Phenomenological Model for Defect Interactions in Irradiated Functional Materials

The ability to tailor the performance of functional materials, such as semiconductors, via careful manipulation of defects has led to extraordinary advances in microelectronics. Functional metal oxides are no exception – protonic-defect-conducting oxides find use in solid oxide fuel cells (SOFCs) and oxygen-deficient high-temperature superconductors are poised for power transmission and magnetic imaging applications. Similarly, the advantageous functional responses in ferroelectric materials that make them attractive for use in microelectromechanical systems (MEMS), logic elements, and environmental energy harvesting, are derived from interactions of defects with other defects (such as domain walls) and with the lattice. Chemical doping has traditionally been employed to study the effects of defects in functional materials, but complications arising from compositional heterogeneity often make interpretation of results difficult. Alternatively, irradiation is a versatile means of evaluating defect interactions while avoiding the complexities of doping. Here, a generalized phenomenological model is developed to quantify defect interactions and compare material performance in functional oxides as a function of radiation dose. The model is demonstrated with historical data from literature on ferroelectrics, and expanded to functional materials for SOFCs, mixed ionic-electronic conductors (MIECs), He-ion implantation, and superconductors. Experimental data is used to study microstructural effects on defect interactions in ferroelectrics.


I. Radiation and Phenomenological Model
Generally, radiation interacts with a material by causing either ionization or displacement events. Lower-energy sourcese.g., X-rays and gamma raystransfer energy to a material via electron interactions, ionizing the material with which they interact to form electron-hole pairs, and at higher energies, vacancy-interstitial pairs. 1,2 Irradiation with massive particles, such as protons and neutrons, transfers energy to a material through both electron interactions, as described previously, and atomic interactions, potentially resulting in displacement of atoms (vacancy-interstitial pairs) and subsequent defect cascades. 3 In a ferroelectric material, radiation-induced ionization and displacement events can potentially increase the stable defect concentrations, including the trapping of charges at preexisting defects, thus modifying the energy associated with such defects' energy, their mobility, and other forms of interaction with the material. Prior work on the effects of radiation on ferroelectric materials has shown a direct correlation between total radiation dose and degradation of functional properties. [4][5][6][7][8][9][10][11] Specifically, trapped charges generated as a result of X-ray, proton, gamma and neutron irradiation can result in degradation of polarization, dielectric, and electromechanical response, primarily through changes to the energies of defects of the ferroelectric material. [11][12][13][14][15] In addition to ionizing energy transfer to the material, highenergy electrons and massive energetic particles (protons, neutrons, alpha-particles, etc.) also transfer energy via nuclear interactions that lead to direct atomic displacement. These radiationinduced displacements may degrade the functional ferroelectric response in a fashion that is qualitatively similar to that of ionization-related defects (especially for point defects like Frenkel pairs) or may give rise to unique degradation modes in the case of multi-atom defects (defect clusters) that span over larger volumes. Ultimately, radiation-induced modification of functional response in ferroelectrics is the result of changes in defect concentration and energy in the material. Therefore, irradiation can be leveraged as a method for controlled introduction and/or activation of defects in these material, thus modifying material properties and functional response, to meet the needs of a variety of applications. Such methods stand to benefit from robust quantification and phenomenological modeling to more adequately and conveniently compare material performance as a function of radiation.
Here, we offer such a model, beginning with the relationship describing the change in material volume affected by defect interactions, Vdef, with the number of defects, N (assumed proportional to an external stimulus, such as radiation dose), which is a function of: (1) the volume fraction of the material that is not impacted by defects, Vfree/VT, where VT is the total material volume; and (2) the mean change in material volume impacted per new defect created/activated, VN; When the volume fraction of free material is large, the effective volume impacted per new defect is maximized, due to the fact that there is greater free volume vulnerable to defects and overlap with pre-existing defects is less probable. Conversely, as the volume fraction of free material becomes small (i.e., reduced availability of free volume "sites" to be impacted), the probability of interaction and overlap with pre-existing defects is increased, and thus, the effective volume influenced by a new defect decreases ( Figure 1). To account for material nonlinearities, such as grain boundaries, domain walls, pre-existing defects, etc., we introduce a weighting function, W(N). The weighting function modifies the relationship between a newly-introduced defect and the deviation from the mean volume it will impact, and thus contribute to increased defect interactions. Thus, where VN is a the mean volume pinned per defect introduced into the material and W(N) is the weighting function. Using the relationship we can substitute for the volume of free material, arriving at This expression is difficult to solve analytically without further simplification. We thus convert the expression for volume fraction to a normalized volume, thus eliminating the variable VT and constraining the normalized volume impacted by active defect interactions, Vd to the range from 0 to 1: Substituting this result into expression (S4) requires multiplying the left-most part by VT/VT in order to convert the differential to the appropriate relation of dVd: Dividing VT to the right-hand side of (S6) allows for normalization of the mean change in material volume impacted by defect interactions per new defect created/activated. We divide VN by VT to get φN. The result is Separation of variables and integration results in Fitting the exponential function to degradation trends requires use of functional response data. Changes in functional response data are directly dependent on change of the volume impacted by defect interactions in the sample. Furthermore, testing the boundary condition, a virgin sample with zero defects (N = 0) yields a defective volume of zero. Further evaluation of expression (S8) requires integration of the weighting function (see full article). We use the weighting function where k is a fitting coefficient related to the rate of defect saturation in the material. Solving, we arrive at We can then use this to fit the degradation trend data and extract the φN and k parameters. Supplementary Figure 1 shows the effects of arbitrary changes to the φN and k parameters as a function of a given radiation dose. It is worth noting that fittings are done to the decimal value of degradation in response, e.g. for 5% degradation, a value of 0.05 is used for the fitting. Additionally, degradation is assumed to be positive, as it is the result of increases to the volume impacted by defect interactions. The result in Equation (S10) is robust and capable of fitting both degradation and enhancement data. This derivation assumes that the degradation trend data begins at zero change in response for zero defects/radiation dose. Realistically, materials contain inherent defects prior to irradiation. However, given the nature of percent changes in measured response of control samples, degradation trend data should conceivably show minimal change at zero exposure to radiation.

