1. Electrical Conductivity

2. Surface scattering including grain boundary scattering
Electrical conductivity
The effects of size on electrical conductivity of nanostructures and nanomaterials are complex, since they are based on distinct mechanisms. These mechanisms can be generally grouped into four categories: surface scattering including grain boundary scattering, quantized conduction including ballistic conduction, Coulomb charging and tunneling, and widening and discretization of band gap, and change of microstructures. In addition, increased perfection, such as reduced impurity, structural defects and dislocations, would affect the electrical conductivity of nanostructures and nanomaterials.
4 distinct mechanisms:
Widening of band gap
Surface scattering including grain boundary scattering
Quantized conduction including ballistic conduction, coulomb blockade
Structural Changes (conformational changes, impurities, defects and dislocations)

3. Electrical conductivity
Surface scattering
Electrical conduction in metals or Ohmic conduction can be described by the various electron scattering.
Matthiessen's rule -- the total resistivity = thermal resistivity + defect resistivity
Impurity atoms, defects such as vacancies, and grain boundaries locally disrupt the periodic electric potential of the lattice and effectively cause electron scattering, which is temperature independent.
This one can be further divided into impurity resistivity, lattice defect resistivity, and grain boundary resistivity.
Vibrating atoms (phonons) displaced from their equilibrium lattice positions are the source of the thermal or phonon contribution, which increases linearly with temperature.

4. RESPECTIVE MEAN FEE PATH
Electrical conductivity
Surface scattering
Matthiessen's rule -- the total resistivity = thermal resistivity + defect resistivity
RESPECTIVE MEAN FEE PATH

6. Fermi Energy/Speed How big is the Characteristic Scattering Distance?
Time, Length, Fermi Speed
In metals, the Fermi energy gives us information about the velocities of the electrons which participate in ordinary electrical conduction. The amount of energy which can be given to an electron in such conduction processes is on the order of micro-electron volts (see copper wire example), so only those electrons very close to the Fermi energy can participate. The Fermi velocity of these conduction electrons can be calculated from the Fermi energy.
Element
Fermi Energy eV
Fermi Velocity x 10^6 m/s Li
4.74
1.29 Na
3.24
1.07 K
2.12
0.86 Rb
1.85
0.81 Cs
1.59
0.75 Cu
7.00
1.57 Ag
5.49
1.39 Au
5.53
1.40 Be
14.3
2.25 Mg
7.08
1.58 Ca
4.69
1.28 Sr
3.93
1.18 Ba
3.64
1.13 Nb
5.32
1.37 Fe
11.1
1.98 Mn
10.9
1.96 Zn
9.47
1.83 Cd
7.47
1.62 Hg
7.13 Al
11.7
2.03 Ga
10.4
1.92 In
8.63
1.74 Tl
8.15
1.69 Sn
10.2
1.90 Pb Bi
9.90
1.87 Sb
           

7. Fermi Energy/Speed How big is the Characteristic Scattering Distance?
Time, Length, Fermi Speed
In metals, the Fermi energy gives us information about the velocities of the electrons which participate in ordinary electrical conduction. The amount of energy which can be given to an electron in such conduction processes is on the order of micro-electron volts (see copper wire example), so only those electrons very close to the Fermi energy can participate. The Fermi velocity of these conduction electrons can be calculated from the Fermi energy.
Element
Fermi Energy eV
Fermi Velocity x 10^6 m/s Li
4.74
1.29 Na
3.24
1.07 K
2.12
0.86 Rb
1.85
0.81 Cs
1.59
0.75 Cu
7.00
1.57 Ag
5.49
1.39 Au
5.53
1.40 Be
14.3
2.25 Mg
7.08
1.58 Ca
4.69
1.28 Sr
3.93
1.18 Ba
3.64
1.13 Nb
5.32
1.37 Fe
11.1
1.98 Mn
10.9
1.96 Zn
9.47
1.83 Cd
7.47
1.62 Hg
7.13 Al
11.7
2.03 Ga
10.4
1.92 In
8.63
1.74 Tl
8.15
1.69 Sn
10.2
1.90 Pb Bi
9.90
1.87 Sb
           

