Correlation of Radiation-Induced Interface Traps With Band Edge Energy Through Band Structure-Based Analysis of Electrostatics of UTB SOI Devices

The effect of Radiation on the semiconductor-oxide interface, inducing interface trap states, has generally only been experimentally measured, which makes it difficult to quantify the impact of this radiation on device electrostatics. For an Ultra-Thin-Body (UTB) MOS device, the 1-D Band structure along the direction of confinement, if solved self-consistently with the 1-D Poisson’s equation, while varying the band edge energy <inline-formula> <tex-math notation="LaTeX">$(\Delta E_{edge})$ </tex-math></inline-formula> at the <inline-formula> <tex-math notation="LaTeX">$Si-SiO_{2}$ </tex-math></inline-formula> interface, can enable the quantification of the effect of interface trap states on channel electrostatics, while also accounting for Quantum Confinement Effects. In this work, we present an approach to correlate the radiation dose to the band edge energy <inline-formula> <tex-math notation="LaTeX">$(\Delta E_{edge})$ </tex-math></inline-formula>, thus enabling the channel thickness dependent band structure-based approach to be used to quantify the effect of these radiation-induced traps on the device electrostatics. We show a methodology that co-relates the interface charge induced by <inline-formula> <tex-math notation="LaTeX">$\Delta E_{edge}$ </tex-math></inline-formula> variation and the charge yield, due to different radiating particles, on the <inline-formula> <tex-math notation="LaTeX">$Si-SiO_{2}$ </tex-math></inline-formula> interface. After identifying appropriate values of <inline-formula> <tex-math notation="LaTeX">$\Delta E_{edge}$ </tex-math></inline-formula> for different particles and doses, the degradation due to radiation on the channel electrostatics can be accurately simulated, for a wide range of channel thicknesses with the atomistic band structure-based methodology. We also show an approach to extend this methodology to lower device temperatures, thus effectively quantifying the effect of radiation dose on UTB device electrostatics for a wide range of device temperatures.

from the perspective of establishing semiconductor device (specifically MOSFETs) applicability in various circuit applications, a robust assessment of device reliability, particularly considering the effect of radiation on device behavior, is required.Therefore, comprehending the impact of radiation on semiconductor materials and interfaces is vital for assessing device durability, specifically for silicon MOS devices, given their wide usability in various circuit applications [4], [5].Various studies have shown that radiation induces oxide and interface traps [6], [7], [8] in MOSFETs, adversely impacting the device electrostatics [9], which needs to be quantified.
In MOS devices, on exposure to radiation, electron-hole pairs are generated in the oxide, where the fraction of holes that escape the initial recombination, travel towards the Si-Si0 2 interface [2].A fraction of the holes are trapped at defects (primarily O vacancies) near the Si-Si0 2 interface, resulting in oxide traps, while the remaining holes induce interface trap states in the band gap [10], [11] at the Si-Si0 2 interface.In Modern MOS devices, given the Ultra-Thin nature of the oxide (below 10 nm), the rate of decrease in hole trapping is very significant, resulting in significantly lower radiation induced oxide traps but also lower interface traps [12].However, given that most reports on the study of radiation on MOS devices have been experimental (covered in a review paper [2]), while relatively few simulation frameworks and models exist that take the effect of the radiation-induced defects in electronic material and devices [13], [14].This necessitates the development of a simulation methodology for considering the effects of radiation on the behavior of MOS devices.This requirement is furthermore pressing for Multi-Gate MOS devices [15], particularly Double-Gate (DG) SOI MOS devices, where reports of improved channel electrostatics compared to Bulk MOS devices can be realized in terms of wider applicability of these devices in circuit applications only through assessment of the robustness of these devices to extraneous factors, such as radiation.
Based on the work by Dozier et al. [16] and Paillet et al. [17], the fraction of holes escaping the initial recombination can be calculated by means of the experimental model, which is also used in the TCAD radiation-based models [18].Even though the total trapped charge within the oxide is proportional to TID, this model is not valid anymore for high TID (in the order of several hundreds of 1530-4388 c 2024 IEEE.Personal use is permitted, but republication/redistribution requires IEEE permission.
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Mrad), where it is expected that at high doses, saturation of oxide and interface trap states should occur [19].Therefore, developing a simulation methodology capable of quantifying and analyzing the effect of radiation on MOS devices (specifically the Si-SiO 2 interface), including the impact seen at high doses, remains an important, yet challenging question to answer.
Recently, we have shown a methodology to take the effect of interface traps on the electrostatics of Ultra-Thin-Body (UTB) MOS devices by varying the band edge energy ( E edge ) [20] (which is generally used to passivate the Si-Si0 2 interface [21]) whereby the position and shape of these interface states within the band gap can be controlled by varying the edge energy at the top/bottom channel/oxide (Si−SiO 2 ) interfaces.By including these interface trap states within the boundary conditions of the self-consistent solution of the 1-D Band structure [22] (along the direction of confinement) and Poisson's equation, the device electrostatics was determined for UTB DG (Double Gate) MOS devices for a wide range of bias voltages and device temperatures [23].
In this work, we co-relate the charge yield (fraction of holes that escapes initial recombination, which is a function of the oxide electric field and varies based on the type of radiation [24], [25]), for SiO 2 and the band edge energy ( E edge ), so that the number of unrecombined holes due to radiation can be correlated with the interface trap states seen due to the variation of the band edge energy ( E edge ).This will enable the type and dose of radiation to be related with the band edge energy for a range of E edge values.We use the fraction of unrecombined holes for different radiating particles (such as X-ray, γ radiation, α particles, etc..) and calculate the highest permissible dose (HPD) value, a parameter that reflects the dose value which results in the highest trap state density at the Si/SiO 2 interface, for different channel thicknesses and device temperatures for UTB DG MOS devices.
The rest of the paper is organized as follows.Section IIdiscusses the band structure-based simulation methodology, where the calculation of band structure-related parameters like the density of states, band edge energy ( E edge ) and interface trap states density is discussed.The radiation mechanism for various radiating particles like γ , x-ray, protons and α particles and its behavior with applied bias is discussed in Section III.In the next section, we presented a methodology to consider the effects of radiation in the band structure simulation framework for a wide range of gate voltages, channel thicknesses, and device temperature.Finally, a conclusion based on the work discussed in this paper is presented.

