1 Introduction

Infrared (IR) photodetectors have attracted high research interest over the past few decades because of wide-ranging applications in biomedical imaging, spectral analysis, optical communication, security surveillance, artificial retina, remote sensing, environmental monitoring and gas identification (Rao, et al. 2019; Zha, et al. 2022; Cheng et al. 2022; Wang, et al. 2019; Li et al. 2021). Conventional infrared photodetectors include mercury cadmium telluride (HgCdTe) which is commonly used because of its suitable narrow band gap despite its non-economical fabrication cost (Jiao, et al. 2022; Kopytko and Rogalski 2022; Chen et al. 2021). Also, perovskite material has been used recently in photodetection because of its improved optical and electronic properties and simple manufacturing but it still suffers from stability and photosensitivity issues (Miao and Zhang 2019; Wang and Kim 2017). Additionally, quantum-well infrared photodetectors (QWIPs) have been widely used where the optical bandgap of can be engineered but its main drawback is its insensitivity to the normal incident optical signal and correspondingly, requiring complicated optical systems (Jiang and Bu 2022; Ghosh et al. 2021; Yang et al. 2021).

Quantum dot infrared photodetectors (QDIPs) are considered as a promising alternative for infrared (IR) photodetection rather than the conventional photodetectors in mid-infrared spectrum (wavelength ~ 3–15 µm) (Downs and Vandervelde 2013; Wu and Tong 2021; Youssef et al. 2016; El-Batawy and Feraig 2021; Ameen et al. 2014; Liu and Zhang 2012). QDIP devices generally consists of periodically quantum dots in repetitive barrier layers where the quantum dots confine some electrons and allows inter-band transition. QDIP detect an infrared optical signal based on the electrons transition between band-continuum or band-band in QDs (Liu and Zhang 2012). QDIPs can operate at relatively higher temperatures, as they have longer relaxation time, lower dark current and higher conductive gain than conventional IR photodetectors operating at the same temperature. Moreover, QDIPs have good sensitivity to the normally incident IR radiation (Razeghi 2010).

The dark current is considered as one of the main parameters which define the performance of the photodetector as it describes the noise, especially with maximum temperature limitation. Modeling the dark current of QDIP has been presented is various techniques. Many modeling techniques for the dark current in QDIPs have been developed. In (Liu et al. 2001), the dark current model was based on calculation of the electron density in the QD barrier and the carriers’ drift due to the electric field. While in Stiff-Roberts et al. (2004), field-assisted tunneling was taken into account for more accurate model. Transfer matrix method (TMM) was used to calculate the probability of tunneling in quantum dot considering both tunneling and thermionic emissions (Liu et al. 2015). Besides, Nonequilibrium Green’s Function (NEGF) was presented to study the electron transport in nonequilibrium quantum system to calculate the device dark current (Kumar Gujral et al. 2016; Naser et al. 2008). In QDIPs, the QDs are considered as scatterers of the free carriers and consequently limiting their mobility which will affect the dark current. The impacts of the self-assembled QDs on the electron mobility has been studied for various QD shapes, such as conical (Youssef et al. 2016), truncated conical (El-Batawy and Feraig 2021), and semispherical shapes (El-Batawy and Hosny 2020).

In the presented paper, a theoretical model for the dark current in truncated conical QDIPs has been developed, based on the drift diffusion technique and considering the electron confinement of the QDs. This confinement increases the potential barrier that the free electrons face (Ameen et al. 2014). The mobility carriers of the truncated conical QDs dot has been evaluated based on the solution of Boltzmann transport equation (El-Batawy and Feraig 2021). Both the QD filling and the carrier mobility and in turns the dark current of the QDIP are critically dependent on the operating temperature, the volume of the QD, the QD density and the dimension of the QD, then these effects have been investigated in this work. It is shown that, increasing the temperature and dot volume results in increasing the dark current, whereas increasing the QD density reduces the dark current.

The structure of the paper is as following; the schematic structure of the truncated conical QDIP and the dark current model have been presented in Section two. In Section three, the results of the dark current model have been compared with an experimental published result. A parametric study of the impacts of the operating temperature, the QD volume, the QD aspect ratio and the QD density on the dark current characteristics of the photodetector have been investigated in Section four. Finally, the conclusions are summarized.

2 Methods: structure and dark current modeling

A schematic structure of the studied truncated conical Quantum Dot Infrared Photodetector (QDIP) is illustrated in Fig. 1a and the unit cell is illustrated in Fig. 1b. The QDIP consists of many layers of self-assembled truncated conical InAs QD embedded in a barrier material region of GaAs while both the bottom and the top contacts are heavily doped layers. As shown in Fig. 1b, the main design parameters of the cell are top radius, \({R}_{1}\), the base radius, \({R}_{2}\), and the height \(H.\) For the dark current modeling of this QDIP, the coupling between the QDs is neglected as the dimensions of QDs is much smaller than the distance between them. Also, the effect of the wetting layer is ignored due to its small thickness, hence we can consider the active region as a single medium with embedded QDs.

