Modeling Radiation-Induced Degradation in Top-Gated Epitaxial Graphene Field-Effect-Transistors (FETs)

Abstract

This paper investigates total ionizing dose (TID) effects in top-gated epitaxial graphene field-effect-transistors (GFETs). Measurements reveal voltage shifts in the current-voltage (I-V) characteristics and degradation of carrier mobility and minimum conductivity, consistent with the buildup of oxide-trapped charges. A semi-empirical approach for modeling radiation-induced degradation in GFETs effective carrier mobility is described in the paper. The modeling approach describes Coulomb and short-range scattering based on calculations of charge and effective vertical field that incorporate radiation-induced oxide trapped charges. The transition from the dominant scattering mechanism is correctly described as a function of effective field and oxide trapped charge density. Comparison with experimental data results in good qualitative agreement when including an empirical component to account for scatterer transparency in the low field regime.

1. Introduction

Graphene field effect transistors (GFETs) have received significant interest from the electron-device community in the past few years due to the novel electronic properties of graphene, such as ambipolar field effect, high carrier mobilities and high thermal conductivity [1,2,3]. Due to the lack of a band-gap, GFETs are unsuitable for logic applications. However, the excellent transport properties described above make graphene FETs a good choice for high-speed RF applications where a high I_on/I_off ratio is not required. The work presented here investigates the total dose response of RF GFETs intended for space and other applications that encounter ionizing radiation environments and describes extraction and modeling techniques for radiation induced damage.

As described in [4], there have been a few studies of ionizing radiation effects on graphene and on graphene-based materials. Initial investigations of total ionizing dose (TID) effects on graphene-based devices were done using back-gated GFETs and performed under different conditions (e.g., irradiated and characterized under vacuum or in air) resulting in a different radiation response [4,5,6]. In a typical back-gated GFET test structure the graphene layer is exposed causing the radiation response to be strongly dependent on the experimental ambient condition. As described in [5], different mechanisms can contribute to the radiation response. For example, in [4,7] it was reported that chemical doping (p-type) that results from oxygen adsorption and/or reactions with the graphene layer during X-ray exposure resulted in positive shifts in the current-voltage (I-V) characteristics of devices irradiated in air. In [5], irradiation of back-gated GFETs in a controlled environment (i.e., under static vacuum) confirmed that GFETs are susceptible to oxide charge trapping similar to silicon (Si) metal-oxide-semiconductor FETs (MOSFETs) as manifested by negative voltage shifts in the I-V characteristics. Thus, carefully controlling the experimental environment when investigating the basic radiation mechanisms in back-gated GFETs with exposed graphene layers is crucial [5]. Similar results have been reported for the radiation response of single-walled carbon nanotube thin-film-transistors [8].

As explained in [2], back-gated GFETs are appropriate for proof-of-concept purposes, but cannot be easily integrated with other components. Moreover, the thick dielectric layers in back-gated GFETs are highly vulnerable to radiation-induced charge buildup. As described in [9], practical graphene transistors must be patterned with local back-gates or otherwise have individual top-gate electrodes. In this work, we present evidence of TID effects in top-gated epitaxial GFETs. Oxide trapped charge near the SiO₂/graphene channel interface is identified as the main contributing mechanism to radiation-induced degradation. The buildup of oxide-trapped charges results in negative voltage shifts in the I-V characteristics and degradation in carrier mobility. These effects are determined experimentally by in situ characterization of GFETs exposed to Co-60 gamma rays. Extractions of radiation-induced oxide-trapped charge density (N_ot) are incorporated into a semi-empirical model for GFET effective carrier mobility. The modeling approach is verified through the comparison with GFET effective electron mobility and its radiation response extracted from the degraded I-V characteristics.

