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Applications of Density-Functional Theory:Structure Optimization, Phase Transitions, and Phonons
Christian RatschUCLA, Department of Mathematics
In previous talks, we have learned how to calculate the ground state energy, and the forces between atoms.
Now, we will discuss some important concepts and applications of what we can do with this.
•Structure Optimization•Optimize bond length•Optimize atomic structure of a cluster or molecule•Optimize structure of a surface
•Phase transitions
•Phonons
•Structural and vibrational properties of metal clusters
•Dynamics on surfaces•Molecular dynamics•Use transition state theory: DFT can be used to calculate energy barriers, prefactors.
Outline
Structure optimization; example: Vanadium dimer
F F
F F• Put atoms “anywhere”
• Calculate forces
• Forces will move atoms toward configuration with lowest energy (forces = 0)
My results for vanadium: lbond = 1.80 Å (experimental value: 1.77 Å) 1.80 Å
Algorithms for structure optimization:
•Damped Newton dynamics (option: NEWT)
•Parameters and need to be optimized.
•Quasi-Newton structure optimization (BFGS scheme; option: PORT)
•No parameters needed•In principle, only one iteration needed if Hessian is known, and fully harmonic.•In practise, a few iterations and Hessian updates are needed.•Nevertheless, this is typically the recommended option.
1 nnn ΔxFΔx
nnn FHΔx1
ji
ij xx
EH
2
Hessian:
Bigger vanadium clusters: V8+
Different start geometries lead to very different structures
E=0 eV
E=1.8 eV
• Each structure is in an energetic local minimum (i.e., forces are zero).
• But which one is the global minimum?
• Finding the global minimum is a challenging task
•Sometimes, good intuition is all we need
•But even for O(10) atoms, “intuition” is often not good enough.
•There are many strategies to find global minima.
E=0.8 eV
E=0.4 eV
Surface relaxation on a clean Al(111) surface
Relaxation obtained with DFT (DMol3 code)(Jörg Behler, Ph.D. thesis)
•Top layer relaxes outward
•Second layer relaxes inward (maybe)
12
23
• Semiconductor surfaces reconstruct. Example InAs(100)
• Surface reconstruction is important for evolution of surface morphology which influences device properties
• RHEED experiments show transition of symmetry from (4x2) to (2x4)
Things are more complicated on semiconductor surfaces
In - terminated
[110]
As - terminated
[110]
Which reconstruction ? Use density-functional theory (DFT)
Computation details of the DFT calculations
• Computer code used: fhi98md
• Norm-conserving pseudopotentials
• Plane-wave basis set with Ecut = 12 Ryd
• k summation: 64 k points per (1x1) cell
• Local-density approximation (LDA) for exchange-correlation
• Supercell with surface on one side, pseudo-hydrogen on the other side
• Damped Newton dynamics to optimize atomic structure
Possible structures
(2x4)
(2x4)
(2x4)
(2x4)
(2x4)
(2x4)
We also considered the corresponding (4x2) structures(which are rotated by 90o, and In and As atoms are interchanged)
(4x2)
Phase diagram for InAs(001)
C. Ratsch et al., Phys. Rev. B 62, R7719 (2000).
(2x4)
2 (2x4)
c (4x4)
3 (2x4)
(4x2)
(4x2)
Typical experimental regime
Predictions confirmed by STM images
Low As pressure High As pressure
2(2x4) 2(2x4)
Barvosa-Carter, Ross, Ratsch, Grosse, Owen, Zinck, Surf. Sci. 499, L129 (2002)
lattice constantvolume
Etot
22,11, )()( PVVEPVVE BtotAtot
12
2,1, )()(
VV
VEVEP BtotAtot
• Without pressure, structure B is stable• With pressure, eventually structure A becomes stable
Minimize Gibbs free energy (at T=0):
PVEG tot
Phase transitions
Yesterday, we have learned how to calculate the lattice constant, by calculating E(V).
)()( 21 VGVG BA
Pressure for phase transition is determined by:
Structure A
Structure B
V1 V2
Classical example: Phase transition of silicon
Si has a (cubic) diamond structure, which is semiconducting
Under pressure, there is a phase transition to the tetragonal -tin structure, which is metalic
The pressure of phase transition has been computed from DFT to be 99 kbar (experimental value: 125 kbar)
M.T. Yin, and M.L. Cohen, PRL 45, 1004 (1980)
Historical remark
• In the original paper, the energies for bcc and fcc were not fully converged
• Luckily, this “did not matter” (for the phase transition)
Ying and Cohen, PRL, 1980 Ying and Cohen, PRB, 1982
• Nevertheless, these calculations are considered one of the first successes of (the predictive power) of DFT calculations.
