3. thermal
DESCRIPTION
More information about thermal transport, heat capacity, and otherTRANSCRIPT
3ª SESIÓN
nanopedia.cwru.edu
The physics of low-dimensional solid state matter looks very differentfrom that of the corresponding bulk material.
Characteristic physical length scales will play a determining role, someincreasing when the temperature decreases and entering into competitionwith the sample dimensions.
The main examples are:
• The mean free path Λ of the phonons, treating phonon transport as ballistic.• The dominant wavelength λdom of the phonons.• The mean free path le of the electrons.• The coherence length Lϕ of the electron quantum phase.
CHARACTERISTIC LENGTH SCALES
These physical quantities will determine the thermal and electricalproperties of matter, mainly at low-temperatures
Surface/volume thermodynamic competition (loss of bulk behaviour).
Consequences of boundary conditions on artificial nanostructures.
Effect of phonon confinement on the specific heat and thermalconductance.
Importance of fluctuations in small systems.
Existence of phase transitions specific to small length scales.
SIZE REDUCTION THERMAL CONSEQUENCES
EXAMPLE : PHASE TRANSITION DEPENDENCIES WITH MATERIAL SIZE.
REFERENCE: A.F.Lopeandia, PhD Thesis dissertation (2007)
REFERENCE: Journal of Quantitative Spectroscopy and RadiativeTransfer Volume 112, Issue 3, February 2011, Pages 412–419
Limite del cuerpo negro (far-field)
EXAMPLE: HEAT RADIATION IN NANOSCALE
EXAMPLE : Thermal conductivity reduction on Si nanowires.
REFERENCE: Li et al. Applied Physic Letters. Volume 83, Number 14, October 2003, Pages 412–419
EXAMPLE : THERMAL CONDUCTIVITY REDUCTION ON SI NANOWIRES.
REFERENCE: Li et al. Applied Physic Letters. Volume 83, Number 14, October 2003, Pages 412–419
PROBLEMS IN ENERGY MANAGEMENT AT NANOSCALE
Source DrainGate
Nanowire Channel
Si Nanotransistor(Berkeley Device group)
Hot Spots in Si nanotransistors!
NanoCoolers Energy Harvesting
MACROSCOPIC VISION
PARALLEL TREATMENT.
MICROSCOPIC VISION
Heat Capacity
Ene
rgy
Sto
rage
Ene
rgy
Tran
spo
rt
Radiation
Conduction
Convection
¿How are the actors?
Electrons
Atoms/Molecules
Photons
Phonons
TRANSLATION
ROTATION
VIBRATION
THERMAL ENERGY AS KINETIC ENERGY?
In a Gas…
- ….
KINETIC THEORY OF GASES
MAXWELL BOLTZMANN DISTRIBUTION
4ª SESIÓN
Phonon Gas???
In a CRYSTAL…
PREVIOUS CONCEPTS
- CRYSTALL is periodic arrangement LATTICE +BASIS
- Primitive Lattice Vector.- Primitive cellWinger Zeitz Unit cell.- Bravais Crystall lattices.- Miller indexes.
- RECIPROCAL SPACE (fournier series in K)
- First Brillouin zone
Energy
Distancero
Parabolic Potential of Harmonic Oscillator
Eb
Interatomic Bonding
1-D Array of Spring Mass System
612
)(rr
rV
I. Chain of identical atoms Interatomic potential: Lennard-Jones
ardr
VdaraVrV
2
22
2
)()()(
2
2
1xK
a
Spring constant, g Mass, M
Crystal Vibrations
Energy Quantization and Phonons
h
Energy
Distance
Total Energy of a Quantum
Oscillator in a Parabolic Potential
2
1nn
n = 0, 1, 2, 3, 4…
/2: zero point energy
Phonon: A quantum of vibrational energy,
, which travels through the lattice
Classical Quantum mechanics
Lattice vibrations modes of frequency ω behave as a simple armonic oscillator
Energy levels of atoms vibrating
at a single frequency ω
A transition from a lower energy
level to a higher energy level.