2. Experimental Section

Top-gated GFETs fabricated with epitaxial graphene layers grown on Si-face 6H-SiC substrates via Si sublimation were used in this study [10]. A schematic diagram of the top-gated GFET and a scanning electron microscope image of the test structure are shown in Figure 1a,b [10]. The top-gated GFETs were fabricated using Ti/Pt/Au source and drain contacts as well as Ti/Pt/Au metal gates over a 35 nm SiO₂ dielectric layer deposited via electron beam evaporation [10,11]. Two-finger GFET structures with a 3 μm gate length (L) and various gate widths (W) were selected for total dose experiments. The test structures were exposed to Co-60 gamma rays at the Naval Research Laboratory (NRL) at a dose rate of ~910 rad(Si)/s with a positive gate bias of V_g = 3 V and all other terminals grounded. Electrical characterization was performed following step-stress irradiations up to a total dose of 1000 krad(Si). The characterization consisted of measuring the output characteristics (i.e., I_d-V_ds) by sweeping V_d from 0 to 3 V and stepping V_g from 2 V to −4 V in −1.0 V steps for V_s = 0, and the transfer characteristics (i.e., I_d-V_gs) for V_d = 0.1 and 1 V for V_s = 0. For the transfer characteristics dual gate sweeps were done with V_g being swept from 2 V to −5 V then back to 2 V to observe the effects of gate hysteresis. In the top-gated GFET test structures used in this study, the graphene layer was not directly exposed to the experimental environment (i.e., it is covered by the SiO₂/metal-gate stack; see Figure 1). Nevertheless, half of the samples were irradiated in air and the other half with a continuous flow of ultra pure N₂ purging oxygen, moisture and other contaminants from the irradiation vessel. Both cases revealed negative shifts in the I-V characteristics but significant shifts in the minimum conductivity were only observed for devices tested in air. Additionally, gate hysteresis did not change significantly as a function of TID but consistent shifts in the negative and positive V_g sweeps were observed in the post-irradiation I_d-V_gs characteristics of all devices. These shifts are due to radiation-induced positive oxide trapped charge and were analyzed independently of the mechanisms that cause hysteresis. The results are summarized and discussed in Section 3.

Figure 1. (a) Schematic diagram of top-gated graphene FET (GFET) fabricated with epitaxial graphene layers grown on Si-face 6H-SiC substrates via Si sublimation; (b) Scanning electron microscope (SEM) image of the GFET test structure.

Figure 1. (a) Schematic diagram of top-gated graphene FET (GFET) fabricated with epitaxial graphene layers grown on Si-face 6H-SiC substrates via Si sublimation; (b) Scanning electron microscope (SEM) image of the GFET test structure.

3. Results and Discussion

3.1. Current-Voltage Characteristics

Shown in Figure 2a are the I_d-V_g characteristics for a two-finger GFET with W = 25 μm and L = 3 μm following step-stress irradiations in air up to 1000 krad(Si) and measured using V_ds = 1 V. The measurements in Figure 2a reveal negative shifts in the I-V characteristics as well as an increase in the minimum drain current (I_d-min) as a function of TID. These results indicate a buildup of N_ot near the SiO₂/graphene layer interface shifting the minimum conductivity gate voltage (i.e., the Dirac voltage V_Dirac) negatively as well as increasing the minimum conductivity (σ_min). The increased σ_min results from increasing graphene channel intrinsic charge density (q_int) as a function of dose (i.e., with time during the experiment). In Figure 2a the solid lines are I_d measurements from positive to negative V_g sweeps and the dashed line are from negative to positive V_g sweep measurements as indicated by the arrows in the 0 krad(Si) curve. These are included to indicate hysteresis effects even prior to irradiation. Figure 2b is a plot of the extracted shifts in the Dirac point (i.e., ΔV_Dirac = V_Dirac − V_Dirac0) and the relative increase in I_d-min (i.e., I_d-min/I_d-min0) from the data in Figure 2a. V_Dirac0 and I_d-min0 are respectively the Dirac voltage and I_d at V_g = V_Dirac for 0 krad(Si) (i.e., prior to irradiation). The extractions in Figure 2b indicate a monotonic shift in V_Dirac and an increase in I_d-min with total dose.

Figure 2. (a) Drain current (I_d) plotted as a function of the gate voltage (V_g) for increasing total ionizing dose (TID) up to 1000 krad(Si), V_d = 1 V, V_s = 0 V. Measurements are for a two-finger GFET with W = 25 μm and L = 3 μm; (b) Extracted shifts in the Dirac voltage (ΔV_Dirac) and relative increase in I_d-min plotted as a function of TID.

Figure 2. (a) Drain current (I_d) plotted as a function of the gate voltage (V_g) for increasing total ionizing dose (TID) up to 1000 krad(Si), V_d = 1 V, V_s = 0 V. Measurements are for a two-finger GFET with W = 25 μm and L = 3 μm; (b) Extracted shifts in the Dirac voltage (ΔV_Dirac) and relative increase in I_d-min plotted as a function of TID.