Lattice vibrations: A 1-dimensional monatomic chain of atoms
|2
1sin|2 ka
M
Kk
Dispersion curve for a monatomic chain
N
n
ak
2
nn-1n
n-2 n+1
u(na): displacement of atom n
Equation of Motion: ))1(())1((2 anuanuu(na)K(na)..uM
Mass of atom Spring constant
Assume solution of form: )(),( tknaietnau
Periodic boundary condition requires:
Upon substitution, we get solution
Lattice vibrations of a chain with 2 ions per primitive cell
kaKGGKMM
GKk
effeff
cos21 222
nn-1 n+1
u1(na): displacement of atom n,1
n
u2(na): displacement of atom n,2Spring K Spring G
Coupled equations of motion:
))1(()( 212111 anu(na)uGnau(na)uK(na)..uM
))1(()( 121222 anu(na)uGnau(na)uK(na)..uM
Solution
Optical branch
Acoustic branch
kaKGGKMM
GKk
effeff
cos21 222
Dispersion relation for the diatomic linear chain
•For each k, there are 2 solutions, leading to a total of 2N normal modes.•The normal modes are also called “phonons”, in analogy to the term “photons”, since the energy of the N elastic modes are quantized as
There are N values of k:
)(2
12 khkk
N
n
ak
2
Because for small k, which is characteristic of sound waves
ck
Because long wavelength modes can interact with electromagnetic radiation
Lattice vibrations in 3D with p ions per unit cell
•Analysis essentially the same
•For each k, there are 3p normal modes
•The lowest 3 branches are acoustic
•The remaining 3(p-1) branches are optical
•A “real” phonon spectrum might look slightly different;
•The reason is that interactions beyond nearest neighbors are not included, the potential might not be harmonic, there are electron-phonon coupling, etc.
•More in the talk by Claudia Ambrosch-Draxl
How can we calculate phonon spectrum?
•Molecular Dynamics
•Do an MD simulation for a sufficiently long time•“Measure” the time of vibrations; for example for dimer, this is obvious•For bigger systems, one needs to do Fourier analysis to do this•Very expensive
•DFT perturbation theory
•Frozen phonon calculation
•This is what you will do this afternoon.
Frozen phonon calculations•Choose a supercell that corresponds to the inverse of wave vector k
•More details in the presentation by Mahboubeh Hortamani
/3a /2a /a
•Calculate dynamical matrix D
•in principle, this is done by displacing each atom in each direction, and get the forces acting on all other atoms:
•eigenvalues are the frequencies
•in practise, one exploits the symmetry of the system (need group theory)
•Repeat for several k
j
iij x
FD
M
D
M
diagiii
i
)(
Structural and vibrational properties of small vanadium clusters
Why do we care about small metal clusters?
• Many catalytic converters are based on clusters
• Clusters will play a role in nano-electronics (quantum dots)
• Importance in Bio-Chemistry• Small clusters (consisting of a few atoms) are the smallest nano-particles!
This work was also motivated by interesting experimental results by A. Fielicke, G.v.-Helden, and G. Meijers (all FHI Berlin)
3
456
7
8
14
15
16
17
18
19
20
21
22
23
9
10
11
12
13
Spectra for VxAry+
• Each cluster has an individual signature
• V13+ is the only structure with peaks that are beyond 400 cm-1
• Beginning at size 20, the spectra look “similar”. This suggests a bulk-like structure
Gas flow(~1% Arin He)
metal-rod
Mass-Spectrometer
Experimental setup using a tunable free electron laser
Tunable free electron Laser (FELIX)
Example:
Excitation of V7Ar1 and V7Ar2 at 313 cm-1
Laser Beam: clusters are formed, Ar attaches
DFT calculations for small metal clusters
• Computer Code used: DMol3
• GGA for Exchange-Correlation (PBE); but we also tested and compared LDA, RPBE
• We tested a large number of possible atomic structures, and spin states.
• All atomic structures are fully relaxed.
• Determine the energetically most preferred structures
• Calculate the vibrational spectra with DFT (by diagonalizing force constant matrix, which is obtained by displacing each atom in all directions)
• Calculate the IR intensities from derivative of the dipol moment
What can we learn from these calculations?
