Validity criteria for Fermi’s golden rule scattering rates applied to metallic nanowires

Abstract

Fermi's golden rule underpins the investigation of mobile carriers propagating through various solids, being a standard tool to calculate their scattering rates. As such, it provides a perturbative estimate under the implicit assumption that the effect of the interaction Hamiltonian which causes the scattering events is sufficiently small. To check the validity of this assumption, we present a general framework to derive simple validity criteria in order to assess whether the scattering rates can be trusted for the system under consideration, given its statistical properties such as average size, electron density, impurity density et cetera. We derive concrete validity criteria for metallic nanowires with conduction electrons populating a single parabolic band subjected to different elastic scattering mechanisms: impurities, grain boundaries and surface roughness. *

Resistivity scaling model for metals with conduction band anisotropy

“…The impact is more pro-nounced in case of an isotropic collision time as compared to an isotropic mean free path and this can be understood by having a closer look at the conductivity expressions in Eqs. (11), (20) and (26). In case of an isotropic collision time, both the effective mass along the transport direction and the parameter for grain boundary scattering are renormalized w.r.t.…”

“…While it is hard to guarantee film continuity for the samples at very low thicknesses, several limitations and approximations underlying the Mayadas-Shatzkes model and its extension (including the consideration of a three-dimensional wave vector, the phenomenological treatment of boundary surface scattering and the perturbative treatment of grain boundary scattering) could also be (partially) responsible for a systematic underestimation (already remarked by Choi et al [22]). A recent validity analysis showed that this is certainly the case for a perturbative treatment of boundary surface scattering due to surface roughness and grain boundary scattering for extremely narrow nanowires [11]. Additionally, the consideration of a single (diagonal) effective mass (tensor) could be too approximate for complicated Fermi surfaces such as for Ru.…”

“…Many semiclassical and quantum mechanical resistivity scaling models (see [1][2][3][4][5][6][7][8][9][10][11][12]) have been developed over the last decades and have provided a satisfactory description for the thickness dependent resistivity of metal thin films down to nanometer scale thicknesses [13][14][15][16][17][18]. The thickness dependence can mainly be attributed to grain boundary and boundary surface scattering, whose impact on the resistivity increases when the film thickness is reduced.…”

Theoretical study of scattering in graphene ribbons in the presence of structural and atomistic edge roughness

“…One should of course keep in mind that this is a perturbative approach, which is valid only for smallenough width variations. 62 The resistivity as a function of the roughness correlation length Λ AER is shown in Fig. 6(a) for the three 10-nm-wide GNRs of Fig.…”