Shown in Figure 3a are the I_d-V_d characteristics for V_gs = 0 V, −1 V and −2 V measured at the same dose levels as show in Figure 2. The measurements in Figure 3 indicate an increase in I_d as a function of total dose for all values of V_gs. The increase in I_d is consistent with the negative shifts in I_d-V_g characteristics (i.e., at a given V_g > V_Dirac, I_d will increase as a function of dose for all values of V_d due to the negative voltage shift in the I_d-V_g characteristics). Using the I_d-V_d measurements for all V_g steps (i.e., from 2 V to −4 V), it is possible to represent I_d-V_g curves for any V_ds value contained within the sweep range of V_d (i.e., between 0 and 3 V). For example, shown in Figure 3b are representations of I_d-V_gs for V_ds = 0.1 V and for all dose levels. In Figure 3b symbols are data and solid lines are a least squares fits to the data. Figure 4 is a plot of ΔV_Dirac and Δσ_min obtained from the data in Figure 3b. By comparison, the I_d-V_g representations retain the TID effects observed in Figure 2.

Another TID effect, although not completely apparent from the measurements in Figure 2, Figure 3, is the degradation of carrier mobility. Degradation of carrier mobility is due to increased carrier scattering mechanisms that result from the buildup of oxide-trapped charges near the SiO₂/graphene channel interface [5,12]. The increase in carrier scattering as a function of ionizing radiation is demonstrated in section 3.3 through calculations of effective electron mobility (μ_eff) based on extractions of drain conductance , ΔV_Dirac and Δσ_min. Using ΔV_Dirac and Δσ_min plotted in Figure 4 (extracted from the I_d-V_g representations in Figure 3b) ensures that any contributions from hysteresis effects are equivalent for the extractions of g_ds since the same I_d-V_ds measurements are used for all of the extractions.

Figure 3. (a) Drain current (I_d) plotted as a function of the drain voltage (V_d) for increasing TID up to 1000 krad(Si), V_g = 0, −1 and −2 V, V_s = 0 V. Measurements are for a two-finger graphene field effect transistors (GFET) with W = 25 μm and L = 3 μm; (b) I_d-V_g representation for V_d = 0.1 V and for all dose levels. Symbols are data and solid lines are a least squares fits.

Figure 3. (a) Drain current (I_d) plotted as a function of the drain voltage (V_d) for increasing TID up to 1000 krad(Si), V_g = 0, −1 and −2 V, V_s = 0 V. Measurements are for a two-finger graphene field effect transistors (GFET) with W = 25 μm and L = 3 μm; (b) I_d-V_g representation for V_d = 0.1 V and for all dose levels. Symbols are data and solid lines are a least squares fits.

Figure 4. Shifts in the Dirac voltage (ΔV_Dirac = V_Dirac − V_Dirac0) and minimum conductivity (Δσ_min = σ_min − σ_min0) obtained for V_ds = 0.1 V in Figure 3b. Note: V_Dirac0 = −2.58 V, σ_min0 = 4.7 e²/h (e = 1.6 ×10¹⁹ C, h = 6.63 ×10⁻³⁴ J-s).

Figure 4. Shifts in the Dirac voltage (ΔV_Dirac = V_Dirac − V_Dirac0) and minimum conductivity (Δσ_min = σ_min − σ_min0) obtained for V_ds = 0.1 V in Figure 3b. Note: V_Dirac0 = −2.58 V, σ_min0 = 4.7 e²/h (e = 1.6 ×10¹⁹ C, h = 6.63 ×10⁻³⁴ J-s).

