Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene

Abstract

Analytical expressions are presented for the intraband conductivity tensor of graphene that includes spatial dispersion for arbitrarily wave-vector values and the presence of a nonzero Fermi energy. The conductivity tensor elements are derived from the semiclassical Boltzmann transport equation under both the relaxation-time approximation and the Bhatnagar-Gross-Krook model (which allows for an extra degree of freedom to enforce number conservation). The derived expressions are based on linear electron dispersion near the Dirac points, and extend previous results that assumed small wave-vector values; these are shown to be inadequate for the very slow waves expected on graphene nanoribbons. The new expressions are also compared to results obtained by numerical integration over the first Brillouin zone using the exact (tight-binding) electron dispersion relation. Very good agreement is found between the new analytical expressions and the exact numerical results. Furthermore, a comparison with the longitudinal random-phase conductivity is also made. It is shown analytically that these new expressions lead to the correct value of the quantum capacitance of a graphene sheet and that ignoring spatial dispersion leads to serious errors in the propagation properties of fundamental modes on graphene nanoribbons. DOI: 10.1103/PhysRevB.87.11542

Nonlocal electromagnetic response of graphene nanostructures

“…Coordinate transformation from spatial to spectral domain makes the sophisticated nonlocal equations tractable, i.e.,J(k,ω) = σ (k,ω)Ẽ(k,ω), where the tilde sign denotes quantities in the spectral domain. Graphene's conductivity tensor σ (k,ω) can be obtained by various techniques [22][23][24][25][26][27]. In this work, we employed the Bhatnagar-Gross-Krook (BGK) model [25], which provides us with an analytical form for the longitudinal and azimuthal components of σ (k,ω).…”

“…Graphene's conductivity tensor σ (k,ω) can be obtained by various techniques [22][23][24][25][26][27]. In this work, we employed the Bhatnagar-Gross-Krook (BGK) model [25], which provides us with an analytical form for the longitudinal and azimuthal components of σ (k,ω). According to the BGK model the spatially dispersive conductivity of graphene can be obtained from the following equations:…”

“…Moreover, the electric field polarization allows coupling only to the σ xx term. To first order in the field variations in the graphene sheet (i.e., ∇ · E), σ xx has the following form [25]: (8) where σ 0 is the graphene's conductivity in the local limit. Hence, the dimensionless quantity α…”

“…Rather unexpectedly, a considerable change in the spectra is observed when spatial dispersion effects are included. In general, the nonlocal longitudinal conductivity can include terms quadratic in q x and q y , and beyond [25], where the induced current is affected by the spatial variations transverse to the electric field. From our calculations, the transmission in the dc limit differs as much as 50% in the W = 60 nm array.…”

“…Figures 3(c) and 3(d) quantify the broadening due to nonlocal effects for varying dopings and widths, respectively. Since the dispersive part of the longitudinal conductivity includes terms in various powers of q x and q y [25], nonlocal contribution to the conductivity increases with 1/W β . This is consistent with the increased broadening as W decreases [ Fig.…”