Intrinsic Concentration and Conductivity of Intrinsic Semiconductor

Intrinsic Concentration and Conductivity of Intrinsic Semiconductor

• In an intrinsic semiconductor, at any temperature at any instant, the free electrons and holes are present in equal numbers. This concentration is called intrinsic concentration denoted as n_i.

where

 n=p=n_i              ………..For intrinsic semiconductor

• Using n = p = n_i in conductivity of a semiconductor we get the expression for the conductivity of intrinsic semiconductor denoted as σ_i.

 σ_ị = n_i(μ_n+μ_p) q

• Both n_i and σ_i depends on temperature due to thermal generation.

• As temperature increases,

i) n_i increases

ii) σ_i increases

iii) ρ_i = 1/σ_i = Intrinsic resistivity decreases.

• The temperature dependence of n_i is mathematically given by,

 n_i = BT^3/2 e^‒EG0/2kT

where

B = Constant independent of temperature

T = Absolute temperature in °K

E_G0 = Forbidden energy gap at absolute zero temperature

 k = Boltzmann's constant = 8.62×10^‒5 eV/°K

Ex. 1.7.1 Find the resistivity of an intrinsic silicon at 300 °K if intrinsic concentration of silicon at 300 °K is 1.5×10¹⁰ cm³ per while μ_n = 1300 cm²/V‒sec and μ_p= 500 cm²/V‒sec. Assume q=1.6×10^‒19 C.

Solution:

The given values are, n_i = 1.5×10¹⁰/cm³

n_i = 1.5× 10¹⁰ / 10^‒6 /m³

 = 1.5×10¹⁶/m³

And

μ_n = 1300×10^‒4 m²/V‒sec,

μ_р = 500×10^‒4 m²/V‒sec

Now

 σ_i = n_i (μ_n + μ_p) q

= 1.5 × 10¹⁶ [1300 + 500] × 10^-4 × 1.6×10^‒19

 = 0.000432 (Ω-m)^-1              ………. Conductivity

 ρ = 1/σ_i = 1/0.000432 = 2314.8148 Ω‒m          …….Resistivity

The resistivity is reciprocal of conductivity.

Ex. 1.7.2 Calculate the intrinsic concentration of Germanium in carriers/m³ at a temperature of 320 °K given that ionization energy is 0.75 eV and Boltzmann's constant k = 1.374× 10^‒23 J/°K. Also calculate the intrinsic conductivity given that the motilities of electrons and holes in pure germanium are 0.36 and 0.17 m²/volt ‒ sec respectively.

Solution:

The temperature dependence of n_i is,

n_i  = BT^3/2 e^‒EG0/2kT

B = Constant, E_G0 = Energy gap at absolute zero,

k = Boltzmann's constant

At 300°K,

 n_i = 2.5 × 10¹³/cm³ = 2.5 × 10¹⁹/m³

 k = 1.374 × 10^‒23 J/°K

and

 1 eV = 1.374×10^‒23 / 1.6×10^‒19  eV/°K = 8.58 × 10^‒5 eV/°K

 2.5 × 10¹⁹ = B (300)^3/2 e^‒0.75/2 × 8.58 × 10^‒5 × 300

B = 1.0218 × 10²²

At T = 320°K,

 n₁ = 1.0218 × 10²² × (320)^3/2 e^‒0.75/2 × 8.58 × 10^‒5 × 320 = 6.84 × 10¹⁹/m³

 σ_i = n_i (μ_n +μ_p) q = 6.84 × 10¹⁹ (0.36 + 0.17) × 1.6 × 10^‒19

= 5.8 (Ω-m)^‒1

……. Conductivity at 320°K

Review Questions

1. What is intrinsic concentration? Obtain the expression of conductivity of intrinsic semiconductor.

2. A bar of intrinsic Silicon has a cross‒sectional area of 2.5 × 10^-4 m². The electron density is 1.5 × 10¹⁶ per m³. How long the bar be in order that current in the bar will be 1.2 mA when 9 volts are applied across it? Assume: μ_n = 0.14 m²/V‒sec, μ_p = 0.05 m²/V‒sec.

[Answer: 0.856 mm]