Transcript NEUTRON IMAGING

Nov. 1st, 2011
Multi-scale Heat Conduction
Quantum Size Effect
on the Specific Heat
Hong goo, Kim
1st year of M.S. course
Contents
I.
II.
III.
IV.
V.
VI.
Lattice Vibrational Waves
Lattice Specific Heat
Density of States
Thin Films
Nanocrystals
Carbon Nanostructures
Lattice Wave
 Lattice Waves
- Systemic motions of atoms in periodic lattice structure
- Periodicity is assumed → Fourier series of harmonic function
L
 Dispersion Relation : ω = ω(k)
- 1-to-1 correspondence between frequency and wavevector for
each polarization of lattice vibrational waves
- Can be derived from atomic force constant and lattice geometry
- Slope dω/dk = vg : Group velocity
Lattice Wave
 Harmonic Plane Wave
- Displacement of atoms
r ( x, t )  r0 exp(i )  r0 exp(i(k x x  t ))  r0 exp(ikx x) exp(it )
- Phase velocity vp
exp( i ),   const  d 
v p  (dx / dt) const   / kx


dx 
dt  k x dx  dt  0
x
t
 Fourier Series
- Superposition of harmonic waves
- Spatial period : L


2m
C
kx
kx
exp(ik x x) 
C
m  
m
exp(i
L
x)
Boundary Condition
 Harmonic Lattice Waves
- To determine the wavevectors of Fourier series, boundary
condition is required
exp(i kx x)
 Fixed B.C.
- End nodes(x = 0, L) are fixed → Standing wave solution
exp(ikx 0)  1, exp(ikx L)  1
 Periodic B.C. (Born−von Kármán)
- Simulates the physics of macroscopic periodicity better
than the fixed B.C.
exp(ikx x)  exp(ikx ( x  L))
Periodic B.C.
 Discretized Wavevector (1-D)
- Number of atoms = Nx + 1
- Odd Number of Nx = 2M + 1
k x  0,  (2 / L),  2(2 / L),  3(2 / L), ...,  (M 1)(2 / L),  M (2 / L)
±(Nx − 3)π/Lx
±(Nx − 1)π/Lx
- Even Number of Nx = 2M
k x  0,  (2 / L),  2(2 / L),  3(2 / L), ...,  (M 1)(2 / L), M (2 / L)
±(Nx − 2)π/Lx
±Nx π/Lx
 Independency of Wavevectors
- Number of independent wavevectors are restricted to Nx
- Upper limit can be defined : Cut-off wavenumber KD
Total Number of Modes
 Dependence of wavevectors
- Odd Number : Nx = 2M + 1
 2







M  1x   exp i 2 2M  1  M x   exp i 2 N x as exp  i 2 Mx 
exp i
 Lx

 Lx

 Lx

 Lx

2π/Lx
Nx modes
 2

(N − 3)π/L (N − 1)π/L
 exp  i
Mx 
 Lx

x
− (Nx − 1)π/Lx
x
x
x
(Nx + 1)π/Lx
- Even Number : Nx = 2M
 2

 2

 2

 2








 2M  M x   exp  i N x as exp i Mx 
exp  i
Mx   exp i
 Lx

 Lx

 Lx

 Lx

 2

 exp i
Mx 
 Lx

Total Number of Modes
 Example : Dependence of wavevectors (Nx = 10)
k1 = (Nx + 1)2π/Lx = 22π
k2 = 2π/Lx = 2π
k1 = (Nx + 1)2π/Lx = 22π
k2 = 2π/Lx = 2π
Lattice Vibrational Energy
 Energy for each wave mode (quantum state)
- Phonon energy levels are quantized by  P, K  P, K
- Dispersion relation assigns one quantum state(KP) to one
energy level( P, K ) for each polarization(P)
→ Degeneracy for energy level P, K is one ; gP, K = 1
- Phonons obey Bose-Einstein distribution
N P,K
g P, K
 N P,K
1
1
 f BE 

exp( P, K / k BT )  1 exp(P, K / k BT )  1
- Therefore, energy of each quantum state KP is
 P,K N P,K
1
1
1
  P , K  ( P , K ) f BE   P , K  ( f BE  ) P , K
2
2
2
Lattice Specific Heat
 Total Lattice Vibrational Energy
1
1
U (T )  U 0   ( P , K N P , K  P , K )  U 0   ( f BE  )P , K
2
2
P K
P K
 Lattice Specific Heat (Discrete)
exp(P , K / k BT )  P , K
dU
f BE
Cv (T ) 
  P , K
   P , K
2
2
dT

T
(exp(


/
k
T
)

1
)
k
T
P K
P K
P,K
B
B
 P , K
 k B  
P K  k BT
exp( P , K / k BT )


2
(exp(


/
k
T
)

1
)
1
P,K
B

2
 Lattice Specific2 Heat (Integral)

  P  exp( P / k BT )

