Spatial description theory of narrow-band single-carrier avalanche photodetectors

Runzhang Xie; Runzhang Xie; Qing Li; Qing Li; Peng Wang; Xiaoshuang Chen; Wei Lu; Huijun Guo; Lu Chen; Weida Hu; Weida Hu; Weida Hu

doi:10.1364/OE.418110

1. Introduction

Avalanche photodiodes (APDs) are being widely applied to many areas, which demand low power signal detections, such as remote sensing [1], LIDARs [2], quantum communications [3], and aerospace applications [4,5]. In the band of mid-wavelength infrared and long-wavelength infrared, HgCdTe is an important material for detection for its high quantum efficiency and matured material growth technology [5,6]. HgCdTe APDs have an ideal performance of high internal multiplication gain and low excess noise factor approaching unity owing to its single-carrier multiplication character [7–9]. However, an analytic bridge from connecting quantum foundation to avalanche process composed of numerous random events is still absent [10,11]. Based on the conciseness of the dynamic of the electron in the conduction band of HgCdTe and single-carrier multiplication character, an analytic theory, as a bridge, based on quantum mechanics without any parameter fitting is presented. Discussion about the abnormal raising of excess noise factor (ENF) in single-carrier avalanche is presented.

Avalanche is a series of random events of impact ionization, of which the probability varies with spatial position and time. Beside methods directly dealing with the microscopic mechanism, such as the Monte Carlo method and solving Boltzmann equation numerically [12,13], a reduction of variable is needed. There are two obvious reductions by integrating over spatial position or over time. Most analytic theories belong to the first category by averaging physical quantities over a macroscopic region, of which the spatial position is a macroscopic variable by integrating over a region which is much larger than the scale of the microscopic mechanisms [14,15]. In other words, the spatial position in these theories is a phenomenological variable characterizing the times of impact ionization events. In the device with a thin avalanche region approaching the scale of the microscopic mechanisms, the conventional approaches integrating over spatial coordinate is inappropriate. In this paper, we present the second reduction method and a time average integrating over the time interval much larger than the microscopic mechanisms and comparable with the time interval sensitive to measurement, such as the sampling rate of the measuring equipment, is performed. By taking the time average, the exact value of time becomes insignificant, which changes the physical picture fundamentally, such as the result is invariant when we modify the sequence of events by permuting the order of two existing events, even if there was a causal relation between them. Thus, we could call the first type of method as temporal avalanche description or sequential avalanche description and the second as spatial avalanche description.

History-dependent theory of avalanche solves PDFs of electrons and holes iteratively [16,17]. In history-dependent theory, the energy of the carrier E is determined by the generation position ${x_0}$ and the present position of carriers ${x_1}$, which introduces the collision rate as the function of energy into the model. The dead space is the direct outcome of this modification of the avalanche model [18]. However, the one-to-one correspondence between $({{x_1},{x_0}} )$ and E is a strong assumption, which is equivalent to that the system is one-dimensional and no energy is acquired by cold electrons in the impact ionization process and no energy of hot electrons is left after the impaction ionization process. In our generalized theory of history-dependent theory, we present corrections to describe carriers in three-dimensional system by presenting an overall PDF of the appearance of carriers at a specific ${x_1}$. To include the probability distribution of the separation of energy after impact ionization into consideration, the concept of shadow area is proposed.

In HgCdTe and some other materials, such as InAs, most impact ionization processes take place in the central valley (Γ-valley) and the dynamics around satellite valleys (L-valley and X-valley) could be neglected, as is assumed in this work [10,19]. For materials that the energy of impact ionization occurs are deep into the highly nonuniform band structure, such as GaAs and InP [10], and for materials more energy band should be taken into consideration, such as the case with the bandgap spin-orbit splitting resonance effects [20], under the assumption that the dispersion relations ${E_m}(p )$ of all energy bands under consideration are isotropic, this work could be directly generalized by separating the domain I_m of ${E_m}(p )$ into several monotonic intervals I_mn and defining ${E_{mn}}:{I_{mn}} \to {\mathbb R}$ to ensure the one-to-one correspondence between p and E within each ${E_{mn}}$, which is utilized in both the dynamics in the spatial description theory and the generalized theory of history-dependent theory. The transition behavior should be also generalized to the discussion over indexes of both initial and final states.

The paper is organized as follows. In Sec. 2, we present the theory of dynamics in the spatial description theory, where nonparabolicity of the conduction band, simplification of dynamics in the 3-dimensional system with axial symmetric, and the influence of acute angle scattering on the motion of the electron. In Sec. 3, we discuss the scattering rates of the impaction ionization process and POP scattering process in the spatial description, of which the corrections must be present due to the high alloy scattering in mid-infrared HgCdTe in the spatial description. In Sec. 4, we describe the generalized history-dependent theory and compare its results with the well-established Monte Carlo method and experimental result in others' work, as well as our experimental result. We present our concluding remark in Sec. 5.

