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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Handout 18
Phonons in 2D Crystals: Monoatomic Basis and Diatomic Basis
In this lecture you will learn:
• Phonons in a 2D crystal with a monoatomic basis• Phonons in a 2D crystal with a diatomic basis• Dispersion of phonons• LA and TA acoustic phonons• LO and TO optical phonons
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
1a
x
21 amanRnm
Phonons in a 2D Crystal with a Monoatomic Basis
y
2a
yanxan
yanxan
ˆˆ
ˆˆ
43
21
General lattice vector:
Nearest-neighbor vectors:
Atomic displacement vectors:
tRu
tRutRu
nmy
nmxnm ,
,,
Atoms, can move in 2D therefore atomic displacements are given by a vector:
yaxapyaxap
yaxapyaxap
ˆˆˆˆ
ˆˆˆˆ
43
21
Next nearest-neighbor vectors:
2
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Vector Dynamical Equations
mm
m
RRm
ˆ
12
tRu ,1
tmRutRu ,, 12
mmtRutmRummtRutRudt
tRudM ˆˆ.,,ˆˆ.,,
,11122
12
Vector dynamical equation:
If the nearest-neighbor vectors are known then the dynamical equations can be written easily.
Component dynamical equation:
To find the equations for the x and y-components of the atomic displacement, take the dot-products of the above equation on both sides with and , respectively:x
xmmtRutmRudt
tRudM x ˆ.ˆˆ.,,
,112
12
y
ymmtRutmRudt
tRudM y ˆ.ˆˆ.,,
,112
12
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Vector Dynamical Equations for a 2D Crystal
1a
x
y
2a
21 amanRnm
yanxan
yanxan
ˆˆ
ˆˆ
43
21
General lattice vector:
Nearest-neighbor vectors:
yaxapyaxap
yaxapyaxap
ˆˆˆˆ
ˆˆˆˆ
43
21
Next nearest-neighbor vectors:
4,3,2,12
4,3,2,112
2
ˆˆ.,,
ˆˆ.,,,
jjjnmjnm
jjjnmjnm
nm
pptRutpRu
nntRutnRudt
tRudM
summation over 4 nn
summation over 4 next nn
1
2
3
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
,,2
,,2
,,2
,,2
,,2
,,2
,,2
,,2
,,,,,
42
42
32
32
22
22
12
12
31112
2
tpRutRutpRutRu
tpRutRutpRutRu
tpRutRutpRutRu
tpRutRutpRutRu
tnRutRutnRutRudt
tRudM
nmynmynmxnmx
nmynmynmxnmx
nmynmynmxnmx
nmynmynmxnmx
nmxnmxnmxnmxnmx
Dynamical Equations
4,3,2,12
4,3,2,112
2
ˆˆ.,,
ˆˆ.,,,
jjjnmjnm
jjjnmjnm
nm
pptRutpRu
nntRutnRudt
tRudM
If we take the dot-product of the above equation with we get:x
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
,,2
,,2
,,2
,,2
,,2
,,2
,,2
,,2
,,,,,
42
42
32
32
22
22
12
12
41212
2
tpRutRutpRutRu
tpRutRutpRutRu
tpRutRutpRutRu
tpRutRutpRutRu
tnRutRutnRutRudt
tRudM
nmxnmxnmynmy
nmxnmxnmynmy
nmxnmxnmynmy
nmxnmxnmynmy
nmynmynmynmynmy
Dynamical Equations
4,3,2,12
4,3,2,112
2
ˆˆ.,,
ˆˆ.,,,
jjjnmjnm
jjjnmjnm
nm
pptRutpRu
nntRutnRudt
tRudM
If we take the dot-product of the above equation with we get:y
4
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Solution of the Dynamical Equations
Assume a wave-like solution of the form:
Then:
tiRqi
y
x
nmy
nmxnm ee
qu
qu
tRu
tRutRu nm
.
,
,,
tRue
eequ
que
eequ
qu
tnRu
tnRutnRu
nmnqi
tiRqi
y
xnqi
tinRqi
y
x
jnmy
jnmxjnm
j
nmj
jnm
,
,
,,
.
..
.
