Electrical Conductivity .

Electrical Conductivity

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/ohmmic.html

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)

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

Vibrating atoms (phonons) displaced from their equilibrium lattice positions are the source of the thermal or phonon contribution, which increases linearly with temperature.

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.

Electrical conductivity • Surface scattering • Matthiessen's rule -- the total resistivity = thermal resistivity + defect resistivity

RESPECTIVE MEAN FEE PATH

Fermi Energy/Speed 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 Velocityx 10^6

m/sLi 4.74 1.29Na 3.24 1.07K 2.12 0.86

Rb 1.85 0.81Cs 1.59 0.75Cu 7.00 1.57Ag 5.49 1.39Au 5.53 1.40Be 14.3 2.25Mg 7.08 1.58Ca 4.69 1.28Sr 3.93 1.18Ba 3.64 1.13Nb 5.32 1.37Fe 11.1 1.98Mn 10.9 1.96Zn 9.47 1.83Cd 7.47 1.62Hg 7.13 1.58Al 11.7 2.03Ga 10.4 1.92In 8.63 1.74Tl 8.15 1.69Sn 10.2 1.90

Pb 9.47 1.83

Bi 9.90 1.87

Sb 10.9 1.96

How big is the Characteristic Scattering Distance?


Element

Fermi Energy

eV


m/sLi 4.74 1.29Na 3.24 1.07K 2.12 0.86


Pb 9.47 1.83

Bi 9.90 1.87

Sb 10.9 1.96

accelerating electric field E

electron mass

characteristic time between collisions

How big is the Characteristic Scattering Distance? d = T

Newton

scattering distance


Element

Fermi Energy

eV


m/sLi 4.74 1.29Na 3.24 1.07K 2.12 0.86


Pb 9.47 1.83

Bi 9.90 1.87

Sb 10.9 1.96

accelerating electric field E

electron mass

characteristic time between collisions

2

1F

mv

ne d


scattering distance

d

J Ev

ne ne

Ohms Law

Quantum Mechanics: Fermi Energy

current density

Time, Length, Fermi Speed

Element

Fermi Energy

eV


m/sLi 4.74 1.29Na 3.24 1.07K 2.12 0.86


Pb 9.47 1.83

Bi 9.90 1.87

Sb 10.9 1.96

2

1F

mv

ne d


Number Example for Copper

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

Calculate d

40 nm

Knowing Conductivity

you know relaxation time

and electron mean free path

e is the electronic charge, is the momentum relaxation time, m is effective mass

2

m

ne

e

2

1F

mv

ne d

Element

Fermi Energy

eV


m/sLi 4.74 1.29Na 3.24 1.07K 2.12 0.86


Pb 9.47 1.83

Bi 9.90 1.87

Sb 10.9 1.96

2

1F

mv

ne d

How big is the Characteristic Scattering Distance in Gold?

Calculate d

The measured conductivity of gold at 20°C is

45* 106

Electrical conductivity

• Ok lets talk about Size Effects: Surface scattering

40 nm

2

1

( )F

T

mv

ne d

d = T

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.

If dimension are smaller

than the electron mean

free path!

Electrical conductivity

• Ok lets talk about Size Effects: Surface scattering

40 nmd = T

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 mean free 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.

Conductivity goes up or down?

Inelastic Scattering= Energy Loss

= Resistance Increase

solve an integral: ..........dzd

o

d

o

Mathematical Description (Thomson Model for Thin films)

Relative Film Resistance as a function of film thickness

P being the fraction of elastic surface

0 < d

Not problematicInelastic Scattering

= Energy Loss = Resistance Increase

solve an integral: ..........dzd

o

d

o

Mathematical Description (Thomson Model for Thin films)

Relative Film Resistance as a function of film thickness

P being the fraction of elastic surface

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.

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?

Element

Fermi Energy

eV


m/sLi 4.74 1.29Na 3.24 1.07K 2.12 0.86


Pb 9.47 1.83

Bi 9.90 1.87

Sb 10.9 1.96

2

1F

mv

ne d

The figure illustrates the electrical resistivities of Co thin films 110 nm, 20 nm and 12.5 nm thick

Why do you see an increase in the resistivity as you reduce the film

thickness?

• 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

• Change of electronic structureBand 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.

• Do you expect a reduced electrical conductivity?

Yes, wider band gap result in a reduced electrical conductivity

MaterialElectronEnergy

EffectiveMass Ratio

de BroglieWavelength

Al 11.7 eV 1.0 5.7443 nm

GaAs 0.050 eV 0.067 339.477 nm

GaAs(2D) 0.01 eV 0.067 759.09 nm

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.

When will quantum size effects occur?

• Semiconducting Nanomaterials: A change of bandgap on the optical properties has been discussed on previous slides.

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 metal to-

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.

• Change of electronic structure

Practical Implications:

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

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.

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.

Knowing Conductivityand Fermi velocity

you know relaxation time

and electron mean free path d

e is the electronic charge, is the momentum relaxation time, m is effective mass

2

m

ne

2

1F

mv

ne d

Time, Length, Fermi Speed

Element

Fermi Energy

eV


m/sLi 4.74 1.29Na 3.24 1.07K 2.12 0.86


Pb 9.47 1.83

Bi 9.90 1.87

Sb 10.9 1.96

2

1F

mv

ne d

Number Example for Copper

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

Calculate d

Remember Number Example

40 nm

The single most important parameter is electron Mean Free Path

= |q| · n · µ

bulk mean free path

http://ej.iop.org/links/rKal354Rh/SoPwSTVP2xG49UZ9av5vpA/jfv4i12p2218.pdf

mobility

2

F

ne den

mv

Fmv

de

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

What to expect for the electron velocity as it is funneled through?

Saturation Velocity

Vbias

Vel

ociy

t/st

atur

atio

n ve

loci

ty

one of the first papers

2bias

elocity

eVv

m

2

2elocity

bias

mvbiasEnergy e V kineticEnergy

Electron Mass

It took 20 yearsFrom http://www.bell-labs.com/news/1999/december/6/1.html

Ballistic Transistor Has Virtually Unimpeded Current FlowWASHINGTON (Dec. 6, 1999) -- Researchers at Lucent's Bell Labshave developed a method to significantly improve the flow of current

in nanoscale transistors -- a characteristic that may help the semiconductor industrycontinue making smaller and faster silicon chips.

Equations:

• Boltzmann Transport Equation

http://nina.ecse.rpi.edu/shur/advanced/Notes/23ballistic.pdf

• Theory.... study:

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.

Ballistic ConductionOne of the first Experiments

• 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

Ballistic Conduction in Graphene

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

• 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

Ballistic conductionBallistic 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

• 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

Graphene FETsGraphene 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

• 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

Additional Informationconductivity 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.

Electrical Conductivity .

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