Multiple Quantum Barrier Avalanche Photodiode Based on GaN/AlGaN Heterostructures for Long Wavelength Infrared Detection

A multiple quantum barrier (MQB) avalanche photodiode (APD) structure based on GaN/AlxGa<inline-formula> <tex-math notation="LaTeX">$_{1{-}x}\text{N}$ </tex-math></inline-formula> material system has been proposed in this paper which is capable of detecting infrared (IR) signal up to <inline-formula> <tex-math notation="LaTeX">$6.0~\mu \text{m}$ </tex-math></inline-formula> wavelength. A self-consistent quantum drift-diffusion (SCQDD) model developed by the authors, has been used to determine the current-voltage characteristics under dark and illuminated conditions, spectral response, excess noise properties, signal-to-noise ratio, time and frequency responses. Results show that the proposed MQB APD attains peak responsivity of 60 AW<inline-formula> <tex-math notation="LaTeX">$^{-1}$ </tex-math></inline-formula> at <inline-formula> <tex-math notation="LaTeX">$3.0\, \mu \text{m}$ </tex-math></inline-formula> wavelength. Incorporation of a dedicated thin <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-type GaN layer for avalanche multiplication in between the <inline-formula> <tex-math notation="LaTeX">$p^{+}$ </tex-math></inline-formula>-GaN contact layer and MQB constant-field drift-layer ensures significantly low noise equivalent power under normal operating conditions at room temperature (300 K). Optical pulse response of the device reveals that special restriction over the charge multiplication able to supress the minor peaks of the current response and consequently significantly narrow pulse response can be achieved. Narrow pulse response leads to broad bandwidth of 274.5 GHz, which is significantly broader than the existing IR photo-detectors.


I. INTRODUCTION
Avalanche photodiodes (APDs) are most suitable optical detector for the optical receivers in long-haul optical communication systems [1].The APDs are preferred as optical signal detector over other photo-detectors in both free space The associate editor coordinating the review of this manuscript and approving it for publication was Shuo Sun. and fibre-optic communication systems, except the applications in which the signal-to-noise ratio (SNR)-budget is low.In those cases, low noise p-i-n detector, in combination with the trans-impedance amplifier are preferred.However, high internal gain mechanism of APDs eliminates the burden of trans-impedance amplifiers in case of the applications where SNR-budget is not a major concern.Moreover, high sensitivity and ultra-high speed of APD are the primary reasons behind the superiority of APDs over other photo-detectors [2], [3], [4], [5], [6].The APD-based optical receivers can achieve high signal-to-noise ratio within smallest noise equivalent power among all state-of-the-art photo detectors [7].APDs are most suitable for the detection of very weak ultra-high speed optical signals.Moreover, other than optical communication systems, APDs are also used in various applications like bio-medical engineering, scintillation detection of nuclear radiation, particle detection, optical ranging, astronomical observations, material analysis, military applications, etc.The Silicon (Si) based APDs are most commonly used in various visible wavelength and short-infrared wavelength (e.g., 850, 905, 940 nm) optical signal detection and 850 nm fibre-optic communication due to the lower cost and matured technology of Si [8], [9], [10], [11], [12], [13], [14], [15], [16], [17].
Dark current is the most important factor which determine the sensitivity of the APDs.The Si APD of 3.0 mm diameter produces minimum dark current of 10 nA at room temperature (300K), which can be further reduces to 0.1 nA at -40 • C with appropriate cryogenic cooling arrangements [18].However, some important applications like deep space astronomical observations, composite material analysis, etc., demand very weak signal detection capability (high responsivity over broader range from ultra-violet (UV) spectrum to midinfrared spectrum).Modern day ultra-high speed, long-haul optical communication systems require ultra-narrow impulse response of the optical detectors.These requirements can be fulfilled by ordinary Si APDs.Scaling of the device dimensions in nano scale, the Si APDs (nano-APDs) can fulfil the requirements associated with the ultra-high speed signal detection; but due to the narrow bandgap of Si (E g = 1.12 eV at 300 K) dark current of Si nano-APDs rises extensively due to large amount of band-to-band tunnelling at room temperature.As a result of that its signal to noise ratio (SNR) deteriorated and Si nano-APDs remain no longer suitable for weak optical detection.In the year of 2017, Acharyya et al. proposed the use of multi-quantum well (MQW) or multi-quantum barrier (MQB) structures in p + -i MQB -n + structured APDs for supressing the dark current below nA range, even if the dimensions of the device are scaled down to nm range.They further developed the self-consistent quantum drift-diffusion (SCQDD) model for analysing the spectral response, excess noise properties, time and frequency responses of MQW/MQB APDs.They have shown that the MQB nano-APDs, based on Si∼3C-SiC material system, are most suitable for aforementioned applications demanding very low dark current, high responsivity over broad wavelength range, ultra-high speed, and broad bandwidth.The numerical simulation results are validated by comparing those with the experimental data available in literature [19].
In the present paper the authors have proposed a p +n-i MQB -n + nano-APD structure, based on GaN/Al x Ga 1-x N material system in order to achieve lower dark current, higher responsivity over broader wavelength range, better noise characteristics and better time and frequency responses as compared to the earlier proposed MQB APDs based on Si/3C-SiC heterojunctions.The device structure and possible fabrication steps are briefly discussed in this paper.The effects spontaneous and piezoelectric polarization in GaN and AlGaN layers have been incorporated in the earlier developed SCQDD model, and the modified SCQDD model has been used to study the spectral response, excess noise parameters, time and frequency responses of the proposed APDs structure and the results are presented through appropriate graphs and tables with detailed discussions.Finally, the simulation results are validated by comparing those with the experimental data available in literature.A comparative study has been presented at the end of the paper between the proposed device and other commercially available photodetectors in order to establish the superiority of the proposed device.

