1 Introduction

Spacecrafts in orbit are always working in extreme environments, such as vacuum, irradiation, cryogenic temperature, and low gravity for long time operation, which affecting the reliability of spacecraft. According to statistics analysis, among a series of spacecraft that have been launched, the abnormal events caused by the space environment account for 70% of the total, and the failure events also account for 46% of the total [1]. In the space environment, the energetic particles cause mainly failure of the devices. Usually, there are two ways for energy loss in the interaction of energetic particles with materials, namely, ionization energy loss and non-ionizing energy loss. The ionization energy loss is through the interaction between energetic particles and extranuclear electrons of target atoms, inducing to the ionization of the target to transfer the energy to electrons. The non-ionizing energy loss is through the elastic collision between energetic particles and target atoms, which converts the energy into kinetic energy of target atoms. The enough kinetic energy makes the atoms to leave the lattice position and form a vacancy–interstitial pair. This behavior is called as the displacement effect, which seriously degrades the device’s characteristics [2]. The displacement effect forms point defects in the device, part of which can be recombined under thermal motion and other part remain. These defects enable to capture electrons, which leads to the degradation of device performance [2, 3].

It has examined that the irradiation-induced displacement effect is sensitive to the threshold displacement energy (TDE) of the material, the particle type, and the incident particle energy [4]. The nature of the incident particle is determined by the environment, and the TDE depends on the atomic bond of the material. The strength of TDE can significantly influence the generation and distribution of defects during irradiation. It has been proved that the TDEs of various metals and semiconductor materials have a strong correlation with binding energy. the Young’s Modulus, sublimation energy, Debye temperature, and thermal expansion coefficient of the material are also related to the binding energy [5]. Beeler et al. [6] employed the molecular dynamics (MD) method to study the relationship between the generation probability of vacancy–interstitial pair in different directions and the energy of primary knock-on atoms (PKA) in iron, and found that the TDE is dependent on the bulk strain, and the temperature promotes the formation of vacancy–interstitial pair. Banisalman et al. [7] investigated the TDEs of various metals, such as W, Mo, and V through using molecular dynamic method, found that the TDE is the linear function of bulk strain, and the vacancy–interstitial pair formation energy also performs a similar trend with respect of the strain. Guenole et al. [8] conducted the MD simulation for the energetic particle irradiation on Si and Al to explore the relation of the strain and the TDE.

Meanwhile, GaN has a higher TDE compared to other semiconductor materials, leading to that the GaN-based HEMTs have excellent irradiation resistance. In addition, GaN-based HEMTs promise high power and high frequency operation, which are suitable for the RF power device in spacecraft. While, in space environment, the devices operate in the presence of energetic particles, which will certainly affect the normal work of devices. The proton is the largest number of energetic particles, its energy distribution in several MeV to a few tens of MeV, which will cause serious displacement effect in devices. For example, the daily 1–10 MeV proton flux is about ~ 1011 cm−2 [9]. Thus, the displacement effect caused by proton irradiation on the performance of electronic devices has been extensively studied [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24], which mainly focus on the degradation of device performance after proton irradiation through experiments and theoretical analysis. To name a few, Keum et al. [14] studied the proton irradiation of p-type GaN HEMT, and found that the threshold voltage is negatively shift due to the decrease in hole concentration in p-GaN HEMT under irradiation. They also applied the Silvaco TCAD to verify the radiation degradation coming from the traps introduced by the displacement effect. Kim et al. [17] explored the effect of different isolation methods on radiation degradation of GaN HEMT. They found plasma etching isolation is more sensitive to proton radiation, and the interaction between the pre-existing vacancy and energetic particle could lead to more obvious degradation of the HEMT. Besides of the experimental studies, numerical simulations are also applied to study the degradation behavior of HEMT under proton irradiation [21,22,23,24]. Since irradiation mainly produces supersaturated defects as the traps, these traps can catch the carriers to degrade the device performance. Through introducing the traps to represent irradiation defects in simulation, the degradation response of devices as a function of irradiation can be quantitatively analyzed. For instance, Raut et al. [21] simulated proton irradiation on the Buffer-Free GaN HEMT, and found that Buffer-Free GaN HEMTs have more excellent resistance to proton irradiation than conventional HEMTs. In addition, the low-dose irradiation is also used to improve the device performance [23, 24]. For example, Stockman et al. [24] proposed to use the proton irradiation for adjusting the dynamic conductivity of unintentionally doped (UID) GaN devices.

