Quantum Theory of Solids
Mervyn Roy (S6)
www2.le.ac.uk/departments/physics/people/mervynroy
PA4311 Quantum Theory of Solids
Course Outline
1. Introduction and background
2. The many-electron wavefunction
- Introduction to quantum chemistry (Hartree, HF, and CI methods)
3. Introduction to density functional theory (DFT)
- Framework (Hohenberg-Kohn, Kohn-Sham)
- Periodic solids, plane waves and pseudopotentials
4. Linear combination of atomic orbitals
5. Effective mass theory
6. ABINIT computer workshop (LDA DFT for periodic solids)
Assessment:
70% final exam
30% coursework – mini ‘project’ report for ABINIT calculation
www.abinit.org
PA4311 Quantum Theory of Solids
Last time…
Can’t calculate full 𝑁-electron wavefunction, Ψ0 , for 𝑁 > ~10…
Hohenberg-Kohn
• Can, in principle, determine everything from 𝑛0 (𝒓)
- Simple proof that 𝑛0 (𝒓) uniquely determines 𝑣(𝒓)
• Can, in principle, calculate 𝑛0 (𝒓), by finding the density that
minimises the functional
𝐸 = 𝐹𝐻𝐾 𝑛 + ∫ 𝑣 𝒓 𝑛 𝒓 𝑑𝒓 − 𝜇(∫ 𝑛(𝒓)𝑑𝒓 – 𝑁)
- But, can’t find 𝐹𝐻𝐾 [𝑛] without finding Ψ0 …
PA4311 Quantum Theory of Solids
𝜇 = chemical
potential
Last time…
Kohn-Sham
• Map the difficult interacting problem onto an auxiliary noninteracting problem that has the same 𝑛0 𝒓 (and is easy to
solve)
• Single-particle KS equations for each Kohn-Sham orbital with all
the difficult many-body terms expressed in terms of the
exchange and correlation energy 𝐸𝑋𝐶
PA4311 Quantum Theory of Solids
Kohn-Sham
Want to find 𝑛0 𝒓 by minimising 𝐸 𝑛0 = 𝑇 𝑛0 + 𝑊 𝑛0 + ∫ 𝑛0 𝒓 𝑣 𝒓 𝑑𝒓
Invent auxiliary non-interacting system with same density,
𝐻𝑆 Ψs = 𝐸Ψs
where 𝐻𝑆 =
𝑖
−𝛻𝑖2
2
+ 𝑣𝑠 𝒓𝑖
, Ψs is a single determinant and
𝑁
𝑖
𝜓𝑖
𝐸 𝑛0 = 𝑇 𝑛0 + 𝑊 𝑛0 + ∫ 𝑛0 𝒓 𝑣 𝒓 𝑑𝒓
1
𝑛 𝑟 𝑛 𝑟′
′ + 𝐸 [𝑛 ]
𝐸 𝑛0 = 𝑇𝑠 𝑛0 + ∫ 𝑛0 𝑟 𝑣 𝑟 𝑑𝑟 + ∫ ∫
𝑑𝑟𝑑𝑟
𝑋𝐶 0
2
𝑟 − 𝑟′
Non-interacting
system KE
External
potential
= 𝑛0 (𝒓)
Exchange and
correlation energy
makes up the
difference
Hartree interaction
𝐸𝑋𝐶 𝑛 = 𝑇 𝑛 − 𝑇𝑠 𝑛 + 𝑊 𝑛 − 𝐸𝐻 𝑛