2. Dynamics in the spatial description theory

All scattering processes are described by the quantum theory to avoid any fitting parameters in our work. Impact ionization rate is provided by inverting the Auger process proposed by many works [15,21]. Acoustic phonon scattering and intervalley phonon scattering are small in the mid-infrared HgCdTe APDs and neglected in this paper [11,15,22–25]. Polar optical phonon (POP) scattering contributes to both energy and momentum relaxation, which is the only mechanism causing energy relaxation [26,27]. The time reversal symmetry is assumed before the discussion of the energy relaxation under backward scattering correction is discussed in Sec. 3. Following F. Ma’s work [28], instead of the two modes (${\omega _{\beta ,1}} = 17\;\textrm{meV}$ and ${\omega _{\beta ,2}} = 20\;\textrm{meV}$), one equivalent mode (${\omega _\beta } = 19\;\textrm{meV}$) is considered. Alloy scattering and ionized impurity scattering considered in this paper contribute only to momentum relaxation [29–31]. The scattering-potential model is applied to describe the alloy scattering and the alloy scattering potential is 1.5 eV for electrons [32–34]. The Conwell-Weisskopf approach is used to describe ionized impurity scattering in the multiplication layer and transition to Yukawa potential in the p-region and n⁺-region with the increase of carrier density derived from semiconductor equations [35,36].

2.1 Spatial description theory

As a stochastic process, the evolution of the avalanche process is determined by the current position x_n(t) and the momentum p_n(t) of all carriers in the avalanche region under the classical transport theory, where n denotes the n^th carrier in the avalanche region. Assuming positions of all carriers are different, n could be eliminated by x_n(t) and any physical quantity A could then be expressed as A(x,t), including the momentum and the impact ionization probability distribution discussed below. In the temporal description, the momentum relaxation time ${\tau _m}({{{\textbf p}_0}} )$ for non-degenerate semiconductor could be expressed by

(1)$$\frac{1}{{{\tau _m}({{{\textbf p}_0}} )}} = \sum\limits_{{\textbf p^{\prime}}} {S({{{\textbf p}_0},{\textbf p^{\prime}}} )[{1 - ({{{p^{\prime}} / {{p_0}}}} )\cos \alpha } ]}, $$

where $S({{{\textbf p}_0},{\textbf p^{\prime}}} )$ is the transition rate from the initial state ${{\textbf p}_0}$ to the final state ${\textbf p^{\prime}}$. $\alpha$ is the polar angle between the initial state and the final state. $p^{\prime}$ and ${p_0}$ are the modulus of ${{\textbf p}_0}$ and ${\textbf p^{\prime}}$, respectively. For a group of N carriers injected at ${\textbf x} = {{\textbf x}_0}$ and $t = {t_0}$ with momentum ${{\textbf p}_0}$, the average momentum of these carriers $\left\langle {{\textbf p}(t )} \right\rangle $ decays with the lifetime ${\tau _m}({{{\textbf p}_0}} )$. By rewriting this average into the form of A(x,t), we have

(2)$$\left\langle {{\textbf p}(t )} \right\rangle = \frac{1}{N}\int\limits_{\Omega (t )} {{\textbf p}({{\textbf x},t} )d{\textbf x}}, $$

where $\Omega (t )$ is the region occupied by carriers. As can be seen, the minimum volume of this region is 0 at $t = {t_0}$ and increases with the relaxation of momentum. It should be noticed that the temporal description could not accurately describe the evolution of carriers when the diameter of $\Omega (t )$ is comparable with the width of the avalanche region.

In practical infrared photodetectors, a period of integral time is necessary before the signal processing, which is much larger than the characteristic time scale of the intraband transition and impact ionization. Thus, in the spatial description, the average over spatial coordinates is replaced by the average over time. We have

(3)$$\left\langle {{\textbf p}({\textbf x} )} \right\rangle = \frac{1}{N}\int_{{t_0}}^{{t_0} + \delta t} {{\textbf p}({{\textbf x},t} )dt}, $$

where $\delta t$ is a time much larger than the characteristic time scale of the intraband transition and impact ionization. The momentum relaxation process is discussed by the equation of motion of hot electrons in Sec 2.3.

The difference between the classical temporal description and the spatial description theory could be illustrated in the example shown in Fig. 1(a). There are two electrons generated at the origin $({{x_0},{t_0}} )$. Electron I exhibits a scattering at $({{x_1},{t_1}} )$ inversing its velocity. Electron II arrives at $({{x_2},{t_2}} )$ without scattering. For temporal avalanche description, the scattering leads to the system’s momentum totally relaxed at $t_1^ + $, where $t_1^ + $ is the limit at ${t_1}$ from the right. However, for spatial avalanche description, the momentum of the system at $x_1^ + $ is $2\vec{p}$ as if no scattering occurs. The motion in the range ${x_0} < x < {x_1}$ is folded and extra correction must be presented for physical quantities, such as the scattering rate and equation of motion of the hot electron. This correction is denoted as backward scattering correction because the folded path is due to scattering with angle $\theta \ge {\pi / 2}$.

Fig. 1. Modification of physical quantities. (a) Illustration of two electrons in the electric field. Blue dashed line shows the integration over time for spatial description. Red dot-dash lines show the integration over spatial coordinate for temporal description. For the system composed of the two electrons, the expectation value $\left\langle A \right\rangle $ is degraded to ${A_1} + {A_2}$. (b) Scattering in the pull-back motion starts at ${x_1}$. Cyan curve shows that the trajectory of electron exhibits a scattering at $({{x_2},{y_2},{z_2}} )$. Red curve is the complementary trajectory corresponding to dashed curves ${O_1}$ and ${O_2}$ which is determined by cyan curve before and after the scattering at $({{x_2},{y_2},{z_2}} )$ if there were no scattering. (c) Velocity overshoot and saturation of electron in the nonparabolic conduction band result from the redistribution between components of momentum in spatial description. Corresponding device parameters used throughout in this work are shown in Table 1. Black line shows the velocity saturation due to nonparabolicity when V_bias= 10 V, which is faster than saturation due to momentum redistribution (red line). Black dashed line is the theoretical limit velocity.

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Table 1. Structure and key parameters of the Hg_0.7Cd_0.3Te n⁺/n^-/p homojunction device for corrections and simulations. The temperature is 77 K.

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Assuming there is only one obtuse angle scattering occurred, as is shown in Fig. 1(a), the electron scattered at $({{x_1},{t_1}} )$ is pulled back by the strong electric field. In the spatial description, the point $({{x_1},{t_1}} )$ the scattering occurred, and the point $({{x_1},{{t^{\prime}_1}}} )$ the electron is pulled back could be connected together. Then, the obtuse angle scattering is split into a compound acute angle scattering and a pull-back process, which could be dealt with separately.

2.2 Theorems of expectation of isotropic scattering

As the consequence of time reversal symmetry, theorems of expectation of isotropic scattering could be derived. Then, we could conclude that further scatterings in the pull-back process do not change the expectation of any macroscopic physical quantities. Thus, all scattering causing momentum relaxation in the pull-back process should be ignored. Our assumption that only one obtuse angle scattering occurred is relaxed.

Theorem 1: If all scattering processes are isotropic, any scattering in the pull-back process in a strong homogeneous electric field does not change the expectation of any macroscopic physical quantities.

Proof. Assuming the electron exhibit an obtuse angle scattering at $({{x_1},{y_1},{z_1}} )$ as is shown in Fig. 1(b). We assume that the probability of the electron scattered to the angle ${\alpha _1}$ as $f({{\alpha_1}} )d\Omega $, where $d\Omega $ is the solid angle of the scattered electron occupied. The electron moves along the trajectory ${O_1}$ in the electric field before the next scattering. The angle between the direction of the momentum of the electron and the direction of the electric field $\alpha ({t - {t_0},{\alpha_1}} )$ at any time t is definite when the intensity of the electric field and the trajectory is specified, where ${t_1}$ and ${\alpha _1}$ is the time and the scattering angle the first scattering at $({{x_1},{y_1},{z_1}} )$, respectively. Suppose the electron is scattered again at $({{x_2},{y_2},{z_2}} )$. The probability of electron scattered to angle ${\alpha _2}$ is

(4)$$f({{\alpha_2} - \alpha ({t - {t_1},{\alpha_1}} )+ \delta \alpha } )= f({\Delta \alpha } ). $$

where $\delta \alpha $ is the extra difference of angle derived by the change of directions of the orbital plane for trajectory ${O_1}$ and ${O_2}$, respectively. Assuming the electric field is strong enough, the energy relaxation could be ignored and the conservation of energy is satisfied. Thus, a definite orbit ${O_2}$ could be found corresponding to the process that an electron exhibits an obtuse angle scattering at $(x_1,y^{\prime}_1,z^{\prime}_1)$ with the angle ${\alpha ^{\prime}_1}$ and passes $({{x_2},{y_2},{z_2}} )$ with the angle ${\alpha _2}$ in the of the orbital plane of ${O_2}$, where we have assumed the one to one correspondence between vectors in the tangent space at the given point and trajectories passing through that point, which is fulfilled by the condition that all the scatterings are within the same energy valley. Noticing that all scattering is axisymmetric along the direction of momentum before scattering, the probability density for the complementary trajectory is shown in Fig. 1(b) from ${O_2}$ to ${O_1}$ is

(5)$$f({\alpha ({t - {t_1},{\alpha_1}} )- {\alpha_2} - \delta \alpha } )= f({ - \Delta \alpha } )= f({\Delta \alpha } ). $$

Assuming $\{{{O_2}} \}$ as the set of all trajectories which passes $({{x_2},{y_2} + \delta y,{z_2} + \delta } )$ with the angle ${\alpha _2}$ in the of the orbital plane of ${O_2}$, where $\delta y$ and $\delta z$ are any real numbers. Then, the net transition rate from ${O_1}$ to $\{{{O_2}} \}$ could be written as

(6)$$\begin{aligned} P({\{{{O_2}} \},{O_1}} )&= \int\limits_{{t_0}}^{{t_m}} {2[{f({{\alpha_1}} )- f(\alpha^{\prime}_1)} ]S({E(t )} )f({\Delta \alpha (t )} )dt} \\ &\equiv p({\{{{O_2}} \},{O_1}} )[{f({{\alpha_1}} )- f(\alpha^{\prime}_1)} ], \end{aligned}$$

where ${t_m}$ is the time satisfying $\alpha (t_m,\min \{\alpha_1,\alpha^{\prime}_1 \} )= {\pi / 2}$. $S({E(t )} )$ is the scattering rate when the energy is $E(t )$. $p({\{{{O_2}} \},{O_1}} )$ is a coefficient about ${\alpha ^{\prime}_1}$ and ${\alpha _1}$. Taking isotropic scattering into consideration, we have $f({{\alpha_1}} )- f(\alpha^{\prime}_1)= 0$. Thus, $P({\{{{O_2}} \},{O_1}} )= 0$. This completes the proof. □

Theorem 2: If isotropic scattering is dominant in all scattering mechanisms, any scattering in the pull-back process is equivalent to the reducing of anisotropic scattering and increasing of isotropic scattering.

