Paramagnetic Susceptibility - Advanced Solid State - Exam, Exams of Solid State Physics

This is the Exam of Advanced Solid State which includes Paramagnetism and Diamagnetism, Principle of Operation, Exchange Interaction, Ginzburg-Landau Theory, Phase Transitions, Ferromagnetic Material, Giant Magnetoresistance etc. Key important points are: Paramagnetic Susceptibility, Curie Formula, Diamagnetic Properties, Majority Spin Electrons, Ballistic Conductance, Lattice Structure, Electronic Band Structure, Heisenberg Hamiltonian

Typology: Exams

2012/2013

Uploaded on 02/20/2013

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L A N C A S T E R U N I V E R S I T Y
2011 EXAMINATIONS
Part II
PHYSICS - Paper 4.B ( 2 hours )
An indication of mark weighting is given by the numbers in square brackets following
each part.
Candidates should answer question 1 (40 marks) and TWO questions from questions
A2 to A4 (40 marks each).
PHYSICAL CONSTANTS
Planck’s constant h= 6.63 ×1034 J s
~= 1.05 ×1034 J s
Boltzmann’s constant kB= 1.38 ×1023 J K1
Mass of electron me= 9.11 ×1031 kg
Mass of proton mp= 1.67 ×1027 kg
Electronic charge e= 1.60 ×1019 C
Speed of light c= 3.00 ×108m s1
Avogadro’s number NA= 6.02 ×1023 mol1
Permittivity of the vacuum ²0= 8.85 ×1012 F m1
Permeability of the vacuum µ0= 4π×107H m1
Gravitational constant G= 6.67 ×1011 N m2kg2
Bohr magneton µB= 9.27 ×1024 J T1(or A m2)
Bohr radius a0= 5.29 ×1011 m
Gas constant R= 8.31 J K1mol1
Acceleration due to gravity g= 9.81 m s2
1 standard atmosphere = 1.01 ×105N m2
Mass of Earth = 5.97 ×1024 kg
Radius of Earth = 6.38 ×106m
Density of iron = 7.6×103kg m3
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L A N C A S T E R U N I V E R S I T Y

2011 EXAMINATIONS

Part II

PHYSICS - Paper 4.B ( 2 hours )

  • An indication of mark weighting is given by the numbers in square brackets following each part.
  • Candidates should answer question 1 (40 marks) and TWO questions from questions A2 to A4 (40 marks each).

PHYSICAL CONSTANTS

Planck’s constant h = 6. 63 × 10 −^34 J s ℏ = 1. 05 × 10 −^34 J s Boltzmann’s constant kB = 1. 38 × 10 −^23 J K−^1 Mass of electron me = 9. 11 × 10 −^31 kg Mass of proton mp = 1. 67 × 10 −^27 kg Electronic charge e = 1. 60 × 10 −^19 C Speed of light c = 3. 00 × 108 m s−^1 Avogadro’s number NA = 6. 02 × 1023 mol−^1 Permittivity of the vacuum ≤ 0 = 8. 85 × 10 −^12 F m−^1 Permeability of the vacuum μ 0 = 4 π × 10 −^7 H m−^1 Gravitational constant G = 6. 67 × 10 −^11 N m^2 kg−^2 Bohr magneton μB = 9. 27 × 10 −^24 J T−^1 (or A m^2 ) Bohr radius a 0 = 5. 29 × 10 −^11 m Gas constant R = 8 .31 J K−^1 mol−^1 Acceleration due to gravity g = 9.81 m s−^2 1 standard atmosphere = 1. 01 × 105 N m−^2 Mass of Earth = 5. 97 × 1024 kg Radius of Earth = 6. 38 × 106 m Density of iron = 7. 6 × 103 kg m−^3

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Module 421 - Advanced Solid State & Nanophysics (The time allocated is 120 minutes. Candidates should answer question 1 and TWO questions from questions 2 to 4.)

Compulsory question:

  1. (a) State the Curie formula for the paramagnetic susceptibility of a gas of atoms describing all the parameters. From the following list of atoms and molecules, choose those for which diamagnetic properties are observable: Xe, Fe, Na, Cl, NaCl, FeO 2 , CaO. Explain your choices. [10] (b) With the help of a sketch, explain the terms majority spin electrons, minority spin electrons, and exchange splitting in relation to the band structure of fer- romagnetic metals. Describe how tunnel junctions between two ferromagnetic metals are used in microelectronics. [10] (c) Derive the expression for the ballistic conductance of a quantum wire with only one filled subband. [10] (d) Describe the lattice structure and electronic band structure of graphene. Ex- plain what makes it possible to use this material as a part of a field-effect transistor. [10]
  1. (a) Use a band diagram to explain how a two-dimensional electron gas is formed in GaAs/AlGaAs heterostructures. What factors determine the high mobility of electrons in such systems? [10] (b) Write down the Hamiltonian describing electrons in a homogeneous magnetic field and derive the Landau level spectrum in a two-dimensional electron sys- tem. [14] (c) What maximum density of electrons can be accommodated in a single Lan- dau level? Write down the expression for the filling factor and estimate what electron density is needed to completely fill the lowest Landau level in a het- erostructure at B =10 Tesla [in your estimates, use h/ec ≈ 4 × 10 −^11 cm^2 T]. [8] (d) Sketch and describe the experimental manifestation of the quantum Hall effect phenomenon. What is a possible application of the quantum Hall effect? [8]

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