I find some negative value of real part dielectric function. What does it mean? What is the difference if any material shows negetive value and another material shows positive value?
A negative dielectric constant means the system has a negative refractive index. This usually doesn't happen in naturally-formed materials. However, some meta-materials do show this unique behavior. These materials are of great technological importance for many applications such as perfect (super) lenses microwave circuits, optical switches and modulators, antenna components and etc. They basically enable access to the part of electromagnetic spectrum where conventional materials lack any response.
You have probably calculated the real part of dielectric constant from capacitance dispersions. In that case, you got negative capacitance for specific frequencies.
The idea of negative capacitance (NC) is completely different for ferroelectric and non-ferroelectric materials. For the first kind, an active research has taken place in the past decade already, the roots of which lies in the Landau theory of ferroelectrics, where an unstable equilibrium regime in the energy landscape causes NC. To stabilize this unstable point, there are ways: (i) application of external dc bias to flatten the energy dynamics; and (ii) to fabricate successive assembly of dielectrics (+ve capacitance) and ferroelectrics (-ve capacitance) at specific charge-conditions. Please go through:
https://www.nature.com/articles/nmat4148
and
https://www.nature.com/articles/s41586-018-0855-y
On the other hand, the negative capacitance in rest of the materials is still a distant dream to reach a unified theory. Localization of charges in trap/interface states, contact injection, extrinsic or doping effects, space-charge propagation, slower transients, mixed ionic-electronic hysteresis in the current–voltage dependency, minority carrier migration, sluggish response of heavy dipoles etc can be the reason. For detail, in this category I'd suggest this paper of mine:
Article Negative capacitance switching in size-modulated Fe3O4 nanop...
You may also have a look at the following, which is due to pseudo-inductance:
Article Strain-induced partial phase transition in TiO2 nanoparticle...
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