Logic level mosfet to switch on 3.7v battery
Shopping questions are off-topic. But since several people have already answered, I will pretend you asked "how can I choose a good MOSFET for this application."

Most likely your best choice is going to be to use a P-channel MOSFET (PMOS). You want it to turn on reliably with a voltage as low as, say, 3V, and have a voltage drop of less than, let's say 0.1V when passing 3A.

1) Rds(on). The first thing to consider is Rds(on). When it is on, a MOSFET is like a low-value resistor. The resistance value is given in the datasheet as Rds(on). We want the voltage drop in the PMOS to be less than 0.1V at 3A, so we use Ohm's law to calculate the maximum resistance.

V=I*R
R = V/I
R = 0.1V / 3A = 33mOhm

So we want Rds(on) to be less than 33mOhm. But there is more to it than that. The voltage between gate and source is what determines if the PMOS is on or not. We want to make sure that the 33 mOhm figure is given when gate voltage is 3V or less.

2) Power.
We can also calculate power dissipation. The relevant formula here is:
P = I^2 * R

If the Rds(on) really is 33 mOhms, the calculation is as follows:
P = 3A * 3A * 0.033 Ohms = 300 mW (approximately). You can probably use a SOT-23 package transistor, especially if you don't plan to use 3A very often.

PMOS will turn on when the gate voltage is lower than the source voltage. So the source terminal is your power input, the drain terminal is your power output, and the gate is the control terminal. Here is a circuit that shows how to switch the PMOS on and off.







simulate this circuit &ndash; Schematic created using CircuitLab

You just have to supply the control signal.