2 Thermal Properties of Nanostructures
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8
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 )
Material
Free nanoparticle
Ag
Pd
CdS
7.2
Ar
0.042
2.5
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
Society)
Particle on substrate
Bulk
Reference
6.0
1.7
1.13
1.0–1.5
1.8
0.75
0.75
0.23
[1.46]
[1.47]
[1.48]
[1.49]
[1.50]
1.2
Thermal Properties of Nanostructures
9
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)
10
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
[1.54])
cp vp lp is given by
In a simple model the phonon thermal conductivity κ =
p
the specific heat cp , the group velocity vp , and the mean free path lp of the phonon
−1
+
mode p. The phonon mean free path consists of two contributions: l−1 = lst
−1
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)
1.2
Thermal Properties of Nanostructures
11
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
nanocircuits.
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
12
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
Bulk
References
410
418
383
∼190
∼200
1100
262
371
303
505
505
430
430
234
234
1338
273
[1.62]
[1.64]
[1.65]
[1.66]
[1.67]
[1.68]
[1.60]
[1.69]
[1.70]
[1.71]
230–290
550
555
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
[1.63].
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].
1.2
Thermal Properties of Nanostructures
13
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
Na55
m
where the bulk melting temperature is 371 K. (Reprinted with permission from [1.62]. © 2002
Elsevier)
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.
14
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
2ma2
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
nanostructures.
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.
1.3
Electronic Properties
15
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 =
C
eCg
E+
e2
C
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.
16
1 Introduction and Some Physical Principles
Fig. 1.15 Coulomb
staircases in the I − Vsd
characteristics of a
GaAs/AlGaAs
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
Verlag)
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
Verlag)
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.
1.3
Electronic Properties
17
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
18
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].