department of electronics
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Introductory Nanotechnology ~ Basic Condensed Matter Physics ~. Atsufumi Hirohata. Department of Electronics. Quick Review over the Last Lecture. k y. f(E ). 0. k x. 0. E. 0. Wave / particle duality of an electron :. Fermi-Dirac distribution ( T -dependence) :. - PowerPoint PPT PresentationTRANSCRIPT
Department of Electronics
Introductory Nanotechnology
~ Basic Condensed Matter Physics
~
Atsufumi Hirohata
Quick Review over the Last Lecture
Wave / particle duality of an electron :
Brillouin zone (1st & 2nd) :
Particle nature
Wave nature
Kinetic energy
Momentum€
mv2 2
€
mv
€
hν = hω
€
h λ = hk
kx
ky
0
€
−πa
€
−2π
a
€
πa
€
2π
a
€
2π
a
€
πa
0
€
−πa
€
−2π
a
1
1/2
f(E)
E0
T1
T2
T3
Fermi-Dirac distribution (T-dependence) :
T1 < T2 < T3
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 ?
Second half of the course : Introduction to nanotechnology (nano-fabrication / application)
7. Why does a magnet attract / retract ?
8. What happens at interfaces ?
How Does an Electron Travel in a Material ?
• Group / phase velocity
• Effective mass
• Hall effect
• Harmonic oscillator
• Longitudinal / transverse waves
• Acoustic / optical modes
• Photon / phonon
How Fast a Free Electron Can Travel ?
Electron wave under a uniform E :
€
vg =dω
dk
- -
- -
Wave packet
Group velocity :
Here, energy of an electron wave is
€
E = hν = hωAccordingly,
€
vg =1
h
dE
dkTherefore, electron wave velocity depends on
gradient of energy curve E(k).
-
Phase-travel speed in an electron wave :
€
vp =ω
k
Phase velocity :
Equation of Motion for an Electron with k
For an electron wave travelling along E :
€
dvg
dt=
1
h
d
dt
dE
dk
⎛
⎝ ⎜
⎞
⎠ ⎟=
1
h
d
dk
dE
dk
⎛
⎝ ⎜
⎞
⎠ ⎟dk
dt=
1
h
d 2E
dk 2
dk
dt
Under E, an electron is accelerated by a force of -qE.
In t, an electron travels vgt, and hence E applies work of (-qE)(vgt).
Therefore, energy increase E is written by
€
E = −qEvgΔt = −qE1
h
dE
dkΔt
At the same time E is defined to be
€
E =dE
dkΔk
From these equations,
€
k = −1
hqEΔt
€
∴dk
dt= −
1
hqE
€
∴hdk
dt= −qE
Equation of motion for an electron with k
Effective Mass
By substituting into
€
dvg
dt=
1
h
d 2E
dk 2
dk
dt
€
hdk
dt= −qE
€
dvg
dt= −
1
h2
d 2E
dk 2qE
By comparing with acceleration for a free electron :
€
dv
dt= −
1
mqE
€
m* = h2 d 2E
dk 2
⎛
⎝ ⎜
⎞
⎠ ⎟
Effective mass
x
y
z
B
i
l
Hall Effect
Under an applications of both a electrical current i an magnetic field B :
x
y
z
B
E
i -
- - - -
++++€
F = −q E + v ×B[ ]
€
VHy = −1
qn
ixBz
l
l
E
+
++ +
- - - -€
F = q E + v ×B[ ]
€
VHy =1
qn
ixBz
l+
hole
Hall coefficient
Harmonic Oscillator
Lattice vibration in a crystal :
Hooke’s law :
€
Md 2u
dt 2= −kx
spring constant : k
u
mass : M
Here, we define
€
ω =k
M
€
∴d 2u
dt 2= −ω2u
€
∴u t( ) = A sin ωt +α( )
1D harmonic oscillation
Strain
Displacement per unit length :
