This paper reports on a numerical analysis of methods for current injection into AlInGaAs/InP tunnel-junction electrically pumped vertical-external-cavity surface-emitting lasers. The tunnel junction is patterned to minimize the current crowding effect and support the fundamental transverse modes. Optimization of the tunnel junction radius predicts 9-mW emission and 4% wall-plug efficiency in the single-mode regime. Additional patterning of the tunnel junction in the form of a coaxial ring, without modifying its total radius but reducing its area, maintains or improves the emitted power and in addition enhances wall-plug efficiency by around 60%. A design of E-VECSEL is proposed which offers five times larger emitted power and wall-plug efficiency in the single-mode regime in comparison to recently reported experimental devices.