Multi-scale Heat Conduction

Quantum Size Effecton the Specific Heat

Hong goo, Kim1st year of M.S. course

Nov. 1st, 2011

Contents

I. Lattice Vibrational Waves

II. Lattice Specific Heat

III. Density of States

IV. Thin Films

V. Nanocrystals

VI. 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

- Slope dω/dk = vg : Group velocity

- Can be derived from atomic force constant and lattice geometry

Lattice Wave

m

mk

xk xL

miCxikC

x

x)

2exp()exp(

Fourier Series - Superposition of harmonic waves

- Spatial period : L

Harmonic Plane Wave

)exp()exp())(exp()exp(),( 000 tixikrtxkirirtxr xx

- Phase velocity vp 0 ),exp(

dtdxkdtt

dxx

dconsti x

xconstp kdtdxv /)/(

- Displacement of atoms

Boundary Condition

Fixed B.C. - End nodes(x = 0, L) are fixed → Standing wave solution

Periodic B.C. (Born−von Kármán)

- Simulates the physics of macroscopic periodicity better than the fixed B.C.

Harmonic Lattice Waves - To determine the wavevectors of Fourier series, boundary condi-

tion is required )exp( xi kx

1)exp( ,1)0exp( Likik xx

))(exp()exp( Lxikxik xx

Periodic B.C. Discretized Wavevector (1-D) - Number of atoms = Nx + 1

- Odd Number of Nx = 2M + 1 )/2( ,)/2)(1( ,... ,)/2(3 ,)/2(2 ,)/2( ,0 LMLMLLLkx

)/2( ,)/2)(1( ,... ,)/2(3 ,)/2(2 ,)/2( ,0 LMLMLLLkx

Independency of Wavevectors - Number of independent wavevectors are restricted to Nx

- Upper limit can be defined : Cut-off wavenumber KD

- Even Number of Nx = 2M ±(Nx − 3)π/Lx ±(Nx − 1)π/Lx

±(Nx − 2)π/Lx ±Nx π/Lx

Total Number of Modes Dependence of wavevectors - Odd Number : Nx = 2M + 1

- Even Number : Nx = 2M

MxL

i

MxL

iasNL

ixMML

ixML

i

x

xx

xxx

2exp

2exp

2exp12

2exp1

2exp

MxL

i

MxL

iasNL

ixMML

iMxL

i

x

xx

xxx

2exp

2exp

2exp2

2exp

2exp

− (Nx − 1)π/Lx (Nx + 1)π/Lx

2π/Lx (Nx − 1)π/Lx(Nx − 3)π/LxNx modes

k1 = (Nx + 1)2π/Lx = 22πk2 = 2π/Lx = 2π

Example : Dependence of wavevectors (Nx = 10)

Total Number of Modes

k2 = 2π/Lx = 2π

k1 = (Nx + 1)2π/Lx = 22π

Lattice Vibrational Energy Energy for each wave mode (quantum state) - Phonon energy levels are quantized by KPKP ,,

- Dispersion relation assigns one quantum state(KP) to one energy level( ) for each polarization(P) KP,→ Degeneracy for energy level is one ; gP, K = 1 KP,

- Phonons obey Bose-Einstein distribution

1) / exp(

1

1) / exp(

1

,,,

,

,

TkTkfN

g

N

BKPBKPBEKP

KP

KP

KPBEKPBEKPKPKPKP ffN ,,,,,, )2

1(

2

1)(

2

1

- Therefore, energy of each quantum state KP is

Lattice Specific Heat Total Lattice Vibrational Energy

P K

KPBEP K

KPKPKP fUNUTU ,0,,,0 )2

1()

2

1()(

Lattice Specific Heat (Discrete)

P K BKP

BKP

B

KPB

P B

KP

BKP

BKP

KKP

P K

BEKPv

Tk

Tk

Tkk

TkTk

Tk

T

f

dT

dUTC

1)1)/(exp(

)/exp(

)1)/(exp(

)/exp()(

2,

,

2

,

2,

2,

,,,

Lattice Specific Heat (Integral)