PA4311 Quantum Theory of Solids
2
Trick is that 𝐸𝑋𝐶 is small
Atom
𝐸
𝑇𝑠
∫ n𝟎 𝒓 𝒗 𝒓 𝒅𝒓
𝑬𝑯
𝑬𝑿𝑪
He
-2.83
2.77
-6.63
2.00
-0.97
Ne
-128.23
127.74
-309.99
65.73
-11.71
Ar
-525.95
524.97
-1253.13
231.46
-29.24
Kr
-2750.15
2747.81
-6577.87
1171.72
-91.82
Xe
-7228.86
7225.10
-17159.16
2880.92
-175.71
Total ground state energy, 𝐸 (Hartrees.), calculated within the LDA – CA Ullrich, Table 2.1
PA4311 Quantum Theory of Solids
Exchange correlation potential
1
𝑛 𝑟 𝑛 𝑟′
′
𝐸 𝑛 = 𝑇𝑠 𝑛 + ∫ 𝑛 𝑟 𝑣 𝑟 𝑑𝑟 + ∫ ∫
𝑑𝑟𝑑𝑟
+ 𝐸𝑋𝐶 [𝑛]
2
𝑟 − 𝑟′
where 𝐸𝑋𝐶 𝑛 = 𝑇 𝑛 − 𝑇𝑠 𝑛
𝐻𝑠 =
𝑖
+ 𝑊 𝑛 − 𝐸𝐻 𝑛
−𝛻𝑖2
+ 𝑣𝑠 𝒓𝑖
2
Single particle equations are
where,
𝑣𝑠 𝒓 = 𝑣 𝒓 +
−𝛻2
2
+ 𝑣𝑠 𝒓
𝑛 𝒓
𝑑𝒓′ + 𝑣𝑋𝐶 [𝑛] 𝒓
′
𝒓−𝒓
PA4311 Quantum Theory of Solids
𝜓𝑖 = 𝜖𝑖 𝜓𝑖 ,
Exchange and correlation
potential
Local density approximation
𝑣𝑋𝐶
𝛿𝐸𝑋𝐶
𝑛 𝑟 =
𝛿𝑛
𝑣𝑋𝐶 can be accurately calculated for a uniform electron gas
- exchange part known exactly – 𝑣𝑋 𝑛 𝑟 ∝ 𝑛(𝑟)1/3
𝒓1 , 𝑛1
Use uniform electron gas result at density 𝑛1
for 𝑣𝑋𝐶 (𝒓1 )
𝒓2 , 𝑛2
Use uniform electron gas result at density 𝑛2 for 𝑣𝑋𝐶 (𝒓2 )
Expect LDA should work well for simple metals where density variations are slow
ie. where
𝛻𝑛 𝒓
𝑛 𝒓
≪ 𝑘𝐹 (𝒓)
PA4311 Quantum Theory of Solids
Local density approximation
LDA works surprisingly well in a wide range of materials
• Atomic and molecular ground state energies within ~ 1-5%
• Molecular equilibrium distances and geometries within ~ 3%
• Fermi surfaces of bulk metals within ~ few %
• Lattice constants of solids within ~ 2%
• Vibrational frequencies and phonon energies within ~ 2%
PA4311 Quantum Theory of Solids
Self consistent field (SCF)
Use new 𝑛(𝒓)
Iterative procedure - essentially
𝑖𝑛
𝑛𝑖+1
= 𝛼𝑛𝑜𝑢𝑡 + 1 − 𝛼 𝑛𝑖𝑖𝑛
Pre-condition or ‘seed’ SCF loop
The 𝐾𝑆 single particle
𝑖𝑛
Guess 𝑛 (𝒓)
equations must be solved
Set maximum allowed number of steps
self-consistently
Calculate effective potential
𝑣𝑠 𝒓 = 𝑣(𝒓) + 𝑣𝐻 [𝑛] + 𝑣𝑋𝐶 [𝑛]
1
Solve KS Eq.s - − 2 𝛻𝑖2 − 𝑣𝑠 𝒓𝑖
Calculate new 𝑛𝑜𝑢𝑡 (𝒓) =
No
Self
consistent?