II. DEVICE STRUCTURE AND FABRICATION
The cross-sectional structure diagram and top view of the GaN/Al 0.3 Ga 0.7 N/GaN MQB APD are shown in Figure 1 (a) and (b).Here, the Al mole fraction x = 0.3 in Al x Ga 1-x N has been chosen.The important device dimensions are listed in Table 1.The fabrication technology employed in creating the MQB APD structure depicted in Figure 1 (a) plays a crucial role in attaining optimal device performance, specifically enhancing spectral response and signal-to-noise ratio (SNR).Notably, defects and dislocations introduced during etching steps can elevate dark current and compromise the SNR of the device.Therefore, careful consideration must be given to the technology chosen for fabricating the GaN/Al 0.3 Ga 0.7 N/GaN MQB MQB APD.In this context, two potential process technologies have been discussed, hinging on (i) the molecular beam epitaxy (MBE) growth technique and (ii) the metal-organic chemical vapor deposition (MOCVD) technique.Between these two options, the latter-MOCVD-based device fabrication-proves superior in achieving enhanced device performance, characterized by lower dark current, and improved spectral response and SNR [20].

A. METHOD -1
The initial substance for growing the MQB APD structure is a 400 µm thick, 4-inch diameter single-polished n + -GaN substrate.The n-type doping concentration of the n + -GaN substrate is 2.0 × 10 24 m −3 .The AlGaN/GaN/AlGaN quantum wells can be grown by using molecular beam epitaxy (MBE) technique on the n + -GaN substrate along (0001) direction.At first a very thin (around 10 nm thick) AlN nucleation layer must be grown on the n + -GaN substrate at around 800 • C temperature.After that the substrate temperature must be reduced to 720 • C for growing a 500 nm thick Al 0.3 Ga 0.7 N buffer layer doped with n-type Si dopant of dosage 2.0 × 10 25 m −3 .This buffer layer reduces the growth stress and also used for forming the cathode contact.Next, 9 -10 cycles of Al 0.3 Ga 0.7 N/GaN/Al 0.3 Ga 0.7 N quantum wells are to be grown on the Al 0.3 Ga 0.7 N buffer layer.The thickness   , in order to form a high-low (hi-lo) density profile of the active layer of the device.The hi-lo doping profile helps to constrict the charge multiplication process within a narrow region and conse-quently reduces the effective avalanche zone under reverse bias.Finally, the p + -GaN layer of thickness 300 nm must be grown on the n-GaN layer in order to form the p + -n junction.The p-type doping can be achieved by using Mg dopants and doping concentration of the p + -GaN must be kept around 4.0 × 10 24 m −3 .Therefore, p + -n-i MQB -n + structured APD gets its final form.After getting the final p + -n-i MQB -n + structure, SiO 2 hard masking, electron beam lithography and Cl 2 -based inductively coupled plasma reactive ion etching (ICP-RIE) with moderate etching rate can be used to etch around 415 nm GaN and AlGaN layers to get the diode mesa structure as shown in Figure 1 (a).After another lithography step followed by vacuum evaporation of Ni(30nm) and Au(200nm) can be used to form the square shaped ring type anode contact layer over the p + -GaN layer.The optical window size (see Figure 1 (b)) is L ow ×W ow = 300µm×300 µm.Finally, the third lithography step followed by the thermal evaporation of Ti (30nm), Al (100nm), Ni (30nm) and Au (150nm) can be used to form the cathode ohmic contact over the n + -Al 0.3 Ga 0.7 N buffer layer.

B. METHOD -2
In this approach, MQB APDs are grown on a 400nm thick, 4-inch diameter single-polished n + -GaN substrate.The high-temperature MOCVD system is employed to grow the devices on the n + -GaN substrate along the (0001)-direction.Trimethyl aluminium (TMAl), trimethyl gallium (TMGa), and ammonia (NH 3 ) serve as precursors for Al, Ga, and N, respectively, with hydrogen (H 2 ) chosen as the carrier gas.Si and Mg dopants are used for n-and p-doping of the epi-layers, with silicane (SiH 4 ) and bis-cyclopentadienyl magnesium (Cp 2 Mg) serving as the doping sources, respectively.To enhance the quality of the active layers, an initial very thin (∼100nm) AlN nucleation layer is grown on the n + -GaN substrate at 1300 • C. Subsequently, the substrate temperature is reduced to 1180 • C, and a 500nm thick Sidoped Al 0.30 Ga 0.70 N-buffer layer (dosage: 2.0 × 10 25 m −3 ) is grown on the AlN layer to alleviate growth stress.Following this, 9 -10 cycles of Si-doped (dosage: 10 21 m −3 ) Al 0.30 Ga 0.70 N/GaN/Al 0.30 Ga 0.70 N quantum wells are grown on the Al 0.30 Ga 0.70 N-buffer layer.To achieve a hi-lo doping profile of the active layer of the device, a Si-doped (dosage: 2.0×10 23 m −3 ), 20nm thick n-GaN layer is grown on the final Al 0.30 Ga 0.70 N-barrier layer.The advantage of the hi-lo doping profile is previously mentioned in the method -1.Finally, the growth temperature is reduced to 1080 • C, and the Mgdoped (dosage: 4.0×10 24 m −3 ), 300nm thick p + -GaN anode contact layer is grown.Upon completion of the p + -n-i MQBn + structure growth, the entire structure undergoes ex-situ thermal annealing under N 2 environment at 900 • C to activate the Mg acceptors in the p + -GaN contact layer.Next, a SiO 2 mask layer is applied to the wafers using plasma-enhanced chemical vapor deposition.The device pattern is then transferred to the mask layer through standard photolithography and reactive ion etching (RIE) processes.Subsequently, the wafers undergo etching inside an inductive coupling plasma etching system with a depth of approximately 415 nm, revealing the n-Al 0.30 Ga 0.70 N-buffer layer, as illustrated in Figure 1 (a).Following the removal of the residual SiO 2 mask using a buffered oxide etching solution, the wafers are subjected to a 10% NaOH solution at 80 • C for 10 minutes to passivate the etching defects on the mesa sidewall.The next step involves transferring the n-electrode (cathode) pattern to the wafer via photolithography, followed by the deposition of a Ti(30nm)/Al(100nm)/Ni(30nm)/Au(150nm) electrode using e-beam and thermal evaporation.Subsequent to lift-off, the n-electrode undergoes rapid annealing at 600 • C for 30 seconds in N 2 environment to establish Ohmic contact.Similarly, the p-electrode (anode), composed of Ni(50nm)/Au(200nm), is deposited on the mesa using the same processes.After lift-off, the p-electrode undergoes annealing at 550 • C for 5 minutes in N 2 to form Ohmic contact.Finally, a SiO 2 passivation layer is deposited on the wafers to passivate dangling bonds, thereby reducing leakage current.The SiO 2 layer on electrodes is subsequently removed by reactive ion etching (RIE) for performance testing.
In this work, the characterizations of the device have not been presented.Conducting a comprehensive material characterization of the entire device structure is crucial to assess the crystal quality of the fabricated structure.The estimation of Al content and crystal quality of the epilayers can be achieved through the utilization of a high-resolution X-ray diffractometer (Brucker D8 DISCOVER).For an in-depth investigation of the device cross-section, particularly the multi-quantum well (MQW) region, a high-resolution scanning transmission electron microscope (HR-STEM) should be employed.The mesa and electrodes of the detectors can be observed using an optical microscope (Nikon ECLIPSE LV15ONA).To measure the current-voltage (I-V) curves, the PDA FS-Pro 380 semiconductor analyzer can be employed.Detailed results of the full-structure device characterization will be reported in a forthcoming presentation.

