Computational Solid State Physics

計算物性学特論　第２回

2.Interaction between atoms and the lattice properties of crystals

Atomic interaction

Lennard-Jones potential: for inert gas atoms: He, Ne, Ar, Kr, Xe

Stillinger- Weber potential: for covalent bonding atoms: C, Si, Ge

Lennard-Jones potential (1)

612

4)(rr

rVLJ

r / σ

VLJ/ε

VLJ(r) minimum at

12.126

1

0 rr

)( 0rVLJ

r: inter-atomic distance

repulsive force

attractive force

612

4)(rr

rVLJ

1st term: repulsive interaction caused by Pauli’s principle

2nd term: Van der Waals interaction (attractive)

31

12 )()(r

prErp

6321 1

)(rr

pprVVW

E1: electric field generated by a temporal dipole moment p1

p1

r

p2 (r)

temporal dipole moment

induced dipole moment

Lennard-Jones potential (2)

1-dimensional crystal

612

612

02.100.14

422

1

aa

ananE

n

a xa: lattice constant

04.1)( aE

Energy per atom:

cohesive energy εc=1.04ε

E minimum at a=1.12σ

Bulk modulus

9.669.742

2

2

2

aE

da

daB

NaL

ENdL

dLB

B : Bulk modulusN : the number of atoms in a crystala : lattice constant

Lattice vibration

222

2

22

2

4.5772

2

1)()(

aV

dx

d

xVdx

daVxaV

axLJ

axLJLJLJ

a

xndisplacement

x

Na: length of a crystal

•The first derivative of the inter-atomic potential vanishes because 　　　　　　　　 atoms are located at the equilibrium positions.

•The second derivative of the inter-atomic potential gives the 　spring constant κ between atoms.

assume: neglect the 2nd neighbor interaction

12.1a

Equation of motion for atoms

)()( 112

2

nnnnn xxxx

dt

xdm Nn 1

)()(

)()(

11

11

nnnn

n

nnLJ

n

nnLJn

xxxx

x

xxaV

x

xxaVF

m: mass of an atom

a

xn-1 xn xn+1

Force on the n-th atom:

Equation of motion for atoms:

Solution for equation of motion

2sin4 22

02 ka

Periodic boundary condition:

N

l

ak

lkaN

2

2

)](exp[ tkanixn Assume: )2

()(a

kxkx nn

ak

a

1st Brillouin zone

22

Nl

N N modes

Nnn xx k: wave vector

)()( 12

012

02

2

nnnnn xxxx

dt

xd m

20

Dispersion relation of lattice vibration

2sin2)( 0

kak

ka

ω(k)/ω0 sound velocity: phase velocity at k=0

maa

k

kwv

k

00

)(

acoustic mode

v becomes larger for larger κ and smaller m.

Phonon

)2

1)(()( lkkEl

1)/)(exp(

1))((

Tkkkn

B

Energy quantization of lattice vibration

l=0,1,2,3

Bose distribution function for phonon number:

)())((

k

Tkkn B

TkBfor

2

)()(0

kkE

:zero point oscillation

Role of the acoustic phonon in semiconductors at a room temperature

Main electron scattering mechanism in crystals

Determine the lattice heat capacity Determine the thermal conductivity

Lattice heat capacity: Debye model (1)

32

2

32

3

3

333

3

2)(

63

4

23

4

2

v

V

d

dND

v

V

v

Lk

LN

k

k

3

126

V

N

k

v

k

B

BD

Density of states of acoustic phonos for 1 polarization

Debye temperature θ

32

3

6 v

VN D

Lk

2

N: number of unit cell

Nk: Allowed number of k points in a sphere with a radius k

vkk )( phonon dispersion relation

k

Thermal energy U and lattice heat capacity CV : Debye model (2)

D

D

D

x

x

x

BV

B

B

BV

V

B

e

exdx

TNkC

Tk

Tkd

Tkv

V

T

UC

Tkv

VdnDdU

02

43

02

4

232

2

032

2

)1(9

]1)/[exp(

)/exp(

2

3

1)/exp(23)()(3

3 polarizations for acoustic modes

Dx

x

x

BV e

exdx

TNkC

02

43

)1(9

34

5

12

TNkC BV

BV NkC 3

・ Low temperature T<<θ

・ High temperature T>>θ Equipartition law:

energy per 1 freedom is kBT/2

Debye model (3)

Heat capacity CV of the Debye approximation: Debye model (4)

kB=1.38x10-23JK-1

kBmol=7.70JK-1

3kBmol=23.1JK-1

Heat capacity of Si, Ge and solid Ar: Debye model (5)

cal/mol K=4.185J/mol K

3kB mol=5.52cal K-13TCV

Si and Ge Solid Ar

Thermal conductivity (1)

dx

dTvncj

dx

dTvncv

dx

dTcnvj

E

xxxE

3

2

2

T: temperature

c: heat capacity per particle

n: average number of phonons

v: group velocity of phonon

τ: scattering time

Diffusive energy flux

x

3kBT(x)

vxτ

c vxτdT/dx

Energy

Energy emission

Thermal conductivity (2)

dx

dTKjE

333

22 CvlCvvncK

K is largest for diamond because of the high sound velocity!

C: heat capacity per unit volume,

l=vτ: phonon mean free path

v: sound velocity of acoustic phonon

Thermal conductivity coefficient

Molecular dynamics simulation for atoms

vdt

dr

m

Fa

dt

dv

Equation of motion for atoms:

r: position of an atom

v: velocity

a: acceleration

F: force

t: time

m: mass of an atom 　

(1) velocity Verlet’s method

)()]()([2

1)()(

)()(2

1)()()(

3

32

tOttatattvttv

tOtatttvtrttr

Time evolution for small time interval :t

Proof of (1)

)()]()([2

1)()(

)()()(

2

1

2

1

2

1

)(2

1)()(

3

32222

2

322

2

tOtttatatvttv

tOtt

tattat

dt

dat

dt

vd

tOtdt

vdt

dt

dvtvttv

(2) Verlet method

))(()()(2)1()1(

))(()(6

1)(

2

1)()()1(

))(())(()1()(2)1(

422

2

433

32

2

2

42,

tOtdt

xdnxnxnx

tOtdt

xdt

dt

xdt

dt

dxnxnx

tOtnanxnxnx

iiii

iiiii

xiiii

))((2

)1()1()( 2

, tOt

nxnxnv ii

xi

tnt Time evolution for small time interval t

Temperature

222

,

Tkv

m Bxi

Equipartition theorem

Temperature is determined from the average kinetic energy.

Periodic boundary condition

2-dimensional system

Trajectories of 20 atoms interacting via Lennard-Jones potential

Setting of energy and temperature

triangular crystal

melting

formation of triangular crystal

Time-lapse snapshots of interacting particles (1)

melting

Time-lapse snapshots with increasingTemperatures (2)

Problems 2-1

Calculate two branches of the dispersion relation of the lattice vibration for a diatomic linear lattice using a simple spring model, and describe the characteristics of each branch.

Calculate the dispersion relation for a graphen sheet using a simple spring model between nearest neighbor atoms.

Study the role of the optical phonon in semiconductor physics.

Problems 2-2

Find the most stable 2-dimensional crystal structure, using the Lennard Jones potential.

Find the most stable 3-dimensional crystal structure, using the Lennard Jones potential.

Write a computer simulation program to study the motion of 3 atoms interacting with Lennard-Jones potential. Assume the space of motion to be within a 2-dimensional square region.

Problems 2-3

Study experimental methods to observe the dispersion relation of phonons.

Study the phonon dispersion relations for Si and Ge crystals and discuss about the similarity and the difference between them.

Study the phonon dispersion relations for Ge and GaAs crystals and discuss about the similarity and the difference between them.