  We present an analytical investigation of the exciton optical absorption in a
narrow armchair graphene nanoribbon (AGNR) in the presence of a longitudinal
external electric field directed parallel to the ribbon axis. The two-body 2D
Dirac equation for the massless electron and hole subject to the ribbon
confinement, Coulomb interaction and electric field is employed. The ribbon
confinement is assumed to be much stronger than the internal exciton electric
field which in turn considerably exceeds the external electric field. This
justifies an adiabatic approximation implying a slow longitudinal and fast
transverse electron-hole relative motion governed by the exciton attraction and
accompanied by the external electric field and ribbon confinement effects,
respectively. In the single subband approximation of the isolated
size-quantized subbands induced by the ribbon confinement the exciton
electroabsorption coefficient is determined in an explicit form. The pronounced
dependencies of the exciton peak positions, widths and intensities on the
ribbon width and electric field strength are traced. The electron-hole exciton
attraction modifies remarkably the Franz-Keldysh electroabsorption in the
frequency region both below and above the edges determined by the
size-quantized energy levels. In the double-subband approximation of the
interacting ground and first excited subbands the combined effect of the
exciton electro- and autoionization caused by the electric field and
intersubband coupling, respectively, are explored. Estimates of the expected
experimental values for the typically employed AGNR show that for a weak
electric field the exciton quasi-discrete states remain sufficiently stable to
be observed in optical experiments, while relatively strong fields free the
captured carriers to further restore their contribution to the transport.