II. Application of Phenomenological Model to Literature Data
We apply the phenomenological model developed in this work to data reported in the literature from X-ray, gamma, proton, and neutron irradiation studies of various ferroelectric materials, both in thin films and bulk forms. 6,12,14,[16][17][18][19] The results of fitting the model are tabulated in Supplementary Tables 1 to 6, and results from individual studies are also shown in Supplementary Figures 2 to 7. Notably, by comparing the values of φN within individual publication data from the literature, the phenomenological model is able to consistently and accurately reflect the conclusions of the authors when studying a variety of processing and measurement conditions, including dose rate, bias conditions, and different radiation types. We have also applied the model to the results of chemical doping in various ferroelectric materials (Supplementary Tables 7 and 8, Supplementary Figures 8 and 9), 20,21 as well as ion irradiation of yttrium barium copper oxide (YBCO) superconductors (Supplementary Table 9 Table 11, Supplementary Figure 12). Error bars are reproduced when available.
Studying the effects of various radiation types on ferroelectric thin films from Supplementary Figure 6 and Supplementary Table 5, we note that linear energy transfer (LET) in PZT for Xrays (10 keV), protons (3 MeV), and gamma rays (1.25 MeV) are 0.77 keV µm -1 , 38.25 keV µm -1 , and 0.05 keV µm -1 , respectively. [22][23][24] Notably, the LET for gamma radiation in PZT is much lower than both X-rays and protons. Comparing X-ray and gamma irradiation, the number of incident photons will be greater for gamma rays, but with a greater mean distance between them. Thus, fewer X-rays may impact the sample, but of those that do, more electron-hole pairs are generated. Additionally, at the lower energy of X-rays, there are many photoelectric lines (e.g., K, L, M, etc.) that potentially result in diverse ionization states of affected atoms, compared to simply ejecting outer shell electrons in the case of gamma irradiation.
On the other hand, the LET of protons is two orders of magnitude greater than that of X-rays. A greater dose rate conceivably translates to greater charge generation per unit volume, which could potentially increase the rate of recombination. Evidence of this effect is present in observed local enhancement of functional properties measured by Bastani et al., in samples irradiated with protons (Supplementary Figure 6). 12 Furthermore, smaller values of φN, the effective volume affected by radiation-induced defect interactions, in samples irradiated with protons compared to X-rays, support this hypothesis. Defects that could potentially degrade functional response are annihilated or their charge reduced to less-deleterious states, and the volume they pin is reduced. Additionally, the dose rate of protons is approximately 300 times that of X-rays, meaning 300 times fewer protons impact the surface than X-rays per unit time. Data from Oldham and McLean on irradiated MOS oxides demonstrated that the fractional hole yield for ionizing radiation (gamma, X-ray, electrons) was much greater than that of proton and alpha particle radiation, suggesting a more exaggerated interaction of ionizing radiation with the exposed material compared to particles. 25 The net result is a smaller mean volume affected by radiation-induced defect interactions in samples irradiated with protons compared to X-rays or gamma rays, where the fractional charge yield is greater (than that of proton irradiation) (Supplementary Table 5).
Supplementary Table 1. Extracted φN and k parameters from fitting equation (S10) to degradation data of various parameters from Zhang et al. 16 Note that values of φN are multiplied by three orders of magnitude to make interpretation more manageable.  Table 2. Extracted φN and k parameters from fitting equation (S10) to degradation data of various parameters from Gao et al. 19 Note that values of φN are multiplied by three orders of magnitude to make interpretation more manageable.  Table 3. Extracted φN and k parameters from fitting equation (S10) to degradation data of dielectric permittivity measurements on PbZr0.52Ti0.48O3 and PbTiO3. 17 Note that values of φN are multiplied by three orders of magnitude to make interpretation more manageable.  Table 4. Extracted φN and k parameters from fitting equation (S10) to degradation data of various parameters from Solovev et al. 18 Note that values of φN are multiplied by three orders of magnitude to make interpretation more manageable.  Table 5. Extracted φN and k parameters from fitting equation (S10) to degradation data of various parameters from Bastani et al. 12 Note that values of φN are multiplied by three orders of magnitude to make interpretation more manageable.
Bastani et al.  Figure 6. Application of phenomenological model (curves) to degradation of various functional responses of PZT films subjected to X-ray and proton irradiation. 12