2
1
2
112 nn
2 1
unity
n n
absorption of phonon
a
Spring constant, g Mass, m
u n u n+1 u n-1
Equilibrium Position
Deformed Position
nn
nn
uuC
uuC
1
1
kaCM
2
1sin4 22
112
2
2· nnnn uuuC
dt
udmmaFEquation of motion
)(exp tknaiAun Solution
Force on the nth atom
Considering Nearest neighbour interaction
Dispersion relation
to the left
to the right
Fre
quency,
Wave vector, K0 p/a
II. Chain of two types of atoms
a
Spring constant, K Mass, M
u n u n+1 u n-1
Equilibrium Position
Deformed Position
un-2
Mass, m
211
11
2
2
nnnn
nnnn
uuuCum
uuuCuM
2/1
2
2
2
2
1sin
4·
)·(
Ka
MmMm
mMC
Mm
mMC
Fre
qu
en
cy,
Wave vector, K0 p/a
LA
LO
Optical
Vibrational
Modes
Dispersion Relation
• ω versus k relation for diatomic chain;
• Normal mode frequencies of a chain of two types of atoms.
At A, the two atoms are oscillating in antiphase with their centre of
mass at rest;
at B, the lighter mass m is oscillating and M is at rest;
at C, M is oscillating and m is at rest.
0 л /a 2л /a–л /a k
A
B
C
As there are two values of ω for each value of k, thedispersion relation is said to have two branches;
The difference between transverse optical and acoustic modes in a
diatomic linear chain can be schematically visualized as
+
+ +
+ +- - - -
+
+
+
+ +
- -
- -
k
k
Optical
Acoustic
The acoustic branch has this name because it gives rise to long wavelength
vibrations - speed of sound.
The optical branch is a higher energy vibration (the frequency is higher, and you
need a certain amount of energy to excite this mode). The term “optical” comes from
how these were discovered - notice that if atom 1 is + and atom 2 is -, the charges
are moving in opposite directions. You can excite these modes with EM radiation.
Transverse optical mode for diatomic chain
Amplitude of vibration is strongly exaggerated!
Transverse acoustical mode for diatomic chain
Crystal Vibrations in 3D
III. 3D Crystal
1 atom in the unit cell Existence of. 3 acoustic branches 3N modes
2 atoms in the unit cell Existence of 3 acoustic and 3 optical branches
s atoms in the unit cell 3 acoustic and 3(s-1) optical branches
0 0.2 0.4 0.6 0.8 1.00.20.40
2
4
6
8
(111) Direction (100) Direction XL Ka/p
LA
TATA
LA
LO
TO
LO
TO
Fre
quen
cy
(10
Hz)
12
Phonon Dispersion
in GaAs
Sound velocity
Group Velocity
dK
dvg
Speed of Sound
dK
dv
Ks
0lim
sphonon
hE
• Quanta of lattice vibrations
• Energies of phonons are quantized
~a0=10-10m
phonon
hp
• Quanta of electromagnetic
radiation
• Energies of photons are
quantized as well
photon
hcE
~10-6m
photon
hp
PHONONS PHOTONS
Phonon Gas???
Que energía almacenan?
La energía de una colección de osciladores harmónicos:
Pk
k P
PK
k P
PK nUU ,,, P polarization branchk wave vector
PKn ,
is the ocupation numberin thermal equilibrium
1
1
TBe
n
Energía vibracional de los fonónes
Por conveniencia podemos cambiar el sumatorio sobre todas las K por integral.Y si suponemos que el cristal tiene Dp(w)dw modos de vibración por cada polarización en el rango de w a w+dw:
Bose EinsteinPlanck
p
pTkdD
eTU
B
)(1
/
Densidad de Estados Fonónicos (DosP)
El problema central: D(w)?
En el caso más simple de una cadena unidimesional:
L
Aplicando condiciones de frontera periódicas:
L2πK :que forma de
L
Nπ ,...,
L
8π ,
L
6π ,
L
4π ,
L
2π 0,K
p
p
p
d
dKKL
d
dnD
L
Kn ·
·2
·)(
·2
··3
4
2
23
3
3
De la misma manera en 3D, K estará determinado por las condiciones de frontera sobre N3 celdas de un cubo de lado L. Por tanto habría un valor de K por cada (2p/L)3
en el espacio de ks para cada polarización. Así que si queremos saber el número de modos con vector de ondas inferior a K, dividiríamos la esfera de radio K (4/3·p·K3)por (2p/L)3
Aproximación de Debye
Fre
quency,
Wave vector, K0 p/a
Kvs
-La densidad de estados la obtenemos suponiendo una relación de dispersión líneas.
- Eso es cierto a baja T ya que sólo podremos excitar modos accesibles a kBT
-Consideramos la velocidad promedio del sonido en las tres polarizaciones (L+T+T)
32
23
22
23
··2
··
··2
·)(
ss v
L
d
dK
v
LD
p
p
Dado que el nº de celdas primitivas es N, aparece una frecuencia máxima wD para la que D(w) esta definida:
D(w)
w
Para cada polarización:
Asumimos vs igual para las 3 polarizaciones:
Donde definimos la temperatura de Debye:
- Como el nivel de ocupación de frecuencias altas no será accesible (distribución) a KT en este caso podemos cambiar el límite de integración de D por infinito.
