 Nano Express
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Theoretical Studies on InGaAs/InAlAs SAGCM Avalanche Photodiodes
Nanoscale Research Letters volume 13, Article number: 158 (2018)
Abstract
In this paper, we provide a detailed insight on InGaAs/InAlAs separate absorption, grading, charge, and multiplication avalanche photodiodes (SAGCM APDs) and a theoretical model of APDs is built. Through theoretical analysis and twodimensional (2D) simulation, the influence of charge layer and tunneling effect on the APDs is fully understood. The design of charge layer (including doping level and thickness) can be calculated by our predictive model for different multiplication thickness. We find that as the thickness of charge layer increases, the suitable doping level range in charge layer decreases. Compared to thinner charge layer, performance of APD varies significantly via several percent deviations of doping concentrations in thicker charge layer. Moreover, the generation rate (G_{ btt }) of bandtoband tunnel is calculated, and the influence of tunneling effect on avalanche field was analyzed. We confirm that avalanche field and multiplication factor (M_{ n }) in multiplication will decrease by the tunneling effect. The theoretical model and analysis are based on InGaAs/InAlAs APD; however, they are applicable to other APD material systems as well.
Background
In_{0.53}Ga_{0.47}As (referred to hereafter as InGaAs) avalanche photodiodes (APDs) are the most important photodetectors for shortwave infrared detection. They are significant in traditional fields, such as optical fiber communication, reconnaissance applications, and remote sensing. InP and In_{0.52}Al_{0.48}As (referred to hereafter as InAlAs) have the same lattice spacing with InGaAs and great avalanche breakdown characteristics; therefore, they are the suitable multiplication layer materials of InGaAs APDs in the traditional applications. In recent years, due to the quick development of singlephoton detection in quantum key distribution [1], timeresolved spectroscopy [2], optical VLSI circuit inspection [3], and 3D laser ranging [4], APDs as the key component in these applications have attracted increasing attention [5, 6]. Pellegrini et al. described the design, fabrication, and performance of planargeometry InGaAs/InP devices which were developed for singlephoton detection with the singlephoton detection efficiency (SPDE) 10% at 1550 nm (200 K) [7]. Tosi et al. presented the design criteria of a novel InGaAs/InP singlephoton avalanche photodiode (SPAD) with high SPDE (30%, 225 K), low noise, and low timing jitter [8]. In simulation, a device model based on experimental data was built to predict dark count rate (DCR) and SPDE of InGaAsP/InP SPADs in [9], and an integrated simulation platform that can evaluate the decoystate quantum key distribution performance of InGaAs/InP SPADs was built in [10]. Acerbi et al. presented design criteria for InGaAs/InP singlephoton APDs with a custom SPAD simulator [11]. For InGaAs/InAlAs APDs, a mesa structure SPAD InGaAs/InAlAs was demonstrated to achieve the SPDE of 21% (260 K); however, high DCR was observed and was attributed to excessive tunneling current [12]. Then, [13] used a thick InAlAs avalanche layer in InGaAs/InAlAs APDs to improve the SPDE (26%, 210 K) and decrease the DCR (1 × 10^{8} Hz). In simulation of InAlAsbased APDs, a device model that based on the Monte Carlo method was established to study the basic characterization of InGaAs/InAlAs APDs in [14], and the influence of charge layer and multiplication layer on punchthrough voltage and the breakdown voltage were studied with steadystate 2D numerical simulations in [15].
Compared to InAlAsbased APDs, researches of InPbased APDs are more comprehensive and in depth in theory and simulation. However, InAlAsbased APD is increasingly used in place of InP as it has a larger band gap that can improve the breakdown characteristics both in the APDs and SPADs [16]. The ionization coefficient ratio of electron (α) to hole (β) in InAlAs is larger compared to InP, and, hence, it has low excess noise factor and high gainbandwidth product. Moreover, InAlAs exhibits a large increase in breakdown probability with overbias ratio, making InAlAs SPADs have lower DCR [17]. Some important properties and conclusions regarding InAlAsbased APDs were obtained from previous studies, such as the low excess noise can be achieved in InAlAs structures with both thick and thin avalanche regions [18]. The tunneling threshold electric field in the absorption (InGaAs) is 1.8 × 10^{5} V/cm, and the tunneling current becomes the dominant component of the dark current in the high field [14]. A verticalillumination structure has larger optical tolerance, but it has a more serious tradeoff between the carrier transit time and responsivity [19]. Moreover, theoretical model, structure (thickness and doping), electric field, and other InAlAsbased APD parameters have been studied in [20,21,22]. However, these studies only focused on influences of common APD structure parameters, such as the absorption layer thickness, multiplication thickness, and charge layer doping level. The relationship between the structure parameters and performance of the InAlAsbased APD has not yet been fully understood and optimized.