II. BAND STRUCTURE-BASED SIMULATION METHODOLOGY
To obtain the band structure of a UTB channel Double Gate MOS structure, as shown in Figure 1, we have used the sp 3 d 5 s * semi-empirical Tight Binding approximation (TB), as discussed by Rahman [26] (also discussed in the supplementary material (see supplement S1)).The TB Hamiltonian (Tight-Binding Parameters from Tan et al. [27] used for T = 300 K and Jancu et al. [28] for T = 0 K) is assembled  on each k-point in the irreducible Brillouin zone (BZ), and eigenvalues of the Hamiltonian plotted along k space give the band structure of the channel material.This enables the calculation of various parameters like the band gap, Density of states (g(E)), all of which are critical in calculation of the channel electrostatics.Furthermore, through incorporation of the band edge energy E edge , at the interface in the TB Hamiltonian, the impact of Interface traps states (N it ) on the calculation of electrostatics can also be considered, all of which will be discussed briefly in subsequent subsections.

A. Density of States (g(E)) and E edge Potential
From the band structure (with Si/SiO 2 interface being hydrogen passivated), the density of states g(E) can be calculated by counting the number of occupied electronic states (N) per unit separation of energy, i.e., g(E) = dN dE .The g(E) is then multiplied by the reciprocal area, which is (0.05 × 2π a ) 2 to obtain the g(E) in per eV per unit area.The band structure and the density of states are plotted in Fig. 2(a), where no interface trap states are found to be present within the band gap of silicon.
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It may be noted that the interface and its behavior are implemented in the band structure simulation by changing the basis set of the Hamiltonian from the conventional 10orbital sp 3 d 5 s * set to the set of sp 3 hybridized orbitals aligned along the dangling bond directions [21] (details discussed in supplement S2).
Figure 2 shows the band structure, along with the density of states, for channel thickness of T si = 2 nm, for various band edge energies.Hydrogen passivation at the interface is implemented through the introduction of a band edge energy shift ( E edge ) of 20 eV (greater than 5 eV, where no states are found within the band gap), with this energy shift being high enough to discourage the electrons from occupying the surface atom orbitals, thus eliminating all interface states in the band gap of the channel material (see Figure 2(a)).To see the effects of interface states on the electrostatics, energy shift ( E edge ) of less than 5 eV (a typical of −2 eV) is considered, shown in Figure 2(b).It may be seen that the energy bands change their shape from the conduction band minima and start moving towards the valence band as we vary the edge energy ( E edge ).From Figure 2, we see that a significant increase in the density of interface trap states (D it ) is seen in the channel material (silicon) with a decrease in band edge energies.

