Cense NE 215 Session1 Aug2014

8/12/2019 Cense NE 215 Session1 Aug2014

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NE-215, Applied Solid State Physics, @CeNSE

First Semester, 2014-15

References:a. Solid State Physics, Hook and Hall, 2ndEdn., Wileyb. The Picture Book of Quantum Mechanics, Brandt and Dahmen, 4th Edn.,

Springerc. Introduction to Nanoscience, SM Lindsay, Oxford Univ. Pressd. Others cited as we proceed

And, just because it really is there


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2

Britney SpearsGuide to Semiconductor Physics,http://britneyspears.ac/lasers.htm


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Metals / non-metals

the boundaries can be disputed

simple metals, transition metals, noble metals


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Electrical conductivity of materials

Consider the rangeof values of otherphysical attributesof materials

MUCH narrower


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Basic assumptions of the classical theory

DC electrical conductivity in the Drude model

The Hall effect

Why do metals look shiny?- Plasma Resonance

Thermal conduction / Wiedemann-Franz law

Failures of the Drude model: heat capacity...

Electrical properties of metals:Classical approach (Drude theory)


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Drudesclassical theory

Theory formulated by Paul Drude (Germany) in 1900, a

mere three years after the electron was discovered. Drude treated the (free) electrons in metals as a classical

ideal gas but with electrons colliding elastically with thestationary ions, notwith other electrons.

RMS speed from:

So, at 300 K,

12

=

32

12

=

32

105m/s


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Drudesclassical theory

(average time between scattering events)An average distance between collisions

Relaxation time

Mean free path =

~ 1 nm

~ 105m/s

~ 10-14s

As in the kinetic theory of gases, an interval between collisions,


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Kinetic Theory of Gases: Ideal Gas Assumed

Density of gas molecules at STP:Avogadros Number, NA, of them in 22.4 liters

In 1 cm3, it is

2.69x1019

molecules (LoschmidtsNumber)or, 2.69x1025per m3

In any case, in real gases, the Gas Equation PV=nRTis valid only at high temperatures and low pressures,

i.e., at low gas densities

What about the electron gas?


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Conduction Electron Density

=

Zvis the no. of valence electronsper atom

m/Ais the no. of electrons/unit volume

where m

is the mass density

and A is the atomic weight

MUCH higher than Loschmidtsnumber!


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Drudestheory must surely be wrong, because

At these densities, electrons should interact strongly with

one other. Why dont they?

Electrons should also interact strongly with the lattice ions.Why dont they?

The application of classical statistics, i.e., Maxwell-Boltzmann distribution of energies, which leads to theEquipartition of Energy, cannot be valid for electrons.

This last aspect may be elaborated a bit further(anticipating a later part of this segment)..


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Condition for using classical statistics:

The mean de Broglie wavelength of an electron is:

=

=

2

=

(232 )

At 300 K, ~ 6x10-9m

where is the mean separation between particlesandis the mean de Broglie wavelength

So, classical statisticscan hardly be valid for the

free electron gas


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However,

In a theory which gives results like this, there must

certainly be a great deal of truth

- H.A. Lorentz, a Dutch contemporary of Drude

What are these results?


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Drude theory: electrical conductivityWe apply a DC electric field to a metallic sample.

The equation of motion of an electron in it is:

Upon integration,

Recall that:

and if is the average time between collisions then the

average drift speed isFor ~ 10 V/m, we get

=

= 10-2m/s

105m/s

=

thermal

With ~10-14 s,


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Drude theory: electrical conductivity

we get

Ohms law

Number of electrons passing in unit time

Because

Current density

Current of negatively charged electrons

1

= ()

=

=


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Ohms law

and we can definethe conductivity

and theresistivity

and the

mobility

=

= =

=

=

= = 1/

=

Drift velocity/unit applied field


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Ohms law

Valid for metals

Valid for

homogeneoussemiconductors

Not validforinhomogeneous

(semi)conductors(e.g., a p-n junction)

=


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Measurement ofconductivity / resistivity

A two-point probe can be used but the contact/wireresistance can be a problem, especially if the resistivity of

the sample is low.


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Measurement ofconductivity / resistivity

The problem of contact resistance can be overcome byusing a four-point probe.


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Measurement of Resistivity

Two-point probe Four-point probe

Measures sample only

I

I

V

I

I

V

Rcontact Rcontact

Rsample

I

V

Rcontact Rcontact

Rsample

I

V

I

I

Measures resistance of

(sample + contact + probe)

In four-point probe, negligible current flows through the voltmeter, the onlyvoltage drop measured is across Rsample.

i


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Measurement of Resistivity

Two-point probe Four-point probeMeasures sample only

Rcontact Rcontact

Rsample

I

V

Rcontact Rcontact

Rsample

I

V

I

I

Measures sample + contact

resistance + probe resistance

i


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The van der Pauw method

#1

#2#3

#4

ab

cd

cdab I

V

R ,

fRR

t22ln

41,2334,12

41,23

34,12

RRff


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Collinear Contacts

Typical pellet and contact sizes for collinear contacts:

d s

t

D

d = 1 mm

s= 2 mm

t = 1-2 mm

D= 12.7 mm


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Collinear contacts

I

I

V

s

I

Vs 2


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Collinear contacts

I

I

V

s FI

Vs 2

Fcorrects for sample thickness, sample

diameter, edge effects, and temperature.

D d h l i l d i i


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Supposed to be astraight line =

~ 1 nm

1

2

=

3

2

= (3

)

Discrepancy gets worse at low T


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The Hall Effect

Accumulation of charge leads to Hall field EH.

Hall field proportional both to current density and B field

is called the Hall coefficient = =

The Hall coefficient


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Deduce carrier density from Ohms law?

Not a good idea to depend on estimates of

for the steadyState, we get

= ( )

=

=

= =

Therefore, =

=



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Ohms law contains e2

But for the sign of eis important.

=

=

=

1

not supposed to be negative for a metal!


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What if the current carriers arepositively charged?

=1

=1

electrons holes


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Drude theory: why are metals shiny?

Drudestheory gives an explanation of why metals do not

transmit light, but reflect it.

S l ti f b i ti


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Some relations from basic optics:EM wave propagation in matter

Plane wave

complex indexof refraction

Maxwell relation

all the interesting (material) physics lies in the dielectric function!

(, ) =

()

=2

0 = +

= = ( + )

, = (

)=

(

)

, = 0(

)

Wave damped due to "

Free electron dielectric function


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Free-electron dielectric function

An electron int ime-dependentfield

we write

and get

the dipole momentfor one electron is

and for a unit volume

of solid it is

()

= ()

= =

=

=

=

Harmonic motion

Free electron dielectric function


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Free-electron dielectric function

We have

Therefore,

Thus, we get

pcalled

the plasma frequencyCollective motion of the

Free Electron Gas

= 0 = 0 +

= 1 + 0

= 1

0 = 1

=

0

Meaning of the plasma frequency


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Meaning of the plasma frequency

The dielectric function in the Drude model is

Recalling that

with

real and positive propagating waves,

metal is transparent

real and negative no wave propagation,metal is opaque: skin depth

= 1

=

0

, = (

)=

(

)

<

>

The reflection of visible light accounts for the lustre of metals

Plasma frequency: simple interpretation


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Plasma frequency: simple interpretation

Longitudinal, resonant, collective oscillation mode of the electron gas

values for the plasma energy = (

0) =

Excellent support toDrudes

theory

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Cense NE 215 Session1 Aug2014