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PHY
1 PHY380 TURN OVER
Data Provided: A formula sheet and table of physical constants are attached to this paper.
DEPARTMENT OF PHYSICS & ASTRONOMY AUTUMN SEMESTER 2017-
SOLID STATE PHYSICS 2 HOURS
The paper is divided into 5 questions.
Answer compulsory question 1 , which is marked out of 20.
Answer any two out of the optional questions 2-5 , each of which is marked out of
The breakdown on the right hand-side of the paper is meant as a guide to the marks that can be obtained from each part.
Please clearly indicate the question numbers on which you would like to be examined on the front cover of your answer book. Cross through any work that you do not wish to be examined.
PHY
PHY
CONTINUED
QUESTION 1.
COMPULSORY
a) Show that the Fermi velocity F of conduction electrons in terms of electron
density n is given by the expression F = ħ(3^2 n )1/3/ m , where m is electron mass. [3]
b) Explain the meanings of the terms: (i) Fermi surface [1.5] (ii) Bands and band gaps [1.5] (iii) Brillouin zone [1.5]
c) Describe an experimental technique which measures conduction electron energy distribution in metals. Your answer should include an explanation of how this distribution is obtained. [3.5]
d) Explain the concept of donors and acceptors in semiconductors. Give two examples of donor elements for silicon. Provide an explanation of why these elements act as donors in silicon. [2]
e) It is observed that a frog placed in a strong gradient of magnetic field levitates. Explain the origin of this effect. [2]
f) Write down an expression quantifying the energy U of a magnetic dipole in a
constant magnetic field B. Calculate the energy difference (in electron-volts)
between a magnetic dipole of 1.0 B oriented parallel and antiparallel to the
direction of a magnetic field of 3 T. [2]
g) The ions Cr3+^ and Ni2+^ have the following electronic configurations: Cr3+^ : 1s^2 ; 2s^2 ; 2p^6 ; 3s^2 ; 3p^6 ; 3d^3 Ni2+^ : 1s^2 ; 2s^2 ; 2p^6 ; 3s^2 ; 3p^6 ; 3d^8 Use Hund’s rules to write down the configuration of electrons in the unfilled electronic shells and thereby calculate L, S, and J for both ions. [3]
PHY
PHY
CONTINUED
QUESTION 3.
OPTIONAL
a) Indium antimonide has a dielectric constant =17 and electron effective mass
m = 0.014 m 0 , where m 0 is the free electron mass. Estimate the donor concentration at which the ground state orbits of adjacent donors start overlapping. [3]
b) A sample of GaAs is illuminated with light with a wavelength of 700 nm at 300 K. Calculate the kinetic energy of photoexcited electrons in electron-volts. The electron ( me ) and hole ( mh ) effective masses in GaAs are 0.067 m 0 and 0.45 m 0 , respectively, where m 0 is the free electron mass. The band gap of GaAs is 1.43 eV at 300 K. [4]
c) Explain the concept of a Wannier-Mott exciton in a semiconductor material. How does the radius of a ground state Wannier-Mott exciton compare to that of a ground state electron on donor in the same material? Provide an explanation for your answer. [2]
d) A piece of copper is placed in an external magnetic field H, whose magnitude inside the copper is 10^4 A/m. The magnetic susceptibility of copper is – 0.5×10-5. Calculate the magnetic moment per unit volume in copper. [2]
e) Ferric oxide (Fe 2 O 3 ) is used as a magnetic data storage medium. Fe 2 O 3 has a molecular weight of 159.7 g mol-^1 , a density of 5242 kg m-^3 and a
magnetic dipole moment of 5.9 B per Fe3+^ ion. Calculate the
saturation magnetisation per unit volume of ferric oxide. [4]
PHY
5 PHY380 TURN OVER
QUESTION 4.
