ME 697R: Computational Methods for Nanoscale Energy Transport

Chapter 1: Lattice DynamicsSection 1.1: Lattice Structure

Reading: Kittel, pp. 1-26

Xiulin RuanSchool of Mechanical Engineering

Purdue University

� The building blocks of

these two are identical,

but different crystal

faces are developed

• Kittel, Introduction to Solid State Physics, p. 2

Crystal Structure

http://ultimate-luxury-community.com/the-111-facet-las-vegas-cut-diamond

� A crystal is constructed by the infinite repetition of identical structural units in space.

� Bravais lattice: an infinite array of discrete points whose position vectors can be described as

– : primitive translation vectors (lattice vectors) and – ni: integers.

,332211 anananR����

++=

ia�

Point

Translation Vectors

a1

a3

a2a1, a2 ,a3

Crystal Lattice

� Primitive unit cell is a volume of space that, when translated through all possible s, just fills all space, without overlap or voids. This cell contains ONE lattice point.

� Smallest building block for the crystal structure, “primitive”.� The choice of primitive unit cell is not unique

R�

S

SS

S

example in 2D

Primitive Unit Cell

� The choice of primitive cell is not unique � Wigner-Seitz cell is a special primitive cell and is unique.

� Lattice point is always in the center of the cell.

• Procedure:(1) Draw lines to connect a given

lattice point to all nearby lattice points

(2) At the midpoint and normal to these lines, draw new lines of planes

(3) The smallest volume enclosed in this way is the Wigner-Seitz primitive cell

Wigner-Seitz Primitive Cell

Lattice with a Basis

� Often, we need to describe a crystalline material’s structure by placing a primary atom at each lattice point and one or more basis atoms relative to it– For compound

materials (eg CuO2), this is an obvious requirement

– Also applies to some monoatomic crystals (eg Si)

� There are 14 lattice types in 3D� Most common types (Kittel Table 3):

Cubic: Li, Na, Al, K, Cr, Fe, Ir, Pt, Au etc.

Hexagonal Closed Pack (HCP):Mg, Co, Zn, Y, Zr, etc.

Diamond:C, Si, Ge, Sn (only four)

Lattice Types

a1

a3

a2

a1= a2 =a3

a1 ⊥ a2 ⊥ a3

1. Simple Cubic (SC)

Add one atom at the center of the cubic

2. Body-Centered Cubic (BCC)

Add one atom at the center of each face

3. Face-Centered Cubic (FCC)

Conventional Cell= Primitive Cell

Conventional Cell ≠ Primitive Cell

Cubic Lattices

� SC: One lattice point per conventional cell– (0, 0, 0): fractional coordinate, means (0*a1, 0*a2, 0*a3)

� BCC: Two lattice points per conventional cell– (0, 0, 0), (0.5, 0.5, 0.5)

� FCC: Four lattice points per conventional cell– (0, 0, 0), (0.5, 0.5, 0), (0.5, 0, 0.5), (0, 0.5, 0.5)

� Replicate the unit cell to form a crystal. FCC for example:– (0, 0, 0), (0.5, 0.5, 0), (0.5, 0, 0.5), (0, 0.5, 0.5)– (1, 0, 0), (1.5, 0.5, 0), (1.5, 0, 0.5), (1, 0.5, 0.5): translate a along x– (0, 1, 0), (0.5, 1.5, 0), (0.5, 1, 0.5), (0, 1.5, 0.5): translate a along y– (0, 0, 1), (0.5, 0.5, 1), (0.5, 0, 1.5), (0, 0.5, 1.5): translate a along z

Lattice Points per Conventional Cell

Primitive Cell of BCC

• Rhombohedron primitive cell

109o28’

• Primitive Translation Vectors:

Kittel, Introduction to Solid State Physics

Primitive Cell of FCC

•Angle between a1, a2, a3: 60o

Kittel, Introduction to Solid State Physics

Kittel p. 12

Diamond StructureC, Si, Ge, a-Sn

� FCC structure, 2 atoms per basis, 8 atoms per conventional cell

� Tetrahedral bond arrangement� Each atom has 4 nearest neighbors and 12 second nearest

neighbors

� 1) Find the intercepts on the axes in terms of the lattice constants a1, a2, a3. The axes may be those of a primitive or nonprimitive unit cell.

� 2) Take the reciprocals of these numbers and then reduce to three integers

having the same ratio, usually the smallest three integers. The result enclosed in

parethesis (hkl) is called the index of the plane.

