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We examine the effect of an external guide field and current sheet thicknesson the growth rates and nature of three dimensional unstable modes of anelectron current sheet driven by electron shear flow. The growth rate of thefastest growing mode drops rapidly with current sheet thickness but increasesslowly with the strength of the guide field. The fastest growing mode istearing type only for thin current sheets (half thickness $\approx d_e$, where$d_e=c/\omega_{pe}$ is electron inertial length) and zero guide field. Forfinite guide field or thicker current sheets, fastest growing mode isnon-tearing type. However growth rates of the fastest 2-D tearing mode and 3-Dnon-tearing mode are comparable for thin current sheets ($d_e < $half thickness$ < 2\,d_e$) and small guide field (of the order of the asymptotic value of thecomponent of magnetic field supporting electron current sheet). It is shownthat the general mode resonance conditions for electron-magnetohydrodynamic(EMHD) and magnetohydrodynamic (MHD) tearing modes depend on the effectivedissipation mechanism (electron inertia and resistivity in cases of EMHD andMHD, respectively). The usual tearing mode resonance condition($\mathbf{k}.\mathbf{B}_0=0$, $\mathbf{k}$ is the wave vector and$\mathbf{B}_0$ is equilibrium magnetic field) can be recovered from the generalresonance conditions in the limit of weak dissipation. Necessary conditions(relating current sheet thickness, strength of the guide field and wavenumbers) for the existence of tearing mode are obtained from the general moderesonance conditions.

Effect of guide field on three dimensional electron shear flow instabilities in collisionless magnetic reconnection