B. Interface Traps Determination From Band Edge Method ( E edge )
The nature of the Interface trap (due to the dangling bond at the channel/oxide interface [29]) depends on the energy level of the trap state within the band gap.Trap states with energy level below mid-gap (E i ) exhibit donor-like characteristics, while those states with energy level above E i exhibit acceptorlike characteristics.Also, those trap states that are located below the Fermi level are considered to be negatively charged when trapping an electron.This means that those trap states that lie between the mid-gap (E i ) and the Fermi level (see Figure 1) will trap electrons and reduce the number of free electrons available for conduction.On the other hand for PMOS devices, since negative gate voltages are applied, the acceptor-like trap levels are entirely empty, and the donor-like trap levels are partially filled (between E i and E F ), leading to positively charged traps at the Si/SiO 2 interface.The effect of those donor traps can be considered by using the concept of band edge energy for PMOS devices also but for negative applied gate voltage.
In this study, we focus on the effect of acceptor traps, on device electrostatics for NMOS devices, at positive gate bias where channel inversion is likely to occur.To calculate the number of interface states per unit area, the density of states of the interface trap states that lie in the band gap between the Fermi energy level and the mid-gap energy level is combined with the probability of occupancy of those trap states.The number of interface states is computed as, where, g(E it ) is the density of trap states per eV per unit area (D it ), E F is the equilibrium Fermi level, E it is the discrete interface trap energies.

C. Electrostatics Calculation
Through the modification of the well-known model for temperature-dependent band gap in bulk semiconductors, defined by Varshni [30], Mishra et al. [23] have proposed a temperature-dependent band gap correction term ( E T g (T ch , T)) valid for UTB MOS structures.This band gap correction term is incorporated into the band structure obtained from the TB Hamiltonian assembled at T = 0 K, through which the band structure at any temperature between T = 0 K and 300 K can be determined (see supplement S3).With the selection of significant k-points [22], we selfconsistently solve the band structure on those k-points with the Poisson's equation, as shown in the algorithm in Fig. S2 (see supplement S3).This approach enables the efficient and accurate simulation of the electrostatics of a UTB DG MOS device over a wide range of channel thicknesses and temperatures (0 K to 300 K).
The band structure is solved self-consistently with the Poisson's equation, which is given as where, φ is the electrostatic potential, is the dielectric constant of the channel and ρ(z) is the electron density along the direction of the channel thickness (T si ).It may be noted that, the channel thickness is a function of the number of atomic layers (N), which, for the case of (100) surface, is given as The electron density, ρ(z), used in equation ( 3) is given below where, n is the number of energy levels in the conduction band, the factor F is the Fermi-Dirac probability, g k is the degeneracy factor at each k-point, E F is the Fermi level and ψ k n (z) is the normalized wave function along z direction.For an intrinsic channel the E F will be equal to E g /2.
To see the effect of interface trap states in the electrostatics, the total charge at the semiconductor/oxide interface due to the presence of a trap state is calculated, which can be expressed as, The charge contribution from the presence of interface traps calculated for various E edge is shown in Figure S4 (see Supplement S4) incorporated through the boundary condition Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.at the top and bottom surface, as shown in equation ( 6), in the self-consistent 1-D Poisson's solution shown in Fig. S2, Therefore, in this section, we have discussed how the Full Band Structure can be calculated using the Tight-Binding (TB) approximation, through which the channel electrostatics can be calculated through a self-consistent solution of the Band structure with the 1-D Poisson's equation.Furthermore, we have also discussed how the effect of surface (channel/oxide interface) passivation and surface traps can be considered in the calculation of the channel electrostatics, through suitable modification of the boundary conditions (shown in equation ( 6)).