Cv ~ T3; Funciona muy bien a bajas T, donde sólo los modos acústicos pueden ser excitados!!!
- A altas T, tiende a Dulong-Petit;
Limite a bajas Temperaturas T<<qD:
104
103
102
101
101
102
103
104
105
106
107
Temperature, T (K)
Sp
ec
ific
He
at,
C
(J/m
-K
)3
C T3
C 3kB 4.7 106 J
m3 K
qD 1860 K
Diamond
Classical
Regime
Each atom has
a thermal energy
of 3KBT
Sp
ecific
He
at (J
/m3-K
)
Temperature (K)
C T3
3kBT
Diamond
En general, cuando T << qD,
dlatd TCTUv
,1
d =1, 2, 3: dimension de la muestra
Esto será válido si la relación de dispersión es lineal (válido para muchos materiales)
De forma más general,
Si ω α kδ entonces Cv α Td/δ
Case of 2D graphene Cv α T
Dimensionalidad
behaves as a quasi-2D since the layers are strongly decoupled
Hexagonal symmetry leads to quadratically dispersed acoustic-phononbranches corresponding to the out-of-plane vibration modes of the sheet
Constand D(ω) and linear Cp
However, in graphite: Interlayer coupling can’t beneglected at low frequencies which gives D(ω) α ω and Cp α T2
The 2 in-plane modes also give the same trend
Low T quasi 3DHigh T quasi 2D
2D-3D dimensionality crossoverdue to interlayer coupling
GRAPHITE
CARBON NANOTUBES
TDebye ~ 2400 K
If D(ω) cte. Cp α Twhen T << TDebye
MWCNT (~ 10 nm diameter)
Measurements on bundles of SWCNT demonstrate the linear T dependence except at very high T (Cp α T2-3) due to coupling between them,
restoring the 2D-3D behaviour
W. Yi et al. PRB 1999
Quadratically dispersed branches in 2D ??
The out-of-plane acoustic mode dominates the thermal properties -the two in-plane modes (linearly dispersed) give negligible contribution to Cp
SWNTs ~ 1.2 nm diameter
(Hone et al. Science (2000)
Observation of quantized subbans in PDOS
COMPARISON BETWEEN DIFFERENT C FORMS
Crossover to 3D behavior --- strong coupling betweens tubes
Slope change due to first subband contribution
Cp linear from 2-8 K
T> 8 K slope increases
Evidence of quantized subbands
CAPACIDAD CALORÍFICA ELECTRÓNICA
Suponemos un gas de electrones libres, en 1D:
Si tenemos N partículas, contando que tenemos (dos estados ms +-1/2), el nivel energético de Fermi:
En 3D la solución de la ecuación de Schrödinger:
Con Ki :
De forma que:
FERMI-DIRAC DISTRIBUTION:
Tk
EEEf
B
Fexp1
1
Calor específico del gas de electrones
Cuando calentamos desde 0K esperaríamos:
0
1
EFElectron Energy, E
Occ
up
atio
n
Pro
bab
ility
, f
Work Function,F
T
T= 0 K
kBT
Vacuum Level
EF
F: Work Function
Energy
Vacuum
Level
Band Edge
Visión Clásica, uno esperaría aumentar 3/2 KB cada e-.
Visión Cuántica, sólo lo e- a distancias de KBT del nivel de fermi pueden ser excitados térmicamente.
Element Electron
Density, ne
[1028
m-3
]
Fermi
Energy
EF [eV]
Fermi
Temperature
TF [104 K]
Fermi
Wavelength
F [Å]
Fermi
Velocity
vF [106 m/s]
Work
Function
F [eV]
Cu 8.47 7.00 8.16 4.65 1.57 4.44
Au 5.90 5.53 6.42 5.22 1.40 4.3
Fe 17.0 11.1 13.0 2.67 1.98 4.31
Al 18.1 11.7 13.6 3.59 2.03 4.25
Representative parameters of some metals
0
1
EFElectron Energy, E
Occ
up
atio
n
Pro
bab
ility
, f
Work Function,F
T
T= 0 K
kBT
Vacuum Level
Heat capacity of metals
At T<TF and q23 /; TATCTATC vv
C/T
T2
Specific Heat
for Copper (Bulk)
M. de Podesta. Understanding Properties of Matter
Density of States in Confined Systems
Confinement modifies the Density of States
Specific Heat of Nanocrystalline materials
J.Rupp, et al. PRB 36 (1987) 7888.Chen et al. PRB 52 (1995) 9364.
