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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Handout 17
Lattice Waves (Phonons) in 1D Crystals: Monoatomic Basis and
Diatomic Basis
In this lecture you will learn:
**- Equilibrium bond lengths
- Atomic motion in lattices
- Lattice waves (phonons) in a 1D crystal with a monoatomic basis
- Lattice waves (phonons) in a 1D crystal with a diatomic basis
- Dispersion of lattice waves
- Acoustic and optical phonons**
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
The Hydrogen Molecule: Equilibrium Bond Length
0 x
V r
-d d
EB
EA
2 : E 1 s
1 : EA
1 : E B
2 Vss
The equilibrium distance between the two
hydrogen atoms in a hydrogen molecule is
set by the balance among several different
competing factors:
- The reduction in electronic energy due to
co-valent bonding is 2 V (^) ss . If the atoms
are too far apart, V ss becomes to small
- If the atoms are too close, the positively
charged nuclei (protons) will repel each
other and this leads to an increase in the
system energy
- Electron-electron repulsion also plays a
role
EB E 1 s Vss
EA E 1 s Vss
1 (^) s r dx ˆ H ˆ 1 s r dx ˆ Vss
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A Mass Attached to a Spring: A Simple Harmonic Oscillator
x
0
x
0
u
Equilibrium position Stretched position
Potential Energy:
2
2
PE Vu ku
Kinetic Energy: 2
dt
M du KE
M M
spring constant = k (units: Newton/meter)
Dynamical Equation (Newton’s Second Law):
k u du
dV
dt
du M 2
(^2) Restoring force varies linearly with
the displacement “ u ” of the mass from its equilibrium position
Solution:
u t A cos ^ o t ^ B sin ^ ot where: M
k
o
PE varies quadratically with
the displacement “ u ” of the mass from the equilibrium
position
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A 1D Crystal: Potential Energy
Consider a 1D lattice of N atoms:
a 1 (^) a x ˆ
x
Rn na 1
- The potential energy of the entire crystal can be expressed in terms of the positions
of the atoms. The potential energy will be minimum when all the atoms are at their equilibrium positions.
**- Let the displacement of the atom at the lattice site given by from its equilibrium position be
- One can Taylor expand the potential energy of the entire crystal around its minimum**
equilibrium value:
^ ^ ^
k j
j k
EQ j k
j j EQ j
N EQ
uR uR uR uR
V
uR uR
V
VuR uR uR uR V
2
1 2 3
R n
u Rn
0
Potential energy varies quadratically with the displacements of the
atoms from their equilibrium positions
Atoms can move only in the x-direction
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Dynamical Equation for Nearest-Neighbor Interactions
A1D lattice of N atoms:
j
n j j n
n KR R uR t uR
V
dt
duR t M , ,
2
2
Assume nearest-neighbor interactions:
K R n , Rj j , n 1 j , n 1 2 j , n
u R t u R t u R t u R t
dt
duR t M (^) n n n n n , , , ,
2 1 1
2
This gives:
The constants “^ ” provide restoring forces as if the atoms were connected together
with springs of spring constant “ ”
We have N linear coupled differential equations for N unknowns
u Rn , t n 0 , 1 , 2 ....... N 1
The constant is called “force constant” (not spring constant) in solid state physics
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Solution of the Dynamical Equation: Lattice Waves (Phonons)
A1D lattice of N atoms:
u ^ R t ^ u ^ R t u ^ R t ^ u ^ R t
dt
duR t M (^) n n n n
n , , , ,
2 1 1
2
iqR i t u Rn t uqe e n ^
^ (^) . , Re
Assume a solution of the form:
Represents a wave with
wavevector , frequency , and amplitude
q
u q
iq a iq R i t
iq R i t iq R a i t n
e uq e e
uR t uqe e uqe e
n
n n
^
.. .. 1
1
1 1 ,
Note that:
iq R i t u Rn t uq e e n ^
^ (^) . Or: , Slight abuse of notation
a 1 (^) a x ˆ
Rn na 1
iq a iqR i t
iq R i t iq R a i t n
e uq e e
uR t uqe e uqe e
n
n n
^
.. .. 1
1
,^11
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A1D lattice of N atoms:
u R t u R t u R t u R t dt
duR t M (^) n n n n n , , , ,
2 1 1
2
Plug in the assumed solution:
To get:
M u q u q e u q u q e u q
i q a iqa
^2 ^.^1 .^1
Which simplifies to:
sin
1 cos.