Proof. In the case scattering contains isotropic part and anisotropic part, we could rewrite

(7)$$p({\{{{O_2}} \},{O_1}} )= {p_{iso}} + {p_{aniso}}({\{{{O_2}} \},{O_1}} ), $$

where ${p_{iso}}$ does not depend on ${\alpha ^{\prime}_1}$ and ${\alpha _1}$, and where

(8)$$\int\limits_{{S_1}} {\int\limits_{{{S^{\prime}_1}}} {{p_{aniso}}({\{{{O_2}} \},{O_1}} )d\Omega d\Omega ^{\prime}} } = 0$$

is satisfied. The integration takes over the left hemisphere in the coordinate system defined in Fig. 1(b). Because isotropic scattering is dominant, we have ${p_{iso}} \gg {p_{aniso}}({\{{{O_2}} \},{O_1}} )$. Thus, leaving only the first order of the net transition rate, we have $P({\{{{O_2}} \},{O_1}} )\textrm{ = }{p_{iso}}[{f({{\alpha_1}} )- f(\alpha^{\prime}_1)} ]$. The evolution of $f({{\alpha_1}} )$ could be written as

(9)$$\begin{aligned} \frac{{df({{\alpha_1}} )}}{{dt}} &={-} \int\limits_{{S_1}} {{p_{iso}}[{f({{\alpha_1}} )- f({{{\alpha^{\prime}_1}}} )} ]d\Omega } \\ &={-} 2\pi {p_{iso}}[{f({{\alpha_1}} )- {f_{average}}} ], \end{aligned}$$

where ${f_{average}}$ is the average scattering probability. Thus, when ${p_{iso}} \gg {p_{aniso}}({\{{{O_2}} \},{O_1}} )$ is satisfied, the scattering probability tends to the average scattering probability exponentially. This completes the proof. □

According to the angle $\theta $ between directions of momentum before and after scattering, the scattering could be separated into three parts: small acute angle scattering with $\theta \ll {\pi / 2}$, large acute angle scattering $0 \ll \theta < {\pi / 2}$, and obtuse angle scattering $\theta \ge {\pi / 2}$. In the system discussed above, alloy scattering is isotropic. POP scattering is a small acute angle scattering for hot electrons. Ionized impurity scattering is anisotropic scattering with small acute angle scattering dominant, of which the anisotropic part of obtuse angle scattering is small [24,35–37]. According to the analysis of Conwell and Weisskopf, the impact parameter is greater than ${b_{\min }} = {1 / {2N_I^{ - {1 / 3}}}}$, which is the half-space between impurities. Substituting the impact parameter into the Rutherford scattering equation

(10)$$b = \frac{{Z{q^2}}}{{8\pi {\varepsilon _0}{\varepsilon _r}E}}\cot \left( {\frac{\alpha }{2}} \right), $$

we could come to that the minimum scattering angle for an electron with energy ${E_g}$ is smaller than 0.05 rad when the doping density is less than $3.59 \times {10^{17}}$ cm⁻³,where Z, q, ${\varepsilon _0}$, ${\varepsilon _r}$, E, $\alpha $ are the number of charge on impurity atoms, constant of elementary charge, the permittivity of vacuum, low-frequency dielectric constant, the energy of the electron in the conduction band, and the deflection angle of the scattering, respectively [38]. Thus, the small acute angle scattering is dominant in all three parts of scattering. For example, the doping density in the multiplication area is $5 \times {10^{14}}\;\textrm{c}{\textrm{m}^{ - 3}}$ and the large acute angle scattering and obtuse angle scattering share 0.31% of scattering events. Leaving along small acute angle scattering, ionized impurity has a smaller scattering rate and a larger part of isotropic scattering.

2.3 Equation of the motion of hot electrons

The correction of the dispersion relation of conduction band electron in a strong electric field should be taken into account in a narrow bandgap material, such as mid-infrared HgCdTe in our work [39,40]. The satellite valleys are relatively insignificant in mid-infrared HgCdTe [11,41]. Thus, the lowest order correction of constant nonparabolicity coefficient ${\alpha _c}$ is suitable [42,43]. For simplicity, two effective masses are defined from the relation between momentum and velocity, and between momentum and energy: $\vec{p} = {m^{(P )}}{v_g}(k )$ and ${\vec{p}^2} = 2{m^{(E )}}E$, where ${m^{(P )}}\textrm{ = }m_c^\ast ({1 + 2{\alpha_c}E} )$ and ${m^{(E )}} = m_c^\ast ({1 + {\alpha_c}E} )$. ${v_g}(k )$, $\vec{p}$, and E are group velocity, momentum and energy of the electron, respectively. It could be found that the ratio between the two effective masses $\beta \textrm{ = }{{{m^{(E )}}} / {{m^{(P )}}}}$ tends to $1/2$ when $E \to \infty $. Assuming the electron is in the external electric field ${E_x}$ along the x-direction, we have ${{d{p_x}} / {dt}} = \beta e{E_x}$, which is the modified Newton’s equation for electrons in the nonparabolic conduction band. In the case of the uniform electric field, we have ${{d{p_x}} / {d\hat{t}}} = \beta $, where $\hat{t} \equiv eEt$.

Acute angle scattering and compound acute angle scattering could be combined together and two conclusions is drawn: (I) Only one acute angle scattering need to be considered in transport and the effective scattering rate is the superposition of acute angle scattering and obtuse angle scattering; (II) Momentum is separated into ${p_x}$ and ${p_n}$ which are parallel to and perpendicular to the electric field, and the momentum transfer from ${p_n}$ to ${p_x}$ is not random in direction anymore and its statistical average is nonzero in contract with classical transport theory. The numerical result of electron transport injected into n^--region with $E = 0.11\;\textrm{eV}$ taken momentum redistribution between ${p_x}$ and ${p_n}$ into consideration is shown in Fig. 1(c). With the increase of bias voltage, the velocity saturation due to nonparabolicity is faster than the saturation of momentum redistribution and velocity overshoot is observed, which is different from the classical velocity overshoot due to the inequality of momentum relaxation and energy relaxation.

The equation of motion of hot electrons in classical transport theory is [42,44,45]

(11)$$\frac{d p}{d t}=e E-\frac{p}{\tau_{m}}, \\ \frac{d \varepsilon}{d t}=e E \frac{p}{m^{(P)}}-\frac{\varepsilon-\varepsilon_{0}}{\tau_{\varepsilon}},$$

where ${\tau _m}$ and ${\tau _\varepsilon }$ are the momentum and energy relaxation time. $\varepsilon $ and ${\varepsilon _0}$ are the energy of the electron and average energy of the cold electron in the conduction band. From the dispersion relation in the nonparabolic conduction band derived from equation of motion

(12)$$\frac{{d\varepsilon }}{{dt}} = \beta \frac{p}{{2{m^{(E )}}}}\frac{{dp}}{{dt}}, $$

the effective dissipative force in the equation of energy evolution could be written as $- {p / {\beta {\tau _\varepsilon }}}$. Thus, the equation of $\varepsilon $ could be changed into an equation of the norm of the momentum.

Assuming isotropic scattering, the momentum transferred from ${p_n}$ to ${p_x}$ is always along the direction of electric field force, and an energy transfer term ${\pm} {3^{ - {1 / 2}}}{{{p_n}} / {{\tau _m}}}$ between ${p_x}$ and ${p_n}$ should be added. Thus, the equation of motion could be written as

(13)$$\frac{d p_{x}}{d t}=e E \beta-\frac{p_{x}}{\tau_{m}}-\frac{p_{x}}{\beta \tau_{\varepsilon}}+\frac{\sqrt{3}}{3} \frac{p_{n}}{\tau_{m}}, \\ \frac{d p_{n}}{d t}=\frac{p_{x}}{\tau_{m}}-\frac{p_{n}}{\beta \tau_{\varepsilon}}-\frac{\sqrt{3}}{3} \frac{p_{n}}{\tau_{m}} $$

Assuming $\tan \theta \equiv {{{p_n}} / {{p_x}}}$, and noticing

(14)$$\frac{{dt}}{{dx}} = \frac{1}{{{v_x}}} = \frac{{{m^{(P )}}}}{{{p_x}}} = {m^{(P )}}\frac{{1 + {{\tan }^2}\theta }}{p}, $$

we could derive the equation about $\theta $,

(15)$$\frac{{d\theta }}{{dx}} = \frac{1}{{{x_l}}} + \frac{{\tan \theta }}{{{x_l}}}\left[ {1 - \frac{{\sqrt 3 }}{3}({1 + \tan \theta } )- \frac{{eE{x_l}}}{{2\varepsilon }}\sec \theta } \right], $$

where a characteristic length ${x_l} = {{{\tau _m}p} / {{m^{(P )}}}}$ about momentum transfer is defined. It should be noticed that terms about ${\tau _\varepsilon }$ are eliminated, which permit us to consider the direction and the norm of momentum separately.

The above equation has a singularity at the position that the norm of the momentum of the electron is zero. In the discussion of impact ionization, we could constrain the energy larger than the average energy of cold electrons in the conduction band and we have ${x_l} \approx {{{\tau _m}} / {\sqrt {2m_c^\ast {\alpha _c}} }}$. Then, denoting $\hat{x} = {x / {{x_l}}}$, in the uniform electric field, we have $\varepsilon = eE({x - {x_0}} )$ and the equation of motion could be written as

(16)$$\frac{{d\theta }}{{d\hat{x}}} = 1 + \tan \theta \left[ {1 - \frac{{\sqrt 3 }}{3}({1 + \tan \theta } )- \frac{{\sec \theta }}{{2\hat{x}}}} \right]. $$

This form of the equation of motion does not depend on momentum relaxation time and electric field intensity explicitly. From the form of the equation, we could directly derive the asymptotic behavior that the saturation value of $\theta $ is ${\theta _{sat}} = {\pi / 3}$ when $\hat{x} \to \infty $. Thus, the velocity saturation is at ${{{v_{sat}}} / 2}$ regardless of the value of momentum relaxation time and electric field intensity, which is due to nonparabolicity of the conduction band. As is shown in Fig. 1(c), there is velocity overshoot when the bias voltage is high enough. For example, the distance that the electron accelerates from 0 to $0.9{v_g}$ is 62.5 nm and ${x_l} \approx 285nm$.

