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Classification of Solids, Fermi
Level and Conductivity in
Metals
Dr. Anurag Srivastava
Web address: http://tiiciiitm.com/profanurag
Email: [email protected]
Visit me: Room-110, Block-E, IIITM Campus
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Classification of Solids
On The basis of Geometry and Bonding
(Intermolecular forces)
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AMORPHOUS
CLASSIFICATION OF SOLIDS BASED ON ATOMIC
ARRANGEMENT
QUASICRYSTALS CRYSTALS
Ordered
+
Periodic
Ordered
+
Periodic
Ordered
+
Periodic
There exists at least one crystalline state of lower energy (G) thanthe amorphous state (glass)
The crystal exhibits a sharp melting point
“Crystal has a higher density”!!
A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry.
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AMORPHOUS
CLASSIFICATION OF SOLIDS BASED ON ATOMIC
ARRANGEMENT
QUASICRYSTALS CRYSTALS
ADDITIONAL POSSIBLE STRUCTURES
Modulated structuresIncommensurately
Modulated structures
Liquid crystals
Modulated crystal structures belong to that kind of crystal structures, in which atoms suffer from certain substitutional and/or
positional fluctuation. If the period of fluctuation matches that of the three-dimensional unit cell then a superstructure results,
otherwise an incommensurate modulated structure is obtained.
Incommensurate modulated phases can be found in many important solid state materials. In many cases, the transition to
the incommensurate modulated structure corresponds to a change of certain physical properties. A common feature of
incommensurate modulated structures is that they do not have 3-dimensional periodicity. However incommensurate
modulated structures can be regarded as the 3-dimensional hypersection of a 4- or higher-dimensional periodic structure.
Liquid crystals (LCs) are highly structured liquids, with orientational (nematic,cholesteric) and positional (smectic) order of
constituent molecules . The type of molecular order is controlled by shape and chirality of LC molecules, with over a hundred
known LC phase,
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Commensurate: corresponding in size or degree; in proportion
Incommensurate: out of keeping or proportion with
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THE ENTITY IN QUESTION
GEOMETRICAL PHYSICAL
E.g. Atoms, Cluster of Atoms
Ions, etc.E.g. Electronic Spin, Nuclear spin
ORDER
ORIENTATIONAL POSITIONAL
ORDER
TRUE PROBABILISTIC
Order-disorder of: POSITION, ORIENTATION, ELECTRONIC & NUCLEAR SPIN
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Range of Spatial Order
Short Range (SRO) Long Range Order (LRO)
Class/
example(s)
Short Range Long Range
Ordered Disordered Ordered Disordered
Crystals*/
Quasicrystals
Glasses#
Crystallized
virus$
Gases
Notes:
* In practical terms crystals are disordered both in the short range (thermal vibrations) and
in the long range (as they are finite)
# ~ Amorphous solids
$ Other examples could be: colloidal crystals, artificially created macroscopic crystals
Liquids have short range spatial order but NO temporal order
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When primary bonds are 1D or 2D and secondary bonds aid in the
formation of the crystal
The crystal structure is very complex
Factors affecting the formation of the amorphous state
When the free energy difference between the crystal and the glass is
small Tendency to crystallize would be small
Cooling rate → fast cooling promotes amorphization
“fast” depends on the material in consideration
Certain alloys have to be cooled at 106 K/s for amorphization
Silicates amorphizes during air cooling
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Fundamentals of Electrical EnginPHYring
10
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Fundamentals of Electrical EnginPHYring
11
The theoretical models for metals and
semiconductors use band theory, ionic
materials use lattice models, molecular and
network materials are modelled either
by molecular orbital (MO) theory or by valence
shell electron pair repulsion (VSEPR):
Molecular – Network: Molecular Covalent
Dimensionality
Ionic–Network: Ceramics & Oxides
Metallic–Molecular (not shown on diagram): Cluster
Compounds
Metallic–Ionic: Alloys
Ionic–Molecular: Polar Bonding
Metallic–Network: Semiconductors
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COVALENT
CLASSIFICATION OF SOLIDS BASED ON BONDING
IONIC METALLIC
Molecular
CRYSTALS
Non-molecular
COVALENT
IONIC
METALLIC
Molecule held together by primary
covalent bonds
Intermolecular bonding is Van der walls
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The London dispersion force is the weakest intermolecular force. The London dispersion force is a temporary
attractive force that results when the electrons in two adjacent atoms occupy positions that make the atoms form
temporary dipoles. This force is sometimes called an induced dipole-induced dipole attraction.