III. MODELLING AND SIMULATION
The one-dimensional (1)-D) model of the p + -n-i MQB -n + structured MQB APD based on GaN/Al 0.3 Ga 0.7 N material system is shown in Figure 2.Under reverse bias, the highly doped p + -GaN layer injects electrons into the n-GaN layer.The majority of charge multiplication phenomenon occurs within the high-field n-GaN layer and consequently huge number of electrons entered to the constant field (nearly undoped) MQB drift region (see the simulated electric field profile shown in Figure 2).Therefore, MQB drift-layer does not take part in avalanche multiplication process.As a result of that under optical illumination condition (i.e., when time-varying photon flux density (λ ) (t) of wavelength λ is incident on p + -GaN surface), the wavelengths associated with GaN (E g ≈ 3.4 eV at 300 K) are absorbed in p +and n-GaN regions and get the chance of being multiplied through the avalanche multiplication at n-GaN layer.Therefore, shorter wavelength (UV spectrum) gets the optical gain from avalanche multiplication.However, the longer wavelengths are absorbed at MQB drift-layer due to the inter sub-band transitions in the quantum wells.Several cycles of quantum wells are responsible for obtaining the sufficiently high optical gain at longer wavelengths (IR-spectrum).
A self-consistent quantum drift-diffusion (SCQDD) model developed by the authors has been used to calculate the multiplied responsivity of the MQB APD structure under reverse bias [19], [21].The SCQDD model has been formulated by coupling the classical drift diffusion (CLDD) equations with the time-independent Schrodinger's equations associ- ated with electrons in conduction band and holes in valance band [21].The CLDD equations are as follows: Poisson's Equation: Steady-State Continuity Equations: Current Density Equations: where q = 1.6 × 10 −19 C is the unit electronic charge, space-dependent parameters like r∈ {p, n} are the hole and electron concentrations, G r( ) are hole and electron generation rates due to avalanche multiplication ( ≡ AV), bandto-band tunneling ( ≡ BBT) and trap-assisted tunneling ( ≡ TAT), R r are the Shockley-Hall-Read recombination rates of holes and electrons, µ r are the hole and electron mobility, and D r are the diffusion constants associated with holes and electrons.The space-dependent parameters γ r are the quantum-correction factors associated with holes and electrons having the initial values of unity at the first iteration [21].The parameter N (z) is the space-dependent doping profile of the p + -n-i MQB -n + structured device [21].The space-dependent electric potential (V (z)), electron (n(z)) density, hole (p(z)) density, electron current density (J (total) n (z)) and hole (J (total) p (z)) current density solutions for given biasing and optical illumination conditions at each iteration can be obtained by simultaneously solving the CLDD equations ( 1) -( 3) by using finite difference method (FDM) subject to the following boundary conditions imposed at the depletion layer edges, i.e. at the p + -n junction (z = 0) and at i MQB -n + interface (z = h n + h i ): Electric Field Boundary Conditions: Current Density Boundary Conditions: where the normalized current density is given by NJ = J . In group-III nitride semiconductor materials like wurtzite GaN and its binary compound Al x Ga 1-x N are characterized by polarization properties.These materials are inherently characterized by spontaneous polarization properties under unstrained condition, which constitutes strain-independent built-in electrostatic fields [22].The spontaneous polarization induced electric fields significantly influence the transport and optical properties of the group-III nitride-based nanostructures.On the other hand, both the wurtzite GaN and its binary compound Al x Ga 1-x N are forced to exhibit piezoelectric polarization fields under strained condition [22].In Al x Ga 1-x N/GaN/Al x Ga 1-x N MQB structure, the lattice constants of GaN and Al x Ga 1-x N are significantly different.Thus, in presence of pseudomorphic growth, both the active (GaN) and cladding (Al x Ga 1-x N) layers will be strained and constitute piezoelectric polarization fields [22].The direction of spontaneous polarization is same as the growth direction, i.e. along the (0001) axis.However, the piezoelectric polarization may take any direction.In wurtzite nitrides, the growth direction is always kept along polar (0001) axis; consequently, a piezoelectric polarization is induced along the growth axis due to the existence of non-accommodated in-plane mismatches [23].The sign of the piezoelectric polarization depends on the type of the epitaxial strain, i.e. either tensile or compressive.The polarization-induced electric field in the well (GaN) and barrier (Al x Ga 1-x N) layers of Al x Ga 1-x N/GaN/Al x Ga 1-x N MQB structure are given by  7) and ( 8) b , which is relevant with the MQB structure under consideration (h w = h b = 5 nm).Therefore, in general, for an arbitrary strained condition, the total polarization-induced electric field is given by In case of the pseudomorphically grown Al x Ga 1-x N/GaN/Al x Ga 1-x N MQB structure on GaN substrate, in which h w = h b , the following relation will be maintained Thus, the position-dependent total transverse polarization (P t ) in equation ( 1) can be determined from the relation , where ϵ position-dependent dielectric constant.The spontaneous and piezoelectric polarization components in Al x Ga 1-x N cladding layers can be obtained from [23] where x is the Al mole fraction in Al x Ga 1-x N, S xy = a GaN − a Al x Ga 1-x N a Al x Ga 1-x N is the strain of x-yplane, e 31 , e 33 are the piezoelectric constants, C 13 , C 33 are the elastic constants, a GaN and a Al x Ga 1-x N are the lattice constants of GaN and Al x Ga 1-x N respectively.The model of Fiorentini et al. [22] has been taken into account in the present simulation method with 40% of the polarization charges compensated by defects and interface charges.Other material parameters related to the polarization in wurtzite group-III nitrides (GaN and Al x Ga 1-x N) are taken from the [23].
After obtaining the solutions of CLDD equations under a particular iteration, 1-D time-independent Schrödinger equations associated with conduction and valance bands are solved in order to take into account the bound-states in the quantum wells.The Schrödinger equations corresponding to the valance and conduction bands are given by where ℏ = h 2π is the normalized Planck's constant (h = 6.62 × 10 −31 J s), m * r are the space-dependent effective mass of holes in valance band and electrons in conduction band, ψ Z r r and E Z r r are the wave-function and eigen-energy state solutions associated with the valance and conduction bands respectively, Z r are the number of modes (typically Z r = 5 − 10).The lowest energy state of the conduction band and highest energy state of the valance band in MQB structure can be obtained from where T represents the temperature in Kelvin (K), k B = 1.38 × 10 −23 J K −1 is the Boltzmann constant, Vis the space-dependent electric potential obtained as one of the solutions of CLDD equations, E g is the space-dependent bandgap and N C,V are the effective density-of-states in conduction and valance bands respectively.Now, the quantum hole and electron densities can be calculated from' where N Z r r are the sub-band hole and electron densities in valance and conduction bands of the MQBs respectively.These are determined from the knowledge of the 2-D densityof-states functions and Fermi-Dirac distribution functions associated with the hole and electron densities in valance and conduction bands of the MQBs respectively [21].Finally, the space-dependent quantum-correction factors corresponding to the hole and electron densities are determined from Through the space-dependent parameter γ r , the quantumcorrections are introduced in the hole (p) and electron (n) densities of the CLDD equations ( 1) -( 3) from the next iteration (iteration number ≥ 2).At the end of each iteration, the inter-iteration deviation of V , n, p, J (total) n (z) and J (total) p (z) are individually calculated [21].The self-consistent solutions are assumed to be achieved, when all the inter-iteration deviations are found to be less than the order of 10 −3 [21].The self-consistent current density for given reverse bias and optical illumination condition can be determined from p(SC) dz, (18) where H = h p + + h n + h i + h n + is the effective thickness of the device.
The multiplied responsivity of the MQB APD for a given reverse bias voltage V R and for incident optical power P opt of wavelength λ is given by where A j = L a W a is the cross-sectional area of the p +n junction [21].The term λ is the self-consistent solution of the total terminal current (photocurrent plus dark current) for the bias voltage of V R and incident optical power of P opt of wavelength λ .The details of the algorithm for solving the SCQDD equations subject to appropriate boundary conditions and the method of calculating the self-consistent solution of total terminal current J (total) (SC) λ for a given set of V R , P opt and λ is already reported elsewhere [19], [21].The self-consistent dark current associated with the device for the reverse bias voltage of V R can be obtained from (for incident optical power P opt = 0) and therefore, the total photo current for the same amount of reverse bias voltage (V R ) and for the incident optical power of P opt of wavelength λ is given by In the proposed MQB APD structure the light energy is illuminated on the p + -GaN surface through an optical window of area A ow = L ow W ow .Here, only the photo-generated electrons in the p + -GaN layer injected to the n-GaN avalanche multiplication region and finally drifted through the multiple quantum wells before reaching the positively biased cathode.
Since the charge multiplications only occur in n-GaN and MQB regions, electron current gets enough avalanche multiplication gain.On the other hand, photo-generated holes in the p + -GaN region immediately reach the negatively biased anode and do not get opportunity to be multiplied.Therefore, the photo current in the proposed MQB APD is clearly electron dominated.The random nature of the electron multiplication process causes significant number of unwanted fluctuations in the total photo current; these unwanted fluctuations are known as avalanche noise.The noise performance of an APD is quantified by a parameter named excess noise factor (ENF).The ENF of the proposed MQB APD based on GaN/Al 0.3 Ga 0.7 N hetero-structures can be expressed as [24] where ⟨M ⟩ is the multiplication gain associated with the electron-initiated multiplication process and the expression of ⟨M ⟩ is given by where m is the number of QB plus QW stages.The electron and hole ionization probabilities (P and Q respectively) per stage are given by where ς (MQB) n,p are respectively the ionization rate of electrons and holes in MQB structure.The ratio ς (MQB) n /ς (MQB) p can be calculated as function of electric field (ξ ) by using the method prescribed by Chin et al. [25].The ratio of electron ionization probability and hole ionization probability per stage of the MQWs denoted by K t in equations ( 4) and ( 5) is given by After obtaining the spectral response and noise performance of the proposed MQB APD structure, finally the time and frequency response of that structure must be investigated.In order to obtain the time response a unit rectangular pulse having form where pl is the duration of the pulse and u(t) is the Heaviside step function (i.e, u(t) = 1 for t ≥ 0 and u(t) = 0 for t < 0), is assumed to incident on the p + -GaN surface through the optical window of area A ow = L ow W ow .The unmultiplied time varying photo-generated electron and hole currents having both drift and diffusion components are calculated by considering the time dependent incident photon flux density of wavelength λ where (λ) 0 is the amplitude of the incident optical flux density pulse, i.e.
where c = 3.0 × 10 8 m s −1 is the velocity of light in vacuum, R(λ ) is the reflectance of GaN surface [26], which is given by where n (GaN ) λ and n (air) λ are the refractive indexes of GaN surface [26] and air at the wavelength of λ .Finally, the time-varying drift-diffusion equations such as Poisson's equation, carrier continuity equations and current density equations are simultaneously solved subject to appropriate time-varying boundary conditions at the depletion layer edges, in order to obtain the time and space dependent multiplied electron and hole photocurrents (J (λ) n (x, t)) and From knowledge of the device dimensions as well as electron and hole photocurrent solutions, the current response associated with the incident optical pulse can be determined.It is given by [27] Fourier transform of I (λ) (t) provides the frequency domain current response of the device, which is given by 16028 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Appropriate fast-Fourier transform (FFT) algorithm with sampling frequency f s >10/ pl must be used to obtain I (λ) (f ).

