The effect of electron scattering from disordered grain boundaries on the resistivity of metallic nanostructures

Applied Surface Science, Volume 329, 28 Feb 2015, Pages 184–196.

Claudio Arenas1,2, Ricardo Henriquez3, Luis Moraga4, Enrique Muñoz5, Raul C. Munoz1.

 

  1. Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Blanco Encalada 2008, Casilla 487-3, Santiago 8370449, Chile and
  2. Synopsys Inc., Avenida Vitacura 5250, Oficina 708, Vitacura, Santiago, Chile and
  3. Departamento de Física, Universidad Técnica Federico Santa María, Av. España 1680, Casilla 110-V, Valparaíso, Chile and
  4. Universidad Central de Chile, Toesca 1783, Santiago, Chile and
  5. Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 7820436, Chile

 

Abstract

We calculate the electrical resistivity of a metallic specimen, under the combined effects of electron scattering by impurities, grain boundaries, and rough surfaces limiting the film, using a quantum theory based upon the Kubo formalism. Grain boundaries are represented by a one-dimensional periodic array of Dirac delta functions separated by a distance “d” giving rise to a Kronig–Penney (KP) potential. We use the Green’s function built from the wave functions that are solutions of this KP potential; disorder is included by incorporating into the theory the probability that an electron is transmitted through several successive grain boundaries. We apply this new theory to analyze the resistivity of samples S1, S2, S7 and S8 measured between 4 and 300 K reported in Appl. Surf. Science 273, 315 (2013). Although both the classical and the quantum theories predict a resistivity that agrees with experimental data to within a few percent or better, the phenomena giving rise to the increase of resistivity over the bulk are remarkably different. Classically, each grain boundary contributes to the electrical resistance by reflecting a certain fraction of the incoming electrons. In the quantum description, there are states (in the allowed KP bands) that transmit electrons unhindered, without reflections, while the electrons in the forbidden KP bands are localized. A distinctive feature of the quantum theory is that it provides a description of the temperature dependence of the resistivity where the contribution to the resistivity originating on electron-grain boundary scattering can be identified by a certain unique grain boundary reflectivity R, and the resistivity arising from electron-impurity scattering can be identified by a certain unique IMP mean free path attributable to impurity scattering. This is in contrast to the classical theory of Mayadas and Shatzkes (MS), that does not discriminate properly between a resistivity arising from electron-grain boundary scattering and that arising from electron-impurity scattering, for MS theory does not allow parameters (IMPR) to be uniquely adjusted to describe the temperature dependence of the resistivity data. The same data can be described using different sets of (R,IMP); the latter parameter can be varied by two orders of magnitude in the case of small grained samplesd < , and by a factor of 4 in the case of samples made out of large grains d >  (where  is the bulk mean free path at 300 K). For samples d > the increase of resistivity is attributed not to electrons being partially reflected by the grain boundaries, but to a decrease in the number of states at the Fermi sphere that are allowed bands of the KP potential; hence the reflectivity required by the quantum model turns out to be an order of magnitude smaller than that required by the classical MS theory. For samples d < ℓ, the resistivity increase originates mainly from Anderson localization induced by electron grain boundary scattering from disordered successive grains characterized by a localization length of the order of 110 nm and not from electrons being partially reflected by grain boundaries; the outcome is that the reflectivity required by the quantum theory turns out to be about 4 times smaller than that required by the classical MS theory.

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Additional Information: 

We present a quantum theory (QT) of the resistivity induced by electron-grain boundary scattering; unpublished details as well as previous research leading to this theory are being written as a monograph to be published by Lambert Academic Publishing. Following Mayadas and Shatzkes (MS), grain boundaries are represented by a one-dimensional periodic array of Dirac delta functions separated by a distance “d” giving rise to a Kronig-Penney (KP) potential. Disorder is included by assuming that the positions of the grain boundaries fluctuate around those corresponding to the periodic array according to a Gaussian distribution. The current density is calculated using the solutions of the Schrodinguer equation for the KP potential, so equally spaced grain boundaries give rise to extended states (in the allowed KP bands) that exhibit the translational symmetry of the boundaries and transmit electrons unhindered, without reflections, while electrons in the forbidden KP bands are localized. Consequently the resistivity is controlled by the collective properties of the grain boundaries: (a) by the forbidden KP bands projected onto the Fermi sphere that do not carry any current (that occur when boundaries are equally spaced), and (b) by TN, the transmission probability that describes electrons propagating across N disordered grain boundaries found along a bulk mean free path l that (because of disorder) decays exponentially with distance if N > 1. In MS theory TN =1, hence the resistivity originates from electrons that are reflected upon colliding with each grain boundary, violating the translational symmetry of equally spaced grains assumed by MS. We assess the predictive power of both theories by analyzing the resistivity measured at 20 temperatures 4 K < Ti < 300 K on 4 gold films whose morphology was measured with a STM; in 2 of these films d approximately equal to 11 nm, in the other two d > 110 mn. The Hall effect measured at 4 K confirms that in the 2 films where d approximately equal to 11 nm, electron-grain boundary scattering is the dominant scattering mechanism. The resistivity predicted by both theories agrees to within a few percent with the data. However, the grain boundary reflectivity R and the impurity concentration nIMP appropriate for QT are unique, while those appropriate for MS model are not: The resistivity predicted by MS can be described by several sets of different parameters (R, nIMP) because MS theory does not discriminate beween electron-grain boundary and electron-impurity/point defect scattering. We conclude that the predictive power of MS theory is questionable. Moreover, in samples where d approximately equal to 11 nm, the resistivity is dominated by Anderson localization induced by electron scattering from disordered grain boundaries, the relevant parameter is the degree of disorder. This suggests that Anderson localization may be the dominant mechanism controlling the resistivity of nanometric Cu interconnects planed by ITRS for the next decade.

Figure Legend

Figure Panel (a) illustrates a localized charge carrier. Panel (b) illustrates a weakly localized carrier, which is the mechanism generating resistivity on nanometric metallic structures made up of disordered grains; the weak localization is the result of Anderson localization induced by electron scattering from a disordered 1-D potential. Panel (c) illsutrates a free electron, the description of a charge carrier following the classical theory.

 

The effect of electron scattering from disordered grain  boundaries on the resistivity of metallic nanostructures. Advances In Engineering

 

 

 

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