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Quantum Theory of Solids. Mervyn Roy (S6 ) www2.le.ac.uk/departments/physics/people/mervynroy. Course Outline. Introduction and background The many-electron wavefunction - Introduction to quantum chemistry ( Hartree , HF, and CI methods) Introduction to density functional theory (DFT)
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Quantum Theory of Solids Mervyn Roy (S6) www2.le.ac.uk/departments/physics/people/mervynroy
Course Outline • Introduction and background • The many-electron wavefunction • - Introduction to quantum chemistry (Hartree, HF, and CI methods) • Introduction to density functional theory (DFT) • - Framework (Hohenberg-Kohn, Kohn-Sham) • - Periodic solids, plane waves and pseudopotentials • Linear combination of atomic orbitals • Effective mass theory • ABINIT computer workshop (LDA DFT for periodic solids) • Assessment: 70% final exam • 30% coursework – mini ‘project’ report for ABINIT calculation • www.abinit.org
Last time… • Solve self-consistent Kohn-Sham single particle equations to find for real interacting system • , where, • , and • , where • Know exactly for uniform electron gas – use LDA for real materials • Many different functionals available • In principle, Kohn-Sham and are meaningless (except the HOMO). In practice, often give decent band structures, effective masses etc • DFT band gap problem – extend DFT (GW or TDDFT) to get excitations right
Periodic structures and plane waves • 223 course notes • Solid state text books – e.g. • Tanner, Introduction to the Physics of Electrons in Solids, Cambridge University press • Hook and Hall, Solid State Physics 2nd Ed., John Wiley and Sons • Ashcroft and Mermin, Solid State Physics, Holt-Saunders
grapheneunit cell 1 2 Crystal = Bravais lattice + basis 2 atom basis atoms at: and Primitive cell vectors: nm
2D crystal – many choices for unit cell Hexagonal lattice, 2 atom basis • Primitive • Primitive centred • Non-primitive Wigner-Seitz (primitive)
3D crystal: zinc blende structure (diamond, Si, GaAsetc) FCC 2 atom basis and Primitive cell vectors Volume of cell, wikipedia.org www.seas.upenn.edu
Any function f(r), defined in the crystal which is the same in each unit cell (e.g. electron density, potential etc.) must obey, where, e.g. environment is the same at as it is at
Reciprocal lattice where reciprocal lattice vectors, , satisfy Then, Wigner-Seitz cell in reciprocal space = Brillouin zone ,
FCC Reciprocal lattice = BCC Volume of Brillouin zone = recip Brillouin Zone Léon Brillouin (1889-1969): most convenient primitive cell in reciprocal space is the Wigner-Seitz cell - edges of BZ are Bragg planes.
Question 3.1 Calculate the reciprocal lattice vectors for an FCC structure Show that the FCC reciprocal lattice is body centred cubic Calculate the reciprocal lattice vectors for graphene Construct the graphene BZ, labelling the high symmetry points Show that, in 3 dimensions, hint:
Example band structure for a Zinc Blende structure crystal Dispersion relation, plotted along high symmetry lines in Brillouin zone L-G-X conduction band valence band (heavy holes) band, doubly degenerate band (no spin orbit coupling) filled states,
Fourier representation of a periodic function If then, where, are reciprocal lattice vectors and
Bloch theorem If is an eigenstate of the single-electron Hamiltonian, , then . The Bloch states, , are often written in the form, periodic part - has the periodicity of the lattice so plane wave part Orthogonality - the are orthonormal within one unit cell, the are only orthogonal over the whole crystal
Question 3.2 • If is the crystal volume, show that the spacing between k states is in • a cuboid crystal • a non-cuboid crystal • Show that the number of states in the first BZ for a single band is , where is the number of unit cells in the crystal • If there are atoms in the basis and electrons per atom, show that the band index of the highest valence band is