Young’s law (stress = Young’s modulus strain) :
€
F
S= EY
∂u
∂x€
δ =∂u ∂x( )dx
dx=
∂u
∂x
S : area
x x + dx
u (x) u (x + dx)Here,
€
+F x + dx( ) = F x( ) + ∂F ∂x( )dx +L
For density of ,
€
Sdx∂ 2u
∂t 2=
∂F
∂xdx
€
∴∂2u
∂t 2=EY
ρ
∂ 2u
∂x 2≡ vl
∂ 2u
∂x 2
Wave eqution in an elastomer
Therefore, velocity of a strain wave (acoustic velocity) :
€
vl ≡EY
ρ
Longitudinal / Transverse Waves
Longitudinal wave : vibrations along their direction of travel
Transverse wave : vibrations perpendicular to their direction of travel
Longitudinal Wave
* http://www12.plala.or.jp/ksp/wave/waves/
Propagation direction Amplitude
sparsedense
sparse sparsedense
sparse
Acoustic / Optical Modes
For a crystal consisting of 2 elements (k : spring constant between atoms) :
By assuming,
€
Md 2un
dt 2= k vn − un( ) + vn−1 − un( ){ }
md 2vn
dt 2= k un +1 − vn( ) + un − vn( ){ }
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
mM
vnun vn+1un+1
a x
vn-1un-1
€
un na, t( ) = A exp i ωt − qna( ){ }
vn na, t( ) = B exp i ωt − qna( ){ }
⎧ ⎨ ⎪
⎩ ⎪
€
−Mω2A = kB 1+ exp iqa( ){ } − 2kA
−mω2B = kA exp −iqa( ) +1{ } − 2kB
⎧ ⎨ ⎪
⎩ ⎪
€
∴2k − Mω2
( )A − k 1+ exp iqa( ){ }B = 0
−k exp −iqa( ) +1{ }A + 2k − mω2( )B = 0
⎧
⎨ ⎪
⎩ ⎪
€
∴2k − Mω2 −k 1+ exp iqa( ){ }
−k exp −iqa( ) +1{ } 2k − mω2 = 0
Acoustic / Optical Modes - Cont'd
Therefore,
For qa = 0,
€
ω 2 = k1
M+
1
m
⎛
⎝ ⎜
⎞
⎠ ⎟± k
1
M+
1
m
⎛
⎝ ⎜
⎞
⎠ ⎟2
−4
Mmsin2 qa
2
€
ω+ = 2k1
M+
1
m
⎛
⎝ ⎜
⎞
⎠ ⎟
ω− = 0
⎧
⎨ ⎪
⎩ ⎪
For qa ~ 0,
€
ω+ ≈ 2k1
M+
1
m
⎛
⎝ ⎜
⎞
⎠ ⎟
ω− ≈k 2
M + mqa
⎧
⎨
⎪ ⎪
⎩
⎪ ⎪
For qa = π,
€
ω+ =2k
m
ω− =2k
M
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
* N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson Learning, London, 1976).
€
2k1
M+
1
m
⎛
⎝ ⎜
⎞
⎠ ⎟
€
2k
m
€
2k
M
Optical vibration
Acoustic vibration
Why Acoustic / Optical Modes ?
Oscillation amplitude ratio between M and m (A / B):
Optical mode :
€
m
MNeighbouring 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.
* M. Sakata, Solid State Physics (Baifukan, Tokyo, 1989).
Optical
Acoustic
k =0 small k large k
Photon / Phonon
Quantum hypothesis by M. Planck (black-body radiation) :
€
E =1
2hν + nhν n = 0,1,2,K( )
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
2hω + nhω n = 0,1,2,K( )
Oscillation amplitude : larger number of phonons : larger
What is a conductor ?
Number of electron states (including spins) in the 1st Brillouin zone :
€
2L
2π−π a
π a
∫ dk =2L
a= 2N N =
L
a
⎛
⎝ ⎜
⎞
⎠ ⎟
Here, N : Number of atoms for a monovalent metal
As there are N electrons, they fill half of the states.
E
k0
€
−πa
€
πa
1st
By applying an electrical field E, the occupied states
become asymmetric.
E
k0
€
−πa
€
πa
1st
E
• E increases asymmetry.
• Elastic scattering with phonon / non-elastic
scattering decreases asymmetry.
Stable asymmetry
Constant current flow
Conductor :
Only bottom of the band is filled by electrons
with unoccupied upper band.
Forbidden
Allowed