In this paper, a theoretical study and numerical simulation analysis based on the vertical structure of InGaAs/InAlAs for 1.55μm wavelength detection were investigated. We built a theoretical model to study the influence of structure parameters on device and detailed relationship of each layer in APDs. The design of the charge layer with different multiplication thickness, influence of the thickness on the doping level in charge layer, and the tunneling effect on the avalanche field in the multiplication were analyzed and simulated.
Methods
In this section, a mathematical relationship between the device parameters and electric field distribution in the device was built, which was applied to analyze the charge layer and the tunneling effect. Concurrently, a simulation model that included simulation structure, material parameters, and basic physical models was built. The theoretical analysis model and simulation model was based on the vertical structure of SAGCM InGaAs/InAlAs APD.
Theoretical Model and Analysis of Charge Layer
Device parameters, such as doping level, thickness, materials, and structure, were used to build the mathematical model for calculating the electric field distribution in APD. The basic physical theories that include Poisson equation, depletionlayer model, and PN junction model of semiconductor device can be found in chapters 1, 2, and 4 in [23] and [24]. The junction multiplication factor equation can be found in [25], and material parameters of semiconductor are from [26]. The presented model adopts Poisson equation, tunneling current density equation, depletionlayer model, junction theory model, and the local model of avalanche gain. The simplified mathematical coordinate system of the APD that includes basic structure parameters (materials, thickness, doping, and dielectric constant) is presented in Fig. 1. It is a simplified SACM APD structure that ignores grading layer. The materials of the contact layer, charge layer, and multiplication layer are InAlAs, and the absorption layer is InGaAs. The junctions of layers are separated by X_{ n }, 0, X_{ m }, X_{ c }, and X_{ a } and X_{ p } by the x coordinate. Doping levels are expressed by N_{ 0 }, N_{ 1 }, N_{ 2 }, N_{ 3 }, and N_{ 4 }, the layer thicknesses are expressed by w_{ 0 }, w_{ 1 }, w_{ 2 }, w_{ 3 }, and w_{ 4 }, and dielectric constants are expressed by ε_{ s0 }, ε_{ s1 }, ε_{ s2 }, ε_{ s3 }, and ε_{ s4 } of contact A, multiplication, charge, absorption, and contact B, respectively.
Equation 1 is the Poisson equation, which can solve the electric potential distribution using the charge density ρ. In this equation, ρ is equal to dopant ion N in the depletionlayer model, w is equal to the thickness of depletion layer, and ε is the dielectric constant of the material. In the common PN junction electric field distribution model, ρ is a variable that depends on the depletionlayer thickness w and dopant ion N. In this model, it changes after considering the tunneling effect. However, before considering the tunneling effect, we first built the electric field distribution using a common method.
By solving the Poisson equation with the device parameters, the mathematical expression of the max electric field is obtained. This expression is determined by the penetration thickness variation in the depletion layer shown in Formulas 2 and 3. In this expression, the parameters that include doping levels (N), thicknesses of depletion layer (w), and dielectric constants (ε) of different layers can be found in Fig. 1.
Then, the electric field distribution can be derived in all points using Formulas 4 and 5. The boundary condition ignores the builtin potential V_{br} in Formula 6; therefore, the mathematical relationship between depletion layer thickness and bias voltage can be calculated.
Finally, the mathematical relationship between electric field distribution and bias voltage in the device is obtained using Formulas 7–11:
From the model, once the boundary of the depletion layer reaches the contact region, Formulas 7–11 can be used to analyze the electric field in each layer. In the practical APD, the absorption and multiplication layers are unintentionally doped in intrinsic layers. N_{ 3 } and N_{ 1 } are less than N_{ 2 }. Thus, Formula 9 is approximately equal to Formula 12. It is the reason that charge layer can control the electric field distribution in the device.
