Quantum entropy of two-dimensional extreme charged dilaton black hole
  By using Hawking's treatment as well as Zaslavskii's treatment respectively and the brick wall model, two different values of classical entropy and quantum entropy of scalar fields in the two-dimensional extreme charged dilaton black hole backgrounds have been obtained. A new divergent term emerges in the quantum entropy under the extreme limit for Zaslavskii's treatment and its connection with the phase transition has been addressed. 
Bin Wang, Ru-Keng Su and P.K.N.Yu