Yes
PA4311 Quantum Theory of Solids
𝜓 𝒓𝑖 = 𝐸𝑖 𝜓 𝒓𝑖
𝑖
𝜓𝑖 (𝒓)
2
Set acceptable tolerance for self consistency
e.g. difference in total energies < 10−6 H
Finished, output 𝐸[𝑛] and 𝑛(𝑟)
# Skeleton abinit input file (example for H2)
#Definition of the atoms
ntypat 1
# There is only one type of atom
znucl 1
# atomic number
natom 2
# There are two atoms
typat 1 1
# They both are of type 1, that is, Hydrogen
xcart
# cartesian co-ordinates for each atom
-0.7 0.0 0.0 # Triplet giving the cartesian coordinates of atom 1, in Bohr
0.7 0.0 0.0
# Triplet giving the cartesian coordinates of atom 2, in Bohr
#Definition of the SCF procedure
nstep 10
# Maximal number of SCF cycles
toldfe 1.0d-6 # Will stop when, twice in a row, the difference between two
# consecutive total energies differ by less than toldfe (in Hartree)
diemac 2.0 # precondition the SCF cycle (see documentation)
iscf 7
# default - use Pulay mixing of the density (update info from 7 steps)
PA4311 Quantum Theory of Solids
Beyond the LDA – acronym zoo (ii)
exact
RPA
Unoccupied orbitals
ℎ𝑦𝑏𝑟𝑖𝑑
Hyper-GGA
hybrids
Exact exchange
𝐸𝑋𝐶
= 𝑎𝐸𝑋𝑒𝑥𝑎𝑐𝑡 + 1 − 𝑎 𝐸𝑋𝐺𝐺𝐴 + 𝐸𝐶𝐺𝐺𝐴
e.g. B3LYP
Meta-GGA
𝛻2𝑛 𝑟 , 𝜏
PKZB, TPSS, VSXC etc.
GGA
𝛻𝑛(𝑟)
Hundreds of GGA functionals:
PBE, BLYP, PW91 etc. etc.
LDA
𝑛(𝑟)
Slater 𝑋𝛼, LSDA
Hartree
0
CA Ullrich, Fig. 2.7
PA4311 Quantum Theory of Solids
LDA + U
Kohn-Sham eigenvalues
−𝛻 2
+ 𝑣𝑠 𝒓
2
𝜓 𝑖 = 𝜖𝑖 𝜓 𝑖
𝑛 𝒓
′
𝑣𝑠 𝒓 = 𝑣 𝒓 +
𝑑𝒓
+ 𝑣𝑋𝐶 [𝑛] 𝒓
′
𝒓−𝒓
Eigenvalues, 𝜖𝑖 , have no physical meaning
• except - energy of highest occupied state 𝜖𝑁 𝑁 = −𝐼
Band gap problem with DFT
• KS band gap 𝜖𝑁+1 𝑁 − 𝜖𝑁 𝑁 is not the correct gap
• 𝐸𝑔𝑎𝑝 = 𝜖𝑁+1 𝑁 + 1 − 𝜖𝑁 (𝑁)
• Problem related to lack of derivative discontinuities in 𝑣𝑋𝐶
• Extend DFT (TDDFT, GW etc.) to get excitations right
PA4311 Quantum Theory of Solids
DFT in crystals
A crystal is a periodic structure
Specified by
• types and positions of atoms in one repeat
• rules that describe the repetitions
crystal = Bravais lattice + basis
PA4311 Quantum Theory of Solids
2D crystal
graphene unit cell
2 atom basis
atoms at:
a1
a2
0,0 and
𝑎0
Primitive cell vectors:
𝑎1 =
𝑎2 =
3 1
, 𝑎
2 2 0
3 −1
,
𝑎0
2 2
𝑎0 = 0.246 nm
PA4311 Quantum Theory of Solids
1
,0
3
3D crystal: zinc blende structure (diamond, Si, GaAs etc)
FCC
2 atom basis
(0,0,0) and
1 1 1
, ,
4 4 4
𝑎0
Primitive cell vectors
0.5,0.5,0 𝑎0
0.5,0,0.5 𝑎0
0.5,0,0.5 𝑎0
www.seas.upenn.edu
wikipedia.org
PA4311 Quantum Theory of Solids