In Formula 8, the electric field difference between multiplication and absorption is determined using the product of N_{ 2 } and w_{ 2 }. N_{ 2 } is the doping level in the charge layer and w_{ 2 } is the charge layer thickness. For a suitable electric field distribution in InGaAs/InAlAs APD, the electric field in the absorption layer (InGaAs) should be within the interval values of 50–180 kV/cm that ensure enough velocity for the photoinduced carriers and avoid the tunneling effect in the absorption layer [10]. That is, the avalanche field in multiplication should decrease to 50–180 kV/cm in absorption by the charge layer. Thus, we can use Formula 8 to find optimal calculated doping level and thicknesses of charge layer. When the multiplication layer is 200 nm (the avalanche field E in the multiplication is 6.7 × 10^{5} V/cm while the multiplication layer is 200 nm [27]); the calculated values of doping level and thickness in the charge layer are compared with results from [28,29,30,31,32,33] in Fig. 2. The region of theoretical values is in good agreement with the experimental data. This result proves that Formula 8 can be used to predict the doping level with different thicknesses in the charge layer when the multiplication thickness is certain.
We calculate the optimal doping level for different thicknesses of the charge layer with the multiplication layer of 300, 500, and 700 nm, and the results are presented in Fig. 3. This result illustrates that the tolerance in the doping level in charge layer is related to its thickness and the range of doping level decreases with the thickness increase in charge layer. That is, if we apply a thick charge region, only a small range of doping level in the charge layer would exist to satisfy the optimal electrical filed. As a result, the performance of APD varies significantly via several percent deviations of doping concentrations in the thicker charge layer. In the “Results and Discussion” section, the practical structures of APDs were simulated to study and verify the theoretical analysis, which includes influence of charge layer thickness on doping level range in the charge layer and the variety of performance for different charge layer thickness in APDs.
Theoretical Model with Consideration of Tunneling
The above analysis model is about electric field distribution in the device and based on the premise that ρ is the dopant ion in the depletion layer. If a sufficiently high electric field exists within the absorption layer, the local band bending may be sufficient to allow electrons to tunnel [34]. Therefore, electron tunneling can occur. From the tunneling schematic diagram in Fig. 4, when the absorption layer has a breakdown tunneling, the tunneling effect changes the charge density ρ, the positive charge in absorption increases, and the negative charge in the multiplication and charge layers increases. Thus, ρ is not equal to the dopant ion charge density in the depletion layer while the tunneling effect appears. The formulas that were discussed earlier will change after considering the tunneling effect.
The generation rate G_{ bbt } of bandtoband tunnel is described in Formula 13 [35, 36].
In Formula 13, E_{ g } is the energy band gap of InGaAs, m* (equal to 0.04 m_{ e }) is the effective reduced mass, E_{ p } is the breakdown electric field in the absorption layer, and γ is a userdefinable parameter that is usually restricted to 1~2. The A and B are the characterization parameters. We calculate the G_{ bbt } with different γ, and the results are shown in Fig. 5. It can be found that G_{ bbt } adapts the same order of magnitude for the charge layer doping level while γ is restricted to 1~1.5.
As a result, charge density ρ is a variable and determined by the tunneling effect and the dopant ion in the absorption tunnel. On this occasion, Formula 1 will be changed to Formula 14 and the electric field in the multiplication layer will be described by Formula 15. w_{tunnel} is the effective depletion layer of the tunneling process [35]. Thus, the changing of avalanche field can be described by Formula 16, and the avalanche field will decrease in the multiplication with the tunneling effect.
The electron and hole ionization coefficients are described by Formulas 17 and 18 in [18]. E is the avalanche field in multiplication.
The effect of carrier avalanche is accounted by the impact ionization model. Considering the extremely low carrier density of the multiplication layer compared to charge layer, it is reasonable to assume that the electric field is uniform throughout the multiplication layer. Therefore, the multiplication factor (M_{ n }) can be expressed as the following Eq. 19. Here, w_{ m } is the multiplication layer thickness and k is the impact ionization coefficient ratio defined by α/β. Since k varies very slowly with the electric field, k is approximately constant for a slight variation of w_{ m } [37].