IV. CURRENT-VOLTAGE CHARACTERISTICS AND SPECTRAL RESPONSE
The proposed simulation methodology based on SCQDD model has been initially verified by comparing the simulation results of an AlGaN/GaN multi-quantum well (MQW) APD with the experimentally measured data of Zheng et al. [28].
Here, the simulations have been carried out on the MQW APD structure fabricated and tested by Zheng et al. [28] in the year 2016, under the dark and same optical illumination conditions, i.e. for P opt = 0 and for incident optical power of P opt = 250 pW from an UV light source of 350 nm wavelength respectively.Figure 3 (a) shows the currentvoltage (I-V) characteristics of the MQW APD obtained from both simulation (present work) and experimental measurements (reported in [28]).The Figures 3 (b    range 100 -360 nm (UV spectrum) is achieved due to the photon absorption in p + -and n-GaN layers.Within this UV range the peak responsivity is 4.1 AW −1 is attained at 200 nm wavelength.Direct transitions of electrons from the energy states of valance band to the energy states of conduction band is the primary mechanism of photocurrent generation within this UV wavelength range.The inter sub-band transitions of electrons within the quantum wells of the MQB structure leads to the significant rise in responsivity of the APD within the visible spectrum (400 -770 nm), near-infrared (700 -1400 nm), shortwave-infrared (1400 -3000nm), mid-infrared (3.0 -6.0 µm) spectrums.The peak responsivity of 60 AW −1 is achieved at 3.0 µm wavelength.
The carbon dioxide (CO 2 ) absorption causes a dip in the responsivity at the wavelength of 4.3 µm. the effect of the CO 2 absorption has been included in the simulation through the experimental data of absorption coefficients at room temperature.The spectral response of Si/3C-SiC MQB APD at reverse bias of 9.0 V and for incident optical power of 1 W of wavelength range 100 -4000 nm is also shown in Figure 5. Unlike the GaN/Al 0.3 Ga 0.7 N MQB APD the transition of band-to-band absorption in barrier material (wider bandgap) to quantum well inter sub-band absorption occurs at 450 nm; the peak responsivity of 1.6 AW −1 can be achieved at 260 nm in Si/3C-SiC MQB APD with the detectable UV spectrum of it.The peak responsivity Si/3C-SiC MQB APD occurs at 350 nm which is around 50 AW −1 and its responsivity gradually decays to 15 AW −1 at 4.0 µm wavelength after which sharp decay in the responsivity versus wavelength curve is observed.Therefore, GaN/Al 0.3 Ga 0.7 N MQB APD capable of detecting wavelengths up to 6.0 µm, whereas the maximum detectable wavelength of Si/3C-SiC MQB APD is around 4.0 µm.

