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Electrical Properties
Introduction to Electrical Properties
 We have noted earlier that atomic structure and electromagnetic structure decide
the properties of materials (a simplified view).
 The ‘electromagnetic structure’ can be thought of in a simplified way as the:
spatial and energetic distribution of electrons (taking into account the charge and
spin of the electrons).
 The electron density distribution can be interpreted in the language of bonding.
 The nucleus of an atom usually plays a lesser role in determining the usual
properties we are discussing in this course
(but may become important in describing certain phenomenon).
 Often we are interested in the response of a material to fields and other external
stimuli (like heating).
 Many aspects of this response is governed by the band structure of the solid.
Resistivity range in Ohm m  ~25 orders of magnitude
 Si (20C ) ~ 6.40  102 m
 Ag (20C )  1.59 108 m
Semi-conductors
Metallic materials
10-9
10-7
Ag
Ni
Cu Al
Pb
Au
10-5
10-3
Sb Bi
Graphite
L
R
A
10-1
Ge
(doped)
10-1
Ge
103
Si
Insulators
105
Window
glass
Ionic
conductivity
107
109
Bakelite
1011
1013
Porcelain
Lucite
Diamond
Mica
Rubber
Polyethylene
1015
1017
PVC
SiO2
(pure)
 Fused quartz (20C ) ~ 7  1017 m
Free Electron Theory
 Outermost electrons of the atoms take part in conduction
 These electrons are assumed to be free to move through the whole solid
 Free electron cloud / gas, Fermi gas
 Potential field due to ion-cores is assumed constant
 potential energy of electrons is not a function of the position (constant negative
potential)
 The kinetic energy of the ‘free’ electron is much lower than that of bound electrons in an
isolated atom
Wave particle duality of electrons
h

mv
6.62 x 1034 J s
7.27 x104


m
31
(9.109 x 10 kg ) v
v
Wave number vector (k)
 2
1
k
E  mv 2

2
h2k 2
E
8 2 m
Non relativistic
E →
h2k 2
E
2
8 m
↑→ k↓→E↓
  → de Broglie wavelength
 v → velocity of the electrons
 h → Planck’s constant
 m → mass of electron
Discrete energy levels
(Pauli’s exclusion principle)
k →
Fermi level
 At zero K the highest filled energy level (EF) is called the Fermi level
 If EF is independent of temperature (valid for usual temperatures)
► Fermi level is that level which has 50% probability of occupation by an electron (finite T)
EF 
 2   3 N 
kF 


2m
2m  V 
2
h2k 2
E
8 2 m
2
2
2/3
Constant Energy- EF
 N → total number of orbitals
 V → volume
Total number of orbitals with energy below E
3/ 2
T>0K
P( E ) 
1
 E  EF 
1  exp 

 kT 
1
0K
P(E) →
V  2mE 
N 2 2 
3   
0
I
E →
g
is n
ea
r
nc
T
EF
Conduction by free electrons
 If there are empty energy states above the Fermi level then in the presence of an electric
field there is a redistribution of the electron occupation of the energy levels
E →
Field
 
 
Electric
Field
EF
k →
Force experienced by an electron

F  ma  Ee
 m → mass of an electron
 E → applied electric field
EF
k →
 In the presence of the field the electron velocity increases by an amount (above its usual
velocity) by an amount called the drift velocity.
 The velocity is lost on collision with obstacles. The average time between collisions  →
and the distance travelled during this period is ‘mean free path’.
Collisions
vd
 vd  
F  m   Ee
 
Velocity →
 vd → Drift velocity
  → Average collision time
‘Mean free time’

Ee
vd 
m

time →
The flux due to flow of electrons → Current density (Je)

ne  E
J e  n e vd 
m
2
 n → number of free electrons
Flux (J e )
Conductivity (  ) 
unit potential gradient (E)
n e2 
 
m
Conductivity is directly
dependent on the
‘mean free time’
Je   E
~ Ohm’s law
 Amp   1   V 
 m 2    Ohm m   m 

V  IR
 V 
 Ohm   Amp 

 Amp   V 1 
 m 2    Ohm m 2 

 

Mean free path (MFP) (l) of an electron





l = vd 
The mean distance travelled by an electron between successive collisions
For an ideal crystal with no imperfections (or impurities) the MFP at 0 K is 
Ideal crystal  there are no collisions and the conductivity is 
Scattering centres → MFP↓ , ↓  ↓ , ↑
Scattering centres
Thermal vibration → Phonons
Sources of
Electron Scattering
MFP in Au, Ag ~ 50nm
MFP Cu ~39 nm RT
MFP Al ~15 nm RT
Solute / impurity atoms
Defects
Dislocations
Grain boundaries
Etc.
Thermal scattering
 At T > 0K → atomic vibration scatters electrons → Phonon scattering
T↑→↓→↑
 Low T
 MFP  1 / T3
   1 / T3
 High T
 MFP  1 / T
   1 / T
Impurity scattering
 Resistivity of the alloy is higher than that of the pure metal at all T
 The increase in resistivity is  the amount of alloying element added!
Cu, Cu-Ni alloy
Resistivity () [x 10-8 Ohm m] →
Increased phonon scattering
5
Cu-3%Ni
4
Cu-2%Ni
3
2
1
Impurity scattering (r)
Pure Cu
100
→ 0 as T→ 0K
Mattheissen rule
With low density of
imperfections
200
300
T (K) →
 = T + r
Net resistivity (approx.) = Thermal resistivity + Resistivity due to impurity scattering
Band Structure
A physical picture of the origin of bands
 At the equilibrium atomic separation the outer
electronic energy levels give rise to bands.
 The core levels continue to remain discrete.
 Bands may overlap and fill in parallel over a
range of energy values, as shown in the figure.
 As a range of energy values are allowed in a
band (discrete but closely spaced), any radiative
transition from these outer levels to a core levels
has a broad range of wavelengths.
Density of states  Density of states is defined as the number of available states in a given
interval of energy at each energy level. (available  available to be
occupied by electrons).
1
P( E ) 
 E  EF 
1  exp 