3. Corrections in spatial description theory in mid-infrared HgCdTe

A full quantum description of impaction ionization follows Kinch’s approach which utilizes the Auger recombination theory by reversing the direction of the Auger process [21,22]. Assuming the total angular momentum of the system composed of the hot conduction band electron and the impacting valence band electron invariant in impact ionization, an external overall factor of $1/2$ is added to the product of the density of states borrowed from the semiconductor theory. As is shown in [15], the impact ionization rate could be expressed as

(17)$$ P=\frac{4 \pi^{2}}{h} \sum^{*}_{E_{c}} \sum^{*}_{E_{v}} \frac{1}{2}|H|^{2} g_{c}\left(E_{1}^{\prime}\right) g_{c}\left(E_{2}^{\prime}\right) g_{v}\left(E_{2}\right) $$

where ${g_c}$ and ${g_v}$ are the density of states in the conduction band and valence band, respectively [46]. ${|H |^2}$ is the interaction term of Auger recombination [21,47]. Replacing the summation by integral, and substituting the interaction term of Auger recombination [15,48], we have

(18)$${P_{ii}}({{E_n}} )= \frac{{4{\pi ^2}{q^4}V{E_g}m_c^\ast m_v^\ast \sqrt {8m_c^\ast {E_g}} }}{{{\varepsilon _0}^2{\varepsilon ^2}{h^6}}}{|{{F_1}{F_2}} |^2}\int\limits_1^{{E_n}} {g({x,{E_n}} )dx}, $$

where

(19)$$g({x,{E_n}} )= \frac{{\sqrt {{{[{2({{E_n} - x} )+ 1} ]}^2} - 1} [{2({{E_n} - x} )+ 1} ]\sqrt {{{({2x - 1} )}^2} - 1} ({2x - 1} )}}{{{{({{{{x^2} + {h^2}} / {8m_c^\ast {E_g}{\pi^2}L_D^2}}} )}^2}}}. $$

$\varepsilon $, $m_v^\ast $, ${L_D}$, V, ${F_1}{F_2}$, and $g({x,{E_n}} )$ are the static dielectric constant, effective mass in the valance band, Debye length, system volume, overlap integral of the Auger recombination and the density of state of impact ionization, respectively. ${E_n} = {E / {{E_g}}}$. The energy distribution after impact ionization is

(20)$$f({x,{E_n}} )= {{[{\hat{g}({x,{E_n}} )+ \hat{g}({{E_n} + 1 - x,{E_n}} )} ]} / 2}, $$

where $\hat{g}({x,{E_n}} )= {{g({x,{E_n}} )} / {\int_1^{{E_n}} {g({x,{E_n}} )} }}$ is the normalized density of states. Two terms on the right-hand side are for hot conduction electron and cold impacting valence electron, respectively.

If the energy of the electron exceeds ${E_g}$ when it was scattered, impact ionization is nonzero in the pull-back process. With the theorem of the expectation of isotropic scattering, the momentum in the pull-back could be directly described by ${{d{p_x}} / {d\hat{t}}} = \beta $ without any further assumption being introduced. Denoting the momentum for an electron with energy E and scattering angle $\theta $ as ${p_x}(\hat{t};E,\theta )$, the probability of impact ionization in the pull-back process is written as

(21)$${P_r}({E,\theta } )= \textrm{exp} ({B[{{P_{ii}}} ]({E,\theta } )} ), $$

where $B[F ]$ is a functional of any function F with variables E and $\theta $ defined as

(22)$$B[F ]({E,\theta } )={-} \frac{2}{{eE}}\int_0^{{{\hat{t}}_x}} {F\left[ {\frac{{{p_x}{{({\hat{t};E,\theta } )}^2}}}{{2{m^{(E )}}{E_g}}} + \frac{E}{{{E_g}}}{{\sin }^2}\theta } \right]d\hat{t}}, $$

and where ${\hat{t}_x}$ is the solution of ${p_x}(t;E,\theta ) = 0$. Thus, the correction of impaction due to obtuse scattering is written as

(23)$$P_{ii}^{(r )}(E )= {P_{back}}(E )\int_0^{{\pi / 2}} {{P_r}({E,\theta } )\sin \theta d\theta }, $$

where ${P_{back}}$ is the total obtuse angle scattering rate of all scattering mechanisms, as is shown in Fig. 2(a). With the increase of the electric field, the correction is reduced due to the decreased time spending in the pull-back process.

Fig. 2. Corrections in spatial description theory in mid-infrared HgCdTe. (a) Nonparabolic band modified POP scattering rate (black) and correction term due to backward scattering in respective electric field intensity. (b) Energy of electrons at different positions generated at different injecting positions. (c) Forward impact ionization rate and backward correction terms in respective electric field intensity. (d) Probability density function of impact ionization of electrons generated at different injecting positions.

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As is shown in Fig. 2(b), correction of energy relaxation in the pull-back process cannot be neglected. The influence of nonparabolicity on POP scattering is derived by substituting the dispersion relation into Fermi’s golden rule. Noticing that the case that impact ionization occurred in the pull-back process should be excluded, the energy dissipation to the lowest order is written as

(24)$$\Delta {E_{pop}} ={-} \hbar {\omega _\beta }{P_{back}}(E )\int_0^{{\pi / 2}} {B[{{P_{pop}}} ]({E,\theta } )({1 - {P_r}({E,\theta } )} )\sin \theta d\theta }, $$

where ${P_{pop}}(E )$ is the POP scattering rate in the nonparabolic conduction band. With the increase of electron energy, the part of the electron could back to the point obtuse scattering occurred is reduced, and the energy correction is decreased.