Fig. 1
figure 1

Schematic of truncated cone QDIP: a total QD layers, b unit cell

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The dark current has been calculated using the generalized drift diffusion model, where the electron transports due to the drift and diffusion mechanisms are calculated considering the quantum dots’ confinement of some electrons. This confinement increases the potential barrier facing the injected electrons between the two contacts and in turns limits the dark current. The potential main change is in the axial direction of the QD, so that Poisson’s equation can be considered as one-dimension as follows:

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

where q is the electron charge, \(V\) is the potential, \({N}_{D}^{+}\) is the ionized donor concentration, \(\epsilon\) is the barrier material permittivity, \({n}_{dot}\) is the QD density per unit volume, and \(\langle N\rangle\) is the average QD filling.

To estimate the electron concentration ‘n’ in the barrier region, degenerate semiconductor equation is used as heavily doped contact regions makes the fermi level \({E}_{f}\) to be too close to the conduction band edge EC (Razeghi 2009). Then, the free electron concentration is expressed as(Razeghi 2009):

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

where \(K\) is Boltzmann’s constant, \(T\) is the operating temperature, \({N}_{c}\) is the effective density of states in the conduction band. In the above equation, \({F}_{1/2}\) is Fermi–Dirac integral that is given by Razeghi (2009):

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

In the proposed model, the dark current density \(J\) is calculated based on the drift–diffusion modeling as follows:

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

where \({\mu }_{n}\) is the mobility of electrons and \({D}_{n}\) is the diffusion coefficient of electron which expressed as:

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

Taking into account the fact that there is no illumination and the uniform distribution of the dark current density all over the photodetector, this current density can be expressed in terms of the quasi Fermi level \({\varepsilon }_{f}\) as (Ameen et al. 2014):

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

where the fermi level ‘\({E}_{f}\)’ is obtained from the continuity equation as follows:

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

The number of the confined electrons in QD is obtained from both the number of bound states in each quantum dot the number of electrons in each of them. First, the bound states obtained from Hamiltonian diagonalizing of the effective mass of the QD. Considering the symmetry of the basic cell, the Hamiltonian takes the following form:

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

where \(U\left(r,z\right)\) is the QD potential in the basic cell which equals zero inside the QD and equals to \({V}_{b}\) outside the QD in the barrier region. This implies that the bound states are confined in the QD region and decay outside it. In (8), \({m}_{z}\) and \({m}_{r}\) are the effective masses in the growth and the lateral directions, respectively, where \(l\) is the quantum number.

In the above equations, the carrier mobility is not constant as it is highly impacted by the scattering events occurred due to the QDs, so it depends on both structure dimensions and the operating temperature (Youssef et al. 2016; El-Batawy and Feraig 2021).

According to Matthiessen’s rule (Schwerer et al. 1969), the average total number of scattering electrons per unit time ‘\(\tau\)’ depends on the scattering times by some microscopic scattering mechanisms, and it can be expressed as

$$\frac{1}{\tau }=\frac{1}{{\tau }_{QD}}+\frac{1}{{\tau }_{imp}}+\frac{1}{{\tau }_{Opt phon}}+\frac{1}{{\tau }_{la}},$$
(9)

where \({\tau }_{\mathrm{QD}}\), \({\tau }_{\mathrm{imp}}\), \({\tau }_{\mathrm{Opt phon}}\) and \({\tau }_{la}\) represent the mean free time corresponding to the scattering from QDs, impurities, optical phonons and lattice vibrations, respectively. In QD structures, the scattering due to the QDs is the dominant factor in affecting the electron mobility (Youssef et al. 2016), as the scattering due to the optical phonons and lattice vibrations have negligible effects on the mobility at low operating temperatures.

Then, the carrier mobility is calculated by solving Boltzmann transport equation (BTE) where the distribution function of the electrons is calculated. In these calculations, the impacts of different scattering mechanisms and the external fields are taken into consideration (Ridley 1999). As the active region of the photodetector is considered homogeneous where the QDs act as the scatterers of the free carriers, then the scattering term in BTE is

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

where \({f}_{{p}^{^{\prime}}}\) and \({f}_{p}\) are the probabilities of occupation of the energy levels \({\varepsilon }_{{p}^{^{\prime}}}\) and \({\varepsilon }_{p}\), respectively, while \({S}_{{p}^{^{\prime}},p}\) and \({S}_{p,{p}^{^{\prime}}}\) are rate of the transitions from \({p}^{^{\prime}}\) to \(p\) and from \(p\) to \({p}^{^{\prime}}\), respectively, according the following:

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

where \({V}_{{p}^{^{\prime}},p}\) is the matrix element of the perturbation \(V\left(r\right)\) between the unperturbed states. The solution of BTE is based on finite difference time domain method (FDTD) (Kunz and Luebbers 1993), while the electorn drift velocity is determined averaging the velocity over the obtained steady state distribution function.

