Negative density of states: screening, Einstein relation, and negative   diffusion
  In strongly interacting electron systems with low density and at low temperature the thermodynamic density of states is negative. It creates difficulties with understanding of the Einstein relation between conductivity and diffusion coefficient. Using the expression for electrochemical potential that takes into account the long range part of the Coulomb interaction it is shown that at negative density of states Einstein relation gives a negative sign of the diffusion coefficient D, but under this condition there is no thermodynamic limitation on the sign of D. It happens because the unipolar relaxation of inhomogeneous electron density is not described by the diffusion equation. The relaxation goes much faster due to electric forces caused by electron density and by neutralizing background. Diffusion coefficient is irrelevant in this case and it is not necessarily positive because process of diffusion does not contribute to the positive production of entropy. In the case of bipolar diffusion negative D results in a global absolute instability that leads to formation of neutral excitons. Graphene is considered as an example of a system, where the density relaxation is expected to be due to electric force rather than diffusion. It may also have a negative density of states. 
A. L. Efros