Since there exits plenty of the unique heterostructures in the HEMTs, the mismatch strain is inevitably introduced during forming these heterostructures in devices [25, 26], which affects the radiation resistance of HEMTs. The influence of the stress or strain on the electrical properties of semiconductor materials has also attracted a lot of attentions. For example, Roisin et al. [27] studied the optical properties of silicon under uniaxial tension and compression, and found that the strain engineering is particularly interesting for infrared applications. Guin et al. [28] established a general continuum model that couples the mechanical, electrical, and electronic responses in a finitely deformable semiconductor. Cao et al. [29] studied the effect of stress on GaN effective mass and 2DEG electron mobility through first principles, and found that the effective mass decreases with the increase of stress. Yeo et al. [30] calculated the valence sub-band structures of uniaxial-strained wurtzite GaN/AlGaN quantum wells using multiband effective-mass theory. Therefore, the radiation degradation behavior of GaN-based HEMTs under strain becomes crucial for optimization design. However, there are few studies to quantitatively explore the strain effects on the radiation degradation of HEMTs yet.

In this work, the simulation and modeling are carried out to investigate the strain dependence of the TDE for GaN, defect distribution in GaN films as well as the radiation degradation in the GaN-based HEMTs. The relationship between the strain and the defect density of the material under proton irradiation was obtained by molecular dynamics and the Monte Carlo method, then TCAD simulation is performed to analyze the proton irradiation degradation of the strained device. Based these simulations, a theoretical model is also proposed to predict the strain-independent radiation damage in GaN-based HEMTs. The simulations demonstrate that the existence of strain could change the irradiation resistance of HEMT, which will be useful for the application of stress/strain engineering to the optimal design of space devices.

2 Methodology

Molecular dynamics simulation is employed to study the TDE of GaN under different strains. Here, give the wurtzite GaN crystal as an example. One can set the initial kinetic energy of PKA in the simulated unit, as shown in Fig. 1a. Then, the dynamic process of the lattice atoms after the elastic collision can be simulated. The tersoff potential is used to describe the interatomic interactions. The system is a 16 × 16 × 16 supercell of wurtzite include 32,768 atoms. The size of the box needs to be determined with the actual situation, the larger the energy of PKA, the larger size the box. The periodic boundary conditions are used in the simulations. The dichotomy is used to determine the TDE in each direction, and the Voronoi package is employed to judge the formation of defects. The initial kinetic energy is given for a PKA atom and then its motion is simulated in matter. When a defect occurs, the energy of PKA is considered to be greater than or equal to TDE. The TDEs of 1000 random directions are simulated with NVE ensemble. The simulated TDE of Ga is concentrated around 40 eV, and the minimum TDE of the GaN is around 23 eV which is close to the experimental result of 19 ± 2 eV [31, 32].

Fig. 1
figure 1

a Schematic diagram of atomic displacement; b schematic diagram of simulated irradiated material structure; c range of protons with different energies in GaN material

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Once the relationship between strain and TDE is obtained, the defect distribution in strained GaN-based multilayer structure under irradiation can be simulated by the Monte Carlo method combined with binary collision. Through setting displacement threshold energies under strain, the defect densities of GaN film with different strains under 1.8 MeV proton incidence are simulated by SRIM software [11, 21]. The interatomic actions are the main random variable. The probability of interatomic actions is determined according to the reaction cross section, and then random sampling is used to determine whether each reaction occurs. For ensuring accuracy, the number of incidence particles is set as 106. Considering the energy loss in the electrode to the incident ions, the target structure adopts Au (200 nm)/Ni (20 nm)/AlGaN (10 nm)/GaN (250 nm) [33], as shown in Fig. 1b. The project range of protons with different energies in GaN is depicted in Fig. 1c. Due to that the project range of 1.8 MeV protons in GaN exceeds the thickness of the structure, it can be considered that most of the protons leave the structure, and only a few protons stop in the material. Thereby, the effect of the protons interstitial atoms can be ignored. Since the displacement effect degrades the HEMT through the formation of traps [14], the influence of the displacement effect can be represented by setting the spatial distribution and the corresponding energy level of vacancies which affects device performance as a traps in the device. According to the first-principles calculation, the defect energy level formed by Ga vacancy is located at 0.86 eV from the top of the valence band. Ga vacancy is an acceptor trap, whose electron capture cross section is σn = 2.7 × 10−21 cm2, the hole capture cross section is σp = 2.7 × 10−14 cm2 [34–32].

