In the case of n-type semiconductor

 

From the charge neutrality condition, the electron concentration in the freeze-out range is expressed as

.                           (1)

When we define  as

,                                                    (2)

Eq. (1) can be rewritten as a quadratic equation:

,                                                     (3)

where

,                                               (4)

,                                                         (5)

and

.                                                          (6)

The solution of Eq. (3) is

                                                  (7)

or

.                                           (8)

When

                                                      (9)

that is

,                                          (10)

the solution is

                                                         (11)

that is

.                                  (12)

Therefore,  is derived as

.                      (13)

Since

,                                           (14)

we obtain the following relationship:

.                           (15)