(^21)
1
2
q a
M
q a M
Or:
sin
4 q a 1
M
a 1 (^) a x ˆ
Rn na 1
q qx x ˆ
iqR i t u Rn t uqe e n ^
^ (^) . ,
Solution of the Dynamical Equation: Lattice Waves (Phonons)
Since is always positive, the negative sign is chosen when the
sine term is negative
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A1D lattice of N atoms: a 1^ ax ˆ
Rn na 1
Solution is: (^) and iq R i t u Rn t uqe e n ^
^ (^) . ,
sin
4 q a 1
M
Solution of the Dynamical Equation: Lattice Waves (Phonons)
x
x
a
u Rn
- The lattice waves are like the compressional sound waves in the air
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
q x
sin
4 qa
M
x
a
a
Phase and Group Velocities
First
BZ
Case I: For q x ≈ 0 (i.e. q x a << ):
q a M
x
Linear
dispersion
Phase velocity and group velocity of lattice waves
are defined as:
x dq
d v q q
x q
q q
q v q
x
D g q
x
D p
1
1
p a v g M
v
Case II: For q x a = :
M
g
p
v
a
M
v
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Periodic Boundary Conditions
In the solution:
allowed values of the wavevector depend on the boundary conditions
iq R i t u Rn t uq e e n ^
^ (^) . ,
A1D lattice of N atoms: a 1^ ax ˆ
Rn na 1
Periodic Boundary Condition:
0 N-1 1
The N -th atom is the same as the 0-th atom
This implies:
integer 2
0
0
.
. 0
.
N
m
N
m
N a
m q
e
e
uR t uq e e uR t uqe e
x
iqNa
iq R R
iq R i t iq R i t N
x
N
N
^
q x
a
a
First BZ
N a
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Counting and Conserving Degrees of Freedom
A1D lattice of N atoms: a 1^ ax ˆ
Rn na 1
- We started with N degrees of freedom which were related to the motion in 1D of N
different atoms
- The dynamical variables were the amplitudes of the displacements of N different
atoms
u^ ^ R , t ^ n 0 , 1 , 2 ....... N 1
n
We then ended up with lattice waves:
- There are N different lattice wave modes
corresponding to the N different possible
wavevector values in the first BZ
- The dynamical variables are the amplitudes of
the N different lattice wave modes
2 2
and
N
m
N
Na
m u qt qx
The number of degrees of freedom are the same before and after – as they should be!
x q
a
a
First
BZ
N a
n^ n
iq R i qt iqR n uR t uqe e uqt e
^ (^) . . , Re Re ,
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Counting and Conserving Degrees of Freedom
The atomic displacements,
taken together provide a complete description of the motion of all the atoms in the
crystal
In general, one can expand the atomic displacements in terms of all the lattice wave
modes (resembles a Fourier series expansion):
u Rn , t n 0 , 1 , 2 ....... N 1
(^) (^)
in FBZ
.
.**
inFBZ
.
.**
inFBZ
.
, Re
q
iqR iqR
q
iqR i qt iqR i qt
q
iq R i qt n
n n
n n
n
e
u qt e
uqt
e e
u q e e
uq
uR t uqe e
Therefore, the lattice wave amplitudes also provide a complete description of the motion of all the atoms in the crystal
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Plug the solutions in the dynamical equations to get:
M u q u q e u q u q e u q
i q n iq n
2
. 2 1 1
. 1 1 2 1
(^2 2 )
M u q u q e u q u q e u q
i qn iq n
1
. 1 2 2
. 2 2 1 2
2 (^) 1 2
Write the equations in a matrix form:
u q
u q
M
M
u q
u q
e e
e e iqn iqn
iqn iqn
2
1
2
2 1
2
1
1 2
. 2
. 1
. 2
. 1 2 1 0
0
1 2
1 2
x
M 1 M 2
n 1
n 2
This is a 2x2 matrix eigenvalue equation that needs to be solved for each value of
the wavevector to get the dispersion of the lattice waves
Lattice Waves (Phonons) in a 1D crystal: Diatomic Basis
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
x
M 1 M 2
n 1
n 2
2^ 1
The Dynamical Matrix
Or: (^)
u q
u q M u q
u q D q
2
2 1
2
1
