Computational Material Science Karsten Held, Jan Kunes TU Wien Bloch wave function - what is it? In most 'real life' cases, the wave functions are hidden somewhere in the memory of our computer. However, looking at their shape sometimes can provide a better intuition about 'how the theory actually works'. This is such an example. Image Transition metal dichalcogenides (e.g. MoS2): source Wikipedia Unitary transformation (very common trick for periodic systems): Bloch theorem for lattice models periodicity lattice site coincides with the unit cell (not the most general case) Example: 1D chain with nn hopping Image Hopping matrix (Hamiltonian): Dispersion: Diagonalize h directly for N=3. What eigenstates do you get? Generalize the problem for square and cubic lattices. Basis transformation: Example: Honeycomb lattice Type to enter text (0,0) (1,0) (0,1) (-1,0) a b (0,-1) After FT: Example: Honeycomb lattice After FT: Image Image At each k-point k=(ka,kb) we have a 2x2 matrix to diagonalize: Finally we get the dispersion relation: Example: Honeycomb lattice After FT: Image Image At each k-point k=(ka,kb) we have a 2x2 matrix to diagonalize: Image Finally we get the dispersion relation: Image Image Transition metal dichalcogenides (TMDs) Type to enter text (0,0) (1,0) (0,1) (-1,0) a b (0,-1) At K and K' the off-diagonal element vanishes: How does the wave function at K look? Wave function at K Type to enter text Image How does the wave function at K look? mathbf_K_=(_frac.pdf a b h(_mathbf_K_)=_D.pdf Type to enter text Image a b Ga Gb Are different K points equivalent? Type to enter text Image mathbf_K_=(_frac.pdf a b Ga Gb Type to enter text Image a b Ga Gb mathbf_K_=(-_fra.pdf Type to enter text Image a b Ga Gb mathbf_K_=(_frac.pdf Image How does the wave function at K look? Type to enter text mathbf_K'_=(-_fr.pdf Wave function at K' psi(_mathbf_R_=m.pdf Image How does the wave function at K look? Type to enter text mathbf_K'_=(-_fr.pdf Wave function at K' psi(_mathbf_R_=m.pdf psi(_mathbf_R_=m.pdf Is this an eigenstate? = 0 H Σ Image How does the wave function at K look? psi(_mathbf_R_=m.pdf Type to enter text * . Are function at K and K' mutually orthogonal? Σ sites How does the wave function at K look? psi(_mathbf_R_=m.pdf . = Type to enter text Are function at K and K' mutually orthogonal? Σ = 0 sites Σ sites How does the wave function at K look? psi(_mathbf_R_=m.pdf Type to enter text + = Type to enter text Linear superposition of |K > and |K'>? Optical transitions (at K) Type to enter text mathbf_K_=(_frac.pdf Type to enter text Peierls substitution: A is related to the vector potential of the elmag. field light or Optical transitions (at K) Type to enter text mathbf_K_=(_frac.pdf Type to enter text Image t_rightarrow_te^.pdf Peierls substitution: When driven by external field electron picks a direction dependent phase on any given bond: A is related to the vector potential of the elmag. field x y langle_K_+_|d|K_.pdf Optical transitions (at K) Type to enter text mathbf_K_=(_frac.pdf Type to enter text t_rightarrow_te^.pdf &iA_x-iA_x_tfrac.pdf Peierls substitution: When driven by external field electron picks a direction dependent phase on any given bond: A is related to the vector potential of the elmag. field x y langle_K_+_|d|K_.pdf Type to enter text mathbf_K_=(_frac.pdf Type to enter text &iA_xe^i_tfrac_2.pdf Optical transitions (at K) t_rightarrow_te^.pdf Peierls substitution: When driven by external field electron picks a direction dependent phase on any given bond: A is related to the vector potential of the elmag. field x y Type to enter text langle_K_+_|d|K_.pdf Type to enter text mathbf_K_=(_frac.pdf Type to enter text Image &iA_xe^-i_tfrac_.pdf Optical transitions (at K) t_rightarrow_te^.pdf Peierls substitution: When driven by external field electron picks a direction dependent phase on any given bond: A is related to the vector potential of the elmag. field x y langle_K_+_|d|K_.pdf Type to enter text mathbf_K_=(_frac.pdf Type to enter text Image langle_K_+_|d|K_.pdf langle_K_+_|d|K_.pdf |K_+_rangle.pdf excites optical transition at K does not excite optical transition at K circular polarization: You can show that at K' the role of circular polarizations is exchanged. => You can choose the valley K or K' by using a circularly polarized light. Optical transitions (at K and K') Summary •Continuous and lattice models of solids •Translational symmetry (unit cell, basis vectors) •Bloch theorem •Simple lattice models and their diagonalization •Density of states its calculation Next time •Basics of the density functional theory Project no. 1 Calculate the band structure and density of states for 2D Kagome and 3D pyrochlore lattices. Are there any bands with special properties? How does the the band structure change when we add a sublattice site potentials?