 kT 
dN
V  2m 

 (E) 


dE 2 2   2 
3/ 2
E
1
2
At T > 0K → (E) = (E, 0K).P(E)
Close to top of the band
V  2 | m* | 
 (E)  2  2 
2   
3/ 2
( EV  E )
1
2
Typical of divalent metals
The number of electrons occupying the range dE
dn( E )  f ( E ). ( E ).dE
Transition metals
 Often a simplified schematic version of the band
structure is shown in elementary texts. These
simplified figures can often lead to a misleading
interpretation that the number of energy levels in
a given energy width (dE) is constant. This value
(the density of states (N(E) or (E)) is a function
of the energy and can vary quite 'wildly'. A
schematic of a 3d band is shown in .
Schematic showing the 'wild' variation possible in the density
Metals
Classification
based on
Band structure
of states of a 3d band. (E) is also written as N(E).
Semi-metals
Semi-conductors
Insulators
Metal
Insulator
Semiconductor
Conduction Band
Conduction Band
2-3 eV
Valence Band
> 3 eV
Valence Band
Electrical Properties of Nanomaterials
Conduction in nanoscale materials
 In bulk materials conduction electrons are delocalized and travel ‘freely’ till they
are scattered by phonons, impurities, grain boundaries etc.
 In nanoscale conductors two effects become important:
 Quantum effect: Continuous (‘nearly’) bands are replaced with discrete energy
states
 Classical effect: mean free path (MFP) for inelastic scattering becomes
comparable to the size of the system (can lead to reduction in scattering events).
 In metals: change in DOS on reduction of size of the system plays a major role
(along with change in electronic and vibrational energy levels).
 In semiconductors quantum confinement of both the electron and hole leads to an
increase in the effective band gap of the material with decreasing crystallite size.
 These effects can lead to altered conductivity in nanomaterials.
Quantum confinement: particle in a box problem
 In the confined direction the energy levels are discretized.
L
If the length of the box is L
n
n L k
L
2

k
2

1
E  mv 2
2
2
2
nh
E
8mL2

h
mv
n → integer (quantum number)
Quantization of Energy levels
h2k 2
E
8 2 m
2D nanomaterials
 Electrons are free in 2D and restricted in 1D → electrons in 1D potential well (quantum
well).
 If the 2D structure is polycrystalline (especially nanocrystalline) → then grain boundary
scattering will play an important role.
  2  2 n 2    2 k F2 
En  


2 
2
2
m
L
m

 

  h / 2
k F2  k x2  k y2

V  2m 
 (E)  2  2 
2   
3D
3/ 2
E
1
2
 E1/ 2
2D
m*
(E)  2

For each
quantum state
 E 0  Constant
1D nanomaterials
 Electrons are free in 1D and restricted in 2D → electrons in 2D potential well.
 Two principle quantum numbers characterize the system → nx and ny.
2 2 2
  2  2 nx2     n y    2 k F2 
En  



2 
2 
m
L
m
L
m
2
2
2


x 
y 

 
1/ 2
1  m* 
1D
 (E) 


   2E 
E
 12
Conduction in carbon nanotubes
 In ballistic transport of electrons, scattering does not lead to loss of kinetic energy and
electrons can move unimpeded.
 Ballistic transport is observed when the mean free path of the electron is bigger than the
size material (bounding box)→ electron transport mechanism changes from diffusive to
ballistic.
Ballistic transport is coherent in wave mechanics terms.
 (If the length of the wire is further reduced to the Fermi wavelength scale, the conductance
is quantized as 2e2/h → does not depend on the length of the conductor).
 Optical analogy to ballistic transport is light transmission through a waveguide.
 In metallic carbon nanotubes (a 1D nanostructure), the transport along the length can
become ballistic → current density ~ 109 A/cm2
(in contrast Cu has a current density of ~106 A/cm2).
 Ballistic conduction may be observed in Si nanowires at very low temperatures (~2-3 K).
0D nanomaterials
 All energy levels are discrete and no delocalization occurs.
 Density of states similar to an atom!
 Metals behave as insulators.
2 2 2

 n y    2  2 nz2 

  n 
En  


2 
2 
2 
 2 m Lx   2 m Ly   2 m Lz 
2
2
2
x
 0D (E)   (E)
  (E)
Electron tunneling & Coulomb blockade
 Tunnel junction behaves as a resistor (constant resistance ohmic resistor).
 The resistance increases exponentially with the barrier thickness (~nm).
 The conductor|insulator(dielectric)|conductor configuration has resistance and
capacitance.
 Current through the tunnel junction passes one electron at a time (co-tunneling of two
electrons also possible).
 The tunnel junction capacitor is charged with one tunnelling electron → leading to a
voltage buildup → can prevent another electron from tunnelling.
For an additional charge ‘q’
introduced into a conductor, work
has to be done against the electric
field of preexisting charges residing
on the conductor. Charging an island
with capacitance C, with an electron
of charge ‘e’ requires energy (EC):
e2
EC 
2C
 At low bias voltages current is suppressed! → Coulomb blockade.
 Coulomb blockade is observed at low temperatures → so that the energy required to
charge the junction with one electron is larger than the thermal energy of the charge
carriers. Else thermal excitation can transport the electron (instead of tunnelling).
 Single electron transistor and Coulomb blockade thermometer are based on the above
concepts.
End
Example: quantum dot based field effect transistor (FET)