Question

What are energy-bands in solids ? Differentiate conductors, insulators and semiconductors on the basis of energy-bands and explain the effect of temperature on these.

Hint:

The key distinguishing factor is the size of the forbidden energy gap (\(E_g\)). For conductors \(E_g \approx 0\), for semiconductors \(E_g\) is small, and for insulators \(E_g\) is large. This gap size dictates how easily electrons can become charge carriers.

Answer 1

1. Energy Bands in Solids:
In an isolated atom, electrons occupy discrete energy levels. When atoms are brought close together to form a crystalline solid, the electrons in the outer shells of an atom are influenced by the electrons and nuclei of neighboring atoms. This interaction causes the discrete energy levels of the isolated atoms to split and form a collection of a very large number of closely spaced energy levels, which are effectively continuous. These ranges of allowed energies are called **energy bands**. The two most important bands are: \begin{itemize} \item Valence Band (VB): The energy band containing the valence electrons. It is the highest occupied energy band at 0 K. \item Conduction Band (CB): The lowest unoccupied energy band. Electrons in this band are free to move and contribute to electrical conduction. \item Forbidden Energy Gap (\(E_g\)): The energy range between the top of the valence band and the bottom of the conduction band, which contains no allowed energy levels for electrons. \end{itemize} 2. Differentiation based on Band Theory: \begin{center} \begin{tikzpicture} % Conductor \node at (0, 3) {Conductor}; \draw[fill=blue!30] (-1,0) rectangle (1,2); \draw[fill=red!30] (-1,1.5) rectangle (1,2.5); \node at (0, 0.75) {Valence Band}; \node at (0, 2) {Conduction Band}; \node at (0, 1.5) [right] {Overlap}; % Insulator \node at (4, 3) {Insulator}; \draw[fill=blue!30] (3,0) rectangle (5,1); \draw[fill=red!30] (3,2) rectangle (5,3); \draw[<->] (5.2, 1) -- (5.2, 2) node[midway, right] {$E_g > 3$ eV}; \node at (4, 0.5) {Valence Band (Filled)}; \node at (4, 2.5) {Conduction Band (Empty)}; % Semiconductor \node at (8, 3) {Semiconductor}; \draw[fill=blue!30] (7,0) rectangle (9,1); \draw[fill=red!30] (7,1.5) rectangle (9,2.5); \draw[<->] (9.2, 1) -- (9.2, 1.5) node[midway, right] {$E_g \approx 1$ eV}; \node at (8, 0.5) {Valence Band}; \node at (8, 2) {Conduction Band}; \end{tikzpicture} \end{center} \begin{itemize} \item Conductors: In conductors (metals), the valence band and the conduction band overlap. There is no forbidden energy gap between them. Due to this overlap, a large number of free electrons are readily available in the conduction band to conduct electricity, even at zero Kelvin. \item Insulators: In insulators, the valence band is completely filled, and the conduction band is completely empty. There is a very large forbidden energy gap (\(E_g > 3\) eV) separating them. It is almost impossible for an electron to gain enough energy to jump from the valence band to the conduction band. \item Semiconductors: In semiconductors, the band structure is similar to that of insulators, but the forbidden energy gap is much smaller (\(E_g < 3\) eV, e.g., \(\sim 1.1\) eV for Silicon). At 0 K, they behave like insulators. However, at room temperature, some electrons can gain enough thermal energy to jump the gap into the conduction band, allowing for limited conductivity. \end{itemize} 3. Effect of Temperature: \begin{itemize} \item Conductors: When the temperature of a conductor is increased, the thermal vibrations of the metal ions (lattice) increase. This leads to more frequent collisions between the free electrons and the ions, which increases the electrical resistance. \item Insulators: Due to the large band gap, an increase in temperature has very little effect on the conductivity of insulators, as very few electrons can be thermally excited to the conduction band. \item Semiconductors: When the temperature of a semiconductor is increased, more valence electrons gain sufficient thermal energy to jump across the small forbidden gap into the conduction band, creating electron-hole pairs. This increases the number of charge carriers, which significantly increases the conductivity (and decreases the resistance). \end{itemize}