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### Transcript of Chapter 6. Free Electron Theory of Metals - YUyu.ac.kr/~hyulee/ssp_lecture/Chap6.pdf¢ ...

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

82 2

2

-19

1 2

2

2222

n L

En mL

nh m k

En ×

≅=== h

kxAA sin=ψ

kxAS cos=ψ

2 0

2

LL x −=

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

Electronic Properties of Materials Hee Young Lee

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)

(X 1 0

- 1 9

e V

)

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

828 222

2

19 222

2

2

2

222

2

22

zyx

zyxzyxn

nnn

nnn L

nnn mL h

mL n

mL nh

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

( ) EECEm h

ENEZ ∝=== 2/3 3

2 4

)]( )[( 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 2 4

)(

2/3* 3

2/3* 3

EEm h

EN

EEm h

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

i ii

i i

∑

∑ =

===

1)kT/)EEexp(( 1

)E(F F +−

=

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”

Electronic Properties of Materials Hee Young Lee

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

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

Electronic Properties of Materials Hee Young Lee

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

Electronic Properties of Materials Hee Young Lee

(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

Electronic Properties of Materials Hee Young Lee

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 Pauli exclusion principle.

text) of 6.23 (equ E

k6

dT

Ed C

F

2

V = ><

=

Electronic Specific Heat

Electronic Properties of Materials Hee Young Lee

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

Thermionic, Schottky, Field Emission and Photoemission

Electronic Properties of Materials Hee Young Lee

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 the value of EF

3/22

2/3 3

2/3

0

2/1

0 3

2/3

8 3

2

3 2)2(4

)2(4 )( )(

=

=

== ∫∫ ∞

π

π

π

N m

h E

E h m

dEE h m

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

Electronic Properties of Materials Hee Young Lee

eV1 )104)(1085.8(16

106.1 U

distance)ic (Interatom 4xsay

eV in x16

q U

x16

q U

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

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