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Conductivity is the ability of a material to conduct electric current. In semiconductors, the current flow is mainly due to the movement of electrons and holes (missing electrons). These carriers of electric current can move between energy bands in the semiconductor called the valence band and the conduction band. The valence band is fully occupied, while the conduction band is partially filled or empty. When sufficient energy is provided (like in the form of heat), electrons can jump from the valence band to the conduction band, thereby causing an increase in the current flow.

As the temperature of a semiconductor material increases, the thermal energy provided by this increase in temperature causes the electrons in the material to gain enough energy to jump from the valence band to the conduction band. As a result, the number of electrons available for current flow in the conduction band increases, leading to an increase in the conductivity of the semiconductor material.

The difference in the temperature-dependence of conductivity between metals and semiconductors can be attributed to the difference in their energy band structures. In metals, the valence band and conduction band overlap, meaning that there are always electrons available in the conduction band for current flow. In semiconductors, however, there is an energy gap between the valence band and the conduction band, which prevents electrons from freely moving between the bands at low temperatures. As a result, the conductivity of metals decreases with increasing temperature due to increased lattice vibrations or scattering of electrons, while the conductivity of semiconductors increases with increasing temperature as more electrons become available for current flow.

It's important to note that the increase in conductivity of semiconductors with temperature is true only up to a certain point. At extremely high temperatures, the thermal energy may cause significant lattice vibrations or the material could undergo a phase change, thereby reducing the conductivity. The specific temperature range at which the conductivity increases can vary depending on the type and composition of the semiconductor material.