3.2. Experimental Ambient Conditions

Shown in Figure 5a,b are the I_d-V_g and the I_d-V_d characteristics for a GFET device with W = 12 μm and L = 3 μm irradiated with a continuous flow of ultra pure N₂ purging the testing vessel. The response observed in Figure 5a,b reveals negative voltage shifts consistent with positive trapped charge as seen for devices irradiated in air. However, the devices irradiated with the N₂ purge have less hysteresis and show negligible shifts in σ_min. These differences indicate that molecular adsorption at the SiO₂ surface near the graphene channel (resulting from moisture and contaminants in the experimental environment) does have an effect, albeit reduced, on the electrical characteristics of top-gated GFET test structures. Therefore, it is possible that any increase in σ_min might be due to contaminants in the testing environment and not an effect of ionizing radiation exposure. In that case, the observed increase in σ_min for devices tested in air might be the result from accumulated adsorption of contaminants in the testing environment. For devices exposed with flowing N₂, the concentration of the contaminants was significantly reduced and therefore σ_min did not change significantly. Figure 6 plots extractions of hysteresis width (h) as a function of TID for both devices irradiated in air and with flowing N₂. Hysteresis width is extracted as the difference in the V_g intercept at a constant current for the negative and positive sweeps. The extractions of h reveal larger widths for devices tested in air, as expected due molecular adsorption at the surface of the SiO₂ contributing to charge trapping mechanisms that cause hysteresis [13]. However both experiments show negligible changes in hysteresis with TID. The radiation-induced degradation of effective carrier mobility is analyzed independently of the mechanisms that cause hysteresis in Section 3.3 based on these observations.

Figure 5. (a) Drain current (I_d) plotted as a function of the gate voltage (V_g) for increasing TID up to 1000 krad(Si), V_d = 1 V, V_s = 0 V. Measurements are for a two-finger GFET with W = 12 μm and L = 3 μm; (b) Drain current (I_d) plotted as a function of the drain voltage (V_d) for V_s = 0 V and for increasing TID up to 1000 krad(Si).

Figure 5. (a) Drain current (I_d) plotted as a function of the gate voltage (V_g) for increasing TID up to 1000 krad(Si), V_d = 1 V, V_s = 0 V. Measurements are for a two-finger GFET with W = 12 μm and L = 3 μm; (b) Drain current (I_d) plotted as a function of the drain voltage (V_d) for V_s = 0 V and for increasing TID up to 1000 krad(Si).

Figure 6. Extraction of hysteresis width (h) as a function of TID for devices tested in air and with flowing N₂ purge. Symbols are data and dashed line is the mean hysteresis width.

Figure 6. Extraction of hysteresis width (h) as a function of TID for devices tested in air and with flowing N₂ purge. Symbols are data and dashed line is the mean hysteresis width.

3.3. Mobility Degradation

The radiation-induced degradation of carrier mobility is determined through calculations of effective electron mobility in the graphene layer (μ_eff) as a function of the effective vertical field (E_eff). The effective electron mobility is given by

(1)

where Q_gc is the graphene channel charge density [14]. The effective vertical field in the graphene channel is estimated as [10]

(2)

and the graphene channel charge density is calculated as

(3)

Here, q_int is the graphene channel intrinsic charge density that results from charge inhomogeneity in the graphene layer typically described as electron-hole puddles that arise from the random charged impurity potential fluctuations and contribute to the non-zero minimum conductivity at the Dirac point [15,16,17,18].

Shown in Figure 7a is g_ds plotted as a function of V_g for all dose levels. Calculations of μ_eff using Equations 1–3 are plotted in Figure 7b as a function of E_eff and for increasing TID. In the calculations, q_int is estimated by σ_min/μ_FE, μ_FE is the peak field effect mobility and g_ds is extracted for V_ds = 0.1 V. The results in Figure 7b reveal a decrease in μ_eff with increasing total dose as can be observed at E_eff ~ 1 MV/cm. Deviations from the pre-irradiation μ_eff(E_eff) characteristics are due to the increased contribution of scattering mechanisms that results from radiation-induced oxide trapped charges. The deviations from the pre-irradiation μ_eff(E_eff) curve are expected to continue towards lower values of E_eff (i.e., towards lower carrier densities n ∝ E_eff) where scattering due to charged impurities is the dominant mechanism. Other scattering mechanisms such as short-range scattering dominate for large carrier densities (n) [15,19] (i.e., for large E_eff) and the μ_eff(E_eff) characteristics converge to a universal response. In Figure 7b this convergence appears for E_eff > 2 MV/cm.

