Scanning Tunnelling Microscope - 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: Scanning Tunnelling Microscope, Operating Principles, Curie Susceptibility, Pauli Paramagnetic Susceptibility, Band Structures, Metal Tunnel Junctions, Polarisation States, Curie Temperature

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
2010 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 ǫ0= 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

2010 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 ǫ 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) Describe the operating principles of a scanning tunnelling microscope (STM), and state its main applications. [12] (b) Write down (without derivation) expressions for the two contributions to the magnetic susceptibility of a metal: the Pauli paramagnetic susceptibility due to the Fermi sea of electrons and the Curie susceptibility of nuclear spins. Define any symbols used. For an alkali metal where the nuclei have spin 1/2 and the Fermi energy of electrons is EF ∼ 8 eV , estimate at what temperature the contribution from nuclei exceeds the contribution from electrons. [10] (c) With the aid of a sketch, describe the difference between the band structures of ferromagnetic and normal metals. Describe how this leads to magnetore- sistance in ferromagnetic metal tunnel junctions, stating appropriate formula for the conductances of the two polarisation states. State an application of ferromagnetic metal tunnel junctions in microelectronics. [18]

Answer two of the following three questions:

  1. (a) Write down the Heisenberg Hamiltonian describing a magnetic solid and ex- plain which sign of the exchange constant J in the Heisenberg Hamiltonian leads to ferromagnetic rather than antiferromagnetic ordering. Without deriva- tion, state the relation between the critical temperature of the ferromagnetic phase transition (Curie temperature, TC ), the exchange constant J and the number of closest neighbours N for magnetic atoms on the lattice. [8] (b) Using the Ginzburg-Landau theory of ferromagnetic phase transitions, derive the formulae which describe: (i) the magnetisation of a ferromagnetic material at temperatures just below the Curie temperature TC ; (ii) the magnetic susceptibility of such a material just above TC. [15] (c) Describe the influence of crystalline anisotropy on the magnetisation of ferro- magnetic materials and state the corresponding terms in the Ginzburg-Landau expansion. Explain how anisotropy influences the formation of domains during ferromagnetic phase transitions and how these domains lead to hysteresis of the magnetisation when reversing the direction of an external magnetic field. [10] (d) Use the Ginzburg-Landau theory to determine how many different energeti- cally favourable equivalent orientations of the magnetisation are possible in a ferromagnetic material with cubic lattice symmetry. [7]