  Some very interesting solutions of the field equations of Einstein's general
theory of relativity have been constructed in the framework of nonlinear
electrodynamics. In particular, magnetically charged black hole solutions in
the framework of exponential nonlinear electrodynamics have been obtained in
general relativity in Kruglov (2017). Using this approach a magnetically
charged non-singular black hole spacetime in the framework of exponential
electrodynamics in some modified theory of gravity has been constructed in this
letter. The metric describing asymptotic non-singular magnetized black hole is
worked out in terms of the parameter of our model. When this parameter vanishes
our solution reduces to the above mentioned black hole solution. Thermodynamics
of the resulting solution is also discussed by calculating the Hawking
temperature and heat capacity when the magnetic charge is constant. We also
find out the point where the first order phase transition induced by
temperature changes takes place. The quantum radiations from this black hole
are also discussed and the mathematical expression for the rate of energy flux
of these radiations has been obtained.
