Determine the thermal-equilibrium concentrations of electrons and hole

Correct answer: To solve these problems, we'll need to apply some basic concepts from semiconductor physics.

Activity 1.1

(a) Thermal-equilibrium concentrations of electrons and holes in silicon at room temperature (T = 300 K)

The intrinsic carrier concentration (ni) in silicon at room temperature is given by:

ni = √(Nc \* Nv) \* exp(-Eg/2kT)

where:
- Nc = 2.8 × 10^19 cm^-3 (effective density of states in the conduction band)
- Nv = 3.1 × 10^19 cm^-3 (effective density of states in the valence band)
- Eg = 1.12 eV (bandgap energy of silicon at 300 K)
- k = 8.617 × 10^-5 eV/K (Boltzmann constant)
- T = 300 K (room temperature)

First, calculate the intrinsic carrier concentration:

ni = √(2.8 × 10^19 \* 3.1 × 10^19) \* exp(-1.12/2 \* 8.617 × 10^-5 \* 300)
= √(8.68 × 10^38) \* exp(-1.12/0.0259)
= √(8.68 × 10^38) \* exp(-43.31)
≈ 1.45 × 10^10 cm^-3

Since the Fermi energy level is given as 0.215 eV above the valence-band energy, we can assume that the material is p-type (since the Fermi level is closer to the valence band). However, for intrinsic silicon, the electron and hole concentrations are equal:

n = p = ni ≈ 1.45 × 10^10 cm^-3

(b) Ratio of ni at T = 400 K to that at T = 250 K

To find the intrinsic carrier concentration at different temperatures, we use the same formula:

ni = √(Nc \* Nv) \* exp(-Eg/2kT)

Assuming Nc and Nv are temperature-independent (which is a simplification), and using Eg = 1.12 eV for silicon:

At T = 400 K:
ni(400K) = √(2.8 × 10^19 \* 3.1 × 10^19) \* exp(-1.12/2 \* 8.617 × 10^-5 \* 400)
= √(8.68 × 10^38) \* exp(-1.12/0.0345)
= √(8.68 × 10^38) \* exp(-32.48)
≈ 4.83 × 10^13 cm^-3

At T = 250 K:
ni(250K) = √(2.8 × 10^19 \* 3.1 × 10^19) \* exp(-1.12/2 \* 8.617 × 10^-5 \* 250)
= √(8.68 × 10^38) \* exp(-1.12/0.0216)
= √(8.68 × 10^38) \* exp(-51.62)
≈ 1.13 × 10^5 cm^-3

The ratio of ni at T = 400 K to that at T = 250 K is:
ni(400K) / ni(250K) ≈ 4.83 × 10^13 / 1.13 × 10^5 ≈ 4.28 × 10^8

(c) Intrinsic carrier concentration in GaAs at T = 400 K and T = 250 K

For GaAs, Eg = 1.42 eV. Using the same formula:

At T = 400 K:
ni(400K) = √(Nc \* Nv) \* exp(-Eg/2kT)
= √(4.45 × 10^17 \* 7.0 × 10^18) \* exp(-1.42/2 \* 8.617 × 10^-5 \* 400)
= √(3.12 × 10^36) \* exp(-1.42/0.0345)
= √(3.12 × 10^36) \* exp(-41.13)
≈ 1.12 × 10^6 cm^-3

At T = 250 K:
ni(250K) = √(Nc \* Nv) \* exp(-Eg/2kT)
= √(4.45 × 10^17 \* 7.0 × 10^18) \* exp(-1.42/2 \* 8.617 × 10^-5 \* 250)
= √(3.12 × 10^36) \* exp(-1.42/0.0216)
= √(3.12 × 10^36) \* exp(-65.74)
≈ 1.47 × 10^-3 cm^-3

Activity 1.2

(a) Electron and hole densities in phosphorus-doped silicon at T = 300 K and T = 600 K

Given: Doping density (ND) = 10^16 atoms/cm^3

At T = 300 K:
Since the doping density is much higher than the intrinsic carrier concentration, we can assume that all phosphorus atoms are ionized, and the electron concentration is approximately equal to the doping density:
n ≈ ND = 10^16 cm^-3

The hole concentration can be found using the mass-action law:
p \* n = ni^2

Substituting the values:
p \* 10^16 = (1.45 × 10^10)^2
p ≈ (1.45 × 10^10)^2 / 10^16
≈ 2.10 × 10^4 cm^-3

At T = 600 K:
First, calculate the intrinsic carrier concentration at T = 600 K:
ni(600K) = √(2.8 × 10^19 \* 3.1 × 10^19) \* exp(-1.12/2 \* 8.617 × 10^-5 \* 600)
= √(8.68 × 10^38) \* exp(-1.12/0.0517)
= √(8.68 × 10^38) \* exp(-21.67)
≈ 2.25 × 10^16 cm^-3

Since the doping density is comparable to the intrinsic carrier concentration at this temperature, we cannot assume that all phosphorus atoms are ionized. However, for simplicity, let's assume that the electron concentration is still approximately equal to the doping density:
n ≈ ND = 10^16 cm^-3

The hole concentration can be found using the mass-action law:
p \* n = ni^2

Substituting the values:
p \* 10^16 = (2.25 × 10^16)^2
p ≈ (2.25 × 10^16)^2 / 10^16
≈ 5.06 × 10^16 cm^-3

(b) Density of electrons in silicon at T = 300 K and T = 600 K

At T = 300 K:
We already calculated the electron concentration:
n ≈ 10^16 cm^-3

At T = 600 K:
We also calculated the electron concentration:
n ≈ 10^16 cm^-3

Note that the electron concentration remains approximately the same at both temperatures, since the doping density is much higher than the intrinsic carrier concentration at T = 300 K, and comparable to the intrinsic carrier concentration at T = 600 K.