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Phonon Engineering:Phonon Engineering: an introduction – part 1an introduction part 1
C M Sotomayor Torres, P.-O. Chapuis, F Alzina, D Dudek, J Cuffe and L Schneider
Catalan Institute of Nanotechnology (CIN2-ICN-CSIC)
NISP Summer School on “Energy Harvesting at the micro- and nano-scale”
1
NISP Summer School on Energy Harvesting at the micro- and nano-scaleAvigliano Umbro TR, Italy, 1st-8th August 2010
Collaborators• N Kehagias, V Reboud - ICNg ,
• J Ahopelto, M Prunnila – VTTp ,
• E H El Boudouti – U Oujda & IEMN, U Lillej ,
• B Djafari-Rouhani - IEMN, U Lillej ,
•A Zwick, J Groenen, F Poinsotte, A Mlayah – U Paul , , , ySabatier (LPST now part of CNRS-CEMES)
2
SupportSupport
www.nanopack.org
3www.tailphox.org
The Phononic and Photonic N t t G (P2N)Nanostructures Group (P2N)
Vi tFrancesc VincentReboud
NikolaosKehagias
TimothyKehoe
FrancescAlsina Oliver
ChapuisDamian Dudek
And
Clivia M Sotomayor
TorresJohn Yamila Lars Naomi Emigdio
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TorresCuffeGarcia Schneider Baruch
gChavez
OutlineOutline• MotivationMotivation• Taxonomy of phonons• Phononic crystals• Phononic crystals• Phonon cavities
Ph tifi• Phonon rectifiers• Inelastic light scattering
- Confined acoustic phonons• Thermal transport (part 2-O Chapuis)p (p p )
– the 3-omega methodConclusions
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Conclusions
Phonons in solidsKey excitation in energy and momentum relaxation in, e.g, semiconductor quantum wells wires and dots (1 THz ~ 4 meV)
(meV)
semiconductor quantum wells, wires and dots (1 THz 4 meV)
Typical dispersion relations
TOLO
65-30of optical phonons and
TALA
TO
acoustic phonons TA
up to~30
p
/2a k in bulk material
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Focus on acoustic phononsFocus on acoustic phononsReasons to study THz acoustic phonon engineering:
- Imaging with nm resolution acoustic wavelength ~ 5nm at 1 THz5nm at 1 THz
- Thermal dissipation- Thermoelectricityy- Optoelectronic devices modulation at high speed- Phonon caustics
C t l f d h- Control of decoherence- Detrimental role in nanoelectronic devices (mobility)- NanometrologyNanometrology- Design of density of phonon states- A candidate for a non-charge state variable
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Phonon anharmonic decayPhonon anharmonic decay
Optical phonons Acoustic phonons(f V)(10’s of meV) (few meV)
optac ~ 5 ps in Si
e-opt ph~ 100s fsOptical phonon
Acoustic hOptical phonon
emissionhigh-field Joule heating
phonons carry heat away from hot spots
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hot spots
“Phonon bottleneck” in deep etched quantum dots
- Intrinsic model of luminescence yield in decoupled 1- and 0-D semiconductors.
wires and dots- wires and dots decoupled from surrounding (unlikesurrounding (unlike self-organised ones)
9 H Benisty, C M Sotomayor Torres & C Weisbuch, Phys Rev B 44 (19) 10945 (1991)
Effect of phonon confinement on ZT of quantum wells
A Balandin and K L Wang 1998
See also, M.S. Dresselhaus et al, Adv Mat 19, 1043
10
(2007).
Electronic, Photonic and Phononic crystals
Common key parameters: periodicity contrast filling factor geometry
11 Adapted from Sigalas et al Z. Kristallogr, vol. 220, pp. 765–809, 2005
Common key parameters: periodicity, contrast, filling factor, geometry
OutlineOutline• MotivationMotivation• Taxonomy of phonons• Phononic crystals• Phononic crystals• Phonon cavities
Ph tifi• Phonon rectifiers• Inelastic light scattering
- Confined acoustic phonons• Thermal transport p
– the 3-omega methodConclusions
12
Conclusions
Phonons in Low Dimensional Systems– Interface phonons – Zone-folded acoustic phononso e o ded cous c p o o s– Geometrically confined surface modes– Localised and or confined phononsLocalised and or confined phonons– Surface acoustic waves– Acoustic cavity modesAcoustic cavity modes
– …
Treat as standing waves (eg, confined) or propagating waves (eg., zone-folding in superlattices)
“Phonon Raman scattering in semiconductors, quantum wells and superlattices: Basic results and applications “ Tobias Ruf, Springer (Berlin, New York) 1998.