III. RADIATION MECHANISM
Figure 1 shows the schematic of a typical Ultra-Thin Body (UTB) Double Gate (DG) MOS device, where the effect of radiation on the MOS device is seen in terms of the creation of electron-hole (e-h) pairs.This figure also shows the energy band diagram depicting the effect of radiation on inducing trap states at the Si/SiO 2 interface, likely to be seen in the band gap of the channel material (Silicon).The mechanisms by which device degradation occurs are depicted in Figure 3.When a MOS transistor is exposed to high-energy ionizing irradiation, electron-hole pairs are created in the oxide.Immediately after electron-hole pairs are created, most of the electrons will rapidly drift (within picoseconds) toward the gate, and holes will drift toward the interface.However, even before the electrons leave the oxide, some of the electrons will recombine with holes.The fraction of electron-hole pairs that escape recombination is called the electron-hole yield or charge yield, shown in Figure 4.
Those holes that escape "initial" recombination will transport through the oxide toward the interface by hopping through localized states in the oxide.As the holes approach the interface, some fraction will be trapped, forming a positive oxide-trap charge.It is believed that hydrogen ions (protons) are likely released as holes "hop" through the oxide or as they are trapped near the interface.The hydrogen ions can also drift to the interface, where they may react to form interface traps (N it ).The process is very sensitive to applied electric Fig. 4. The fraction of holes that escape initial recombination (charge yield) for x-rays, low-energy protons, gamma rays, and alpha particles [2].field, temperature, and oxide thickness [31].An interface state can be formed if a P b center (i.e., three-coordinated silicon atoms with an unpaired electron) traps a hole or a H + ion (i.e., proton) [32].Experimental data [33] indicates that most interface states are generated in reactions with the participation of hydrogen.
In the next section, we discuss how the type and dose of radiation can be correlated with the interface trap states seen at the surface.

IV. CORRELATION METHODOLOGY
In order to correlate the effect of radiation on the Si-SiO 2 interface with the band structure based simulation of interface states, we now consider two independent scenarios.In the first scenario, we consider a passivated Si-SiO 2 interface (interface passivation with hydrogen species) where due to the effect of radiation of Dose (D) on SiO 2 , there is a certain number of unrecombined holes which is shown in equation (7).(7) where N h is the number of unrecombined holes at the Si − Si0 2 interface, f (E ox ) is the experimentally determined hole yield or the fraction of unrecombined holes as a function of oxide electric field, D is the dose (rad), t ox is the oxide thickness (in cm units), g 0 is a material-dependent parameter that gives the initial charge pair density per rad of dose (g 0 = 8.1x10 12 pairs per cm −3 per rad for Si − Si0 2 [24]).It is seen from Figure 4 that as the hole yield is different for different types of radiation, the effect of each of these radiations, for the same radiation dose, will be quantifiably different on the Si-SiO 2 interface (some radiations such as α and protons are heavier than others such as γ ).Amongst the holes that escape re-combination in the oxide, if g 1 represents the fraction of those holes that do not contribute to oxide traps, but instead contribute to interface traps, then the number of radiation induced interface traps can be expressed as g 1 × N h .This implies that (1−g 1 )×N h contributes to oxide traps (due to radiation).Due to the Ultra-Thin nature of the oxide (T ox = 1 nm, in this entire study) and given the high quality gate-oxide deposition techniques used in Modern MOS devices, including Multi-Gate MOS devices such as DG SOI MOS devices, the effects of radiation on oxide trap generation Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.can be neglected, because of which the parameter g 1 can be considered to be 1.Amongst the number of unrecombined holes that contribute to traps at the Si/SiO 2 interface, let us assume that the traps are uniformly distributed in the channel material (Si) band gap.Considering that interface traps that lie between the Fermi Energy level (E F ) and the intrinsic/midgap energy level (E i ) are non-neutral acceptor traps, with f int (shown in equation ( 8)), representing the distribution function of these acceptor traps: where E c and E v are conduction band minima and valence band maxima energy levels respectively.The fraction of acceptor type interface traps (f int ) is plotted as a function of gate voltage in Figure 5, for different SOI channel thicknesses and device temperatures.Based on this the radiation induced interface traps can be quantified as shown in equation (9).
We now consider the second scenario, where an unpassivated Si-SiO 2 interface is simulated through a band structure simulation by varying the band edge energy parameter ( E edge ) at the interface, resulting in interface trap states (N it ) being seen in the band gap, as shown in Figure 2. By equating radiation induced trap states with band structure based simulation of interface trap states (N it ), we correlate the impact of the radiation dose and the type of radiation on interface trap state density.
By replacing N h with the expression shown in equation ( 7) and the experimentally determined charge yield function (f, only dependent on E ox and hence V g ) with a E edge dependent charge yield function (f edge also dependent on V g ), we obtain the expression shown in equation (10).Note that f edge and Dose (D) are both unknown parameters and need to be determined.Hence, the product of dose and f edge is shown in equation (10).(10) In equation (10), it may be noted that a fraction of the unrecombined holes generated due to radiation that did not result in oxide traps have resulted in the generation of interface traps.However, this when viewed from the channel material (Silicon) perspective implies that from band structure based simulation, through band edge energy variation, these interface traps would be equal to the radiation induced interface traps, as seen in equation (10).Such an equalization of the radiation induced interface traps and the band structure based approach of simulating interface traps is made possible by replacing f (E ox ) in equation ( 7) with the mapping function (f edge ), which enables the translation of the unrecombined holes due to radiation to non-neutral acceptor traps at the Si/SiO 2 interface, that is dependent on the band edge energy and gate voltage.Therefore, f edge is critical in enabling the correlation of the radiation Dose and oxide thickness to the interface traps simulated through the band structure based approach.
For a particular value of N it (obtained from equation ( 1)) equation (10) shows that the product of the radiation dose (D) and the Band Edge Energy dependent charge yield mapping function (f edge ) is constant.This basically means that knowing the charge yield and the type of radiation will enable the determination of the radiation dose or vice-versa.Additionally, it may be noted that for a specific band edge energy, this product of the radiation dose (D) and f edge is a function of gate voltage and has identical functional behavior to N it .For a particular band edge energy, we see that the value of N it reaches a maximum value for a specific gate voltage [20], which is where the product of the radiation dose (D) and f edge also reaches a maximum value.We then determine the highest permissible dose (HPD), for a particular band edge energy by dividing the maximum value of the product of the radiation dose (D) and f edge with the maximum value of the experimentally determined charge yield function (f sat , shown in Figure 4), as shown in equation (11).

