PA34 flight instructor experienced a runway excursion due to brake failure after returning to the airport for an unlatched door. The instructor had made the landing and transfered control to the student who had the only set of brake peddles.

Narrative

After departure I noticed instructor side door was partially open due to upper latch not securing properly; made a left hand pattern to return for a stop and go in order to secure the latch. During final all systems were normal made a normal landing. During this process I the instructor was in control of the aircraft; and performing the landing; however; this airplane does not have brakes on the right side only on the left side; therefore; after touchdown I requested my student apply brakes so that we can make a full stop on the runway. Student tapped the brake the first time brakes seemed to work normally; I then requested to apply firm braking in order to come to a complete stop. Student proceeded to apply brakes firmly as he did aircraft veered to the left very quickly; my first reaction was to apply brakes on my side in order to regain control but did not have any; so I immediately told the student to 'stop' (meaning to stop applying brakes) realizing that this was not the correct thing to say a shouted out 'let go' in order to regain control; student however tried applying more brake on the right side in order to change the direction of the aircraft since we were turning to the left. Right brakes however did not seem to make any effect; the airplane kept moving towards the left. We had already slowed down by the thousand foot markers; power was at complete idle; but since the brakes were still being applied we kept turning; we slowly ran off the runway down the slop to rest in a small ditch. Left engine prop struck the ground causing the engine to stop and bending the tip on one of the blades. No other damage was found right engine was intact.In order to prevent this situation from happing again: First; good communication is paramount; since I did not use proper verbiage student did not react correctly; proper phraseology would have been 'I have the controls' not sure why I did not say this in the first place.Second; there are no brakes installed on the instructor's side (right side) in this model of aircraft since it's an optional item; therefore; I could not assist with braking of the aircraft. Since this model is being used for training brakes need to be installed on the right side.Third; aircraft is equipped with a parking brake I could have used it in order to help assist the breaking action; however; I was too concerned in over braking and causing a complete lock on one of the brakes that could have blown out a tire. This could have actually aggravated the situation; but since we had slowed down quite a bit if used carefully; it may have helped. All this happed so fast that there was very little time to react which is why applying the above recommendations could have prevented this incident.