Transcript Surface Morphology and Physical Structure

Surface Morphology and Physical Structure
Morphology: Macroscopic from or shape of a surface
Structure: microscopic, atomic geometry arrangement
Atomic structure is important to determine the morphology
Surface Tension and Macroscopic Shape
The existence of the surface breaks the continuity of the bulk, the
missing bonds between the some neighboring atoms will contribute to
the excess surface free energy g (per unit are). The total variation of the
free energy for a surface is:
dFS = -SSdT – PdVS + g dA + SmidniS
which SS, VS, P, niS, T and mi are the entropy, volume, pressure,
particle number, temperature and chemical potential per particle for ith
kind of particle in the interface region, A is surface area.
FS = – PVS + g A + SminiS
with dT = 0
Surface tension
Force/length=energy/area
g is around half of the
energy to melt an
atom, since to form
surface it needs to
break about half
bonds of an atom in
the cubic. For metal g
about 0.16 DHmelting
Wulff construction
Liquids and amorphous solids have isotropic g. Therefore they have
spherical shape.
For crystalline material, g depends on the orientation (hkl) of the surface,
while the broken bonds and charge compensation per unit area may be
different. g (n) is then the function of orientation of the surface, n. It
must have:
When we plot g (n) as function of n (Wulff plot), theoretically one can
determine the equilibrium shape of a solid.
minimum
g(hkl) as function of q (the angle
between the normal directions of the
surface to the {hkl} planes. The
Wulff plot gives the shape of the solid
(dashed dotted) as the inner envelope
of the Wulff planes (broken lines).
Samples of Wulff plot
minimum
Strong anisotropic case
Weakly anisotropic case
How does Wulff plot come?
1
q
cosq
a
na
a) g(q) can be write as g0cos(q) + g1sin(q)/a
g0 is surface
tension of (01) plane, g1 is the first order term (proportional to step
density) contribution to the surface tension from the existence of the
steps.
b) Therefore g(q) can be shown as 2Rcos(q-Q) = 2R [cosqcosQ +
sinqsinQ]. Considering 4 possible quadrants in the plane, a Wulff plot
can be shown as in b). With higher order of step density, the plot
becomes more complicated as shown before.
Conclusion
The Wulff plots consists of a number of circles/spheres, leading to a
polygon/polyhedron shape of the crystals, with surfaces of low g
preferentially exposed.
Any defect effects change vacancy formation energy or mixing
defect sites with prefect sites will change g and the plot. g plot will
become more spherical when melting T is approached.
Crystal growth often occurs under highly non-equilibrium
conditions so the equilibrium shape rarely achieved.
Surface tension has enormous importance for surface
inhomogeneities, such as faceting. Therefore Surface tension plays
a important role in the morphology of different crystal surfaces.
Roughening transition
Thermal fluctuations is root mean square deviation of the
position with respect to the average position.
Finite
smooth
or
diverge
rough
Some cases, rough at any T>0 K (for example 1 dimensional
line). Some cases, rough when T>Tr (roughening transition
temperature). For example, surface exhibits such a transition
at Tr depends on detailed structure. Therefore, for the same
element there will be different Tr for different orientation. This
implies different g plot at different temperatures too. (See page
15 to 43 of Concepts in Surface Physics ).
Crystal Shape at different temperatures
Lead
T  323K, crystal is bound by (111),
(001), (110) and (112)
323K < T < 393K, (112) facets
disappear, (111), (001) and (110) left
T > 393K, (110) facets disappear, (111)
and (001) left, which are the most
closed-packed with highest
coordination number of surface atoms
Relaxation, reconstruction, Defects
Due to lack of the neighboring
atoms for the surface atoms, there
are characteristic rearrangement
of atoms on the surface. The
change of surface structure of
semiconductor is more dramatic
than metal.
Three kinds of this rearrangement
can be seen in the shown figure:
relaxation, reconstruction and
missing row reconstruction.
Relaxation
The in-plane lattice same as the bulk with out of-plane one changed. It
involves a few layers into the bulk, but often a compensating expansion
between the 2nd and 3rd layers can happen with smaller amplitude.
The Lower the atomic packing density of the surface, the larger the
inward contraction.
Low index surfaces of metals (100) and (111) are very close to be simple
bulk terminations of the crystal. The more open (110) of almost all
metals such as Al(110) usually shows an inward relaxation at the first
layer and oscillatory relaxation at next several layers.
Almost all nonpolar semiconductor (Si, GaAs…) surfaces show outward
expansion of anion and inward contraction of cation: bond rotation
theory.
Insulator show little relaxation.
The relaxation in metal
Cu(410)
Tables for relaxation of some clean metal surfaces
The relaxation of wurtzite compounds
inward
outward
inward
outward
One material called TCC with wurtzite structure
reconstruction
Semiconductor’s
covalent
bonding
is
strongly
directional. For GaAs (110)
there is tilt in the surface GaAs bond of about 27O to the
surface and only small
change in the bond length.
reconstruction
Much more strong reconstruction
involves bonds breaking and
reforming of new bonds is found in
Si(111)
surface.
All
these
reconstruction is to lower the total
energy at the surface.