PPP

BP

BP

B

PBv dD

Tk

Tk

TkkTC

0

2

)(1)/exp(

)/exp()(

Density of States Distribution of Quantum States

3

2

1

ωω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8 ω9

g

iiiDg

ω

iD

ω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8 ω9

) ( 1 iii

Density of States

Density of States as a Continuous Function D(ω)

iD

dDdg )(

ωω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8 ω9

D(ω)

iiiDg

3-D Case

Density of States : 3-D

ky

kx

kz

2π/Ly

2π/Lx

2π/Lz

32

2

2

3

2

2

14

8

1

1 4

222

1

1

1)(

a

aa

a

zyx

K

K

v

vv

V

V

vK

LLL

V

d

dK

dK

dV

dV

dg

V

d

dg

VD

K3-space Spherical shell in K3-space

Linear dispersion relationω/K = dω/dK = va

Obtained from Periodic B.C

Density of States : 2-D, 1-D

2-D Case

2

2

2

12

4

1

1 2

22

1

1

1)(

a

aa

a

yx

K

K

v

vv

A

A

vK

LL

A

d

dK

dK

dA

dA

dg

A

d

dg

AD

1-D Case

a

a

x

x

K

Kx

x

v

v

L

L

d

dK

dK

dL

dL

dg

L

d

dg

LD

1

1 2

2

1

1

1)(

K

LK = 2K

K1-space

kx

Thin Film : Concept

Thin Film − Confined in 1-D

. 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

- Thickness : 1 nm ~ 100 μm

- Application : thermal barrier, optical/electrical device

Assumption (Example 5-4)

- Fabricated microstructure with thickness much smaller than lateral dimenstions

Thin Film : Specific Heat Specific Heat of Thin Film

z

D

z

z

D

z

z

Da

za B

B

z

zD

B

B

z

zD

B

B

z

zD

zD

zyD

zyD B

B

z y x

B

B

z y x

k

x

x x

x

a

BB

k

x

x

B

ax

x

B

k

kv

kva

Tk

Tk

BB

k

kk

Tk

Tk

BB

k

kk

Tk

Tk

BB

yx

k

kk

kk y

kkk

kkk

yx

xTk

Tk

BB

k k kTk

Tk

BB

BE

P k k kv

dxe

ex

v

Tkk

A

xdxTk

ve

exk

A

dve

e

Tkk

A

de

e

Tkk

A

dde

e

Tkk

LL

dkL

dkL

e

e

Tkk

e

e

Tkk

T

fTC

23

2

2

222

2

22/

/2

2

0 2/

/2

2

2

0 0 2/

/2

2

2/

/2

2/

/2

)1(2

3

12

)1(4

3

12

)1(4

3

2)1(4

3

)1(4

3

22)1(3

)1(3)(

22

22

22

22

222

222

22222yxz kkkk

dvd a222

)( 222222zaa kvkv

xTkB

dxTk

d B

Dkk 0

222zyx kkkk

Thin Film : kD

Debye Wavevector kD

- Upper limit of absolute value of wavevector which includes all the vibrational modes in the 1st Brillouin zone

- Number of modes in z-direction = Number of 1-atom layers q

- Total number of modes = Total number of atoms N

Thin Film : kD

Debye Wavevector kD : 3-D Bulk

kx

kz

kD

2π/Lx

2π/Lz

3

3

4Dk kV

33

83

4

222

V

kLLL

VN Dzyx

k

30

1

LV

N

30

223 166

LV

NkD

0

3 26

LkD

Thin Film : kD

Debye Wavevector kD : Thin Film

kx

kz

kD

Δkx=2π /Lx

Δkz=2π/Lz

yxz LLL , zyx kkk ,

422

22 Akk

LLAN zD

yxkkz

zz

zk

zDk

k

AkkNN

422

z

z

kzD

kzD

kqL

k

kA

N

qk

220

22

14

41

300

1

LAqL

N

V

N

20L

q

A

N

kz

22zDk kkA

Thin Film : Quantum Size Effect

z

D

zk

x

x x

x

a

BBvv dx

e

ex

v

Tk

qL

kTC

VTc

2

32

0 )1(

2

3)(

1)(

Spec

ific

hea

t,

c v(T

)

Temperature, T [K]

Bulk

( ~T3 )

Single layer ( ~T2 )

q = 1q = 2

q = 20

q = 7

....