8. How big is the Characteristic Scattering Distance? d = lT
Fermi Energy/Speed
Time, Length, Fermi Speed
Element
Fermi Energy eV
Fermi Velocity x 10^6 m/s Li
4.74
1.29 Na
3.24
1.07 K
2.12
0.86 Rb
1.85
0.81 Cs
1.59
0.75 Cu
7.00
1.57 Ag
5.49
1.39 Au
5.53
1.40 Be
14.3
2.25 Mg
7.08
1.58 Ca
4.69
1.28 Sr
3.93
1.18 Ba
3.64
1.13 Nb
5.32
1.37 Fe
11.1
1.98 Mn
10.9
1.96 Zn
9.47
1.83 Cd
7.47
1.62 Hg
7.13 Al
11.7
2.03 Ga
10.4
1.92 In
8.63
1.74 Tl
8.15
1.69 Sn
10.2
1.90 Pb Bi
9.90
1.87 Sb
Newton
           
accelerating electric field E
scattering distance
electron mass
characteristic time between collisions

9. Fermi Energy/Speed Quantum Mechanics: Fermi Energy Ohms Law
How big is the Characteristic Scattering Distance? d = lT
Fermi Energy/Speed
Time, Length, Fermi Speed
Quantum Mechanics: Fermi Energy
Element
Fermi Energy eV
Fermi Velocity x 10^6 m/s Li
4.74
1.29 Na
3.24
1.07 K
2.12
0.86 Rb
1.85
0.81 Cs
1.59
0.75 Cu
7.00
1.57 Ag
5.49
1.39 Au
5.53
1.40 Be
14.3
2.25 Mg
7.08
1.58 Ca
4.69
1.28 Sr
3.93
1.18 Ba
3.64
1.13 Nb
5.32
1.37 Fe
11.1
1.98 Mn
10.9
1.96 Zn
9.47
1.83 Cd
7.47
1.62 Hg
7.13 Al
11.7
2.03 Ga
10.4
1.92 In
8.63
1.74 Tl
8.15
1.69 Sn
10.2
1.90 Pb Bi
9.90
1.87 Sb
accelerating electric field E
scattering distance
electron mass
characteristic time between collisions
           
Ohms Law
current density

10. Fermi Energy/Speed Quantum Mechanics: Fermi Energy Ohms Law
How big is the Characteristic Scattering Distance? d = lT
Fermi Energy/Speed
Time, Length, Fermi Speed
Quantum Mechanics: Fermi Energy
Element
Fermi Energy eV
Fermi Velocity x 10^6 m/s Li
4.74
1.29 Na
3.24
1.07 K
2.12
0.86 Rb
1.85
0.81 Cs
1.59
0.75 Cu
7.00
1.57 Ag
5.49
1.39 Au
5.53
1.40 Be
14.3
2.25 Mg
7.08
1.58 Ca
4.69
1.28 Sr
3.93
1.18 Ba
3.64
1.13 Nb
5.32
1.37 Fe
11.1
1.98 Mn
10.9
1.96 Zn
9.47
1.83 Cd
7.47
1.62 Hg
7.13 Al
11.7
2.03 Ga
10.4
1.92 In
8.63
1.74 Tl
8.15
1.69 Sn
10.2
1.90 Pb Bi
9.90
1.87 Sb
accelerating electric field E
scattering distance
electron mass
characteristic time between collisions
           
Ohms Law
current density

11. 40 nm How big is the Characteristic Scattering Distance? d = lT
Time, Length, Fermi Speed
Number Example for Copper
Calculate d
Element
Fermi Energy eV
Fermi Velocity x 10^6 m/s Li
4.74
1.29 Na
3.24
1.07 K
2.12
0.86 Rb
1.85
0.81 Cs
1.59
0.75 Cu
7.00
1.57 Ag
5.49
1.39 Au
5.53
1.40 Be
14.3
2.25 Mg
7.08
1.58 Ca
4.69
1.28 Sr
3.93
1.18 Ba
3.64
1.13 Nb
5.32
1.37 Fe
11.1
1.98 Mn
10.9
1.96 Zn
9.47
1.83 Cd
7.47
1.62 Hg
7.13 Al
11.7
2.03 Ga
10.4
1.92 In
8.63
1.74 Tl
8.15
1.69 Sn
10.2
1.90 Pb Bi
9.90
1.87 Sb
The Fermi energy for copper is about 7 eV, so the Fermi speed is
           
