Investigation of Incident Angle Dependence of Single Event Transient Model in MOSFET

Abstract

As the manufacturing process level of semiconductor devices continues to improve, the device size gradually decreases, and the devices are affected by the single event effect more and more severely. In this paper, the physical process of single particle incident N-channel Metal-Oxide-Semiconductor Field-Effect Transistor (NMOSFET) is simulated. By changing the particle incidence position, incidence angle, LET value, and temperature, the transient current variation with time is obtained, and the susceptibility of the device to single event effect under the action of four factors is analyzed, which provides the basis for the next research of the device against single event effect and ideas for the radiation hardening of semiconductor devices. In addition, in these experiments, there were bimodal-shaped current pulses that were different from single peak transient current pulses. Therefore, a new model describing the bimodal single event transient (SET) current waveform was proposed, which can be used as a current source model in circuit simulation and help to predict the ability of the circuit to resist the single event effect.

1. Introduction

When energetic ionized particles pass near a sensitive P-N junction, a single event transient (SET) may occur [1,2,3], resulting in a change in the shape of the transient current in the transistor and a corresponding voltage perturbation in the circuit response. Due to the dynamic interplay between charge deposition, charge collection, and voltage perturbation, a distinct platform region can be observed in the transient current of NMOS transistors when SET is integrated into Complementary Metal Oxide Semiconductor (CMOS) inverters [4]. Furthermore, SET pulses are correspondingly wider or narrower as they pass through the circuit due to hysteresis or charge sharing effect [5,6]. Pulse width has now become a key indicator of circuit response as transient pulse widths get closer to the clock period [7], and it is critical to accurately predict the performance of a circuit. However, in some cases, the SET pulse width is fixed in the range of tens to hundreds of picoseconds [2]. Therefore, accurate modeling of SET pulse widths based on radiation, device, and circuit parameters is required [8] to assess the radiation resistance of applied circuits.

There has been a lot of research conducted by many researchers for the modeling and estimation of SET current shapes. Expressions based on double exponential functions are commonly used as a model source for approximating the fast rise and fall times of transient current waveforms [9], and peak currents and time constants can be extracted by simulation of the device [10,11]. However, in some cases, modeling in circuit simulation using only an ideal independent double exponential current source is inaccurate. Therefore, D. A. Black et al. [12] proposed to model single-particle transient current waveforms by connecting two double exponential current sources in parallel, one for excitation charge collection and one for maintenance charge collection. The model can be used to perform SET simulations to reduce the overestimation of the critical charge. In addition, A. M. Francis et al. [13] have proposed a calculation of current effects based on a piecewise linear current source (PWL) and an additional SPICE component. Messenger has proposed an expression resolution model for the transition shift function over time [14].

In addition to model sources based on function expressions, several researchers have conducted studies based on circuit models. In [15], a geometry-aware biased SET current source model is proposed to capture the current platform region effect in CMOS inverters, as well as in [16], D.G. Mavis et al. proposed an equivalent circuit model (ECM), which is also used to capture the platform effect. Also, in [17], a predictive model for estimating the SET pulse duration in CMOS inverters was proposed. Y. Aneesh et al. proposed a physical bias-based SET current model for dual-gate MOSFETs [18]. J.S. Kauppila et al. proposed a single-particle compact model for capturing dynamic charge collection features with very few calibration parameters to accurately capture bias-based single-particle current [19]. A compact model for SET based on FDSOI MOSFET that includes physical modeling of bipolar amplification was previously presented in [20].

This paper presents a model to describe the single event transient current waveform. The model divides the SET current waveform generated after a heavy ion strike on a MOSFET into two parts. The first part is modeled based on a pulse function expression, while the second part is fitted by the Asymmetric Double Sigmoidal (Asym2Sig) function, which has a wide range of applications, such as in dynamics and architecture. The model can be used as a current source model in circuit simulations to help predict the ability of a circuit to resist a single event effect.

