2}t}{n_{0}*\tau_{mc}},$$

(17)

where $q$ represents the electronic charge, $n_{i}$ is the intrinsic carrier density of the material, $n_{0}$ is the electron concentration which is in the majority, and $t$ is the thickness of the active region, and $\tau_{mc}$ is the minority carrier lifetime. The lifetime of the carriers depends on the various recombination phenomena, which are Shockley-Read-Hall(SRH), radiative, and Auger recombination. The SRH recombination process is due to the defect levels present in the material, which restricts their flow as the carriers get trapped in the trap level. The minority SRH lifetimeBlakemore (2002); Bandara et al. (2011) due to an electron in the p-type absorber is given by

$$\tau(SRH)_{n}=\frac{\tau_{n_{0}}(p_{0}+p_{1}+\Delta p)+\tau_{p_{0}}(n_{0}+n_{1}+\Delta n)}{p_{0}+n_{0}+\Delta n},$$

(18)

where, $\Delta n$ and $\Delta n$ are the excess carrier concentration($\Delta n$=$\Delta p$) and the $n_{1}$ and $p_{1}$ can be expressed as

$$n_{1}=n_{i}exp(\frac{E_{t}-E_{i}}{k_{B}T}),p_{1}=n_{i}exp(\frac{E_{i}-E_{t}}{k_{B}T}),$$

(19)

where $E_{i}$ is the intrinsic energy level, and the $E_{t}$ is the trap level, respectively. The SRH lifetime Höglund et al. (2013) depends on the electron and hole capture constants which are given by

$$\tau_{n0}=\frac{1}{\sigma_{n}v_{n}N_{t}},\tau_{p0}=\frac{1}{\sigma_{p}v_{p}N_{t}},$$

(20)

where, $\sigma_{n,p}$ are the capture cross-section area for the electron and hole defect, respectively, and $v_{p,n}$ is the thermal velocity of the carriers.
The radiative recombination coefficient which involves the band-band transitioning from the conduction band to the valence band Trupke et al. (2003); Azarhoosh et al. (2016) is calculated by the following equation where $\alpha(\hbar w,T)$ is the absorption coefficient at frequency w and temperature T, and $n_{r}$ is the refractive index of the superlattice material.

Reduced SRH, Auger, and radiative recombination dark currents that led to higher operating temperature can be achieved owing to the spatial confinement of carriers in the InAs/GaSb T2SL(holes in GaSb well and electrons in InAs well)Kopytko et al. (2015); Li et al. (2019); Downs and Vandervelde (2013). In this work, we want to design an absorber for the XBp barrier-based IR detectors, where most of the depletion region falls in the barrier(B) region, and due to its higher bandgap, the SRH dark current component reduces. The barrier further reduces the band-to-band dark tunneling currentsLi et al. (2019); Singh et al. (2022); Kopytko et al. (2015). To achieve higher quantum efficiency, we investigate a p-type doped absorption region almost at equilibrium (the depletion region shifts more toward the barrier region).

An XBp photodetector consists of three regions: contact, barrier, and active or absorber region. Introducing a barrier in these detectors reduces the SRH and band-to-band tunneling currents; therefore, they perform better than traditional PIN detectorsDelmas et al. (2017). To restrict the flow of carriers from the contact to the active region, which may contribute to the thermionic currents. If not appropriately treated, it may become equal to the diffusion current. Hence, there should be a proper band offset, which prohibits this flow. Not just the band offsets, the doping and the thickness of the barrier corresponding to the absorber region play a crucial role in the dark current flow and carriers’ collection to their respective contacts. Here, we plot the density-of-states(DOS) for the 6ML/7ML InAs/AlSb hole barrier and for the 5ML/1ML/2ML/1ML SWIR MSL absorber in Fig.23. Here we can observe that the hole barrier provides a valence band offset of around 0.369$eV$ for the holes and is almost negligible i.e. 6$meV$ conduction band offset w.r.t to the absorberMir and Frensley (2013).