Week 3 Electrons tightbinding models Electron orbitals tightbinding

Week 3, Electrons, tight-binding models Electron orbitals, tight-binding models, creation/annihilation operators, second quantization, graphene band structures, density of states

Atomic orbitals of carbon atom Density (gm cm-3) carbon 3. 52 12. 011 6 Lattice constant (Å) Melting temperature 1 s 22 p 2 3. 57 diamond (4300) 1860 C Mass Atomic number Debye temperature (K)

Potential Bonding VA VB E 0+t E 0 -t

pz (or ) orbital in graphene Each atomic orbital can be occupied by two electrons with spin up and down. 1 s -> 2 electrons 3 sp 2 -> 3 electrons from each atom (shared occupation of the bond) pz -> 1 electron each atom (half filling)

Graphene band structure D 6 h D 3 h K M

Focus on one site (spinless) 0 1

Quantum dot Hamiltonian energy

Many electrons, property of identical fermions

Matrix representation of fermionic operators : Jordan-Wigner

Tight-binding model for graphene

Choice of k values ky b 2 M K’ b 1 K kx

Solution of graphene model

Dirac model near K points

Quantum field

Many-body ground state EF=0

Statistical mechanics Problem: Compute the internal energy and heat capacity at constant volume for graphene at low temperature when chemical potential is at the Dirac point ( =0).

More general tight-binding models •

Calculating band structure of graphene, steps 1 to 3, run pw. x • Step 1, use calculation='vc-relax’, in the &CONTROL namelist to find the optimal lattice parameters, if it is not yet optimal (e. g. , 0 stress). You need an ion_dynamics in this case. You can skip this if lattice parameter is already optimal • Step 2, do a self-consistent calculation='scf’ • Step 3, do a non-self-consistent, band structure calculation with calculation='bands’, other parameters remain the same as in step 2. Specify the k-point path now with a data card as K_POINTS crystal_b 4 0. 00000 -0. 33333 0. 66667 0. 00000 0. 500000000 0. 000000000 50 50 ! ! G K M G

Calculating band structure of graphene, step 4, run band. x to get data for plotting • Step 4, run mpirun -np 8 bands. x<bands. in >bands. out to generate from the raw data for plotting in the data file band. dat with an input of a single namelist: &bands prefix = 'graphene’ filband = 'bands. dat’ / (as used in other input files)