V. EXCESS NOISE CHARACTERISTICS
The valance band offset (VBO) in GaN/Al 0.3 Ga 0.7 N heterojunctions is E v = 0.3x eV, whereas conduction band offset (CBO) of those is E c = E g (x) -E v -E g ; where E g (x) = 6.13x + 3.42(1-x) -1.0(1-x)x eV is the bandgap of Al x Ga 1-x N at room temperature and E g = 3.42 eV is the bandgap of GaN at room temperature (300K) [30].Therefore, the values of VBO and CBO in the type-I band alignment of Al 0.3 Ga 0.7 N are E v = 0.09 eV and E c = 0.513 eV respectively.Thus, in GaN/Al 0.3 Ga 0.7 N heterojunction, the discontinuity in conduction band ( E c = 0.513eV) is significantly larger as compared to the discontinuity in valance band ( E v = 0.09 eV).The ionization rate of holes is larger than of electrons in GaN, while ionization rate of electrons is larger than holes in AlN [31], [32], [33].Although the ratio of ionization rate of holes to ionization rate of electrons (ς p /ς n ) is greater than one in bulk GaN, due to the greater conduction band discontinuity in GaN/Al 0.3 Ga 0.7 N heterojunctions the ratio (ς p /ς n ) reduces slightly in GaN/Al 0.3 Ga 0.7 N MQB structures [30].As a result of that elimination of secondary carrier multiplication cannot be achieved like Si/3C-SiC structures, in which E v > E c .Therefore, in order to reduce the excess noise in GaN/Al 0.3 Ga 0.7 N MQB APD an extra n-GaN dedicated layer has been introduced to restrict the electron dominated carrier multiplication within a narrow region (nearly around 20 nm thick).By providing the spectral restrictions to the avalanche multiplication phenomenon, better excess noise characteristics can be achieved in GaN/Al 0.3 Ga 0.7 N MQB APD, having p + -n-i MQB -n + structure as compared to the Si/3C-SiC having p + -i MQB -n + structure.The gain (⟨M ⟩) versus bias voltage (V R ) curve of MQB APD based on GaN/Al 0.3 Ga 0.7 N heterojunctions is shown in Figure 6

