The QM model of Solids - Carleton Universitytjs/4700/lec09/lec9.pdf · Fig 4.29 From Principles of...

Fig 4.29

From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

The QM model of Solids• Thermocouples

• Thermionic emission

• Phonons

Fig 4.29


Seebeck Effect/Thermocouples• An effect that is best understood using the Fermi energy and electron

distribution is the Seebeck effect which is exploited in thermocouples

• Heat one end of rod the “hot electrons” will diffuse to the cold side producing

a charge distribution as the “solid” at the hot end is now not charge neutral.

• Two few electrons.

• A metal with have Seebeck coefficient that quantifies this effect.

• Not simple because the electron MFP is fct of temperature.

• In the hot end of the rod the electrons can get “trapped” by the higher phonon

concentration (more scattering! Smaller MFP)

• The Coefficient can be negative!

• Use for thermocouples by making a ring of two different materials.

Fig 4.30



Seebeck Effect

Seebeck effect (thermoelectric power)

is the built-in potential difference ∆V across a material due to a temperature difference ∆T across it.

T

VS

∆∆=

Sign of S

is the potential of the cold side with respect to the hot side; negative if

electrons have accumulated in the cold side.

Fig 4.31


H C

E n e r g y

x

( a ) S p o s i t i v e

′

H C

E n e r g y

x

( a ) S n e g a t i v e

′

Consider two neighboring regions H (hot) and C (cold) with widthscorresponding to the mean free paths and ′ in H and C. Half theelectrons in H would be moving in +x direction and the other halfin −x direction. Half of the electrons in H therefore cross into C,and half in C cross into H.


Seebeck coefficient for metals

xeE

TkS

FO3

22π−≈

Mott and Jones thermoelectric power equation

x = a numerical constant that takes into account how various charge transport parameters, such as the mean free path , depend on the electron energy.

x values are tabulated in Table 4.3• Parameter x is very material dependent


Fig 4.32


(a) If same metal wires are used to measure the Seebeck voltage across themetal rod, then the net emf is zero. (b)The thermocouple from two differentmetals, type A and B. The cold end is maintained at 0 ¡C which is thereference temperature. The other junction is used to sense the temperature. Inthis example it is heated to 100 ¡C.

M e t a l

1 0 0o

C

C o l dH o t

µ V

0

M e t a l

M e t a l

0o

C

(a)

C o l dH o t

µ V

0M e t a l

t y p e B

M e t a l

t y p e B

M e t a l

t y p e A

(b)

1 0 0o

C 0o

C


Thermocouple

We can only measure differences between thermoelectric powers of materials.

When two different metals A and B are connected to make a thermocouple,

then the net EMF is the voltage difference between the two elements.

( ) ∫∫ =−=T

T AB

T

T BAABoo

dTSdTSSV


Thermionic emission• Vacuum tubes were base on thermionic emission (ie temperature activated

electron emission) from a surface.

• Now used in modern integrated MEMs structures

• If for electron at the surface E > EF + phi it will leave the surface.

• So hot electrons can be emitted

• As Maxwellian tail is appropriate we get an e-E/kt type relationship for current

• Can modulate the current by changing the barrier height with an applied

voltage

• Non-linear device I-V curve exponential.

Fig 4.34


C a t h o d e

F i l a m e n t

I

V a c u u m

P l a t e o r A n o d e

(a )

I

V

S a t u r a t i o n c u r r e n t

(b)

(a) Thermionic electron emission in a vacuum tube.(b) Current-voltage characteristics of a vacuum diode.

Fig 4.35


Fermi Dirac function, f(E) and the energy density of electrons, n(E),(electrons per unit energy and per unit volume) at three differenttemperatures. The electron concentration extends more and more to higherenergies as the temperature increases. Electrons with energies in excess ofEF+Φ can leave the metal (thermionic emission).

0

T3

T2

n ( E ) = g ( E ) f ( E )

E

T1

E l e c t r o n c o n c e n t r a t i o n

p e r u n i t e n e r g y

E

EF

+ Φ

EF

0 1 . 0f ( E )0

P r o b a b i l i t y

T3

T2

T1

F r e e E l e c t r o n


Thermionic Emission

Richardson-Dushman thermionic emission equation

Φ−=

kTTBJ o exp2

Φ−=

kTTBJ e exp2

where Be = effective emission constant due to electron reflection at the surface

(impedance mismatch)

Bo=4πemek2/h3 = 120×106 A m-2 K-2

Richardson-Dushman constant

w

Fig 4.36


I m a g e P E

0

EF

+ Φ

x x

N e t P E

x

EF

+ Φ

A p p l i e d P E

EF

+ Φe f f

( a ) ( b ) ( c )

(a) PE of the electron near the surface of a conductor,(b) Electron PE due to an applied field e.g. between cathode and anode(c) The overall PE is the sum.