“Ph i S i d t N t t ” Ed J P L b t J P l d C M
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“Phonons in Semiconductor Nanostructures”, Eds J-P Leburton, J Pascual and C M Sotomayor Torres, Kluwer Publishing, The Netherlands, 1993
Folded acoustic phonons in superlattices
F iFor a review on vibrations in superlattices, see B. Jusserand and M. Cardona, in Light Scattering in Solids, edited by M Cardonaby M. Cardona and G. Güntherodt(Springer-Verlag,
14 From Lecture notes of B Jusserand, Son et Lumiere 2006
( p g gHeidelberg, 1989), p. 49.
Surface Phonons in semiconductor cylinders 1Surface Phonons in semiconductor cylinders 1(nh)2/()2=( m nh)/( m nh),
El t t ti ti d l f R i d
with nh in terms of modified Bessel functions and derivatives.
Electrostatic continuum model of Ruppin and Engelman (1970) with geometry determined by boundary conditions, neglecting retardation effects.
GaAs pillar 100 nm diameter 700 nm high
15 M Watt et al Semicond Sci Technol 5, 285-290 (1990)
700 nm high.
Surface Phonons in semiconductor cylinders 2GaAs cylinders of 80 nm diameter and 250 nm high.
Contribution of surface phonons to the RamanContribution of surface phonons to the Raman signal appear between the TO and LO phonons as expected.
Pillars coated with SiN: surface phonon frequencies
16
surface phonon frequencies decrease.
M Watt et al Semicond Sci Technol 5, 285-290 (1990)
OutlineOutline• MotivationMotivation• Taxonomy of phonons• Phononic crystals• Phononic crystals• Phonon cavities
Ph tifi• Phonon rectifiers• Inelastic light scattering
- Confined acoustic phonons• Thermal transport p
– the 3-omega methodConclusions
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Conclusions
Phononin crystal types
Typical phonon frequency range in solids:
Phononin crystal types
Typical phonon frequency range in solids:(0 – 45 THz)
– Sonic crystals (≤ kHz)y ( )– Ultrasonic crystals (kHz – MHz)
Hypersonic crystals (GHz THz)– Hypersonic crystals (GHz - THz)– Mixed optical and phononic crystals (several
TH )THz)
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Phononic crystals– Acoustic and elastic analogues of photonic crystals
y
– ‘stop bands’ in phonon spectrum (phonon mirrors); – ‘negative refraction’ of phonons (phonon caustics)g p (p )– Good theory available: Multiple scattering theory
for elastic and acoustic waves See for example:for elastic and acoustic waves. See, for example:Kafesaki & Economou PRB 60, 11993 (1999), Li t l PRB 62 2446 (2000)Liu et al PRB 62, 2446 (2000)Psaroba et al PRB 62, 278 (2000).
A d f d t b 2006 2008And for a database 2006-2008:http://www.phys.uoa.gr/phononics/PhononicDatabase.html
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2D infinite phononic crystal: air holes in siliconmatrix (B Djafari-Rouhani Y Pennec IEMN U Lille)matrix (B Djafari Rouhani, Y Pennec, IEMN, U Lille)
Sq are H l H b
y 1.0triangular, f=0.6
y 1.0square, f=0.3
y 1.0honeycomb, f=0.3
Square Hexagonal Honeycomb
ed fr
eque
ncy
0.4
0.6
0.8
ed fr
eque
ncy
0.4
0.6
0.8
ed fr
eque
ncy
0.4
0.6
0.8
wavenumber
redu
ce0.0
0.2
X J Xwavenumber
redu
ce
0.0
0.2
M X Mwavenumber
redu
ce
0.0
0.2
X J Xwavenumber
ncy
0 8
1.0triangular, f=0.85
wavenumber
ncy
0 8
1.0square, f=0.6
wavenumber
ency
0.8
1.0honeycomb, f=0.6
uced
freq
ue
0 2
0.4
0.6
0.8
uced
freq
ue
0 2
0.4
0.6
0.8
duce
d fre
que
0 2
0.4
0.6
20 wavenumber
red
0.0
0.2
X J Xwavenumber
red
0.0
0.2
M X Mwavenumber
red
0.0
0.2
X J X
Imaging with phononic crystals - I
21 S Yang, J Page, et al, PRL 93, 024301 (2004)
Imaging with phononic crystals - II
22 S Yang, J Page, et al, PRL 93, 024301 (2004)
OutlineOutline• MotivationMotivation• Taxonomy of phonons• Phononic crystals• Phononic crystals• Phonon cavities
Ph tifi• Phonon rectifiers• Inelastic light scattering
- Confined acoustic phonons• Thermal transport p
– the 3-omega methodConclusions
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Conclusions
Length scales in SiLength scales in SiPhonon MPFi b lk Si 41 @ RT D b d lin bulk Si = 41 nm @ RT Debye model