Cv (T )  k B   
D( P )d P
P 0  k BT  exp( P / k BT )  1
Density of States
 Distribution of Quantum States
g
3
2
1
ω1 ω2
Di
ω1 ω2
ω3
ω 4 ω5 ω6 ω7
ω8
ω9
ω
gi  Di i ( i  i 1  i )
ω3
ω 4 ω5 ω6 ω7
ω8
ω9
ω
Density of States
 Density of States as a Continuous Function D(ω)
Di
D(ω)
ω1 ω2
ω3
gi  Di i
ω 4 ω5 ω6 ω7
ω8
ω9
dg  D( )d
ω
Density of States : 3-D
 3-D Case
kz
K3-space
2π/Lz
1 dg
D ( ) 
Spherical shell in K -space
V d
1 dg dVK dK Linear

dispersion relation
V dVK dK d ω/K = dω/dK = v
3
a
ky
2π/Lx
kx
2π/Ly
1

V
 Lx L y Lz 

 4K 2
 2 2 2 


 1

 va



Obtained from Periodic B.C
 
1 V

4  
3
V 8
 va 
2

3
2
2 va
2
 1

 va



Density of States : 2-D, 1-D
 2-D Case
 1-D Case
1 dg
D( ) 
A d
1 dg dAK dK

A dAK dK d
1 dg
D ( ) 
Lx d
1 dg dLK dK

Lx dLK dK d
1
1  Lx Ly 
2K  
 
A  2 2 
 va
   1
1 A

2  
2
A 4
 va  va


2
2va






1

Lx
 Lx

 2
1

va
  1
2
  va



LK = 2K
K1-space
K
kx
Thin Film : Concept
 Thin Film − Confined in 1-D
- Fabricated microstructure with thickness much smaller than
lateral dimenstions
- Application : thermal barrier, optical/electrical device
- Thickness : 1 nm ~ 100 μm
 Assumption (Example 5-4)
.
.
.
.
.
.
Monatomic solid thin silicon film
Confined in z-direction
Infinite in x-, y-direction
Film thickness L
Number of monatomic layers q ( = L / L0 )
Acoustic speed va : independent of T
Thin Film : Specific Heat
 Specific Heat of Thin Film
2
  
f BE
e  / k BT
  / k T
Cv (T )   
 3k B  
B
T
 1) 2
P kz k y kx
k z k y k x  k BT  ( e
 3k B  
kz
k D2  k z2

 k D2  k z2 
2
  
e  / k BT
 Lx   L y 


dky

dkx 
k D2  k y2  k z2  k T  (e  / k BT  1) 2 2
2


 

 B 
k D2  k y2  k z2
2
 / k BT
2
k D2  k z2   
 3 Lx L y 
e



 
k
2  B  0 0
 k T  (e  / k BT  1) 2 dd
4


 kz
 B 
 3A 
  2 k B  
 4  k z 0
k D2  k z2
2
2
xD
ex
1  k BT 
 3A 
2
  2 k B   x
2

 xdx
x
2
2 
xz
4

(
e

1
)
v


 kz

a 
2

kz
0  k  kD
 2  k 2  kz2  kx2  k y2
2
  
e  / k BT

  / k T
2d
2
B
k
T
(
e

1
)
 B 
va k D   
e  / k BT
1
 3A 
  / k T
  2 k B   
2

d
2
2
B
v
k
a
z
 1)
va
 4  k z
 k BT  ( e
 3 A   k BT 

k B 
2


  va  
k  k x2  k y2  k z2
xD
xz
ex
x x
dx
(e  1) 2
3
 2  va2k 2  va2 ( 2  kz2 )
2d  2va2d
k T

 x d  B dx

k BT
Thin Film : kD
 Debye Wavevector kD
- Upper limit of absolute value of wavevector which includes all
the vibrational modes in the 1st Brillouin zone
- Total number of modes = Total number of atoms N
- Number of modes in z-direction = Number of 1-atom layers q
Thin Film : kD
 Debye Wavevector kD : 3-D Bulk
kz
Vk 
kD
kx
2π/Lz
2π/Lx
4 3
kD
3
 Lx Ly Lz  4 3 V
 
N  Vk 
kD 3
8
 2 2 2  3
N 1
 3
V L0
3
2 N
2 1
k D  6
 6 3
V
L0
6 2
kD 
L0
3
Thin Film : kD
 Debye Wavevector kD : Thin Film
Lz  Lx , Ly k x , k y  k z
 Lx Ly 
A
2
2