Schottky effect

When a positive voltage is applied to the anode with respect to the cathode, the electric field at the cathode helps the thermionic emission process by lowering the PE barrier Φ by an amount βSE1/2. The current density in field assisted thermionic emission is

−Φ−=kT

TBJ Se

2/12 exp

Eβ

Schottky coefficientMetal’s work function


Field Assisted (tunneling)• If the field is very large (sharp points)

• The barrier becomes very thin and QM tunneling can occur

• Not temperature dependent

• Get Exp. Relationship with applied field ~ exp(-Ec/E)

• Used in MEMS structures for display technology (many many very small

CRT’s)

Fig 4.37


Vo

e -

x = 0 x = xF

EF

(b)

E

C a t h o d e

G r i d o r A n o d e

H V V

(c)

(a) Field emission is the tunneling of an electron at an energy EFthrough the narrow PE barrier induced by a large applied field. (b)For simplicity we take the barrier to be rectangular. (c) A sharp pointcathode has the maximumfield at the tip where the field-emission ofelectrons occurs.

P E ( x )

x

EF

+ Φe f f

xF

M e t a l V a c u u m

EF

00

(a)

Field assisted emission is field assisted tunneling from the cathode


Field-assisted Tunneling

Field-assisted tunneling: the Fowler-Nordheim equation

( )

ΦΦ−≈Ee

mp e

2/1eff22

exp

−≈− E

EE cCJ exp2emissionfield

Field-assisted tunneling probability

Effective work function due to the Schottky effect

Applied field at the cathode

Constants


Fig 4.38


(a) Spindt type cathode and the basic structure of one of the pixels inthe FED.(b) Emission (anode) current vs, gate voltage (c) Fowler-Nordheim plot that confirms field emission.

3 0 4 0 5 0 6 0 6 55 54 53 5

0

1 0 0

2 0 0

3 0 0

4 0 0( b )

Anode current, IA

Gate Voltage VG

0 . 0 2 0 0 . 0 2 4 0 . 0 2 6

- 1 8

- 2 0

- 2 20 . 0 2 2

( c )- 1 6

1/VG

In(I

A/V

G2 )

Emission tip

Phosphor

Conductingglass

Gate

Dielectric

VG

(a)

Substrate

+

+

-


CNT (Carbon NanoTube)

A carbon nanotube (CNT) is a very thin filament-like carbon molecule whose diameter is in the nanometer range but whose length can be quite long, e.g., 10-100 microns, depending on how it is grown or prepared.

Fig 4.39


Wrapped graphite sheetCapped end with half Buckyball

(a)

(a) A carbon nanotube (CNT) is a whisker-like very thin and longcarbon molecule with rounded ends; almost a perfect shape as anelectron field-emitter. (b) Multiple CNTs as electron emitters. (c) Asingle CNT as an emitter.|SOURCE: Courtesy of Professor W.I. Milne, University of Cam-bridge; G. Pirio et al, Nanotechnology, 13, 1, 2002.

( c )( b )

Fig 4.39


Phonons• A number of times I have mentioned phonons. A phonon is the QM way of

describing heat energy in the lattice.

• Modeling the crystal a “balls on springs” is essentially using a simple

harmonic oscillator model of the solid.

• The simplest way of understanding phonons is to use QM simple harmonic

oscillator rather than a classical one.

• Parabolic PE in the SCE.

– Quantization of the E and p arises

– E separated by h-bar omega

– Zero point energy


Quantum Harmonic Oscillator

Harmonic potential energy

2

2

1)( xxV β=

02

12 222

2

=

−+ ψβψ

xEM

dx

d

ω

+=

2

1nEn

Schrodinger equation for the harmonic oscillator

Energy of the harmonic oscillator

Constant

Angular vibrational frequency of the oscillator.

ω =(β/M)1/2.

Quantum number = 0, 1, 2, …

Fig 4.40


( a ) H a r m o n i c v i b r a t i o n s o f a n a t o m a b o u t i t s e q u i l i b r i u m p o s i t i o na s s u m i n g i t s n e i g h b o r s a r e f i x e d . ( b ) T h e P E c u r v e V ( x ) v s . d i s p l a c e m e n tf r o m e q u i l i b r i u m , x . ( c ) T h e e n e g y i s q u a n t i z e d .

M

x

O X

O+ x− x

V ( x )

( b )

O+ x− x

E n

E 0

E 1

E 2

E 3

( a )

( c )

Fig 4.40


Types of phonons• Two types of Phonons: quantized vibrational state of energy

– Acoustic – vibrations of acoustic frequencies (heat)

• Longitudinal

• Transverse

– Optical – vibrations at optical frequencies (not in book)

• Only in materials with two atoms/unit cell

• Very high frequency – excited by optical signals

– Physics very similar to electrons in lattice (periodic structure)

– Bosons (each state can have many phonons)

– Cut off frequency no phonons with omega > 2(beta/M)1/2

Fig 4.41


ur

ur

( a )

( b )

( c )

0 a 2 a r a

x r = r a N o v i b r a t i o n s

( N 1 ) a

x

L - w a v e

T - w a v e

(a) A chain of N atoms through a crystal in the absence of vibrations. (b)Coupled atomic vibrations generate a traveling longitudinal (L) wave along x.Atomic displacements (ur) are parallel to x. (c)Atransverse (T) wave travelingalong x. Atomic displacements (ur) are perpendicular to the x-axis. b and c aresnapshots at one instant.