....

x

x

x

dxe

ex0 2

3

)1(

2

3

)1( x

x

e

ex

x

Thin Film : Quantum Size Effect

- Quantum size effect becomes more significant at lower temperature and for smaller film thickness q

T 2 Dependence of Specific Heat

- Specific heat for thinner film increases due to q−1 dependence and contribution of planar modes (kz = 0)

z

D

zk

x

x x

x

a

BBv dx

e

ex

v

Tk

qL

kTc

2

32

0 )1(

2

3)(

- Specific heat at lower temperature converges to zero slowly due to T 2 instead of T 3 dependence

Nanocrystal Cubic Solid L3 (L = qL0)

0

3 26

LkD

- Debye wavevector

- Fraction of planar modes ~ q -1

kD

Δk z=

2π /q

L0

Δkx=2π /qL0

qL

qL

k

k

k

kk

V

V

DD

D

k

planar

627.3

64

29

4

9

34

3

03 2

0

3

2

- Fraction of axial modes ~ q -2

220

3/22

20

22

2

2

3

2

69.11

62

49

2

9

34

23

qL

Lq

k

k

k

kk

V

V

DD

D

k

axial

K3-space

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

- At lower temperature, axial mode ( ~T 1) contribution increases

kx = 0 or ky = 0 or kz = 0

kx = ky = 0 or ky = kz = 0 or kz = kx = 0

- Temperature dependence of specific heat (general form)

2

1

11

2

20

3

3)(L

Ta

L

Ta

L

TaTcv

Nanocrystal Second Size Effect − Extremely Low T

- Only the lowest vibrational modes are excited

- Results in reduction of specific heat

)2

,0 ,0( ),0 ,2

,0( ),0 ,0 ,2

(LLL

k

xxBxx

x

Bk k k

Tk

Tk

BBv ee

xk

ee

exk

e

e

TkkTC

z y x

B

B

2

1812

18)1(

3)(2

2

2

2/

/2

integer : 2

min 222 p,q,r L

kkrkqkp zyx

k

T

b

T

a

e

xk

ee

xkTC

TxB

xxxB

xv

Texplim

18lim

2

18lim)(lim

20

22

0

Converges to ‘0’ faster than T3

R. Lautenschläger (1975)

Nanocrystal Second Size Effect − Lead Grains

Departure from bulk solid specific heat

Carbon : Graphite / Graphene Graphite - Layers of hexagonal plane (graphene)

structure

- Weakly bonded between layers: Van der Waals bond

- Covalent bond of neighboring atoms within a layer

- 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 1 dependence of specific

heat at low temperature

Twisting Mode - Rigid rotation around nan-

otube axis

- 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)

~T 1

~T 2

- Bounded between graphene and graphite

~T 2.3

Carbon Nanotube Coupling of In-Plane and Out-of-Plane Modes

M. S. Dresselhaus and P. C. Ecklund(2000)

Conclusion Lattice Vibrational Waves

Quantum size effect in nanocrystals

2

1

11

2

20

3

3)(L

Ta

L

Ta

L

TaTcv

Density of States

d

dg

VD

1)(

Quantum size effect in thin films

- Spatial periodicity of lattice

- Superposition of harmonic waves

- Periodic boundary condition : Discretized wavevectors

- Dimensionality of the crystal structure should be considered

- Departure from bulk behavior at low temperature and thickness (T 2 dependency )

Lattice Specific Heat

z y x

B

B

k k kTk

Tk

BBv e

e

TkkTC

2/

/2

)1(3)(

- Lattice vibrational energy : Summation of phonon energy over quantum states

Thin Film : Quantum Size Effect

z

D

zk

x

x x

x

a

BBvv dx

e

ex

v

Tk

qL

kTC

VTc

2

32

0 )1(

2

3)(

1)(

- T 2 dependence : 2~)( TTcv

- At low temperature :

z y x

B

B

k k kTk

Tk

BBv e

e

TkkTC

2/

/2

)1(3)(

- Specific heat