Transcript 4471 Session 4: Numerical Simulations

Summary of 4471 Session 4:
Numerical Simulations
Methods for representing interactions between atoms:
•
•
Simple empirical methods (e.g. pair interactions)
First-principles techniques (e.g. Hartree-Fock, Density Functional Theory)
Methods for extracting information once the atomic interactions
are known:
•Static calculations (minimise total energy)
•Molecular dynamics (follow Newton’s laws)
• Monte Carlo methods (use random sample to mimic equilibrium ensemble)
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4471 Session 5:
Surfaces
• Simulation methods:
– Extracting information from Monte Carlo
– Comparison of Monte Carlo vs MD
• Interatomic interactions beyond the pair-interaction
approximation
• Ordered surfaces and their structures
• [Break]
• Reactivity and reconstructions of surfaces: the (001)
surface of silicon as an example
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Making use of Monte Carlo
• To think about: given a sequence of configurations
generated by the Metropolis algorithm, how would you
find
–
–
–
–
–
The mean energy of the system?
An estimate for the statistical error in the energy?
The pair correlation function?
The heat capacity?
The free energy?
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Comparison of MD and Monte
Carlo
Thermal averages?
Other ensembles?
Time dependence?
Get trapped in regions of
phase space?
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Molecular dynamics
Monte Carlo
Yes (microcanonical if
dynanics ergodic,
canonical if include an
additional `thermostat')
Constant-pressure or
constant stress OK;
constant chemical
potential difficult
Yes
Yes (if obey accessibility
and detailed balance)
Yes
Can design algorithms to
avoid it
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Yes
No
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More on interatomic interactions
• Often need some `middle ground’ between a simple pair
potential and a full first-principles calculation
• Especially useful for semi-quantitative understanding of
structural properties and trends (as in this course)
• Make use of approximations to density functional theory
(can also derive some approximations to Hartree-Fock
theory)
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Approximate charge densities
and tight-binding models
• Approximate full charge density by superposition of
reference atomic-like charge densities (with error (r)):
(r)
=
+
r
• Then one can show, using the variational principle of
density functional theory, that the effective interatomic
potential can be approximated to order by ()2 by
Veff   n  U IJ
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n
I ,J
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Approximate charge densities
and tight-binding models
• ….where the one-electron energies n come from solving a `tightbinding model’ of the electronic structure for the outermost (valence)
electrons….
n   i i n   ci (n) i ; Hˆ   Ei i i  t ij i j
i
i
Localised (fixed) basis
functions
i
On-site energy of
state i

H n  n n
i j
Hopping amplitude
between states i and j
• ...and the UIJ are repulsive pair interactions between the atomic cores.
• NOTE: in this formula we do not need to correct any error from the
double counting of electron-electron interactions (see references for
why not)
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The Moments Theorem
• Now relate changes in the tight-binding eigenvalues
directly to the local bonding environment of the atoms
• Define a total density of states, and its projection on a local
atomic state i, by
N ( E )    ( E   n ); ni ( E )   i n  ( E   n )
2
n
n
• Moments of this density of states defined by
M (i ) n   ni ( E ) E n dE   i n  n n i
n
  i Hˆ n n n i  i Hˆ n i  H ij H jk ...H zi
n
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The Moments Theorem (2)
• Suppose H contains only
near-neighbour
interactions
• Counts the number of
ways of starting at the site
of i and hopping from
neighbour to neighbour to
return there after exactly n
hops
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4 routes to nearest
neighbours and back so
second moment = 4t2
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The Moments Theorem (3)
N(E)
• The ‘standard deviation’
(measure of width of a
band) is given by
2
  [ E 2  E ]1/ 2
E
 [M 2  M1 ]
2 1/ 2
EF
• For a partially filled band
(metallic structure),
cohesive energy depends
on (number of
neighbours)1/2
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Filled states
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Empty states
Wider band  filled states
lower in energy relative to
average, so more stable
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Embedded atom potentials
(metals)
• One route to incorporate
this: embedded atom
potentials in which
‘Embedding
term’
Pair potential
Veff   ( RIJ )   F (  I )
I ,J
I
I
 I    atomic( RIJ )
J I
F (  I )  a I  b I
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1/ 2
Atomic density
from J on I site
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Many-body potentials
(semiconductors)
• By a similar route can derive angularly-dependent
potentials for covalent sp3-bonded semiconductors:
Veff   [( RIJ )  2h( RIJ )IJ ]  U promotion
I
Core repulsion
J I
Bond energy:
h = hybridization energy between
hybrid orbitals on I and J,
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=bond order (excess of bonding
over antibonding character) sensitive to atomic environment
(e.g. bond angles as well as bond
lengths) 4471 Solid-State Physics
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Energy to ‘promote’
atom to sp3 bonding
configuration
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The promotion energy for Si
• Free atom has one s state
(containing two electrons)
and three p states
(containing two electrons
between them), energy 2Es
Ep
Es
+ 2Ep
• In ‘bonding state’ has four
sp3 hybrid orbitals each
with energy (Es+3Ep)/4,
so total energy is (Es+3Ep)
• Promotion energy = Es-Ep
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(Es+3Ep)/4
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Surfaces
• Why are surfaces important?