We take the above solution form and plug it into the dynamical equations
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Dynamical Matrix and Phonon Bands
qu
qu
M
M
qu
quqD
y
x
y
x
0
02
Compare with the standard form:
Solutions:
X M
FBZ
a2
a2
kg 2x10
N/m 100
N/m 200
26-
2
1
M
coscos122
sin4sinsin2
sinsin2coscos122
sin42
22
12
222
1
qu
quM
qu
qu
aqaqaq
aqaq
aqaqaqaqaq
y
x
y
x
yxy
yx
yxyxx
5
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transverse (TA) and Longitudinal (LA) Acoustic Phonons
242
22
42
222
2
12
222
2
2
1
qu
quM
qu
qu
aqaqaq
aqaq
aqaqaqaqaq
y
x
y
x
yxy
yx
yxyxx
Case I: 0,0 yx qq
0
121 Aqu
quaq
Mq
xy
xxxxLA
Longitudinal acoustic phonons: atomic motion in the direction of wave propagation
Transverse acoustic phonons: atomic motion in the direction perpendicular to wave propagation
1
02 Aqu
quaq
Mq
xy
xxxxTA
TA
LA
X
M
FBZ
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transverse (TA) and Longitudinal (LA) Acoustic Phonons
242
22
42
222
2
12
222
2
2
1
qu
quM
qu
qu
aqaqaq
aqaq
aqaqaqaqaq
y
x
y
x
yxy
yx
yxyxx
Case II: qqqqq yxyx 0,0
1
14 21 Aqu
quaq
Mq
y
xLA
Longitudinal acoustic phonons: atomic motion in the direction of wave propagation
Transverse acoustic phonons: atomic motion in the direction perpendicular to wave propagation
1
11 Aqu
quaq
Mq
y
xTA
TA
LA
X
M
FBZ
TA
LATA
LA
6
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transverse (TA) and Longitudinal (LA) Acoustic Phonons
TA
LATA
LATA
LAIn general for longitudinal acoustic phonons near the zone center:
y
x
y
x
q
q
qA
qu
qu
And for transverse acoustic phonons near the zone center:
x
y
y
x
q
q
q
Aqu
qu
In general, away from the zone center, the LA phonons are not entirely longitudinal and neither the TA phonons are entirely transverse
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transverse (TA) and Longitudinal (LA) Acoustic Phonons
LATA
TA and LA
7
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Periodic Boundary Conditions in 2D
tiRqi
y
x
nmy
nmxnm ee
qu
qu
tRu
tRutRu nm
.
,
,,
Our solution was:
Periodic boundary conditions for a lattice of N1xN2 primitive cells imply:
22
where
21
21
where22
integer an is where2.
1
,,
11
1
11
1111
1111
.
..11
11
11
N
mN
-Nm
-mN
mmaNq
e
eequ
qutRuee
qu
qutaNRu
aNqi
tiRqi
y
xnm
tiaNRqi
y
xnm
nmnm
21 amanRnm
General lattice vector:
21,21 212211 bbq
General reciprocal lattice vector inside FBZ:
Similarly:
22 where 2
22
2
22
Nm
N-
Nm
FBZ
1b2b
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Counting Degrees of Freedom
In the solution the values of the phonon wavevector are dictated by the periodic boundary conditions:
2211 bbq
22 where 11
1
11
Nm
N-
Nm
22 where 2
22
2
22
Nm
N-
Nm
There are N1N2 allowed wavevectors in the FBZ(There are also N1N2 primitive cells in the crystals)
There are N1N2 phonon modes per phonon band
Counting degrees of freedom:
• There are 2N1N2 degrees of freedom corresponding to the motion in 2D of N1N2 atoms
• The total number of different phonon modes in the two bands is also 2N1N2
FBZ
8
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Phonons in a 2D Crystal with a Diatomic Basis
Atomic displacement vectors:
tdRu
tdRu
tdRu
tdRu
tdRu
tdRu
nmy
nmx
nmy
nmx
nm
nm
,
,
,
,
,
,
22
22
11
11
22
11
The two atoms in a primitive cell can move in 2D therefore atomic displacements are given by a four-component column vector:
1a
x
y
2a
1
2
21 amanRnm
2
ˆˆ
2
ˆˆ2
ˆˆ
2
ˆˆ
43
21
yaxah
yaxah
yaxah
yaxah
1st nearest-neighbor vectors (red to blue):
yanxan
yanxan
ˆˆ
ˆˆ
43
21
2nd nearest-neighbor vectors (red to red):
3rd nearest-neighbor vectors (red to red):
yaxapyaxap
yaxapyaxap
ˆˆˆˆ
ˆˆˆˆ
43
21
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Diatomic Basis: Force Constants
plus
1a
x
y
2a
1
The force constants between the 1st
2nd and 3rd nearest-neighbors need to be included (at least)
1a
x
y
2a
2
3
9
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
4,3,2,111113
4,3,2,111112
4,3,2,1111212
112
1
ˆˆ.,,
ˆˆ.,,
ˆˆ.,,,
jjjnmjnm
jjjnmjnm
jjjnmjnm
nm
pptdRutpdRu
nntdRutndRu
hhtdRuthdRudt
tdRudM
summation over 4 1st nnsummation over 4 2nd nn
Diatomic Basis: Dynamical Equations
4,3,2,122223
4,3,2,122222
4,3,2,1222212
222
2
ˆˆ.,,
ˆˆ.,,
ˆˆ.,,,
jjjnmjnm
jjjnmjnm
jjjnmjnm
nm
pptdRutpdRu
nntdRutndRu
hhtdRuthdRudt
tdRudM
summation over 4 1st nnsummation over 4 2nd nn
Dynamical equation for the red(1) atom:
Dynamical equation for the blue(2) atom:
summation over 4 3rd nn
summation over 4 3rd nn
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Diatomic Basis: Dynamical Equations
tiRqi
dqiy
dqix
dqiy
dqix
nmy
nmx
nmy
nmx
nm
nm ee
equ
equ
equ
equ
tdRu
tdRu
tdRu
tdRu
tdRu
tdRunm
.