The measured conductivity of copper at 20°C is
                                   
The mean free path of an electron in copper is                                                                                                                                       
40 nm

12. 40 nm How big is the Characteristic Scattering Distance? d = lT
Time, Length, Fermi Speed
Number Example for Copper
Calculate d
Element
Fermi Energy eV
Fermi Velocity x 10^6 m/s Li
4.74
1.29 Na
3.24
1.07 K
2.12
0.86 Rb
1.85
0.81 Cs
1.59
0.75 Cu
7.00
1.57 Ag
5.49
1.39 Au
5.53
1.40 Be
14.3
2.25 Mg
7.08
1.58 Ca
4.69
1.28 Sr
3.93
1.18 Ba
3.64
1.13 Nb
5.32
1.37 Fe
11.1
1.98 Mn
10.9
1.96 Zn
9.47
1.83 Cd
7.47
1.62 Hg
7.13 Al
11.7
2.03 Ga
10.4
1.92 In
8.63
1.74 Tl
8.15
1.69 Sn
10.2
1.90 Pb Bi
9.90
1.87 Sb
The Fermi energy for copper is about 7 eV, so the Fermi speed is
           
The measured conductivity of copper at 20°C is
                                   
The mean free path of an electron in copper is                                                                                                                                       
40 nm

13. Knowing Conductivity you know relaxation time and electron mean free path
e is the electronic charge, t is the momentum relaxation time, m is effective mass

14. Knowing Conductivity you know relaxation time and electron mean free path
e is the electronic charge, t is the momentum relaxation time, m is effective mass

15. How big is the Characteristic Scattering Distance in Gold?
Calculate d
Element
Fermi Energy eV
Fermi Velocity x 10^6 m/s Li
4.74
1.29 Na
3.24
1.07 K
2.12
0.86 Rb
1.85
0.81 Cs
1.59
0.75 Cu
7.00
1.57 Ag
5.49
1.39 Au
5.53
1.40 Be
14.3
2.25 Mg
7.08
1.58 Ca
4.69
1.28 Sr
3.93
1.18 Ba
3.64
1.13 Nb
5.32
1.37 Fe
11.1
1.98 Mn
10.9
1.96 Zn
9.47
1.83 Cd
7.47
1.62 Hg
7.13 Al
11.7
2.03 Ga
10.4
1.92 In
8.63
1.74 Tl
8.15
1.69 Sn
10.2
1.90 Pb Bi
9.90
1.87 Sb
The measured conductivity of gold at 20°C is
                                   
45* 106

16. If dimension are smaller than the electron mean free path!
Electrical conductivity
Ok lets talk about Size Effects: Surface scattering
40 nm
d = lT
If dimension are smaller than the electron mean free path!
In nanowires and thin films, the surface scattering of electrons results in reduction of electrical conductivity. When the critical dimension of thin films and nanowires is smaller than the electron mean-free path, the motion of electrons will be interrupted through collision with the surface.

17. 40 nm d = lT Ok lets talk about Size Effects: Surface scattering
Electrical conductivity
Ok lets talk about Size Effects: Surface scattering
40 nm
d = lT
Conductivity goes up or down?
If dimension are smaller than the electron mean free path!
Elastic and Inelastic Scattering
The electrons undergo either elastic or inelastic scattering. In elastic, also known as specular, scattering, the electron reflects in the same way as a photon reflects from a mirror. In this case, the electron does not lose its energy and its momentum or velocity along the direction parallel to the surface is preserved. As a result, the electrical conductivity remains the same as in the bulk and there is no size effect on the conductivity. When scattering is totally inelastic, or nonspecular or diffuse, the electron mean­free path is terminated by impinging on the surface. After the collision, the electron trajectory is independent of the impingement direction and the subsequent scattering angle is random. Consequently, the scattered electron loses its velocity along the direction parallel to the surface or the conduction direction, and the electrical conductivity decreases. There will be a size effect on electrical conduction.

