# Influence of Impurity Scattering on Surface Plasmons in Graphene in the Lindhard Approximation
## 1 Introduction  
The building blocks of all contemporary digital electronic devices, communications, and circuits, which are employed to process, transport, and store information, are made up of metal-oxide-semiconductor field-effect transistors (MOSFETs) [1, 2, 3]. As many issues in the vast majority of scientific fields and economy rely on very fast transistors nowadays, the demand for low-cost and high-speed MOSFETs with lower power consumptions is very strong. The clock speed of information processing in transistors directly pertains to the size, miniaturization, and density of comprised elements in them [4]. In contrast, smaller transistors have high speeds but there is a strong challenge related to the interconnect number. By reducing the size of the latter, the delay time increases and imposes critical limitations on the clock speed frequencies [3]. Fiber-optic cables are another promising candidate for communication and the transportation of information due to their high transparency [2, 5]. They are capable of carrying information about one thousand times faster compared to electronic circuits. In addition, in fiber-optic cables, light packets carry information as they travel over long distances [3].  
These fiber-optic features suggest a synergy between these two technologies [6]. However, because the strength of light-matter interactions in these dielectric materials is not sufficient to exhibit nonlinear behaviour, they require a high power density and volume and lead to limitations in their integration with electronic circuits [7]. The major obstacle is a consequence of the diffraction limit, which does not allow light to squeeze in a region smaller than its wavelength [2, 3, 8]. One promising candidate to face this challenge is surface plasmons (SPs) [9, 10, 11, 12, 13]. An SP is a sub-wavelength evanescent electromagnetic wave that propagates at the interface of a metallic-dielectric medium [14, 15]. The SP frequency in the noble metal is in the order of femtoseconds, reflecting higher clock speeds compared with state-of-the-art transistors [1]. SPs enable us to investigate the light-matter interaction in the sub-wavelength area, which is otherwise inaccessible. SPs allow one to manipulate and route light at the nanoscale [3, 6, 12, 13, 16]. They have a plethora of prospective applications such as optical switches, spacers, near-field optical microscopy, bio-sensing and nanosensors, improving efficiency of light-emitting diodes (LEDs), solar cells, cancer treatment, biomedical tests, quantum sensing, and teleportation [17, 18, 19, 20, 21, 22, 23, 24, 25, 26].  
One of the most interesting materials that supports SPs at room temperature is graphene. In contrast to bulk-dielectric systems that support only transverse magnetic (TM) SPs [27], graphene also supports transverse electric (TE) and TM SPs [28]. In addition, SP losses in graphene are lower than those in noble metals. Scattering, e.g., by impurities, can profoundly affect SPs. In the literature, these effects has been regarded phenomenologically by heuristically introducing a Drude-form conductivity in the long-wavelength limit [29; 30]. In this paper, we borrow some results from Ref. [31] and evaluate TM and TE SPs in graphene in the presence of impurity scattering.  
In Section 2, we present some general expressions for the Lindhard polarization in the presence and absence of impurities as well as the transmission and reflection amplitudes. In Section 3, we investigate TM SPs in graphene for intra-band and inter-band transitions, renamed, for clarity and brevity, as SB and TB transitions, in the presence and absence of impurity scattering. In Section 4, we evaluate the quality factor (QF) of these SPs. In Section 5, we evaluate transverse electric surface plasmons for SB and TB transitions. Our summary follows in Section 6.