The article is structured as follows: Section 2 describes the principle of MOSFET and an introduction to the SET current model. Section 3 analyses and discusses the experimental results. Finally, Section 4 concludes the paper.

2. MOSFET Physics and SET Model

2.1. Transmission Theory

Figure 1 shows a brief schematic of the equivalent model for current transfer in an N-type field effect tube. The left side represents the source of the NMOSFET, connected to GND and holding a potential of 0 V. The middle represents a channel of length L, and the right indicates the drain of the NMOSFET. When the source and drain reach thermal equilibrium, the probability of the different energy levels on the source and drain being occupied by electrons, respectively, can be described by the Fermi–Dirac distribution, as in Equations (1) and (2):

$$f_S(E)=\frac{1}{{1+\mathrm{e}}^{(E-E_{FS})/k_{B}T}}$$

(1)

$$f_D(E)=\frac{1}{{1+\mathrm{e}}^{(E-E_{FD})/k_{B}T}}$$

(2)

where E_FS and E_FD are the Fermi energies of the source and drain, respectively. E represents the energy of the carrier. k_B is the Boltzmann constant of about 1.38 × 10⁻²³ J/K. Then the electrons pass from the source through the channel and are then transferred to the drain, producing a drain current as in Equation (3):

$$I=\frac{2q}{h}{\displaystyle \int \Gamma (E)M(E)(f_S(E)-f_D(E))}dE$$

(3)

where 2 represents the electron spin number and q is the charge of the electron. h is the Planck constant, which is 4.14 × 10¹⁵ eV·s. Γ(E) is the probability of electron transfer from source to drain, between 0 and 1. M(E) is the number of channels representing the transfer of electrons from source to drain. The source and drain are assumed to be large reservoirs, very close to thermodynamic equilibrium, so that a small perturbation from the equilibrium process can drive the current flowing through the MOSFET. This equation can be obtained from the Boltzmann transport equation.

In one case, assuming that L is much smaller than the mean free path for backscattering, most of the electrons can be pulled apart and approximate a vacuum. In this ballistic case, there is no scattering. Electrons from the source, injected into the channel, can all be transported to the drain, mainly by ballistic transport, as shown in Figure 2a.

Nowadays, some MOSFETs have trench lengths that are very close to these ballistic limits. In this case, the channel length is many times the mean free path. The electrons injected from the source, due to scattering caused by lattice vibration, impurities, and other defects, randomly hit the electrons in all directions, thus changing the direction of electron motion. Therefore, that part of the electrons returns to the source, resulting in only part of the electron transfer from the source to the drain, making the transfer rate Γ(E) < 1. If there is a lot of scattering, this results in Γ(E) ≪ 1. This is the principle of diffusion transmission, as shown in Figure 2b, where the Transmission rate Γ(E) is solved by Equations (4) and (5).

$$\Gamma (E)=\frac{\lambda (E)}{\lambda (E)+L}$$

(4)

$$\lambda (E)=2v(E)\tau (E)$$

(5)

where λ is the mean free path for backscattering, λ(E) is then equal to two times the velocity multiplied by the scattering time.

Electrons can move through the channel at a certain speed, thus forming a current, the value of which is calculated as shown in Equation (3). M(E) represents the number of electron flow channels and is solved by Equation (6). D(E) is the density of states. In the 3D model, the density of states increases with the square root of the energy E. From Equation (7), the velocity of the electron is also proportional to the square root of the energy. Thus, the number of electron flow channels M(E) is linearly proportional to the energy of the electron.

$$M(E)=\frac{h}{4}{v}_x^{*}(E)D(E)$$

(6)

$$v(E)=\sqrt{2(E-E_C)/m^{*}}$$

(7)

2.2. Physical Equations for Single Particle Incidence

The high-energy particles incident on the NMOSFET ionizes with the silicon material (mainly) in it, generating a large amount of charge. These charges are collected on their trajectory by the drain of the NMOSFET, eventually generating a pulsed current. This physical process can be described by a set of fundamental equations which are derived by combining Maxwell’s set of equations with the solid-state physical properties of the semiconductor. The main ones are Poisson’s equation, the carrier continuity equation, and the carrier transport equation.