Nanocrystalline material showing differentinteratomic spacings in the grain boundaries.
Samples of Cu: 6 nm and Pd: 8 nm and Cp measured over 150-300 K.
Nanocrystalline Cu and Pd
For Pd nanoparticles enhancement variesfrom 29-53% and for Cu between 9-11%.
In Cu and Pd no electronic or magnetic contribution to Cp between 150-300 K.
Thus, increase in Cp due to variation in vibrational and configurational entropy of the material
400 500 600 700
10
100
CP
(nJ/K
)
T (K)
Effect of nanoparticle size on Cp
Cp increases with decreasing size
Specific heat of Zr90Al10 nanocrystallites
Specific Heat of Ni thin films
A.F.Lopeandia et al. APL (2008)
1 10 100
0.45
0.50
0.55
0.60
0.65
Cesp a
t 3
50
K (
J/g
K)
Grain size (nm)
U. Herr et al. Phylosophical Magazine A,77:3(1998)p646
Nanocrystalline FeBai et al. JAP 79 (1996) 361-.
2/ TTCp
Phonon contribution term, , reduced by 40%(Contribution of Debye oscillators decreases)
Electronic contribution, , to Cp reduced by 50%
Follows Debye-Sommerfeld theory
Nanoparticles of 40 nm
TBA
nnnpn eT
ATTATC /
2
032
Due to surface modes of lattice vibrations
Vibrational Density of States: From Nanoparticles to Nanocrystalline materials
Meyer et al. PRB 68 (2003) 104303.
- The VDOS of clusters (nanoparticles) and nanocrystalline materials are different.
- VDOS nanoparticles dominated by large number of surface atoms and
homogeneous capillary force
- VDOS nanocrystallime material dominated by large volume proportion of grain
boundary atoms
Rp
2
- VDOS obtained from molecular-dynamics simulations using potential of Cleri and
Rossato for the calculation of the interatomic forces.
Phonon energy (meV)
Capillary pressure small shift of LO peak towards higher energies
Nanocrystalline Cu
(96% relative density)
Comparison cluster, nanocrystalline
and crystalline Cu
Core contribution: 2 peaks:
LA and TA modes of the fcc VDOS
Surface contribution only one peak
below TA modes
Similar to VDOS of flat surfaces
Cu791 cluster
VDOS inside grains similar to
crystalline Cu
Most important contribution
coming from GB
Enhancement at high and low
energies of nc-Cu
Both nc similar despite different
internal surface area
Gradual transition from GFCC to SURF
Phonon density of states
Specific heat of the cluster, nanocrystalline and crystalline Cu calculated with the VDOS
In agreement with experiments:excess specific heats observed in nanocrystalline materials
The excess Cp is maximum for the cluster due to theStronger increase of the VDOS at low energies.
CALORIMETRYSuperconductivity
Crystallization
Melting
Phase formation
Spin transitions
Order-disorder transitions
Glass Transition
Under suitable conditions or proper calibration calorimetry measures Heat
Capacity, which gives acces to theoretical and fundamental quantities,
such as entropy & internal energy and therefore to Free Energy.
dt
)t(dTC
R
T)t(T)t(P P
th
block
Calorimetry is a general technique that measures the heat
absorbed or released by a sample during a given transformation
heat leaks heat capacity
The Step forward towards measurement on low-dimensional systems
NANOCALORIMETRY
APPLICATIONS
- Analysis of heat capacity in nanoscale systems. Influence of interfaces and
low-dimensionality.
- Phase trans. related to size-dependent phenomena. Thermodynamic
properties of thin films and nanoparticles. NanoThermodynamics
- Glass transition in thin or ultra-film polymeric systems.
- Thermal conductivity in low-dimensional systems.
Measurement of thermal properties at the nanoscale requires a
highly sensitive method
Conventional calorimetry in materials science (DSC or DTA)
is limited to large amounts of sample (mg scale).
Often in thin films and nanoscopic systems the amount of energy released
during a phase transformation is too small to be measured and analyzed
with accuracy.
Microfabrication techniques associated to Si technology allows the development
of miniature reactors with heaters and sensors and very low thermal masses
Fast heat transfer between sample and heater/sensors
NEED OF A NANO (MICRO) CALORIMETRIC TECHNIQUE?