Index System for Crystal Planes (Miller Indices)

Crystal Planes

ME 697R: Computational Methods for Nanoscale Energy Transport

Chapter 1: Lattice DynamicsSection 1.2: Reciprocal Lattice

Reading: Kittel, pp. 28-50

Xiulin RuanSchool of Mechanical Engineering

Purdue University

Fourier Transform of the Lattice

� Consider the mass density ρ(x) of a 1D atomic lattice. The primitive translation vector is

� Lattice periodicity dictates

� The Fourier series of density is

� We claim that Gn is a point in the reciprocal lattice of the crystal. The primitive lattice vector for the reciprocal lattice is:

a

x

( ) ( )x ma xρ + = ρ

{ }

{ } { } { }

( ) exp

( ) exp ( ) exp exp

n nn

n n n n nn n

x iG x

x ma iG x ma iG x iG ma

ρ = ρ

ρ + = ρ + = ρ

∑

∑ ∑{ } 2

exp 1 (given that is an integer)n n

niG ma G m

a

π⇒ = → =

2ˆ

a

π=g x

ˆa=a x

a

Brillouin zones are Weigner-Seitz primitive cell in the reciprocal lattice

For 1D lattice, the first Brillouin zone is [-π/a, π/a]

• Real lattice

• Reciprocal lattice

0 2π/a 4π/a-2π/a-4π/a-6π/a

x

-π/a π/a

• First Brillouin zone

Reciprocal Lattice in 1D and Brillouin Zone

Generalization to 3D� Lattice periodicity dictates…

� The Fourier series of density is

� Therefore

1 1 2 2 3 3

( ) ( )

wherem

m m m m

ρ + = ρ= + +

r R r

R a a a

{ }

{ } { } { }

{ }

( ) exp

( ) exp ( ) exp exp

exp 1 (given that is an integer)

n nn

m n n m n n n mn n

n m

i

i i i

i m

ρ = ρ ⋅

ρ + = ρ ⋅ + = ρ ⋅ ⋅

∴ ⋅ =

∑

∑ ∑

r G r

r R G r R G r G R

G R

2 integern m• = π×G R

Reciprocal Lattice in 3D� Note

Then

� The simplest choice of the g’s are

where

1 1 2 2 3 3n n n n= + +G g g g

( ) ( )1 1 2 2 3 3 1 1 2 2 3 3

2 integern m n n n m m m• = + + ⋅ + +

= π×G R g g g a a a

2j i ij⋅ = πδg a

1 for

0 forij

i j

i j

=δ = ≠

Reciprocal Lattice in 3D� The solution for g’s are

� A plane oblique lattice and its corresonding reciprocal lattice

2 3 3 1 1 21 2 3

1 2 3 1 2 3 1 2 3

2 ; 2 ; 2 ;× × ×= π = π = π

⋅ × ⋅ × ⋅ ×a a a a a a

g g ga a a a a a a a a

1 2 2 1; ;⊥ ⊥g a g a

Reciprocal Lattice to Simple Cubic Lattice

� The primitive translation vectors of the simple cubic lattice are:

� The primitive translation vectors of the reciprocal lattice are

� The reciprocal lattice of a simple cubic lattice is still a simple cubic lattice.

1 2 3ˆ ˆ ˆ; ;a a a= = =a x a y a z

1 2 3ˆ ˆ ˆ(2 / ) ; (2 / ) ; (2 / )a a a= π = π = πg x g y g z

� The primitive translation vectors of the lattice are:

� The primitive translation vectors of the reciprocal lattice are

� The reciprocal lattice of a BCC lattice is an FCC lattice.

� The first Brillouin zone is a rhombic dodecahedron.

Reciprocal Lattice to BCC Lattice

1

2

3

1ˆ ˆ ˆ( );

21

ˆ ˆ ˆ( );21

ˆ ˆ ˆ( )2

a

a

a

= − + +

= − +

= + −

a x y z

a x y z

a x y z

1 2

3

ˆ ˆ ˆ ˆ(2 / )( ); (2 / )( );

ˆ ˆ(2 / )( )

a a

a

= π + = π += π +

g y z g x z

g x y

The first Brillouin zone

The BCC Lattice

Reciprocal Lattice to FCC Lattice� The primitive translation vectors of

the lattice are:

� The primitive translation vectors of the reciprocal lattice are

� The reciprocal lattice of an FCC lattice is a BCC lattice.

