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Recent angle resolved photoemission spectrascope (ARPES) experiments onstrongly underdoped Bi$_2$Sr$_{2-x}$La$_x$CuO$_{6+\delta}$ cuprates havereported an unusual gap in the nodal direction. Transport experiments on thesecuprates found variable range hopping behavior observed. These cuprates haveboth electron and hole doping which has led to proposals that this cuprate isanalogous to a partially compensated semiconductors. The nodal gap thencorresponds to the Efros-Schklovskii(ES) gap in such semiconductors. Wecalculate the doping dependence and temperature dependence of a ES gap modeland find support for an Efros-Schklovskii model.

Variable range hopping theory for the nodal gap in strongly underdoped cuprate Bi$_2$Sr$_{2-x}$La$_x$CuO$_{6+\delta}$