**- The matrix is called the dynamical matrix of the medium
- For any medium, in any dimension, the dispersion relations for the lattice**
waves (phonons) are obtained by solving a similar matrix eigenvalue equation
D q
u q
u q
M
M
u q
u q
e e
e e iqn iqn
iqn iqn
2
1
2
2 1
2
1
1 2
. 2
. 1
. 2
. 1 2 1 0
0
1 2
1 2
Or: (^)
u q
u q
u q
u q M Dq
2
2 1
2
1 1
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Optical and Acoustic Phonons
x
M 1 M 2
n 1
n 2
The frequency eigenvalues are:
16 sin 2
(^21)
1 2
12 2
2 (^2 ) q a
M M MM
q r r
1 2
M M M
r
a 1 (^) a x ˆ
q x
a
a
- The two frequency eigenvalues for each wavevector value in the FBZ give two phonon
bands
- The higher frequency band is called the optical
phonon band
- The lower frequency band is called the acoustic phonon band
Optical
Acoustic
FBZ
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Optical and Acoustic Phonons: Special Cases
x
M 1 M 2
2^ 1
a 1 (^) a x ˆ
Case I: q 0
qa M M
q (^) x 1 2 1 2
12 0
Acoustic band:
2
1 A u q
u q
iq R d i t n
iq R d i t n
u R d t Ae e
u R d t Ae e
n
n
2
1
. 2 2
. 1 1
M r
q 1 2 0
Optical band:
2 1 2
M M
A
u q
u q
iq R d i t n
iq R d i t n
e e M
M
u R d t A
u R d t Ae e
n
n
2
1
.
2
1 2 2
. 1 1
^
q x
a
a
Optical
Acoustic
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Optical and Acoustic Phonons: Special Cases
x
M 1 M 2
a 1 (^) a x ˆ
Case II: ˆ, 1 2 , 1 2 Acoustic Mode
x M M M a
q
q x
a
a
Optical
Acoustic
M
x a
. 1
2
iq n e
A
x a
u
x a
u
iq R d i t
iq n iq R d i t n
iq R d i t n
Ae e
u R d t Ae e e
u R d t Ae e
n
n
n
1
1 2
1
.
.. 2 2
. 1 1
^
x
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Optical and Acoustic Phonons: Special Cases
x
M 1 M 2
2^ 1
a 1 (^) a x ˆ
Case II: Optical Mode , 1 2 , 1 2
x M M M a
q
q x
a
a
Optical
Acoustic
M
x a
1
. 1
2
1 1
iq n e
A
x a
u
x a
u
x
iq ^ R d ^ i t
iqn iq R d i t n
iq R d i t n
Ae e
u R d t Ae e e
u R d t Ae e
n
n
n
1
1 2
1
.
.. 2 2
. 1 1
^
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Counting and Conserving Degrees of Freedom
- We started with 2 N degrees of freedom which were related to the motion in 1D of 2 N
different atoms
- The dynamical variables were the amplitudes of the displacements of 2 N different
atoms
u 1 R (^) n d 1 , t u 2 R (^) n d 2 , t n 0 , 1 , 2 ....... N 1
We then ended up with lattice waves:
- There are N different modes per phonon band
corresponding to the N different possible wavevector
values in the first BZ
- There are 2 phonon bands and therefore a total of 2 N
different phonon modes
- The dynamical variables are the amplitudes of the 2 N
different phonon modes
The number of degrees of freedom are the same before and after – as they should be!
q x
a
a
FBZ
N a
x
M 1 M 2
a 1 (^) a x ˆ
Optical
Acoustic
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Counting and Conserving Degrees of Freedom
The atomic displacements,
taken together provide a complete description of the motion of all the 2 N atoms in the crystal
In general, one can expand the atomic displacements in terms of all the lattice wave
modes – all wavevectors and all bands:
^
A Oq
iqR iqd
iqd iqR iqd
iqd
A Oq
iqR i qt iqd
iqd iqR i qt iqd
iqd
A Oq
iqR i qt iqd
iqd
n
n
n n
n n
n
e u qte
u qte e u qte
u qte
e e u qe
u qe e e u qe
u qe
e e u qe
u qe
uR d t
uR dt
, inFBZ
*. . 2
*.
. 1 . 2
. 1
, inFBZ
*. . 2
*.
.^1 . 2
. 1
, inFBZ
. . 2
. 1
2 2
1 1
2
1
2
1
2
1
2
1
2
1
,
,
2
1
,
,
2
1
2
1
2
1
Re ,
,
Therefore, the lattice wave amplitudes also provide a complete description of the
motion of all the atoms in the crystal
, , 0 , 1 , 2 ....... 1 1 1 2 2 u R d t u R d t n N n n
