Transcript Computational Solid State Physics 計算物性学特論 第２回 2.Interaction between atoms and the

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)
  12   6 
VLJ (r )  4      
 r  
 r 
r: inter-atomic distance
VLJ/ε
repulsive
force
attractive
force
VLJ(r) minimum at
1
6
r  r0  2   1.12
VLJ (r0 )  
r/σ
Lennard-Jones potential (2)
  12   6 
VLJ (r )  4      
 r  
 r 
1st term: repulsive interaction caused by Pauli’s principle
2nd term: Van der Waals interaction (attractive)
r
p1
temporal
dipole
moment
p2 (r)
induced
dipole
moment
p1
p2 (r )  E1 (r )   3
r
p1 p2
1
VVW (r )  3   6
r
r
E1: electric field generated by a
temporal dipole moment p1
1-dimensional crystal
a
x
Energy per atom:
12
6

1
 
  
E   2  4       
2
 an  
n 
 an 
12
6



 
  
 4 1.00   1.02  
a
 a  

a: lattice constant
E minimum at a=1.12σ
E (a)  1.04
cohesive energy
εc=1.04ε
Bulk modulus
2
d
B  L 2 EN
dL
L  Na
B : Bulk modulus
N : the number of atoms in a crystal
a : lattice constant
d


B  a 2 E  74.9  66.9
da
a

2
Lattice vibration
displacement xn
Na: length of a crystal
x
a
assume: neglect the 2nd
neighbor interaction
1 d2
2
VLJ (a  x)  VLJ (a) 
V
x
LJ x  a
2 dx 2
d2


  2 VLJ x a  72 2  57.4 2
dx
a

a  1.12
•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.
Equation of motion for atoms
xn-1
xn
xn+1
m: mass of an atom
a
Force on the n-th atom:
VLJ (a  xn 1  xn ) VLJ (a  xn  xn 1 )
Fn  

xn
xn
  ( xn 1  xn )   ( xn  xn 1 )
Equation of motion for atoms:
d 2 xn
m 2   ( xn 1  xn )   ( xn  xn1 )
dt
1 n  N
Solution for equation
of motion
d 2 xn
2
2
 0 ( xn 1  xn )  0 ( xn  xn1 )
2
dt

0 
m
2
2
xn ( k )  xn ( k 
)
a
Assume: xn  exp[ i(kan  t )]
2
2
2 ka
k: wave vector
  40 sin
2
Periodic boundary condition: xn  xn  N
kaN  2l
2 l
k
a N


k

a
a
N
N
 l 
2
2
1st Brillouin zone
N modes
Dispersion relation of
lattice vibration
ka
 (k )  20 sin
2
acoustic mode
sound velocity:
phase velocity at k=0
ω(k)/ω0
w(k )

v
 0 a  a
k k 0
m
ka
v becomes larger for larger κ
and smaller m.
Phonon
Energy quantization of lattice vibration
1
El (k )   (k )(l  )
l=0,1,2,3
2
 ( k )
E0 ( k ) 
2
:zero point oscillation
Bose distribution function for phonon number:
1
n( (k )) 
exp(  (k ) / k BT )  1
k BT
for k BT  
n( (k )) 
 (k )
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)
3
V 3
 L  4 3  L  4
Nk  
k 
 2 3


3
6 v
 2  3
 2  3v
dN k
V 2 Density of states of acoustic
D( ) 
 2 3
d 2 v phonos for 1 polarization
3
3
 (k )  vk
k
2
k 
L
Nk: Allowed number of k
points in a sphere with a
radius k
phonon dispersion relation
Debye temperature θ
3
 D  k B
V
N
D
2 3
6 v
N: number of unit cell
v  6 2 N 

  
kB  V 
1
3
Thermal energy U and lattice heat
capacity CV : Debye model (2)
3 polarizations for acoustic modes
D
V 2

U  3 dD( )n( )  3  d 2 3
2 v exp(  / k BT )  1
0
2

U
3
V



CV  
 
2 3
2
 T  V 2 v k BT
T 
CV  9 Nk B  
 
3 xD
D
 4 exp(  / k BT )
0 d [exp(  / k BT )  1]2
x 4e x
0 dx (e x  1) 2
Debye model (3)
T 
CV  9 Nk B  
 
3 xD
4 x
xe
0 dx (e x  1) 2
・Low temperature T<<θ
3
4
12
T 
CV 
Nk B  
5
 
・High temperature T>>θ
CV  3NkB
Equipartition law:
energy per 1 freedom
is kBT/2
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)
Si and Ge
cal/mol K=4.185J/mol K
3kB mol=5.52cal
K-1
Solid Ar
CV  T
3
Thermal conductivity (1)
Diffusive energy flux
dT
dT
2
jE    nvx c
vx  nc  vx  
dx
dx
nc  v 2   dT
jE  
3
dx
Energy
T: temperature
3kBT(x)
vxτ
c: heat capacity per particle
Energy emission
c vxτdT/dx
n: average number of phonons
v: group velocity of phonon
τ: scattering time
x
Thermal conductivity (2)
dT
jE   K
dx
Thermal conductivity coefficient
nc  v 2   Cv 2 Cvl
K


3
3
3
C: heat capacity per unit volume,
l=vτ: phonon mean free path
v: sound velocity of acoustic phonon
K is largest for diamond because
of the high sound velocity!
Molecular dynamics simulation
for atoms
Equation of motion for atoms:
dr
v
dt
dv
F
a
dt
m
r: position of an atom
v: velocity
a: acceleration
F: force
t: time
m: mass of an atom
(1) velocity Verlet’s method
Time evolution for small time interval t :
1 2
3
r (t  t )  r (t )  tv(t )  t a(t )  O(t )
2
1
v(t  t )  v(t )  t[a(t )  a(t  t )]  O(t 3 )
2
Proof of (1)
2
dv
1d v 2
3
v(t  t )  v(t )  t 
t  O(t )
2
dt
2 dt
1 d 2 v 2 1 da 2 1 a (t  t )  a (t ) 2
3
t 
t 
t  O(t )
2
2 dt
2 dt
2
t
1
 v(t  t )  v(t )  [a (t )  a(t  t )]t  O(t 3 )
2
(2) Verlet method
Time evolution for small time interval
t
t  nt
xi (n  1)  2 xi (n)  xi (n  1)  ai , x (n)( t ) 2  O(( t ) 4 )

3
dxi
d
xi
1 d 2 xi
1
2
3
4
xi (n  1)  xi (n) 
(t ) 
(

t
)

(

t
)

O
((

t
)
)
2
3
dt
2 dt
6 dt
d 2 xi
xi (n  1)  xi (n  1)  2 xi (n)  2 (t ) 2  O(( t ) 4 )
dt
xi (n  1)  xi (n  1)
vi , x (n) 
 O(( t ) 2 )
2t
Temperature
Equipartition theorem
m 2
k BT
 vi , x 
2
2
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
melting
triangular crystal
Time-lapse snapshots of interacting
particles (1)
formation of triangular crystal
Time-lapse snapshots with increasing
Temperatures (2)
melting
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.