SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

110 CHAPTER 3. CHARGE TRANSPORT IN MATERIALS

while ifNv>Ncwe get

pmax=Nv

This gives us for the maximum density: i) for Si, 2. 78 × 1019 cm−^3 ii) for GaAs,

7. 72 × 1018 cm−^3. Based on these numbers we can calculate the maximum conductivity:

For Si

σmax=2. 78 × 1019 × 1. 6 × 10 −^19 ×1000 = 4. 45 × 103 (Ω cm)−^1

For GaAs

σmax=7. 72 × 1018 × 1. 6 × 10 −^19 ×400 = 4. 9 × 102 (Ω cm)−^1

To find the minimum conductivity we need to find the minima of the expression

σ = neμn+peμp

=

n^2 i

p

eμn+peμp

To find the minimum we take the derivative with respect topand equate the result to zero.

This gives

p=ni

√

μn

μp

This then gives for the minimum conductivity

σmin=nie[μn

√

μp

μn

+μp

√

μn

μp

]

For Si this gives upon plugging in numbers

σmin=2. 8 × 10 −^6 (Ω cm)−^1

and for GaAs

σmin=1. 05 × 10 −^9 (Ω cm)−^1

Note that these values are lower than the values we get in the the previous problem for the

undoped cases. This example shows the tremendous variation in conductivity that can be

obtained in a semiconductor.

High field transport: velocity–field relations

In most electronic devices a significant portion of the electronic transport occurs under strong
electric fields. This is especially true of field effect transistors. At such high fields (∼ 1 −
500 kV/cm) the electrons get “hot” and acquire a high average energy. The extra energy comes
due to the strong electric fields. The drift velocities are also quite high. The description of