18. solve an integral: Mathematical Description
(Thomson Model for Thin films)
Inelastic Scattering
= Energy Loss
= Resistance Increase
solve an integral: p o d
Relative Film Resistance as a function of film thickness
P being the fraction of elastic surface

19. solve an integral: Mathematical Description
(Thomson Model for Thin films)
Inelastic Scattering
= Energy Loss
= Resistance Increase
Not problematic
l0 < d
solve an integral: p o d
Relative Film Resistance as a function of film thickness
P being the fraction of elastic surface

20. Results (Practical Implications)
It should be noted that although the surface scattering discussed above is focused on metals, the general conclusions are equally applicable to semiconductors.
An increased surface scattering would result in reduced electron mobility and, thus, an increased electrical resistivity.
Increased electrical resistivity of metallic nanowires with reduced diameters due to surface scattering has been widely reported.

21. The figure illustrates the electrical resistivities of Co thin films 110 nm, 20 nm and 12.5 nm thick
Element
Fermi Energy eV
Fermi Velocity x 10^6 m/s Li
4.74
1.29 Na
3.24
1.07 K
2.12
0.86 Rb
1.85
0.81 Cs
1.59
0.75 Cu
7.00
1.57 Ag
5.49
1.39 Au
5.53
1.40 Be
14.3
2.25 Mg
7.08
1.58 Ca
4.69
1.28 Sr
3.93
1.18 Ba
3.64
1.13 Nb
5.32
1.37 Fe
11.1
1.98 Mn
10.9
1.96 Zn
9.47
1.83 Cd
7.47
1.62 Hg
7.13 Al
11.7
2.03 Ga
10.4
1.92 In
8.63
1.74 Tl
8.15
1.69 Sn
10.2
1.90 Pb Bi
9.90
1.87 Sb
How big is the Effective Scattering Distance of Cobalt at Room temperature for the three different films?
Why does the resistivity reduce approximately proportionally with temperature?
Why do you see an increase in the resistivity as you reduce the film thickness?

22. Reduced Thermal Conductivity?
Surface inelastic scattering of electrons and phonons would result in a reduced thermal conductivity of nanostructures and nanomaterials as well.
Very little research has been reported so far.
Theoretical studies suggest that thermal conductivity of silicon nanowires with a diameter less than 20 nm would be significantly smaller than the bulk value.
inelastic
20 nm

23. When will quantum size effects occur?
Change of electronic structure
Band splitting: A reduction in characteristic dimension below a critical size, i.e. the electron de Broglie wavelength, would result in a change of electronic structure, leading to widening and discrete band gap.
Semiconducting Nanomaterials:
A change of bandgap on the optical properties has been discussed on previous slides.
Do you expect a reduced electrical conductivity?
Yes, wider band gap result in a reduced electrical conductivity
When will quantum size effects occur?
For nanostructures one of the critical parameters is the lattice constant of the crystal structure relative to the de Broglie wavelength of the electrons in the structure.
Material
Electron Energy
Effective Mass Ratio
de Broglie Wavelength Al
11.7 eV
1.0 nm
GaAs
0.050 eV
0.067 nm
GaAs(2D)
0.01 eV nm

24. Practical Implications:
Change of electronic structure
Practical Implications:
Some metal nanowires may undergo a transition to become semiconducting as their diameters are reduced below certain values, and semiconductor nanowires may become insulators. Such a change can be partially attributed to the quantum size effects, i.e. increased electronic energy levels when the dimensions of materials are below a certain size
Examples (for references see)
Single crystalline Bi nanowires undergo a metal­to-
semiconductor transition at a diameter of ~52nm and below.
GaN nanowires of 17.6 nm in diameter was found to be still semiconducting,
Si nanowires, however, of ~ 15 nm became insulating.

25. Describe a mechanism by which semiconducting nanowires can become insulating as you shrink down the size.

26. Quantum transport
Quantum transport in small devices and materials has been studied extensively.
Only a brief summary is presented below including discussions on ballistic conduction, Coulomb charging and tunneling conduction.

