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A5: Nanoscale and low dimensional effects
The electron states of graphene nanoribbons depend strongly on their size and geometry. For the armchair nanoribbons [1] the boundary conditions for the wave function are vanishing on both sublattices at the edges, and two Dirac valleys are mixing. The armchair nanoribbons are metallic under the critical widths and insulating otherwise. The motion of the armchair nanoribbon carriers on the wire is coupled with the transverse motion only through the relative phases of the wave functions.
The elastic scattering by a screened potential of a charged impurity and the carrier relaxation time τ(ε) have been calculated in the Born approximation for the wave functions of armchair nanoribbon and spectrum with energy gap Δ (ε=±[(h/2π)2u2py2+Δ2]1/2, u is the Fermi velocity). With the help of the kinetic Bolzmann equation we have found the expressions of the electric conductivity σyy and the thermoelectric coefficient βyy for an arbitrary temperature. We have obtained the termoelectric power Syy=-βyy/σyy for different values of the chemical potential μ, temperature T and the energy gap Δ. Maxima |Syy| increase with the size of the gap Δ. Due to the one-dimensional nature of movement the max|Syy| becomes larger than in the case of two-dimensional motion in the unrestrected graphene [2] (e.g. twice at T=5K and Δ=50K). For nonzero Δ the sign of Syy is inverse to the sign of μ, and Syy=0 at μ=0. At Δ=0 and |μ|>>T the thermopower Syy=-(π2/3e)T/μ is the same as in ordinary metals.
In zigzag nanoribbons the thermoelectric effect may be only conditioned by the inelastic electron transitions.
[1]. L.Brey and H.A.Fertig. Phys.Rev. B73, 235411 (2006). Phys.Rev. B75, 125434 (2007).
[2]. S.G.Sharapov, A.A.Varlamov. Phys.Rev. B86, 035430 (2012).