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Condensed Matter Physics

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Condensed Matter Physics Condensed Matter Physics Crystal Structure Basic crystal structure Crystal: an infinite repeated pattern of identical groups of atoms, consisting of a basis and a lattice Basis: a group of atoms which forms the crystal structure Lattice: the three-dimensional set of points on which the basis is attached to form the crystal Primitive cell: the cell of the smallest volume which can be translated to map out all the points on the lattice. The primitive cell is not unique for a given crystal. There is always exactly one lattice point per primitive cell Conventional cell: a more convenient unit which also translates to map out the entire lattice, often a cube Wigner-Seitz primitive cell: this is a special form of the primitive cell constructed by connecting a given lattice point to all its neighbours, then drawing new lines as the mid-points of these lines 1 Lattice point group: a collection of symmetry operations which map the lattice back onto itself, such as translation, reflection, and rotation Miller indices Miller indices are an indexing system for defining planes and directions in a crystal. The method for defining them is as follows: Find the intercepts of the plane with the axes in terms of lattice constants Take the reciprocal of each intercept, then reduce the ratio to the smallest possible integer ratio A particular plane is denoted A set of planes equivalent by symmetry is denoted The direction perpendicular to is denoted by The reciprocal lattice If we have a repeating lattice we can represent the electron density by a periodic function, which we can write as a Fourier series: Where the factors of ensure that has the required periodic relation . Points in the reciprocal lattice are then given by , for positive integer coefficients. In three dimensions the lattice is defined by: 2 The Fourier transform of the original lattice series gives the series of the inverse lattice: In three dimensions this becomes (for cell volume ): The reciprocal lattice vector is given by: Where are the primitive vectors of the crystal lattice and are the primitive vectors of the reciprocal lattice. Diffraction conditions The total amplitude of a scattered wave from one point in the crystal in direction is proportional to: Where is the electron density at a given point, and the exponential term is the phase factor difference between the incident wave with wavevector and the reflected wave with wavevector . Integrating over the entire crystal we get the scattering amplitude : Substituting in the Fourier decomposition of we have: 3 The exponential will be negligible if is much different from , so is only really non-zero when (the Bragg scattering condition), meaning that we have: The diffraction condition can now be found starting with: Using the elastic scattering condition that we can write this as: The distance between two parallel lattice planes normal to is . Thus if if is the angle between (the incident wavevector) and (the crystal lattice), and noting that is perpendicular to so the angle between and is , then we can write this as: Using the fact that we have: 4 Likewise and . Substitute and we arrive at the Laue equations: For diffraction to occur, must simultaneously satisfy all three of these equations. Each equation is satisfied for a conic region, so this corresponds to lying at the centre of a cone about each of the directions , meaning that must lie at the intersection of three cones at once. Brillouin zones The Brillouin zone is just the Wigner-Seitz primitive cell in the reciprocal lattice. It is formed in the same way as in the real lattice: draw a vector from the origin to each neighbouring reciprocal lattice point, then at the midpoint of this line draw a second line perpendicular to each of these lines, and shade the central area bounded by perpendicular lines. This central area is the Brillouin zone. Only incident waves whose wavevector drawn from the origin terminates on a bounding surface of the Brillouin zone will be diffracted by the crystal. To see that any such vector will satisfy the diffraction condition, observe that: 5 Note that a simple cubic is its own reciprocal lattice, while bcc and fcc lattices are the reciprocals of each other. Structure factor When the scattering condition is met, the scattering amplitude for an array of N cells can be written: We call the structure factor, which is the electron density multiplied by the phase, integrated over a single cell with directed from the center of the cell. We can recast this as an integral over each of the atoms in the cell if we write for the electron concentration at each of the atoms in the basis. Then we have: We thus define the atomic form factor as: This is a measure of the scattering power of the th atom in the unit cell. Using this definition, the structure factor of the basis then has the form: Substituting in the lattice coordinates of atom : We then have: 6 Theory of Phonons Classical theory of phonons Total kinetic energy (in one dimension) of a chain of atoms each displaced a distance from equilibrium is given by: The potential energy is similarly given by: Where is the restoring force constant.The Lagrangian for atom then is: We use this Lagrangian with the Euler-Lagrange equation: To arrive at the equation of motion: Consider the solution (where is the wavenumber): Using the identity , we find the dispersion relation: The wavenumber is effectively the ‘distance’ in the reciprocal lattice, so we can plot these dispersion relations as a function of the frequency of phonons at various positions in the reciprocal lattice: 7 In the case where there are two atoms with masses and and spring constants and , the dispersion relation becomes: Using solutions: We arrive at the equations: In matrix form: Which has the solution: 8 The negative solution gives the acoustic phonon mode, while the positive solution gives the optical phonon mode. In the full three-dimensional case, there will be degrees of freedom for a unit cell with different atoms, leading to three acoustic and optical branches. 9 Quantum theory of phonons While normal modes are wave-like phenomena in classical mechanics, they have particle-like properties in the wave-particle duality description of quantum mechanics. For the full quantum mechanical description the atomic displacements and atomic momenta become operators: For a particle in a parabolic potential we use the Hamiltonian: We now define quantum raising and lowering operators: We use the standard canonical commutation relations: The Hamiltonian then becomes: The number operator gives the excitation level of normal mode . The eigenstates of this operator are denoted: 10 Photon emission or absorption follows the rule: Total energy is then: Thermal statistics of phonons Each phonon has energy , yielding a total lattice energy of vibrational mode is: The probability distribution of the number of phonons in level is: This is the Bose-Einstein distribution, and allows us to calculate the average phonon number in equilibrium: We thus have for mode : 11 Thus we see that the average number of phonons depends both on the temperature and on the phonon frequency . The average energy of the entire lattice at temperature is then given by: It would be more convenient to write this as an integral: If we use the simple dispersion relation for for constant velocity then this becomes: The density of states is the number of phonon modes per unit frequency per unit length. It is a way of expressing the number of different values of that can be generated by various combinations of etc, divided by . It is usually not an easy function to evaluate, but becomes simpler for small values of . This is the approximation used above. We need it to convert the sum to an integral. Note that is the Debye frequency, the cutoff frequency representing the highest frequency that can propagate within the lattice (depends on atomic separation a). We can compute it by noting the integral over the density of states is just the volume density of states, and so: 12 We also note that each of the phonon branches (LA, TA, and TO) has its own set of frequencies and its own Debye frequency, so we need to multiply the answer by three. We can now use this integral result to calculate the thermal energy of the lattice per unit volume at temperature : Neglecting the factor of : Let , then we have: By the definintion of , we are led to the definition of a natural temperature scale: We can then write the specific heat capacity as: At low temperatures we can use the approximation that the upper limit goes to infinity: 13 Hence the specific heat capacity becomes: Phonon anharmonicity The potential energy Taylor series expansion around the equilibrium position takes the form: For the harmonic oscillator case . Including higher order terms leads to anharmonicity. This means that photons will decay over time and can interact to create a third phonon. 14 The Origin of Band Gaps Introduction to free electron theory The free electron model assumes that the valence electrons of atoms in the crystal lattice become conduction electrons that can move freely throughout the volume of the metal without interacting with either the ion cores or each other (Pauli exclusion principle). The energy levels of free electrons in one dimension are given by the TISE: To solve this we can consider both fixed boundary conditions where and periodic boundary conditions where . The solution for periodic boundary conditions is: In one dimension, the Fermi energy is the energy of the uppermost filled electron state in the ground state configuration of an N electron system.
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