What is the correct carrier concentration equation to use?

Question

In this book I am reading, it gives the intrinsic carrier concentration as:

\$ n_i = \sqrt{N_c\cdot N_v} \cdot e^{({\frac{-E_g}{2*K*T}})} \$

While another book says the carrier concentration is:

\$ \ n_i = BT^{\frac{3}{2}}\cdot e^{({\frac{-E_g}{2*K*T}})} \$

Where "B" is the coefficient related to the specific semiconductor material. I have searched around and cannot find anything on this topic. How can I actually find "B" of other semiconductor materials? Note: B = 5.23e15 for Si, B = 1.66e15 for Ge, and B = 2.10e14 for GaAs.

Where do they get the values for B from?

The \$ \ n_i = BT^{3/2}*e^{(-E_g/2*K*T)} \$ equation comes from this book called, "Microelectronics Circuit Analysis and Design" This equation is also in another textbook called Microelectronic Circuits by Sedra/Smith.

Where this equation \$ n_i = \sqrt{N_c\cdot N_v} \cdot e^{({\frac{-E_g}{2*K*T}})} \$

comes from almost any textbook in "Solid State Electronic Devices" My specific book is by Streetman Banerjee.

Answer 1

Intrinsic carrier density depends on temperature, and it is shown in both equations, though there is also a second temperature factor in the equation.

Please read Physics of Semiconductor Devices by Kwok Kwok Ng, Simon Sze, and Yiming Li, page 32.

Nc defined in Eq. 10 is the effective density of states.
Nv is effective density of states in the valence band.

In your first eq. it is not shown, that Nc and Nv implicitly depend on temperature.
In equation with B, temperature is shown explicitly as a parameter that changes the effective density of states.
Usually Nc and Nv are given for room temperature in various books.

eg. Nv ≡ 2 (2π m_p kT/h²)³/2

B factor is the above equation without T.