1 Introduction

Due to their numerous applications in environmental and biomedical imaging, spectrum analysis, remote sensing, security surveillance and optical communication, infrared (IR) photodetectors have recently gained a lot of research interests (Li et al. 2021; Zha et al. 2022). Despite having a non-economical manufacturing cost, Mercury Cadmium Telluride (HgCdTe) is frequently utilized for conventional infrared photodetection because of its appropriate narrow band gap (Kopytko and Rogalski 2022; Chen et al. 2021; Jiao et al. 2022). Due to its enhanced electrical and optical characteristics and ease of manufacture, perovskite material has recently been employed in IR sensing, however reliability and photosensitivity problems still exist (Miao and Zhang 2019; Wang and Kim 2017). Quantum Well Infrared Photodetectors are also utilized extensively as the optical bandgap is engineered; nonetheless, their fundamental drawback is that they are insensitive to the normal incident illumination, necessitating complex optical systems (Ghosh et al. 2021; Yang et al. 2021).

In contrast to these conventional photodetectors in the mid-infrared range, Quantum Dot Infrared Photodetectors (QDIPs) are viewed as a good candidate for IR sensing, where the photodetection process depends on the electrons band to continuum transitions or electrons band to band transitions in QDs (Nasr and Mashade 2006; Dehdashti Jahromi et al. 2011). QDIPs typically have repeating barrier layers of periodically spaced quantum dots, which serve to confine some electrons and enable inter-band transitions. Since QDIPs have a lower dark current, a longer relaxation time and a larger gain than conventional IR photodetectors, they can operate at substantially higher temperatures. Additionally, QDIPs are highly sensitive to IR radiation that is normally incident.

The dark current, which characterizes the noise, is one of the key operating parameters for photodetectors, especially when operating temperatures are restricted. There is a number of strategies that have been developed for modelling QDIP's dark current characteristics (Ryzhii et al. 2004, 2001; Nasr et al. 2007; Kochman et al. 2003; Lim et al. 2005). The QD barrier's electron density and the carriers' drift as a result of the electric field are calculated as the basis for the dark current model in Liu et al. (2001), whereas the field-assisted tunnelling is taken into account in Stiff-Roberts et al. (2004). The likelihood of tunnelling in quantum dots has been determined using the transfer matrix method, which takes into account both thermionic emissions and tunnelling (Mahmoodi et al. 2015; Liu and Zhang 2012; Liu et al. 2015). Additionally, nonequilibrium Green's function is developed to analyze electron transport in a nonequilibrium QD structure and determine the photodetector dark current (Naser et al. 2008; Kumar Gujral et al. 2016). In the presented model, the lateral variations of the potential can be safely ignored, and only its variation in the direction of growth need be considered where the electrostatic problem becomes one dimensional because the potential variation is very small for the typical size dot with 50 nm interspace and even smaller for multilayers self-assembled dots where a randomness in the lateral dot positions is expected. Besides, we believe that the confined electrons in the dots are in local thermal equilibrium with those in the conduction band of the wide band-gap material like GaAs, which makes purely drift–diffusion model of transport can give an acceptable estimate of the dark current.

As free carriers' scatterers in QDIPs, QDs constrain their mobility and have an impact on the dark current. Numerous QD shapes, including semispherical (El-Batawy and Hosny 2020), truncated conical (El-Batawy and Feraig 2021) and conical (Youssef et al. 2016) structures have been investigated for their effects on the electron mobility.

In the work that is being presented, we have developed a dark current model for semispherical QDIP that is based on the drift diffusion approach and takes into account the electron confinement in QDs. The potential barrier that the free electrons must overcome is increased by electron confinement (Ameen et al. 2014). By resolving the Boltzmann Transport Equation, the electron mobility in the derived model is determined. The QD density, the operating temperature and the QD dimensions have a significant impact on both the carrier mobility and the QD filling, which in turn affects the dark current density of the QDIP. So, in this paper, these consequences have been studied.

This study is organized as follows; Sect. 2 contains the schematics for both the semispherical QDIP structure and the proposed dark current model. In Sect. 3, the outcomes of the dark current model are compared to the truncated conical QDIPs that have been previously published. In Sect. 4, a parametric analysis of the effects of QD dimensions and operating temperature and on the photodetector's dark current characteristics has been done. The conclusions are then briefly summarized.

