153 © 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License The Effect of Scattering of Phonons, Size and Grain Boundary on Electrical Properties for Ruthenium Nano Metals Reda F. Hanon Almajedi1* , May A. S. Mohammed2 , and Haider FA. Abdul Amir 3 1,2 Department of Physics ,College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad ,Baghdad, Iraq. 3University of Algoma, Canada. *Corresponding Author. Received: 11 March 2024 Accepted: 2 June 2024 Published: 20 January 2025 doi.org/10.30526/38.1.3957 Abstract The study examines the impact of thickness on the electrical resistance of Ruthenium at room temperature. By applying the Fuchs-Sondheier and Mayadas Shatzkces models, the study establishes a linear relationship between thickness and grain boundary scattering. The M.S. model is crucial in calculating the size impact, accounting for all types of scattering affecting grain boundaries. On the other hand, the F.S. model focuses on explaining conduction electron scattering on material surfaces, particularly on tiny grains. The study's equation, derived from these two models, considers surface scattering and metal resistance to determine an experimental thickness that depends on metal resistivity. The Boltzmann Equation can be utilized to solve this equation. The study highlights the significance of Ruthenium as a common component of electrical and electronic circuits in producing electronic chips due to its excellent electrical conductivity. Keywords: Phonons, electrical properties, size, grain. 1. Introduction Ruthenium (Ru) is a versatile material that finds applications in various fields. In the electronic and electrical industry, Ru is used for the production of electronic chips due to its excellent electrical conductivity. This property of Ru can also be utilized in the production of solar cells to generate solar energy. It is worth noting that Russian chemist Karl Karlovich Klaus (1796-1864) discovered Ruthenium. He provided positive evidence for new elements derived from platinum and suggested naming the newly discovered element (Ruthenium) after the name of the ancient Russian country (Ruthenia) (1) . When materials are at the nanoscale level, resistance increases due to more scattering centers that conduct electrons, primarily at the boundaries between grains and surfaces. As the size approaches the average electron-free route, more resistance is observed. Electrons only collide with surfaces and grain boundaries at that point, becoming significant https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ https://doi.org/10.30526/38.1.3501 https://orcid.org/0009-0002-5009-5872 mailto:reda.hanon1704b@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0003-3883-402X mailto:may.as@ihcoedu.uobaghdad.edu.iq https://orcid.org/0009-0003-5974-1444 mailto:haider@salam.uitm.edu.my IHJPAS. 2025, 38 (1) 154 compared to their collision with other lattice errors such as impurities, point defects, and vacancies. Understanding the difference between surface and grain boundary contributions is essential when studying the resistive behavior of polycrystalline thin films (2) . Theoretical studies estimate the size effect by applying the F.S. model to surface electron dispersion, characterized by the surface effect coefficient (p), and the M.S. (3) . It is observed that electrical resistivity increases as thickness decreases because resistivity thickness depends on Fermi surface area values and metal- free paths (4,5). The uneven surface of the Grain boundary reflection in the F.S. and M.S. Models allows for the identification of different metals. To match the F.S. and M.S. theories to the experimental data, a simple equation was developed. This equation enables quick fitting of the two-theory model to the experimental data(6). 