Download Chap. 41: Conduction of electricity in solids

Chap. 41:
Conduction of electricity
in solids
Hyun-Woo Lee
41-1 What Is Physics?
 Q:
Why certain materials conduct
electricity?
 Q: Why certain materials do NOT conduct
electricity?
 Solid

Many many electrons and atoms
 Solid

material
state physics
Application of quantum physics to solids
41-2 Electrical Properties of
Solids
 Crystalline

solids
Lattice structure
• Repetition of unit cells
 Classification


criteria
Resistivity  at room temperature (m)
Temperature coefficient of resistivity  (K-1)
  (1 /  )( d / dT )

Number density of charge carriers n (m-3)
• Can be found from Hall effect measurement

 Metals, semiconductors, insulators
Insulators, semiconductors
& metals
 Insulators


Extremely large 
Ex: Diamond diamond/copper~1024
 Semiconductors

vs Metals
insulator >> semiconductor >> metal
• silicon=3103 m, copper=210-8 m

semiconductor <0, metal>0
• silicon= -70 10-3 K-1, Copper= +410-3 K-1

nsemiconductor << nmetal
• nsilicon=11016 m-3, ncopper=91028 m-3
41-3 Energy Levels
in a Crystalline Solids
 Single

1s2 2s2 2p6 3s2 3p6 3d10 4s1
 Two

atoms
Tunneling between two atoms
 Three

atom (Ex: Cu Z=29)
atoms
More tunneling
Tunneling effects
 Two

wells
 Level splitting into two levels
Tunneling effects in solids
 N()


wells
Energy level splitting into N levels
 Energy bands & energy gaps
41-4 Insulators
 No
partially filled bands
 For
a current to exist,

Kinetic energy must increase
•  Electrons must move to higher-energy levels

Pauli exclusion principle
• Transition to filled state is prohibited

Energy gap (Ex: Eg=5.5 eV in diamond)
•  Large energy supply needed

 Current flow strongly suppressed
Thermal fluctuation effects
 Thermal

excitations
Finite probability to jump Eg
 Probability

P for the jump
For Eg=5.5 eV, T=300K
P ~ exp( Eg / kT )  e213  3 1093
• cf: # of electron in 1 cm3 ~ 1023
41-5 Metals
 Partially

Easy to induce energy “jump”
 Fermi




level EF
Highest occupied level at T=0K
Ex: EF=7.0 eV for copper
 Fermi

filled bands
speed vF
Electron speed at EF
Ex: vF=1.6106 m/s for copper
No relaxation of vF due to Pauli exclusion
principle
How Many Conduction
Electrons Are There?
 Number
n
density n
number of conduction electrons in sample
sample volume V
 number of conduction   number of atoms  number of valence 

  


in sample
 electrons in sample  
 electrons per atom 
 Ex:
Magnesium w/ volume 2.0010-6 m3
 number of atoms 

  8.6110 22
in sample



Bivalent

  number of conduction 

  1.72 1023
 electrons in sample 
Conductivity Above Absolutely
Zero
 Ex:


T=1000 K
kT=0.086 eV
cf: EF=7.0 eV in copper

# of charge carriers extremely
insensitive to T
41-6 Semiconductors
 No
partially filled bands
 But small energy gap


Ex: Eg=1.1 eV for silicon
cf: Eg=5.5 eV for diamond
 Valence

band
Highest filled band
 Conduction

band
Lowest vacant band
Number Density of Charge
Carriers
 Probability
P for jump
P ~ exp( Eg / kT )  e
 Charge

42.6
 4 10
19
carriers
Electrons
• Conduction band

Holes
• Valence band

# of charge carriers extremely sensitive to T
Motion of charge
carriers
 Electrons
in conduction band
E
 Holes


in valence band
E
Efficient description in terms of holes
 Effective charge of hole: +e
Resistivity 
 silicon
/ copper = 1.51011
 Classical estimation
  m / e 2 n

Difference between silicon and copper mainly
from carrier density n
Temperature Coefficient of
Resistivity 

:

1 d
 dT
 Temperature

dependence
Classical estimation   m / e 2 n
 Semiconductor

n increases as T increases   < 0
 Metal

(Ex: silicon)
(Ex: copper)
 decreases as T increases   > 0
More about metals
How Many Quantum States Are
There?
 Too
many states to list all states
 Density of states N(E)


N(E)dE : # of states between E and E+dE per
volume
Near lower edge of partially filled band
8 2m3 / 2 1/ 2
N (E) 
E
3
h
(m -3 J -1 )
How Many Quantum States Are
There ? (continued)
 Ex:
Metal w/ V=210-9 m3 at E=7 eV
 # of states 

