MSE/ECE 310
MSE/ECE 310
Phonons
Phonons
Longitudinal Lattice Wave (phonon)
Phonon Dispersion Relation
for a monatomic lattice
vg =
s+p = specific plane
us+p = displacement
Knowlton
∂ω
= 0 at the Brillouin zone boundaries.
∂k
p = whole number
Kittel, Introduction to Solid State Physics, 6th Ed (Wiley, 1986) Ch. 4
1
Knowlton
Kittel, Introduction to Solid State Physics, 6th Ed (Wiley, 1986) Ch. 4
2
1
MSE/ECE 310
MSE/ECE 310
Phonons
Phonons
Phonon Dispersion Relation for Diatomic Linear Lattice
1
⎡ ⎛ 1
1 ⎞⎤ 2
⎢ 2c ⎜ +
⎟⎥
m
m
2 ⎠⎦
⎣ ⎝ 1
2c
m2
2c
m1
−
π
π
a
a
McKelvey, Solid State Physics for Engineering & Materials Science, (Krieger, 1993) Ch. 3
Knowlton
Kittel, Introduction to Solid State Physics, 6th Ed (Wiley, 1986) Ch. 4
3
Knowlton
4
2
MSE/ECE 310
MSE/ECE 310
Phonons
Phonons
u
k
u & k Are perpendicular
for a transverse phonon
Blakemore, Solid State Physics, 2nd Ed (Cambridge, 1985) Ch. 2
‰ Whether a vibrational mode is longitudinal or transverse, a
given atomic amplitude of motion requires much more
energy for a long wave optical than acoustic mode.
‰ Why? B/c optical modes minimize changes in 2nd n.n.
separation by maximizing separation between n.n.’s.
‰ N.n. interaction greater than any other n.n. interaction,
therefore, energy required for n.n. separation greater than
any other separation.
Knowlton
‰ U = dF/dx integrate under curves to get energy.
5
Knowlton
6
3
MSE/ECE 310
MSE/ECE 310
Phonons Dispersion Curves
Phonons
P.Y. Yu & M. Cardona, Fundamentals of Semiconductors, (Springer, 1996) Ch. 3
Knowlton
Blakemore, Solid State Physics, 2nd Ed (Cambridge, 1985) Ch. 2 7
Knowlton
8
4
MSE/ECE 310
MSE/ECE 310
Phonon Imaging
Phonons
‰ Phonon Transmission study:
Phonon Combination Bands
(two phonon overtone bands)
9 65% transmission of ballistic phonons
fast transverse
<111>
slow transverse
<110>
unbonded
border
side
between top & bottom
unbonded
side
Blakemore, Solid State Physics, 2nd Ed (Cambridge, 1985) Ch. 2
Dark Matter Detector Sensing Phonon Using
Ge:Au:Ge Eutectic Bonding
Knowlton
9
Knowlton
10
5
MSE/ECE 310
MSE/ECE 310
Phonon Imaging
Phonon Imaging
‰ Phonon Transmission study:
‰ Phonon Transmission study:
9 Sample configuration
9 Time of flight
2750
Integration: FTA Spectrum - Side Without Interface
Area
Peak at
Width
Height
377.64547 1.57617
0.21094 1269
2500
Ballistic
900μm
Integration: FTA Spectrum - Side With Interface
Area
Peak at
Width
Height
693.59764 1.49414 0.19922 2515
2000
1750
% Phonon Transmission =
Area1/Area2 x 100 = 55%
1500
Laser Pulse
3mm
4mm
Transition
Edge
Detector
Phonon
PhononImage
ImageSide
Side
380μm
E
In utec
te ti
rfa c
ce
Bolometer Intensity (arbitrary units)
2250
1250
1000
FTA TOF - No Bond (A1)
FTA TOF - With Bond (B1)
750
500
250
0
-250
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
Time (μs)
•
•
•
•
Laser
LaserPulse
PulseScan
ScanSide
Side
Dark Matter Detector Sensing Phonon Using
Ge:Au:Ge Eutectic Bonding
Knowlton
11
Knowlton
Fast tranverse optical phonon
Slow transverse optical phonon
If we examine TOF data from phonon imaging study
Notice that time at peak max is no more than 1 μs
Dark Matter Detector Sensing Phonon Using
Ge:Au:Ge Eutectic Bonding
12
6
MSE/ECE 310
MSE/ECE 310
Phonon Imaging
Phonons – Local Vibrational Modes
‰ Phonon Transmission study:
Si
9 Prototype Dark Matter Detector
9 Really a phonon detector
Oi
O interstitial
Si
[111]
Si
Si
Si
Energy (meV)
62
100
150
200
250
300
350
400
450
496
λ (μm)
20 15
10
5
2.5
4000
Transmittance (arb. units)
6000
Transmittance (arb. units)
5000
4000
3500
3000
2500
2000
1100
1110
1120
1130
1140
1150
1160
1170
3000
2000
1000
1136.4 cm
-1
0
Knowlton
Dark Matter Detector Sensing Phonon Using
Ge:Au:Ge Eutectic Bonding
500
Knowlton
LVM of O
13
Knowlton
1000
1500
2000
2500
-1
3000
3500
4000
Wave Number (cm )
Knowlton, U.C. Berkeley, ~1995
14
7