Finally, the QuanFINE GaN HEMT, which has better radiation resistance than the conventional device, is given as an example to simulate the device degradation behavior under an irradiation environment [33]. The QuanFINE GaN HEMTs are shown schematically in Fig. 2a. SiC is used as the substrate. The epitaxial structure consists of 60 nm AlN nucleation layer, 250 nm undoped GaN layer, 1 nm AlN spacer layer, 10 nm AlGaN barrier layer, and 2 nm GaN cap. The Al components of AlGaN are 0.3. The gate length is 200 nm, the gate–source distance is 1 μm, the gate–drain distance is 1.75 μm, and the drain field plate is 0.25 μm. Schottky gate contact work function is set to 5.1 eV. The carrier transport Eq. (1) is used to simulated the degradation of proton Proton-Irradiated GaN-HEMT by Sentaurus TCAD. The doping mobility model, SRH recombination model, and high-field model are employed. The acceptor interface trap density in AlN/GaN is adopted as 1.34 × 1012 cm−3. Considering the traps generated by the epitaxial growth of thin undoped GaN, acceptor traps of 5.4 × 1016 cm−3 are defined in the GaN buffer layer [33]. The effects of irradiation are manifested by setting traps equivalent to defects in the device. Through Sentaurus TCAD[36] simulation, the results of Id–Vd without irradiation are compared with the experimental results, as shown in Fig. 2b. One can note that the simulations is consistent with the experimental results:

$$\left\{ {\begin{array}{*{20}c} {J_{{\text{n}}} = nq\mu_{{\text{n}}} E + qD_{{\text{n}}} \nabla n} \\ {J_{{\text{p}}} = pq\mu_{{\text{p}}} E + qD_{{\text{p}}} \nabla p.} \\ \end{array} } \right.$$
(1)
Fig. 2
figure 2

a Schematic diagram of the GaN HEMT structure; b comparison of the Id–Vd curve of the simulation with the experimental results [33]

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Here, \(J_{{\text{n}}} ,J_{{\text{p}}}\) are the current densities of the electron and hole, respectively. \(n,p\) are the electron and hole densities, respectively. \(q\) is the electronic charge. \(\mu_{{\text{n}}} ,\mu_{{\text{p}}}\) are the mobility of electron and hole, respectively. \(D_{{\text{n}}} ,D_{{\text{p}}}\) are the diffusion coefficients of electron and hole. E is the electric field. Due to the presence of a large number of electrons in 2DEG, the diffusion current is small, and the current can be approximated as

$$I_{{\text{d}}} = qW\mu_{{\text{n}}} n,$$
(2)

where W is the gate width.

Due to the spontaneous polarization and piezoelectric polarization of GaN-based structures, two-dimensional electron gas (2DEG) is formed at the heterojunction interface of HEMT device. Considering the presence of two-dimensional electron gas, currents are mainly generated by electrons in GaN. Under the source–drain voltage Vds, the transverse electric field is formed inside the device. Electrons moves horizontally along the heterojunction interface to form current Ids under the action of the electric field. The electrons in the gate region are depleted under the gate voltage Vgs, to realize the function of NO and OFF. Considering the acceptor defects induced by proton irradiation, combined with the current density equation, the 2DEG can be expressed as [10]

$$qn_{{\text{s}}} = \sigma_{{{\text{pol}}}} + \frac{{\varepsilon_{1} }}{d}\left[ {\frac{{qN_{{\text{d}}} }}{{2\varepsilon_{1} }}d^{2} - \frac{{qN_{{\text{A}}} d^{2} }}{{2\varepsilon_{1} }} - \phi_{{\text{b}}} + V_{{{\text{gs}}}} - \frac{{\Delta E_{F} }}{q} + \frac{{\Delta E_{{\text{C}}} }}{q}} \right] - qN_{{\text{A}}} w,$$
(3)

where \(\varepsilon_{1}\) is the dielectric constant of AlGaN at the heterojunction interface, \(d\) is the thickness of AlGaN, \(N_{{\text{d}}}\) is the doping concentration, \(q\phi_{{\text{b}}}\) is the Schottky barrier height, \(\Delta E_{{\text{C}}}\) is the energy between the device conduction band at the heterojunction and the Femi level, \(\Delta E_{{\text{F}}}\) is the distance between the bottom of the GaN potential well conduction band and the Femi level. \(\sigma_{{{\text{pol}}}}\) is the polarization charge density generated by the polarization in the heterojunction region, \(N_{{\text{A}}}\) is the acceptor trap introduced by proton irradiation and the acceptor trap in the material itself, \(w = \sqrt {2\varepsilon_{1} \left( {E_{{\text{g}}} /2 + \Delta E - \Delta E_{{\text{F}}} } \right)/\left( {qN_{{\text{A}}} } \right)}\). In combination with Eqs. (2) and (3), one can simulate device degradation under proton irradiation.

3 Results and discussion

3.1 Displacement effect of strained GaN

The TDEs of GaN under different strains are determined from MD simulations. The cloud atlas of the TDEs with respect to the direction angle for different strains are illustrated in Fig. 3. The different points in the figures represent the relative position of each atom in 8 supercells (surround PKA) with respect to the PKA. The coordinate value indicates the angle between the PKA and other atoms, and the size of the points indicates the distance between the atoms and the PKA. The red point represents the Ga atom and the brown one is the N atom. The size of the point represents the reciprocal of the distance between each atom and PKA. The larger the point is, the closer the distance is. Figure 3a shows that in the range of π/3 < φ < 2π/3, the distribution of the TDE is related to atomic distance and distribution. There is a larger displacement threshold energy in the direction of atoms, and the maximum TDE usually occurs in the position between atoms. Figure 3b, c plots the distributions of TDE under tensile strain and compressive strain, respectively. One can find that the TDE of GaN is decreased for the tensile strain and increased for the compressive strain significantly. The TDE distributions with respect of the angle under strains are similar with the one free of the strain, all of which are dependent on the atomic distribution.

Fig. 3
figure 3

Cloud diagram of displacement threshold energy a without strain; b 1% tensile strain; c 1% compressive strain

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Figure 4a depicts the distribution histograms of the displacement threshold energy of GaN material under three different strains. The curves are fitted from the logarithmic normal distribution function based on these distribution histograms. It can be seen that the displacement threshold energy of GaN is concentrated in the range of 30–60 eV, which is because of the different bond strength of atoms in different directions, resulting in obvious anisotropy of the TDE. One also can find that the TDE of GaN is close to the logarithmic normal distribution. The tensile strain makes the peak of the normal distribution shift negatively, and the peak of the normal distribution shift positively under compressive strain. The influence of tensile strain on TDE is greater than that of compressive strain. Due to the randomness of the collision direction angle, the average TDE is adopted from the log-normal distribution in Fig. 4a as the TDE of GaN for Monto Carlo calculation of defect distribution. The relationship between the TDE of GaN and the strain then can be determined, as shown in Fig. 4b. It can be found that the TDE of GaN decreases with the increase of the strain. It is because that the binding energy [37], which will affect the properties of materials, such as TDE, the Young’s Modulus, etc. [5], will decrease when the strain increase, resulting in the decrement of the TDE. The average TDE of GaN Ed,avg is linearly related to the strain. The relationship between TDE and strain is obtained by fitting as follows:

$$E_{{{\text{d}},{\text{avg}}}}^{{{\text{strained}}}} = E_{{{\text{d}},0}} \left( {1 - \alpha \varepsilon } \right),$$
(4)

where Ed,0 is the average TDE without strain, α is the fitting parameter, and ε is the strain.

