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

Transverse Wave

* http://www12.plala.or.jp/ksp/wave/waves/

Propagation direction

Amplitude

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