3 Results and discussion

3.1 Comparison with experimental results

To check the validation of the model of the dark current for truncated conical QDIP, it has been compared with experimental results published in Harrison et al. (2006). The fabricated cell consists of 10 layers of InGaAs truncated conical shaped QDs with base diameter of \(20 \mathrm{nm}\) and height \(\sim 7 \mathrm{nm}.\) The QD layer is followed by barrier layer of GaAs with height of \(50 \mathrm{nm}\). This comparison is shown in Fig. 2, where high agreement between the model results and these experimental J–V characteristics has been achieved confirming the validity of the presented dark current model. As the bias increases, it results in a reduction in the potential barrier till it vanished completely, which leads to a bias limitation of the model, as shown in Fig. 2, for high biasing.

Fig. 2
figure 2

Dark Current Characteristics Comparison with Exp. Results of (Harrison et al. 2006)

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3.2 Temperature effects on the dark current characteristics

In this section, the effects of the operating temperature on the J–V characteristics of the dark current have been investigated. As shown in Fig. 3, increasing the operating temperature results in an effective increase of the associated dark current for any operating biasing. As the temperature increases, less barrier will lead to more free electrons in the conduction band, besides, the shot-noise increases with the temperature (Liao et al. 2006).

Fig. 3
figure 3

Dark current Characteristics for different operating temperatures

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It is worthy to be mentioned that the temperature is limited because the thermal energy ‘\(KT\)’ should not exceed the charging energy \({E}_{C}\) or else the electrons will be able to pass the quantum dot via thermal excitation as the associated thermal energy will be enough to ionize the donor atoms. The charging energy, which is corresponding to the change in the electrostatic potential, results from the extra electron addition due to the quantum dot confinement.

3.3 QD density effects on the dark current

As shown in Fig. 4, increasing the QD density results in a lower dark current and this effect appears clearly for higher biasing. Increasing the QD density will be associated with an increase of the scattering events that results in a decrease of the carrier mobility and hence the dark current decreases. Besides, adding more QDs will confine more electrons and correspondingly increase the potential barrier (Youssef et al. 2016).

Fig. 4
figure 4

Dark Current for different QD densities

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3.4 QD volume effects on the dark current

As shown in Fig. 5, increasing the QD volume while the QD density constant is kept constant results in higher surface area of the QD, more scattering events and correspondingly lower mobility and lower dark current. This matches with the previously reported QD mobility decrease with the QD size increasing for cone shape(Youssef et al. 2016). Besides, at low temperatures (\(\sim 77 \mathrm{K}\)), the emission of electrons from the QD is dominated by phonon assisted tunneling emission rather than thermal emission which explains the reduction of dark current for larger QD where less confinement (Rajanna and Ghosh 2021).

Fig. 5
figure 5

Dark Current Characteristics for different QD volumes

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3.5 QD aspect ratio effects on the dark current

Figure 6 and Table 1 show the change of the J–V characteristics of the photodetector due to the changing the aspect ratio of the truncated conical QD (that is defined by (R1/R2)) while keeping the volume and the height of the dot constant. As this aspect ratio of the dot decreseas, the dark current increases as the carrier mobility increases. The dominant feature of this behavior is that when the QD structure is closer to the cone shape leads to larger dark current than in the case of the cylindrical shaped QD, as shown in Table 1.

Fig. 6
figure 6

J–V Characteristics for different aspect ratio (R1/R2) for H = 7 nm

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Table 1 Dark current densities for different aspect ratio (\({R}_{1}/{R}_{2}\)) at constant height
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4 Conclusions

In this paper, a theoretical model for the dark current of the truncated conical QDIPs has been presented, where the truncated conical shaped QD is the most likely shaped similar to the self-assembled QDs. The presented model is based on a generalized drift diffusion model where the confinement of electrons in the quantum dot has been taken into account, where the mobility of the carriers is calculated by solving Boltzmann transport equation using time-domain finite difference method. The results of the model for the dark current characteristics have been compared with published experimental measurements showing a good agreement. The effects of the operating temperature, QD density, QD volume and QD aspect ratio on J–V characteristic of the dark current of the device have been investigated. Increasing the operating temperature results in an increase of the Dark current, while the dark current decreases by increasing the QD density. Increasing the volume of the QD results in severe decrease of the dark current due to the increase of the scattering events and the decrease of the associated carrier mobility. Also, the dark current density increases as the aspect ratio of the truncated QD decreases.