Introductory Nanotechnology ~ Basic Condensed Matter Physics ~ Atsufumi Hirohata Department of Electronics Quick Review over the Last Lecture 3 states of matters : solid gas liquid density ( large ) ( large ) ordering range ( long ) ( short ) rigid.

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Introductory Nanotechnology
~ Basic Condensed Matter Physics ~
Atsufumi Hirohata
Department of Electronics
Quick Review over the Last Lecture
3 states of matters :
solid
gas
liquid
density
(
large
)
(
large
)
ordering range
(
long
)
(
short
)
rigid time scale
(
long
)
(
short
)
(
small
)
4 major crystals :
soft
( van der Waals crystal )
solid
(
metallic crystal
(
) (
covalent crystal
ionic crystal
)
)
Contents of Introductory Nanotechnology
First half of the course :
Basic condensed matter physics
1. Why solids are solid ?
2. What is the most common atom on the earth ?
3. How does an electron travel in a material ?
4. How does lattices vibrate thermally ?
5. What is a semi-conductor ?
6. How does an electron tunnel through a barrier ?
7. Why does a magnet attract / retract ?
8. What happens at interfaces ?
Second half of the course :
Introduction to nanotechnology (nano-fabrication / application)
What Is the Most Common Atom
on the earth?
Phase diagram
•
Free electron model
•
Electron transport
•
•
Electron Potential
•
•
•
Degeneracy
Brillouin Zone
Fermi Distribution
Abundance of Elements in the Earth
Ca Al
Ni S
Na
Mg (12.70 %)
Fe (34.63 %)
Si (15.20 %)
O (29.53 %)
Mason (1966)
Only surface 10 miles (Clarke number)
Mantle :
Fe, Si, Mg, ...
Can We Find So Much Fe around Us ?
Iron sand (Fe3O4)
Iron ore (Fe2O3)
Desert (Si + Fe2O3)
"Iron Civilization" Today
Iron-based products around us :
Buildings
(reinforced concrete)
Qutb Minar :
Pure Fe pillar (99.72 %), which has
never rusted since AD 415.
Bridges
* Corresponding pages on the web.
Major Phases of Fe
Fe changes the crystalline structures with temperature / pressure :
56
T [K]
Liquid-Fe
Fe :
Most stable atoms in the universe.
1808
-Fe
bcc
1665
Phase change
-Fe (austenite)
fcc
1184
Martensite Transformation :
(-Fe)
’-Fe (martensite)
1043
-Fe
-Fe (ferrite)
bcc
hcp
1
p [hPa]
Electron Transport in a Metal
Free electrons :
-
r = 0.95 Å
+
Na
+
3.71 Å
r~2Å
Na bcc :
3s electrons move freely
a = 4.28 Å
among Na+ atoms.
Uncertainty principle :
xp 
(x : position and p : momentum)
When a 3s electron is confined in one Na atom (x  r) :
p  p 
r
E  p 2m   r 2m 
2


2
2
2mr2
(m : electron mass)
For a free electron, r  large and hence E  small.
Effective energy for electrons
Inside a metal
-
free electron
decrease in E
Free Electron Model
Equilibrium state :
-
+
For each free electron :
+
- i
+
+
-
- +
v0i +
-
+
+
-
i
+
thermal velocity (after collision) : v0i
-
acceleration by E for i
v i  v 0i  
-
Average over free electrons :
collision time : 

Free electrons :
vd  
q
E
m
mass m, charge -q and velocity vi
Using a number density of electrons n,
Equation of motion along E :
current density J :
m
dvi
 qE
dt

Average over free electrons :

q
E i
m
drift velocity : vd

J  v d qn  q 2
n
E
m
Free Electron Model and Ohm’s Law
Ohm’s law :
V  iR  i

S
For a small area : V  i

V
i


 E  J
S

S
where  : electric resistivity (electric conductivity :  = 1 / )
By comparing with the free electron model :



n
E
m
1
n
   q2 

m
J  q2



Relaxation Time
Resistive force by collision :
+
+
+
v
+
If E is removed in the equation of motion :
-

+
+
+
m
+
For the initial condition :
Equation of motion :
with resistive force mv / 
m
dv
m
 qE  v
dt

For the initial condition :
v = 0 at t = 0

v
q
E1 exp t  
m
For a steady state (t >> ),

v 
q
E  vd
m
 : collision time
dv
m
 v
dt


v = vd at t = 0
v  v d expt  
Number of N
0

non-collided
Electrons
N t   N 0 expt / 
N


 : relaxation time  t
time
Mobility
Equation of motion under E :
m
dv
 qE
dt
Also,
t
For an electron at r,

collision at t = 0 and r = r0 with v0
1 qE 2
r  r0  v0 t 
t
2 m

therefore, by taking an average
over non-collided short period,
qE 2
r  r0  v0 t 
t
2m
Since t and v0 are independent,


v0 t  v0 t
Here, v 0 = 0, as v0 is random.


 t 
exp d t  2 2
  

 t2
0
Accordingly,

Here, ergodic assumption :
temporal mean = ensemble mean
2
qE 2

m
q

E  vd
m
r  r0 
r  r0

 Finally, vd = -E is obtained.