Fig. 4
figure 4

a Distribution of Ga atom displacement threshold energy in GaN material; b relationship between displacement threshold energy and strain

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After determining the quantitative relationship between displacement threshold energy and strain, Monte Carlo method is employed to study the defect distribution of GaN-based structure caused by irradiation under different strains. Figure 5a shows the distribution of proton irradiation in GaN-based structure. White points are protons and other colors correspond to different material atoms. It can be seen that most of the protons in the figure pass straight through the material, and a small amount of protons are deflected due to collisions and electric field forces. At the same time, due to the existence of surface binding energy at the interface between two different the materials, the recoil particles may be blocked at the interface. Figure 5b shows the defect density generated in the GaN layer under different strain states. Under compressive strain, the defect density generated is effectively reduced, while the defect density increases significantly under tensile strain. These point defects generated by the displacement effect will affect the performance of the device as traps.

Fig. 5
figure 5

a Cloud diagram of particle distribution; b defect density distribution in GaN material under different strains

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3.2 Degradation behavior of proton irradiated devices under strain

Furthermore, to explore the effect of strain-induced changes in irradiation defects on the performance degradation behavior of devices, assuming that the high-energy protons irradiate evenly on the device surface, the Id–Vd output characteristic curves of the device under different irradiation doses are shown in Fig. 6a. With the increase of the radiation dose, the saturation drain current decreased. At low doses the saturation drain current degradation is negligible, but at high doses, the device saturation drain current degrades significantly. As shown in Fig. 6b, as the irradiation dose increases, the threshold voltage shift increases, and the threshold voltage changes significantly at lower doses, while the maximum transconductance is little affected at lower doses, only when the dose reaches 1013 cm−2 will occur obvious degradation.

Fig. 6
figure 6

Degradation of HEMT when Vg = 0 V under different proton irradiation doses. a Id–Vd output characteristic curves and b variation of threshold voltage and maximum transconductance with irradiation dose

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Then, we investigate the effect of stress on device performance. Due to the existence of strain, the defect distribution of the device caused by irradiation could be influenced, which affects the irradiation degradation of the device. Figure 7 demonstrates the degradation of the device under different strains at different irradiation doses. Figure 7a plots the output characteristic curves of the device. It can be seen that the device degradation changes under strain. Degradation of saturation drain current decreases under compressive strain and increases under tensile strain. Figure 7b shows the device transfer characteristic curves, in which the device current increases under compressive strain, and decreases under tensile strain. Figure 7c presents the trans-conductance curves of the device in which the trans-conductance increases under compressive strain and decreases under tensile strain. Figure 7d gives the variation of the maximum trans-conductance and threshold voltage with irradiation and strain. Under proton irradiation, the threshold voltage is positively shift, and the maximum transconductance decreases. The tensile strain will increase the variation, while the variation decreases under compressive strain. The results show that the strain has a significant effect on the radiation resistance of the device. Compressive strain can effectively improve the radiation resistance, and tensile strain will weaken the radiation resistance of the device.

Fig. 7
figure 7

Degradation of HEMT under different strains. a Id–Vd output curves and b Id–Vg transfer curves when Vg = 0 V, c trans-conductance curves and d variation of maximum trans-conductance and threshold voltage with irradiation dose

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Proton irradiation mainly affects the performance of the device by generating point defects as traps to trap electrons inside the device [19]. Figure 8 shows the distribution of electrons in the device under different irradiation doses and different strains. Figure 8a, b clearly shows that with the increase of the irradiation dose, the electron density inside the device decreases. According to the current density equation, with the decreases in electron density, the saturation drain current of the device degrades, resulting in a significant degradation of the device performance. The results in Fig. 8b–d show that the strain can effectively change the electron density inside the device. That is, the compressive strain increases the electron density, while the tensile strain reduces the electron density. Through changing the strain, the radiation resistance performance of the device could be modified. At the same time, these defects also can affect the impurity scattering to reduce the electron mobility, and the performance of the device together with the change of the electron density. Figure 9a, b shows the distribution of electron density with Y-axis and X-axis under an irradiation dose of 5 × 1013 cm−2 with different strains. The position in Y-axis close to 0.07 μm represents the region generating the two-dimensional electron gas (2DEG), which forms electron aggregation due to the heterojunction structure and polarization of the device. The position in X-axis from 0.75 to 1.5 μm is the gate position. Due to the Schottky barrier, the electron concentration at this position is less than that around. The saturation drain current of the device can be controlled by changing the voltage. It is clear that the electron density in the device decreases significantly under tensile strain, and increases under compressive strain. To quantify the strain effect on the irradiation characteristics of the device, the change rate CR defined in following expressions are employed to represent the strain effect of the device:

$${\text{CR}} = \frac{{I_{{{\text{dstrained}}}} - I_{{{\text{dunstrain}}}} }}{{I_{{{\text{dunstrain}}}} - I_{{{\text{d}}0}} }}.$$
(5)
Fig. 8
figure 8

Cloud diagram of electron density under different irradiation doses: a 0 cm−2, b 5 × 1013 cm−2 without strain, c 5 × 1013 cm−2 with − 5% compressive strain, and d 5 × 1013 cm−2 with 5% tensile strain

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Fig. 9
figure 9

Electron density under the irradiation dose of 5 × 1013 cm−2 a distribution along the Y-axis; b distribution along the X-axis

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Figure 10a shows the variation of the saturation drain current with the irradiation dose. The saturation drain current of the device decreases significantly with the increase of the irradiation dose, and the degradation of the device will change under the action of strain. Performance of device is deteriorated under tensile strain, and can be improved under compressive strain. Figure 10b shows that the strain effect of the device under irradiation dose can effectively improve the irradiation resistance of the device, and can effectively affect the degradation of the device. The strain effect is more obvious under tensile strain. It is due to the defect density is inversely proportional to the TDE of material. Under compressive strain, the irradiation resistance of the device can be improved by nearly 20%, and the service life of the device can be effectively prolonged.

Fig.10
figure 10

Degradation of saturation drain current under strain: a degradation rate of saturation drain current and b change Rate of saturation drain current under strain

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For studying the changes of devices under different irradiation doses and strains, we extracted the degradation of devices under different irradiation doses and strains. Figure 11 shows the irradiation change rate of the device under different strains and different irradiation doses. The change rate of the device under irradiation has been significantly improved by the strain, but the strain effect is gradually weakened under high irradiation doses. When the irradiation dose is high, the trap inside the device is the supersaturated. The electron concentration is less than the defect density, which leads to the incomplete ionization of the irradiation defect. Although the strain can change the defect density generated by the irradiation, the supersaturated trap can still reduce the electron density to low level, thus weakening the effect of strain. At medium and high doses in Fig. 11, the strain still has a certain contribution on the performance of the device, which is due to the reduction of impurity scattering and improving the carrier mobility inside the device.

Fig. 11
figure 11

Change rate of HEMT varied with the strain for different irradiation doses

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3.3 Theoretical model for predicting radiation degradation

Figure 12 gives the changes of electron density and electron mobility with the increase of irradiation dose. At low doses, there is no obvious degradation of electron density and electron mobility, but with the increase of irradiation dose, the electron density decreases rapidly. The internal electrons of the device are captured by the defects, which makes the free carrier density decrease and affects the performance of the device. The influence of irradiation on electron mobility is mainly through the effect of irradiation defects on impurity scattering, thus affecting the electron mobility. The electron mobility changed little at low dose, and decreased significantly at high dose. It is worth noting that there is a significant difference in the irradiation dose between the obvious degradation of electron mobility and the obvious change of electron density.