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Fundamentals of Electrical EnginPHYring
14
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AlloysA substance that contains mixture of elements and has
metallic properties
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METALLIC
Positive ions in a free electron cloud
Metallic bonds are non-directional
Each atoms tends to surround itself with as many neighbours as possible!
Usually high temperature (wrt to MP) → BCC (Open structure)
The partial covalent character of transition metals is a possible reason
for many of them having the BCC structure at low temperatures
FCC → Al, Fe (910 - 1410ºC), Cu, Ag, Au, Ni, Pd, Pt
BCC → Li, Na, K , Ti, Zr, Hf, Nb, Ta, Cr, Mo, W, Fe (below 910ºC),
HCP → Be, Mg, Ti, Zr, Hf, Zn, Cd
Others → La, Sm Po, α-Mn, Pu
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CLOSE PACKING
A B C
+ +
FCC
=
Note: Atoms are coloured differently but are the same
FCC
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AB
+
HCP
=
A
+
Note: Atoms are coloured differently but are the same
HCPShown displaced for clarity
Unit cell of HCP (Rhombic prism)
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Note: diagrams not to scale
Atoms: (0,0,0), (⅔, ⅓,½)
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PACKING FRACTION / Efficiency
Cell of Volume
atomsby occupied VolumeFraction Packing
SC* BCC* CCP HCP
Relation between atomic radius (r)
and lattice parameter (a)
a = 2r a = 2r
Atoms / cell 1 2 4 2
Lattice points / cell 1 2 4 1
No. of nearest neighbours 6 8 12 12
Packing fraction
= 0.52 = 0.68 = 0.74 = 0.74
ra 43 ra 42
6
8
3
6
2
3
24rc
6
2
* Crystal formed by monoatomic decoration of the lattice
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ATOMIC DENSITY (atoms/unit area)
SC FCC BCC
(100) 1/a2 = 1/a2 2/a2 = 2/a2 1/a2 = 1/a2
(110) 1/(a22) = 0.707/a2 2/a2 = 1.414/a2 2/a2 = 1.414/a2
(111) 1/(3a2) = 0.577/a2 4/(3a2) = 2.309/a2 1/(3a2) = 0.577/a2
Order (111) < (110) < (100) (110) < (100) < (111) (111) < (100) < (110)
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Fundamentals of Electrical EnginPHYring
24
Enrico Fermi (1901–1954)
Citizenship Italian (1901–54)
American (1944–54)
Alma mater Scuola Normale Superiore, Pisa
Sapienza University of Rome
Known for •Demonstrating first self-sustaining Nuclear chain reaction
•Fermi–Dirac statistics
•Fermi paradox
•Fermi method
•Fermi theory of beta decay
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Fundamentals of Electrical EnginPHYring
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Fermi Energy (EF) and
Fermi-Dirac Distribution Function f(E)
"Fermi level" is the term used to describe the top of the collection of
electron energy levels at absolute zero temperature. This concept comes
from Fermi-Dirac statistics. Electrons are fermions and by the Pauli
exclusion principle cannot exist in identical energy states. So at absolute
zero they pack into the lowest available energy states and build up a "Fermi
sea" of electron energy states.
At higher temperatures a certain fraction, characterized by the Fermi
function, will exist above the Fermi level. The Fermi level plays an
important role in the band theory of solids. In doped semiconductors, p-
typeand n-type, the Fermi level is shifted by the impurities, illustrated by
their band gaps. The Fermi level is referred to as the electron chemical
potential in other contexts.
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Fermi Energy (EF)
Fermi Energy is the energy of the state at which the
probability of electron occupation is ½ at any temperature
above 0 K.
It is also the maximum kinetic energy that a free
electron can have at 0 K.
The energy of the highest occupied level at absolute
zero temperature is called the Fermi Energy or Fermi Level.
Fermi Energy (EF) and
Fermi-Dirac Distribution Function f(E)
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Fundamentals of Electrical EnginPHYring
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Fundamentals of Electrical EnginPHYring
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Important
Definitions:
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Fundamentals of Electrical EnginPHYring
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Effect of Temperature on f(E)
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