Condensed Matter - Semiconductor

In an n-type semiconductor, which of the following is true of impurity atoms?

1. They accept electrons from the filled valence band into empty energy levels just above the valence band.
2. They accept electrons from the filled valence band into empty energy levels just below the valence band.
3. They accept electrons from the conduction band into empty energy levels just below the conduction band.
4. They donate electrons to the filled valence band from donor levels just above the valence band.
5. They donate electrons to the conduction band from filled donor levels just below the conduction band.

(GR9677 #78)

Solution:

Semiconductor:

n-type:

• n for negative-charge
• atoms have extra electron
• donors to the conduction band

p-type:

• p for positive-charge (holes)
• atoms have missing electrons
• acceptors in the valence band

Source: physics.udel.edu

Answer: E 

Condensed Matter - Conductor and Semiconductor

Which of the following statements concerning the electrical conductivities at a room temperature of a pure copper sample and a pure silicon sample is NOT true? 

1. The conductivity of the copper sample is many orders of magnitude greater than that of silicon sample.
2. If the temperature of the copper sample is increased, its conductivity will decrease.
3. If the temperature of the silicon sample is increased, its conductivity will increase.
4. The addition of an impurity in the copper sample always decreases its conductivity.
5. The addition of an impurity in the silicon sample always decreases its conductivity.

(GR8677 #23)

Solution:

A. TRUE.
Copper → good conductor
Silicon → semi conductor

B. TRUE.
Conductivity is the inverse of resistivity, 
Copper → good conductor → resistivity, → , decreases.

C. TRUE.
Silicon → semi conductor → resistivity, → , exponentially increases

D. TRUE.
For conductor, doping will decrease conductivity.

E. FALSE.
For semiconductor, doping increases the charge carrier density (electron or electron holes) in the conduction band → increasing conductivity

Answer: E

Thermal Physics - Einstein solid

One feature common to both the Debye theory and the Einstein theory of the specific heat of a crystal composed of N identical atoms is that the

1. Average energy of each atom is 3kT
2. Vibrational energy of the crystal is equivalent to the energy of 3N independent harmonic oscillators
3. Crystal is assumed to be continuous for all elastic waves
4. Speed of the longitudinal elastic waves is less than the speed of the transverse elastic waves
5. Upper cutoff frequency of the elastic waves is the same

(GR8677 #51)

Solution:

Einstein solid:

• Thermal properties of crystal
• Highly idealized model focusing only the vibrational modes of crystal
• Assumption: Each of N atoms is free to vibrate around its equilibrium position in any of 3 coordinate directions by a harmonic force with a natural frequency ω₀.

Einstein solid equipartition energy:

• Number of oscillator N = N₀,
• 3-D Harmonic Oscillator, f = 6
• U = (6/2) N₀kT = 3N₀kT

Answer: B

Condensed Matter - Bravais Lattice

The conventional unit cell of a body-centered cubic Bravais lattice is shown in the figure. The conventional cell has volume a³. What is the volume of the primitive unit cell?

A. a³/8
B. a³/4
C. a³/2
D. a³
E. 2a³

(GR0177 #52)

Solution:

The body-centered cubic Bravais lattice has 2 atoms:
1/8 part of atom at each of its 8 corners + 1 atom in the middle = 2 atoms

Thus, the volume of the primitive unit cell: a³/2

Answer: C

Condensed Matter - Semiconductor

Which of the following best represents the temperature dependence of the resistivity of an undoped semiconductor? 

(GR0177 #53)

Solution:

ρ ~ e^(1/T)

As temperature increases, resistance decreases, in contrast to metals.

Answer: B

Condensed Matter - Effective mass

Lattice forces affect the motion of electrons in a metallic crystal, so that the relationship between the energy E and wave number k is not the classical equation E = ħ²k²/2m, where m is the electron mass. Instead, it is possible to use an effective mass m* given by which of the following?

A.

B.

C.

D.

E.

(GR9277 #97) 

Solution: 

F = m*a

F = dp/dt = ħ dk/dt     (momentum, p = ħk)

a = dv_g/dt

Group velocity, v_g = dω/dk

Energy of electron, E = hf = ħω 
ω = E/ħ





F = m*a