2 Modeling of the dark current

The studied semispherical QDIP is schematically represented in Fig. 1 as layers of self-assembled hemisphere shaped InAs QDs implanted in a GaAs barrier substance, with layers heavily doped at the bottom and top serving as the contacts. The radius R of the QD serves as the primary design factor. Because the diameter of the QDs is far less than their separation, the coupling between them is ignored. Additionally, the very thin wetting layer's effects are disregarded, allowing the active region to be considered as a single medium with implanted QDs.

Fig. 1
figure 1

Schematic of Semispherical QDIP

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Drift–Diffusion approach is utilized to compute the dark current density for the proposed QDIP, taking into account the confinement of some electrons within quantum dots while calculating electron transports by both the drift and diffusion mechanisms. The potential barrier that the injected electrons must cross to reach the two contacts is increased by the QD confinement, which lowers the dark current in the photodetector. The axial direction of the QD experiences the largest change in the potential, making Poisson's equation one-dimensional as follows:

$$\frac{{d}^{2}V}{d{z}^{2}}=\frac{q}{\epsilon } \left(n-{N}_{D}^{+}+{n}_{dot}\langle N\rangle \right),$$
(1)

where \(V\) denotes the voltage, \(\epsilon \) denotes the barrier material permittivity, \(q\) denotes the electron charge, \({N}_{D}^{+}\) denotes the concentration of ionized donors, \({n}_{dot}\) denotes the volumetric QD density, and the average QD filling is \(\langle N\rangle \). The electron concentration ‘n’ in the barrier area is calculated using the degenerate semiconductor equation, as the Fermi level \({E}_{f}\) is too near to the conduction band edge EC. As a result, ‘n’ can be stated as (Razeghi 2009):

$$n={N}_{c}{F}_{1/2}\frac{{E}_{f}-{E}_{c}}{KT},$$
(2)

where \({N}_{c}\) is the actual density of states in the conduction band, T is the operating temperature, K is Boltzmann's constant, \({F}_{1/2}\) is the Fermi–Dirac integral (Razeghi 2009):

$${F}_\frac{1}{2}(x)=\frac{2}{\sqrt{\pi }} {\int }_{0}^{\infty }\frac{\sqrt{y}}{1+{e}^{(y-x)} } dy.$$
(3)

Then, the drift–diffusion approach is utilized to compute the QDIP dark current density J as follows:

$$\underline{J}=-q {\mu }_{n} n \underset{\_}{\nabla } V+ q {D}_{n} \underset{\_}{\nabla } n,$$
(4)

where \({D}_{n}\) and \({\mu }_{n}\) stand for the electrons diffusion coefficient and mobility, respectively. \({D}_{n}\) can be expressed as:

$${D}_{n}=\frac{{F}_{1/2}\left(\frac{{E}_{f}-{E}_{c}}{{K}_{B}T}\right)}{{F}_{-1/2}\left(\frac{{E}_{f}-{E}_{c}}{{K}_{B}T}\right)}\left(\frac{{K}_{B}T}{q}\right){\mu }_{n}.$$
(5)

While considering that there is no illumination and that the dark current density is uniformly distributed throughout the photodetector, then the dark current density '\(J\)' can be expressed in terms of Fermi level \({E}_{f}\) (Ameen et al. 2014) as follows:

$$J= {\mu }_{n} n\frac{d{E}_{f}}{dz},$$
(6)

where \({E}_{f}\) is calculated using the following continuity equation:

$$\frac{dJ}{dz}={\mu }_{n}{N}_{c}\frac{d}{dz}\left({F}_\frac{1}{2}\left(\frac{{{E}_{f}-E}_{c}}{KT}\right)\frac{d{E}_{f}}{dz} \right)=0.$$
(7)

The number of electrons that are confined in a QD is determined by both the total number of bound states in the QD and the electron filling of each state. The bound states are computed from the Hamiltonian diagonalizing of the QD's effective mass, that has the following form because of the symmetry of the QD structure's fundamental cell:

$$\widehat{H}=U\left(r,z\right)-\frac{{\hslash }^{2}}{2}\left(\frac{-{l}^{2}}{{r}^{2}{m}_{r}}+\frac{1}{r}\frac{\partial }{\partial r}\left(\frac{r}{{m}_{r}}\frac{\partial }{\partial r}\right)+\frac{\partial }{\partial z}\left(\frac{1}{{m}_{z}}\frac{\partial }{\partial z}\right)\right),$$
(8)

where the quantum number is \(l\) and the effective masses in the lateral and growth directions are denoted by \({m}_{r}\) and \({m}_{z}\), respectively. The QD potential \(U\left(r,z\right)\), equals \({V}_{b}\) in the barrier area outside the QD and disappears inside the QD, are both used in the equation above. This means that the bound states degrade outwardly and are constrained within the QD region. The electron mobility varies on both the QD structure's size and the operating temperature, as is significantly affected by the scattering events brought on by the QDs (Septianto et al. 2020).

