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2 Thermal Properties of Nanostructures

2 Thermal Properties of Nanostructures

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1 Introduction and Some Physical Principles

seconds. This is particularly important to applications of nanomachines and molecular motors. As these nanomachines become smaller, the probability that they will

run thermodynamically in reverse inescapably increases [1.45].

1.2.2 Surface Energy

The surface energy of nanoparticles has been measured to be higher than that of bulk

solids. This is essential for processes such as melting and evaporation. This effect

depends on whether the particles are free or deposited on a substrate (see Table 1.1;

Fig. 1.9).

The surface energy γ can, e.g., be determined according to the Kelvin equation

ps /ps0 = exp[4γ M/(ρp RTon dM )]

by measuring the onset temperature Ton of evaporation in dependence of the particle

size dM (see Fig. 1.9) where M is the molecular weight, ρ p the particle mass density,

R the gas constant, ps the vapor pressure of the nanoparticle, and ps0 the vapor pressure of a plane surface. The increase of γ with decreasing dM , which is supported

Table 1.1 Surface energies of nanoparticles

Surface energy γ (J/m2 )


Free nanoparticle








Fig. 1.9 Onset temperature

Ton of the evaporation from

Ag nanoparticles in

dependence of the particle

size dM . (Reprinted with

permission from [1.46].

© 2003 American Physical


Particle on substrate

















Thermal Properties of Nanostructures


by molecular dynamics studies [1.50] and which is attributed to a weak dilatation of

the nanoparticle surface, is discussed in a simple model in terms of a net increase of

the inward cohesive force which is reduced in a particle on a substrate or in a matrix

giving rise to a reduction of γ (see Table 1.1).

1.2.3 Thermal Conductance

In nanosystems the classical picture of a diffusive heat flow mechanism is often not

applicable because the phonons or electrons that carry heat have mean free paths

similar to or larger than the nanoscale feature size. This is a challenge for heat

removal in microelectronic devices which already involve features with sizes of the

order of the mean free path.

The thermal conductance κ(Vg ) of electrons in a semiconductor quantum wire at

low temperatures shows a quantized behavior in dependence of a gate voltage Vg

(Fig. 1.10). This originates from the plateaus in the electrical conductance G(Vg )

quantized in units of GO = 2e2 /h [1.51]. In the case that charge and energy are

transported by electrons the Wiedemann–Franz relation

κ/GT = π 2 kB2 /3e2 = Lo

applies with LO the Lorenz number. From this relation it is expected that the electrical conductance plateaus in units of GO are matched by a thermal conductance

˜ A (i, ii, iii) and of the elecFig. 1.10 (a) Quantized step-like behavior of the thermal conductance G

trical conductance GA of a semiconductor quantum wire at 0.27 K obeying the Wiedemann–Franz

relation. (b) Close-up of (a) with a half plateau κ = LO T(GO /2) for G < GO . (Reprinted with

permission from [1.51]. © 2006 American Physical Society)


1 Introduction and Some Physical Principles

quantized in units of LO TGO = π 2 kB2 2T/3 h = 1.89 · 10−12 W/k2 T. This is in

agreement with the data in Fig. 1.10 for G > GO .

The temperature dependence of the phonon thermal conductivity K(T) has been

measured for individual multiwalled carbon nanotubes (MWNTs) [1.52] with a high

value of over 3000 W/K m at room temperature (Fig. 1.11) in the range of theoretical

expectations of 3000–6000 W/K m [1.53] or of diamond or graphite (2000 W/K m


cp vp lp is given by

In a simple model the phonon thermal conductivity κ =


the specific heat cp , the group velocity vp , and the mean free path lp of the phonon



mode p. The phonon mean free path consists of two contributions: l−1 = lst


lum where lst and lum are the static and umklapp scattering lengths, respectively.

At low temperatures, the umklapp freezes out, l = lst , and therefore the κ of the

MWNT follows the temperature dependence of cp with k ∝ T 2.5 similar to that of

3D graphite [1.55] and which therefore indicates the 3D features of MWNTs.

