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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Transcript 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.
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 :
xp
(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
E1 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 expt
Number of N
0
non-collided
Electrons
N t N 0 expt /
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 qE 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
expE 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
gE 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 EgEdE
T=0
g(E)
f(E)
EF
0
E
T≠0
g(E)
f(E)
n(E)
0
EF
E