Poisson’s equation represents the generation of electron –hole pairs and relates the concentration of the generated electron–hole pairs to the electrostatic potential and electric field. The carrier continuity equation refers to the increase in charge within a semiconductor device. By continuity, it means the spatiotemporal continuity of the charge concentration, i.e., the cause of the increase in charge within a volume, which generally means that there is a net inflow of charge into the volume and a net production of charge within the volume. The carrier continuity equation is shown in Equation (8):

$$\frac{\partial n}{\partial t}=\frac{1}{q}\mathrm{di}v{J}_n-U_n$$

(8)

where n is the concentration of electrons, J_n is the current density of electrons, and U_n represents the net compounding rate of electrons. When U_n > 0, it represents net complexation, and when U_n < 0, it represents net production. The Expression for U_n is as in Equation (9):

$$U_n=R_n-G_n$$

(9)

where G_n is the rate of electron production, and R_n is the rate of electron recombination.

Poisson’s equation and the carrier continuity equation provide the general framework for semiconductor device simulation, but further secondary equations are required to specify a specific physical model J_n, using a drift-diffusion model to represent the transport equation. The transport equation is also known as the current density equation and is mainly determined by the drift current density as well as the diffusion current density [21]. The continuity equation and the transport equation describe the way in which the electron density evolves with the transport process, the generation process, and the recombination process. The expression is given in Equation (10):

$$J_n=qn{\mu }_n{E}_n+q{D}_n{\nabla }_n$$

(10)

where µ_n is the charge mobility (cm²/V·s), q is the charge, D_n is the diffusion coefficient of charge (cm²/s), and ∇_n is the change in charge concentration with respect to distance.

2.3. SET Model

When an incident particle strikes a MOSFET device, the change in the device’s current characteristics can be analyzed by TCAD simulation. However, due to the complexity of the 3D TCAD model, large-scale circuit-level simulations cannot be supported. Therefore, in order to implement circuit-level simulations of single event effect, circuit-level modeling based on TCAD simulation results is required. This section introduces two commonly used models for SET current pulse analysis.

2.3.1. Double Exponential Function Based Model

The double exponential model is a common analytical model for describing SET current waveforms. The model can be converted into an equivalent current source model to approximate the transient currents caused by incident particles striking the MOSFET, thereby supporting circuit-level simulations of the single event effect. The expression for the double exponential model is shown in Equation (11):

$$I(t)=\{\begin{array}{l}0;t<t_{d1}\\ I_{peak}\times (1-\exp (-\frac{t-t_{d1}}{{\tau }_1}));t_{d1}<t<t_{d2}\\ I_{peak}\times (\exp (-\frac{t-t_{d2}}{{\tau }_2})-\exp (-\frac{t-t_{d1}}{{\tau }_1}));t>t_{d2}\end{array}$$

(11)

where t_d1 is the time at which the current starts to rise, t_d2 is the time at which the current starts to fall, τ₁ is the rise time constant, and τ₂ is the fall time constant. I_peak is an approximation to the maximum current generated in a single event effect, which is associated with the injected charge Q_dep. The expressions are shown in Equations (12) and (13):

$$Q_{dep}=1.035\times {10}^{-2}\times LET[\mathrm{Mev}\cdot {\mathrm{cm}}^2\cdot {\mathrm{mg}}^{-1}]$$

(12)

$$I_{peak}=Q_{dep}/[t_1+{\tau }_2+(t_{d2}-t_{d1})-{\tau }_1\times \exp (-\frac{(t_{d2}-t_{d1})}{{\tau }_1})]$$

(13)

2.3.2. Bias-Based SET Model

The structure of the bias-based single-particle model is shown in Figure 3. The model consists of a total of four branches, where the I_SRC is a current source describing single-particle charge deposition. The C_S is a conventional capacitive model for describing the charge collection process. The G_REC is a controlled current source for modeling carrier complexation in the circuit node. The G_RAD current is a function of the voltage across the C_S and is used to model the single-particle current injected into the junction.