• Confirm the observed spectra
• Determine the structure of the clusters
• Is the spectrum the result of one or several isomers?
Structure determination for V8+
experiment
theory E=0
E=0.4eV
E=0.8eV
E=1.8eV
Structure determination for V9+
experiment
theory
S=0
S=1
S=0
S=1E=0
E=0.01eV
E=0.06eV
E=0.08eV
Open issues:
• Sometimes neutral and cationic niobium have similar spectrum, sometimes they are very different
• Cationic Nb is sometimes like cationic V, sometimes different.
Niobium
Niobium 7
neutral
cationic
Experimental Spectra
neutral
cationic
Calculated Spectra oflowest energy structure
Niobium 6
neutral
cationic
Experimental Spectra
neutral
cationic
Calculated Spectra oflowest energy structure
Molecular dynamics
Once we have the forces, we can solve the equation of motion for a large number of atoms, describing the dynamics of a system of interest.
But there is the timescale problem:
10-13 s
One way to “speed up dynamics” is to use transition state theory (TST)
Even with big computers, we can’t describe dynamics beyond microseconds, for up to ~ 106 atoms
More about molecular dynamics in the talk by Karsten Reuter.
Transition state theory (TST) to calculate microscopic rate parameters
Transition state theory (Vineyard, 1957): )/exp(0 kTED d
13
1
*
3
10 N
j j
N
j j
Attempt frequency(using harmonic approximations)
j *j,
normal mode frequencies at adsorption and transition site
j
*j
adtransd EEE Energy barrier
•Finding transition state is a big challenge
•Sometimes, intuition is enough
•Often, sophisticated schemes are needed (nudged elastic band method, dimer method, …. )
adE
transEdE
Model system: Ag/Ag(111) and Ag/Pt(111)(Brune et al, Phys. Rev. B 52, 14380 (1995))
100 Ao
T = 65 K, Coverage = 0.12 ML
Ag/Pt(111) Ag/ 1ML Ag/Pt(111) Ag/Ag(111)
System: Ag on Ed (meV) 0 (s
-1)Pt(111) 157 (10) 1 x 1013 (0.4)
Ag/Pt(111) 60 (10) 1 x 109 (0.6)
Ag(111) 97 (10) 2 x 1011 (0.5)
Nucleation Theory: N ~ (D/F)-1/3
Results and comparison: Diffusion barrier
Lowered diffusion barrier for Ag on Ag/Pt(111) is mainly an effect of strain. Ratsch et al., Phys. Rev. B 55, 6750 (1997)
• Tensile strain: less diffusion• Compressive strain: increased diffusion
How to calculate the prefactor
13
1
*
3
10 N
j j
N
j j
Attempt frequency
• Calculate force constant matrix by displacing each atom in x,y,z-direction, and by calculating the forces that act on all atoms
• Eigenvalues of this matrix are the normal mode frequencies
• Important question: how many degrees of freedom need to be included? j *
j,
0.710.821.55Prefactor 0 (THz)
99153# degrees of freedom
adatom and 2 layers, 4x4 cell
adatom and top layer, 2x2 cell
only adatom
2x2 cell
Convergence test for Ag/Ag(111)
Ratsch et al, Phys. Rev. B 58, 13163 (1998)
Results and comparison: Prefactor
System: Ag on Experiment Ed (meV)
Experiment
0 (THz)
Pt Ag (stretched) Ag Ag (compressed) 1 ML Ag/Pt
157 97 65
10 0.2 0.001
Compensation effect:
Higher barrier
Higher prefactor
• Prefactor is O(1 THz) for all systems
• Compensation effect is not confirmed
• Explanation for experimental result:• Long range interactions are important for systems with small barrier (Fichthorn and Scheffler, PRL, 2000; Bogicevic et al., PRL, 2000),
• Simple nucleation theory does not apply any longer but can be modified (Venables, Brune, PRB 2002)
TheoryEd (meV)
Theory
0 (THz)
106816063
0.250.821.3~ 7.0
Conclusion and summary
•DFT calculations can be used to optimize the atomic structure of a system
•DFT calculations can be used to calculate the pressure of a phase transition. This will be part of the exercises this afternoon!
•DFT calculations can be used to calculate a phonon spectrum. This will also be part of the exercises this afternoon.
•DFT calculations can be used to obtain structural and vibrational properties of clusters
•DFT calculations can be used to obtain the relevant microscopic parameters that describe the dynamics on surfaces.