1 DSMT theory

Impact ionization can occur when the electrons and holes possess sufficient kinetic energy of motion. The minimum distance that a newly generated carrier must travel in order to build up enough energy to become capable of initiating ionization is called dead space. The effect of dead space on the excess noise and mean gain is determined using a recurrence method.

For an electron, it travels in the opposite direction of electric field and ionize and form two electrons and a hole after a random distance. Similarly, a hole travels in the direction of electric field and ionize and form an electron and two holes after a random distance. Electrons and holes repeat this process until they reach the edge of the multiplication layer. The parameters in the DSMT numeric models are as follows.

1.1　dead space

The expression for dead space is as follows:

where ε (x) is electric field at x location, E _ie (x) and E _ih (x) are impact ionization threshold energy of electrons and holes at location x respectively, d _e (x) and d _h (x) denote the dead space of electrons and holes at location x respectively, q is the electronic charge.

1.2　Probability density function(PDF) of distance to impact ionization

After a random distance an impact ionization occurs.^[5] hξ|x  is the probability density function of the impact ionization of electrons and holes that locate at x and move to location ξ and impact ionize at location ξ, impact ionization can only occur when the motion distance ξ is larger than the dead space of carriers at location x. PDF is given by,

α(ξ|x) and β(ξ|x) are impact ionization rates at location ξ of electrons and holes that come from location x without initial energy respectively, that are calculated by,

1.3　The recurrence equations of gain and excess noise

For double-carrier-multiplication avalanche photodiodes, Z(x) is used to denote the random number of all electrons and holes progeny produced by an electron at location x and itself. Similarly, Y(x) is used to denote the random number of all electrons and holes progeny produced by a hole at location x and itself. Assume the range of multiplication layer is from 0 to w, and electrons moves along direction of positive x, holes moves along opposite direction. The random gain generated by only an electron injecting is [Z(0)+Y(0)]/2, clearly Y(0)=1, so the random gain is [Z(0)+1]/2. The mean values of Z(x) and Y(x) are denoted by z(x) and y(x) respectively, and the mean squared values of Z(x) and Y(x) are denoted by z₂ (x) and y₂ (x) respectively. The recurrence equations are as follows. ^[4]

Original values of z(x), z ₂(x), y(x) and y ₂(x) at boundaries are z(0)=1,z ₂(0)=1,y(w)=1 and y ₂(w)=1, original values at other locations are zero.

After getting steady solutions from the recurrence equations, the mean gain and excess noise can be calculated.

2 The modified DSMT theory(MDSMT)

Different from DSMT theory, the modified DSMT theory considers that the carriers gained initial energy when they go through non-zero electric field region before entering multiplication layer, thus the energy needed to obtain from the multiplication layer for the first impact ionization is reduced. ^[2]

If the injecting carrier’s initial energy is E0, then the initial dead space d _e0 is given by:

Clearly, if the carrier’s initial energy before entering the multiplication layer is larger than the ionization threshold energy in the boundary of multiplication layer, then d _e0=0.

In addition, the pdf of the distance to the first impact ionization for the injecting carrier is:

Z ₀(x) is used to denote the random number of all electrons and holes progeny produced by an electron at location x and itself. The probability density function of the first impact ionization is he0ξ . Once the first impact ionization occurred for the injecting electron, the energy of newly generated electron and hole is zero, and independent of the energy of parent electron. Averaging all possible situations in ξ, z₀(x)=<Z₀(x)>:The following formula can be obtained.

Similarly, the average value of the square of Z₀(x),

Putting the steady solutions from the DSMT theory recurrence equations into the above two equations, the mean gain and excess noise can be calculated, the results are as follows.

3 Van Vliet theory

If the ionization multiplication processes including two types of carriers, in low gain, it must consider the discrete nature of the ionization process for both carriers. The excess noise value from Van Vliet is lower than that from McIntyre formula in sufficiently low gain.^[6] In multi-gain-stage APD including both hole-ionization and electron-ionization, the excess noise is given by:

where P _e is the electron-ionization probability per gain stage, J is the number of gain stages, k _s is the ratio of the hole-ionization probability per gain stage to P _e. The mean gain is given by:

4 Multi-gain-stage single photon avalanche photodiode

Multi-gain-stage single photon avalanche photodiodes adjust the doping concentration to change the electric field profile in the multiplying junction, thus further control the carrier’s energy by accelerating it in high electric field region and releasing its energy in the forms like phonon scattering. The impact ionization threshold energy is changed by adjusting the alloy composition in multi-gain-stage single photon avalanche photodiodes.

The randomness of multiplication process can be reduced by means of suppressing the hole-ionization and enhancing electron-ionization, meanwhile spatially localizing the latter.

The structure of one gain stage of multi-gain-stage single photon APD is designed as figure 1. Electrons can be pre-heated before injecting into impact ionization layer and holes can be injected cold by modulating the electric field profile.The low-field carrier relaxation regions in which holes and electrons lose their accumulated energy by random phonon scattering are used to separate the gain stages. Thus, the holes injected cold cannot imapact ionize.Compared to two-carrier ionizations, ionizations only including electron-ionization decrease the number of impact ionization chains, furthermore narrow the gain distribution. Therefor, the multi-gain-stage single photon avalanche photodiodes suppress the excess noise.

The material of high-field multiplication layers in this structure is Al_0.336In_0.523Ga_0.141As, which lattice match with In_0.52Al_0.48As and In_0.53Ga_0.47As and are is composed by of 70% In_0.52Al_0.48As and 30% In_0.53Ga_0.47As.

The impact ionization threshold energies energy of electrons and holes in different materials are shown as Tabel 1.^[7,8,9] Where E _ie are threshold energies of electrons and E _ih are threshold energies of holes.

The material parameters A, B and m to calculated the impact ionization rates in the Eq. 5 are shown in table 2. ^[7,8,9] The results are obtained by fitting the experimental values of ionization coefficients to the model given by Eq. 5.

5 Results and discussion

In the process of calculating dead space and ionization rates, the effect of phonon scattering is taken into consideration, that is the energy of a carrier is reseted to zero after travelling a certain distance.^[2,5,10] In addition, the effect on ionization rates results from the bandedge discontinuity of multilayer structures is ignored.^[11] The results below are obtained in the photodiodes with each gain stage the same as structure in Fig. 1 unless otherwise specified.

Fig. 1 Epitaxial layer structure of an multi-gain-stage single photon avalanche photodiode

图1 级联倍增结光电探测器外延结构

Figure 2 shows the electric filed of multiplication layers and the dead space distribution of a 10-stage APD at an average gain of M=690. Figure 3 plots the non-localized ionization rates and the scattering-aware ionization rates for electrons and holes. Taking phonon scattering in the energy relaxation layers into consideration centralizes the ionization events, electrons can ionize immediately upon entering the high field multiplication layers, meanwhile, the holes ionization rates decrease in each gain stage.

Fig. 2 the electric field and deadspace profile of multiplication layer in a 10-gain-stage APD(M=690)

图2 10级级联倍增APD（M=690）电场和弛豫空间分布

Fig. 3 ionization rates(scattering-aware and no scattering-aware)

图3 离化率（考虑声子散射和未考虑声子散射）

The excess noise factors are calculated therotically for 3-gain-stage, 7-gain-stage and 10-gain-stage APD using DSMT theory, as shown in Fig.4. The result of conventional InAlAs APD with 200nm multiplication width is caculated for comparison. The results show that the excess noise factor of gain stage APD is better than that of conventional APD and the excess noise factor can be reduced significantly by increasing the number of gain stages.

Fig. 4 the excess noise factor in 3-gain-stage, 7-gain-stage, 10-gain-stage APD and conventional InAlAs APD

图4 3级、7级和10级级联APD和常规APD过剩噪声因子

The effects on excess noise of different width of high field multiplication layer and electrons pre-heating layer, doping concentration of charge layer are analyzed and compared, the results are shown in Fig. 5,6,7 respectively.

Fig. 5 the excess noise factor of structure with the width of multiplication layer at 10 nm, 15 nm, 20 nm and 35 nm

图5 倍增层厚度分别为10 nm，15 nm，20 nm和35 nm时过剩噪声因子

Fig. 6 the excess noise factor of structures with width of pre-heating layer at 0 nm, 40 nm and 100 nm

图6 预加热层厚度分别为0 nm，40 nm和100 nm时过剩噪声因子

Fig. 7 The excess noise factor of different doping of charge layer(10-stage)

图7 电荷层不同掺杂浓度时过剩噪声因子（10级）

Figure 5 shows the excess noise predicted by the DSMT numeric model for a 10-stage APD with the width of high field multiplication layer at 10 nm, 15 nm and 35 nm respectively. The numeric results show that the excess noise firstly decreases and then increases with the decreasing of the width of multiplication layer, for example, the excess noise of APD with width at 10 nm is lager than that at 15 nm. In addition, if the multiplication layer is too narrow, the maximum gain can be suppressed.

Fig.6 shows the DSMT numeric model results of excess noise factor for a 10-stage, 20nm-width multiplication layer APD with the width of electron pre-heating layer at 0nm, 40 nm and 100nm respectively. Clearly, the excess noise is larger when the width of electron pre-heating layer is larger at lower gain, the reason is that the dead space of electrons and holes are large in this condition. While the effect of the width of electrons pre-heating layer on the excess noise is small, because the electron dead space is small, in this condition the maximum gain is affected, such as decreasing the width of electron pre-heating layer to zero, the maximum gain is lower than 70.

Figure 7 compares the excess noise of two 10-stage structures, one gain stage of one structure is as Fig.1, the other’s single gain stage is modulating the doping of three charge layer in Fig.1 to 7e17 cm^-3,7e17 cm^-3,8.2e17 cm^-3 and to 5e17 cm^-3,5e17 cm^-3,5.85e17 cm^-3. It is obvious the excess noise factor and the maximum gain are affected.

Figure 8 shows the MDSMT theoretical numeric results of excess noise of a 10-stage cascaded multiplier APD with carrier initial energy of 0, 0.2 times and 1 times of threshold energy, respectively. The excess noise will reduce when the intial energy is taken into consideration, because the initial energy furthermore localize the carriers impact ionization.

Fig. 8 The excess noise factor with the initial energy of 0, 0.2Eth and Eth

图8 载流子初始能量为0，0.2Eth和Eth时过剩噪声因子

Figure 9 shows the results from DSMT numeric model, McIntyre model formula and Van Vliet model formula, DSMT numeric result of a 10-gain-stage APD with each gain stage as Fig.1 is approximately in accordance with results from McIntyre formula at k=0.086 and M>100. Furthermore, it is in accordance with the results of a 10-stage cascaded multiplier APD with single gain stage as Fig.1 from Van Vliet formula at k=0.086.

Fig. 9 Results from DSMT numeric theory(10-stage), Van Vliet model(10-stage,k _s=0.086) and McIntyre model(k=0.086)

图9 DSMT数值理论模型（10级），Van Vliet模型（10级，ks=0.086）和McIntyre 模型（k=0.086）结果

By modulating the width of multipliction layer of Fig.1 to 17 nm, the width of pre-heating layer to 30 nm, the doping of charge layer to 6e17cm^-3,10e17cm^-3,9.4e17cm^-3, a 10-stage cascaded multiplier APD is obtained, the excess noise factor of which is shown in Fig.10 and it is in accordance with the result from Van Vliet model at k _s =0.057.

Figure 10 results of 10-stage DSMT(by modulating the structure in Fig.1, the width of multiplication layer at 17 nm, the width of pre-heating layer at 30 nm, the doping of charge layer at 6e17cm^-3,10e17cm^-3,9.4e17cm^-3 ) and Van Vliet(10-stage,k _s=0.057)

图10 DSMT 10级级联倍增模拟（模拟结构如图1，倍增层厚度为17 nm，预加热层厚度30 nm，电场层掺杂浓度分别为6e17cm-3,10e17cm-3,9.4e17cm-3）和Van Vliet模拟（10级，ks=0.057）

6 Conclusions

The DSMT theoretical numeric results indicate the existence of better width values of high field multiplication layer and electron pre-heating layer and better doping values of charge layer to optimize the excess noise and multiplication gain. By modulating the structure parameters, a 10-stage cascaded multiplier APD is obtained, the excess noise factor of which is in accordance with the result from Van Vliet model at k _s =0.057. DSMT theoretical numeric model can guide the structure design of the multi-gain-stage single photon photodiodes.

Reference