260 nm considering dispersion300 nm (Ju & Goodson APL 1999)300 nm (Ju & Goodson, APL 1999)
(cf Electron MFP = 7.6 nm)Dominant phonon wavelength
/ fd = vs / fd in Si d = 1.4 nm @ RT = 4000 nm @0.1
velocity ofsound 1.48 /kBT
To confine phonons in the strong regime at RT need structures with
24From A Balandin, UC Riverside
sound~ 1-10 nm lateral dimensions
Phonons in MEMS and NEMS
Predicted in-plane dispersion curves for dilatational modes in a 10 nm Si freedilatational modes in a 10 nm Si free
standing membrane Example of a MEMS / NEMS
90cm-1
A Erbe et al Appl Phys Lett 77
0
A. Erbe et al., Appl. Phys. Lett 77 (2000) 3102.
25 (Balandin & Wang PRB 581554 (1998))
Confined Acoustic Phonons: Phonon cavity approachConfined Acoustic Phonons: Phonon cavity approach
q in plane
q
q// in-plane (small)
SOISOIair
vsound
qz
+ SOI
BOX
V / Vtransverse longitudinal
qz
2V2/1V1
idensityPhonons in SOI/BOX Longitudinal acoustic mismatch = 0.69 (significant)
Vi=sound velocity
Transverse acoustic mismatch = 0.99 (almost negligible)
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Cavity interference effects: phonons and electronsCavity interference effects: phonons and electrons
Interaction via deformation potential p
SurfaceDisplacement field
Dis
0
d
stance to surface
electrons planePlane Waves
Standing Waves
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Waves WavesHuntzinger et al 2000, Cazayous et al 2002
Observation of confined modes: Raman scattering
EEi
Coupled optical and EsEip
phonon cavities
105
M. Trigo et al., PRL 89, 227402 (2002)
28
g , , ( )P. Lacharmoise et al., APL 84, 3274 (2004)N. D. Lanzillotti-Kimura, PRL 99, 217405 (2007)
From: A. Fainstein – presented at ICREA Workshop - May 2010 – Barcelona
Opto-mechanical structures
29 M Eichenfield et al. Optomechanical Crystals, Nature 462, 78-82 (2009
Optical forces control mechanical modes prospects for cooling, heating, …
OutlineOutline• MotivationMotivation• Taxonomy of phonons• Phononic crystals• Phononic crystals• Phonon cavities
Ph tifi• Phonon rectifiers• Inelastic light scattering
- Confined acoustic phonons• Thermal transport p
– the 3-omega methodConclusions
30
Conclusions
31
Considerations
Thermal rectification requires asymmetry and non-linearity.
Asymmetry as in pyramids in the plane or 2D arrays of offset trianglestriangles. Also coupled phonon cavities with mismatched modes.
Non-linearity: distorted distribution functions
Model by ray tracing or Monte Carlo techniques or displaced Bose-Einstein distribution or, better still, by Landauer-Buettiker theoryBuettiker theory.
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Solid state rectifier
Maximum rectification = 7% for a mass ratio of 5
H=heater, s= sensorL=low massH hi hH= high mass
33 C W Chang et al, Science 314, 1121 (2006)
OutlineOutline• MotivationMotivation• Taxonomy of phonons• Phononic crystals• Phononic crystals• Phonon cavities
Ph tifi• Phonon rectifiers• Inelastic light scattering
- Confined acoustic phonons• Thermal transport p
– the 3-omega methodConclusions
34
Conclusions
Raman scattering in semiconductorsRaman scattering in semiconductorsRaman scattering is one type of inelastic light scattering.
cb
i id tScattered light
gES
Sincident
lightlScattered light
Bandgap luminescence
vbOthers: Rayleigh,
Bandgap luminescence,Unwanted in Raman scatt experiments
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Others: Rayleigh, Brillouin scattering
Considerations
Momentum conservation: Typical laser line = 4880Åincident 4880Å
4.33(GaAs);2 nnk
Thus15106.5 cm
maximum momentum transfer is twice thismaximum momentum transfer is twice this,i.e., 1.1 x 106cm-1
The extent of the Brillouin Zone /awith a = lattice parameter = 5.65Å for GaAs
B.Z. boundary ~ 6 x107cm-1
Th i f B ZThus maximum wavevector transfer << B.Z. extent First order approximation: Raman scattering
probes excitations with q0
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probes excitations with q0.
Thin SOI film sample cross-section
Native oxide 3 nm
Buried (thermal) oxide (SiO2) 400 nm
SOI 40 nm
28 nmBuried (thermal) oxide (SiO2) 400 nm
Base Si wafer CZ p-type <100> 525 micrometer
37
Sample fromSotomayor Torres et al phys stat sol c , 1, 2609 (2004)
Raman spectra of thin SOI films
1500
1750
2000
ity (u
a)
Silicon thickness 34nm
500
750
1000
1250R
aman
Inte
nsity
Silicon thickness 34nm
Black - experimentRed simulation
300 K, 413.1
0 10 20 30 40 50 60 700
250
500
Ra Red - simulation
413.1 nm, // polarisation 500
600
700
sity
(ua)
200
300
400
Ram
an In
tens
ity
Silicon thickness 30.5nm
0 10 20 30 40 50 60 70
Δω (cm -1
)
0
100R
38
Δω (cm )J Groenen et al PRB 77, 045420 (2008)
Confined LA phonons vs. thin film thicknessConfined LA phonons vs. thin film thickness
39Sotomayor Torres et al unpublished
Simulations Raman spectra SOI thin filmS u at o s a a spect a SO t
• Photoelastic model 2)(Photoelastic model for scattering by LA phonos
* )().(...)(
zzzpEEdzI SLqz
EL (ES) : laser (scattered) fieldp(z) : photoelastic constant
Φ1(z) oxide
p(z) : photoelastic constantΦ(z) : phonon displacementΦ2(z) silicon
Φ3(z) oxide
Silicon buffer
40
J Groenen et al PRB 77, 045420 (2008)
• Vibrational part
- phonons displacement and stress boundary conditions )()(
)()(
/2
2/1
1
/2/1
SiOSiO
SiOxSiOx
zCzC
zz
p
{stress boundary conditions )()( /2/1 SiOxSiOx zz
Czz
C
ii- Hypothesis
{Phonons stationary waves
Free surface
ziqziq eBeAz 11111 )(
0)(1C Free surface
Dispersion relation
0)( /1
1 Oxairzz
C
vqsound velocity
Vac(oxide) =5970 m s-1
• Electronic part
Infinite silicon buffer
0)(zP
aczqz vq . Vac(oxide) =5970 m.s-1
Vac(silicon) =8433 m.s-1
{• Electronic part1)(0)(
zPzP
Si
Ox{41
J Groenen et al PRB 77, 045420 (2008)
Si l i f Sili hi k i iSimulations of Silicon thickness variationSOI simulations
silicon thickness variation
1500
300
(ua)
1500
man
(ua)
100
200
Inte
nsité
Ram
an (
ua
1000
Inte
nsité
Ram
a
0 10 20 30 40 50
Δω (cm-1
)
0
500
In
-50 -40 -30 -20 -10 0 10 20 30 40 500
42
Δω (cm-1
)F Poinsotte et al unpublished
6000
SOI simulationoxide thickness variationSimulation of oxide thickness variation
5000
6000
oxide : 0nm
4000
5000
(ua)
2nm
4nm
3000
sité
Ram
an (u
a
6nm
8nm
2000Inte
nsité 10nm
12nm
14nm
1000
14nm
16nm
18nm
-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70
(cm-1
)
0
18nm
20nm
43
Δω (cm-1
)F Poinsotte et al unpublished
SOI membranes and configuration
Back-scattering500 m
g
Laser spot
Forward scatteringspot
44Sotomayor Torres et al phys stat sol c , 1, 2609 (2004)
Si membrane
30 nm SOI + 400 nm BOX
45
30 SO 00 O
Examples of raw datap
Wavenumber (cm-1)
46
Wavenumber (cm )
Sotomayor Torres et al, unpublished
Simulations of Raman spectra of SOI membranes
T t SOI l it f ti h i
Simulations of Raman spectra of SOI membranes
Treat SOI layer as a cavity for acoustic phonons, ie, confined since longitudinal vs in Si = 8433 m/s (cf. 332 m/s in air at 0 C)m/s in air at 0 C).
Displacement field of acoustic vibrations in a slab of thi k t i ti l t
)(n thickness t is proportional to:
)cos( zt
n order of the confined frequencies can be derived fromn order of the confined frequencies can be derived from LA dispersion branch, considering discrete wave vectors q = n /t
47
q n /t
Simulations of Raman spectra of SOI membranes- cont.
t)(zE),(
)(tzu
ptzP zAcoustic vibration periodici i f i l i i
Simulations of Raman spectra of SOI membranes cont.
t)(z,E),( izptzP z
S variation of strain polarisation
field in presence of em wavep photo-elastic coefficient of slab
tzu ),( *),(
tzuz
ps photo elastic coefficient of slab
P(z,t) OK for anti-Stokes part.Obtain Stoke part by changing by
ztzuz
),( ),(
zzObtain Stoke part by changing by
Thus, scattered field:
2
2
22
2
2
2
2
2 ),(1),(),( tzPtzEsntzEs
22
0222 tctcz
Where n = slab index of refraction.
48
Raman spectra of 31.5 nm thick SOI membrane
Forward scatteringscattering
BackBack scattering
Wavenumber cm-1
49
J Groenen et al PRB 77, 045420 (2008)
Data from five SOI membranes
50Sotomayor Torres et al phys stat sol c , 1, 2609 (2004)
15 nm Si QW, expect shear, dilatation & flexural modes
2nd dilatation2 dilatation mode
3rd dilatation mode
51
A Balandin 2000 in-plane V-group 0 at cusps
Raman & Brillouin scattering Inelastic Light Scattering (I vs. )
vs k [~ Phononic Dispersion Relation vs. k [~ Phononic Dispersion Relation
Frequency Range
0.2 – 500 GHz> 90 GHz0.0067 – 16.6 cm-1> 3 cm-1
5282.7 ev – 2.07 meV> 0.37 meV
Conclusions and trends• Phonon engineering possible with phononic crystals, cavities, coupled cavities., p
• Phonon sources are needed for progress in the field
• A wealth of device-relevant physics expected• A wealth of device-relevant physics expected
• Nanofabrication (in 3D) and nanometrology developments neededneeded.
• “Heterogeneous” coupled cavities need better description with e g quantum mechanics and elasticity theorywith, e.g., quantum mechanics and elasticity theory.
• Phonon coherence studies in confined structures are expected to become important soon (Impulsive Ramanexpected to become important soon (Impulsive Raman scattering, see work by R Merlin, U Michigan)
53
References (see, for example, in addition to refs in slides)
An early work:D F Sheahan and R A Johnson, Crystal and mechanical oscillators, IEEE Trans Circuits and Systems, Vol CAS-22, pp 69-89, February 1975
On 2D Phononic Crystals:M.S. Kushwaha et al, Phys. Rev. Lett. 71, 2022 (1993)J. Vasseur et al, J. Phys.: Cond. Matter, 7, 8759 (1994)J Vasseur et al J Phys : Cond Matter 9 7327 (1997)J. Vasseur et al J. Phys.: Cond Matter, 9, 7327 (1997)Y. Pennec et al, Optics Express (2010)
On Light scattering:Book series «Topics in Applied Physics” volumes on “Light Scattering in Solids” Ed MBook series «Topics in Applied Physics , volumes on Light Scattering in Solids , Ed M. Cardona, Springer.
On modeling of light scattering and acoustic structures:E H El Boudouti et al Surface Science Reports 64 (2009) 471594E.H. El Boudouti et al., Surface Science Reports 64 (2009) 471594.
On phonon-photon interaction:Lecture Notes of Summer Schools Son et Lumiere 2006, 2008 and 2010, http://prn1.univ-lemans fr/prn1/siteheberge/Son et lumiere/index php?page=1lemans.fr/prn1/siteheberge/Son_et_lumiere/index.php?page=1M Maldovan and EL. Thomas. Appl Phys Lett, 88(25):251907, 2006.
On thermal rectifiers: C Dames J Heat Transfer vol 131 061301-1 to 061301-7 (2009)
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C Dames, J Heat Transfer, vol 131, 061301-1 to 061301-7 (2009)