VI. SIGNAL-TO NOISE RATIO AND OPTIMUM GAIN
The avalanche multiplication process typically begins when a thermally generated electron is present within the avalanche 16030 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.zone, and when combined with other non-optical sources of noise, it results in a multiplied dark current as follows: The multiplied dark current adds excess noise, and the shot noise component of this multiplied dark current is expressed as: Considering both the multiplied dark and photo currents, the expression of total output current (signal) and output voltage are expressed as: The shot-noise component due to the total output current (signal) can be expressed as: To characterize the noise characteristics of an APD, we illustrate a standard read-out circuit for an APD in Figure 7 [34].This circuit comprises of a steady voltage source for providing reverse bias (V bias ) to the APD and a trans-impedance amplifier, as depicted in Figure 7.The trans-impedance amplifier serves to convert and amplify the overall output current from the APD into a corresponding output voltage V out .Under a first-order approximation, the output voltage resulting from the signal power is given by Additional standard assumptions regarding the amplifier include a 3-dB bandwidth (B n ) that aligns with the rise time of the photocurrent pulse, a linear response, and the absence of stray capacitances.Moreover, the amplifier's noise contribution is characterized by the inclusion of a noise current spectral density ⟨I n ⟩ at its output.The corresponding amplified voltage, V n , obtained by multiplying it by the square-root of the bandwidth (B n ), is expressed as: The resistor R L determines the gain of the trans-impedance amplifier and introduces a Johnson noise current of The capacitor C L serves as phase compensation, reducing gain peaking in the read-out circuit at higher frequencies [35].
Taking into account the 3-dB bandwidth B n (i.e., the amplifier's noise bandwidth), the expression for the Johnson noise voltage is as follows: The shot noise voltage component due to the signal is given by Summing up the previously mentioned noise currents in a quadrature manner and converting them into a voltage results in the overall noise voltage at the amplifier's output.This total noise voltage at the amplifier's output can be expressed as: The Signal-to-Noise Ratio (SNR) of the APD and the read-out circuit is defined as the ratio of the signal output voltage to the total noise voltage and is given by [34] and [35], (42), as shown at the bottom of the next page.A typical transimpedance amplifier with B n = 50 MHz and amplification noise voltage spectra density ⟨V n ⟩ = 1.414 × 10 −7 V Hz −1/2 terminated with a resistance of R L = 1.0 K at an effective input noise temperature T = 300 K [34] has been considered for the SNR calculation and determination of optimum optical gain of the MQB APD based on GaN/Al 0.3 Ga 0.7 N heterojunctions.Figure 8  Therefore, nearly around 3.5 times higher bandwidth is achievable in p + -n-i MQB -n + structured MQB APD based on GaN/Al 0.3 Ga 0.7 N material system as compared to its p +i MQB -n + structured Si/3C-SiC counterpart at their respective wavelengths corresponding to the responsivity peaks (i.e., 3.0 and 0.85 µm respectively).

VIII. PERFORMANCE LIMITATIONS
The MQB APD proposed here features a vertical illumination structure, where the photo-generated charge carriers are generated at the top layer (p + -GaN) of the APD.Electrons and holes then travel along the same axis (z-axis) parallel to the axis of the incident light.This vertical illumination design comes with an inherent trade-off between carrier transit time and responsivity.The convenient optical coupling is a notable advantage of vertical illumination structure.Lateral illumination APD structures induce the optical signal perpendicular to the carrier's travel direction, allowing for a relaxation of the trade-off between carrier transport distance and optical path length.This, in turn, eases the trade-off between carrier transit time and responsivity.Nevertheless, a trade-off still exists between the resistance-capacitance (RC) time constant and optical path length in such lateral structures.The proposed device structure, as depicted in Figure 1, incorporates vertical illumination and a field-confinement configuration.It's worth noting that in the structure shown in Figure 1, 16032 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.the total thickness of the MQB absorption layer (95 nm) and the doping density of this layer (10 21 m −3 ) determine the thickness of the depletion layer.Any deviation in this thickness will result in changes in responsivity, device capacitance, and operational speed.Therefore, for practical APDs employing this structure, precise control over the thickness of each quantum well (GaN), barrier (Al 0.3 Ga 0.7 N) layer, and their respective n-type doping concentrations is crucial.
In addition to the limitations mentioned earlier, there exists another factor in APDs that significantly impacts their performance in terms of responsivity, bandwidth, and noise characteristics.This factor is known as avalanche build-up time or avalanche response time, and it is closely tied to the properties of the avalanche layer, including its mate- rial composition and thickness.The processes of avalanche multiplication have an inherent finite response time; time required to attain the multiplication gain ∼10 6 is generally referred as avalanche response time [37], [38].When the ionization coefficients of both electrons and holes are nearly identical and these carriers travel in opposite directions within the avalanche layer, creating new electrons and holes in the process.This behaviour introduces a delay in effectively sweeping out the ionized carriers from the avalanche layer.Since the avalanche multiplication and transport processes involve inherent randomness, this delay contributes to an increase in noise current.Consequently, a higher gain typically leads to a longer delay in carrier transit time and greater noise.To quantify this delay and noise impact, two Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.essential metrics are considered: the gain-bandwidth product (GBP) and the avalanche response time (τ A ) [37].These metrics vary for different materials, and they can even differ based on the type of injection carriers [37].Hence, selecting the appropriate material for the avalanche layer is a critical decision.In this context, an n-GaN avalanche layer with a doping density of 2.0 × 10 23 m −3 has been chosen.This n-type GaN layer exhibits an exceptionally low avalanche response time (approximately 10 −16 seconds [37]) and a notably high GBP (around 300 GHz [39]).It is worth noting that thinning the avalanche layer, a phenomenon known as the ''dead-space effect'' [40], can further improve GBP and τ A .However, it's essential to strike a balance between the material composition and thickness of the avalanche layer, while considering the trade-off between GBP, τ A , and tunnelling dark current, as a thin avalanche layer can increase the latter [29].