Fig. 1.11 The thermal conductance of an individual multiwalled carbon nanotube (MWNT) of

a diameter of 14 nm. The solid lines represent linear fits of the data in a logarithmic scale at

different temperatures with the slopes 2.50 and 2.01. Lower inset: the solid line represents κ(T) of

an individual MWNT (d = 14 nm). Broken and dotted lines represent small (d = 80 nm) and large

bundles (d = 200 nm) of MWNTs, respectively. Upper inset: SEM image of suspended islands

with an individual MWNT. Scale bar: 10 μm. (Reprinted with permission from [1.52]. © 2001

American Physical Society)


Thermal Properties of Nanostructures


For 50 K < T < 150 K, a κ ∝ T 2 behavior is found (Fig. 1.11) and at

higher temperatures κ decreases due to the onset of phonon umklapp processes

and a rapidly decreasing lum . From the peak value of κ(T) where lst ∼ lum a

T-independent value lst ∼ 500 nm for MWNTs can be estimated. This means

that below room temperature phonon transport in the 2.5 μm long MWNT is

nearly ballistic with only a few scattering events. The ratio of the thermal conductances κel /κphon of electrons and phonons increases rapidly when the temperature is

lowered [1.56].

The thermoelectric power of MWNTs [1.52] is found to be linearly increasing

with T which indicates a hole-like major carrier.

What happens, however, to the thermal conductivity of systems that are effectively one dimensional, such as a single-wall carbon nanotube (SWNT), a nanowire,

or a DNA molecule [1.57, 1.58]? By analytical calculations it is estimated that the

thermal conductivity diverges with a one-third power law as the length of a 1D

system increases. This would be a very promising feature to use in the application of SWNTs, such as the design of components that dissipate heat efficiently in


It should be mentioned here that for the investigation of heat conductivity in

confined dimensions extremely sensitive calorimeters are developed [1.59] with a

low-temperature heat capacity of c ≈ 103 kB [1.59]. This may lead to an energy

sensitivity sufficient to count individual thermal phonons at 10–100 mK and observe

the particle nature of phonons [1.59].

1.2.4 Melting of Nanoparticles

According to simple theories for spherical nanoparticles the bulk melting temperature Tb is decreased in terms of the Gibbs–Thompson equation by

Tm = 4γ ν0 Tb /Ld

with d the particle diameter, γ the surface energy of the solid, L the latent heat of

fusion, and ν 0 the molar volume of the solid [1.60]. This qualitatively applies to

nanoparticles embedded in matrices or deposited on substrates (see Table 1.2 and

Fig. 1.12). However, the melting behavior of embedded particles may be strongly

affected by the characteristics of the embedding material.

For free Na clusters, Tm is found to be lower than the bulk Tb (see Table 1.2) but

roughly independent of cluster size (Fig. 1.13).

For free Ga clusters and free Sn clusters an increase of Tm compared to the bulk

materials is observed (see Table 1.2) which is attributed to more covalent bonding

in the cluster in contrast to covalent-metallic bonding of the bulk material [1.61].

The latent heat of fusion (0.012 eV/atom) as well as the entropy of fusion (0.5 kB )

of Na clusters is lower than the bulk values (0.025 eV/atom; 0.85 kB ) [1.62].

The evaporation of atoms from nanoparticles is facilitated when the particle size


Table 1.2 Melting

temperatures Tm of free

nanoparticles, of particles in

matrices or on substrates, and

of the bulk materials

1 Introduction and Some Physical Principles

Tm (K)

Material, size Free

Na, 2–3 nm

Ga, 1.2 nm

Sn, 1.0 nm

Sn, 5.0 nm

In, 5.0 nm

In, 5.0 nm

Hg, 7 nm

Hg, 7 nm

Au, 5 nm

H2 O, 4 nm

In matrix,

on substrate

































Fig. 1.12 Melting temperatures of In particles in CPG glass or Vycor glass matrices as a function

of particle size d. (Reprinted with permission from [1.68]. © 1993 American Physical Society)

decreases. This is concluded from the observation that the temperature at which

free PbS nanoparticles start to evaporate decreases with decreasing particle size


1.2.5 Lattice Parameter

The lattice parameter of Pd nanoparticles in a polymer matrix decreases by about

3% when the particle size decreases from 10 to 1.4 nm [1.72]. A similar behavior is

observed for Ag nanoparticles [1.73].


Thermal Properties of Nanostructures


Fig. 1.13 Melting temperatures of Na+ clusters versus the cluster size. The oscillations do not correlate with electronic (dotted lines) or geometric (dashed lines) shell closings. The smallest cluster

+ which has an icosahedral structure exhibits the highest melting temperature T = 290 K



where the bulk melting temperature is 371 K. (Reprinted with permission from [1.62]. © 2002


1.2.6 Phase Transitions

Phase transitions in confined systems differ, as shown in the case of the melting

transition above, from those of bulk materials and strongly depend on particle size,

wetting, as well as the interaction of a nanoscale system with a matrix or a substrate

[1.60]. Phase transitions in nanosized systems have been investigated for superfluidity where the critical behavior of superfluid He in an aerogel deviates from that of

bulk He [1.74]. In superconductivity the superconducting transition temperatures of

Ga [1.75] or In nanoparticles [1.76] in vycor glass are shifted to values higher than in

the bulk materials. The Curie temperature TC of ferromagnetic nanolayers decreases

with decreasing layer thickness [1.77]. Liquefaction [1.78] in nanopores or order–

disorder phase transitions near surfaces [1.79] was also found to differ from that of

bulk systems. Two examples of solid–solid phase transitions on the nanoscale will

be sketched in the following.

Under elevated pressures PbS undergoes a B1-to-orthorhombic structural phase

transition. For PbS nanoparticles in a NaCl matrix, the transition is shifted to higher

pressures when the particle size is reduced [1.80].

The stability of crystal structures at ambient conditions depends on the size of

ZrO2 nanocrystals which exhibit a tetragonal structure for small sizes and the bulk

orthorhombic structure for larger sizes [1.81, 1.82]. In this oxide, the lattice strain

varies with the grain sizes giving rise to a variation of the Landau free energy so

that for small grain sizes the tetragonal phase is stabilized whereas for grain sizes

d > 54 nm the bulk orthorhombic phase appears.


1 Introduction and Some Physical Principles

1.3 Electronic Properties

1.3.1 Electron States in Dependence of Size and Dimensionality

Standard quantum mechanical texts show (see [1.83]) that for an electron in an

infinitely deep square potential well of width a in one dimension, the coordinate

x has the range values − 12 a ≤ x ≤ 12 a inside the well, and the energies are given by

En =

π2 2 2

n = E0 n2


where E0 = π 2 2 /2ma2 is the ground-state energy and the quantum number n

assumes the values n = 1, 2, 3, . . . This shows that, when the free electrons in

a solid are confined to particle sizes smaller the Fermi wavelength (quantum confinement), the electron energy shifts to higher values (blue shift) with shrinking size

a of the well which is of importance for many electronic or optical properties of


In addition to quantum confinement effects, quantization effects appear in

nanoparticles because the charge of the electron is quantized in units of e [1.84,

1.85]. When the electron with the quantized charge tunnels to an island with the

capacitance C (see Fig. 1.14a) the electrostatic potential of the island changes by

discrete values of the charging energy Ec = e2 /C. This can be detected at low

temperatures when the energies fluctuations

kB T << e2 /C

are small, and when the resistance Rt of the tunneling barriers is sufficiently high.

From the charging time t = Rt · C of the island and the Heisenberg uncertainty

principle E · t = (e2 /C)Rt C > h it can be estimated that

Rt >> h/e2

should be much higher than the resistance quantum h/e2 = 25.813k .

These conditions can be met by small dots with low C values and weak tunneling

coupling. For a quantum dot sphere with a 1 μm diameter the value Ec ∼

= 3 meV

is found [1.84] which can be easily resolved at low temperatures. If a voltage Vg

is applied to the gate capacitor cg (Fig. 1.14a) a charge is induced on the island

which leads to the so-called Coulomb oscillations of the source-drain conductance

as a function of Vg at a fixed source-drain voltage Vsd (Fig. 1.14b). In the valleys

between the oscillations, the number of electrons on the dots is fixed to the integer N

with zero conductance (Coulomb blockade). Between the two stable configurations

N and N+1 a “charge degeneracy” (see Fig. 1.14b) appears where the number of

electrons can alternate between N an N+1. This produces a current flow and results

in the observed current peaks.


Electronic Properties


Fig. 1.14 (a) Schematics of a

quantum dot (island)

connected to three terminals:

source, drain, and gate. The

terminals and the island are

separated by thin insulating

layers. (b) Coulomb

oscillations for illustrating the

effect of single electronic

charges on the macroscopic

conductance I/Vsd . The

period in the gate voltage is

about e/Cg . (Reprinted with

permission from [1.84].

© 1999 Springer Verlag)

An alternative measurement with fixed gate voltage Vg and varying source-drain

voltage shows nonlinear current–voltage characteristics exhibiting a Coulomb staircase (see Fig. 1.15). A new current step occurs at a threshold voltage e2 /C at which

an extra electron is energetically allowed to enter the island. This threshold voltage is periodic in the gate voltage (see Fig. 1.15) in accordance with the Coulomb

oscillations of Fig. 1.14(b).

Quantum confinement with spacing E between the energy levels and effects of

charge quantization can be observed simultaneously. Electrons in a semiconductor

hetero-interface quantum dot [1.84] with a diameter of 100 nm yield a level spacing of ∼0.03 mV which can be detected at dilution refrigeration temperatures. The

change of the gate voltage Vg (see Fig. 1.14a) between current oscillations in a

quantum dot is given by

Vg =






where Cg is the gate capacitance (Fig. 1.14a). By sweeping Vg , a peak structure in

the current is observed for kB T << E << e2 /C (see Fig. 1.16) where the peakto-peak distance is ascribed to the addition energy e2 /C + E and the separation of

the minipeaks to the level spacing E.


1 Introduction and Some Physical Principles

Fig. 1.15 Coulomb

staircases in the I − Vsd

characteristics of a


hetero-structure. The different

curves which are shifted

vertically for clarity ( I = 0

occurs at Vsd = 0) are taken

for five different gate voltages

to illustrate the periodicity in

accordance with the

oscillations in Fig. 1.14(b).

(Reprinted with permission

from [1.84]. © 1999 Springer


Fig. 1.16 Coulomb peaks at

B = 4T measured at different

Vsd = 0.1, 0.4, and 0.7 mV

from bottom to top. The

peak-to-peak distance

corresponds to the addition

energy, the distance between

the shoulders within a peak to

the excitation energy E

(level spacing) for a constant

number of electrons on a dot.

(Reprinted with permission

from [1.84]. © 1999 Springer


1.3.2 The Electron Density of States D(E)

The electron density of states D(E) depends dramatically on the dimensionality

of nanostructures (see Fig. 1.17). Whereas for bulk systems a square-root dependence of energy prevails, a staircase behavior is characteristic for 2D-quantum well

structures, spikes are found in 1D quantum wires, and discrete features appear in

0D quantum dots. Since many solid-state properties are dominated by the electron density of states, these properties, such as the electronic specific heat Cel ,

the Pauli conduction electron magnetic susceptibility χel , the thermopower, the

superconducting energy gap, etc., are sensitive to dimensional changes.


Electronic Properties


Fig. 1.17 Electron density of states D(E) for conductors in dependence of dimensionality.

(Reprinted with permission from [1.83]. © 2003 Wiley Interscience)

1.3.3 Luttinger Liquid Behavior of Electrons in 1D Metals

In 3D metals the conduction electrons are well described by the Fermi liquid theory

of non-interacting quasi-particles, owing to the fact that the Pauli exclusion principle

strongly quenches the electron–electron interaction (see [1.86]). In contrast to that,

the electron system in 1D metals (quantum wires) is strongly affected by electron–

electron interaction, essentially because the electrons cannot avoid each other (see

[1.87]). These correlated electrons are better described by the Luttinger liquid (LL)

theory [1.86, 1.88–1.90]. The correlation gives rise to enhanced backscattering of

electrons leading to zero conductance at low temperatures [1.91].

The main consequences of the LL theory are (1) the spin and charge degrees of

freedom of electrons are separated, (2) the charge velocity (vc ) is higher than the

Fermi velocity (vF ), while the spin velocity is close to vF , and (3) the density of

states is expected to exhibit a power law suppression N(ω) ∼ |ω|α near the Fermi

level [1.92] where α = 2 + g + g−1 /8 with g the Luttinger parameter [1.93].

Efforts have been made to investigate the short-range physics determining the Luttinger liquid by calculating the ground state energy density ε(n) of 1D wires [1.94].

Several experimental observations of Luttinger liquid behavior have been

reported: (1) power laws of the current–voltage and the conductance–temperature

characteristics of the edge tunneling in the fractional quantum Hall regime [1.94,

1.95], (2) momentum-resolved electron tunneling between two nanowires [1.96], or

(3) power-law dependence of the electronic density of states of carbon nanotubes

[1.97] (see Fig. 1.18).

1.3.4 Superconductivity

In superconducting solids the conduction electrons condense at low temperatures

into Cooper pairs with the characteristic size given by the coherence length ξ . This

macroscopically coherent quantum mechanical state is described by a single wave


1 Introduction and Some Physical Principles

Fig. 1.18 Photo-emission

spectra of bulk Au and of

carbon nanotube bundles at

different temperatures. The

Au spectra show at the

chemical potential ε = μ the

Fermi function behavior

characteristic for a Fermi

liquid in 3D metals. The

carbon nanotubes at low

temperatures exhibit,

however, at energies close to

μ a power-law behavior with

a 0.46 power as characteristic

for a Tomonaga–Luttinger

liquid behavior. (Reprinted

with permission from [1.97].

© 2004 Wiley-VCH)

function ψ = |ψ| eiρ where the magnitude |ψ|can change on scales larger than ξ

and the phase ρ is related to the supercurrent density j ∼ ∇ρ.

In a superconducting 1D wire with transverse dimensions d < ξ there is always

a finite probability of thermal fluctuations driving instantly a fraction of the wire

into a normal state. This process can be described as a thermally activated jump of

the system from on local potential minimum into a neighboring one separated by

±2π in the ρ space (so called phase slip) in analogy to a “classical” jump of a particle over an energy barrier F provoked by the thermal energy kB T. However, taking

into account the quantum mechanical nature of the superconducting system an alternative mechanism has been suggested where tunneling through the barrier occurs

[1.98, 1.99]. The phase difference between two remote points of the system might

change in time due to quantum phase fluctuations. This purely quantum mechanical

phenomenon of phase tunneling, also called quantum phase slip (QPS), should provide an additional channel of energy dissipation and increase of the resistance in a

current-carrying system [1.100].

Superconductivity in quantum wires depends on whether the normal resistance

RN of the wire is lower (superconducting) or higher (normal conducting) than the

quantum resistance Rq = h/(2e)2 ≈ 6.5/k for Cooper pairs. This has been demonstrated for MoGe quantum wires of diameters of ∼15 nm and lengths of ∼150 nm

[1.101] with a coherence length (size of Cooper pairs) ξ = 8 nm, where the normal

resistance RN increases with decreasing diameter so that superconductivity disappears at diameters below 14 nm (see Fig. 1.19). This normal conducting behavior at

small diameters is ascribed to quantum phase slip tunneling [1.102]. A non-zero

low-temperature resistance has also been observed for an 11 nm diameter wire

(RN = 9k ) of Al which is a bulk superconductor [1.100].

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