# Thermal Transport in Nanocrystalline Silicon

Document Sample

```					Structure of Amorphous Materials

 Crystalline vs. amorphous materials
 Free volume and the glass transition
 Radial distribution function, short range
order and structure factor
 Voronoi polyhehra
 Medium-range order and nanocrystalline
materials
Crystalline vs. amorphous

1. There is long-range order (LRO)in crystals - a
unit repeats itself and fills the space
2. There is no LRO in amorphous materials
Free volume and the glass transition
Free volume = specific volume (volume per unit mass) - specific
volume of the corresponding crystal

At the glass transition temperature, Tg, the free volume increases
leading to atomic mobility and liquid-like behavior. Below the glass
transition temperature atoms (ions) are not mobile and the material
behaves like solid
Glass transition
Within the free volume theory it is understood that with large enough
free volume mobility is high, viscosity is low. When the temperature is
decreased free volume becomes “critically” small and the system “jams
up”
The glass transition is not first order transition (such as melting),
meaning there is no discontinuity in the thermodynamic functions
(energy, entropy, density).
Typically Tg is ~ 50-60% of the melting point
The effective glass transition temperature, is a function of cooling
rate - higher rate  higher Tg. It is also called the fictive temperature
 Sometimes the glass transition it is a first order transition, most
prominently in Si where the structure changes from 4 coordinated
amorphous solid to to ~ six coordinated liquid. The same applies to
water (amorphous ice)
Characterizing the structure - radial distribution function,
also called pair distribution function

Gas, amorphous/liquid and crystal structures have very different radial
distribution function

1.   Carve a shell of size r and r + dr
dr             around a center of an atom. The
r                  volume of the shell is dv=4r2dr
2.   Count number of atoms with
centers within the shell (dn)
3.   Average over all atoms in the
system
4.   Divide by the average atomic
density <>

1 dn(r,r  dr)
g(r) 
 dv(r,r  dr)


Properties of the radial distribution function

For gases, liquids and amorphous solids
g(r) becomes unity for large enough r.
The distance over which g(r) becomes
unity is called the correlation distance
which is a measure of the extent of so-
called short range order (SRO)
The first peak corresponds to an
average nearest neighbor distance
Features in g(r) for liquids and
amorphous solids are due to packing
(exclude volume) and possibly bonding
characteristics
Radial Distribution Function - Crystal and Liquid
1
Q(r)  g(r) 1 ~ sin(r /d  )exp(r / )
r
15                                                        6
f it       numerical data crystal
f it simulations               NaCl melt
10                                  T=1000K               4
T=1000K
5                                                        2                                       *=0.28


rQ(r)

rQ(r)
0                                                            0

-5                                                       -2

-10                                                       -4

-15                                                       -6
0      1   2      3       4   5   6     7                0    1     2     3      4      5     6     7   8
r []                                                     r []

Liquid/amorphous g(r), for large r exhibit oscillatory exponential decay
Crystal g(r) does not exhibit an exponential decay (  ∞)
Radial distribution functions and the structure factor
• The structure factor, S(k), which can be measured experimentally
(e.g. by X-rays) is given by the Fourier transform of the radial
distribution function and vice versa

4         
S(k)  1
k
 r[g(r) 1]sin(kr)dr
0

Radial distribution functions can be obtained from experiment
and compared with that from the structural model

More detailed structural characterization - Voronoi
Polyhedra
• Draw lines between a center of an
atom and nearby atoms.
• Construct planes bisecting the
lines perpendicularly
• The sets of planes the closest to
the central atom forms a convex
polyhedron
• Perform the statistical analysis of
such constructed polyhedrons,
most notably evaluate an average
number of faces
• For no-directional bonding promoting packing number of
faces is large ~ 13-14 (metallic glasses)
• For directional bonding (covalent glasses) number of faces
is small
• Ionic glasses - intermediate

• In all cases the number of faces is closely related to the
number of nearest neighbors (the coordination number)
Medium range order and radial distribution
function

Radial distribution functions (and also X-ray) of amorphous silicon and model Si with ~
2 nm crystalline grains are essentially the same - medium range order difficult to see by
standard characterization tools. Such structure is called a paracrystal.
Sensitive enough to see medium range order and crystal size

g(r) 1 ~ sin(r / d  )exp(r / )
10                                                         10

Para3 - = 5.3 [Å]
1                                                          1
CRN - = 2 Å
abs[g(r) -1]*r

abs[g(r) -1]*r
0.1                                                        0.1

0.01                                                       0.01

0.001                                                      0.001
0      5        10       15   20                           0   5     10       15         20
]
r [                                           ]
r [
Behavior of g(r), for large r clearly shows differences for CRN and paracrystal models,
and also provide a measure of paracrystal size
Nanocrystalline material

14
12
nanocrystalline
4nm grain size
10

G(r)
8
6
4
2
0
0   0.3      0.6      0.9       1.2
r [a 0]

Nanocrystalline materials shows clear crystalline peaks with some background coming
from the grain boundary

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 19 posted: 12/24/2009 language: English pages: 13
How are you planning on using Docstoc?