In the approach presented here, the calculations of Q_gc (and E_eff) capture the contribution of radiation-induced oxide trapped charges through the shifts in V_Dirac. Therefore, this approach allows determining the radiation-induced degradation of carrier mobility in GFET devices without ambiguities introduced by voltage shifts in the I_d-V_g characteristics and the changes in σ_min. These ambiguities exist, for example, when plotting μ_FE as a function of V_g at various levels of TID. Nonetheless, it should be noted that discrepancies in the calculations of μ_eff could result from the approximations made in Equations 1–3. Interface states have not been included in the extractions presented here, but can be incorporated by including an interface trap charge density (bias-dependent) term Q_it(V_g, N_it) in Equation 3. Additionally, non-uniform buildup of N_ot may result from non-uniform field distributions in the gate oxide (e.g., in regions not covered by the top gate). These conditions would require a more sophisticated calculation of Q_gc, but the basic methodology presented here would still apply. The following section describes a semi-empirical modeling approach for the degradation of effective carrier mobility based on the experimental observations of μ_eff(E_eff) and its response to ionizing radiation.

Figure 7. (a) Drain conductance (g_ds) plotted as a function of the gate voltage at V_d = 0.1 V for increasing dose levels up to 1000 krad(Si); (b) Effective electron mobility (μ_eff) plotted as a function of the effective vertical field (E_eff) for increasing total dose levels up to 1000 krad(Si).

Figure 7. (a) Drain conductance (g_ds) plotted as a function of the gate voltage at V_d = 0.1 V for increasing dose levels up to 1000 krad(Si); (b) Effective electron mobility (μ_eff) plotted as a function of the effective vertical field (E_eff) for increasing total dose levels up to 1000 krad(Si).

4. Modeling

This section presents a semi-empirical modeling approach for describing effective carrier mobility in GFET devices and its degradation due ionizing radiation exposure. The radiation-induced shifts in V_Dirac can be directly related to N_ot as

ΔV_Dirac = −qΔN_ot/C_ox.

(4)

In (4), N_ot accounts for the buildup of radiation-induced trapped charges that image charge into the graphene layer, C_ox is the oxide capacitance per unit area and q is the electronic charge. The effective electron carrier mobility is expressed using Matthiessen’s rule as

(5)

where E_eff is calculated using Equations 2 and 3 and V_Dirac as given by Equation 4 (i.e., V_Dirac = V_Dirac0 − qΔN_ot/C_ox). In Equation 5, μ_C is the Coulomb scattering limited effective carrier mobility and is modeled as a function of N_ot and Q_gc as

(6)

In Equation 5, μ_SR corresponds to the short-range scattering that dominates for large n and is given by

(7)

Equations 6 and 7 are adapted from a semi-empirical model for inversion layer mobility in Si MOSFETs described in [20]. Figure 8a shows calculations of μ_eff using Equations 5–7 and plotted as a function of E_eff for increasing N_ot. For the calculations in Figure 8a, α = 0.005 V-s/cm², μ_SR0 = 1000 cm²/V-s, β = 2, a = 0.75, N₀ = 10¹² cm⁻², Q₀ = 10⁻⁶ C/cm², E₀ = 10⁶ V/cm, E_eff is in units of V/cm, Q_gc in units of C/cm² and N_ot is in units of cm⁻². The calculations in Figure 8a correctly describe the reduction in μ_eff with N_ot (i.e., μ_eff ∝ 1/N_ot) for low n and the merging with μ_SR (dashed lined) for large n. In the model, parameters a, μ_SR0, α and β can be treated as fitting parameters and Q₀, E₀ and N₀ are constants that allow a simpler interpretation of the units for E_eff, Q_gc and N_ot.

An increase in μ_eff for decreasing E_eff < ~1 MV/cm can be modeled empirically adding μ₀ = A(E₀/E_eff)^b to the calculation of μ_eff given by Equation 5. This allows obtaining good agreements with the experimental results. This increase in mobility for decreasing n has been reported in [19] and attributed to scatterer transparency occurring when carrier wavelengths exceed the spacing between scatterers. In this work, the μ_eff dependence to scatterer transparency in the low n limit is modeled independently from charged impurity and short-range scattering using the aforementioned empirical form. Shown in Figure 8b are calculations of μ_eff compared with extractions from experimental data from irradiated GFETs. For the calculations in Figure 8b the fitting parameters are α = 0.08 V-s/cm², μ_SR0 = 2000 cm²/V-s, A = 340 cm²/V-s, β = 4, a = 0.85, b = 1.08, N₀ = 10¹² cm⁻², Q₀ = 10⁻⁶ C/cm² and E₀ = 10⁶ V/cm. As shown in Figure 8b, the model calculations fit the experimental data with good qualitative agreement validating the presented approach. Additional experimental observations of scatterer transparency and the merging of μ_eff(E_eff) at large n for short range limited scattering would further validate the modeling of radiation-induced degradation as an independent mechanism utilizing Matthiessen’s rule. Nevertheless, the modeling approach presented here can be directly applied to calculations described in [21,22,23] for model radiation-induced degradation in GFET I-V characteristics.