 = q / m : mobility
Degeneracy
For H - H atoms :
Total electron energy
Unstable molecule
1s state energy
isolated H atom
2-fold degeneracy
Stable molecule
r0
H - H distance
Energy Bands in a Crystal
For N atoms in a crystal : Total electron energy
2p
2p
6N-fold degeneracy
2s
2s
2N-fold degeneracy
Forbidden band :
Electrons are not allowed
Allowed band :
Electrons are allowed
1s
1s
2N-fold degeneracy
Energy band
r0
Distance between atoms
Electron Potential Energy
Potential energy of an isolated atom (e.g., Na) :
Na
For an electron is released from the atom :
vacuum level
0
Distance from the atomic nucleus
3s
2p
2s
1s
Na 11+
Electron potential energy : V = -A / r
Periodic Potential in a Crystal
Potential energy in a crystal (e.g., N Na atoms) :
Vacuum level
Distance
3s
2p
3s
2p
2s
2s
1s
1s
Na 11+
Na 11+
Na 11+
Electron potential energy
• Potential energy changes the shape inside a crystal.
• 3s state forms N energy levels  Conduction band
Free Electrons in a Solid
Free electrons in a crystal :
Total electron energy
Total electron energy
Energy band
Wave number k
Wave / particle duality of an electron :
Wave nature of electrons was predicted by de Broglie,
and proved by Davisson and Germer.
Ni crystal
Kinetic energy
electron beam

h
h

p mv
Momentum
(h : Planck’s constant)

Particle nature
Wave nature
mv 2 2
mv
h  
h  k
 h 2  ,   2 , k  2 



Brillouin Zone
Bragg’s law : n  2d sin

In general, forbidden bands are
a

k
For  ~ 90° ( / 2),
n  2a

k
2


n
a


n
 kn n = 1, 2, 3, ...
d sin
Total electron energy
 reflection
Therefore, no travelling wave for
2

n = 1, 2, 3, ...
Allowed band
 Forbidden band

Allowed band :


a
k
Forbidden band

a
Allowed band
 1st Brillouin zone
Forbidden band
Allowed band


2
a

2nd

a
0
1st

a
2
a
2nd
k
Periodic Potential in a Crystal
E
Allowed band
Forbidden band
Allowed band

2
a


a
1st
2nd



a
0

2
a
Forbidden band
Allowed band
k
2nd

Energy band diagram
(reduced zone)
 extended zone
Brillouin Zone - Exercise
Brillouin zone : In a 3d k-space, area where k ≠ 0.
For a 2D square lattice,
2nd Brillouin zone is defined by
nx = ± 1 , ny = ± 1
ky
 ± kx ± ky = 2/a
a
kx
kx n x  ky n y 

a

nx 2  ny 2

nx, ny = 0, ± 1, ± 2, ...

Reciprocal lattice :
1st Brillouin zone is defined by
2
a
nx = 0, ny = ± 1  kx = ± /a
nx = ± 1, ny = 0  ky = ± /a
ky

a

0


Fourier transformation
= Wigner-Seitz cell


kx

a
2
a 2

a


a
0

a
2
a
3D Brillouin Zone
* C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, New York, 1986).
Fermi Energy
Fermi-Dirac distribution :
E
T≠0
T=0
EF
Pauli exclusion principle
At temperature T, probability that one energy state E is occupied by an electron :
f E 
1
expE   kBT 1
 : chemical potential
(= Fermi energy EF at T = 0)

kB : Boltzmann constant
f(E)
1
T=0
T1 ≠ 0
1/2
T2 > T1
0

E
Fermi-Dirac / Maxwell-Boltzmann Distribution
Electron number density :
Fermi sphere :
sphere with the radius kF
Fermi surface :
surface of the Fermi sphere
Decrease number density
classical
Maxwell-Boltzmann distribution
quantum mechanical
Fermi-Dirac distribution
(small electron number density) (large electron number density)
* M. Sakata, Solid State Physics (Baifukan, Tokyo, 1989).
Fermi velocity and Mean Free Path
Fermi wave number kF represents EF :
Fermi velocity : v F 
kF 
m
2EF
m
2mEF
vF 

Under an electrical field :
 Electrons, which can travel, has an energy of ~ EF with velocity of vF
For collision time , average length of electrons path without collision is
 v F
Mean free path
g(E)
Density of states :
 Number of quantum states at a certain
energy in a unit volume
gE  2
1
2 
3
32
4  2m 
EdE
 2 
2  
0
E
Density of States (DOS) and Fermi Distribution
Carrier number density n is defined as :
n
 f EgEdE
T=0
g(E)
f(E)

EF
0
E
T≠0
g(E)
f(E)
n(E)
0
EF
E

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