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Introductory Nanotechnology
~ Basic Condensed Matter Physics ~
Atsufumi Hirohata
Department of Electronics
Quick Review over the Last Lecture
Wave / particle duality of an electron :
Particle nature
Wave nature
mv 2 2
mv
h  
h  k
Kinetic energy
Momentum
Brillouin zone (1st & 2nd) :
Fermi-Dirac distribution (T-dependence) :



k
y
f(E)
2
a
1
T2

a

1/2
0





T1
kx
T3

0
a
2
a 2

a




a
0
a


2
a

T1 < T2 < T3
E
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)
How Does an Electron Travel
in a Material ?
Group / phase velocity
•
Effective mass
•
•
Harmonic oscillator
•
•
Hall effect
Longitudinal / transverse waves
•
Acoustic / optical modes
•
Photon / phonon
How Fast a Free Electron Can Travel ?
Electron wave under a uniform E :
Phase-travel speed in an electron wave :
-
-
Wave packet
-
-
Phase velocity :
Group velocity :
d
vg 
dk
vp 
Here, energy of an electron wave is
E  h  


Accordingly,
vg 

1 dE
dk
Therefore, electron wave velocity depends on
gradient of energy curve E(k).


k
Equation of Motion for an Electron with k
For an electron wave travelling along E :
1 d dE  1 d dE dk 1 d 2 E dk

 
  
dt
dt dk 
dk dk  dt
dk 2 dt
dv g
Under E, an electron is accelerated by a force of -qE.

In t, an electron travels vgt, and hence E applies work of (-qE)(vgt).
Therefore, energy increase E is written by
E  qEvg t  qE
1 dE
t
dk
At the same time E is defined to be

E 
dE
k
dk
From these equations,

1
k   qEt

dk
1
  qE
dt

dk
 qE
dt
 Equation of motion for an electron with k



Effective Mass
dv g 1 d 2 E dk
dk
 qE into

By substituting
dt
dt
dk 2 dt
dv g
1 d 2E
 2
qE
2
dt
dk
 comparing withacceleration for a free electron :
By
dv
1
  qE

dt
m
d 2 E 
*
2
m 
 2 
dk 

 Effective mass

Hall Effect
Under an applications of both a electrical current i an magnetic field B :
z
F  qE  v  B
----
x
V Hy  
i
V Hy 
-
y

++++
E
z

F  qE  v  B
++++
x
hole
+
----

E
B
1 i x Bz
qn l
l
B
y
i
1 i x Bz
qn l
l

Hall coefficient
Harmonic Oscillator
Lattice vibration in a crystal :
spring constant : k
mass : M
u
Hooke’s law :
d 2u
M 2  kx
dt
Here, we define



d 2u
 2   2 u

dt
ut   A sint  
k
M
 1D harmonic oscillation

Strain
Displacement per unit length :
u xdx u



dx
x
Young’s law (stress = Young’s modulus  strain) :

F
u
 EY
S
x
S : area




x + dx
x
Here,
Fx  dx  Fx  F xdx 
u (x)
For density of ,
 2 u F
Sdx 2 
dx
x
t
 2 u EY  2 u
 2u
 2 
v
2

t
x
x 2
 Wave eqution in an elastomer
Therefore, velocity of a strain wave (acoustic velocity) :
v 
EY

u (x + dx)
Longitudinal / Transverse Waves
Longitudinal wave : vibrations along their direction of travel
Transverse wave : vibrations perpendicular to their direction of travel
Transverse Wave
Amplitude
Propagation direction
* http://www12.plala.or.jp/ksp/wave/waves/
Longitudinal Wave
sparse
dense
sparse
Propagation direction
sparse
dense
Amplitude
sparse
* http://www12.plala.or.jp/ksp/wave/waves/
Acoustic / Optical Modes
For a crystal consisting of 2 elements (k : spring constant between atoms) :
 d 2 u
n
 kv n  u n   v n1  u n 
M
2

dt

 d 2 v n
m
 ku n 1  v n   u n  v n 
2

 dt
un-1

M 2 A  kB1 exp iqa 2kA


2

m B  kAexp iqa 1 2kB

 2k  M 2 A  k 1 exp iqa B  0

 

 
kexp iqa  1A  2k  m 2 B  0



2k  M 2
kexp iqa 1


k1 exp iqa
2k  m
2
0
vn
un+1
vn+1
m
a

u n na, t   A exp i t  qna


v n na,t   B exp i t  qna

un
M
By assuming,

vn-1
x
Acoustic / Optical Modes - Cont'd
 1 1 
 1 1 2
4
qa
sin 2
Therefore,   k   k    
M m 
M m  Mm
2
2
For qa = 0,

 1 1 



2k
  
  
M m 

  0
For qa ~ 0,


 1 1 
   2k   

M m 


k 2


qa
 

M m
For qa = ,


2k
  

m

2k






M
Optical vibration
 1
1 
2k   
M m 

2k
m
2k
M


Acoustic vibration
* N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson Learning, London, 1976).
Why Acoustic / Optical Modes ?
Oscillation amplitude ratio between M and m (A / B):
Optical mode :
m
M
Neighbouring atoms changes their position in opposite directions,
of which amplitude is larger for m and smaller for M,
however,
the cetre of gravity stays in the same position.

Acoustic mode : 1
All the atoms move in parallel.
k =0
small k
large k
Optical
Acoustic
* M. Sakata, Solid State Physics (Baifukan, Tokyo, 1989).
Photon / Phonon
Quantum hypothesis by M. Planck (black-body radiation) :
E
1
h  nh
2
n  0,1,2, 
Here, h : energy quantum (photon)

mass : 0, spin : 1
Similarly, for an elastic wave, quasi-particle (phonon) has been introduced
by P. J. W. Debye.
E
1
n 
2
n  0,1,2, 
Oscillation amplitude : larger  number of phonons : larger

What is a conductor ?
Number of electron states (including spins) in the 1st Brillouin zone :
 a


a
2
L
2L
dk 
 2N
2
a

L 
N




a 
E
Here, N : Number of atoms for a monovalent metal

As there are N electrons, they fill half of the states.
By applying an electrical field E, the occupied states

become asymmetric.
• E increases asymmetry.

0
a

a
k
1st
 E

• Elastic scattering with phonon / non-elastic
Forbidden
E
scattering decreases asymmetry.
 Stable asymmetry
Allowed
 Constant current flow

 Conductor :

Only bottom of the band is filled by electrons
with unoccupied upper band.

0
a
1st


a
k