27. Ballistic Conduction
Ballistic conduction occurs when the length of conductor is smaller than the electron mean-free path.
In this case, each transverse waveguide mode or conducting channel contributes
to the total conductance.

28. Knowing Conductivity and Fermi velocity you know relaxation time and electron mean free path d
e is the electronic charge, t is the momentum relaxation time, m is effective mass

29. 40 nm Remember Number Example Number Example for Copper Calculate d
Time, Length, Fermi Speed
Number Example for Copper
Calculate d
Element
Fermi Energy eV
Fermi Velocity x 10^6 m/s Li
4.74
1.29 Na
3.24
1.07 K
2.12
0.86 Rb
1.85
0.81 Cs
1.59
0.75 Cu
7.00
1.57 Ag
5.49
1.39 Au
5.53
1.40 Be
14.3
2.25 Mg
7.08
1.58 Ca
4.69
1.28 Sr
3.93
1.18 Ba
3.64
1.13 Nb
5.32
1.37 Fe
11.1
1.98 Mn
10.9
1.96 Zn
9.47
1.83 Cd
7.47
1.62 Hg
7.13 Al
11.7
2.03 Ga
10.4
1.92 In
8.63
1.74 Tl
8.15
1.69 Sn
10.2
1.90 Pb Bi
9.90
1.87 Sb
The Fermi energy for copper is about 7 eV, so the Fermi speed is
           
The measured conductivity of copper at 20°C is
                                   
The mean free path of an electron in copper is                                                                                                                                       
40 nm

30. The single most important parameter is electron Mean Free Path
bulk mean free path
s  =  |q| · n · µ
mobility

31. If the device dimensions are smaller than the electron mean free path in the respective material the transport can become ballistic!

32. Velociyt/staturation velocity
What to expect for the electron velocity as it is funneled through?
one of the first papers
Electron Mass
Velociyt/staturation velocity
Saturation Velocity
Vbias

33. Ballistic Transistor Has Virtually Unimpeded Current Flow
It took 20 years
From
Ballistic Transistor Has Virtually Unimpeded Current Flow
WASHINGTON (Dec. 6, 1999) -- Researchers at Lucent's Bell Labs
have developed a method to significantly improve the flow of current
in nanoscale transistors -- a characteristic that may help the semiconductor industry
continue making smaller and faster silicon chips.

34. Equations: Boltzmann Transport Equation Theory.... study:

35. No Energy Dissipated in Heat!
Another important aspect of ballistic transport is that no energy is dissipated in the conduction and there exist no elastic scattering.
The latter requires the absence of impurity and defects.
When elastic scattering occurs, the transmission coefficients, and thus the electrical conductance will be reduced which is then no longer precisely quantized.

36. Ballistic Conduction One of the first Experiments

37. Ballistic conduction of carbon nanotubes was first demonstrated by Frank and his co-workers.
The conductance they recorded was one unit of the conductance quantum G0 and no heat dissipation was observed.
Extremely high current densities have been observed

38. Ballistic Conduction in Graphene
Single layer of graphite
Unrolled carbon nanotube
Some devices use ultrathin graphite sheets
Can be manufactured using standard microelectric techniques
Nanotube properties due to graphene not tube shape
Dangling bonds allow donor or acceptor molecules to tune graphene devices

39. Ballistic conduction e- flow unimpeded (no resistance)
Superconductivity without Meissner effect
Caused by lack of impurities in graphene
e- act as massless particles
Possible applications to quantum computing

40. Graphene FETs University of Manchester ~1-3 atoms thick
Electron conduction/hole conduction
Determined by gate voltage
Ballistic conduction at submicron distances
Graphene as zero gap semiconductor

41. End

42. Additional Information conductivity and mean free path
You need to use the Fermi velocity instead of the thermal velocity to get the correct mean free path!
In the classical model, the mean free path is calculated from
electrons are assumed to move at thermal velocity
however this gives a mean free path of a few lattice constants using a typical room temperature value for which is incorrect.
The free-electron fermi-gas model uses the fermi velocity and provides the correct mean free path.