Based on the analysis of the performance parameters of Si-APD above and considering the characteristics of efficient near-ultraviolet absorption, low operating voltage, and high avalanche multiplication coefficient for the Si-APD, preliminary parameters for each layer are set as shown in
Table 1. Based on the parameters set in
Table 1, two-dimensional simulation calculations of Si-APD were performed by the TCAD simulator to analyze the optoelectronic characteristics of the device. The simulation was performed with an incident light wavelength range of 0.3 to 0.8 μm, the power density of the incident light at each wavelength was set to 1 W/cm
2, and the light is vertically incident on the photosensitive surface of the Si-APD with the diameter of 10 μm. Meanwhile, the surface reflectivity is assumed to be the natural reflectivity of silicon material. To improve the accuracy of the simulation results, various physical models are employed, including the Shockley–Read–Hall recombination [
24,
25], Fermi–Dirac carrier distribution [
26,
27], Selberherr’s carrier impact ionization [
28,
29], and carrier mobility [
30,
31], etc.
3.1. Electric Field Distribution and Responsivity of Si-APD
Based on the structural parameters shown in
Table 1, the internal electric field distribution curves of Si-APDs at different applied bias voltages (without considering the effect of impurity concentration on the relative permittivity of the material, the relationship between the relative permittivity of silicon and the doping concentration is given by the literature [
24]) as well as the spectral response curves of Si-APD under applied bias voltages of 0.95
Vbr-apd (
Vbr-apd ≈ 21.87 V) have been calculated, as shown in
Figure 5.
As can be seen from
Figure 5a, at low applied bias (
Vapd < 0.5
Vbr-apd) the device is not yet pulled through and the electric field strength in the depletion region is small, with further increase in applied bias (
Vapd > 0.5
Vbr-apd) the device is fully pulled-through and the thickness of the depletion region and the electric field strength increase, and the maximum electric field strength in the multiplier layer is 5.33 × 10
5 V/cm when the applied bias is equal to the avalanche breakdown voltage (
Vbr-apd) of the device, the maximum electric field strength in the multiplication layer is 5.33 × 10
5 V/cm. As can be seen in
Figure 5b, the peak response wavelength of the APD is in the blue wavelength band (4.51 A/W@ 470 nm), but the responsivity in the UV wavelength band is still low (0.86 A/W@ 360 nm). In the following, the photoresponsivity in the near-UV band is improved by optimizing the structural parameters of the surface layer, the multiplier layer, and the absorber layer.
3.2. Optimization of Si-APD Structural Parameters
The heavily doped surface non-depletion layer is mainly used to form ohmic contacts, and the doping concentration has a minor impact on the responsivity of the Si-APD. Therefore, we only investigated the influence of the surface non-depletion layer thickness on the responsivity. Based on the parameters listed in
Table 1, the spectral responsivity curves with different surface non-depletion layer thicknesses at the applied bias voltage of 0.95
Vbr-apd(
Vbr-apd ≈ 21.87 V) are calculated, as shown in
Figure 6. It can be observed that the responsivity increases in the near-ultraviolet wavelength range as the surface non-depletion layer becomes thinner. According to the relationship between incident light wavelength and penetration depth shown in
Figure 1a, shorter-wavelength incident light is mainly absorbed near the shallow surface of the APD. With the increase of the surface non-depletion layer thickness, the optical absorption loss increases, resulting in a decrease in the number of optically generated carriers entering the depletion region and an increase in the number of carriers being recombined in the surface layer. So, the thickness of the surface non-depletion layer is selected as 0.05 μm.
The multiplication layer is the primary region where the carrier multiplication effect occurs. The thickness and doping concentration of the multiplication layer significantly affect the carrier avalanche multiplication effect and determine the breakdown voltage of the Si-APD to some extent. Firstly, we investigated the influence of the multiplication layer thickness on the spectral responsivity under applied bias voltages of 0.95
Vbr-apd based on the optimization of the surface non-depletion layer. The
Vbr-apd of APDs with different multiplier layer thicknesses are shown in
Table 2. The influences of the thickness of the multiplication layer on the responsivity in the near-ultraviolet and visible light range are different in
Figure 7a. According to Formula (7) of the internal electric field distribution within the APD, the carrier avalanche multiplication process primarily occurs in the multiplication layer. As the thickness of the multiplication layer changes, the width of the depletion region and the electric field intensity in both the surface non-depletion layer and the multiplication layer undergo significant alterations. Additionally, since the penetration depth of photons in silicon is related to the wavelength, the multiplication coefficient produced by the photo-generated carriers varies for different wavelength ranges.
