The Pierre Auger Observatory is exploring the potential of the radio detection technique to study extensive air showers induced by ultra-high energy cosmic rays. The Auger Engineering Radio Array (AERA) addresses both technological and scientific aspects of the radio technique. A first phase of AERA has been operating since September 2010 with detector stations observing radio signals at frequencies between 30 and 80 MHz. In this paper we present comparative studies to identify and optimize the antenna design for the final configuration of AERA consisting of 160 individual radio detector stations. The transient nature of the air shower signal requires a detailed description of the antenna sensor. As the ultra-wideband reception of pulses is not widely discussed in antenna literature, we review the relevant antenna characteristics and enhance theoretical considerations towards the impulse response of antennas including polarization effects and multiple signal reflections. On the basis of the vector effective length we study the transient response characteristics of three candidate antennas in the time domain. Observing the variation of the continuous galactic background intensity we rank the antennas with respect to the noise level added to the galactic signal.