Engineering Physics

LESSON 20. Law of Mass Action

Law of Mass action

We know that the concentration of electrons in conduction band is

\[{n_o}=2{\left( {{{2\pi m_e^*{k_B}T} \over {{h^2}}}} \right)^{{3 \over 2}}}e{^{\left( {{{{E_f} - {E_g}} \over {{K_B}T}}} \right)}}\] …………………(1)

Similarly, hole concentration in valance band is   

\[{n_h} = 2{\left( {{{2\pi m_h^*{k_B}T} \over {{h^2}}}} \right)^{{3 \over 2}}}e{^{\left( {{{ - {E_f}} \over {{K_B}T}}} \right)}}\]  ………………….(2)

Multiplying equation (1) and (2) we have the useful equation

\[{n_e}{n_h}\]  = 4 \[{\left( {{{2\pi m_e^*{k_B}T} \over {{h^2}}}} \right)^{{3 \over 2}}}e{^{\left( {{{{E_f} - {E_g}} \over {{K_B}T}}} \right)}}\] \[{\left( {{{2\pi m_h^*{k_B}T} \over {{h^2}}}} \right)^{{3 \over 2}}}e{^{\left( {{{ - {E_f}} \over {{K_B}T}}} \right)}}\]………….. (3)

This is very useful relation since for the given semiconductor (i.e given effective masses and energy gap) at a given temperature the product of electron and hole concentration is constant.

We have assume that the semiconductor is intrinsic; the only assumption which has been is that the distance of the Fermi level from the edges of both the bands should be large in concentration with K_BT . Due to this consideration, the results of equation (3) will also apply to extrinsic semiconductor. Thus introducing an impurity in an intrinsic semiconductor to increase n , say will decrease ρ as the product must remain constant. Thus the equation (3) governs the relative concentrations of electrons and holes in a given semiconductor and is sometime called the law of mass action.