Figure 7b reveals that the responsivity gradually increases with the increasing thickness of the multiplication layer when the thickness of the multiplication layer is less than 0.25 μm. This phenomenon can be attributed to the insufficient absorption of 0.4 μm incident light by the surface non-depletion layer and the multiplication layer, resulting in a lower number of photo-generated carriers and an incomplete avalanche multiplication process. However, as the thickness of the multiplication layer increases beyond 0.25 μm, the responsivity gradually decreases. This results from the increasing voltage on both sides of the multiplication layer with the increasing thickness of the multiplication layer, resulting in a gradual decrease in the voltage and electric field strength across the absorption layer. Consequently, the drift velocity of photogenerated carriers in the absorption layer decreases, leading to a reduction of drift current. Therefore, a multiplication layer thickness of 0.25 μm is chosen as it exhibits higher responsivity at a wavelength of 0.4 μm.
Next, we investigated the impact of multiplication layer doping concentration on the spectral responsivity with the fixed thickness of the multiplication layer of 0.25 μm. The
Vbr-apd of APDs with different multiplier layer doping concentrations are shown in
Table 3. Based on the spectral responsivity curve of the device at an applied bias voltage of 0.95
Vbr-apd in
Figure 8a, it can be observed that the smaller doping concentration of the multiplier layer could lead to the higher responsivity of the device in the near-ultraviolet wavelength range.
Figure 8b shows the relationship between responsivity and multiplication layer doping concentration at a wavelength of 0.4 μm. As the doping concentration of the multiplication layer in the device increases from 1 × 10
15 cm
−3 to 5.0 × 10
16 cm
−3, the responsivity gradually decreases from 5.30 A/W to 3.70 A/W. Through the electric field distribution within the APD, as described by Formula (8), an increase in the doping concentration of the multiplication layer could lead to a decrease in the electric field strength across both the surface depletion layer and the multiplication layer, resulting in a reduction in the carrier ionization rate and a decrease in the avalanche multiplication current. As depicted in
Figure 8b, a higher responsivity is observed at the doping concentration of 1.0 × 10
15 cm
−3 for an incident light wavelength of 0.4 μm. Therefore, the multiplication layer doping concentration of 1.0 × 10
15 cm
−3 is selected.
To achieve high spectral responsivity of Si-APD over a wide spectral range, the doping concentration and thickness parameters of the absorption layer are optimized. Firstly, the impact of the doping concentration of the absorption layer on the spectral responsivity under applied bias voltages of 0.95
Vbr-apd is investigated based on the optimization parameters of the multiplication layer. The
Vbr-apd of APDs with different absorption layer doping concentrations are shown in
Table 4.
Figure 9a reveals that the variation of spectral responsivity is minimal when the doping concentration of the absorption layer increases from 5.0 × 10
13 cm
−3 to 1.0 × 10
15 cm
−3. According to the electric field distribution within the APD as described by Formula (7), the absorption layer is fully depleted at an applied bias voltage of 0.95
Vbr-apd. Meanwhile, due to the relatively low doping concentration of the absorption layer, the electric field intensity in both the absorption layer and the multiplication layer remains essentially unchanged with increasing the doping concentration of the absorption layer. Consequently, the responsivity remains nearly constant. However, when the doping concentration of the absorption layer continues to increase beyond 1.0 × 10
15 cm
−3, the responsivity rapidly decreases. This can be attributed to the decreasing electric field intensity of the absorption layer as the doping concentration further increases, resulting in a reduction in the drift velocity of photogenerated carriers within the absorption layer and a subsequent decrease of the drift current. The absorption layer doping concentration of 3.0 × 10
14 cm
−3 corresponds to a higher responsivity at an incident light wavelength of 0.4 μm in
Figure 9b. Therefore, the doping concentration of the absorption layer is selected as 3.0 × 10
14 cm
−3.