Figure 8. (a) Calculations of the effective electron mobility (μ_eff) using Equations 5–7 and plotted as a function of the effective vertical field (E_eff) for increasing values of N_ot. The dashed line corresponds to the short-range mobility given by Equation 7; (b) Comparison of μ_eff extracted from data in Figure 5 (symbols) and model calculations (lines) including scatterer transparency.

Figure 8. (a) Calculations of the effective electron mobility (μ_eff) using Equations 5–7 and plotted as a function of the effective vertical field (E_eff) for increasing values of N_ot. The dashed line corresponds to the short-range mobility given by Equation 7; (b) Comparison of μ_eff extracted from data in Figure 5 (symbols) and model calculations (lines) including scatterer transparency.

5. Conclusions

In this work, we have investigated TID effects in top-gated epitaxial GFETs. Oxide charge trapping near the SiO₂/graphene channel interface has been identified as the main contributing mechanisms to radiation-induced degradation. The effects of the radiation-induced oxide trapped charges are manifested as negative voltage shifts in the I-V characteristics and carrier mobility degradation. These effects have been observed for devices exposed in air as well as for devices exposed with flowing ultra pure N₂ purging moisture contaminants from the irradiation vessel. Extractions of shifts in the Dirac voltage (ΔV_Dirac) indicate buildup of oxide trapped charge densities (ΔN_ot) of ~ 10¹² cm⁻² at dose levels of 1000 krad(Si). The experimental results also reveal a significant increase in minimum conductivity (σ_min) for devices tested in air. Since σ_min was initially low in these samples (i.e., σ_min0 = 4.7 e²/h) the increase may be due to compensation and/or reorganization of the potential fluctuations in the graphene layer as a result of molecular adsorption at the SiO₂ surface near the graphene channel [5].

The results reported in this work confirm that GFET devices with top-gated structures are also susceptible to TID effects that result from the buildup of oxide-trapped charges. Mitigation of these degradation mechanisms is required for the successful implementation of GFET devices in space-based applications. An example of such a mitigation technique based on local etching of the dielectric layer in the active regions of back-gated GFETs is described in [5]. However, GFET devices with independent top-gates are desirable for the fabrication of integrated circuits. The devices used in this work were research test structures with relatively thick gate dielectrics. Alternative mitigation approaches such as the using thin top-gate dielectric layers may be available soon as advances in the processing of graphene devices allows further scaling.

An approach for extracting the radiation-induced degradation of the effective carrier mobility from the degraded I-V characteristics of irradiated GFETs is presented in this paper. This approach accounts for shifts in V_Dirac as well as changes in the minimum conductivity by introducing an intrinsic charge density component (i.e., q_int) into the calculations of the total graphene layer charge (i.e., Q_gc). The extractions of effective electron mobility (μ_eff) indicate deviations from the pre-irradiation μ_eff(E_eff) dependence that are due to the increased contribution of scattering mechanisms resulting from oxide-trapped charges. For large values of effective vertical field (E_eff) the extractions merge at the pre-irradiation curve as mechanisms other than charged impurity scattering become dominant for large carrier densities (i.e., n ∝ E_eff). Similar results were observed recently for Cu-CVD back-gated GFETs irradiated with Co-60 gamma rays [24]. Additionally, an increase in μ_eff is observed for E_eff < 1 MV/cm that is consistent with the observations made in [19] and attributed to scatterer transparency that occurs as carrier wavelengths becomes larger than scatterer spacing. The presented approach can be extended to incorporate interface traps, non-uniform buildup of N_ot and the effects of contact resistance as needed based on further experimental investigation.

Additionally, this work introduces a semi-empirical approach for modeling radiation-induced degradation of μ_eff based on calculations of E_eff and Q_gc that incorporate the buildup of N_ot. The degradation of μ_eff due to charge impurity scattering is modeled as an independent scattering mechanism according to Matthiessen’s rule. The appearance of scatterer transparency is modeled empirically as a function of E_eff resulting in good qualitative agreement with the experimental extractions of μ_eff and its radiation response.