N k z  Ak 
 k D  k z 

4
 2 2 
A
2
2
N   N k z   k D  k z 
4
kz
kz
kz

Ak   kD2  k z2

kD
kz
kx
N
N
1

 3
V qL0 A L0
N q
 2
A L0

1  4N
2
k  
  k z 
q A
kz

2
D
Δkz=2π/Lz
Δkx=2π /Lx
4 1
2
kD 

k
 z
L20 q k z
Thin Film : Quantum Size Effect
Specific heat, cv(T)
1
3k B  k BT 


cv (T )  Cv (T ) 
V
2qL0  va 
.... ....
q=1
q=2
2

kz
xD
xz
x 3e x
dx
x
2
(e  1)
Single layer
( ~T2 )
Bulk
( ~T3 )
q=7
q = 20
Temperature, T [K]
Thin Film : Quantum Size Effect
 T 2 Dependence of Specific Heat
3k B
cv (T ) 
2qL0
 k BT 


 va 
2

kz
xD
xz

0
3 x
xe
dx
x
2
(e  1)
x
x 3e x
dx
(e x  1) 2
x 3e x
(e x  1) 2
x
- Quantum size effect becomes more significant at
lower temperature and for smaller film thickness q
- Specific heat for thinner film increases due to q−1 dependence
and contribution of planar modes (kz = 0)
- Specific heat at lower temperature converges to zero slowly
due to T 2 instead of T 3 dependence
Nanocrystal
 Cubic Solid L3 (L = qL0)
6 2
kD 
L0
3
- Fraction of planar modes ~ q -1
3k D2 k 9k


3
4k D 3 4k D
V planar
Vk

9  2 qL0
4  6
3
2
3.627

q
L0
- Fraction of axial modes ~ q -2
Vaxial 3  2k D k 2 9k 2


3
Vk
4k D 3
2k D2

9  4 2 q 2 L20
 
2  6
2 2/3
11.69
 2
2
q
L0
K3-space
Δkz=2π /qL0
- Debye wavevector
Δkx=2π /qL0
kD
Nanocrystal
 Quantum Size Effect of Nanocrystals
- Quantum size effect of nanocrystal becomes significant as
size parameter q decreases
- At low temperature, planar mode ( ~T 2) contribution increases
kx = 0 or ky = 0 or kz = 0
- At lower temperature, axial mode ( ~T 1) contribution increases
kx = ky = 0 or ky = kz = 0 or kz = kx = 0
- Temperature dependence of specific heat (general form)
T3
T2
T1
cv (T )  a3 0  a2 1  a1 2
L
L
L
Nanocrystal
 Second Size Effect − Extremely Low T
- Only the lowest vibrational modes are excited
min k 
k  (
 pk x 2  qk y 2  rk z 2
 k 
2
L
 p,q,r : integer 
2
2
2
, 0, 0), (0, 
, 0), (0, 0, 
)
L
L
L
- Results in reduction of specific heat
2
  
e  / k BT
x 2e x
x2
  / k T
Cv (T )  3k B  
 18k B 2 x
 18k B x
2
x
B
 1)
e  2e  1
e  2  e x
k z k y k x  k BT  (e
18k B x 2
18k B x 2
a
 b
lim Cv (T )  lim x

lim

lim
exp
 
x
2
T 0
x  e  2  e  x
x 
T

0
e
T
 T
Converges to ‘0’ faster than T3
Nanocrystal
 Second Size Effect − Lead Grains
Departure from bulk
solid specific heat
R. Lautenschläger (1975)
Carbon : Graphite / Graphene
 Graphite
- Layers of hexagonal plane (graphene)
structure
- Covalent bond of neighboring atoms
within a layer
- Weakly bonded between layers: Van der Waals bond
- Lattice vibrational modes have 2-D characteristics
- T 2 dependence of specific heat (Debye Theory)
 Graphene
- T 1 dependence at low temperature due to dominant contribution
of out-of-plane(perpendicular) mode: ω ~ k2
→ Transition to T 2 dependence at higher temperature (2-D)
Carbon Nanotube
 Specific Heat
~T 2
1
- T dependence of specific
heat at low temperature
- Bounded between graphene
and graphite
 Twisting Mode
~T 1
- Rigid rotation around
nanotube axis
~T 2.3
- Coupling of in-plane and
out-of-plane modes due to
curvature by rolling up the
graphene sheet
M. S. Dresselhaus and P. C. Ecklund(2000)
Carbon Nanotube
 Coupling of In-Plane and Out-of-Plane Modes
M. S. Dresselhaus and P. C. Ecklund(2000)
Conclusion
 Lattice Vibrational Waves
- Spatial periodicity of lattice
- Superposition of harmonic waves
- Periodic boundary condition : Discretized wavevectors
 Density of States

D ( ) 
1 dg
V d
- Dimensionality of the crystal structure should be considered
2
  
e  / k BT

  / k T
Lattice Specific Heat Cv (T )  3k B 
B
 1) 2
k z k y k x  k BT  ( e
- Lattice vibrational energy : Summation of phonon energy over quantum states
 Quantum size effect in thin films
- Departure from bulk behavior at low temperature and thickness (T 2 dependency )
 Quantum size effect in nanocrystals
T3
T2
T1
cv (T )  a3 0  a2 1  a1 2
L
L
L
Thin Film : Quantum Size Effect
1
3k B  k BT 


cv (T )  Cv (T ) 
V
2qL0  va 
- T 2 dependence :
2

kz
xD
xz
x 3e x
dx
x
2
(e  1)
cv (T ) ~ T 2
- At low temperature :
- Specific heat
2
  
e  / k BT
  / k T
Cv (T )  3k B  
2
B
k
T
(
e

1
)
kz k y kx  B 