HPD =
max Df edge (V g ) f sat (11) This value obtained from equation (11) refers to the highest permissible dose (HPD), of a particular radiation type, experienced by SiO 2 , for a specific band edge energy.The HPD is defined as that value of dose which results in the highest density of non-neutral acceptor trap states at the interface (Si/SiO 2 ) for an NMOS device, in-turn resulting in the highest degradation at the interface, for a particular band edge energy.Any further increase in radiation dose beyond this value does not change the interface trap state density.The relation of Dose with HPD and N it is shown in the definition in equation ( 12), where N it(max) refers to the maximum degradation in the interface due to radiation: In equation (11), it may be noted that the value of f sat is determined at a high oxide electric field (where the function, f (E ox ), saturates), with the effect of temperature on hole yield being negligible and hence f sat is independent of temperature [34].By using HPD from equation (11) and substituting this in equation ( 10), the f edge (the number of unrecombined  holes) is determined as a function of gate voltage for different band edge energies, as shown in equation (13).
The steps described above are encapsulated by the algorithm outlined in Figure 6, which can be applied to different types of radiations, at different SOI channel thicknesses (2 nm and 7 nm) and device temperatures (15 K, 300 K), through which we obtain the highest permissible dose (HPD) as a function of band edge energy and the charge yield (fraction of unrecombined holes) as a function of gate voltage, as shown in Figure 7 and Figure 8, respectively.In Figure 7, the HPD of a specific type of radiation is shown for different band edge energies where it is seen that HPD is higher for heavy particles like α compared to lighter particles like γ .For room temperature, the value of HPD (at E edge = −2 eV), at T si = 2 nm, is higher than that seen for T si = 7 nm, as shown in Figure 7(a) and (b).This is due to strong Quantum Confinement Effects (QCEs), seen for T si = 2 nm, resulting in increased band gap (increased energy separation between the conduction band energy (E c ) and fermi energy (E F )) and hence higher N it values (please see Supplement S4), compared to T si = 7 nm.Also, from Figure 7, it is seen that there is a clear shift seen in the range of E edge ((−3 eV < E edge < 3 eV) for 300 K and (−4 eV < E edge < 2 eV) for 15 K, which could be because of further increase in band gap at lower device temperatures.However, for T si = 7 nm, when the device temperature is reduced from 300 K to 15 K, the range of E edge values appears to have narrowed down, due to a possible compensation of the decrease in band gap because of reduction in confinement with the increase in band gap due to reduction in device temperatures.
In addition to the HPD, f edge plays an extremely important role in determining the number of radiation induced surface trap states (N it ).The band edge energy dependent charge yield function (f edge ) is shown as a function of gate voltage for different SOI channel thicknesses, device temperatures and E edge values in Figure 8.Here it may be seen that, for E edge = −2 eV, for T = 15 K (see Figures 8(c) and (d)), f edge increases nearly linearly and reaches the maximum value only at the highest gate voltage applied, while on the other hand, for T = 300 K, both for T si = 2 nm and 7 nm (see Figures 8(a) and (b)), f edge reaches the maximum value at lower gate voltages and nearly saturates.On the other hand, for E edge = 2 eV, the increase in f edge is relatively more linear with gate voltage, irrespective of device temperature and SOI channel thickness.However, for T = 15 K, the increase in f edge is delayed (commences at a higher gate voltage) and has a sharper rise with gate voltage compared with the room temperature case.This shows that the band edge energy dependent charge yield for T = 15 K is lower than that for T = 300 K, for most gate voltages.It may also be seen from Figure 8, that f edge for heavier particles such as α is much lower than that of lighter particles such as γ (Co-60), in all the cases.The highest permissible dose (HPD in Mrad) as a function of (a) interface trap fraction of holes (g 1 ) for the case of α particle (b) maximum interface trap density for various radiating particles for T si = 2nm and 7 nm, where g 1 = 1.In the determination of HPD, an important aspect to note is the factor g 1 , which represents the fraction of unrecombined holes due to radiation dose that contribute to the interface traps.The value of g 1 lies between 0 and 1, where 0 represents that the unrecombined holes contribute only to oxide traps, while a value of 1 contributes only to interface traps.The effect of g 1 therefore impacts the HPD values, with a lower value of g 1 reducing the number of radiation induced interface traps, thus increasing the HPD values, while these values tend to decrease, with increasing value of g 1 (where the number of radiation induced interface trap density increases), as shown in Figure 9(a).Also, with greater structural confinement, for T si = 2 nm, the values of HPD tend to be higher, indicating greater robustness of the DG SOI device to the effects of radiation.From the definition of HPD included earlier in equation ( 12), the increase in HPD results in an increase in the maximum interface trap state density, seen in Figure 9(b).Additionally, it may be noted from Figure 9(b) that the HPD values are nearly identical for lighter radiation particles with a slightly higher HPD being seen for heavier particles, when T si = 2 nm.Also, from the definition of HPD, it becomes clear that even when the radiation dose exceeds the value of HPD, the impact of radiation on the interface behavior and device electrostatics tends to saturate.
Additionally, we show how the impact of radiation induced interface traps can be quantified on the C-V (gate capacitance versus gate voltage) characteristics of the UTB DG SOI MOS device, for different SOI channel thickness, in Figure 10.Here we have considered three cases, corresponding to a passivated interface, a slightly and highly degraded interface, respectively, where it can be clearly seen that with increase in radiation dose, due to increase in interface trap state density, the C-V characteristics tend to shift and stretch.It may also be noted that the effect of radiation induced interface traps is found to be more significant for T si = 2 nm (see Figure 10(a)), where strong QCEs are seen, compared to T si = 7 nm (see Figure 10(b)).
Another parameter of interest in the analysis of the channel electrostatics is the sub-threshold swing, which is shown as a function of radiation Dose (HPD) for a typical radiation type (2 MeV α particle), as shown in Figure 11(a).The results obtained using the band structure simulation approach presented in this work are compared with results from Jazaeri et al. [19], for T si = 2 nm and 7 nm.The results by Jazaeri et al. are SOI channel thickness independent and agree well with results from the dose-dependent band structure simulation results at low radiation doses.While the model for Sub-Threshold swing given by Jazaeri et al. continues to predict a near monotonic increase in Sub-Threshold swing with radiation Dose, on the other hand, in this study, we present the concept of HPD indicating the maximum possible degradation in the interface trap density (N it(max) ), thus also resulting in the saturation of the Sub-Threshold swing [19].Therefore a deviation is seen between the values predicted by Jazaeri et al. and the present approach, when the interface trap density induced due to radiation exceeds N it(max) , for a particular band edge energy.Further, we also model the HPD, which can be related with E edge through a polynomial fit, (non-linear equation) shown in equation (14).where, e represents E edge in equation ( 14) for simplicity.Here the coefficients p 1 , p 2 ,. . ., p 6 are critical in ensuring good agreement between the HPD obtained from equations ( 11) and ( 14), for different band edge energies and SOI channel thicknesses, as shown in Figure 11(b).The coefficients for different SOI channel thicknesses and device temperatures are shown in Supplement S5, where the values of the coefficients are different for different radiation particles.Through using the model, shown in equation ( 14), and utilizing the algorithm shown in Figure 6, the channel electrostatics of a DG SOI device can be determined for different SOI channel thicknesses, device temperatures, radiation dose and radiation types.