Assuming constant w_{ m } and bias voltage, differentiation of M_{ n } with respect to electron ionization coefficients is in Formulas 20 and 21.
In Formulas 20 and 21, δα/δE is positive. It is assumed that 20% of a total depletion absorption layer is w_{tunnel} and the absorption layer is 400 nm thick. By solving Formula 16, the relationship between the δE and the absorption field with different γ is presented in Fig. 6. It can be found that δE adapts the same order of magnitude for the avalanche field in the multiplication. Thus, the tunneling effect has an influence on the avalanche field and the M_{ n } will decrease with the tunneling effect. In the analysis, we assumed that the negative charge is nonmultiplied in the multiplication and the model will be more rigorous after taking it into consideration. To verify and analyze the influence of tunneling effect on practical structure of APDs, we simulated the relationship between the tunneling effect and multiplication avalanche field in details in the “Results and Discussion” section.
Structure and Simulation Model
A semiconductor device simulation of TCAD was used for simulation and analysis. This simulation engine defines physical models in simulation, and the results have a physical meaning [20]. The basic physical models were presented as follows. The driftdiffusion models, including the Poisson and carrier continuity equations, were used to simulate the electric field distribution and diffusion current I_{DIFF}. Bandtoband tunneling model was used for the bandtoband tunneling current I_{B2B}, and the trapassisted tunneling model was used for trapassisted tunneling current I_{TAT}. The generationrecombination current I_{GR} was described by the Shockley–Read–Hall recombination model, and the Auger recombination current I_{AUGER} was described by the Auger recombination model. The dark current was described clearly by those mechanisms [38]. Avalanche multiplication was described by the Selberherr impact ionization model. Other basic models, including the FermiDirac carrier statistics, carrier concentrationdependent, low field mobility, velocity saturation, and raytracing methods, were used for the simulation model, and a rigorous simulation model was built.
Device structures in the simulation were similar to the experimental structures in [13]. The schematic crosssection of the topilluminated SAGCM InGaAs/InAlAs APD is shown in Fig. 7. The structures from top to bottom are sequentially named as InGaAs contact layer, InAlAs cladding layer, InAlGaAs grading layer, InGaAs absorption layer, InAlGaAs grading layer, InAlAs charge layer, InAlAs multiplication layer, InAlAs cladding layer, InP contact layer, and InP substrate. The thickness and doping of each layer are also presented in Fig. 7. To avoid the influence of thickness on simulation results, we choose two simulation structures. One simulation structure is named as APD1 (multiplication and absorption layers are 800 and 1800 nm, respectively), and the other simulation structure is named as APD2 (multiplication and absorption layers are 200 and 600 nm, respectively).
To test the simulation model, the experiment data in [13] were compared with the simulation results. In this simulation, we used the same structure in the reference, and the currentvoltage characteristics of the device were given. Figure 8 shows our simulation results and the experiment results in the reference. They have the similar punchthrough voltage V_{pt} and breakdown voltage V_{br}. Moreover, the simulation and experiment results correspond well. Therefore, the model in our simulation is accurate. The parameters mentioned above are listed in Table 1.
Results and Discussion
In this section, the theoretical analysis and conclusions were studied by simulation in details. First, the influence of charge layer thickness on doping level tolerance in charge layer was studied in the “Influence of Charge Layer Thickness” section. Then, relationship between the tunneling effect and multiplication avalanche field was analyzed and verified in the “Tunneling Effect on the Electric Field Distribution” section.
Influence of Charge Layer Thickness
From [14], a suitable field distribution in InGaAs/InAlAs APD should comply with those rules. The guarantee V_{pt} < V_{br} and V_{br} − V_{pt} should have a safety margin for processing variations in temperature fluctuations and operation range. In the absorption layer, the electric field should be larger than 50–100 kV/cm to ensure enough velocity for the photoinduced carriers. Concurrently, the electric field must be less than 180 kV/cm to avoid the tunneling effect in the absorption layer. Electric field distribution greatly influences the device performance. The choice of electric field in the absorption layer has a balancing of the tradeoff between small transit time, dark current, and high responsivity for the practical requirement.
In the simulation, we used the structure of APD1 (multiplication is 800 nm thick) and adjusted the charge layer thickness from 50 to 210 nm to study the influence of charge layer thickness on doping level range and verify the theoretical conclusions in analytical model. In the simulation, we selected different doping level ranges in the charge layer so that the electric field distribution complies with the rules. The simulation results on the relationship between thickness and doping level range in the charge layer are presented in Fig. 9a. As the charge layer thickness increases, the suitable doping level range in charge layer decreases. A relatively large doping level range exists in the thin charge layer, and under this doping level range, the device will have a suitable electric field distribution. Apparently, the doping level range is determined by charge layer thickness. The simulation result of APD2 (with a thickness of multiplication of 200 nm) is presented in Fig. 9b, which has a similar result. Moreover, it can be found that the calculated results of Fig. 2 and simulation results of Fig. 9b match well as shown in Fig. 9c. The small difference between the calculated results and simulation results is caused by the different values of avalanche field in the simulation and calculation. The avalanche field in simulation engine is used 6.4 × 10^{5} V/cm, while in the calculation, we use the value of 6.7 × 10^{5} V/cm from [27].
The charge layer thicknesses of 210 and 50 nm (APD1) were selected to show the simulation details and the influence of doping level on the electric field distribution. Figure 10a, c shows the current simulation results of different doping levels in thicknesses of 210 and 50 nm, respectively. Figure 10b, d shows the electric field distribution simulation results using the same structure. The simulation results show that thicknesses of 210 and 50 nm have doping level ranges of 1.0 × 10^{17}–1.3 × 10^{17} cm^{−3} and 3.9 × 10^{17}–5.7 × 10^{17} cm^{−3}, respectively.
Clearly, the device with a charge layer thickness of 210 nm only has a relatively narrow and suitable doping level. A minimal change in the doping level has greatly influence the currentvoltage characteristic and electric field distribution. As a result, the performance of APD varies significantly via several percent deviations of doping concentrations in the thicker charge layer. This conclusion is the same as the theoretical analysis. Concurrently, when designing APD structures, choosing a thin charge layer will give a high level of doping tolerance, as well as confer APD with good controllability.
Finally, the relationship between charge layer and multiplication thickness was simulated. Figure 11a presents the avalanche field with multiplication region thicknesses of 100, 200, and 300 nm in the APD2 structure (with a charge layer thickness of 70 nm). Figure 11b presents the charge layer doping range with different multiplication thicknesses at the suitable electric field distribution condition. The charge layer thicknesses are 50, 70, and 90 nm. Clearly, a high avalanche field exists in the thin multiplication layer. As the multiplication region thickness decreases, the electric field difference between multiplication and absorption layers increases. As a result, a thin multiplication layer needs a high product of the charge layer doping level and thickness to reduce the high avalanche field.
Tunneling Effect on the Electric Field Distribution
The simulation in this part will study the tunneling effect on the electric field in the device. From the theoretical analysis, the tunneling effect has an influence on the avalanche field in multiplication. Thus, the simulation will design to study the influence of electric field in the absorption layer that exceeds the tunneling threshold value. First, group A, with the structure of APD1, charge layer thickness of 90 nm, and different charge layer doping levels of 1.4 × 10^{17}–2.4 × 10^{17} cm^{−3}, was simulated for electric field distribution when the device avalanche breaks down. The result is shown in Fig. 12a. When the charge layer doping level exceeds 2.0 × 10^{17} cm^{−3}, the field in the absorption layer becomes lower than the tunneling threshold field and the avalanche field in the multiplication layer reaches the same value. However, when the doping level is less than 2.0 × 10^{17} cm^{−3}, the field in the absorption layer exceeds the tunneling threshold field and the avalanche field in the multiplication layer decreases with the decrease of the doping level in charge layer. Similar results were observed in the APD2 structure (with a charge layer thickness of 90 nm and doping level of 2.2 × 10^{17}–3.6*10^{17} cm^{−3}) (Fig. 12b). That is, if the electric field in the absorption layer exceeds the tunneling threshold value at or over the breakdown voltage, then the breakdown electric field in the multiplication will decrease.
Groups B (APD1 thickness of 90 nm, doping level of 2.4 × 10^{17} cm^{−3} in charge layer and APD2 thickness of 90 nm, doping level of 3.6 × 10^{17} cm^{−3}) were designed to demonstrate the relationship between the threshold electric field in the absorption layer and avalanche field in the multiplication layer. The multiplication and absorption electric fields vary with the bias voltage on the device. As shown in Fig. 12c, d, when the electric field in the absorption layer reaches the tunneling threshold value, the avalanche breakdown electric field in the multiplication gradually decreases. Moreover, when the absorption field exceeds the tunneling threshold, the avalanche breakdown electric field in the multiplication layer plummets. Furthermore, the absorption field slope increases when the electric field in the absorption layer exceeds the tunneling threshold.
The phenomenon in Fig. 12 can be explained by the theoretical analysis that tunneling has an influence on the charge density in the “Methods” section. When the electric field reaches the tunneling threshold value in the absorption layer, the charge density ρ becomes unequal to the dopant ion. The multiplication field will decrease as the negative ion increases, and the absorption field will increase as the positive ion increases. Concurrently, the absorption field slope will increase due to the tunneling effect. As a result, the electric field in the absorption should be less than the tunneling threshold value to maintain the high field in the multiplication layer and the low dark current when the device avalanche breaks down.
Conclusions
In summary, we have presented a theoretical study and numerical simulation analysis involving the InGaAs/InAlAs APD. The mathematical relationship between the device parameters and electric field distribution in the device was built. And the tunneling effect was taken into consideration in the theoretical analysis. Through analysis and simulation, the influence of structure parameters on the device and the detailed relationship of each layer were fully understood in the device. Three important conclusions can be obtained from this paper. First, the doping level and thickness of the charge layer for different multiplication thicknesses can be calculated by the theoretical model in the “Methods” section. Calculated charge layer values (doping and thickness) are in agreement with the experiment results. Second, as the charge layer thickness increases, the suitable doping level range in charge layer decreases. Compared to the thinner charge layer, the performance of APD varies significantly via several percent deviations of doping concentrations in the thicker charge layer. When designing APD structures, choosing a thin charge layer will give a high level of doping tolerance, as well as confer APD with good controllability. Finally, the G_{ btt } of tunneling effect was calculated, and the influence of tunneling effect on the avalanche field was analyzed. We confirm that the avalanche field and multiplication factor (M_{ n }) in the multiplication will decrease by the tunneling effect.
Abbreviations
 2D:

Twodimensional
 APD:

Avalanche photodiode
 DCR:

Dark count rate
 SACM APDs:

Separate absorption, charge, and multiplication avalanche photodiodes
 SAGCMAPDs:

Separate absorption, grading, charge, and multiplication avalanche photodiodes
 SPAD:

Singlephoton avalanche photodiode
 SPDE:

Singlephoton detection efficiency
 SRH:

Shockley–Read–Hall
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Acknowledgements
The authors acknowledge Xinjing Hou, Xiuli Li, Junying Zhang, and Yongwang Zhang for valuable discussions.
Funding
This work was supported in part by the National Key R&D Program of China (2017YFF0104803), the National Natural Science Foundation of China (Grant no. 61675195, 11504155), the National Thousand Talents Program of China, the Major State Basic Research Development Program of China (Grant no. 2013CB632103), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
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The datasets supporting the conclusions of this article are included within the article.
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SYC initiated the research, built the theoretical model, carried out the simulation, and supervised all the work. SYC, YZ, SR, CBL, and SF drafted the manuscript. SYC, YHZ, LCZ, BWC, and QMW contributed to the data analysis. All authors read and approved the final manuscript.
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Cao, S., Zhao, Y., ur Rehman, S. et al. Theoretical Studies on InGaAs/InAlAs SAGCM Avalanche Photodiodes. Nanoscale Res Lett 13, 158 (2018). https://doi.org/10.1186/s1167101825595
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Keywords
 Avalanche photodiodes
 Theoretical analysis
 Simulation
 Charge layer
 Tunneling effect