1 2

3

1 1ˆ ˆ ˆ ˆ( ); ( );

2 21

ˆ ˆ( )2

a a

a

= + = +

= +

a y z a x z

a x y

1

2

3

ˆ ˆ ˆ(2 / )( );

ˆ ˆ ˆ(2 / )( );

ˆ ˆ ˆ(2 / )( )

a

a

a

= π − + += π − += π + −

g x y z

g x y z

g x y z

The first Brillouin zone

The FCC lattice

ME 697R: Computational Methods for Nanoscale Energy Transport

Chapter 1: Lattice DynamicsSection 1.3: Crystal Binding and Interatomic

Potentials

Reading: Kittel, pp. 54-79

Xiulin RuanSchool of Mechanical Engineering

Purdue University

� Chemical bonds are responsible for the formation of a crystal structure.

� Interaction exists for any atom pairs. It can be bonding interaction (such as ionic bonds, covalent bonds) or non-bonding interaction (such as van der Waals interaction and Coulomb interaction).

� The bond energy U is the energy released by the atomic pair when separating to infinite distance. Usually,

� U is equally shared between the two atoms.

R

1 2

1 2

( )

1

2( ) 0

U U R

U U U

U R

=

= =

=∞ =( ) 0U R = ∞ =

Chemical Bond in a Crystal

van der Waals Interaction

Ar

+ Ar -+ Ar -

ArR

Kittel, Page 63

Attraction

Repulsion

� Examples: inert gases, molecular crystals� Attraction: dipole-dipole interaction; act at long distance� Repulsion: Pauli exclusion principle; act at short distance

� Lennard-Jones (L-J) potential model:

�

� Bond energy: ε ~ 0.01 eV (weak): low melting point

12 6

( ) 4U RR R

σ σε = −

AttractiveRepulsive

where ε and σ are constantsfor argon, σ = 3.4 Å, ε = 0.01042 eV

min

( ) 0 when

( ) 0 when

( ) at 1.12

U R R

U R R

U R R

σσ

ε σ

< >> <

= − =Kittel, Page 64

Lennard-Jones (L-J) potential model

� The cohesive energy is the total potential energy of all atoms in a crystal.

� Consider an FCC argon lattice with N (very large) atoms, the potential energy of atom i is:

12 6

1 ,

1(4 ) ,

2ij N j i ij ij

Up R p R

σ σε= → ≠

= −

∑

12 6

1 ,

1(4 ) ,

2totj N j i ij ij

U Np R p R

σ σε= → ≠

= −

∑

Cohesive Energy of the Lattice

� pijR: the distance between atom i and atom j, expressed in terms of the nearest neighbor distance R. pij = 1 for nearest neighbors, and pij> 1 for second nearest and further neighbors.

� The cohesive energy of this N-atom crystal is

� For fcc crystal (12 nearest neighbors and 6 second nearest neighbors)

� Evidently the lattice cohesive energy is a function of R. For the lattice to be at equilibrium, the cohesive energy should be at its minimum.

� Now you start to see nano size effect on R0!� The process of minimizing the lattice energy to find the equilibrium lattice

configuration is called “energy minimization” or “geometry optimization”.

12 6

13 7

0

,min 0

0 2 (12)(12.13) 6(14.45)

1.09

( ) (2.15)(4 )

tot

tot tot

dUN

dR R R

R

U U R N

σ σε

σε

= = − −

=⇒ = = −

for fcc crystal

( )( )

1212 12

1 ,

66 6

1 ,

12 1 6 2 ... 12.13188

12 1 6 2 ... 14.45392

ijj N j i

ijj N j i

p

p

−− −

= → ≠

−− −

= → ≠

= × + × + =

= × + × + =

∑

∑

Equilibrium Lattice Constant

� Potential model for short-range interaction :

� One electron of the Na atom is removed and absorbed to the Cl atom

� Short-range interaction: Pauli exclusion principle; repulsive

� Long-range interaction: Coulomb force; attractive or repulsive

� Bonding energy: 1-10 eV (strong): high melting point

Na+

Cl-

Cl-

Cl-

Cl-

Cl-

Na+

Na+

Na+

, exp ijij sh

rU A

ρ−

=

where A and ρ are constants, and rij is the distance between the two atoms.

