Introduction:
The specific heat is the amount of heat energy required to raise the temperature of a substance by a certain amount. In the case of electrons in a system, the electronic contribution of specific heat refers to the heat capacity due to the movement of electrons. The electronic contribution of specific heat varies with temperature, and in some cases, it can vary linearly.

Explanation:
The electronic contribution of specific heat varies linearly with temperature for the following systems:

1. 3-dimensional electron gas:
In a 3-dimensional electron gas, the electrons are able to move freely in three dimensions. This can occur in materials such as metals, where the valence electrons are not tightly bound to individual atoms. In this case, the electronic contribution of specific heat can be described by the Fermi-Dirac distribution, which is a function that describes the probability of finding an electron at a given energy level. At low temperatures, the linear behavior of the specific heat is due to the linear energy dependence of the Fermi-Dirac distribution near the Fermi energy.

2. 2-dimensional electron gas:
In a 2-dimensional electron gas, the electrons are confined to move only in two dimensions. This can occur in materials such as thin films or at the interface between two materials with different electronic properties. In this case, the electronic contribution of specific heat can also vary linearly with temperature. The linear behavior arises from the fact that the density of states for a 2-dimensional system is constant, resulting in a linear energy dependence of the specific heat.

3. 1-dimensional electron gas:
In a 1-dimensional electron gas, the electrons are restricted to move along a line. This can occur in certain nanowires or carbon nanotubes, where the electronic motion is confined to one dimension. Similar to the 2-dimensional case, the electronic contribution of specific heat in a 1-dimensional electron gas can also exhibit a linear temperature dependence. The linear behavior arises from the linear energy dependence of the density of states for a 1-dimensional system.

Conclusion:
In summary, the electronic contribution of specific heat can vary linearly with temperature for 3-dimensional, 2-dimensional, and 1-dimensional electron gases. This behavior is a result of the energy dependence of the Fermi-Dirac distribution and the density of states in these systems. Understanding the specific heat behavior in different electron gas systems is important in the study of thermal properties of materials and can have implications in various technological applications.