Key Point is to reduce the thermal mass of the
calorimeter
Small heat capacity of the addenda
Si
Si p-type
Thermal SiO2
Low-stress SiyNx
Alumina
Titanium
Platinum
Si
Si
B: Backside photolithography + RIE (SF6);
Si
A: 4 inches Si (100) wafer; Thermal oxidation (50 nm) + 180 nm coating layer of LPCVD SiNx
D: KOH attack + membrane thinning;
C: Frontside photolithography (IR alignment) + EB-PVD + Lift off; Annealing at high temperatures;
Microdevice Processing
SiNx :180 nm.
CP (Tamb) 150 nJ/K
NANOCALORIMETERS
8 mm
ADIABATIC NANOCALORIMETRY
dt/dT
)t(PCp
Principle of Operation
U-shaped heater.
Four point measurement
Temperature
Calibration (TCR)
Power supplied by a
constant current pulse
Four-point probe
measurement of V(t)
R(t)=
V(t)·I
P(t)=V·I T(t)Power Temperature
Resistance
Phase transitions of low-dimensional materials
Thermodynamics
Melting behaviour of nanoparticles
Continuous transitions
Thermodynamics
Phase Transitions: Classified according to the lowest-order derivative of the
thermodynamic potential which changes discontinously at the transition.
First order
Second order
Lambda Transition
M. de Podesta. Understanding Properties of Matter
MELTING BEHAVIOUR OF NANOPARTICLES
The Lindemann Criterion
Thermodynamic approach
T T T rT
H rm
bulk
mm
bulk
m
bulk
s
( )2
Pawlow in 1909 derived an expression based on Gibbs-Duhem and Laplace equation of the form
The parameter determines the specific model:
i) Homogenous melting and growth model (HGM): Considers equilibrium between entirely solid and liquid particles
ii) Liquid shell model (LSM): Presence of a liquid layer of thickness r0 in equilibrium at the surface of the solid particle
iii) Liquid nucleation and growth model (LNGM): melting starts by the creation of a liquid layer at the surface and moves into the solid as a slow process with a definite Eact
3/2)/( lslvsvHGM
)/1(
1 0lslv
slLSM
r
r
)(
2
3
l
slvsvLNGsl
Indium nanostructures PRL 87 (2001)Peters, PRB, 57 (1998)
Buffat and Borel, PRA , 13 (1976)
Couchman, Nature 269 (1977)
T T T rT
H rm
bulk
mm
bulk
m
bulk
s
( )2
Depresion melting models
INDIUM NANOPARTICLES
Allent et al, PRB 62 (2000)
Melting point depression in CdS nanocrystals
Alivisatos et al. Science 1992
Silica encapsulated gold nanoparticles
APPLICATION ‘Smart bombs’
Gold Nanoparticles
Buffat and Borel, PRA
Dick et al, JACS 124 (2002)
HGM model
Caruso et al. Advanced Materials, 16 (2004)
APPLICATIONS: INFLUENCE OF PARTICLE SIZE ON SINTERING
ZnO nanoparticles sinter faster and the solid reaches a higher density
Continous Transitions
Ferromagnetic-Paramagnetic transition: Ni thin films
1 10 1000.0
0.2
0.4
0.6
0.8
1.0
0.92
0.94
0.96
0.98
1.00
0.0
0.2
0.4
0.6
0.8
1.0
Grain Size (nm)
C
esp(J
/gK
)T
C/T
bu
lkC
(n-N
) /n
400 500 600 700
10
100
1nm
2nm
3nm
5nm7nm
10nm
50nm
100nm
CP (
nJ/
K)
T (K)
200nm
)1)/((3
2exp
00 DDR
CD
T
T esp
c
c
A. F. Lopeandia et al. APL (2008)
CoO
salt rock type
5 1 / n m
1
2
3
4
Spot# d-Spacing (nm)
1 0.2214
2 0.2214
3 0.1579
4 0.1576
Extremely difficult to characterize AFM materials at the nanoscaleµ of antiferromagnets cancel. No net magnetization and magnetic averaging techniques notsuitable.
Nanomagnetism using Heat capacity measurements
150 180 210 240 270 300 330 360
0,40
0,45
0,50
0,55
0,60
0,65
0,70
0,75
0,80
0,85
Sp
ecific
he
at
(J·K
-1·g
-1)
Temperature (K)
20nm
10nm
8nm
6nm
4nm
3nm
2nm
1.5nm