Ionic Bond and Short Range Interaction

Coulomb Interaction and Potential Model

� Coulomb interaction is important when atoms are charged

� Coulomb potential model:

– qi and qj: atomic charges for atoms i and j, including the sign

– ε0: dielectric function

� Coulomb interaction is very long range since it decays as 1/r.

Na+

Cl-

Cl-

Cl-

Cl-

Cl-

Na+

Na+

Na+

,04

i jij Coulomb

ij

q qU

rπε=

04i j

ij

q q

rπε

exp ijrA

ρ−

ijU

Kittel, Page 72

Total Potential Energy for an Ionic Bond

� The total potential energy for a pair:

, ,

0

exp4

ij ij sh ij Coulomb

ij i j

ij

U U U

r q qA

rρ πε

= +

− = +

� Consider an ionic lattice with 2N atoms, where R is the nearest neighbor distance

� The potential energy of atom i is

� The lattice energy is

0

0

exp (nearest neighbors)4

(otherwise)4

i j

iji j

ij

q qRA

RU

q q

p R

ρ πε

πε

− + =

nearest neighbors second neighbors0 0

1exp ...

2 4 4i j i j

ij j ij

q q q qRU A

R p Rρ πε πε= =

− = + + +

∑ ∑

nearest neighbors second neighbors0 0

(2 )

exp ...4 4

tot i

i j i j

j j ij

U N U

q q q qRN A

R p Rρ πε πε= =

=

− = + + +

∑ ∑

Lattice Energy

� The lattice will be equilibrium if the lattice energy is at its minimum.

� The evaluation of the electrostatic terms is tricky. Take NaCl as an example, qNa = +e, qCl = -e.

– z: number of nearest neighbors– α: Madelung constant defined as

( )

j ijpα ±≡∑

0 ,min0 andtottot

dUR U

dR= ⇒

nearest neighbors second neighbors0 0

2

0

( )( ) ( )( )exp ...

4 4

exp4

totj j ij

R e e e eU N A

R p R

R eN zA

R

ρ πε πε

αρ πε

= =

− − + + + = + + +

−= −

∑ ∑

Equilibrium Lattice Constant

• For 1D

1 1 1 1 1 1 12 2 2 2 ... 2 1 ... 2 ln 2

1 2 3 4 2 3 4α = × − × + × − × + = − + − + =

Madelung Constant

• For NaCl

• Other materials

• Electrostatic interaction is a long-range interaction.

1 1 16 12 8 ... 1.747565

1 2 3α = × − × + × − =

Material αCsCl 1.762675

ZnS 1.6381

• Buckingham form:

• Compared to the original form, the term is added to account for the van der Waals attraction.

, 6exp ij ij

ij shij

r CU A

rρ−

= −

Buckingham Potential for Short-Range Interaction

6

ij

ij

C

r−

Covalent Bond

• Two atoms share a pair of electrons

• Bonding energy: ~1-10 eV (strong)

• Examples: C, Ge, Si, H2

C

C C

C C

H H+ HH

pair angular torsional

Multi-body Potentials for Covalent Bonds

� Since electrons pairs are shared, the changes in bond angles and plane angles will result in potential energy change.

� Typically multi-body interactions need to be considered.

� Born harmonic model

– k: spring (force) constant – r0: reference length or equilibrium bond length

� Expanded harmonic model

– k3 and k4 are optional– It is actually a Taylor series of a more complicated function

Pair Potentials

20

1( ) ( )

2U r k r r= −

2 3 42 0 3 0 4 0

1 1 1( ) ( ) ( ) ( ) ...

2 6 24U r k r r k r r k r r= − + − + − +

r

� Morse model:

– De: the depth of potential well

– a = , where ke is the force constant at the minimum of the well

– re: equilibrium bond length

� Morse potential can take into account the anharmonic effect when ris much larger than re.

Morse Potentials

2( )( ) 1 ea r reU r D e− − = −

/ 2e ek D

� A convenient model for the potential energy of a diatomic molecule.

� It is a better approximation for the vibrational structure of the molecule than the quantum harmonic oscillator

� It explicitly includes the effects of bond breaking, such as the existence of unbound states.

� It also accounts for the anharmonicity of real bonds.