OPTIONAL
a) Derive the following formula for electron concentration ne in the conduction band as a function of temperature T in a semiconductor doped with donors of concentration ND. 𝑛𝑒 = √𝑁𝑐𝑁𝐷 exp (
Here ED is the binding energy of an electron to a donor, and Nc is the electron effective density of states in the conduction band. In your derivation you may use the following expressions for electron concentration ne in the conduction band and Fermi-Dirac distribution f(E):
𝑛𝑒 = 𝑁𝐶 exp (
1 + exp (
Here EF is the Fermi level and EG is the band gap of the material. In your derivation you may assume that | EG – ED – EF |>> kBT. [7]
b) (i) Explain how and why the phenomenon of impurity compensation may arise in a semiconductor initially doped only with donors or acceptors. [2]
(ii) A semiconductor is heavily doped with acceptors with density Na and donors with density Nd , where Nd < Na. What will be the resultant carrier type and free carrier density in the material at temperatures significantly greater than the impurity binding energies? [2]
c) The Lanthanide element holmium (Ho) undergoes a transition from a ferromagnetic to a paramagnetic state at a temperature of 19 K. Explain: (i) What is meant by a ferromagnetic and a paramagnetic state, [2] (ii) why a change of temperature is responsible for driving a transition between these states. [2]
PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE
Physical Constants
electron charge e = 1. 60 × 10 −^19 C electron mass me = 9. 11 × 10 −^31 kg = 0. 511 MeV c−^2 proton mass mp = 1. 673 × 10 −^27 kg = 938. 3 MeV c−^2 neutron mass mn = 1. 675 × 10 −^27 kg = 939. 6 MeV c−^2 Planck’s constant h = 6. 63 × 10 −^34 J s Dirac’s constant (ℏ = h/ 2 π) ℏ = 1. 05 × 10 −^34 J s Boltzmann’s constant kB = 1. 38 × 10 −^23 J K−^1 = 8. 62 × 10 −^5 eV K−^1 speed of light in free space c = 299 792 458 m s−^1 ≈ 3. 00 × 108 m s−^1 permittivity of free space ε 0 = 8. 85 × 10 −^12 F m−^1 permeability of free space μ 0 = 4π× 10 −^7 H m−^1 Avogadro’s constant NA = 6. 02 × 1023 mol−^1 gas constant R = 8. 314 J mol−^1 K−^1 ideal gas volume (STP) V 0 = 22. 4 l mol−^1 gravitational constant G = 6. 67 × 10 −^11 N m^2 kg−^2 Rydberg constant R∞ = 1. 10 × 107 m−^1 Rydberg energy of hydrogen RH = 13. 6 eV Bohr radius a 0 = 0. 529 × 10 −^10 m Bohr magneton μB = 9. 27 × 10 −^24 J T−^1 fine structure constant α ≈ 1 / 137 Wien displacement law constant b = 2. 898 × 10 −^3 m K Stefan’s constant σ = 5. 67 × 10 −^8 W m−^2 K−^4 radiation density constant a = 7. 55 × 10 −^16 J m−^3 K−^4 mass of the Sun M = 1. 99 × 1030 kg radius of the Sun R = 6. 96 × 108 m luminosity of the Sun L = 3. 85 × 1026 W mass of the Earth M⊕ = 6. 0 × 1024 kg radius of the Earth R⊕ = 6. 4 × 106 m
Conversion Factors
1 u (atomic mass unit) = 1. 66 × 10 −^27 kg = 931. 5 MeV c−^2 1 Å (angstrom) = 10−^10 m 1 astronomical unit = 1. 50 × 1011 m 1 g (gravity) = 9. 81 m s−^2 1 eV = 1. 60 × 10 −^19 J 1 parsec = 3. 08 × 1016 m 1 atmosphere = 1. 01 × 105 Pa 1 year = 3. 16 × 107 s
Polar Coordinates
x = r cos θ y = r sin θ dA = r dr dθ
∇^2 =
r
∂r
r
∂r
r^2
∂^2
∂θ^2
Spherical Coordinates
x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dV = r^2 sin θ dr dθ dφ
∇^2 =
r^2
∂r
r^2
∂r
r^2 sin θ
∂θ
sin θ
∂θ
r^2 sin^2 θ
∂^2
∂φ^2
Calculus
f (x) f ′(x) f (x) f ′(x)
xn^ nxn−^1 tan x sec^2 x ex^ ex^ sin−^1
(x a
√ a^2 −x^2 ln x = loge x (^1) x cos−^1
(x a
− √a (^21) −x 2 sin x cos x tan−^1
(x a
) (^) a a^2 +x^2 cos x − sin x sinh−^1
(x a
√x (^2) +a 2 cosh x sinh x cosh−^1
(x a
√ x^2 −a^2 sinh x cosh x tanh−^1
(x a
) (^) a a^2 −x^2 cosec x −cosec x cot x uv u′v + uv′ sec x sec x tan x u/v u
′v−uv′ v^2
Definite Integrals
∫ (^) ∞
0
xne−ax^ dx =
n! an+^
(n ≥ 0 and a > 0 )
∫ (^) +∞
−∞
e−ax 2 dx =
π a ∫ (^) +∞
−∞
x^2 e−ax
2 dx =
π a^3
Integration by Parts:
∫ (^) b
a
u(x)
dv(x) dx
dx = u(x)v(x)
b a
∫ (^) b
a
du(x) dx
v(x) dx
Vector Calculus
A · B = AxBx + AyBy + Az Bz = Aj Bj
A×B = (AyBz − Az By)ˆi + (Az Bx − AxBz )ˆj + (AxBy − AyBx) kˆ = ijkAj Bk A×(B×C) = (A · C)B − (A · B)C A · (B×C) = B · (C×A) = C · (A×B)
grad φ = ∇φ = ∂j φ =
∂φ ∂x
ˆi + ∂φ ∂y
ˆj + ∂φ ∂z
kˆ
div A = ∇ · A = ∂j Aj =
∂Ax ∂x
∂Ay ∂y
∂Az ∂z
curl A = ∇×A = ijk∂j Ak =
∂Az ∂y
∂Ay ∂z
ˆi +
∂Ax ∂z
∂Az ∂x
ˆj +
∂Ay ∂x
∂Ax ∂y
k^ ˆ
∇ · ∇φ = ∇^2 φ =
∂^2 φ ∂x^2
∂^2 φ ∂y^2
∂^2 φ ∂z^2 ∇×(∇φ) = 0 and ∇ · (∇×A) = 0 ∇×(∇×A) = ∇(∇ · A) − ∇^2 A
