ab initio Lattice Vibrations:
Calculating the Thermal Expansion Coeffcient
Felix Hanke & Martin Fuchs
June 30, 2009
This afternoon’s plan
introductory talk
• Phonons: harmonic vibrations for solids
• Phonons: how
• Thermodynamics with phonons
practical exercise
• Obtain & assess phonons for Si
• Calculate & understand thermal expansion
Recap: Molecular vibrations (non-periodic)
Newton’s equations for small displacements
of atom !, coordinate i about PES minimum:
= Fα,i ({xβ,j })
!
∂2E
≈ 0+
δxβ,j
∂(δxα,i )∂(δxβ,j )
mα ẍα,i
mα ẍα,i
β,j
In symmetric matrix form
!
√
1
∂2E
√
−ω [ mα δxα,i ] =
[
mj δxβ,j ]
√
mα mβ ∂(δxα,i )∂(δxβ,j )
2
β,j
The harmonic lattice
Translational symmetry (PBC’s) leads to k-dependence
One eigenvalue problem for each k,
giving 3Natoms modes
−ωn2 (k)ξn (k) = D(k)ξn (k)
D is the dynamic matrix, contains information from
all supercells R, all pairs of atoms ! and ", and all pairs
of coordinates i and j:
Dαβ,ij
! eik·R
∂2E
=
√
mi mj ∂r(R=0),αi ∂r(R),βj
R
Phonon dispersion relation
Diamond
fcc conventional cell
2 atoms per primitive cell
= 6 phonon branches
a = 3.5Å
Recall diatomic molecules:
2 atoms = 6 degrees of freedom
• 3 translations ! = 0
• 2 rotations ! = 0
• 1 vibration ! = !vib
How about solids?
Phonon dispersion relation
Diamond
fcc conventional cell
2 atoms per primitive cell
= 6 phonon branches
a = 3.5Å
1500
1500
1000
1000
500
500
-1
Frequency " (cm )
optical branches
0
!
X
W
K
k-vector
acoustic branches
!
L
DOS
0
Phonon density of states
DOS = number of phonon modes per
unit frequency per unit cell
"
#
2
!
!
(ω − ωn (k))
g(ω) =
δ(ω − ωn (k)) ≈
exp −
2σ 2
n,k
n,k
smearing # makes it feasible to
plot (same as electronic DOS!)
Converge smearing vs
k-point sampling!
numerical computation:
frequency interval [$+%$/2,$-%$/2] with discrete k-point sampling
#
$
"
wk ! ω+∆ω/2
(ω − ωn (k))2
g(ω) = √
dω exp −
2σ 2
2π n,k ω−∆ω/2
The Direct Method
approximation:
finite interaction distance, use finite supercells
Dαβ,ij
! eik·R
∂2E
=
√
mi mj ∂r(R=0),αi ∂r(R),βj
R
Harmonic approximation: Thermodynamics
Approximate thermal effects with independent
harmonic oscillators: one for each mode
F
!
=
ln Z(k, N )
k,N
! " !ωk,N
=
k,N
2
#
+ kB T ln[1 − exp(−!ωk,N /kB T )]
Practically, one tends to use the density of states:
=
!
"
#
!ω
dω g(ω)
+ kB T ln[1 − exp(−!ω/kB T )]
2
Quasiarmonic approximation: Changing
lattice constants
Alternative to explicit treatment of each
phonon at all a’s
•Anharmonic properties implicitly through
dependence on lattice constant
•Treat many lattice constants harmonically &
minimize free energy over all a’s
Lattice Constants: Zero Point Vibrations
Density of states
how does the DOS change as fct of lattice constant?
does this have any effect on the lattice?
a = 3.3 Å
a = 3.5 Å
a = 3.7 Å
a
diamond
0
1000
500
-1
1500
Phonon frequency ! (cm )
Lattice Constants: Zero Point Vibrations
Decreasing phonon free energy at larger a:
Lattice constant including ZPE is larger than T=0 result
1
dω g(ω)
!ω
2
diamond
3.532 Å
0.5
no ZPE
ZPE only
sum
3.519 Å
1
0
3.4
nergy (eV)
Energy (eV)
ZPE =
!
3.5
3.6
Lattice
0.5constant (Å)
3.7
Lattice constants: Temperature effects
Murnaghan fits of Ftot(a,T) at different temperatures
lattice constant & Bulk modulus
a (Å)
3.54
diamond
quasiharmonic
3.53
no phonons
B0 (MBar)
3.52
no phonons
4.7
4.6
quasiharmonic
4.5
0
100
200
300
400
500
Temperature (K)
600
Thermal expansion coefficient
need to minimize F(a,T) at a given
temperature to find lattice constant
F (a, T ) = Eel (a) + Fph (a, T )
1 da
Differentiate to find α(T ) =
a dT
-6
Watch the scale!!!
-1
" (K )
4!10
3!10
2!10
1!10
-6
-6
-6
0
0
diamond
100 200 300 400 500 600 700
Temperature (K)
700
Heat capacity: cv
Computed from free energy
cv (kB/unit cell)
!
!
dS !!
∂ 2 F !!
cv (T ) = T
= −T
dT !V
∂T 2 !V
"
exp(!ω/kB T )
(!ω)2
=
dω g(ω)
kB T 2 (exp(!ω/kB T ) − 1)2
Grüneisen parameter:
coupling to !
α=
γcv
3B
6
4
2
0
0
diamond
1000
2000
Temperature (K)
3000
Breakdown of the harmonic approximation
• High temperatures: anharmonic excitations
• “soft modes” - dynamically stabilized structures
!