2. Materials and Methods The grain boundary reflection coefficient (R) represents the limit of electron scattering(3). Moreover, the statistical study of the conduction electron distribution created by varying the collision mechanism's enforced rate of change forms the basis of the (F.S.) and (M.S.) theories. When two fields, an electric field (E) and a magnetic field (H), exist simultaneously, Boltzmann's equation takes the following form(7-9): − e m∗ (V̅ + H̅ + E̅) . grad v̅ f + v̅ . grad r ̅ f = ( ∂f ∂t ) coll. = − f − f0 τ (1) f1(V, Z) = e E τ0 m∗ ∂f0 ∂VX [1 + F(V)exp ( −Z τ0 VZ )] (2) In this context, f0 represents the Fermi Dirac function, while f represents the distribution function of the charge carriers. Meanwhile, τ represents the relaxation time, e represents the charge of the electron, m* represents the effective electron mass, t represents the integration variable, and v represents the electron velocity. It is worth noting that two electron distribution functions exist: one with VZ > 0 and another with f1- for electrons with VZ < 0.(9,10) f1 +(V , Z) = e E τ0 m∗ ∂f ∂VX {1 − exp ( −Z τ0VZ )} VZ > 0 (3) f1 −(V , Z) = e E τ0 m∗ ∂f0 ∂VX {1 − exp ( d − Z τ0VZ )} VZ < 0 (4) J(Z) = −2 e ( m∗ h ) 3 ∫ ∫ ∫ VX f1 dVX dVy dVZ (5) The Fermi surface (F.S.) integral is typically denoted as A(Z, P) and is a function of the atomic number (Z) and momentum (P). The integral is used to calculate various properties of metals, including their electrical conductivity, and Planck's constant (h). (11,12) : A (Z, P) = ∫ ( 1 t3 − 1 t5 ) ∞ 1 1 − exp (−Z t) 1 − p exp (−Z t) dt (6) Where is B (Z, P, ∝) the integral of M.S. as given by [11][9][13] : B (Z, p, ∝) = ∫ d∅ ∫ cos2 ∅ H (t , ∅) [ 1 t3 − 1 t5 ] X 1 − exp [−Z t H (t, ∅)] 1 − p exp[−Z t H (t, ∅)] ∞ 1 π 2⁄ 0 dt (7) IHJPAS. 2025, 38 (1) 155 elucidation of the resistivity rate ( ρf /ρ0 ) to the bulk metal (9,14) : 𝜌𝑓 𝜌0 = 3 8𝑍 (1 − p) + 1 (8) In this context, ρ0 represents the bulk resistivity, while ρf represents the thin resistivity. Z denotes the rate, which is equal to (Z = d/l0), where d represents thickness and l0 represents the mean free path. p represents the surface scattering coefficient. To find the total resistivity of metals, the condensed solution based on the (F.S) and (M.S) models computes the surface scattering and grain boundary reflection coefficient (15,16). When calculating the electrical resistivity of conducting electrons as a function of thickness, the scattering at grain boundaries is crucial. (17-19). ∝= l Z R 1 − R (9) The parameters (R) are utilized to determine the coefficient of reflection at the Ru grain boundary, with (∝) equaling R = 0.043 for Ru (13), to make it easier to evaluate resistivity (16)and grain size at different grain sizes, while analyzed, goes on (20,21). The electron surface scattering-related Fuchs-Sondheier estimate in Eq (8) is used to determine the resistance of metals, while M.S. electron scattering for grain boundaries is determined from Eq 10 (22). 𝜌𝑓 𝜌0 = 1 + 3 8𝑍 (1 − 𝑝) + 3 2 ∝ Z ≫ 1 ∝≪ 1 (10) The two models show that Ru is one of the conductive metals (23) . Its dependent temperature in terms of electrical resistance increases with decreased thickness(4,24). Grain boundary scattering is dependent on the average linear distance between grain boundaries, whereas surface scattering is directly related to film thickness and scattering (25). Grain-boundary scattering is achieved by solving a (Boltzmann equation) with background and electrical resistance(7). The concentration of a material influences the electrical properties of metals (26,27). Lastly, it was discovered that the temperature of all conductor metals affects electrical resistivity (28,29). has an inverse relationship with thickness(30-32). which equal (𝜌0 = 7.8 × 10−8Ωm), mean free path (MFP) equal (𝑙0 = 6.6 nm), (𝜌0𝑙0 = 51.48 × 10−17Ω𝑚2 ) (33). 