Features of Morse Potential

Angular (Three-Body) Potentials

� Expanded harmonic model:

– k2, k3 and k4: angular spring constant– k3 and k4 are optional– θ0: equilibrium bond angle

� Cosine model

angular

2 32 0 3 0

44 0

1 1( ) ( ) ( )

2 61

( ) ...24

U k k

k

θ θ θ θ θ

θ θ

= − + −

+ − +

θ

20( ) (cos cos )U Aθ θ θ= −

Torsional (Four-Body) Potentials

� Harmonic model:

– φ1234: angle between planes 123 and 234– φ1234,0: equilibrium angle

Plane 123 defined by atoms 1, 2, and 3

1234 1234,0( ) 1 cos( )U A n φ = + φ − φ

Plane 234 defined by atoms 2, 3, and 4

Stillinger Weber Potential

� One of the first attempts to model a semiconductor.

� Includes two- and three-body terms.

� Two-body term:

where A, B, p, q, a are positive.

1

2

( ) exp ( ) ,( )

0 ,

p qA Br r r a r af r

r a

− − − − − < = ≥

Stillinger Weber Potential

� Three-body term:

where

– i, j, k: labels of atoms– rij: Length of the bond between atoms i & j– θijk: Bond angle between bonds ij & ik

� The parameters for silicon: – A = 7.049556277, B = 0.6022245584– p = 4, q = 0, a = 1.80, λ = 21.0, γ = 1.2

3( , , ) ( , , ) ( , , ) ( , , )i j k ij ik jik ji jk ijk ij ik jikr r r h r r h r r h r rf θ θ θ= + +

( ) ( )2

1 1 1( , , ) exp * cos

3ij ik jik ij ik jikh r r r a r aθ λ γ γ θ− − = − + − +

Tersoff Potential

� Originally developed to the Group IV elements (e.g. carbon, silicon, germanium). Later extended to other metallic bonding.

� Form

ij

1, ( )[ ( ) ( )] ;

2i ij c ij R ij ij A ij

i i j

E E V V f r f r b f r≠

= = = +∑ ∑

( ) exp( ), ( ) exp( );

1,

1 1( ) cos[ ( ) ( )], ,

2 20,

R ij ij A ij ijij ij ij ij

ij ij

c ij ij ij ij ij ij ij ij

ij ij

f r A r f r B r

r R

f r r R S R R r S

r S

λ µ

π

= − = − −<

= + − − < <

>

Tersoff Potential

– i,j,k: labels of atoms– rij: length of the bond between atoms i & j– θijk: bond angle between bonds ij & ik– fA: pair wise attraction term– fR: pair wise repulsion term– fc: cutoff function– bij: bond order

( )( )( )

1

,

22 2 2 2

1 2

1 , ( ) ( )

( ) 1 [ ];cos

2, ,

, , , ,

ni

k i

ijk

ij ij ij c ij ij ijk

ijk i i i i

ij i

ij i j

n ni ib f r gi ij

g c d c d hi

X X Xj for X

X for X A B R SX X

χ ζ ω θβ ζ

θ θ

λ µ

−

≠=

= + ∑

= + − +

= + =

= =

−

Tersoff Potential Parameters

C Si Ge

A(ev) 1.396X103 1.8308X103 1.769X103

B(ev) 3.4674X102 4.7118X102 4.1923X102

λ(Ang-1) 3.4879 2.4799 2.4451

μ(Ang-1) 2.2119 1.7322 1.7047

β 1.5724X10-7 1.1X10-6 9.0166X10-7

n 7.2751X10-1 7.8734X10-1 7.5627X10-1

c 3.8049X104 1.0039X105 1.0643X105

d 4.384 1.6217X101 1.5652X101

h -5.7058X10-1 -5.9824X10-1 -4.3884X10-1

R(Ang) 1.8 2.7 2.8

S (Ang) 2.1 3.0 3.1

REBO Potential

� Reactive Empirical Bond Order (REBO) potential

where A, Q, B1, B2, B3, β1, β2, β3 are fitting parameters.

� The values of the parameters for Carbon are:

3

1

1( ) 1 .

2i j n ijr r

c ij ij nbi j n

QE f r A e b B e

rα β− −

≠ =

= + −

∑ ∑

B1=12388.79197798 ev β1=4.7204523127Ang-1 Q=0.313460296

B2=17.56740646509 ev β2=1.43320132499Ang-1 A=10953.54416217 ev

B3=30.71493208065 ev β3=1.3826912506Ang-1 α=4.7465390606595 Ang-1

Dmin=1.7 Dmax=2.0

Metallic Bonds

� Metallic bonding is the type of bonding found in metal elements. This is the electrostatic force of attraction between positively charged ions and delocalized outer electrons.

