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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
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)
(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”
Electronic Properties of Materials Hee Young Lee
0K => all states < Ef occupiedall 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>Efsome 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 Pauliexclusion principle.
text) of 6.23 (equ E
k6
dT
EdC
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 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
Electronic Properties of Materials Hee Young Lee
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
Electronic Properties of Materials Hee Young Lee
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
Electronic Properties of Materials Hee Young Lee
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
Electronic Properties of Materials Hee Young Lee
(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
Electronic Properties of Materials Hee Young Lee
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
Electronic Properties of Materials Hee Young Lee
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.
Electronic Properties of Materials Hee Young Lee
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
Electronic Properties of Materials Hee Young Lee
(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
Electronic Properties of Materials Hee Young Lee
Thermionic emission
Schottky emission
Field emission
N(E)
This applies to all solid, emitting into vacuum or another solid
solid vacuum
Electronic Properties of Materials Hee Young Lee
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
Electronic Properties of Materials Hee Young Lee
1Φ2Φ