3. Results and Discussion The experimental portion of Ru is shown in Figure 1, from which the data for the surface reflection coefficient was obtained by using equation (8) for (F.S) It seems that the theory developed by Fuchs-Sondheimer provides a way to obtain numerical values for the surface scattering coefficient in the range of (0-0.9), as well as the lowered resistivity based on equations with varying thicknesses. This measurement is dependent on the Fuchs-Sondheimer theory, which also takes into account conduction electron scattering on metal surfaces, particularly background scattering and grain boundaries (33,34). IHJPAS. 2025, 38 (1) 156 Figure 1. Resistance versus thickness for Ruthenium. The intersection point of Ruthenium, originally (𝜌𝑑), and the slopes of Ruthenium, determined using values between (ρ and d), where d is the (x-axis) and ρ is the (y-axis), are obtained from the preceding Figure 1, Then Intersection point for (Ru= 68.03 * 10−17Ω𝑚2), while slope of 𝜌0 Ru (7.878 µ𝛺. cm ) as a following Figure 2. It seems that in the theoretical study shown in the Figure 1, the intersection point represents the resistance of the metal. However, due to defects, the difference in resistance at that point is very small. Figure 2. Point of intersection Ruthenium with thickness. Then as indicated in Equation (8) application. Determine the surface scattering coefficient (𝑝) for Ru, where (𝑝𝑅𝑢= 0.6) (33,34). based on total scattering, as shown in the figures below Figure 3. 0 5 10 15 20 25 0 10 20 30 40 50 𝜌 ( µ 𝛺 . c m ( d (nm( Ru ρ.orginal y = 7.878x + 68.03 0 50 100 150 200 250 300 350 400 450 500 0 10 20 30 40 50 𝜌 𝑑 ( µ 𝛺 . 𝑐 𝑚 ^2 ) d (nm) Ru IHJPAS. 2025, 38 (1) 157 Figure3. Theoretical changing of the resistivity and thickness of Ruthenium. Using the M.S. model, equation (10) calculated the reflection coefficient for Ruthenium at grain boundaries. Additionally, the value of bulk resistivity for Ru (𝜌0 = 6.8 × 10−8 Ωm), (𝑙0 = 6.6 nm), ln (𝜌0𝑙0 = 51.48 × 10−17Ω𝑚2) (33-37). As the result shown in the staying Figure 4, we computed the new resistivity by applying the surface scattering coefficient (p) and reflection coefficient (R) in Equation 10 for Ru. The Mayadas-Shatzkes model takes into consideration three scattering mechanisms, namely background scattering, grain-boundary scattering, and external surface scattering. Background scattering refers to the scattering of particles in the same direction in which they radiate after passing through a thick material. Grain-boundary scattering, on the other hand, is caused by binary defects in the crystalline structure of a solid that reduce electrical conductivity. Finally, external surface scattering results in a reduction of electrical conductivity as the scattered electron loses its velocity along the direction parallel to the surface of the conduction direction (5). Figure 4. Resistivity and thickness of Ruthenium. 0 5 10 15 20 25 30 0 10 20 30 40 50 𝜌 ( µ 𝛺 . 𝑐 𝑚 ) d (nm) Ru ρ.p=0.6 ρ.orginal 0 5 10 15 20 25 0 10 20 30 40 50 𝜌 ( µ 𝛺 . c m ) d (nm) Ru ρ.p=0.6 ρ.orginal ρ.R=0.043 IHJPAS. 2025, 38 (1) 158 4. Conclusion The electrical resistivity of Ru metal was calculated in this study using the Boltzmann equation's solution due to its strong electrical conductivity and numerous other advantages, the most important being its use in electrical and electronic circuits to produce electronic chips. Utilizing a Fermi surface, Fuchs-Sondheimer observed a phenomenon known as the size effect hypothesis for the free electron model. The statistical analysis used in their theory is based on the Boltzmann equation, which describes the distribution of conduction. As a result, F.S. theory is also used to describe conduction electron scattering on metal surfaces, focusing on background scattering and grain boundaries. The Fuchs-Sondheier model-based Equation 8 was used to calculate the surface scattering coefficient. In contrast, Equation 10 was used to obtain the grain- boundary reflection coefficient, which is based on the F.S and M.S models. 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