The solution relations for each parameter in the model are shown in Equations (14)–(16):

$$I_{SRC}+\frac{C_{S}dV(C_S)}{dt}=G_{REC(t)}+G_{RAD(t)}$$

(14)

$$G_{REC(t)}=f(V(C_S),C_S,RecombParameter)$$

(15)

$$G_{RAD(t)}=f(V(C_S),C_S)\times Fermi(V(P,N))$$

(16)

where the RecombParameter composite parameter is approximated as the reciprocal of the minority carrier lifetime, which is a combination of the Shockly–Reed–Hall (SRH) and Auger lifetimes. The production of excess carriers in the single event effect is affected by this value. V(P, N) is the bias voltage from P to N. The Fermi model in Equation (16) will suppress the node current glitch by G_RAD during the reverse biased PN junction glitch. The Fermi energy function, whose expression is shown in Equation (17), where the parameter F determines the slope and range of the current drop, the value of which can be obtained by TCAD simulation [22].

$$Fermi(V(P,N))=\frac{1}{1+\exp (\frac{V(P,N)}{F})}$$

(17)

3. Results and Discussion

In this chapter, the 3D model of NMOSFET is constructed by the Silvaco TCAD simulation platform and the physical process of single particle incident NMOSFET is simulated by the CVT integrated model, Shockley–Read–Hall composite model (SRH), and Auger composite model (Auger). The effect of the single event effect on the device is studied by observing the changes in drain transient current and charge collection.

3.1. Incident Angle Dependence Investigation

Different incidence positions of energetic particles may have different effects on the device, so the effect of incidence position on the device was investigated before studying the effect of the incidence angle of energetic particles on the device. The control variable method is used, specifying the LET value of 20 MeV-cm²/mg and the particle vertical incidence θ = 0°. Three different representative points in the device are selected: the center of the source, the gate, and the drain, and then a single particle incidence simulation is performed for the three points.

The simulation results are shown in Figure 4. It can be seen that the transient current at the center of the source is almost zero, the transient current at the center of the gate is 6.58 mA, and the transient current at the center of the drain is 12 mA. The peak transient current at the incident point at the drain is significantly larger than the transient currents at the source and gate. When high-energy particles are injected into the source, it is difficult for the electrons to be collected by the drain at the other end because of the potential energy barrier between the source and the PN junction formed in the channel, and the electrons can only slowly diffuse to the drain through the concentration gradient. However, the farther away from the drain, which is a sensitive area, the farther the electrons diffuse to the sensitive area, and the more they compound with holes and other positive ions in the middle, resulting in less charge being collected at the drain. Therefore, it is concluded that the sensitive area of the NMOSFET is near the drain, and the closer the particle incident position is to the sensitive area, the higher the transient current.

The MOSFET sensitive region is near the drain, so the center of the drain is chosen as the incidence point, and a LET value of 20 MeV-cm²/mg is specified to investigate the effect of different incidence angles of the particles on the device. As shown in Figure 5, the angle between the normal of the drain center and the incident ray on the right side of the normal is specified as the angle of incidence θ. The selected incident angles and the corresponding exited areas are shown in

Table 1. Selection of incident angles and corresponding emitted areas.

Incidence Angles/Deg	Exited Areas
0, 10, 15, 20, 30, 40, 45	Bottom of substrate
50, 60, 70, 80	Source edge
100, 110, 120, 130, 140, 150, 160, 170	Drain edge

.

By changing the coordinates of the incident and exit points, transient current waveforms with the above different incidence angles were obtained, and these transients were classified into three categories according to the different exited areas, as shown in Figure 6.

It can be found that the transient currents at different angles are also different. When the incident angle is 0–45°, the exited area is at the bottom of the substrate, and the transient current pulses are all bimodal in shape. When the incident angle is 50–80°, the exited area is at the edge of the source, and most of the transient current pulses are single-peaked in shape, and only when the incidence angle is 50°, the transient currents are double-peaked pulses. When the incidence angle is 100–170°, the exited area is at the edge of the drain, and most of the transient current pulses are single-peaked current pulses, and only when the incidence angle is 160° and 170° do the pulses show a double-peaked shape. The specific data are shown in

Table 2. Data of transient currents at different angles.