• Structure of crystalline surfaces and an experimental
method of structure determination
– Low Energy Electron Diffraction (LEED)
• Driving forces for surface reconstructions - the Si(001)
surface as an example
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Importance of surfaces
• Surfaces are reactive and important as heterogeneous
catalysts (e.g. in
• Surfaces are important in the growth of materials (e.g.
silicon growth from the melt in the semiconductor
industry)
• Surfaces are important in the processing of materials
• Surfaces provide a foundation for the fabrication of
nanoscale structures and devices
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‘Clean surfaces’ - a caution
• Surfaces tend to be strongly reactive
• Even in a very good vacuum, atoms or molecules in the
gas phase will tend to attach to the surface
• To get remotely ‘clean’ surfaces must work in Ultra High
Vacuum (pressure ~10-13 atmospheres)
• Be careful (obviously) about extrapolating any results from
this region to surfaces in contact with ordinary gas or
liquid pressures
• Also, defects from the bulk will often diffuse preferentially
to the surface, so even a ‘clean’ surface is seldom ideal
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Structure of Surfaces
• Simplest model of the surface of a crystal: cut the material
along a lattice plane and remove half the atoms, leaving the
others behind:
(hkl)
plane
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Structure of crystal surfaces
• Simplest model of the surface of a crystal: cut the material
along a lattice plane and remove half the atoms, leaving the
others behind
• If the plane (hkl) is rational, then the resulting surface is
still periodic
• Lattice vectors are the same as those of the truncated bulk
structure, but these may be increased by displacements of
the atoms (‘reconstructions’)
• Stoichiometry may not be the same as that of truncated
bulk structure: can lose (or gain) atoms to or from bulk
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Notation for surface structures
• Reconstructions occur because the surface environment for
atoms is very different from the bulk
Rotation of surface
• Usual notation:
unit cell with
(hkl)  N  M  R
Lattice plane used
to create surface
respect to bulk by
angle 
Factors by which lattice vectors
increase on reconstruction
• Works provided the angles of the surface and bulk lattices
are the same; if not, must use a more general matrix
notation (see Zangwill)
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Two-dimensional crystallography
• Five primitive surface nets
are possible
Lattice
Angle
Lattice vectors
Square
Choose two shortest mutually
90
perpendicular vectors; a=b
Rectangular
Choose two shortest
mutually perpendicular
vectors; ab
Usually choose two
shortest mutually
perpendicular vectors; ab
Choose two shortest
vectors making angle of
120; a=b
ab
Centred rectagular
Hexagonal
Oblique
Square
Primitive rectangular
90
90
Hexagonal
120
Centred rectangular
90
Oblique
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Low energy electron diffraction
(LEED)
• One of the most natural
surface-sensitive
scattering techniques
• Low energy electron beam
(energy 20 - 300 eV)
• Surface sensitive because
– inelastic mean free path
very short
– very strong backscattering
by surface layers
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Electron gun
Retarding
grids
Sample
Detecting
screen
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Low energy electron diffraction
(LEED)
• Negative potential
difference between first
and second grid
decelarates electrons
• Allows only elastically
scattered electrons to
penetrate to screen
• Screen held at positive
potential to accelerate
electrons onto it
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Electron gun
Retarding
grids
eSample
Detecting
screen
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Diffraction conditions for LEED
• An ordered surface still
has two-dimensional
crystalline order
• Component of scattering
vector in the surface plane
must be a reciprocal lattice
vector G of the surface
structure
• No constraints on
scattering vector normal to
surface plane
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k out  k in  K
k in
(00) (01)(02)
Ewald
sphere
k out  k in  K
K
Reciprocal
lattice ‘rods’
k in
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Exercises
• Why is the centred square net not listed as a separate type?
