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In this paper, using Hamilton-Jacobi method, we address the tunnelling of fermions in a 4-dimensional Schwarzschild spacetime. Based on the generalized uncertainty principle, we introduce
the influence of quantum gravity. After solving the equation of motion of the spin-1/2 field, we
derive the corrected Hawking temperature. It turns out that the correction depends not only on
the black hole’s mass but also on the mass (energy) of emitted fermions. It is of interest that, in
our calculation, the quantum gravity correction decelerates the temperature increase during the radiation
explicitly. This observation then naturally leads to the remnants in black hole evaporation.
Our calculation shows that the residue mass is

Hawking radiation is described as a quantum tunnelling effects of particles at horizons of black holes [

Various theories of quantum gravity predict the existence of a minimum measurable length [

[

[

[

[

We start with the Dirac equation in curved spacetime,

In this section, we address the tunnelling behavior of spin-1/2 fermions across the event horizon of the Schwarzschild black hole. Effects of quantum gravity are taken into account. The metric is given by

Our task is to find the solutions of (

It is of interest to note that in (

In this work, we modified the Dirac equation in curved spacetime to include the quantum gravity influence. To fulfill this purpose, we employed the generalized uncertainty principle model. This model is derived from the existence of minimal length which arises when combining quantum and gravity. We calculated the radiation of spin-1/2 particles in the 4-dimensional Schwarzschild spacetime with Hamilton-Jacob method. The tunnelling rate and Hawking temperature were presented.

We found that the quantum gravity correction is related not only to the black hole's mass but also to the mass (energy) of emitted fermions. More interestingly, our result shows that the quantum gravity correction explicitly retards the temperature rising in the black hole evaporation. Therefore, at some point during the evaporation, the quantum correction balances the traditional temperature rising tendency. This leads to the existence of the remnants. We showed that the remnants is

We use the 4-dimensional Schwarzschild metric in this work. It is known that the WKB type of approximation is basically the same as working with a 1+1-dimensional spacetime [

Deyou Chen and Houwen Wu are very grateful for Professor S. Q. Wu and Prof. Professor Wang for their useful discussions. This work is supported in part by the NSFC (Grants nos. 11205125, 11175039, and 11375121), Sichuan Province Science Foundation for Youths (Grant no. 2012JQ0039), and China West Normal University Foundation (Grant no. 11B005).