You're on the right track, but what N-FET did you use?
First: how much power?
The article here suggests you might need to switch 500 mA. ESP8266EX has a typical transmit current of about 170 mA and no listed maximum.
You don't say how much time you'll have the ESP-05 enabled nor how much power you can allow during its off-time.
Second: What threshold voltage?
Fundamentally an nFET is a voltage controlled switch. The level at which it switches is the VGS(thresh) and you need one which will switch at 3.3V, often called a "3.3V logic-level MOSFET" (watch out for those which are good for 5V logic). Many FETs, especially those for larger currents and voltages, have a much higher switching threshold.
Two questions about this specifically:
Sparkfun shows modules for which might immediately do what you need, if you don't mind using a ready-made module. Seach for
MOSFET on that site.
- One product there uses FDS6630A which has VGS(thresh) of min=1.0V, typ=1.7V, max=3.0V.
Once you've chosen a MOSFET, you look for examples for the details of its required current limit and bias resistors. (You won't need the diode of your example circuit, that's for the back-EMF of the inductive load.) The FDS6630A example above uses
Source: Sparkfun documentation
SLEEP The ESP8266EX has various sleep modes its various sleep modes is 15 mA down to 0.5 μA according to its datasheet. You can enable these with
AT commands, and perhaps that will work for your application.
RESET Some ESP-05 models have a 5-pin connector with
RESET signal. If you assert this, I'll guess the power consumption will be very low (but couldn't find a value in the documentation) and perhaps that would work for you.