  4 1019 eV -1
 per eV at 7 eV 
# of states N



  11017
 in range 0.003 eV at 7 eV 
The Occupancy
Probability P(E)
 Maxwell

distribution
Not applicable due to Pauli exclusion principle
 Fermi-Dirac
P( E ) 
 At


statistics
1
e
( E  E F ) / kT
1
E=EF
P(E)=1/2 regardless of T
 Useful way to define EF at T>0
How Many Occupied States Are
There?
 Density

of occupied states N0(E)
N0(E)=N(E)P(E)
Calculating the Fermi Energy
 At

T=0,
Due to Pauli exclusion principle
EF
n   N ( E )dE
0

With N(E)  E1/2
8 2m 3 / 2 2 EF3 / 2
n
h3
3
 3 
EF  

 16 2 
2/3
h 2 2 / 3 0.121h 2 2 / 3
n 
n
m
m
More about semiconductors
41-7 Doped
Semiconductors
 Doping


Introducing a small number of replacement
atoms (impurities) into semiconductor lattice
~ 1 out of 107 atoms replaced
n-Type Semiconductors
 Pure

silicon: Si (Z=14) 1s2 2s2 3p6 3s2 3p2
Valence number: 4
 Doping


by P (Z=15, valence=5)
One extra el. n(egative)-type
5th el. in the “conduction band”
Extra electron & proton
 w/o
 w/

extra proton
extra proton
Weakly bound donor levels
At room temperature
 Thermal



Ed =0.045 eV for phosphorous doping
cf: Eg=1.1 eV
 Excitations from donor levels to conduction
band much easier
 Majority

carriers
Electrons in conduction band
 Minority

excitations
carriers
Holes in valence band
Doping level
 Pure

silicon
# density of conduction el. at room temp
• (n0)no-doping ~ 1016 m-3
 Q:




Doping for (n0)doping=106  (n0)no-doping
(n0)doping= (n0)no-doping + nP
 nP  1022 m-3
cf: nSi  51028 m-3
 nP
1
nSi

5  10
6
p-Type Semiconductors
 Doping


One missing el  p(ositive)-type
Missing el in “valence band”
 w/

by Al (Z=13)
missing proton
Weakly bound acceptor levels
At room temperature
 Thermal



Ed =0.067 eV for aluminium doping
cf: Eg=1.1 eV
 Excitations from valence band to acceptor
levels much easier
 Majority

carriers
Holes in valence band
 Minority

excitations
carriers
Electrons in conduction band
41-8 The p-n Junction
 Junction
of p-type and n-type semicond.
Junction plane
 Upon
contact, …(no bias yet)
Motions of the Majority
Carriers
 Diffusion

-e
Diffusion current Idiff
Idiff
 Space
charge
+e
Depletion zone
Contact potential difference V0
 Idiff = 0
Motions of the Minority
Carriers
 Minority


Drift current Idrift
Idrift
Space charge somewhat relaxed
 Majority

carriers
& minority carriers
Balance of Idiff & Idrift
41-9 The Junction Rectifier
I
vs. V
 p-n

junction as a rectifier
AC  DC conversion
Forward bias
 Reduce
V0
Reduce V0
Narrower depletion zone
Backward bias
 Enhance
V0
Enhance V0
Wider depletion zone
41-10 The Light-Emitting Diode (LED)
 LED
 Light
emission from p-n junction
Photon or lattice vibration Forward bias
p-n junction as LED
 Forward

biased p-n junction
Photon wavelength
c
c
hc
 

f Eg / h Eg
 Commercial


LEDs
 in visible range
Ex: Gallium (valence 3) doped with arsenic
(valence 5, 60%) and phosphorous (valence 5,
40%) atoms
• Eg=1.8 eV (red color)
The Photo-Diode
 Photo-diode

= (LED)-1
Photon  Current
Photon-induced
transition

Ex: TV remote control
• Remote control : LED

Generate a certain sequence of infrared photons
• TV : Photo-diode

Photon detection  Electric signal
The Junction Laser
 Stimulated
emission in p-n junction
Mirror
Mirror

 Junction laser
• Ex: Laser head in compact disc (CD) players
41-11 The Transistor
 Transistor


Intentional control of on-off
Application: Amplifier
 FET
(Field Effect Transistor)
 Integrated circuits



Transistors
Capacitors
Resistors etc.
Intel Pentium chip
(w/ ~7 million transistors)
MOSFET
(Metal-Oxide-Semiconductor-FET)
 MOSFET


High speed on-off
~500 nm in length
 Gate





voltage VGS
Negatively charge gate
 Repel el. in n-channel down into substrate
 Wider depletion zone between p and n
 n-channel width reduced
 Larger resistance (off realized)
The End