X. CONCLUSION
A GaN/Al x Ga 1-x N/GaN MQB APD structure has been proposed and studied in this paper in order to investigate its potentiality to detect long wavelength IR signals up to 6.0 µm wavelength.A self-consistent quantum drift-diffusion (SCQDD) model developed by the authors has been used to determine the current-voltage characteristics under dark and illuminated conditions, spectral response, excess noise properties, time and frequency responses.Results show that the proposed MQB APD attains peak responsivity of 60 AW −1 at 3.0 µm wavelength.Incorporation of a dedicated thin n-GaN layer for avalanche multiplication in between the p + -GaN contact layer and MQB constant field drift layer ensures significantly low noise equivalent power under normal operating conditions at room temperature (300 K).Optical pulse response of the device reveals that restricting the charge multiplication within a narrow avalanche zone near the p + -n junction able to supress the minor peaks of the current response and consequently significantly narrow pulse response can be achieved.Narrow pulse response leads to broad bandwidth of 274.5 GHz, which is significantly broader than the existing IR photo-detectors.The performance of the proposed GaN/Al 0.3 Ga 0.7 N/GaN MQB APD has been compared with various state-of-the-art near IR mid IR photo detectors available in literature, and found to be highly promising for long wavelength IR detection up to 6.0 µm wavelength.

FIGURE 1 .
FIGURE 1.(a) Structure diagram of the GaN/Al 0.3 Ga 0.7 N/GaN Multiple Quantum Barrier avalanche photodiode for infrared detection.(b) Top view of the device structure.

FIGURE 2 .
FIGURE 2. One-dimensional model of the GaN/Al 0.3 Ga 0.7 N MQB APD and simulated electric field profile (blue lines) under reverse bias.The electric field profile is presented here for the reverse bias of 10.2 V under the dark condition ( (λ ) 0 = 0); electric field penetration depths in p + -and n + -layers are found to be δ p+ = 5.46 nm and δ p+ = 0.68 nm respectively.
where ϵ w,b are the dielectric constants, and h w,b are the thicknesses of the well and barrier layers, χ ∈ {sp, pz} are the polarization type, i.e. spontaneous (sp) and piezoelectric (pz) polarizations, and ⃗ P (χ) w,b are the χ-type polarization in GaN-well and Al x Ga 1-x N-barrier.It is noteworthy from the equations ( ) and (c) show the simulated variations of the dark current (I d ) and photo-current (I ph ) with the applied reverse bias (V R ) ranging from 0 to 40 V and from 40 to 80 V respectively; those figures also illustrate the amount of deviation in the simulated values of I d and I ph with respect to the corresponding experimentally measured values at each data point.It can be observed from Figures 3 (a) -(c) that the simulation results are very close in agreement with the experimental data of Zheng et al. [28] and therefore, it validates the proposed simulation model for the further simulation study of the proposed Al 0.3 Ga 0.7 N/GaN/Al 0.3 Ga 0.7 N MQB APD structure.The I-V characteristics of the Al 0.3 Ga 0.7 N/GaN/Al 0.3 Ga 0.7 N MQB APD have now been obtained from the numerical simulations.The IV characteristics of the APD have been obtained under dark condition (P opt = 0) and for different incident optical power (P opt = 10, 100, 500 and 1000 mW) of wavelengths ranging from 100 -2500 nm.Breakdown of the device occurs at around 10.3 -10.4V at different illumination conditions.The dark current of the GaN/Al 0.3 Ga 0.7 N MQB APD is found to be around 1.4 × 10 −13 A at the reverse bias of 10.1V (V R < breakdown voltage (V B )), which is significantly smaller than the simulated dark current of Si/3C-SiC MQB APD (0.24 -0.40 nA at 9.0V of reverse bias)[29].Breakdown characteristics of GaN/Al 0.3 Ga 0.7 N MQB APD are considerably sharper than those of Si/3C-SiC MQB APDs.The I-V characteristics of GaN/Al 0.3 Ga 0.7 N MQB APD under dark condition and for incident optical power of 10, 100, 500 and 1000 mW of wavelength 3.0 µm have been shown in Figure4.For reverse bias of 10.2V just below the breakdown voltage, the photocurrent through the device is observed to be increased from 1.1pA to 0.29nA for the increment of the incident optical power from 10 mW to 1.0 W at λ = 3.0 µm.The spectral response of an APD majorly depends on these factors such as (i) thickness of the absorption region (nlayer and total MQB layers) (ii) anti-reflection coating on the

FIGURE 3 .
FIGURE 3. I-V characteristics of AlGaN/GaN MQW APD under the dark and optical illumination of 350 nm UV light with incident optical power of 250 pW; I-V curves: (a) experimental measured data of Zheng et al. [28] (solid black star markers (⋆) represent I d , solid black circular markers (•) represent I ph ), and simulated data using the proposed model (solid blue line (-) represents I d , and solid red line (-) represents I ph ); deviation of simulated data with the experimental data between the reverse bias voltage ranging from (b) 0 to 40 V and (c) 40 to 80 V (blue error bars represent I d , and red error bars represent I ph ).

FIGURE 4 .
FIGURE 4. The I-V characteristics of the GaN/Al 0.3 Ga 0.7 N MQB APD under dark condition and for different incident optical powers of wavelength 3.0 µm.

FIGURE 5 .
FIGURE 5. Variations of spectral responsivity of MQB APDs based on GaN/Al 0.3 Ga 0.7 N (reverse bias voltage of 10.2 V, and incident optical power of 1 W) and Si/3C-SiC (reverse bias voltage of 9.0 V, and incident optical power of 1 W) heterojunctions with wavelength.
(a).The Figure 6 (a) also shows the ⟨M ⟩ versus V R characteristics of Si/3C-SiC MQB APDs with different number of quantum barriers (N b ), such as N b = 1 in APD1, N b = 3 in APD2, and N b = 5 in APD3 [24].It is observed from Figure 6 (a) that at the bias voltage of 9.0 V, the gain of Si/3C-SiC MQB APDs vary from 102.29 to 118.65; whereas the peak gain of 92.69 is achieved in GaN/Al 0.3 Ga 0.7 N MQB APD at the bias voltage of 10.2 V.The variation of excess noise factor (ENF) of GaN/Al 0.3 Ga 0.7 N MQB APD with multiplication gain (⟨M ⟩) has been shown in Figure 6 (b).It is noteworthy from Figure 6 (b) that the ENF of GaN/Al 0.3 Ga 0.7 N MQB APD varies from 1.62 to 5.95 for the multiplication gain range of 1 -100, while the lowest achievable ENF in Si/3C-SiC MQB APD is ranging from 1.58 -10.9 for the same multiplication gain range.Therefore, clearly the GaN/Al 0.3 Ga 0.7 N MQB APD is less noisy as compared to its Si/3C-SiC counterpart.

FIGURE 6 .
FIGURE 6. Variations of (a) gain with reverse bias voltage, and (b) excess noise factor with gain in MQB APDs based on GaN/Al 0.3 Ga 0.7 N and Si/3C-SiC [24] heterojunctions.

FIGURE 7 .
FIGURE 7. Schematic of the read-out circuit.

FIGURE 8 .
FIGURE 8. Noise characteristics of the MQB APD based on GaN/Al 0.3 Ga 0.7 N heterojunctions.

) 0 = 70 . 15 ×
(a) illustrates the fluctuations in several key parameters: the output signal voltage (V out ), the shot noise voltage component (⟨V L ⟩), the Johnson noise voltage (⟨V Th ⟩), and the noise voltage at the amplified output (V n ), all in relation to the multiplication gain (⟨M ⟩) associated with the electron-initiated multiplication process.In contrast, Figure 8 (b) depicts the variation of the SNR as a function of ⟨M ⟩.Notably, Figure 8 (b) reveals that the SNR reaches its peak within a multiplication gain range of 40 to 50.Consequently, this range of 40 to 50 can be considered as the optimal gain values (⟨M ⟩ opt ), maximizing SNR at the output of the trans-impedance amplifier.It's worth noting that enhancing the trans-impedance amplifier circuit can further improve the output SNR, but it won't affect the range of the optimal gain.VII.TIME AND FREQUENCY RESPONSESThe pulse responses of GaN/Al 0.3 Ga 0.7 N and Si/3C-SiC MQB APDs are shown in Figure9.The pulse width pl = 0.4 ps and height (λ10 30 m −2 s −1 are kept fixed, but the wavelengths are selected separately for each APD.The wavelength of the incident optical pulse is selected for each APD for which their respective responsivity peaks are attained.Therefore, 3.0 and 0.8 µm wavelengths are selected for GaN/Al 0.3 Ga 0.7 N and Si/3C-SiC MQB APDs respectively.The current responses of the APDs associated with the similar input optical pulse show that significantly narrower current response can be achieved in the proposed GaN/Al 0.3 Ga 0.7 N MQB APD as compared to Si/3C-SiC MQB APD.Primarily the current response possesses multiple peaks (i) first or major peak is associated with the original injected electrons and (ii) subsequent minor peaks are associated with the secondary generation of the electron-hole pairs due to charge multiplication phenomenon at the active regions.Narrow down of the avalanche zone due to the introduction of n-type dedicated multiplication layer in GaN/Al 0.3 Ga 0.7 N MQB APD the secondary electron-hole pair generation is supressed in MQB layers.As a result of that the minor peaks of the current response are significantly supressed in GaN/Al 0.3 Ga 0.7 N MQB APD as compared to the Si/3C-SiC MQB APDs.Thus GaN/Al 0.3 Ga 0.7 N MQB APDs have significantly shorter response time (τ rs = 2.4 ps) as compared to the Si/3C-SiC MQB APDs (τ rs = 4.2 ps).The normalised frequency responses of the APDs have been shown in Figure 10, which are obtained from the FFT of corresponding time responses shown in Figure 9. Shorter time response of GaN/Al 0.3 Ga 0.7 N MQB APD leads to significantly broader 3-dB bandwidth (BW = 274.5 GHz) as compared to the Si/3C-SiC MQB APD (BW = 78.4GHz).

FIGURE 9 .
FIGURE 9. Pulse response currents of MQB APDs based on GaN/Al 0.3 Ga 0.7 N (bias voltage V R = 10.2V) and Si/3C-SiC (bias voltage V R = 9.0 V) heterojunctions as function of time due to the incident rectangular pulse of photon flux density of 70.15 × 10 30 m −2 s −1 and pulse width of 0.4 ps.

TABLE 1 .
Dimensions of the structural parameters.

TABLE 2 .
Comparison between the proposed APD with the earlier reported group-III nitride-based APDs.

TABLE 3 .
Comparison between the proposed APD and other state-of-the-art near-IR to mid-IR photodetectors.