basics of dft applications to solids and surfaces
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7/29/2019 Basics of DFT applications to solids and surfaces
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Peter Kratzer
Physics Department, University Duisburg-Essen, Duisburg, Germany
E-mail: Peter.Kratzer@uni-duisburg-essen.de
Periodicity in real space and reciprocal space
Example: honeycomb lattice
Lattice vectors a1, a2, a3Unit cell volume
Crystallographic basis consisting of two atoms
Reciprocal-lattice vectors b1, b2, b3 , each
perpendicular to a pair of lattice vectors
real space
Reciprocal (wave-vector)
space
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From molecules to solids
Electronic bands as limit of bonding and anti-bonding combinations of
atomic orbitals
R. Hoffmann,
Solids and Surfaces - A chemists view of bonding in extended structures, VCH Publishers, 1998
Blochs theorem
In an infinite periodic solid, the solutions of the Kohn-Sham equations must
behave like
under translations by a lattice vector
Consequently, we can write
with a lattice-periodic part uj,k(r).
The index kis a vector in reciprocal space taken
from the first Brillouin zone.
The number of different k-indices equals the
number of unit cells in a crystal.
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Brillouin zones
in two dimensions
for a square lattice
for a hexagonal lattice
in three dimensions
for a face-centered cubic (fcc)lattice
Band structure - schematic
The orbital energies are smoothfunctions ofk
Example: chain of Pt-R4 complexes
Adopted from: R. Hoffmann, Solids and Surfaces - A chemists view of bonding in extended
structures, VCH Publishers, 1998
k=0 k=/a
0 k /a
z2xy
xz,yz
x
2
-y
2
z
energy
z
xy
xz
z2
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Brillouin zone sampling
Charge densities (and other quantities) are
represented by Brillouin-zone integrals
Integrand smooth (for non-metals):
replace integral by sum over sampling points
H.J. Monkhorst and J.D. Pack, Phys. Rev.B 13, 5188 (1976); Phys. Rev. B 16, 1748 (1977)
16 8 MP grid
In a metal, some (at least one) energy bands
are only partially occupied by electrons.The Fermi energy Fdefines the highest
occupied state(s). Plotting the relation
in reciprocal space yields the Fermi surface(s).
The grid used ink-space must be suffficiently fine to accurately sample the Fermi surface.
Fermi surfaces
http://www.phys.ufl.edu/fermisurface/
Fermi
surface
of Cu
Fermi
surface ofFe (spin )
Fermi
surface ofFe (spin )
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Band structure of a magnetic material
A. Yamasaki & T. Fujiwara, J. Phys.
Soc. Japan 72, 607 (2003)
Spin-resolved band
structure for fcc iron
GW method (full)
LSDA (dashed)
Correct description of
magnet properties
requires dense k-spacesampling !
F
How to treat metals in DFT
Fermi distribution functionfFenters Brillouin zone integral:
Simplest method: smearing of the Fermi function
artificially increased kBTel~ 0.2 eV
Extrapolation of the total energy to Tel= 0
Alternatives:
Methfessel-Paxton distribution [ Phys. Rev. B 40, 3616 (1989) ]
Marzari-Vanderbilt ensemble-DFT method [ Phys. Rev. Lett. 79, 1337 (1997) ]
Tetrahedron method
Fermi surface is approximated by a polyhedron consisting of small tetrahedra in
each plaquette [P. E. Blchl et al., Phys. Rev. B 49, 16223 (1994)]
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Total-energy correction for finite Tel
-60
-40
-20
0
20
40
60
80
0.01
0.03
0.05
0.07
0.09
electronic temperature [eV]
E[meV]
extrapolated to 0
E(T)
substitutional adsorption of Na on Al(111)
J. Neugebauer and M. Scheffler,
Phys. Rev. B 46, 16067 (1992)
Indium adatom on GaAs(001)
c(4x4) surface
E. Penev, unpublished
Ezero = (F(T)+E(T)) + O(T3) = F(T) Tel Sel(T) + O(T3)
E
Ezero
F
Density of states (DOS)
Information about the single-particle contribution to the total energy
Projected density of states (PDOS)
is the atomic orbital with angular momentum lat atomI
Recovery of the chemical interpretation in terms of orbitals
Qualitative analysis tool; ambiguities must be resolved by truncating
the r-integral orby Lwdin orthogonalization of the
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energy
Ferromagnetic half-metal: Co2MnSi
DOS (states / eV)
SiMn
Cominority spin band structure
Surfaces: geometric andelectronic structure
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atomic structure: relaxation and reconstruction
