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Computational%20Solid%20State%20Physics%20???????%20?2?

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zero point oscillation. Role of the acoustic phonon in semiconductors at a room temperature ... Thermal energy U and lattice heat capacity CV : Debye model (2) ... – PowerPoint PPT presentation

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Title: Computational%20Solid%20State%20Physics%20???????%20?2?


1
Computational Solid State Physics ??????? ?2?
  • 2.Interaction between atoms and the lattice
    properties of crystals

2
Atomic interaction
  • Lennard-Jones potential
  • for inert gas atoms He, Ne, Ar, Kr, Xe
  • Stillinger- Weber potential
  • for covalent bonding atoms C, Si, Ge

3
Lennard-Jones potential (1)
r inter-atomic distance
VLJ/e
VLJ(r) minimum at
r / s
4
Lennard-Jones potential (2)
1st term repulsive interaction caused by
Paulis principle 2nd term Van der Waals
interaction (attractive)
E1 electric field generated by a temporal dipole
moment p1
5
1-dimensional crystal
a lattice constant
Energy per atom
E minimum at a1.12s
cohesive energy ec1.04e
6
Bulk modulus
B Bulk modulus N the number of atoms in a
crystal a lattice constant
7
Lattice vibration
xn
displacement
Na length of a crystal
assume neglect the 2nd neighbor
interaction
x
  • 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 ? between
    atoms.

8
Equation of motion for atoms
m mass of an atom
Force on the n-th atom
Equation of motion for atoms
9
Solution for equation of motion
k wave vector
Periodic boundary condition
1st Brillouin zone
N modes
10
Dispersion relation of lattice vibration
acoustic mode
sound velocity phase velocity at k0
?(k)/?0
v becomes larger for larger ? and smaller m.
ka
11
Phonon
Energy quantization of lattice vibration
l0,1,2,3
zero point oscillation
Bose distribution function for phonon number
12
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

13
Lattice heat capacity Debye model (1)
Density of states of acoustic phonos for 1
polarization
phonon dispersion relation
Debye temperature ?
Nk Allowed number of k points in a sphere with a
radius k
N number of unit cell
14
Thermal energy U and lattice heat capacity CV
Debye model (2)
3 polarizations for acoustic modes
15
Debye model (3)
Low temperature Tltlt?
High temperature Tgtgt?
Equipartition law energy per 1 freedom is kBT/2
16
Heat capacity CV of the Debye approximation
Debye model (4)
kB1.38x10-23JK-1 kBmol7.70JK-1 3kBmol23.1JK-1
17
Heat capacity of Si, Ge and solid Ar Debye model
(5)
cal/mol K4.185J/mol K 3kB mol5.52cal K-1
18
Thermal conductivity (1)
Diffusive energy flux
T temperature c heat capacity per particle n
average number of phonons v group velocity of
phonon t scattering time
19
Thermal conductivity (2)
Thermal conductivity coefficient
K is largest for diamond because of the high
sound velocity!
20
Molecular dynamics simulation for atoms
Equation of motion for atoms
21
(1) velocity Verlets method
Time evolution for small time interval
22
Proof of (1)
23
(2) Verlet method
Time evolution for small time interval
24
Temperature
Equipartition theorem
Temperature is determined from the average
kinetic energy.
25
Periodic boundary condition
2-dimensional system
26
Trajectories of 20 atoms interacting via
Lennard-Jones potential
27
Setting of energy and temperature
melting
triangular crystal
28
Time-lapse snapshots of interacting particles (1)
formation of triangular crystal
29
Time-lapse snapshots with increasing Temperatures
(2)
melting
30
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.

31
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.

32
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.
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