V. CONCLUSION
In conclusion, we show how an atomistic band structurebased approach can be utilized to assess the impact of radiation on UTB MOS devices, by correlating radiationinduced interface traps with the band edge energy ( E edge ).We demonstrated how different radiating particles and doses impact the band edge energy ( E edge ) and hence the interface charge and device electrostatics.We observed that the interface undergoes lesser degradation due to heavier particles like α as compared to lighter particles such as γ (Co-60), for the same radiation dose.Furthermore, through the band structure based approach, the worst case effect of radiation dose on the interface and the electrostatics are quantified for a wide range of device temperatures, channel thicknesses and particles.Through this kind of band structure based approach and electrical characterization, we are able to assess the robustness against radiation of the UTB SOI MOS device, which is an important measure of the reliability of such devices.

Fig. 1 .
Fig. 1.Typical schematic of Ultra-thin body (UTB) double gate (DG) MOS device with Si channel atoms (100) at oxide/channel un passivated interface, i.e., E edge < 5eV with the impact of radiation (Top) along with the impact of radiation on energy band diagram and band structure (Bottom).

Fig. 2 .
Fig. 2. The variation of band structure and density of states of Si channel, for T si = 2 nm, V g = 0 V and T = 300 K, when E edge takes values of: (a) 20 eV (passivated) (b) −2 eV (unpassivated).

Fig. 3 .
Fig. 3. Band diagram of a MOS capacitor with a positive gate bias illustrating the main processes for radiation-induced charge generation inside the SiO 2 oxide [2].

Fig. 5 .
Fig. 5.The fraction of traps found between mid gap energy level and fermi energy level (f int ) as a function of gate voltage for T si = 2 nm and 7 nm at low and room temperature.

Fig. 6 .
Fig. 6.Algorithm for determining dose and fraction of unrecombined holes, for different E edge and radiations.

Fig. 7 .
Fig. 7.The Highest Permissible Dose (HPD) in Mrad for various particles as a function of E edge (eV) for (a) T si = 2 nm (b) T si = 7 nm at 300 K (c) T si = 2 nm (d) T si = 7 nm at 15 K.

Fig. 9 .
Fig. 9.The highest permissible dose (HPD in Mrad) as a function of (a) interface trap fraction of holes (g 1 ) for the case of α particle (b) maximum interface trap density for various radiating particles for T si = 2nm and 7 nm, where g 1 = 1.

Fig. 10 .
Fig. 10.Comparison of the gate capacitance fully passivated case (No traps) with low and high doses of (a) T si = 2 nm, (b) T si = 7 nm at 300 K, where g 1 = 1.

Fig. 11 .
Fig. 11.(a) Comparison of the SS from the proposed approach with the model shown by Jazaeri et al. [19] and (b) Validation of the proposed model with the simulation, for T si = 2 nm and 7 nm, at T = 300 K, for 2-MeVα particle.