Combining backward corrections, the energy, and impact ionization rate for electron generated at the different injecting position at the given electron position when the bias voltage is −4V is shown in Fig. 2(c) and (d). Energy is saturated for electrons to move enough distance in n^--region. Dead space could be observed directly in Fig. 2(d).

4. Generalized history-dependent theory

Reducing the dark current and the ENF are vital to HgCdTe APDs [7,49]. The analysis of the reduction of the dark current and ENF by introducing new structures relies on accurate and fast simulation technologies [50,51]. Analytic methods include one or more fitting factors based on experimental results, which is beneficial to the fast repetition of the experiment data and separation of factors acting together on the experiment data. By using fitting factors, some microscopic processes may be binding together and the problem of separation of factors itself may be an ill-posed problem due to the similarity of behavior between different factors. Monte Carlo method has the advantage of reducing the dependence on fitting factors by discussing the microscopic processes directly [10,11,28]. However, there are two limitations the Monte Carlo method exhibits. Firstly, the Monte Carlo method is hard to solve problems at the device scale independently, especially for optimizing device structure, which is because the Monte Carlo method is based on random sampling, which prevents reusing the results of previous steps. Secondly, the Monte Carlo method is based on statistical averaging. Physical quantities, such as distribution of impact ionization and the ENF, are based on statistical estimation. History-dependent theory presents an insightful and compact framework that determines electron energy by its position and generation position, which could be generalized to the spatial description [16,17]. However, the assumption that the energy of electron is determined by the generation position and current position limits its direct generalization to the case taking different final states of both cold and hot electrons into consideration. As is shown in Fig. 3, the discussion on the final states is presented by introducing the concept of shadow area. It should be noticed that many dark-current related carriers are generated in the junction area. Under the analysis of Fig. 4, the avalanche of this part of carrier should be isolated out, which has not been reported by other works as far as we know. In this section, we present the generalized form of history-dependent theory taking the shadow area and the case of partially avalanche.

Fig. 3. Illustration of shadow area and calculated shadow area correction. (a) Impact ionization event at (x₁, t₁). (b) Schematic of shadow path. (c-d) Calculated $P_{route}^{(n )}(x^{\prime}_0,x_1,x_0)$ and ${\alpha ^{(n )}}(x^{\prime}_0,x^{\prime}_1,x_0)$ with ${V_{bias}} ={-} 4\,\textrm{V}$, n = 2, and ${x_0} ={-} 0.5\;\mathrm{\mu }\textrm{m}$. (e-g) Impact ionization generation rate for different times of generations. Corresponding bias voltage is $- 4\,\textrm{V}$.

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Fig. 4. Dark current analysis of serval low ordered cases. (I) Electron injected from p-contact. (II-IV) Electron-hole pair generated at p-region, n⁺-region, and n^--region, respectively. (V-VI) Example of cases including more electron-hole pair generation. Right half shows the simplification for the avalanche process. Because Hg_0.7Cd_0.3Te exhibit single carrier avalanche, Case (III) and (VI) does not lead to avalanche. Case (IV) only leads to avalanche in the region to the right of the position electron-hole pair generated.

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Fig. 5. Calculated and experimental results gain and ENF of planar homojunction p-i-n Hg_0.7Cd_0.3Te avalanche photodiodes. (a) Light current gain and leakage current gain. Experimental gain and Monte Carlo simulation gain are the result in work [11] for comparison. (b) Probability of impact ionization for each time of impact ionization. Red part represents impact ionization occurs. Blue part represents the part no further impact ionization occurs. (c) ENF of light current and leakage current. Experimental and Monte Carlo simulation result is the result in work [11] for comparison. The experimental result from Shanghai Institute of Technical Physics (SITP) shows the measured typical ENF. (d) Spatial distribution of different times of impact ionization when the bias voltage is 4V. (e) Dark current and dominant components [52]. BTB for band to band tunneling. SRH for Shockley-Read-Hall recombination. Radiative recombination and Auger recombination are too small to be shown in the figure. The distribution of normalized dark generation is shown in the inset. (f) ENF considers both light and leakage current noise. Ratio between light and leakage current is 4×103 and 4×104 for the larger and smaller dark current, respectively, when V_bias = 0.6V.

