1. Introductory Nanotechnology ~ Basic Condensed Matter Physics ~
Atsufumi Hirohata
Department of Electronics

2. Quick Review over the Last Lecture
3 states of matters :
solid
liquid
gas
density
( large )
( large )
( small )
ordering range
( long )
( short )
rigid time scale
( long )
( short )
4 major crystals :
soft
solid
( van der Waals crystal )
( metallic crystal )
( covalent crystal )
( ionic crystal )

3. 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)

4. What Is the Most Common Atom
on the earth?
Phase diagram
Free electron model
Electron transport
Degeneracy
Electron Potential
Brillouin Zone
Fermi Distribution

5. Abundance of Elements in the EarthCa Al Na S Ni
Mg (12.70 %)
Fe (34.63 %)
Si (15.20 %)
O (29.53 %)
Mantle :
Fe, Si, Mg, ...
Only surface 10 miles (Clarke number)

6. Can We Find So Much Fe around Us ?
Desert (Si + Fe2O3)
Iron sand (Fe3O4)
Iron ore (Fe2O3)

7. "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.

8. Major Phases of Fe
Fe changes the crystalline structures with temperature / pressure :
56 Fe :
Most stable atoms in the universe.
T [K]
Liquid-Fe
1808
-Fe
bcc
1665
Phase change
-Fe (austenite)
fcc
1184
Martensite Transformation :
’-Fe (martensite)
(-Fe)
1043
-Fe
-Fe (ferrite)
bcc
hcp 1
p [hPa]

9. Electron Transport in a Metal
Free electrons : +
　　-
　　-
r = 0.95 Å Na
3.71 Å
r ~ 2 Å
Na bcc :
a = 4.28 Å
3s electrons move freely
among Na+ atoms.
Uncertainty principle :
(x : position and p : momentum)
When a 3s electron is confined in one Na atom (x  r) :
(m : electron mass)
For a free electron, r  large and hence E  small.
Effective energy for electrons
Inside a metal
decrease in E
free electron
　　-

10. Free Electron Model - + - - - - - - - - - Equilibrium state :
For each free electron :
thermal velocity (after collision) : v0i
acceleration by E for i
　　- +
v0i
　　-
　　- i
　　- + -
　　-
　　-
　　-
　　-
　　-
　　- i
Average over free electrons :
collision time : 
Free electrons :
mass m, charge -q and velocity vi
Equation of motion along E :
Using a number density of electrons n,
current density J :
Average over free electrons :
drift velocity : vd

11. Free Electron Model and Ohm’s Law
For a small area :  
where  : electric resistivity (electric conductivity :  = 1 / )
By comparing with the free electron model :

12. Relaxation Time + - Resistive force by collision :
If E is removed in the equation of motion : v 
　　-
For the initial condition :
v = vd at t = 0
Equation of motion :
with resistive force mv / 
Number of
non-collided
Electrons N
time 
For the initial condition :
v = 0 at t = 0 N0
For a steady state (t >> ),
 : relaxation time
 : collision time

13. Mobility Equation of motion under E : Also, For an electron at r,
collision at t = 0 and r = r0 with v0
Accordingly,
Here, ergodic assumption :
temporal mean = ensemble mean
therefore, by taking an average
over non-collided short period,
Finally, vd = -E is obtained.
 = q / m : mobility
Since t and v0 are independent,
Here, = 0, as v0 is random.

14. 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

15. 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

16. Electron Potential Energy
Potential energy of an isolated atom (e.g., Na) : Na
For an electron is released from the atom :
vacuum level
Distance from the atomic nucleus
Electron potential energy : V = -A / r 3s 2p 2s 1s
Na 11+

17. Periodic Potential in a Crystal
Potential energy in a crystal (e.g., N Na atoms) :
Vacuum level
Distance 3s 3s 2p 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

18. Free Electrons in a Solid
Free electrons in a crystal :
Total electron energy
Wave number k
Total electron energy
Energy band
Wave / particle duality of an electron :
Wave nature of electrons was predicted by de Broglie,
and proved by Davisson and Germer.
electron beam
Ni crystal
Particle nature
Wave nature
Kinetic energy
Momentum
(h : Planck’s constant)

19. Brillouin Zone Bragg’s law :  In general, forbidden bands are a
Total electron energy k
 reflection
Therefore, no travelling wave for
n = 1, 2, 3, ...
Allowed band
 Forbidden band
Allowed band :
Forbidden band
 1st Brillouin zone
Allowed band
Forbidden band
Allowed band
2nd
1st
2nd

20. Periodic Potential in a Crystal
Allowed band
Forbidden band
Allowed band
Forbidden band
Allowed band k
2nd
1st
2nd
Energy band diagram
(reduced zone)
 extended zone

21. 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
 ± kx ± ky = 2/a a kx ky
nx, ny = 0, ± 1, ± 2, ...
Reciprocal lattice : kx ky
1st Brillouin zone is defined by
nx = 0, ny = ± 1  kx = ± /a
nx = ± 1, ny = 0  ky = ± /a
Fourier transformation
= Wigner-Seitz cell

22. 3D Brillouin Zone
* C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, New York, 1986).

23. 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) E
T = 0 1
 : chemical potential
(= Fermi energy EF at T = 0)
kB : Boltzmann constant
T1 ≠ 0
1/2
T2 > T1 

24. 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
quantum mechanical
Maxwell-Boltzmann distribution
(small electron number density)
Fermi-Dirac distribution
(large electron number density)
* M. Sakata, Solid State Physics (Baifukan, Tokyo, 1989).

25. Fermi velocity and Mean Free Path
Fermi wave number kF represents EF :
Fermi velocity :
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
Mean free path
g(E) E
Density of states :
Number of quantum states at a certain
energy in a unit volume

26. Density of States (DOS) and Fermi Distribution
Carrier number density n is defined as : E
T = 0
f(E)
g(E) EF E
T ≠ 0
f(E)
g(E)
n(E) EF