• What are the surface periodicities of a (001) surface of
– a bcc structure?
– an fcc structure?
• If the cubic lattice constant is a in each case, sketch the surface
reciprocal lattices, and hence the LEED patterns you would expect.
• What LEED pattern would you expect from a centred rectangular net
with rectangular cell sides a and b?
• Prove the conditions on K stated in the previous slide, on the
assumption that the scattered electrons experience a potential with the
same periodicity of the surface.
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The Si(001) surface: an example
• The most technologically
important of all semiconductor
surfaces (used in most
semiconductor growth)
• Unreconstructed surface has
two ‘dangling bonds’ on each
atom from the removal of the
next layer up, each having one
unpaired electron
• Give rise to two surface bands
with one electron per band, so
surface is metallic
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[001]
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The Si(001) surface: an example
[001]
• This situation is highly unstable
• Atoms gain energy by
dimerising - each atom ‘uses
up’ one of its dangling bonds to
form a covalent bond with a
neighbour
• The bonding energy gained
more than compensates for the
strain energy involved
[110]
• Get a reconstruction with ‘rows’
of dimers along the surface: the
Si(001)-21 reconstruction
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[110]
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The Si(001) surface: an example
• LEED pattern changes
from (11) to (21)
• In reality almost always
have multiple domains on
surface, in which the
dimer rows run in
different directions
• Resulting LEED pattern is
a superposition of those
from the separate domains
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Unreconstructed
surface
Single-domain
(21) surface
Multi-domain
(21) surface
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Difficulties with interpretation of
LEED
• Multiple electron scattering events are very likely in LEED
(in contrast to the situation for X-ray or neutron scattering)
• This means the single-scattering theory (‘Born
approximation’) used for other diffraction techniques
cannot be applied to LEED
• Need a full ‘dynamical’ theory of the electron scattering
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The Si(001) surface: refinements
• If the dimers are flat, the
dangling bonds remaining on
the two atoms are equivalent
(and therefore degenerate)
• Could lower overall energy if
one dangling bond acquired
lower energy, so both electrons
reside on it (even at expense of
raising energy of other state)
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The Si(001) surface: refinements
• This happens if the dimers tilt
• The ‘up’ atom moves closer to
the ideal tetrahedral bonding
pattern of sp3 hybridisation
• Its dangling bond becomes
lower in energy, and the atom is
negatively charged
• The ‘down’ atom moves closer
to sp2 bonding, so its dangling
bond rises in energy and it
becomes positively charged
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The Si(001) surface: refinements
• Neighbouring dimers along a
row prefer to tilt in opposite
directions
• Gives rise to two structures
when coupling to the next
dimer row is included: the
(22) and the centred (42)
[110]
[110]
=‘up atom’
=‘down atom’
[110]
[110]
[110]
(22)
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[110]
c(42)
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Observations of tilted dimers
• Observe this in scanning tunnelling microscopy (STM)
near defects or at low temperatures
• To think about: why is it not seen on a defect-free
region of surface at room temperature?
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Conclusions
• Possible to derive relatively simple interatomic potentials,
beyond the pair-potential approximation, by using modern
techniques of electronic structure theory
• Structure of ordered surfaces and its determination by
LEED
• Driving forces for reconstruction: the Si(001) surface as an
example
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