Lattice Waves: PhononsTraveling-wave-type lattice vibrations along x

( )[ ]tKxjAu rr ω−= exp

υω hE ==phonon

Kp =phonon

= Ka

M 2

1sin2

2/1βω

Phonon Energy

Phonon Momentum

Dispersion Relation

Phonon frequencyPhonon wavevector

Fig 4.42


0

ω

π / a− π / a

ω m a x

0

v g

π / a

v g m a

x

(a) (b)

(a) Frequency ω vs. wavevector K relationship for lattice(b) Group velocity vg vs. wavevector K.

AB

− K K K

• Omega max is largest freq. In system


Group Velocity

The velocity at which traveling waves carry energy

== Kaa

Mdk

dvg 2

1cos

2/1βω

2/1

≈ρY

vg

Y = elastic modulus (Example 1.5), ρ = density

Fig 4.43


Four examples of standing waves in a linear crystal corresponding toq = 1, 2 and 4. q is maximum when alternating atoms are vibratingin opposite directions. A portion from a very long is crystal shown.

q = 1

q = 2

q = 4

q = N

Fig 4.43


Debye Temperature• As the materials temperature is increase more phonon frequencies are

excited.

• At the Debye temperature we have all frequencies present in the material.

• Increasing beyond this creates more phonons, not higher frequencies.

• By using a density of states fct g(E) and average energy of the phonons we

can obtain the Debye temperature and the heat capacity.

• Classical heat capacity is for materials above the Debye temperature when all

modes are excited.

Fig 4.44


0 1 2 3 4 5

ω ( 1 0 2 3 r a d i a n s / s ) ω m a x

Den

sity

ofSt

ates

g(ω)

Density of states for phonons in copper. The solid curve is deduced fromexperiments on neutron scattering . The broken curve is the three-dimensionalDebye approximation, scaled so that the areas under the two curves are thesame. This requires that ω max = 4.5 × 1013 radians s-1, or a Debyecharacteristic temperature ΤD = 344 K.


Debye frequency and temperature

Debye frequency: maximum vibration (angular) frequency in the crystal

( ) 3/12max /6 VNv Aπω ≈

kTD

maxω=

Debye temperature: all vibrations are fully excited up to ωmax

Mean velocity of lattice waves

Crystal volume

Avogadro’s number


Debye heat capacity

∫=

−

≈ DTT

x

x

Dm e

dxex

T

TRC

0 2

43

)1(9

RCm 3≈3

∝

Dm T

TC

High temperatures

Low temperatures

Heat capacity per mole

Fig 4.45


0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

0 . 9

1

0

5

1 0

1 5

2 0

2 5

0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1

C u

S i ( T D = 6 2 5 K )

C m

J K - 1 m o l e - 1C m / ( 3 R )

T / T D

C m = 3 R

Debye constant-volume molar heat capacity curve. The dependence of the molar heat capacity Cm on temperature with respect to the Debye temperature:

Cm vs. T/TD. For Si, TD = 625 K so that at room temperature (300 K), T/TD =

0.48 and Cm is only 0.81(3R).



Thermal conductivity

(non metals)

• Heat diffuses (phonons diffuse)

• As with any diffusion path thermal diffusivity or conductivity is ~ to the MFP

• Scattering by defects, impurities and other phonons!

• Low temperature phonon-phonon collisions negligible.

• High temperatures important.

Fig 4.47


H o t C o l d

U n h a r m o n i c

i n t e r a c t i o n

D i r e c t i o n o f h e a t f l o w

1

2

3

Phonons generated in the hot region travel towards the cold regionand thereby transport heat energy. Phonon-phonon unharmonicinteraction generates a new phonon whose momentum is towardsthe hot region.


Thermal Conductivity

Thermal conductivity κ

Measures the rate at which heat can be transported through a medium per unit area per unit temperature gradient.

phph3

1vCv=κ

Thermal conductivity due to phononsPhonon mean free path

Phonon velocityHeat capacity per unit

volume

Fig 4.48


Thermal conductivity of sapphire and MgO as a function of temperature.

Fig 4.48


Phonons and electrical transport• Our model of electron transport assumed 100% randomization of the electron

at every scattering occurance.

• Now we know more about phonons ( the main source of scattering) we can

deduce a few things.

– Most electrons in transport have E ~ Ef

– At high temperatures most phonons are at wmax

cause 100%

randomization and we have 1/T dependence of the conductivity

– At low temperature (< TD) scattering by phonon only causes a low angle

event. We need many events to randomize, less phonons, 1/T5

dependence

Fig 4.49


p f

p i

θe −

I n i t i a lm o m e n t u m

p i

p f

P h o n o n

x

E F i n a l m o m e n t u m

K

Low angle scaterring of a conduction electron by a phonon


Electrical Conductivity

Electrical conductivity T > TD

Tn

11

ph

∝∝∝τσ

5ph

1

Tn

NN ∝∝∝ τσ

Electrical conductivity T < TD

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