Incident Angle/Deg	Primary Peak Current/mA	Inflection Current/mA	Secondary Peak Current/mA	Cumulative Charge/fC
0	12	3.99	4.64	323
10	13.6	4.92	5.86	358
15	13.8	5.34	6.56	365
20	14.1	5.81	7.31	374
30	14.7	7.12	9.12	400
40	16.2	9.6	11.6	455
45	17.2	11.3	13	493
50	14.3	12.1	15.2	501
60	19.1	-	-	434
70	19.7	-	-	403
80	17.9	-	-	367
100	0.261	-	-	41.6
110	0.397	-	-	48.3
120	0.582	-	-	54.2
130	0.869	-	-	61.8
140	1.3	-	-	73
150	2	-	-	91.8
160	0.64	0.589	3.32	131
170	3.59	2.41	5	247

.

Figure 6a shows the transient current pulses at the incidence angle of 0–45° and the exited area at the bottom of the substrate. It can be seen that these current pulses are double-peaked current pulses, the specific data in Table 2. As the angle increases, the primary peak current, the inflection point current, and the secondary peak current of the transient current increases, and the primary peak current is higher than the secondary peak current. Therefore, the larger the angle, the higher the transient current pulse, the more charge collection is obtained by integrating the current, and the more it is affected by the single event effect.

Figure 6b shows the transient current pulses at the incidence angle of 50–80° and the exited area at the edge of the source. It can be seen that only the current pulse at θ = 50° is bimodal in shape. Unlike the double-peaked waveform at 0–45°, where the primary peak current is lower than the secondary peak current, the other three current pulses are all single-peaked in shape. In terms of the peak current, it also does not increase exactly with increasing angle. Therefore, it is not possible to determine the magnitude of the charge collection directly from the waveform of the current pulse. The charge collection is obtained by integrating the current, as in Table 2. It can be seen that the charge collection decreases as the angle increases.

Figure 6c shows the transient current pulses at the incidence angle of 100–170° and the exited area at the edge of the drain. From θ = 100° to θ = 150°, the current pulses are single-peaked in shape. At θ = 160° and θ = 170°, the current pulses are bimodal in shape, and the primary peak current is lower than the secondary peak current. The specific data are shown in Table 2. Therefore, the larger the angle, the higher the transient current pulse, the more charge is collected, and the more it is affected by the single event effect.

Figure 7 and Figure 8 show the peak values of the transient currents at different angles and the time at which the peak occurs. It can be seen that the peak current increases at the incident angle when the exited area is the bottom of the substrate, with an upward trend, where the maximum peak current is 17.2 mA at θ = 45°, and the peak time is basically the same, stable at 5 ps. The peak current also generally shows an increasing trend at the incident angle when the exited area is the edge of the source. The largest peak current is 19.7 mA at θ = 70°, which is not very different from the previous peak current. At the same time, there is an increasing shortening in the time to peak. The peak current generally shows an increasing trend at the incident angle when the exited area is the drain edge. The maximum peak current is 5 mA at θ = 170°, a difference of 14.7 mA and a decrease of 74.6% from the previous peak current. There is an overall increasing length of time in which the peak occurs.

A summary of the single event transient current pulses at different angles in the three different exited areas (bottom of the substrate, source edge, and drain edge) mentioned above reveals that the amount of charge collected is related to both the incidence angle and the exited area, as shown in Figure 9. When the emitted area is the bottom of the substrate, the amount of charge collected increases as the angle increases, as in Figure 9a; when the emitted area is the source edge, the amount of charge collected decreases as the angle increases, as in Figure 9b; when the emitted area is the drain edge, the amount of charge collected increases as the angle increases, as in Figure 9c.

