MOSFETs can be placed in push-pull mode, just like PNP and NPN transistors.
They must be connected correctly to prevent damage.
In the following circuit you can see the transistors and MOSFETs have been connected incorrectly.
For the PNP/NPN transistor circuit, as the input changes from high to low or low to high, both transistors are turned on during the transition. Only one transistor is turned on when the line is high and only the other transistor is turned on when the line is low, but during the transition, BOTH are turned on.
The same applies with the MOSFETs. When the input is at mid-rail, a voltage between gate and source will be produced for both MOSFETs. Since a MOSFET can handle many amps, this will put a short-circuit across the power rail and will cause a lot of damage.
The correct placement for the NPN and PNP transistors is shown in the diagram below. The output will rise and fall in harmony with the input, however there will be a small 1v2 gap at mid-rail where the output will not respond as this represents 0.6v for the base-emitter voltage of each transistor. You should not connect two MOSFETs as shown the gap will be 6v as the gate to source voltage for each transistor is about 3v, but you cannot connect the gates of the two MOSFETs because each MOSFET will turn off when the gate-to-source voltage is less than about 3v across these two terminal. This means the output will be 3v less than rail voltage and not go below 3v above 0v rail. Both MOSFETs will not turn on during any part of the cycle and no short circuit will occur, but the output will be less than full rail-voltage swing and the MOSFETs are not being supplied with a gate-to-source voltage that has a guaranteed fast rise and fall time (and the MOSFETs may heat up). This is an unreliable design.
Transistors and MOSFETs will produce short-circuit
MOSFET output is less than rail voltage
The solution is shown in the diagram below. The transistor configuration will work on ANY rail voltage but the MOSFET "totem-pole configuration" will only work up to 5v. This is due to the characteristics of a MOSFET. The MOSFETs used in this arrangement have a gate-to-source characteristic of slightly more than 3v and do not turn on when the voltage across these two terminals is 3v. This means the supply can be 6v and when the input is at mid-rail, 3v will be across each gate-to-source and neither will be turned on. That's why TTL logic is limited to 5v operation. The output will be extremely close to rail-to-rail for the MOSFET configuration.http://www.talkingelectronics.com/projects/MOSFET/MOSFET.html