V (ξ)
“imaginary” frequency
ξ!
• Quantum effects, eg in hydrogen bonded crystals
Beyond the quasiharmonic approximation
MD to describe complete PES
beyond quasiharmonic approximation
F = Eel + Fqh + Fanh + Fvac
thermodynamic integration: slowly
switching on anh terms from qh solution
Determine from self-consistently
optimizing vacancy volume
Grabowski, Ismer, Hickel & Neugebauer, Phys. Rev. B 79 134106 (2009)
Today’s tutorial: Thermal properties of Silicon
this talk
afternoon
exercise
Schedule for this tutorial
(A) simple phonon dispersion
relation
45 min
(B) converging the supercell
45 min
(C) zero point vibrations
80 min
(D) thermal expansion
40 min
Exercise A: calculations of phonons with FHI-aims
use Si (diamond structure)
lattice constant = 5.44 Å, light species defaults
Supercell size 2x2x2
Compute:
phonon dispersion relation for the directions
"-X-W-K-"-L
density of states, specific heat cv
converge DOS
see also: FHI-aims manual, section 4.5
Exercise A: calculations of phonons with FHI-aims
Phonon calculation works similar to vibrations:
run aims.phonons.workshop.mpi.pl in the directory
containing your control.in and geometry.in files.
Output
phonon_band_structure.dat - same format as e-bands
phonon_DOS.dat - frequency, density of states
phonon_free_energy.dat - T, F(T), U(T), cv(T), Svib(T)
output stream - status reports
phonon_workdir/ - working files & restart info
Attention
Please specify a SINGLE PRIMITIVE CELL ONLY!
The script does all the necessary copying & displacing.
Exercise A: calculations of phonons with FHI-aims
Phonon dispersion calculation completely driven by phonon
keyword in control.in (see FHI-aims manual, section 3.5)
supercell size - phonon supercell 2 2 2 (for now)
DOS specification - phonon dos 0 600 600 5 20
Free energy - phonon free_energy 0 800 801 20
k-grid: hand set to match supercell size, i.e. k_grid 6 6 6
Band structure:
phonon band <start> <end> <Npoints> <sname> <ename>
working directory:
/usr/local/aimsfiles/tutorial6/exercise_A
-1
Frequency " (cm )
Exercise A: Solutions
500
400
400
300
200
200
100
0
!
cv (kB/unit cell)
X
W
K
k-vector
L
!
DOS
0
6
4
2
0
0
200
400
600
Temperature (K)
800
Exercise B: Supercell convergence
Compute phonon dispersion and DOS for the
supercell sizes 4x4x4, 6x6x6
Use the converged DOS settings from the last exercise.
All necessary files are in directory
/usr/local/aimsfiles/tutorial6/exercise_B
NOTE: We have provided partial control.in and
geometry.in files as well as ALL of the required DFT
output for the remainder of this tutorial
500
2x2x2
250
0
500
4x4x4
250
0
500
6x6x6
250
0
!
X
W K
!
k-vector
Exercise B: solutions
DOS (arb units)
Phonon frequency ! (cm-1)
Exercise B: solutions
0
2x2x2
4x4x4
6x6x6
200
400
-1
Phonon frequency (cm )
L
Exercise C: zero point vibrations
Using data for the lattice constants provided in the directory
/usr/local/aimsfiles/tutorial6/exercise_C+D
For each lattice constant provided, calculate the total energy of a single
unit cell using a 12x12x12 k-point grid. Store the result in the format
4x4x4_a*.***/output.single_point_energy
Again, for each lattice constant, calculate the phonon free energy &
specific heat for a temperature range from 0K to 800K in 801 steps (***)
Extract the ZPE from the T=0 free energies from
file phonon_free_energy.dat in each directory
produce a Murnaghan fit with and without ZPE and plot the two fits &
the resulting lattice constants
(***) These
exact settings are important for exercise D
Exercise C: solutions
ZPE
E(a), no ZPE
E(a), incl ZPE
Energy (meV)
60
40
20
0
5.30
5.35
5.40
5.45
5.50
Lattice constant a (Å)
5.55
Exercise D: compute a(T), cv & !
Again, using the data provided directory
/usr/local/aimsfiles/tutorial6/exercise_C+D
use the script eval_alpha.sh to calculate the lattice
constant at different temperatures & the thermal
expansion coefficient
eval_alpha.sh has to be started in the directory
exercise_C+D and requires the EXACT specifications
of phonon free_energy from the last exercise.
find cv(T) at the optimal lattice constant a=5.44Å
Compare with previously calculated values
Exercise D: solutions
Specific heat at constant volume
cv (kB/unit cell)
6
4
2
0
0
200
400
600
Temperature (K)
800
Exercise D: solutions
a (Å)
lattice constant a(T) & thermal expansion coefficient !
5.432
5.428
-1
" (K )
5.424
2!10
-6
0
0
100
200
300
400
500
Temperature (K)
600
700
0
a = 5.40 Å
a = 5.45 Å
a = 5.50 Å
frequency
reordering
250
nsity oof states (arb units)
Density oof states (arb units)
Density of states
500
-1
Phonon frequency ! (cm )
a = 5.40 Å
a = 5.45 Å
a = 5.50 Å
-1
Phonon frequency " (cm )
Phonon band structure
600
500
400
300
200
100
0
!
X W K
k-vector
!
L
DOS
Negative thermal expansion coefficient: Why
• optical phonon
bands become softer
• acoustic bands
become harder in
some part of the BZ