At different emitted areas, the amount of charge collected either increases or decreases sequentially with increasing angle. Using the example of the exited area being at the bottom of the substrate, as in Figure 10a,b, the green vertical channel in Figure 10a represents the path of the particle vertical incident, and the blue tilted channel in Figure 10b represents the path of the particle oblique incident, where the oblique path is obviously longer than the vertical path. Therefore, at the same LET value, the longer the injection path of the particle, the more carriers are generated by colliding with the device, and the more charge is collected (as in Figure 9a). The possibility of generating the transient current is greater, and the device is more sensitive to the single event effect. By the same token, it can be analyzed that the larger the angle of incidence, the shorter the injection path of the particle, and the less charge is collected when the emitted area is at the edge of the source. When the emitted area is at the edge of the drain, the greater the angle of incidence, the longer the injection path of the particle, and the greater the amount of charge collected.

By changing different incident angles and observing the amount of charge collected by the drain and the length of the incident path, this conclusion can be drawn: at the same LET value, the longer the incident path of the particle, the more carriers are generated by colliding with the device, and the more charge is collected. Thus, the greater the likelihood that the charge threshold will be reached, and the greater the susceptibility to the single event effect.

3.2. Double-Peaked SET Current Pulse Model

In the process of single-particle incident NMOSFET, the device is observed to be affected by the single event effect by changing the incident position, incident angle, LET value, and temperature of the high-energy particles. In the experiments in Section 3.1, Section 3.3.1 and Section 3.3.2, it was found that the pulse waveform of single event transient current differs from previous studies in that the waveform will have double peaks, and the SET model with double exponential function is no longer valid. Therefore, this paper proposes a new model to describe the single event transient current waveform by fitting the SET to the experimentally obtained double-peaked current pulse pattern.

Taking the experimental results of the single particle incident vertically at the drain center with LET values of 20 MeV-cm²/mg and 70 MeV-cm²/mg as examples, the obtained single event transient current waveforms are shown in Figure 11. The figure shows a double-peaked current pulse, which is fitted in two stages. The experiment was completed within 100 ps, so the first peak of the waveform from 0–30 ps was taken as the first stage, and the second peak of the waveform from 30–100 ps was taken as the second stage. The waveforms of these two stages are fitted separately.

The results of the first stage fitting of the transient current waveform are shown in Figure 12, where an impulse function is used for the fitting, and the fitting equation is as in Equation (18):

$$y=y_0+A{(1-e^{-\frac{x-x_0}{t_1}})}^p{e}^{-\frac{x-x_0}{t_2}}$$

(18)

where y₀ is the current value corresponding to t = 30 ps, x₀ is the time corresponding to the current value y₀ during the waveform rise phase, t₁ is the pulse rise time, and t₂ is the pulse fall time.

The results of the second stage fitting of the transient current waveform are shown in Figure 13. The Asym2Sig function has a wide range of applications, such as in dynamics and architecture, where the Asym2Sig function is applied for fitting the peak function. Therefore, the Asym2Sig function was used to fit the current waveform in this stage, and the fitting formula is shown in Equation (19):

$$y=y_0+A\cdot \frac{1}{1+e^{-\frac{x-x_c+w_1/2}{w_2}}}\cdot \left(1-\frac{1}{1+e^{-\frac{x-x_c-w_1/2}{w_3}}}\right)$$

(19)

where y₀ is the corresponding current value at t = 30 ps, x_c is the peak time, w₁ is the full width of half maximum, w₂ is the variance of the low-energy side, w₃ is the variance of the high-energy side, and A is the amplitude.

The two stages of the transient current waveform are summarized to obtain a new model describing the single event transient current waveform, as in Equation (20). The model focuses on the transient current pulse with a bimodal shape and divides the SET current waveform generated after the heavy ion strikes the MOSFET in two phases. The first stage is modeled based on the pulse function, while the second part is fitted by the Asym2Sig function. The model effectively fills the gap in modeling the bimodal transient current pulse generated by MOSFET subjected to single event effect and can be used as a current source model in circuit simulation to help predict the circuit’s ability to resist single event effect.

$$I(t)=\{\begin{array}{l}y=y_0+A{(1-e^{-\frac{x-x_0}{t_1}})}^p{e}^{-\frac{x-x_0}{t_2}},0<t<30\\ y=y_0+A\cdot \frac{1}{1+e^{-\frac{x-x_c+w_1/2}{w_2}}}\cdot \left(1-\frac{1}{1+e^{-\frac{x-x_c-w_1/2}{w_3}}}\right),30<t<100\end{array}$$

(20)

3.3. Incident Energy and Temperature Dependence Investigation

3.3.1. Incident Energy Dependence Investigation

Typically, space-emitting particles are complex and diverse, and LET can deliver the relevant particle energy to the device at a precise degree and a stable state. Therefore, LET values are used to represent particles of different energies and study the effect of particle incidence on the device for different LETs. According to the previous experiments, the center of the drain is still specified as the incidence point. The LET value of the particles we come into contact with in daily life is not greater than 20 MeV-cm²/mg, while the LET value of particles in the space universe can reach 90 MeV-cm²/mg. Therefore, six LET values were selected for the simulations on LET: 20 MeV-cm²/mg, 30 MeV-cm²/mg, 40 MeV-cm²/mg, 50 MeV-cm²/mg, 60 MeV-cm²/mg, and 70 MeV-cm²/mg. Simulations are performed in four cases with incidence angles of 0°, 30°, 60°, and 120°.

The simulation results are shown in Figure 14. As can be seen from Figure 14, the peak transient current and pulse width increase with increasing LET at the four angles, and the charge collected at the drain increases as well. The detailed data are shown in

Table 3. Peak transient current and charge collection of different LETs at four angles.

Incidence Angles/Deg	LET/MeV-cm2/mg	Peak Current/mA	Charge Collection/fC
0	20	12	323
	30	17.4	484
	40	22.7	646
	50	27.9	807
	60	33.1	969
	70	38.3	1130
30	20	14.7	400
	30	21.1	600
	40	27.4	800
	50	33.6	999
	60	39.8	1200
	70	45.9	1400
60	20	19.1	434
	30	27.2	652
	40	34	869
	50	39.9	1090
	60	45.1	1300
	70	49.6	1520
120	20	0.58	54.2
	30	1.04	81.3
	40	1.52	108
	50	2.02	135
	60	2.51	163
	70	2.99	190

.

LET represents the energy per unit length of the particle deposited in the impact path of the material in MeV-cm²/mg. Therefore, the number of electron–hole pairs deposited per unit length of the high-energy particle incident on the silicon material is shown in Equation (21):

$$\frac{dN}{dx}=\frac{dP}{dx}=\frac{\rho }{3.6 \mathrm{eV}/\mathrm{pair}}\times LET$$

(21)

It can be seen from Equation (21) that as the value of LET increases, the number of electron–hole pairs generated is significantly elevated and the greater the charge collected at the drain, and that the relationship is linear. The data in Table 3 are studied and analyzed to obtain Figure 15. It can be visualized that the peak transient current grows approximately in a straight line as the LET value increases, and the collected charge also grows approximately in a positive proportion as the LET value increases.

By changing the LET value and observing the amount of charge collected by the drain and the magnitude of the current peak, this conclusion can be drawn: within a certain range, the peak transient current and the charge collected at the drain increases with increasing LET, and that the relationships are linear.

3.3.2. Temperature Dependence Investigation

Devices are affected differently in different external environments. For semiconductor devices, temperature is one of the most important external environments to pay attention to, which is closely related to the operating state and energy consumption of semiconductor devices. Therefore, when simulating the single-particle effect of semiconductor devices, in addition to considering the impact of incident position, incident angle, and LET value on particle incidence, the variable of temperature also needs to be studied. In this section, the center of the drain is still selected as the incidence point, and the LET value is specified as 20 MeV-cm²/mg. Three different temperatures are selected: 100 K, 200 K, and 300 K. Simulations are performed in four cases with incidence angles of 0° and 30°.