change of interlayerdistances near the surface
response of the surface
atoms to an adsorbate
change of the symmetry
atomic rearrangement, formation of new bonds
relaxation reconstruction
oxygen on Ru(0001)
supercell
adatom(+superficial periodic images)
Supercell approach to surfaces
Approach accounts for the lateral periodicity
Sufficiently broad vacuum region to decouple theslabs
Sufficient slab thickness to mimic semi-infinite
crystal
Semiconductors: saturate dangling bonds on theback surface
Inequivalent surfaces: use dipole correction
With some codes, slabs with inversion symmetry(for metals) are computationally more efficient
Alternative: cluster model
Points to consider:
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LM
Surface Brillouin zone
L
U
XZW
K
Y'
Y
K 6
RQ
kz
kx
ky
example: fcc crystal, (111) surface
M
K
Surface band structure of Cu(111)
(111)
Shockley surface state
Tamm surface state
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Surface states (I)
gap
2 |VG|
/a k
i
(k) matching condition
potential
exp[ i(k+ i) z]
~exp[z]
Shockley surface state
complex band structure
approximation of the nearly-free-electron metal
Surface states (II)
In the tight-binding
(localized orbital) picture,
surface states mayappear due to dangling
orbitals split off from the
band edge Tamm surface state
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CoCo
MnSi
CoCo
MnMn
Out-of-plane
dorb.In-plane dorb.Minority
bands
Spin-down components of d
orbital ofsurface Mn andsubsurface Co
Surface states derived
mainly from Co d3r2z2 in
subsurface layer
Orbital mixing of Co orbitals withMn dxz dyz orbitals pushes Co
states out of the gap.
Co2MnSi: spin-polarized surface state
P. Kratzeret al., J. Appl.
Phys. 101, 081725 (2007)PDOS (states/eV)
Dimerization at (001)-surface of group IV-elements
top view
side view
[001]
bulk-terminated atomic structure
reconstructionside view
[1 1 0]
[1 1 0]
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Stabilization of dimerized Si(100) surface
sp3 sp3 sp3 sp3
pz
s-like
-bond re-hybridisation and charge transfer
sp2-backbondsp-backbondsJahn-Teller-like effect enhances the
splitting of the surface state.
[see, e.g., J. Dabrowski and M. Scheffler,
Appl. Surf. Sci. 56-58, 15 (1992)]
Different reconstructions of group-IV (001)-surfaces
-bonding
Jahn-Teller effect
(buckling)
Jahn-Teller effekt
(buckling)
C
Si
Ge
P. Krger & J. Pollmann, Phys. Rev. Lett. 74, 1155 (1995)
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Surface reconstruction of Si(001)
top view of Si(001)
A. Ramstad, G. Brocks, and P. J. Kelly,
Phys. Rev. B 51, 14504 (1995).
Thermodynamics ofsurfaces
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n
_a
[001]
a(O)
O
surface free energy
definition:
(n) = free energy per unit arearequired to create a surface
Depends on orientation of the planein a crystal, and on temperature andpressure
(n)
Simple metal: potassium
M. Timmer, unpublished
DFT calculations at T=0 K: A = Eslab - Natom Ebulk
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Quantum size effects: Al(110)
A. Kiejna, J. Peisert and P. Scharoch, Surf. Sci. 432 (1999) 54
LDA, 20Ry plane wave cut-off,
16 k-points in IBZ
EF
binary material: GaAs
Gibbs free enthalpy G(p, T, NGa, NAs)
Surface (free) energy
equilibrium with bulk GaAs fixes one
chemical potential: Ga + As = GaAs
remaining variable ( for instance, As )
determined by gas pressure
coexistence with elemental bulk phases setsbounds for As(p,T)
=[G(p,T,NGa, NAs) NGa Ga NAs As]/A
As crystal face
(001)
Ga crystal face
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Surface relaxation: GaAs(001) 2 (2x4)
LDA, 10 Ry plane-wave cut-off,
2 x 4 k-points in full BZ
Slabs with n layersEbulk(n) := [Eslab(n) Eslab(n 2)]/Natsurface energy(n)A := Eslab(n) (NAs+NGa) Ebulk(n) (NAs-NGa)As(p,T)
As2 pressure determines the stable structure
Under the conditions
commonly used in
molecular beamepitaxy, the 2
surface recon-
struction is stable.
Only at lower
temperatures and
higher As2 pressures,the c(4 4) structure
prevails (terminating
As double layer).
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Summary & Acknowledgements
Bjrn Hlsen,FHI
Sung Sakong,
UDE
Matthias Timmer,
UDE
S. Javad Hashemifar,
Isfahan University of
Technology
The chemical physics of solid surfaces and interfaces is an active
and exciting field of current research !
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