X-rays Protons
Supplementary Table 6. Extracted φN and k parameters from fitting equation (S10) to dielectric degradation data as a function of neutron irradiation in PZT thin films by Graham et al. 14 Note that values of φN are multiplied by three orders of magnitude to make interpretation more manageable, and that fitting was done with neutron flux × 10 15 cm -2 .  Table 7. Extracted φN and k parameters from fitting equation (S10) to dielectric degradation data as a function of Ho2O3 dopant concentration in BaTiO3 ceramics by Paunovic et al. 21 Note that values of φN are multiplied by three orders of magnitude to make interpretation more manageable.  Figure 9. Application of phenomenological model (curves) to historical data on dielectric permittivity as a function of Fe2O3 dopant concentration in PZT ceramics. 20 Supplementary Table 9. Extracted φN and k parameters from fitting equation (S10) to degradation of conductivity in superconducting yttrium barium copper oxide (YBCO) by Clark et al. 26 Notably, the range of exposure doses for As ions is shorter than that of the O ions, but results in greater degradation and higher φN, suggesting that As ion bombardment is likely associated with a different degradation mechanism compared to O ions. Note that values of φN are multiplied by three orders of magnitude to make interpretation more manageable.  Table 10. Extracted φN and k parameters from fitting equation (S10) to current density and polarization response degradation data as a function of He 2+ -ion bombardment in ferroelectric thin films by Saremi et al. 27 Note that values of φN are multiplied by three orders of magnitude to make interpretation more manageable.  Figure 11. Application of phenomenological model (curves) to data on He 2+ -ion bombardment of epitaxial ferroelectric thin films. 27 Supplementary Table 11. Extracted φN and k parameters from fitting equation (S10) to degradation of total conductivity of In dopant yttrium barium cerate, compiled by Medvedev. 28 Note that values of φN are multiplied by three orders of magnitude to make interpretation more manageable. approximately 85 to 120 seconds, due to the continuous set of data at that location. Data has been normalized to the first point of the selected set, and the percent change from that point as the baseline plotted. The model fits well in (a), but does encounter some difficulty due to the starting point at 85 s and resulting discontinuity from 0 to 85 s. By shifting the initial point to 0 s (b), the model provides a better fit. We note that the model requires fitting the degradation/reduction of a property as a positive value (as explained in the manuscript), thus the negative trends here represent an increase to the electric current.

