## Abstract

In this paper we present a novel long wave length infrared quantum dot photodetector. A cubic shaped 6nm GaN quantum dot (QD) within a large 18 nm $A{l}_{0.2}G{a}_{0.8}N$ QD (capping layer) embedded in $A{l}_{0.8}G{a}_{0.2}N$ has been considered as the unit cell of the active layer of the device. Single band effective mass approximation has been applied in order to calculate the QD electronic structure. The temperature dependent behavior of the responsivity and dark current were presented and discussed for different applied electric fields. The capping layer has been proposed to improve upon the dark current of the detector. The proposed device has demonstrated exceptionally low dark current, therefore low noise, and high detectivity. Excellent specific detectivity (D*) up to ~3 × 10^{8}
*CmHz*
^{1/ 2}/*W* is achieved at room temperature.

© 2010 OSA

## 1. Introduction

The most advanced III-V mid and long wavelength infrared (MWIR and LWIR) detectors, to date, is the quantum well infrared photodetectors (QWIPs) which utilize intersubband or subband to continuum transitions in quantum wells [1,2]. QWIPs have demonstrated excellent imagery performance and also extremely uniformity across a large area, which increases the pixel operability in a focal plane array without the reliance on correction algorithms needed for MCT detectors. However, QWIPs require lower operating temperature, owing to their higher thermionic emission rates. The operating temperature for QWIPs is lower than for MCT detectors, because the thermionic emission in MCT, for equivalent device parameters, is approximately five orders of magnitude less than in a QWIP [3]. Another serious drawback is the fact that, the n-type QWIPs cannot detect normal incidence radiation, due to the polarization selection rules [1]. Consequently, QWIPs require the addition of light couplers, such as surface gratings, which add to the cost and complexity.

Recently, quantum dot infrared photodetectors (QDIPs) have been emerged as a potential alternative to MCT and QWIPs [4–7]. The advantages of QDIPs, can mainly categorize in three parts, (i) The three dimensional quantum confinement of the carriers, which results in the *δ*-like density of states, and sensitivity to the normal incident radiation, without the use of a grating or corrugations, as is often done in QWIPs [8–10], (ii) reduced electron-phonon scattering, so long excited state lifetime, and high current gain [7, 11, 12]. (iii) The QDIP technology is believed to be promising for high-temperature operations [13, 14].

In order to improve the performance of these detectors, different structures and materials have been investigated [4–8, 15, 16]. It has been shown that a current blocking layer can be effectively used to reduce the dark current. Lin et al. [15] and Stiff et al. [9, 17] have reported QDIPs with a single AlGaAs blocking layer on one side of the InAs/GaAs QD layers. Wang et al. [4] introduced a thin AlGaAs barrier layer between the InAs QDs. This layer filled the area between the dots but left the top of the dots uncovered. An improvement in the detectivity relative to similar devices without this barrier layer has been deduced in these papers. On the other hand, in the last few years, the III-nitride QDs have been extensively studied for their potential use in transistors, lasers and light emitting diodes [18–20]. GaN and its alloys with AlN have strange properties such as larger saturation velocity, wide band gap and higher thermal stability, in comparison to the usual and prevalent III-V materials. But unfortunately, they still suffer from a certain lack of knowledge in terms of fundamental material parameters, and they are in their early stage. Here we tried to investigate special kind of QDs with these materials. The Eigen functions, Eigen values, oscillator strength and other physical parameters calculated in the first stage. Then, the detector parameters such as responsivity and dark current were evaluated precisely, by considering their temperature dependence. Specific detectivity used as figure of merit, and its peak was calculated at function of temperatures for different applied bias.

## 2. Model derivation

To model the device, a cubic shaped 6nm GaN QD within a large 18 nm $A{l}_{0.2}G{a}_{0.8}N$ QD embedded in $A{l}_{0.8}G{a}_{0.2}N$ layer is assumed. The proposed structure has been shown in Fig. 1 .

Five layers with QD density of $Nd={10}^{24}{m}^{-3}$ are used as active region of the device. We have assumed the large QDs are very close to each other.

In order to study the electronic structures, different methods have been experienced [21–24]. The single band method is used in this study. In the frame work of the envelope function, and the effective mass theory, the Hamiltonian can be written as [22]:

In which ${m}^{*}$ is the electron effective mass and is given by:As the system needs an applied electric field to operate and also has a strong built in electric field, one has to take into account the total fields effect in the Hamiltonian:

The Built in electric field which applied in the equations is [26]:

The spontaneous polarization for $A{l}_{x}G{a}_{1-x}N$ is Al molar fraction dependent and is given by: ${P}_{sp}=(-0.052x-0.029)$.

To solve the Schrödinger equation, assuming that the wave functions are expanded in terms of the normalized plane waves [22]:

As reported in [28], the attraction of the normalized plane wave approach is the fact that there is no need to explicitly match the wave function, across the boundary of the barrier and QD. Hence this method is easy to apply to an arbitrary confining potential problem. As more plane waves are taken, more accurate results are achieved. We used thirteen normalized plane waves in each direction to form the Hamiltonian matrix (i.e. ${n}_{x},{n}_{y},\text{\hspace{0.17em} \hspace{0.17em}}and\text{\hspace{0.17em} \hspace{0.17em}}{n}_{z}$ from −6 to 6) and we formed 2197*2197 matrix. It was found that using more than 13 normalized plane waves in each direction takes significantly long computational time and only about 1 meV more accurate energy eigenvalues. By substituting the Eq. (8) in Schrödinger equation, eigenfunctions and eigenvalues are calculated. The energy Eigenvalues of the considered structure have been demonstrated in Fig. 2 .

The transition matrix element, ${\overrightarrow{r}}_{if}=<{\psi}_{i}\left|r\right|{\psi}_{f}>$ can be calculated using obtained wave functions. In this relation ${\psi}_{i}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{\psi}_{f}$are the initial and final transition states, respectively. The oscillator strength, ${f}_{if}$, of a given transition is one of the most important factors in the absorption coefficient $\alpha (\omega )$and is given by:

The absorption coefficient can be expressed as [16]:

*ε*are the permeability of free space and the medium, respectively. ${n}_{i}$ and ${n}_{f}$ are the occupation probabilities of the initial and final states. The occupation probability can be defined as:

Figure 3 indicates the behavior of optical absorption of the structure with different QD size for the transition indicated as “a” in Fig. 2. It is obvious that by increasing the size of QD, the peak of the absorption increases and there is a red shift which can be related to increasing of the oscillator strength, and decreasing of the energy levels difference, respectively.

As long as there are unoccupied excited states available, the electrons in the lower states can participate in photon induced intraband transitions. However, with further increasing of the temperature, the electrons occupy the excited states and consequently the absorption coefficient decreases and dark current increases. It should be mentioned that the strongest photonic transitions are usually the ones which are energetically directly above each other, with an s-symmetry to p-symmetry change and in our calculation the “a” transition not only are in the range of 8-12 μm, but also is a transition from a state with s-symmetry to p-symmetry.

## 3. Results and discussion

The insertion of the capping layer is supposed to change the transport properties of the carriers. In this paper, we introduce a structure, which has a low dark current and high responsivity and therefore have a good signal to noise ratio. The main parameters of the detector which discussed in this paper by details are the device responsivity, dark current and detectivity.

#### 3.1 Responsivity

The responsivity is one of the most important parameters of the photodetectors and defined as the ratio of its output electrical signal, either a current ${I}_{out}$ or a voltage${V}_{out}$, to the input optical signal. It is given by

Where g is the gain and defined as the ratio of the recombination time over the transit time:where C_{be}is the quantum mechanical capture rate into the QD excited state. Estimates for the C

_{be}in the literatures, are in the range of $~{10}^{11}-{10}^{12}$ Hz for shallow excited states, which are reachable by acoustic phonon emission, and is about $~{10}^{10}$Hz for deep levels [16]. µ is the mobility of the electron, which has been successfully demonstrated in our previous work by considering all scattering mechanisms, and the effects of temperature and electric fields [29].

*η*is the quantum efficiency and is defined as:

As shown in Fig. 4 with increasing the temperature until 100-170K the responsivity increases and further increasing of the temperature decreases the responsivity. To explain this effects, as can be deduced from the relation (12), there are two main sources for temperature dependence of the responsivity; current gain, and quantum efficiency. So, the increasing of the temperature increases the current gain as well the responsivity. With further increasing the temperature ${n}_{i}$ starts to decreasing, therefore the absorption coefficient and quantum efficiency decreases and it make a reduction in the responsivity.

Also, the logarithm of the normalized responsivity versus applied fields for different temperatures has been illustrated in Fig. 5 . We have not considered the temperature dependency of the life time $1/{\nu}_{0}(T)$ and assumed it as a constant. As shown in this figure, with increasing the applied electric field the responsivity increases. The reason for this behavior is that the increasing of the applied bias increases significantly the current gain as well as the photocurrent.

#### 3.2 Dark current

In the absence of any incident light and the existence of applied fields there is an unwanted electrical current which is well-known as dark current. At high temperature range, the dark current originates from thermionic emissions and for low temperatures, sequential resonant tunneling and phonon-assisted tunneling are probably the dominant components of the dark curve. This important parameter has been discussed in several articles [4,5,9].

Considering the most important factors dark current could be written as [30]:

It should be mentioned that the proposed structure has a very low dark current in comparison to the structures introduced in Ref [16, 30, 31]. It can be deduced from the relations, where the high values for C_{be} not only will decrease the gain, but also the dark current. So, structures with high densities of QDs might be useful and have a better performance in suppressing the dark current effects. Having high densities of QDs has another benefit. It gives hope to engineering the band structure in order to enhance the tunneling of the photoexcited carriers from the large dots (capping layer). Therefore we can have high barriers in order to suppress the thermionic term in dark current without having presentiment about collecting the photoexcited carriers.

#### 3.3 Detectivity

The detectivity is one of the most important factors of detectors and is considered as figure of merit in most of the literatures. Specific detectivity, is defined as

As mentioned before, with increasing the temperature the dark current increases, thus the responsivity decreases and these behaviors will affect the detectivity. For a given applied bias and temperature, the responsivity in our proposed structure lower than the other structures without capping layer. On the other hand, the dark current is much lower in our proposed structure. This may because an enhancement in impact ionization, which is enabled by the increased operating voltage that results from the lower dark current [15].Consequently, a net improvement in the signal to noise ratio is expected. The specific detectivity as function of temperature for different applied bias are shown in Fig. 7 .

The results are representative of high values for specific detectivity in compared to structures which have been studied previously. Xuejun Lu et al reported in [32], Peak specific photodetectivity D* of $3.8\times {10}^{9}$ $cmH{z}^{1/2}/W$ and $1.3\times {10}^{8}$ $cmH{z}^{1/2}/W$ at the detector temperature T = 78K and T = 170 K, respectively. Zhengmao Ye give an account that for the photoresponse peaked at 6.2 mm and 77 K for −0.7 V bias, the responsivity was 14 mA/W and the detectivity, was ${10}^{10}$ $cmH{z}^{1/2}/W$ [33]Bhattacharya et all, reported the some deal high detectivity, about $8.6\times {10}^{6}cmH{z}^{1/2}/W$, in $17\mu m$ wave length for 300k temperature [34] and in the other work they reported $6\times {10}^{9}\le {D}^{*}(CmH{z}^{1/2}/W)\le {10}^{11}$ for temperatures$100K\le T\le 200K$ [18,22]. Here we present appropriate results in comparison and the device which introduced has a good potential to be compared with the structures, have been presented in [34,35].

## 4. Conclusion

As stated in this paper and detailed in numerous publications, owing to their unique material characteristics, III-N QDIPs have the potential for superior performance as infrared detectors in the LWIR. In this article we report on the photodetector characteristics related to QDIPs. The amount of the dark current which calculated is exceptionally perfect and the specific detectivity of the devise is appreciable in high temperatures, even room temperature. Therefore the proposed structure will be considered a proper alternative to the mature technologies that have been widely deployed. The structure studied is sufficiently general, so covers a large rang of possible device types. Due to better 3-D confinement of carriers, operating in higher temperatures was observed. High density of QDs was suggested to solve the collecting difficulties of the carriers. Also the results indicate that there is hope for band structure engineering for further improve of the detector parameters at high temperatures.

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