� The metallic bond is weaker than the ionic and the covalent bonds.

� A metallic bond result from the sharing of a variable number of electrons by a variable number of atoms. A metal may be described as a cloud of free electrons.

+

+

+

+

+

+

+

+

+

Embedded Atom Model (EAM)

� The potential energy for an atom i is:

– Fα: embedding function that represents the energy required to place atom i of type α into the electron cloud

– ρα: contribution to the electron charge density from atom j at the location of atom i

– φαβ: pair-wise potential function

� Tabulated functions– Fα : as a function of ρ– ρα : as a function of rij

– φαβ : as a function of rij

1( ) ( )

2i ij iji j i j

E F r rα α αβρ φ≠ ≠

= +

∑ ∑

Kaviany, Heat Transfer Physics, Cambridge University Press, 2008.

Available Interatomic Potentials in GULP

Kaviany, Heat Transfer Physics, Cambridge University Press, 2008.

Available Interatomic Potentials in GULP

http://projects.ivec.org/gulp/help/gulp_30_manual/gulpnode8.html

ME 697R: Computational Methods for Nanoscale Energy Transport

Chapter 1: Lattice DynamicsSection 1.4: Dynamical Matrix and Phonon Dispersion

Reading: Kittel, pp. 98-107

Xiulin RuanSchool of Mechanical Engineering

Purdue University

Harmonic Approximation

� Harmonic approximation: near the potential well minimum. This approximation works especially well for low temperature (is exact at 0 K).

� Chemical bonds can be assumed as elastic springs.

Energy

Distancero

Parabolic Potential of Harmonic Oscillator

Eb

� Equilibrium and deformed positions

� Force and Equation of motion for the sth atom

Crystal Vibration for Monatomic Chain

a

Spring constant, g Mass, m

xn xn+1xn-1

Equilibrium Position

Deformed Position

us: displacement of the sth atom from its equilibrium position

us-1 us us+1

( )

( )

1 1

2

1 12

( ) ( ) 1

( 2 ) 2

s s s s s

ss s s

F C u u C u u

d uM C u u u

dt

+ −

+ −

= − + −

= + −

equilibrium coordinates

displacements

� Assume a wave solution in the form

� Then

� The equation of motion (2) becomes

� Note

Solution of the Wave Equation

( )( )

1

0

exp

exp( )exp

s s

s

u u iKa

u u i t iKsaω± = ±

= −

0( , ) exp( )u x t u i t iKxω= − +

22

2s

s

d uu

dtω= −

( )21 1( 2 ) 3s s s sM u C u u uω + −− = + −

� The Equation (3) becomes

� Identity

� The ω-K relation is called the dispersion relation

Solution of the Wave Equation

( )[ ] [ ] ( ){ }

( ) ( )

20

0

2

exp

exp ( 1) exp ( 1) 2exp

exp exp 2

Mu iKsa

Cu i s Ka i s Ka iKsa

M C iKa iKa

ω

ω

− =

+ + − −

= − + − −

( )1/2

2 2 4 11 cos sin

2

C CKa or Ka

M Mω ω = − =

( ) ( )2cos exp expKa iKa iKa= + −

Phonon Dispersion Relation

� The dispersion relation has a period of 2π/a, so the range of independent K values is [- π/a, π/a]. This is the first Brillouinzone of 1-D lattice.

-π/a<K< π/aKmax = π/aK = 2π/λλmin = 2a

Polarization and Group Velocity

� Group Velocity:

� Speed of Sound:

� Zone edge (boundary):

– Standing wave

Fre

quen

cy,ω

Wave vector, K0 π/a

dK

dvg

ω=

dK

dv

Ks

ω0

lim→

=

/lim 0g

K a

dv K

a dKπ

π ω→

= = =

Longitudinal Acoustic Phonons

Long wavelength

Zone edgeStanding wave

Two Atoms per Basis

� Equation of motion:

Lattice Constant, a

xs ysys-1 xs+1

M2 M1

us vsvs-1 us+1

equilibrium coordinates:

displacements:

( )

( )

2

1 12

2

2 12

( 2 ); 1

( 2 ); 2

ss s s

ss s s

d uM C v v u

dt

d vM C u u v

dt

−

+

= + −

= + −

� Assume a wave solution in the form

� Then the equations of motion becomes

� The homogeneous linear equations have a solution only if

Solution of the Wave Equation

0

0

exp( )exp( )

exp( )exp( )s

s

u u isKa i t

v v isKa i t

ωω

= −= −

[ ][ ]

