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.