Fig. 12
figure 12

Changes of electron density and electron mobility with the increase of irradiation dose

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Note that the saturation drain current of the device is proportional to the electron density and electron mobility. Considering the irradiation dose of 1012 ~ 5 × 1013 cm−2, the defects of the device is assumed to be completely ionized. Due to the low irradiation dose, the degradation phenomenon of the mobility with the increase of the acceptor trap is ignored. Since the TDE of the material depends on the strength of the interatomic bond, the number of defects resulting from the displacement effect can be approximately described by the Norgett–Robinson–Torrens (NRT) model [4]. It can be expressed as

$$N_{{\text{d}}} = \left\{ {\begin{array}{*{20}c} 0 \\ 1 \\ {0.8T_{{\text{d}}} /E_{{\text{d}}} } \\ \end{array} \begin{array}{*{20}l} {T_{{\text{d}}} < E_{{\text{d}}} } \\ {E_{{\text{d}}} < T_{{\text{d}}} < 2E_{{\text{d}}} /0.8} \\ {2E_{{\text{d}}} /0.8 < T_{{\text{d}}} < \infty ,} \\ \end{array} } \right.$$
(6)

where Nd is the density of defects, Td is the damage energy related to the incident particles, Ed is the TDE, and 0.8 in the formula is the correction coefficient introduced by considering the defect recombination caused by the local thermal peak. According to the relationship between strain and electron density in Eq. (3), displacement threshold energy in Eq. (4), and NRT model in Eq. (6), the electron density under irradiation can be written as

$$qn_{{\text{s}}} = \sigma_{{{\text{pol}}}} + \frac{{\varepsilon_{1} }}{d}\left[ {\frac{{qN_{{\text{d}}} }}{{2\varepsilon_{1} }}d^{2} - \phi_{{\text{b}}} + V_{{{\text{gs}}}} - \frac{{\Delta E_{{\text{F}}} }}{q} + \frac{{\Delta E_{{\text{C}}} }}{q}} \right] - q\left( {N_{{{\text{A}},0}} + \frac{{0.8T_{{\text{d}}} \varphi }}{{\left( {1 + \alpha \varepsilon } \right)E_{{{\text{d}},0}} }}} \right)\left( {w + \frac{d}{2}} \right),$$
(7)

where \(\varphi\) is the irradiation dose, \(N_{{{\text{A}},0}}\) is the acceptor traps in the device.

Combined with the source–drain current Eq. (2) calculation formula, the relationship of device irradiated degradation rate under strain can be obtained as

$$\frac{{I_{{{\text{ds}}}} \left( {\varepsilon ,\varphi } \right)}}{{I_{{{\text{ds}},0}} }} \approx 1 - \eta \frac{\varphi }{{\left( {1 - \alpha \varepsilon } \right)}},$$
(8)

where η is he numerical parameter. Figure 13 shows the comparison between the model predictions and simulation results. It can be found that the proposed model can well describe the numerical results in Sect. 3.2. Then, one can further predict the irradiation degradation of GaN HEMT devices based on the proposed model described in above. Figure 14 gives the predicted radiation degradation for various gate voltages, strains and the irradiation dose. One can notice that the gate voltage will significantly affect the degradation rate of the device under irradiation. The closer the gate voltage is to the threshold voltage, the more obvious the device degradation is. The larger the gate voltage is, the smaller the degradation rate is. In practical applications, large gate voltage can be selected within the power limit, which has reduced the impact of irradiation on device degradation.

Fig. 13
figure 13

Comparison between the predictions based on the proposed model and the simulation results of this work

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Fig. 14
figure 14

a Relationship between the irradiation dose and the degradation rate under strain of 2%. b Relationship between the gate voltage and the degradation rate under strain of 2%. c Relationship between the strain and the degradation rate under the irradiation dose of 5 × 1013 cm−2. d Relationship between gate voltage and degradation rate under the irradiation dose of 5 × 1013 cm−2

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4 Conclusion

In this paper, atomic-scale simulation, Monte Carlo method, and carrier transport theory are employed to systematically study the irradiation damage characteristics of GaN-based HEMTs under strains, and a force–electric coupling irradiation degradation theoretical model is established. The simulation results demonstrated that the strain enables to affect the defect density generated by the irradiation displacement effect, thereby weakening/enhancing the performance degradation caused by irradiation. At low doses, the degradation of the device is mainly due to the defect capture caused by electron irradiation. However, at a high dose, the defects generated by irradiation of the device enhance the impurity scattering, thus affecting the carrier mobility and affecting the device performance together with the decrease of electron density. According to the results, the degradation amplitude of the device can be improved by nearly 20% under low dose and strain. This work can provide a theoretical basis for the application of stress/strain engineering in the optimization design of space devices.