The average number of scattering electrons per unit time "\(\tau \)" depends on the microscopic scattering mechanisms, and is represented as follows using Matthiessen's rule (Schwerer et al. 1969),

$$\frac{1}{\tau }=\frac{1}{{\tau }_{QD}}+\frac{1}{{\tau }_{\mathrm{phn}}}+\frac{1}{{\tau }_{\mathrm{im}}}+\frac{1}{{\tau }_{lat}},$$
(9)

where QD, optical phonons, impurities, and lattice vibrations are each represented by the mean free time that corresponds to their respective scattering τQD, τphn, τim and τlat, respectively. As optical phonon and lattice vibration scattering have minor impacts at low temperatures, the QD-related scattering is the primary component impacting the electron mobility in QD structures (El-Batawy and Hosny 2020).

The distribution function of the electrons is calculated by solving the Boltzmann Transport Equation using finite difference time domain approach, allowing for the computation of the carrier mobility (Kunz and Luebbers 1993). In these calculations, the photodetector's active region is modelled as homogenous, with QDs serving as the free carriers' scatterers, and the impacts of the scattering mechanisms and the external fields are taken into consideration (Ridley 1999). Consequently, the scattering term is expressed as follows:

$${\left(\frac{\partial {f}_{p}}{\partial t}\right)}_{\mathrm{scat}}=\iiint \left[{f}_{{p}^{\prime}}\left(1-{f}_{p}\right){W}_{{p}^{\prime},p}-{f}_{p}\left(1-{f}_{{p}^{\prime}}\right){W}_{p,{p}^{\prime}}\right]{d}^{3}{k}^{\prime}\frac{\Omega }{{\left(2\pi \right)}^{3}}$$
(10)

where \({W}_{p,{p}^{\prime}}\) and \({W}_{{p}^{\prime},p}\) and are rate of the transitions from \(p\) to \({p}^{\prime}\) and from \({p}^{\prime}\) to \(p\), respectively, and the probabilities of occupation of the energy levels \({\varepsilon }_{p}\) and \({\varepsilon }_{{p}^{\prime}}\) are donated by \({f}_{p}\) and \({f}_{{p}^{\prime}}\), respectively. \({W}_{{p}^{\prime},p}\) and \({W}_{p,{p}^{\prime}}\) are computed as:

$${W}_{{p}^{\prime},p}={W}_{p,{p}^{\prime}}= {\frac{2\pi }{\hslash }\left|{V}_{{p}^{\prime},p}\right|}^{2}\delta \left({\varepsilon }_{p}- {\varepsilon }_{{p}^{\prime}}\right),$$
(11)

where \({V}_{{p}^{\prime},p}\) is the matrix element that expresses the perturbation between the unperturbed states. The electron mobility is then computed using the slope of the averaging velocity over the acquired steady state distribution function versus the applied electric field.

3 Parametric study and results

3.1 Comparison with truncated conical QDIP and QD volume effect

This section compares the derived model of the semispherical QDIP dark current to the published results of the truncated conical QDIP (Ali and El-Batawy 2023). The QDIP is composed of 10 layers of QDs of InAs those are implanted in GaAs layer, and a 50 nm GaAs barrier layer follows the QD layer, where both structures have the same QD volume and base diameter. For this comparison, the truncated conical structure is assumed to have a constant aspect ratio R1/R2 = 0.25.

Figure 2a compares the dark current for both semispherical and truncated conical QDs for a conduction band offset of 0.259 eV, while Fig. 2b compares the dark current of these structures for a conduction band offset of 0.341 eV, that is more appropriate for InAs/GaAs structure (Ghosh et al. 2000; Zieliński 2013). It is worthy to be mentioned that the conduction offset changes with the atomistic strain as the lattice strain depend on the shape and dimensions of quantum dot (Ilatikhameneh et al. 2015; Ameen et al. 2018). Comparing Fig. 2a and b, increasing the conduction band offset results in more potential barrier and correspondingly less current.

