Summary of 4471 Session 4: Numerical Simulations

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Summary of 4471 Session 4Numerical 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 Newtons laws)
• Monte Carlo methods (use random sample to mimic
  equilibrium ensemble)

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4471 Session 5Surfaces

• 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?

4
Comparison of MD and Monte Carlo
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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)

6
Approximate charge densities and tight-binding
models

• Approximate full charge density by superposition
  of reference atomic-like charge densities (with
  error ??(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

?(r)


r
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Approximate charge densities and tight-binding
models

• .where the one-electron energies ?n come from
  solving a tight-binding model of the electronic
  structure for the outermost (valence) electrons.
• ...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)

Localised (fixed) basis functions
Hopping amplitude between states i and j
On-site energy of state i
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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
• Moments of this density of states defined by

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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

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
• For a partially filled band (metallic structure),
  cohesive energy depends on (number of
  neighbours)1/2

E
EF
Empty states
Filled 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
I
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

Bond energy h hybridization energy between
hybrid orbitals on I and J, ?bond order (excess
of bonding over antibonding character) -
sensitive to atomic environment (e.g. bond angles
as well as bond lengths)
Core repulsion
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 2Ep
• In bonding state has four sp3 hybrid orbitals
  each with energy (Es3Ep)/4, so total energy is
  (Es3Ep)
• Promotion energy Es-Ep

Ep
Es
(Es3Ep)/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
• Usual notation
• Works provided the angles of the surface and bulk
  lattices are the same if not, must use a more
  general matrix notation (see Zangwill)

Rotation of surface unit cell with respect to
bulk by angle ?
Lattice plane used to create surface
Factors by which lattice vectors increase on
reconstruction
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Two-dimensional crystallography

• Five primitive surface nets are possible

Primitive rectangular
Square
Hexagonal
Centred rectangular
Oblique
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Low energy electron diffraction (LEED)
Electron gun

• 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

Retarding grids
Sample
Detecting screen
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Low energy electron diffraction (LEED)
Electron gun

• 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

Retarding grids
e-
Sample
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

(01)
(00)
(02)
Ewald sphere
Reciprocal lattice rods
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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
001

• 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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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
• Get a reconstruction with rows of dimers along
  the surface the Si(001)-2?1 reconstruction

110
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

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
110
(2?2)
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