Electronic Properties of Materials Hee Young Lee

Chapter 6.

Free Electron Theory of Metals

Electronic Properties of Materials Hee Young Lee

Electron Energy Levels in a Rigid Well

(Source: D.A. Neamen, Semiconductor Physics and Devices, Irwin, 1997)

textof (6.1) and (3.42) (eV) 103.75

822

2

-19

12

2

2222

nL

EnmL

nhmk

En×

≅===h

kxAA sin=ψ

kxAS cos=ψ

2 0

2

LLx −=

Electronic Properties of Materials Hee Young Lee

Calculated Energy levels for 20 Å wide 20 eV deep potential well

Eel = 0.314, 1.80, 2.83, 7.26, 7.79, 14.95 and 16.97 eV � Quantized Levels

L = 20 Å

�electrons confined

around an atom or a molecule

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What Happens If L Increases to 20 mm?

Eel : many energy levels � Nearly Continuous Distribution � “Free”

L = 20 mm

�electrons confined

within a metal piece

(mm)

(X10

-19

eV

)

Electronic Properties of Materials Hee Young Lee

Assumption: consider valence electrons in a metal to be confined or bound within three dimensional potential well

Allowed energy levels for an electron are quantized, i.e. discontinuous. For an infinite potential well,

( ) ( )

integers. positive are and ,, where

eV 1075.3

828222

2

19222

2

2

2

222

2

22

zyx

zyxzyxn

nnn

nnnL

nnnmLh

mLn

mLnh

EE ++×

≅++===≡−πh

Free Electron Theory of Metals

Electronic Properties of Materials Hee Young Lee

Notes :

a) Several electrons with different wave functions can have the same

energy (=degeneracy)

b) For “normal” sizes (L>1 mm), these levels are very closely spaced (~continuous).

kTEeEF /)( −=

The probability that a M-B gas particle has energy E, from kinetic theory

It is treated by Maxwell-Boltzmann (M-B) statistics. Its velocity distribution is

)]2/(exp[)( 222/3 kTmvvTvN −∝ −

Electronic Properties of Materials Hee Young Lee

In fact, there is a certain “density of energy states” function, Z(E) which determines how many particles can have certain energies.

eunit volumper particles of # )( )(

and

)( )()(

0∫∞

==

=

dEEZEFN

EZEFEN

These concepts also apply to the electron gas in the metal, I.e.

Electronic Properties of Materials Hee Young Lee

dE Z(E)F(E) g(E) 1/N g(E)

:determined becan g(E)energy of

functionany of n valueexpectatioor average theLikewise, 5.

gas) B-M clasicalfor (3/2)kT(

Z(E)dEF(E) E 1/N E 4.

eunit volumper particles of #dE Z(E)F(E)N 3.

derived.) be also will(This

F(E) states thoseof occupation ofy Probabilit 2.

shortly.) derived be will(This

Z(E) states available ofDensity 1.

0

0

0

∫

∫∫

∞

∞

∞

=><

=

>=<

==

≡

≡

Electronic Properties of Materials Hee Young Lee

Density of States Function

( ) EECEmh

ENEZ ∝=== 2/33

24

)]( )[(π

Using infinite square well approximation, the distribution of allowed energy states in a metal can be derived, and the result is given below.

where C is the proportionality constant. The above equation can also be applied to semiconductors and insulators at the bottom of the conduction band and the top of the valence band,simply replacing m with m*, i.e. the effective mass.

Electronic Properties of Materials Hee Young Lee

Density of States Function (cont’d)

( )

( ) 9)-(1 24)(

8)-(1 24

)(

2/3*3

2/3*3

EEmh

EN

EEmh

EN

Vhh

Cee

−=

−=

π

π

where me* and mh* are the effective mass values of an electron near the bottom of the conduction band and a hole near the top of the valence band, respectively.

Electronic Properties of Materials Hee Young Lee

Usually drawn to represent surface, as follows

Electronic Properties of Materials Hee Young Lee

Fermi-Dirac statistics

(a) The Fermi-Dirac distribution function for a Fermi energy of 2eV and for

temperatures of 0K, 600K and 6000K

(b) The classical Maxwell-Boltzmann distribution function of energies for the

same temperatures.

Electronic Properties of Materials Hee Young Lee

.E level of "degeneracy"the called is Eenergy at states SThe (

energy) of ion(conservat ENE

N)constantelectrons(# NN :Conditions

iii

iii

ii

∑

∑=

===

1)kT/)EEexp((1

)E(FF +−

=

Definig EF =-α/β, and requiring that the above function approach the classical Maxwell-Boltzmann distribution at high temperatures, we can write this as

Which is the “Fermi-Dirac distribution function”

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0K => all states < Ef occupiedall states > Ef empty

(Source:http://jas.eng.buffalo.edu/applets/education/semicon/fermi/functionAndStates/functionAndStates.html)

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0>K => some e- have E>Efsome holes exist E

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(Source:http://jas.eng.buffalo.edu/applets/education/semicon/fermi/functionAndStates/functionAndStates.html)

Electronic Properties of Materials Hee Young Leehttp://jas.eng.buffalo.edu/applets/

(Source:http://jas.eng.buffalo.edu/applets/education/semicon/fermi/functionAndStates/functionAndStates.html)

Electronic Properties of Materials Hee Young Lee

(Source: B.G. Streetman, Solid State Electronic Devices, Prentice-Hall, 1980, p.71.)

Fermi Function

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The free electron theory of metal can largely account for the following phenomena:

• Specific heat (and thermal conductivity)

•Thermoelectric effect

•Thermionic emission

•Schottky effect and field emission

•Photoemission

•Contact potential

Electronic Properties of Materials Hee Young Lee

The specific heat is the amount of energy required to raise temperature by 1 degree Kelvin (or Celsius).

)text of 22.6 equ( E

3(kT)kT)

2

3(

E

2kT E

F

2

F

=≅><

The specific heat of a metal (electron contribution) is some 1/100 times less than expected, all because of the Pauliexclusion principle.

text) of 6.23 (equ E

k6

dT

EdC

F

2

V =><

=

Electronic Specific Heat

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Let’s now consider several phenomena where electrons are actually emitted from metals ;

Thermionic, Schottky, Field Emission and Photoemission

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The work function is defined as the energy required to take an electron out of metal, form EF (I.e. the minimum energy remove an electron).Φ is more critical, therefore, in emission processes, than is thevalue of EF

3/22

2/33

2/3

0

2/1

0 3

2/3

83

2

32)2(4

)2(4 )( )(

=

=

== ∫∫∞

π

π

π

Nm

hE

Ehm

dEEhm

dEEFEZN

F

F

EF

Electronic Properties of Materials Hee Young Lee

Estimate of φ

The potential energy developed between these two charges, when the electron is removed, equals the work function

x+ -

Image Charge

surface

vacuum

metal

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eV1)104)(1085.8(16

106.1U

distance)ic (Interatom 4xsay

eV in x16

qU

x16

qU

1012

19

0

00

00

2

≅××

×=≡

≅

=

−=

−−

−

π

πε

πε

φ

Å

This is a little on the small side, but is the right order of magnitude.

Electronic Properties of Materials Hee Young Lee

used. wasdNeVdJ where

dE}kT/)EE(exp{EV )m2(e4)h/A(

dNeVAI

:escape can Eabove Electron

}kT/)EE(exp{)E(F

E)m2(h4

)E(Z

xx

FE

2/1x

3/23

x

F

F

2/12/33

F

=

−−=

=

+−−=

=

∫∫

∞

+Φ

Φ

π

Thermionic Emission

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zyx2

2z

2y

2x

2

xx

dpdpdpdpp4

m/dp pdE

)ppp)(m2/1(m2/pE

:momentum over intergrate

and momentum toenergy convert

,m2/pV Let

=

=

++==

=

π

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23

2216

3

2

0

kT/2'0

kT/20

kT/2

3

2

kA/cm120

kA/m 102.1h

emk4 A

eTAeT)r1(AJ

eTh

mek4JA/I

=

×==

=−=

==

−−

−

π

π

ΦΦ

φ

The text has include an electron reflection coefficient (r)

Richardson(-Dushman) equation

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(a) Potential at metal-vacuum interface.

(b) Potential changed by image charge field.

(c) Potential due to applied anode voltage in vacuum region.

(d) Total potential field showing reduction in height of the potential barrier compared with (a)

Schottky Effect

Electronic Properties of Materials Hee Young Lee

When an electron field exists normal to an emitting surface(which is the cathode), the work function is reduced.

The work function is not reduced until the electron’s “image potential” is included in the above picture. First consider what the image potential is

Electronic Properties of Materials Hee Young Lee

x

0

2/3

0

x00

x20

2

E4

q)U(x

E16q

x

0qEx16

q

dx

dU

πε

πε

πε

−=

=⇒

=−=

There is now a potential energy maximum in front of the surface

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eV in 4

qE

Joulesin E4

q

0

'

0

2/3'

πεΦΦ

πεΦΦ

−=

−=

This represents the reduction in φ indicated above, I.e.

In the case of Thermionic emission in a solid system

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The Schottky effect obviously occurs only for large fields

The work function is reduced by 0.1eV and K=9

cm/V10m/V10E

E)3/108.3(1.068

2/15

=≅

×= −

The Richardson equation for thermal emission must include the reduced work function if a field is present, and large enough.

The “Schottky effect” applies to interfaces and junctions.

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This is not as common as Schottky emission because the required fields are so large

Tunneling current though narrow forbidden region

metal large field

vacuum

Field Emission

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(a) Potential energy diagram helpful to understanding Thermionic emission(b) The same but for Field emission

Source: G.B ,Solid State Physics Academic press,Inc. p.705

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Thermionic emission

Schottky emission

Field emission

N(E)

This applies to all solid, emitting into vacuum or another solid

solid vacuum

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max

2mV)2/1(hf += Φ

eV/m24.1/hc

/hchf

0

00

ΦµΦλ

Φλ

==⇒

==

Good Photoemitters have small work functions.

Fhotoemission

Electronic Properties of Materials Hee Young Lee

When two solid are in contact, a potential exists between them if their work function differ.(thermocouples, for example, are based on the fact that this voltage is a function of temperature.If two junctions are at different temperatures, a net voltage isgenerated =>Seebeck effect)

Contact Potential

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1Φ2Φ