.2
.2
.1
.1
22
22
11
11
22
11
2
2
1
1
,
,
,
,
,
,
Assume a solution of the form:
qu
ququ
qu
M
M
M
M
qu
ququ
qu
qD
y
x
y
x
y
x
y
x
2
2
1
1
2
2
1
1
2
2
2
1
1
000
000
000
000
To get a matrix equation of the form:
10
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
The Dynamical Matrix
aqaq
aq
yx
x
coscos12
2sin42
3
221
2cos
2cos2 1
aqaq yx
2sin
2sin2 1
aqaq yx
2cos
2cos2 1
aqaq yx
2sin
2sin2 1
aqaq yx
2cos
2cos2 1
aqaq yx
2sin
2sin2 1
aqaq yx
2sin
2sin2 1
aqaq yx
2cos
2cos2 1
aqaq yx
qu
ququ
qu
M
M
M
M
qu
ququ
qu
qD
y
x
y
x
y
x
y
x
2
2
1
1
2
2
1
1
2
2
2
1
1
000
000
000
000
aqaq
aq
yx
y
coscos12
2sin42
3
221
aqaq
aq
yx
x
coscos12
2sin42
3
221
aqaq
aq
yx
y
coscos12
2sin42
3
221
aqaq yx sinsin2 3
aqaq yx sinsin2 3
aqaq yx sinsin2 3
aqaq yx sinsin2 3
qD
The matrix is:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Diatomic Basis: Solution and Phonon Bands
N/m100
N/m200
N/m300
kg1042
3
2
1
2621
MM
For calculations: Opticalbands
Acousticbands
One obtains:
- 2 optical phonon bands (that have a non-zero frequency at the zone center)
- 2 acoustic phonon bands (that have zero frequency at the zone center)
TA
LA
LO
TO
TA
LA
LO
TO
X M
FBZ
a2
a2
X
TA
LA
TOLO
11
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Longitudinal (LO) and Transverse (TO) Optical Phonons
Case I: 0,0 yx qq
0
0
1
20
21
2
2
1
1
1MM
A
qu
ququ
qu
Mq
xy
xx
xy
xx
rxLO
212
2
1
1
101
0
20
MM
A
qu
ququ
qu
Mq
xy
xx
xy
xx
rxTO
Longitudinal optical phonons: atomic motion in the direction of wave propagation and basis atoms move out of phase
Transverse optical phonons: atomic motion in the direction perpendicular to wave propagation and basis atoms move out of phase
X
M
FBZ
21
111MMMr
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Longitudinal (LA) and Transverse (TA) Acoustic Phonons
Case I: 0,0 yx qq
0
10
1
?0
2
2
1
1
A
qu
ququ
qu
q
xy
xx
xy
xx
xLA
1
01
0
?0
2
2
1
1
A
qu
ququ
qu
q
xy
xx
xy
xx
xTO
Longitudinal acoustic phonons: atomic motion in the direction of wave propagation and basis atoms move in phase
Transverse acoustic phonons: atomic motion in the direction perpendicular to wave propagation and basis atoms move in phase
X
M
FBZ
12
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Counting Degrees of Freedom and the Number of Phonon Bands
2211 bbq
22 where 11
1
11
Nm
N-
Nm
22 where 2
22
2
22
Nm
N-
Nm
There are N1N2 allowed wavevectors in the FBZThere are N1N2 phonon modes per phonon band
Counting degrees of freedom:
• There are 4N1N2 degrees of freedom corresponding to the motion in 2D of 2N1N2 atoms (2 atoms in each primitive cell)
• The total number of different phonon modes in the four bands is also 4N1N2
Periodic boundary conditions for a lattice of N1xN2 primitive cells imply: