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