III. Crystallographic Phase Analysis
Crystallographic phase analysis was performed to confirm the texture of PZT samples prepared via 2-methoxyethanol-based (2-MOE) and acetic acid-based inverted mixing order (IMO) chemical solution deposition (CSD) processes (see Experimental Section). Supplementary Figure 14 shows baseline X-ray diffraction (XRD) phase analysis of the films deposited using 2-MOE-based and IMO-prepared solutions, and the resulting columnar and equiaxed grain morphologies, respectively.
Noteworthy are the large 100-and 200-textures in the columnar sample, indicating highlytextured samples; and the relatively large 101-peak in the equiaxed sample compared to the 100-and 200-peaks, indicating their more equiaxial nature and random orientation.
Supplementary Figure 14. X-ray diffraction crystallographic phase analysis comparing representative samples with columnar (top) and equiaxed (bottom) grain structures prepared using 2-MOE and IMO PZT solutions, respectively. Note the large relative intensity of the 100-peak for samples with columnar grains while the opposite behavior for the 101-peak is found for samples with equiaxed structure.

IV. TEM/TKD and Statistical Analysis
Cross-sectional transmission electron microscopy (TEM) and transmission Kikuchi diffraction (TKD) were performed at North Carolina State University (NC State) to observe domain and grain structure in the samples (Figure 3b, 3e). The regions of mottled contrast, primarily visible in the sample with columnar grains (Figure 3b), indicate potential nanodomains or striated 90 domain walls. 30,31 TKD and TEM images were analyzed to calculate statistical values of several grain size characteristics, including mean in-plane grain size, mean out-of-plane grain height, and mean size of multi-grain regions of similar dimension (Supplementary Table 12). These statistics aid in analysis of samples with columnar vs. equiaxed grains.

V. Functional Response Characterization
Dielectric, polarization, and electromechanical responses of the samples were fully characterized at Georgia Institute of Technology both before and after irradiation, including measurements of low-field permittivity and DC electric field-dependent electromechanical response, followed by irradiation and repetition of experiments (see Experimental Section). A summary of these measurements for samples with both columnar and equiaxed grain structures as a function of radiation dose is shown in Supplementary Table 13. A 600 second poling step at 10 V, approximately five times the coercive voltage, Vc, was performed directly before the electromechanical measurements in both pre-and post-irradiation measurement sets in order to eliminate anisotropic polarization contributions to the electromechanical response. All measurements were performed on the same sample/electrode both before and after irradiation in order to monitor precise changes in response behavior. Low-field dielectric permittivity (εr) measurements were conducted at 100 mV and 1 kHz using an Agilent 4284A precision LCR meter. Measurements of the converse, effective longitudinal piezoelectric response (d33,f) were performed on an aixACCT double beam laser interferometer (DBLI) measurement system up to 300 kV cm -1 DC bias with an overlapping AC signal VAC ≈ 0.5 Vc. All measurements reported are subject to experimental error up to 3-5%, due to sample variability. From the data in Supplementary Table 13, degradation trends were extracted by calculating the percent change in response from the virgin control sample. This data has been tabulated in Supplementary  Table 14. Supplementary

VI. Electron-Hole Pairs Generated by Gamma Irradiation
Exposure to gamma radiation results in the formation of electron-hole pairs (ehp) in ferroelectric PZT. Work by Leray et al., showed that radiation dose in PZT is 1.23 times that of the dose felt in Si, i.e. 1 Mrad(Si) = 1.23 Mrad (PZT). 33 Using a mean density of PZT as 7.6 g cm -3 and 100 rad = 1 Gy = 1 J kG -1 we can calculate the total energy deposited: