Nanophononics: Phonons in
Nanostructures and Novel Materials
Alexander A. Balandin
Nano-Device Laboratory
Department of Electrical Engineering
University of California – Riverside
http://ndl.ee.ucr.edu/
Institut des NanoSciences de Paris
University of California – Riverside
UCR
UCR Engineering – II
Building
UC-Riverside Botanic Gardens
UCR Bell Tower
Joshua Tree Park, California
City of
Riverside
2
Outline
Š Motivations
Š Phonon Engineering Concept
z
GeSi/Ge QDS
Phonon dispersion and scattering modification
Š Semiconductor Quantum Dot Superlattices
z
z
z
Š
Applications
Measurements of thermal and electrical conductivity
Theoretical interpretation: Callaway’s model vs phonon hopping
30 nm
Hybrid Bio-Inorganic Nanostructures
z
z
z
Plant viruses as nano-teplates
Optical phonon modes
Phonon confinement effects on electron mobility
Š Optical Phonons in ZnO Nanostructures
z
Not every shift is confinement shift
Š Phonon Scattering GaN Heterostructures and Devices
z
z
Thermal conductivity measurements
Self-heating and ambient temperature effects on the device
performance
metal-coated virus
3
Understanding and Controlling Phonons
Property
Electrons
Photons
Phonons
Tuning
Parameters
EG of
constituents
εr of
constituents
ζ=ρVs of
Scale
0.5 nm -10 nm
0.1 μm – 1 cm
5 nm – 1 cm
Waves
Ψ electron de
Broglie wave
EM waves or
light
u vibrational or
sound waves
Schrodinger
Maxwell
Elastic
Continuum
Governing
Equation
Bulk Limit
h2k 2
E=
2m
ω=
c
ε
k
constituents
ω = c l ,t k
4
Nanophononics and Phonon Engineering
Š
Phonons affect all thermal, electrical and
optical phenomena in semiconductors
Š
„
Heat is carried by acoustic phonons
„
Electron mobility is limited by phonons
„
Optical response is affected by phonons
„
Spin coherence is influenced by phonons
„
Noise is affected by phonon
Nanostructures offer a novel way for tuning
Band-gap
engineering:
mismatch of EG
EG
Phonon engineering:
mismatch of Ζ=ρVsound
phonon transport other than via
temperature or surface roughness
„
Z1
Modification of phonon dispersion
Acoustic Impedance:
Z=ρVs [Pa s/m] or [kg/m2s]
Z2
ρ – density of the material [kg/m3]
Vs – velocity of sound [m/s]
Balandin Group
5
Confined Phonons Bibliography
Š Lamb (1880s)
z
z
Lamb’s constants; Lamb modes: breathing spheroidal, torsional and
ellipsoidal
Interpretation of Raman spectra: (breathing-longitudinal modes
Š Rytov (1950s)
z
z
Folded phonons in superlattices
Interpretation of Raman spectra
Š M. Stroscio, V. Mitin, S. Bandyopadhya (1980s)
z
Confined electron – confined phonon scattering rates: some effect
Š N. Nishiguchi (1980s)
z
Confined electron – confined phonon scattering rates: no effect
Š N. Perrin (1990s)
z
Anderson localization of phonons
Š L.G. Rego, G. Kirczenow, M.L. Roukes (1990s)
z
Quantum of thermal conductance: Ko=(π2/3)(kB2/h)T
Note: the list is not conclusive
6
Length Scale Considerations
why phonon confinement effects are important now?
Length Scale
Phonon Dispersion
Dominant
Scattering
Processes
Acoustic phonon MFP in bulk
crystalline Si (T=300K)
41 nm - 260 nm
L >> MFP
ŠBulk dispersion
ŠUmklapp
ŠPoint defects
Comparison: electron MFP in Si: 7.6 nm
λ<< L~< MFP
ŠBulk dispersion
ŠUmklapp
ŠPoint defects
ŠBoundary
Dominant phonon wavelength in Si
λ ~<L<< MFP
ŠModified dispersion
ŠMany branches
populated
ŠUmklapp
ŠPoint defects
ŠBoundary
L< λ
ŠModified dispersion
ŠFew branches populated
ŠBallistic
1.4 nm at T=300 K or 4 μm at T=0.1 K
modified from IBM picture
Dominant phonon wavelength λd is defined as λd=(VS/fd), where
VS is the sound velocity and the dominant frequency given by
−1
f d = 4.25(kB / h)T ≈ (90GHz ⋅ K )T .
Balandin Group
Device Feature Sizes (Year 2005)
CMOS gate length 50 nm
CMOS gate oxide thickness 1.2 nm
Superlattice period: 1.5 nm
7
Bulk vs Confined Phonons
Bulk Semiconductor
Si: 64.3 meV in Γ
40
35
Ultra-Thin Free-Standing Films
Si quantum well: W=10 nm
free surface boundary
ENERGY (meV)
30
25
20
15
10
π/a ~ 6.3 (1/nm) Æ
bulk mode: dashed line
5
0
0.0
Calculation of confined phonon modes:
r
r
∂ 2u
2 2r
2
2
(
)
(
)
=
s
∇
u
+
s
−
s
grad
div
u
t
l
t
∂t 2
Si: VL=8.4-9.0 km/s and VT=5.3-5.8 km/s
0.2
0.4
0.6
0.8
1.0
PHONON WAVE VECTOR (1/nm)
TA Æ SM (shear)
LA Æ DM (dilatational)
FM (flexural)
8
Terminology Issues
optical phonon confinement vs optical phonon localization vs acoustic phonon
confinement vs. folded acoustic phonons
Confinement of Optical Phonons
Folded Acoustic Phonons in Quantum Well Superlattices
ω (rad/s)
ω (rad/s)
ω=ω1
ω=Vq
q
wave vector q (1/nm)
ω (rad/s)
π/D
wave vector q (1/nm)
π/a
Duplets in Raman spectra
ω=ω2
wave vector q (1/nm)
bulk optical modes can be confined to
certain layers, acoustic phonons in bulk
case are not confined.
Rytov (1956) Model for Thinly Laminated Medium
⎛ ωD ⎞ ⎛ ωD2 ⎞ 1 + ς 2
⎛ ωD ⎞ ⎛ ωD2 ⎞
⎟⎟ −
⎟⎟,
cos(qD) = cos⎜⎜ 1 ⎟⎟ cos⎜⎜
sin⎜⎜ 1 ⎟⎟ sin⎜⎜
V
V
V
V
2
ς
1
2
1
2
⎝
⎠ ⎝
⎠
⎝
⎠ ⎝
⎠
Vi=(C/r)1/2 is the sound velocity in
each layer, and ζ=ρ2V2/ρ1V1 is the
acoustic mismatch between the
layers, D=D1+D2 is the period of the
superlattice.
9
Phonon Confinement and Quantization in
Nanowires
Dilatational phonon energy spectrum as the
functions of the phonon wave vector for the
free-surface
boundary conditions. phonon
ground levels are given for all polarizations D,
F1, F2 and S.
Balandin Group
The equation of motion for elastic vibrations
in an anisotropic medium:
∂ 2U m ∂σ mi
ρ 2 =
∂t
∂xi
10
Phonon Engineering in Nanowires with
Elastically Dissimilar Barriers
Equation of motion for elastic vibrations in
anisotropic medium [Landau-Lifshits]:
Nanowires with
Acoustically Mismatched
Barrier Shells
∂ 2U m ∂σ mi
ρ 2 =
∂t
∂xi
Tuning parameters:
Crystalline structure
Dimensions
r
U = (U1,U 2 ,U 3 ) - displacement vector
Sound velocity
σ mi = cmikjU kj
Acoustic Impedance
- elastic stress tensor
U kj = (1/ 2)((∂U k / ∂xi )
+(∂U i / ∂xk ))
Mass density
What we want to achieve:
- strain tensor
Change in thermal
conductivity
Cylindrical nanowire along Z direction:
U r ( r , ϕ , z ) = A ( q )u r ( r ) cos mϕ e
Change in electron – phonon
scattering rates
i ( ω t − kz )
Specific phonon mode
energies
U ϕ ( r , ϕ , z ) = A ( q )uϕ ( r ) sin mϕ e i (ω t − kz )
U z ( r , ϕ , z ) = A ( q )u z ( r ) cos mϕ e i ( ω t − kz )
11
Phonon Engineering in Coated Cylindrical
Nanowires
Phonon dispersion for breathing modes (m=0). The
results are shown for GaN nanowire with the
“acoustically soft” barrier layer (R1(GaN)= 6 nm and
R=10 nm). Plastic: VL=2 km/s, VT=1 km/s, ρ=1 g/cm3.
Number of modes S~R/2a
Averaged phonon group velocity for breathing
modes in GaN nanowire with the “acoustically
soft” barrier layer (R1(GaN)= 6 nm, R=10 nm)
and the uncoated GaN nanowire (R=6 nm).
E.P. Pokatilov, D.L. Nika and A.A. Balandin, JSM, 38, 168 (2005).
Balandin Group
12
Phonon Engineering in Coated Nanowires
Phonon dispersion in GaN nanowires coated with
the acoustically mismatched barrier layers.
Vs(AlN)/Vs(GaN)≈1.5 (1.3) for the trans. (long.)
(z(GaN)/z(AlN)≈1.3).
VS(GaN)≈4 km/s (8 km/s) for for the trans. (long.)
Balandin Group
Plastic or organic materials are used as
acoustically soft barriers: phonon buffer
layers
Vs=1000 m/s.
13
Redistribution of Phonon Amplitudes in
Rectangular Coated Nanowires
Displacement vector components in the cross-sectional plain of
the free-standing GaN nanowire coated with acoustically soft
material. The cross-section is 4nm x 6nm size with the GaN core of
2nm x 3nm core (q=0.4 nm-1). Æ
Note: larger
amplitudes are
in the
acoustically
soft coating
layers. Æ
Å Displacement vector magnitude for
4nm x 6nm free-standing GaN nanowire
embedded into acoustically soft material.
E.P. Pokatilov, D.L. Nika and A.A. Balandin, Phys. Rev. B, 72, 113311 (2005).
14
Confined Electron – Confined Phonon
Scattering Rate Engineering
Theoretical study of the possibility of the scattering rate suppression in
acoustically mismatched hetero- and nanostructures
AlN
Plastic
GaN
D
D<< Phonon MFP
Acoustically mismatched hetero- and
nanostructures: combining acoustically hard
and acoustically soft materials
E.P. Pokatilov, D.L. Nika and A.A. Balandin, J. Appl. Phys., 95, 5626 (2004).
E.P. Pokatilov, D.L. Nika and A.A. Balandin, Appl. Phys. Lett., 85, 825 (2004).
Balandin Group
15
Electron – Acoustic Phonon Scattering in the
Acoustically Mismatched Heterostructures
Fermi’s golden rule for the structure is written as
τ n−,1n ' (ε ) =
∞
(α )
s
d
2
( N + m )dq
1
( ∫ Gnβ,n,α' , s (q, x3 )dx3 )2
m⊥,n ' ∫
∑
2
α
α
m
(
)
(
)
2
2π h k α ,β , s
d
0 ρ s (q)ωs (q) 1 − (Δ )
1
2
1
2
−
2
h2k 2 1
1
h2q2
−
(Δ ) =
{ε − ε +
(
)+
m hωs(α ) ( q )}
2
kqh
2 m⊥ , n ' m⊥ , n
2m⊥ ,n '
m
m⊥ , n '
0
n'
0
n
Gnβ,n,α' , s (q, x3 ) = Φ (sα ), β (q, x3 )ψ n*' ( x3 )ψ n ( x3 )
Scattering mechanisms included:
Deformation potential
Piezoelectric potential
Ripple scattering
16
Thermal Conductivity of Nanowires
J. Zou and A. Balandin, J.
Appl. Phys. (2001).
BULK
Æ Predicted decrease of the thermal conductivity from 148 W/m-K in bulk to
about 13 W/m-K in 20 nm Si crystalline nanowire at T=300 K, 2001.
Æ Experimental study: ~ 9 W/m-K in 22 nm nanowire at T=300K; strong diameter
dependence; deviation from Debye T3 law at low T, Majumdar Group, UCB, 2003.
17
Quantum Dot Superlattices:
Real and Ideal
Periodic variation of the elastic constants and/or mass density
GexSi1-x
quantum
dots
Si
layer
p-type Si
substrate
In-plane regimentation of
quantum dot is not implied
by the term QDS. Periodicity
of the layers along the
growth direction is implied.
100 nm
Si layer
30 nm
Schematic of Ge/Si QDS.
Cross-sectional TEM of
MBE grown Ge/Si QDS.
y Schematic of regimented
QDS used in modeling.
Regimented quantum dot array
grown by electrochemistry.
Balandin Group
H
L
D
L
H
A.A. Balandin, et al., Appl. Phys. Lett., 76, 137 (2000).
18
Applications of Quantum Dot Superlattices
Thermoelectric Figure of Merit Enhancement in Regimented Si/Ge Quantum Dot Superlattices
ZTQDC/ZTB
QDS with reduced lattice thermal
conductivity of 15 W m-1K-1
102
Figure of Merit:
10
α 2σ T
ZT =
κ
1
10-1
10-2
10-3
mini-band
transport
regime
-0.3
-0.2
α - Seebeck coefficient
QDS with bulk
lattice thermal
conductivity of
156 W m-1K-1
-0.1
0
Fermi Energy (eV)
σ – electrical conductivity
κ – thermal conductivity
0.1
0.2
T – absolute temperature
Z – figure of merit
A.A. Balandin and O.L. Lazarenkova, Appl. Phys. Lett.,
82, 415 (2003).
Other Applications: photodetectors, photovoltaic cells,
quantum-dot electronic architectures, etc.
Balandin Group
19
Modeling Regimented Quantum Dot
Superlattices
Anisotropic Continuum Approximation for
Regimented Ge/Si QDS
Phonon Stop-Bands in Quantum Dot Superlattices
4.5
4
3.5
3
2.5
2 Si
1.5
Ge
1
0.5
0
0 0.2 0.4 0.6 0.8 1
q[[100]]
Effective medium
approximation
0.0055
0.0050
0.0045
0.0040
Energy (eV)
Energy (meV)
0.0060
0.0035
0.0030
0.0025
0.0020
0.0015
0.0010
0.0005
0.0000
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
-1
q[[111]] (nm )
1.2 1.4 1.6
(nm-1)
Dispersion is from the
anisotropic continuum.
Appearance of the phonon stop-band along
[[111]] direction around the energy of 2.7 meV
in the quantum dot superlattice with tuned
parameters.
O.L. Lazarenkova and A.A. Balandin,
Phys. Rev. B, 66, 245319 (2002).
20
Structural Characterization of GeSi/Si
Quantum Dot Superlattices
Raman Spectroscopic Microscopy
X-Ray Microanalysis of QDS
Interaction
volume: 500 nm
30 nm
λ=488 nm
From the weights of Ge and Si lines: Ge is 18.10%
and Si is 81.90%.
Æ Ge dot layers were not under very strong strain:
comparison of Si and Ge peak positions in Ge/Si QDS
with those in bulk Si (520.4 cm-1) and Ge (301 cm-1).
Æ distinguish the strain-stress effects on
the electrical and thermal conduction from
those due to the quantum dots
21
Balandin Group
Electron Mobility in Ge/Si Quantum Dot
Superlattice
Hall Mobility Measurements Results
dot density 3.5-30.0 x 108 cm-2; dot base: 40 nm –
120 nm; aspect ratio: ~10; samples: J.L. Liu (UCR).
μH=|RHσ|, where RH=(p-nb2)/[e(p+nb)2], and b=μe/μh – ratio
of drift mobilities; RH>0 – p-type conduction; B=0.37 T
Æ band-type rather than hopping type
electron conduction:
Æ μ~T-3/2 not G~Goexp{-(To/T)x}
Y. Bao, A.A. Balandin, J.L. Liu and Y.H. Xie, Appl. Phys. Lett., 84, 3355 (2004).
22
Thermal Conductivity Measurement by
3ω Method
Principle of Measurement:
1ω current→ 2ω heating → 2ω ΔT → 2ω ΔR→ 2ω ΔR×1ω current →3ω ΔV
V(3ω)=I(ω)*αΔT(2ω), α: thermal coefficient of resistor
3ω voltage signal
Ie
ΔT (r ) = ( P / lπΛ ) K o (qr )
Ie
1ω current input
iω t
iω t
1 / q = ( D / i 2ω )1 / 2
|1/q| - wavelength of the diffusive thermal wave or thermal
penetration depth
ΔT – temperature oscillations
Reduces black-body radiation error
Suitable for small samples
Fast
Λ – thermal conductivity of half-volume
P/l - amplitude of power per unit length generated at frequency 2ω
Ko – zero-order modified Bessel function
23
3ω-Method: Details
Principle of Measurement:
1ω current→ 2ω heating → 2ω ΔT → 2ω ΔR→ 2ω ΔR×1ω current →3ω ΔV
V(3ω)=I(ω)*αΔT(2ω), α: thermal coefficient of resistor
Ie i ω t
Temperature oscillations at the
substrate surface:
∞
P
sin 2 (kb)
ΔTS =
dk
lπK ∫0 (kb) 2 (k 2+ q 2 )1 / 2
⎛ 2iω ⎞
q2 = ⎜
⎟
⎝ D ⎠
Thin film leads to the
addition of the term:
ΔT f =
2b – width of the metal line
P
t
lK film 2 b
l – length of the line
t – film thickness
P – power amplitude
Kfilm < KS
Measured temperature
oscillations:
ΔT = 4
dT R
V3
dR V
R – average resistance of
the metal line
V – voltage across the
metal line at ω
V3 – voltage at 3ω
K – thermal conductivity
ΔT ~ -(1/2)ln(ω)-(1/2)ln(ib2/D)+const
24
Measurement of Thermal Conductivity
Experimental Setup
Heater-Thermometer
Temperature Rise (K)
Measured Temperature Rise
1.2
Semi-inifinite Sapphire
Exp.
Num.
k-sapphire=29W/mK
k-GaN=125W/mK
1.0
k-SiO2=1.292W/mK
0.8
Semi-inifinite
Sapphire
0.6 +18.5um GaN
Trise-SiO2
0.4
0.2
300K
Semi-inifinite GaN
3
10
4
10
Frequency (Hz)
W.L. Liu and A.A. Balandin, Appl. Phys. Lett.,
85, 5230 (2004).
25
Balandin Group
Thermal Conductivity Measurements in QDS
SiN insulation
Ge quantum
dots
Si substrate
Schematic of the quantum
dot superlattice structure
Partially Regimented Quantum Dot Superlattice
Measured thermal conductivity
of the SiN insulation layer
Thermal Condutvity (W/mK)
metal line
test results for the
hardware and software
0.8
0.7
0.6
0.5
0.4
0.3
0.2
PECVD SiN 40nm
0.1
0.0
0
50
100
150
200
250
300
350
400
Temperature (K)
Fig. 2. Cross-sectional
TEM of Ge/Si QDS with
partial dot regimentation.
Sample: J.L. Liu (UCR).
Quantum dot parameters: base ~ 100 nm;
height ~ 14 nm; Si layer thickness ~ 20 nm; Ge
layer thickness ~ 1.5 nm; density ~ 4 x 109 cm-2.
30 nm
26
Balandin Group
20
Thermal Condutvity (W/mK)
Thermal Cnductivity (W/mK)
Measured Thermal Conductivity
Si 20 nm /Ge 1.2 nm
Si 20 nm /Ge 1.5 nm
Si 20 nm /Ge 1.8 nm
18
16
14
12
10
8
6
4
14
12
10
8
6
C: Si 20 nm /Ge 1.8 nm
B: Si 20 nm /Ge 1.5 nm
A: Si 20 nm /Ge 1.2 nm
4
2
0
2
0
0
100
200
300
400
Temperature (K)
50
100
150
200
250
300
350
400
Temperature (K)
Observations:
Order of magnitude decrease compared to bulk
Significant shift of the thermal conductivity peak to higher temperature
How to describe the
phonon transport in
these nanostructures?
Changed T-dependence: K~T0.7 – T0.9 in the low-T region
27
Thermal Conductivity of Bulk Crystals
Callaway’s thermal conductivity: K=K1+K2
3
⎛ k ⎞ kB
K1 = ⎜ B ⎟
T3
2
⎝ h ⎠ 2π Vg
3
⎛ k ⎞ kB
K2 = ⎜ B ⎟
T
2
⎝ h ⎠ 2π Vg
θD / T
∫
0
τ C ,B x 4e x
(e
x
− 1)
2
dx
⎧⎪θD / T
⎫⎪
−2
4 x
x
−
τ
/
τ
x
e
e
1
dx
(
)
( ) ⎬
⎨ ∫ C ,B N
⎪ 0
⎭⎪
3 ⎩
θD / T
∫ (τ
2
/ τ Nτ R ) x 4 e x ( e x − 1) dx
−2
C ,B
0
τ C−1 =
Vg
LF
+ Aω 4 + ( B1 + B2 )ω 2T 3
J. Callaway, Phys. Rev., 1959
Klemens’ scattering rates (second-order perturbation
theory):
V oω 4
=
Γ,
τ
4π V g3
1
Γ =
∑
1
τ DC
a
= η N D 42 ω 3
Vg
f i [1 − ( M i / M )] 2
Temperature (K)
Si has Γx104=2.64 for three natural isotopes
Æ add an extra term to account for quantum dots
28
Modified Klemens-Callaway Model for QDS
100
Acoustic phonon scattering rate on quantum dots:
DOT RADI US
σV – total phonon scattering cross section in volume V
Vg – phonon group velocity
Incoherent scattering:
σV ~Nσ, σ – scattering cross-section of a single dot
For rigid spherical dots:
σ~5.6(ka)4a2 for ka<<1 (long wave limit)
σ~2πa2
THERMAL CONDUCTIVITY (W/m K)
1/τD=VgσV/V
A
B
C
D
80
E
1
0.8
0.4
0.2
-
nm
nm
nm
nm
m as s-difference limit
60
A
40
B
C
D
20
for ka>>1 (short wave limit)
1/τD~ω4/Vg3 (ka<<1) – pint defect scattering
1/τD~Vg (ka>>1) – boundary scattering
ÆReasonable prediction of the thermal conductivity
behavior around room temperature.
Æ Assumption: good crystal quality and low disorder
E
0
0.00
0.00
0.02
0.02
0.04
0.04
0.06
0.06
0.08
0.08
0.10
0.10
0.12
0.12
0.14
0.14
Ge DOT VOLUME FRACTION
A. Khitun, A.A. Balandin et al., J. Appl. Phys.,
88, 696 (2000).
29
Phonon Hopping Transport
Polycrystalline –
Granular Materials
Thermal conductivity in the phonon-hopping
transport model*:
Application of the phonon-hopping
model to quantum dot superlattices:
1
θ
K int B ( x)tS Φ
K = k BT ∫ −1
dx
hk B K int a 2 d + k BTD B ( x)tS Φ
0
9
x4e x ⎛
1⎞
B( x ) = θ 4 x
x− ⎟
2 ⎜
2 ( e − 1) ⎝
θ⎠
rB
2
T is temperature
Phonon-Current
Resistances Network
GeSi/Si QDS
30 nm
TD is Debye temperature
a is lattice constant
rhop
Si host
Ge dot
d is average grain size
S is mean area of interface grain boundary
Φ is disorder factor
t is semi-empirical parameter for the
transparency of the inter- grain boundary
*
Braginsky, et. al. Phys. Rev. B, 66, 134203 (2002).
n periods
Si substrate
30
Thermal Conduction in QDS as Phonon
Hopping Transport
Transition to the Bulk Limit
0
t=0.232
12
10
d=10μm
t=0.178
K/Kbulk
Thermal Conductivity (W/mK)
Measured and Calculated Thermal Conductivity
t=0.151
8
4
0
Sample A (Ge 1.8 nm)
Sample B (Ge 1.5 nm)
Sample C (Ge 1.2 nm)
0
100
200
300
400
Temperature (K)
Note: good agreement between experimental and
calculated results over a very wide T range.
-1
10
d=100nm
-2
10
100K-mod.
300K-mod.
100K-exp.
300K-exp.
-3
10
0.0
0.2
0.4
0.6
0.8
1.0
Hopping Parameter t
Bulk limit: tÆ very large or d Æ very large
First-principle estimates of t for Si (MA=28) and
Ge (MA=72) interface: t~0.15
M. Shamsa, W.L. Liu and A.A. Balandin, Appl. Phys. Lett., 87 (2005) – in print.
Balandin Group
200K-mod.
400K-mod.
200K-exp.
400K-exp.
31
Hybrid Virus-Inorganic Nanostructures
Plant Viruses as Nano-Templates
Nanofabrication Benefits:
suitable dimensions
small size dispersion
selective attachment
W. Shenton, T. Douglas, et al., Adv.
253 (1999).
Mater., 11,
C. E. Flynn, et al., Acta Materialia, 51, 5867
(2003).
W.L. Liu, A.A. Balandin, et al., Appl. Phys. Lett.,
86, 253108 (2005).
SEM of a pure TMV and TMV end-to-end assembly (left); nanowire
“interconnect” made of metal coated TMV assembly (right).
32
Nanofabrication Using Virus Nano-Templates
Nanostructure Growth:
University of California –
Riverside (UCR), 2005
Pl
Balandin Group
TEM micrograph of the pure TMV and metal coated TMV. Scale
bar is 50 nm. Nano-Device Laboratory (NDL), UCR, 2005.
X-Ray Characterization
33
Analysis of Optical Phonons in Hybrid BioInorganic Nanostructures
Measured spectra under 488 nm excitation;
room temperature; backscattering configuration.
10000
Note: water is strong infrared (IR) absorbing
medium, and generally Raman is better than
Fourier transform infrared (FTIR) methods.
800
1000
1400
-1
-1
1200
Amide I (1655cm )
C-H def (1332cm )
-1
TMV
-1
0
C-H def (1454.5cm )
TMV-Pt
Phe res (1005cm )
Intensitty (a.u.)
TMV-Au
1600
-1
1800
Raman spectra of TMV, Pt coated TMV and Au
coated TMV: the Amide I line at 1655cm-1, C-H
deformation lines at 1454.5cm-1 and 1332cm-1,
and the phenylalanine residue line at 1005cm-1.
The Amide I lines of TMV-Pt and TMV Au are at
1664cm-1 and 1672cm-1 respectively.
Raman Shift (cm )
Balandin Group
Amide I line is related to TMV coat protein capsid, the line shift
indicates the change of vibrational modes due to the binding of
metal with certain functional group in the shell protein .
34
Calculation of Phonon Modes in
Biological Templates
Radial modes of the lowest
frequencies with m = 0 and k = 0
for a cylindrical virus in air (a-c) and
in water (d-f). The viruses without
(a, d) and with (b-c, e-f) an axial
canal are considered. The length or
arrows is proportional to the
magnitude of displacement vector
u(r,φ,z,t)=wm,k(r)exp(imφ+ikz-iwm,kt)
Elastic parameters of viruses
(lysozyme protein crystal):
Longitudinal sound velocity
VL=1817 m/s
Poisson’s ration σ=0.33
Mass density ρ=1.21 g/cm3
Approach: complex-frequency model (effect of the exterior medium)
V.A. Fonoberov and A.A. Balandin, phys. status solidi (b), 12, R67 (2004)
A. A. Balandin and V. A. Fonoberov, J. Biomedical Nanotechnology, 1, 90 (2005).
35
Modeling Phonon Dispersion in TMV-Based
Nanotubes
Engineering Phonon Modes in Hybrid Bio-Inorganic Structures
Phonon density of
states (PDOS) for
TMV/silica and
empty silica
nanotubes as a
function of phonon
frequency. Æ
Thickness: H=3 nm
For other m:
|m| x 0.7 cm-1
Dispersion of axially symmetric phonon
modes (m = 0) for TMV/silica and
empty silica nanotubes.
Balandin Group
Å The intensity of red color is
proportional to the probability
of finding a phonon.
V.A. Fonoberov and A.A. Balandin,
Nano Letters, 5, 1920 (2005).
36
Mobility Increase Via Electron – Phonon
Scattering Suppression
Å Log-log plot of the electron-phonon scattering rates (T = 1
K) for TMV/silicon and empty silicon nanotubes as a function
of the electron energy above the band gap.
Phonon Transport Regimes
Low Energy
ω< 3 cm-1
μ=
e
τ
m*
Medium Energy
High Energy
3 cm-1<ω<50 cm-1
ω>50 cm-1
Weak coupling
No coupling
Strong coupling
PDOS
Debye cutoff
Å Log-log plot of the low-field acoustic-phonon limited electron
mobility for TMV/silicon and empty silicon nanotubes.
V.A. Fonoberov and A.A. Balandin,
Nano Letters, 5, 1920 (2005).
Balandin Group
37
AlGaN/GaN Heterostructure Field-Effect
Transistors
Material Parameters:
GaN: wurtzite; direct band-gap EG=3.4 eV; breakdown field EB=4 MV/cm;
saturation velocity Vsat=250 km/s.
For comparison, Si: indirect band-gap EG=1.12 eV; breakdown field EB=0.4
MV/cm; saturation velocity Vsat=100 km/s.
Cut off frequencies: fT>80.4 GHZ
Max frequencies: fm > 80.4 GHz
Power levels: P=30 W/mm at 4GHz
Uniqueness:
The only heterostructure in wideband gap semiconductors with
good electronic properties
Micrographs of GaN/AlGaN HFET
NDL 2004
Very high channel charge due to
polarization effects
n+ AlxGa1-xN (6×1018) 15 nm
AlxGa1-xN, undoped 3 nm
GaN undoped channel layer 50 nm
GaN undoped 1.2 μm
SiC substrate
Schematic of AlGaN/GaN HFET structure
High thermal conductivity
38
Balandin Group
Thermal Conduction in GaN/AlGaN
Heterostructures and Devices
Self-heating became a major issue for the development of GaN technology
Ids (mA/mm)
Discrepancy in reported thermal conductivity
(T=300 K)
HFET A1
HFET A2
500
400
K=4.1 W/cmK – theoretical limit [Witek, 1998]
300
K=1.3 W/cmK [Sichel and Pankove, 1997]
200
K=1.7-1.8; 2.1 W/cmK [Florescu et. al., 2000]
0
K=1.55 W/cmK [Luo et al., 1999]
Vgs=0V
100
0
10
20
30
40
50
Device-structure optimization via modeling
VDS (V)
Motivations:
High power-density involved
Absence of native substrate: Kapitza
resistance
Large defects densities
Breakdown below the predicted VB
Modeling-based device structure
optimization
What model of thermal conductivity
to use (K~1/Tα)?
39
Measured Thermal Conductivity
400
HVPE GaN Film
HVPE Al0.4Ga0.6NFilm
MBE GaN Polycrystaline Film (Ref. 1)
MOCVD Al0.44Ga0.56N Film (Ref. 1)
HVPE GaN Film (Ref. 2)
350
300
250
200
150
100
50
0
50
100
150
200
250
300
350
Temperature (K)
400
200
Thermal Conductivity (W/mK)
Thremal Cnductivity (W/mK)
Thermal Conductivity of AlGaN Thin Films
450
Al2O3
substrate thermal
properties
K~1/T
dependence
100
0
0
100
200
300
400
500
Temperature(K)
Observation:
25 W/mK at 300K with temperature dependence more characteristic for disordered
materials
W.L. Liu and A.A. Balandin, Appl. Phys. Lett., 85, 5230 (2004).
Balandin Group
40
AlxGa1-xN: Alloy Scattering of Acoustic
Phonons
Measured Thermal Conductivity
500
Al0.4Ga0.6N
350
Thermal Conductivity (W/mK)
Thermal Conductivity (W/mK)
400
Al0.33Ga0.67N
300
Al0.23Ga0.77N
250
Al0.09Ga0.91N
200
GaN
150
100
50
0
50
100
150
200
250
300
350
400
450
Mod. 200K
Mod. 300K
Mod. 400K
Exp. 200K
Exp. 300K
Exp. 400K
400
300
200
100
0
0.0
0.2
0.4
0.6
0.8
1.0
Al Mole Fraction (x)
Temperature (K)
Virtual Crystal Model (Abeles, 1963): replace the disordered lattice by the ordered virtual crystal with
randomly distributed atoms of constituent materials; the phonons are scattered by the disorder
perturbation and anharmonicity of the virtual crystal.
virtual a.m.:
M = xM AlN + (1 − x )M GaN
virtual lattice:
δ = xδ AlN + (1 − x )δ GaN
W.L. Liu and A.A. Balandin, J. Appl. Phys., 97, 073710 (2005).
Balandin Group
41
Phonon Scattering on Dislocations in GaN
Thin Films
Thermal conductivity:
3
⎛ k ⎞ kB
K1 ≈ ⎜ B ⎟
T3
2
⎝ h ⎠ 2π V
θD / T
∫
0
τ C ,B x 4e x
(e
x
− 1)
2
dx
Umklapp scattering:
1
Scattering on dislocations (dislocation
core, screw, edge and mixed):
=
THERMAL CONDUCTIVITY (W/cm-K)
τD
1
τ DC
+
1
τS
+
1
τE
+
GaN
LEO
2.0
1.5
1.0
D. Kotchetkov, J. Zou, A.A.
Balandin,et al., Appl. Phys.
Lett., 79, 4216 (2001).
0.5
0.0
8
10
9
10
10
10
23 / 2
S
2
= 7 / 2 ηN D bS γ 2ω
τS 3
1
1
τM
ROOM TEMPERATURE
2.5
τU
k BT ω 2
= 2γ
μ V0 ω D
2
2 2⎫
⎧
2⎡
⎤ ⎪
⎛
⎞
−
23 / 2
1
1
1
2
ν
v
⎛
⎞
E
2 2 ⎪
L
⎜
⎟
=
ηN D bE γ ω ⎨ + ⎜
⎟ ⎢1 + 2 ⎜ ⎟ ⎥ ⎬
τ E 37 / 2
⎝ vT ⎠ ⎥⎦ ⎪
⎪⎩ 2 24 ⎝ 1 − ν ⎠ ⎢⎣
⎭
1
11
10
DISLOCATION LINE DENSITY
12
10
(cm-2)
13
10
Thermal Conductivity (W/mK)
1
τ DC
Vo4 / 3 3
= η ND 2 ω
VG
1
GaN
1/T
free-standing
W.L. Liu, A.A.
Balandin et al.,
PSS Rapid
Research Lett.,
202, R135 (2005).
200
#3
#1
#2
Reference
100
100
200
300
Temperature (K)
42
Phonon Scattering on Point Defects
Model validation using experimental data:
Acoustic phonon scattering rates on point
defects:
THERMAL CONDUCTIVITY (W/cm-K)
2.4
2.2
V0 Γω 4
=
τP
4πv 3
1
SOLID: CORRELATED H AND SI CONCENTRATIONS
DASHED: FIXED H CONCENTRATION
2.0
experimental points indicated
with error bars
1.8
Γ =∑
i
1.6
1.4
Solid curve: increase in Si doping nSi is
accompanied by the increase in the hydrogen nH
impurity concentration.
1.2
1.0
Dashed curve: H impurity concentration is fixed at
nH =2×10 17 cm-3, only Si doping nSi changes.
0.8
0.6
0.4
0.2
2
⎡⎛ M ⎞ 2
⎧
⎛ Ri ⎞⎫ ⎤
i
f i ⎢⎜1 −
⎟ + 2⎨6.4γ ⎜1 − ⎟⎬ ⎥
M ⎠
⎝ R ⎠⎭ ⎥⎦
⎩
⎢⎣⎝
Si doped GaN thin film
Order of magnitude increase in the doping density
leads to about a factor of two decrease in K: from
1.77 W/cm-K to 0.86 W/cm-K.
Impurities: O, H, Si, C
10
17
10
18
10
19
Extracted temperature dependence: K~1/T0.5
-3
DOPING CONCENTRATION (1/cm )
Compare to the regular: K~1/T
J. Zou, D. Kotchetkov, A.A. Balandin, et al., Appl. Phys., 92, 2534 (2002).
43
Kapitza Thermal Boundary Resistance
-3
Thermal resistance at the interface between two
media:
−1
RKapitza
=
1
∑ c1, j Γ1, j
2 j
ω1Debye
∫
hω
dN1, j (ω , T )
0
dT
2
AΔT
Q&
dω
Phonon transmission coefficient:
π
DMM
TBR (cm W/K)
RKapitza =
10
GaN/SiC
GaN/Sapphire
GaN/AlN
-4
10
-5
10
2
Γ1, j = ∫ α1→2 (θ , j ) cos θ sin θdθ
0
100
N1, j (ω , T ) =
αi =
∑c
−2
2, j
j
∑c
i, j
−2
i, j
ω2
⎡ ⎛ hω ⎞ ⎤
⎟⎟ − 1⎥
2π 2 c13, j ⎢exp⎜⎜
κ
T
⎣ ⎝ B ⎠ ⎦
200
300
400
500
Temperature (K)
0
GaN
SiC
Al2O3
AlN
ρ [g/cm3]
6.15
3.21
4.89
3.23
VL [105
cm/s]
8.04
13.1
10.8
10.97
VT [105
cm/s]
4.13
7.1
6.4
6.22
44
Effect of TBR on Transistor Performance
Device structure
200 nm n – GaN Active Layer
3 μm SI GaN Buffer
R bd
Thermal resistance of the device structure including
interface TBR RKapitza
RΣ = RGaN + Rth
Rth = RSUB + RKapitza
- total thermal structure resistance
RGaN = LGaN /KGaN eff
100 - 300 μm Substrate
Drain current vs sourcedrain voltage for gate
biases (0 V, -2 V, -4 V) for
TBR (a) Rth= 0.001
Kcm2/W (solid line) and
Rth= 0.005 Kcm2/W (dash
line) and (b) Rth= 0.010 K
cm2/W. Ambient
temperature T0 = 300 K.
Simulations performed with ISE TCAD software
45
Ambient Temperature Effects
Vg start: +1V
o
25 C
step=-1V
o
250 C
1.00
0.95
10
Isat / Isat,25oC
Drain-Source Current (mA)
15
Drain-Source Current (mA/mm)
Measured IV Characteristics of the SurfacePassivated AlGaN/GaN HFET at Different Ambient
Temperature
5
300
VG = 0, -2, -4 V
o
250
25 C
200
250 C
o
150
100
50
0
0
0.90
2
4
6
8
10
Drain-Source Voltage (V)
0.85
0.80
0.75
0
0
2
4
6
8
10
Drain-Source Voltage (V)
~33% degradation in drain current
Isat/Isat(T=25oC)=1.03-0.0013⋅T
0.70
0
50
100
150
200
250
o
Temperature ( C)
W.L. Liu, V.O. Turin and A.A. Balandin, MRS J. Nitride
Semicond. Research, 9, 7 (2004).
46
Optical Phonons in ZnO Nanocrystals
TEM image of a ZnO QD
Non-resonant Raman scattering spectra of
bulk and ZnO quantum dots
ZnO Parameters:
EG=3.37 eV
379 cm
-1
410 cm
-1
ε=3.7
EB=60 meV
Intensity (a. u.)
me=0.24
Richter model:
Δω~(dω/dk)1/D
3000
439 cm
-1
Laser: 488 nm
bulk ZnO (a-plane)
2000
436 cm
-1
-1
Impurities effect:
582 cm
0.5% impurities
1000
TABLE I. Raman active phonon modes in bulk ZnO
200
ZnO QDs (20 nm)
300
400
500
-1
600
700
Raman shift (cm )
E2(low)
A1(TO)
E1(TO)
E2(high)
A1(LO)
E1(LO)
102
379
410
439
574
591
47
Not Every Raman Shift is the Phonon
Confinement Shift
Intensity (a. u.)
1000
(a) bulk ZnO (a-plane)
Laser: 325 nm; 20 mW
Resonant Raman scattering spectra of bulk ZnO and
ZnO quantum dots.
3 LO
2 LO
-1
-1
574 cm
1 LO
500
500
1000
1500
2000
-1
Raman shift (cm )
(b)
300
Intensity (a. u.)
LO phonon frequency (cm )
2
area = 11 μm
2
area = 1.6 μm
E2(low)=102 cm-1
E2(high)=439 cm-1
565
A1(TO)=379 cm-1
A1(LO)=574 cm-1
E1(TO)=410 cm-1
E1(LO)=591cm-1
560
3 LO
-1
570 cm
1 LO
0
2 LO
5
10
15
20
UV laser power (mW)
LO phonon frequency shift in ZnO QDs vs. excitation power.
Red shift of about ~14 cm-1 is due to local heating.
200
ZnO QDs (20 nm)
Laser: 325 nm; 2 mW
100
570
Bulk ZnO
Peaks
500
1000
1500
-1
Raman shift (cm )
2000
K.A. Alim, V.A. Fonoberov and A.A. Balandin,
Appl Phys. Lett., 86, 053103 (2005).
V.A. Fonoberov and A.A. Balandin,
Appl. Phys. Lett., 85, 5971 (2004).
48
Phonon-Related Research
Nano-Device Laboratory
Thermal Conductivity
Measurements
Theory and Modeling of Phonons
and Electrons at Nanoscale
T: from 4K to 600K
Raman Spectroscopy
phonons in nanostructures, strain,
composition and local heating
Thermal Management
of Nanoscale Devices
Carrier Transport
Enhancement
Design and Fabrication
phonon-engineered structures
Direct Energy
Conversion
Electrical Characterization
I-V, C-V, Hall and drift mobility
Bio and Hybrid Structures
hybrid virus-inorganic nanostructures
Optical
Response
Bio-Inorganic
Interface
49
Balandin Group http://ndl.ee.ucr.edu/
Acknowledgements
Nano-Device Laboratory (NDL) Group Members
Funding Provided by
National Science
Foundation
US Office of Naval
Research
Functional Engineered
Nano Architectonics
From left to right: M. Varshney (GSR), M. Shamsa (GSR), A. Morgan
(Technician), Prof. A. A. Balandin (Group Leader), Dr. V. Turin (PGR),
Dr. V. Fonoberov (PGR), Dr. E.P. Pokatilov (Visiting Researcher), K.
Alim (GSR), Dr. W. L. Liu (PGR) and Y. Bao (GSR).
Semiconductor
Research Corporation
Collaboration:
Prof. K.L. Wang (UCLA), Prof. J. Zou (East Illinois University)
50