Furthermore, the influence of the absorption layer thickness on the spectral responsivity under applied bias voltages of 0.95
Vbr-apd is investigated based on the 3.0 × 10
14 cm
−3 doping concentration of the absorption layer. The
Vbr-apd of APDs with different absorption layer thicknesses are shown in
Table 5. The spectral responsivity of the device in the near-ultraviolet spectral range steadily increases as the thickness of the absorption layer increases from 0.5 μm to 5.0 μm in
Figure 10a. Since the larger breakdown voltage in the device and higher electric field intensity of the multiplication layer as the absorption layer thickness increases. Consequently, the carrier ionization rate becomes large, leading to a larger multiplication current. However, as the absorption layer thickness continues to increase beyond 5.0 μm, the responsivity in the near-ultraviolet spectral range gradually decreases. The reason is that the device is under a non-pull-through state at an applied bias voltage of 0.95
Vbr-apd, and the breakdown voltage reaches a stable value, leading to the decrease of the drift current of photogenerated carriers in the absorption layer.
Figure 10b, the absorption layer thickness of 5.0 μm corresponds to a higher responsivity at an incident light wavelength of 0.4 μm. Therefore, the thickness of the absorption layer is chosen as 5.0 μm.
Based on the mentioned optimization results, the final structural parameters for the Si-APD are summarized in
Table 6. The calculated breakdown voltage
Vbr-apd for the device under these parameters is 21.07 V.
Figure 11a illustrates the spectral responsivity curve of the APD at an applied bias voltage of 0.95
Vbr-apd. It can be observed that the device exhibits a spectral responsivity ranging from 6.79 A/W to 14.51 A/W in the wavelength range of 0.3 μm to 0.4 μm, the responsivity is significantly improved compared to the pre-optimized structure. The device also has a high responsivity in the visible light band, especially in the blue light band with a peak photoresponsivity (14.91 A/W@ 0.47 μm), which is attributed to the wide depletion region thickness after the device is pulled through as shown in
Figure 11b so that high-efficiency absorption of visible light can be realized.
3.3. I-V and Avalanche Multiplication Characteristics of Si-APD
The dark current of the Si-APD primarily consists of the diffusion current of minority carriers (
Idiff), the generation-recombination current of carriers within the depletion region (
Ig,r), and the avalanche multiplication current (
Im) [
32]. The expressions for these currents are as follows:
Here,
Dn and
Dp represent the diffusion coefficients of p-region electrons and n-region holes, respectively,
ni represents the intrinsic carrier concentration,
τn, and
τp represent the diffusion lifetimes of p-region electrons and n-region holes, respectively,
τeff represents the lifetime of the effective carrier,
NA and
ND represent the doping concentrations in the p-region and n-region, respectively,
T,
A and
V represent the lattice temperature, the area of the depletion region boundary, the applied bias voltage, respectively,
Jn and
Jp represent the current densities of electrons and holes, respectively. Considering the parameters provided in
Table 6, the
I-V characteristic curve can be calculated by Equation (8). The calculated results of
Figure 5a are performed using the values given in reference [
28], where
ni = 1.5 × 10
10 cm
−3,
Dn = 25 cm
2s
−1,
Dp = 10 cm
2s
−1,
τn =
τp = 5 × 10
−7 s,
τeff ≈ 2 × 10
−7 s,
T = 300 K,
kT/
q ≈ 0.026 V, and the wavelength of incident light for calculating photocurrent is set to 400 nm. Furthermore, the internal multiplication coefficient curves with different applied bias voltages are calculated and shown in
Figure 12b.
As shown in
Figure 12a, the dark current (
Id) increases with the
Vapd of the Si-APD. When the
Vapd is relatively small, the Id is low (part I in
Figure 12a), and the dark current is mainly formed by the generation–recombination current and the lower diffusion current. With the further increase of the applied bias voltage, the device is fully pulled-through and the thickness of the depletion region and the electric field strength of the multiplication region increase, more carriers generate avalanche multiplication effect in the multiplication region and are rapidly separated under the action of the electric field in the depletion region, and the dark current is increased (corresponding to part II in
Figure 12a). At this time, the multiplication coefficient is increased, as shown in
Figure 12b, and the dark current is mainly formed by the higher avalanche multiplication current and the generation-recombination current. When the applied bias voltage is close to the avalanche breakdown voltage of the APD, the dark current increases sharply (corresponding to part III in
Figure 12a), and the ionization rate of the carriers increases rapidly with the increase of the electric field strength in the multiplication region at this time, resulting in a drastic increase in the dark current and the multiplication coefficient. From
Figure 12, it can also be seen that the device has a high avalanche multiplication gain (≈50) and low dark current (<0.1 nA) at a relatively low applied bias voltage (0.95
Vbr-apd), which also indicates the advantages of the device structure in terms of electrical characteristics.