Week 1: Semiconductor Fundamentals

Question 1

What is the probability of thermal excitation proportional to?

Answer 1

e^(-E_g / k_b * T)

where k_b is Boltzmann constant, T is temperature in Kelvin, and E is the band gap energy

Question 2

What is the conduction band?

Answer 2

The highest unoccupied band

Question 3

What is the valence band?

Answer 3

the highest occupied level at zero Kelvin

Question 4

What is a semiconductor?

Answer 4

A semiconductor is a special kind of insulator whose band gap is small enough (< 2-3eV) to excite electrons across this gap at room temperature.

Question 5

Semiconductors sometimes behave like insulators and sometimes behave like conductors. What is modulated to take advantage of this?

Answer 5

The carrier density

Question 6

What is the density of states used for?

Answer 6

The density of states is used for finding the number of electrons per volume in the conduction band and the number of electrons per volume in the valence band

Question 7

What is the density of states?

Answer 7

The density of states is the number of states that are available for electrons and holes at a given energy, E + dE

Question 8

What is g(E) representing? What happens when you multiply it by an infinitesimally small dE slice of energy?

Answer 8

g(E) represents the number of states per volume per energy; multiplying by dE slice leaves you with the the number of states per volume

Question 9

What is the spin degeneracy?

Answer 9

The fact that for a given state we can have a spin up and a spin down electron.

Question 10

What does the carrier density control?

Answer 10

The carrier density controls the current and is responsible for whether the semiconductor can emit light or not.

Question 11

Describe the probability of occupation in terms of E and f(E).

Answer 11

E is the energy. You can plot E on the y axis and f(E), the probability of finding an electron, on the x axis. At higher energies, the probability f(E) is almost zero. At lower energies the probability increases and at 0 energy, f(E) = 1.

Question 12

Electrons are considered to be fermions. What does this mean for the principle they obey?

Answer 12

This means they obey the Pauli exclusion principle meaning that each electron has to have a unique state.

Question 13

What do holes in the valence band correspond to?

Answer 13

Empty states in the valence band

Question 14

How is the carrier density in a semiconductor obtained?

Answer 14

By integrating the product of the density of states and the probability density function over all possible states

Question 15

How would you calculate the electron density at a given temperature for a semiconductor?

Answer 15

N(T) = integralE_c to inf

g_c(E): density of states in conduction band
f(E): probability of occupation

Question 16

How would you calculate the hole density at a given temperature for a semiconductor?

Answer 16

P(T) = integral-inf to E_v

g_v(E): density of states in valence band
f(E): probability of occupation

Question 17

What is the probability of finding an electron (from the Fermi-Dirac statistics)?

Answer 17

f(E) = 1 / (1 + exp(E - E_f / k_b * T)

where E_f is the fermi level, the place where the probability of finding an electron is 50%

Question 18

Explain the non-degeneracy condition and what it allows us to do.

Answer 18

The non-degeneracy condition means that the fermi level never gets too close to the conduction or the valence band.

E_c - E_f&raquo_space; k_b * T
E_f - E_v&raquo_space; k_b * T

k_b * T is the approximate thermal energy an electron will have at a given temperature.

This allows us to take the fermi dirac statistics and approximate them by classical statistics.

Question 19

What can we approximate f(E) to be with the non-degeneracy condition? What about 1-f(E)? What’s the benefit of this?

Answer 19

We can approximate f(E) to be ~= exp(- (E - E_f) / k_b * T) and this is true for E > E_c.

1-f(E) can be approximated as ~= exp(- (E_f - E)/k_b * T) where E < E_v

This allows us to come up with analytic expressions for our density integrals.

Question 20

How do we use the parabolic bands assumption to simplify the integrals for the density of electrons and density of holes as functions of temperature?

Answer 20

We use the parabolic bands assumption that the carriers are very close to the band edges. Parabolic bands assumption allows us to integrate the N_v(T) and N_c(T) integrals exactly.

We can rewrite them as:

N(T) = N_c(T) exp(- (E_c - Ef)/ k_b * T)

P(T) = N_v(T) exp (- (E_f - E_v)/ k_b * T)

Where N_c(T) is the density of states in the conduction band and N_v(T) is the density of states in the valence band.

Question 21

Using the parabolic band approximation, what is the result of evaluating N_v(T) and N_c(T)?

Answer 21

N_c(T) = (1/4) (2(m_c)k_bT/pi*h_bar^2) ^ (3/2)

N_v(T) = (1/4) (2(m_v)k_bT/pi*h_bar^2) ^ (3/2)

Question 22

What relationship does the law of mass action give us for N(T) and P(T)? How would we find the intrinsic carrier density from this?

Answer 22

N(T)*P(T) = N_c * N_v * exp(- E_g / k_b * T)

The intrinsic carrier density, N_i, is equal to N(T)*P(T) so we have that:

N_i^2 = N_c * N_v * exp(-E_g / 2 * k_b * T)

Question 23

What’s an intrinsic semiconductor and what does it tell us about the electron concentration, hole concentration, and intrinsic carrier concentration?

Answer 23

A semiconductor we have done nothing to. It tells us that N(T) = p(T) = n_i(T), so they’re all equal.

Question 24

Describe two equations that can be used to calculate the fermi level of an intrinsic semiconductor.

Answer 24

E_f = E_v + (1/2)E_g + (1/2)k_bTln(N_v/N_c)

E_f = E_v + (1/2)E_g + 3/4k_bTln(m_v/m_c*)

Question 25

What is an extrinsic semiconductor?

Answer 25

A semiconductor where impurities have been deliberately introduced and the electron densities and hole densities will be determined by the impurity type and density.

Question 26

We treat impurities as an additional charge. What is a donor? What is an acceptor?

Answer 26

A donor is an atom who gives up an electron and leaves itself with positive charge. An acceptor takes an electron creating a hole and leaves itself with a negative charge.

Question 27

What is n doping and how would you calculate the fermi level in the case of n doping?

Answer 27

N-doping is doping with electrons making it electron rich.

E_f = E_i + k_bTln(N_D / n_i)
OR
E_c - E_f = k_bTln(N_c/N_D)

n_i is the intrinsic concentration
N_D is the electron concentration for an n-doped material

Question 28

What is p doping and how would you calculate the fermi level in the case of p doping?

Answer 28

P-doping is making the semiconductor hole rich.

E_f = E_i - k_b * T *ln(N_A/n_i)

E_f - E_v = k_b * T * ln (N_v/N_A)

N_A is the hole concentration
N_i is the intrinsic concentration

Question 29

How would you calculate the donor energy?

Answer 29

E_d = [(m_c*/m_o)(epsilon_o/epsilon_s)^2) * 13.6eV]

epsilon_o: permittivity of free space
epsilon_s: permittivity of the lattice

13.6eV is the energy level of a hydrogen atom and this is what we use to model energy levels of lone electrons.

Question 30

What is the expression for carrier densities at thermal equilibrium? How do we get this expression?

Answer 30

This expression is obtained because at thermal equilibrium, charge neutrality must be obeyed. This means the negative charge in the semiconductor must equal the positive charge.

N_c + N_D + N_A = N_D + P_v + P_A

Question 31

How do you simplify the charge neutrality expression for the case of fully ionized impurities?

Answer 31

If fully ionized impurities we can simplify to:

N_c + N_A = N_D + P_v

Question 32

What region of N do we like to operate in?

Answer 32

We like to operate in the region where the semiconductor is behaving like an extrinsic or doped semiconductor. This is where N = N_D, meaning all donors are fully ionized. This is because in this region, we only need to give the semiconductor a little bit of energy to get it to operate. This is generally the medium temperature zone.

Question 33

How does the intrinsic carrier density compare to the size of the band gap?

Answer 33

The larger the band gap, the smaller the intrinsic carrier density.