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Survival probability ${P_{se}}$ is characterized by ${{d{P_{se}}} / {dt}} ={-} P_{ii}^{tot}{P_{se}}$, where $P_{ii}^{tot} = {P_{ii}} + P_{ii}^{(r )}$. To use the energy distribution after impact ionization by Fermi’s golden rule, the concept of shadow area is introduced to the history-dependent avalanche theory, because the energy at the generation position is a random variable [42]. Electrons after impact ionization at $({{x_1},{t_1}} )$ is replaced by electrons generated at $(x^{\prime}_0,t^{\prime}_0)$ and $(x^{\prime\prime}_0,t^{\prime\prime}_0)$ with E = 0 after assuming no impact ionization is permitted in the area between $(x^{\prime}_0,t^{\prime}_0)$ and $({{x_1},{t_1}} )$ for hot electron and between $(x^{\prime\prime}_0,t^{\prime\prime}_0)$ and $({{x_1},{t_1}} )$ for cold electron, denoted as shadow areas. Two quantities should be defined to characterize the shadow areas: The probability density $P_{route}^{(n )}(x^{\prime}_0,x_1,x_0 )$ that if the energy equal to the electron generated at ${x^{\prime}_0}$ after impact ionization, which is generated at ${x_0}$ and impact ionized at ${x_1}$; The probability ${\alpha ^{(n )}}(x^{\prime}_0,x^{\prime}_1,x_0)$ that if the electron has left its shadow area at ${x^{\prime}_1}$, which is generated at ${x_0}$ with the energy equal to electron generated at ${x^{\prime}_0}$ after impact ionization. The superscript n labels the number of times of impact ionization it performs. Thus, the survival probability of the n^th impact ionization is characterized by ${{dP_{se}^{(n )}} / {dt}} ={-} {\alpha ^{(n )}}P_{ii}^{tot}P_{se}^{(n )}$. As is shown in Fig. 3(c), the probability density of the hot electron and the cold electron is separated because the most probable energy transition is approximately $1.3{E_g}$ [15].

The number of electrons generated in the m^th impact ionization is written as

(25)$${N^{(m )}}({{x_0},{x_m}} )= {2^{m - 1}}g({{x_0}} )P_{se}^{(m )}({{x_{m - 1}},{x_m}} )\prod\nolimits_{n = 1}^{m - 1} {({1 - P_{se}^{(n )}({{x_{n - 1}},{x_n}} )} )} , $$

where $g(x )$ is the normalized distribution of the injected nonequilibrium carrier [7,52]. With $g(x )$ substituted by corresponding distributions, we could discuss leakage current and light current separately. Integration is performed for ${x_{0 < i < m}}$ with ${x_i} < {x_{i + 1}}$ for all indexes. Thus, gain and ENF could then be written as

(26)$$M({{x_0},x} )= \sum\nolimits_m {{N^{(m )}}} ({{x_0},x} ), $$

and

(27)$$F = 1 + \frac{1}{{{M^2}}}{\sum\nolimits_m {({{N^{(m )}}} )} ^2}P_{se}^{(m )}({1 - P_{se}^{(m )}} ), $$

respectively [16,53–55]. As is shown in Fig. 5(a) and 5(c), the gain and ENF for light and leakage current are derived by using the distribution of optical generation and dark generation, respectively. The difference between light and leakage current is that the light generation is mainly in the p-region and the dark generation is mainly in the depletion region shown in the inset of Fig. 5(e). Considering both light and leakage current in the device, the ENF of the signal could be written as

(28)$${F_{tot}} = 1 + \frac{{\textrm{Var}({{M_L}} )+ ({{{{I_d}} / {{I_L}}}} )\textrm{Var}({{M_D}} )}}{{{{\left\langle M \right\rangle }^2}}}, $$

where $\textrm{Var}({{M_L}} )$ and $\textrm{Var}({{M_D}} )$ are the variance of gain of light and leakage current, respectively. ${I_L}$ and ${I_D}$ are the light and the leakage current, respectively [7,16,56]. In the above form of ENF, we have assumed the impact ionization events in light current and that in leakage current are independent with each other. The only coupling between these two parts is presented by Poisson equation, which is solved iteratively [52,57]. According to the analysis of R.J. McIntyre, the ENF of single-carrier avalanche photodiodes tends to a constant value when the gain tends to infinity [55]. As a single-carrier avalanche photodiode [58], mid-infrared MCT APDs exhibit an abnormal rise of ENF for a larger gain [18]. In the device discussed above, the analysis shows that the abnormal rise of ENF shown in Fig. 5(f) is caused by the increase of band to band tunneling shown in Fig. 5(e).

5. Conclusion

In short, we have presented the spatial description theory of narrow-band avalanche phenomenon based on quantum mechanics with a generalized history-dependent theory. Our work directly agrees with the experimental results without using any fitting factors. The abnormal rise of ENF is explained by taking the partially dark current avalanche into consideration. As a paradigm shift, spatial description theory presented here is potentially used in systems with mechanisms that the evolution of different carriers is independent of each other, whereas the characteristic time of sampling in the measurement is much lower than that in microscopic process. We expect this paradigm shift can provide another view on random process in transportation, which is more consistent with the process of measurement.

Funding

National Natural Science Foundation of China (61725505, 11734016, 61521005); Key Research Program of Frontier Science, Chinese Academy of Sciences (QYZDB-SSW-JSC031); Natural Science Foundation of Shanghai (19XD1404100, 18YF1427400); China Postdoctoral Science Foundation (2020TQ0331).

Disclosures

The authors declare no conflicts of interest.

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Region	Doping density (cm⁻³)	Coordinate (µm)	Lifetime (ns)
n⁺-region	2×10¹⁷	0.8 ∼ 2.8	-
n^--region	5×10¹⁴	0.0 ∼ 0.8	10
p-region	3×10¹⁶	−5.0 ∼ 0.0	10

Spatial description theory of narrow-band single-carrier avalanche photodetectors

Abstract

1. Introduction

2. Dynamics in the spatial description theory

2.1 Spatial description theory

2.2 Theorems of expectation of isotropic scattering

2.3 Equation of the motion of hot electrons

3. Corrections in spatial description theory in mid-infrared HgCdTe

4. Generalized history-dependent theory

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (5)

Tables (1)

Equations (28)

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