Although metals rarely have
reconstruction, there are some metals
like Pt(100) with (2X1) with a
missing row of surface atoms and Ir
(100) with (5X1) with a buckled
quasi-hexagonal layer of surface
atoms.
Defects
There are
also defects
like foreign
adatoms as
interstitials
and anti-site
exchange of
atoms for
compound
A defect is a break in peridoicity (lattice or basis). Any real
crystalline material will contain defects so does the real surface.
For the bulk defects in close-packed metals, defects 10-2-10-4 %
at room temp.In some alloys (carbides), vacancy defects may
approach 50 % at room temp. A well-prepared low-index metal
surface contains 0.1-1 % defects. Defects are important to
many properties and plays a predominant role in processes
such as crystal growth, surface chemical reaction, etc.
fcc(775)
Steps
Step is one-dimensional or line defects. Steps are very
important for the formation of the high-index surfaces formed
by small low-index terraces and a high density of steps. Steps
often have interesting electronic properties, for example metal
steps form the dipole moments.
Review of crystal structure
A crystal structure is made up of two
basic elements: lattice and basis. The
basis can be 1 atom and much more
atoms, For example, Si, NaCl have2
atoms basis.
There are generally 7 different Crystal
Systems. For convenience, these are
further divided into 14 Bravais lattices
useful information
2-dimensional lattice, Superstructure, and Reciprocal
space
5 possible 2-dimensional (2D) Bravais Lattice
Centered unit cell can
be also be described by
a primitive cell.
The possible operations
for 2D lattice is 1, 2, 3,
4 and 6-fold rotation
axes perpendicular to
the surface, and mirror
planes normal to the
surface.
Miller Indices
We can define Miller indices (h,k,l) to label crystal planes, which
intersects the unit cell at a/h, b/k and c/l. If the plane I cuts an axis at
infinity, the corresponding index is zero. There can be also a
negative index.
Useful links here!
In cubic systems only,
the [hkl] direction is
perpendicular to the
(hkl) plane.
Miller Indices for Hexagonal system
In the hexagonal system Bravais-Miller Indices may be used by convention
(really only need three), including 4 indices (hkil) with h + k = -i. Three
symmetry-equivalent axes, 120O apart, and fourth c axis perpendicular to
them. i contains no new information and only to identify the structure as
hcp.
Fcc
100
Surface atom density
110
111
Miller Indices Stepped Surface
Characterized by high (hkl) values (Vicinal Surfaces) - (977), (755)
or (533). Terrace and step often resemble simple low index planes,
therefore there are alternate notation:
Miller Indices for Stepped Surface
(610) = 6(100) + 1(110)
This atom
cannot be seen
Useful table
High index steps consists
of low index terrace and
steps.
Notations for Superstructures (super lattice)
The surface reconstruction
will let the surface structure
different from the bulk. In
many
cases,
adsorbed
layers on a surface will also
have different structure.
When
a
different
periodicity is present in the
topmost atomic layer, this
surface lattice is called
superlattice
and
the
structure is superstructure.
Woods or Matrix Notation
is just to tell the
relationship between this
lattice and the bulk lattice.
Matrix Notation
The substrate lattice on the certain surface can be described by a set of 2D
translational vectors:
rmn= m a1 + n a2 (m, n are integers, and a1 and a2 are unit mesh vectors)
The adsorbate unit cell is usually defined by the two vectors b1 and b2. The
relationship between b1 and b2 can be determined by:
b1 = m11 a1 + m12 a2
or
b2 = m21 a1 + m22 a2
detM = | b1 x b2 |/ | a1 x a2 |
detM is an integer
rational number
irrational number
m12
m21
2X2 matrix M
simple superlattice
coincidence lattice
incoherent lattice
Woods Notation
Wood's notation is the simplest and most frequently used method for
describing a surface structure - it only works, however, if the two unit
cells are of the same symmetry or closely-related symmetries (more
specifically, the angle between b1 & b2 must be the same as that
between a1 & a2 ).
In essence, Wood's notation first involves specifying the lengths of the
two overlayer vectors, b1 & b2 , in terms of a1 & a2 respectively - this is
then written in the format :
( |b1|/|a1| x |b2|/|a2| )
In addition, one indicate also the angle through which one mesh is
rotated relative to the other.
Therefore we can have X{hkl}(pxq) – RO or X{hkl}c(pxq) – RO
The possible
Which substrate surface p= |b1|/|a1|
Rotation angle centering
eg. Si(111)….
q= |b2|/|a2|
examples
primitive
Fcc(100) substrate
examples
Substrate : fcc (100)
(2 x 2) overlayer
Substrate : fcc (100)
c(2 x 2) overlayer
2D Reciprocal Lattice
For substrate
For superstructure
or
3 D Reciprocal Lattice
Structural Model of Solid/Solid Interface
The important feature of the
solid/solid interface is the
structure and “abruptness”. The
Thermodynamics can give
rough idea.
Generally the
interface is more complicated
than the surface.
One of the important concept
for the film growth is mismatch:
difference between the lattice
constants of the two materials,
usually (b-a)/a with a for
substrate and b for the overlayer.
The interface depends strongly
on both mismatch and thickness.