Week 1: Semiconductor Fundamentals Flashcards
What is the probability of thermal excitation proportional to?
e^(-E_g / k_b * T)
where k_b is Boltzmann constant, T is temperature in Kelvin, and E is the band gap energy
What is the conduction band?
The highest unoccupied band
What is the valence band?
the highest occupied level at zero Kelvin
What is a semiconductor?
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.
Semiconductors sometimes behave like insulators and sometimes behave like conductors. What is modulated to take advantage of this?
The carrier density
What is the density of states used for?
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
What is the density of states?
The density of states is the number of states that are available for electrons and holes at a given energy, E + dE
What is g(E) representing? What happens when you multiply it by an infinitesimally small dE slice of energy?
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
What is the spin degeneracy?
The fact that for a given state we can have a spin up and a spin down electron.
What does the carrier density control?
The carrier density controls the current and is responsible for whether the semiconductor can emit light or not.
Describe the probability of occupation in terms of E and f(E).
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.
Electrons are considered to be fermions. What does this mean for the principle they obey?
This means they obey the Pauli exclusion principle meaning that each electron has to have a unique state.
What do holes in the valence band correspond to?
Empty states in the valence band
How is the carrier density in a semiconductor obtained?
By integrating the product of the density of states and the probability density function over all possible states
How would you calculate the electron density at a given temperature for a semiconductor?
N(T) = integralE_c to inf
g_c(E): density of states in conduction band
f(E): probability of occupation
How would you calculate the hole density at a given temperature for a semiconductor?
P(T) = integral-inf to E_v
g_v(E): density of states in valence band
f(E): probability of occupation
What is the probability of finding an electron (from the Fermi-Dirac statistics)?
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%
Explain the non-degeneracy condition and what it allows us to do.
The non-degeneracy condition means that the fermi level never gets too close to the conduction or the valence band.
E_c - E_f»_space; k_b * T
E_f - E_v»_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.
What can we approximate f(E) to be with the non-degeneracy condition? What about 1-f(E)? What’s the benefit of this?
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.
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?
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.
Using the parabolic band approximation, what is the result of evaluating N_v(T) and N_c(T)?
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)
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?
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)
What’s an intrinsic semiconductor and what does it tell us about the electron concentration, hole concentration, and intrinsic carrier concentration?
A semiconductor we have done nothing to. It tells us that N(T) = p(T) = n_i(T), so they’re all equal.
Describe two equations that can be used to calculate the fermi level of an intrinsic semiconductor.
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*)