21 0 0 0

22 0 0 0

1 exp( ) 2

exp( ) 1 2

M u Cv iKa Cu

M v Cu iKa Cv

ω

ω

− = + − −

− = + −

[ ][ ]

21

22

2 1 exp( )0

1 exp( ) 2

C M C iKa

C iKa C M

ω

ω

− − + −=

− + −

� The solution is

� Limiting cases:

– K → 0,

– K → π/a,

Dispersion Relation

1/22

2

1 2 1 2 1 2

1/21/22

1 2 1 2 1 2

1 1 1 1 4sin

2

1 1 1 1 4sin (3)

2

KaC C

M M M M M M

KaC C

M M M M M M

ω

ω

= + ± + −

= + ± + −

1/2

LA1 2

,2( )

CKa

M Mω

= +

1/2

LO 01 2

1 12C

M Mω ω

= = +

1/2

LA1

2,

C

Mω

=

1/2

LO2

2C

Mω

=

Dispersion Relation Profiles

Kittel, Figure 7, page 105

Eq. (3)

Dispersion relation for Ge.

� At K = 0– For acoustic mode, , vibrations of neighboring atoms are in

phase

– For optical mode, , vibrations of neighboring atoms are out of phase

0

0

1u

v=

Acoustic and Optical Phonons

0 2

0 1

u M

v M= −

Kittel, Figure 10, page 107

Standing wave

Traveling wave

� Consider the diatomic linear chain case. The homogeneous linear equations have a solution only if

or

� Define

� So

Dynamical Matrix Approach

[ ][ ]

21

22

2 1 exp( )det 0

1 exp( ) 2

C M C iKa

C iKa C M

ω

ω

− − + −=

− + −

( )

( )1 1 2 2

21 2

21 exp

1 0det 0

2 0 11 exp

C CiKa

M M M

C CiKa

MM M

ω

− + −

− = − +

( )

( )1 1 2

21 2

21 exp

( )2

1 exp

C CiKa

M M MD K

C CiKa

MM M

− + −

= −

+

Dynamical Matrix

2det ( ) 0D K Iω − =

Atomic Position Nomenclature in 3D

� An atom in unit cell n and at basis position α has an equilibrium position rnα

� Subscripts i and j denote direction� Subscripts m and β denote the

position of a different atom in the lattice (analogous to n and α)

� Perturbed expression for the potential φ (like a Taylor series expansion)

0 because assumed zero potential and first derivative at the equilibrium position Note: expressions such as

this use implied summationof repeated super/subscripts

Force Constant Matrix

� We define a matrix ΦΦΦΦthat serves the function of a normalized spring constant

� Lattice invariance tells us that this spring constant depends only on the distancebetween positions nαand mβ

� Mathematically, this means that…

Equations of Motion

� The force on the atom at position nα can be expressed in terms of the spatial derivative of potential energy

� Atomic displacement s can be expressed as a Fourier wave

– The wave vector K is an important parameter (inversely proportional to wavelength)

– Wave frequency ω gives the rate of vibration of the wave

� Substituting into the equation of motion, we find

( ) m jn i n in i n i m j n i

n i

F Mβα αα α β α α

α

φ∂ += − = −Φ =∂

r ss s

sɺɺ

[ ]2 1( ) exp ( ) ( )m j

i n i m n ju i uM M

βα α β

α β

ω = Φ ⋅ −K K r r K

( )1( )expn i i nu i t i

Mα α

α

ω= − + ⋅s Κ Κ r

Dynamical Matrix and Dispersion Relation

� The dynamical matrix contains each “spring constant”

– Second equality from translational invariance, and rp is defined as rm– rn

� The euqation of motion becomes

� Where δ is the Kronecker delta (identity matrix).� The dispersion relation defines the relationship between the

wave vector K and frequency ω and emerges from the secular equation,

det[D(K) - ω2I] = 0

( )2 ( ) 0j ji i jD uβ β

α α βω δ− =K

[ ] 0

1 1exp ( ) expj m j p j

i n i m n i pD i iM M M M

β β βα α α

α β α β

= Φ ⋅ − = Φ ⋅ K r r K r

Phonon Dispersion of Silicon

� Phonon dispersion is typically plot along high symmetry lines in the first Brillouin zone.

X