Fig. 2
figure 2

JDark–V Characteristics for Semispherical and truncated conical QD structure for different volumes (Ali and El-Batawy 2023), for a \({V}_{b}=0.259 eV\) b \({V}_{b}=0.341 eV\)

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Figure 2 also illustrates how the QD volume affects the characteristics of the dark current of the proposed photodetector. It is evident that lowering the dark current density arises from boosting the volume of the QD while maintaining the QD density. This can be explained by saying that increasing the volume makes the QD's surface area larger, which leads to more scattering events and, in turn, poorer carrier mobility and dark current, that is consistent with earlier reported that the electron mobility declines as the QD size increases (Li et al. 2020; Ogale and Madhukar 1984). Additionally, at low temperatures (77 K), phonon assisted tunnelling emission rather than thermal emission dominates the electron emission from the QD, which accounts for the decrease in dark current for larger QD with less confinement (Rajanna and Ghosh 2021).

To understand the differences of the JDark -V properties in both semispherical and truncated conical QDIPs, the bound states of both QDIP shapes have been investigated. Semispherical QD exhibits lower dark current for QD volume of 1019 nm3, which corresponds to lower noise in photodetection because the number of semispherical structure's bound states is lower at this volume than that of truncated conical structure. The wave function for the InAs/GaAs QD in Fig. 3 has a 0.341 eV Conduction Band Offset (Ghosh et al. 2000), and the semispherical QD's bound state calculations give rise to the following states; (a) the ground state at energy of 0.158 eV, that is 0.183 eV below the continuum, (b) first excited states with \(n = \pm 1\) and energy of 0.277 eV, that is below the continuum by 0.064 eV. On the other hand, the bound state calculations of truncated conical QD results in the following states; (a) ground state at energy of \(0.1567 eV\) that is below the continuum by 0.184 eV, (b) first excited states with \(n = \pm 1\) and energy \(0.2928 eV\) that is below the continuum by 0.048 eV and (c) another bound state with \(n=0\) at energy of \(0.3352 eV\) as illustrated in Fig. 3e.

Fig. 3
figure 3

Wave function of semispherical QD a ground state, b first excited state and truncated conical QD c ground state, d first excited state, e additional state at n = 0

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As can be seen in Table 1, as the quantum dot dimensions are increased, more bound states are produced as would be expected. Additionally, the bound state will be less constrained if its energy is higher. Figure 3 shows the bounds with the higher energy are less confined in the QD region, and it penetrates further in the barrier region. That explains why, at large dot volumes, the shape of the QD structure has less of an impact on the dark current characteristics, leading to nearly identical dark current densities in semispherical and truncated conical QD systems.

Table 1 Number of Bound States for both semispherical and truncated conical QDs for different QD volumes
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3.2 Temperature impact on the JDark–V characteristics

The impact of operating temperature on the dark current's J–V characteristics for the hemispherical QDIP has been examined in this section. In Fig. 4, it is evident that raising the temperature effectively raises the related dark current density for any operating biasing. More free electrons are allowed to enter the conduction band as the operating temperature rises due to a reduced barrier, and the shot-noise also rises as the temperature does (Liao et al. 2006).

Fig. 4
figure 4

Dark current J–V Characteristics for Semispherical QDIP for different operating temperatures

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It is important to note that the operating temperature range should be constrained as the thermal energy must not be greater than the charging energy, otherwise the electrons pass the QD through thermal excitation because the associated thermal energy is sufficient to ionize the material.

3.3 QD density impact on the JDark–V characteristics

As illustrated in Fig. 5, when the biasing is higher, the impact of raising the QD density results in a lower dark current density. The carrier mobility declines as the QD density rises, which in turn causes an increase in scattering events, that reduces the dark current. Additionally, more QDs confine extra electrons, raising the potential barrier consequently.

Fig. 5
figure 5

JDark–V Characteristics for different QD densities

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4 Conclusions

Based on the generalized drift diffusion approach and considering the confinement of electrons in the semispherical QDs, this paper proposes a complete model for the dark current in semispherical QDIPs, taking into account the effects of the wave functions and the bound states. By employing a finite-difference time-domain technique to solve the Boltzmann transport equation, the electron mobility in the model under consideration is determined. Truncated conical QDIP data that have been published have been compared to the model's predictions for the dark current characteristics for different conduction band offset values, showing the applicability of the model to QDIPS with different materials. Investigations have been done into how the operating temperature and QD volume affect the JDark-V characteristics of the device. While increasing the QD volume causes a drastic decrease in the dark current, raising the temperature causes a rising in JDark. The presented model can be generalized to calculate the photocurrent for the semispherical QDIPs.