The simulation results are shown in Figure 16. At the incidence angle of 0°, the peak transient current decreases as the temperature rises. Compared to the peak transient current at 100 K, the peak transient current at 300 K decreases by 10.1 mA, a decrease of nearly 46.12%. At the incidence angle of 30°, the peak transient current decreases as the temperature rises. Compared to the peak transient current at 100 K, the peak transient current at 300 K decreases by 12.1 mA, a decrease of nearly 44.98%. In addition to the change in peak transient current, it can also be observed that as the temperature increases, the transient current pulse width increases, which is the opposite of the relationship between peak transient current and temperature, as shown in

Table 4. Peak transient currents of different temperatures at two angles.

Incidence Angles/Deg	Temperature/K	Peak Current/mA
0	100 K	21.9
	200 K	15.5
	300 K	11.8
30	100 K	26.9
	200 K	19.4
	300 K	14.8

.

The magnitude of the peak transient current is related to the mobility of the carriers μ. The larger the mobility μ, the larger the peak transient current. Equation (22) shows the relationship between mobility and temperature:

$$\mu ={\mu }_0{\left(T/T_0\right)}^{-\xi }$$

(22)

As can be seen, the temperature is a decreasing function of mobility, with the carrier mobility decreasing as the temperature increases, resulting in a decreasing peak transient current.

The rising phase of the transient current waveform is dominated by charge collection at the drain due to carrier drift by the electric field, while the trailing part is dominated by carrier diffusion, which is then collected by the drain. As can be seen from Figure 12, the transient current reaches the peak at a time that is basically constant, and the pulse width of the transient current changes mainly in the tail part of the current waveform, which means that the change in pulse width of the transient current is mainly related to the diffusion of carriers. Equation (23) shows the relationship between the diffusion coefficient D and temperature:

$$D=\frac{kT}{q}\mu$$

(23)

After analysis and calculation, it can be concluded that the temperature T and the diffusion coefficient D are, in general, a decreasing function of each other. Therefore, as the temperature increases, the diffusion coefficient D decreases, and the diffusion time becomes longer, resulting in a larger pulse width for the transient current as the temperature increases.

Therefore, it can be concluded that as the temperature increases, the carrier mobility decreases, resulting in a decrease in the peak transient current of the device. As the temperature increases, the diffusion coefficient D decreases, and the diffusion time becomes longer, resulting in a larger pulse width of the transient current as the temperature increases.

4. Conclusions

Through the TCAD simulation platform, a 3D model of NMOSFET is constructed, and the physical process of single particle incidence NMOSFET is simulated. Unlike the previous single particle incidence experiments, a bimodal-shaped current pulse different from the single-peaked transient current pulse is found, and then the SET model based on the double-exponential function is no longer valid. Therefore, this paper proposes a new model to describe the single event transient current waveform by fitting the SET current pulse based on the experimentally obtained bimodal current pulse pattern, which can be used as a current source model in circuit simulation and help to predict the ability of the circuit to resist the single event effect. In addition, the effect of a single event effect on the device is observed by varying the incidence position, incidence angle, LET value, and temperature. The sensitivity of the device to the single event effect under the action of each of these four factors is analyzed by the controlled-variable method, which provide the basis for the next research of the device against single event effect and ideas for the radiation hardening of semiconductor devices. This work is helpful for the research and testing of radiation hardening of semiconductor devices.

However, there are still many shortcomings in this paper on the study of the single event effect of semiconductor devices, and there is still much future work on the radiation resistance of semiconductor devices:

(1)

Most of the current research on the single event effect of semiconductor devices is limited to the computer simulation stage, and the conditions for conducting flight tests or ground simulation experiments do not allow for actual measurement data to be obtained yet.

(2)

With the continuous improvement of the semiconductor device fabrication process, the feature size of the device continues to decrease, so further research is needed in the future on the difficult issue of radiation-resistant reinforcement technology for small feature-size devices.