Dielectric quantification of conductivity limitations due to nanofiller size in conductive powders and nanocomposites

0.46 MB·English

by  L. J. Huijbregts

Dielectric quantification of conductivity limitations due to nanofiller size in
conductive powders and nanocomposites
L. J. Huijbregts,1,2 H. B. Brom,1,2,3 J. C. M. Brokken-Zijp,1,2 W. E. Kleinjan,1 and M. A. J. Michels1,2
1Technische Universiteit Eindhoven, POB. 513, 5600 MB Eindhoven, The Netherlands
2Dutch Polymer Institute (DPI), POB. 902, 5600 AX Eindhoven,The Netherlands
3Kamerlingh Onnes Laboratory, Leiden University, POB. 9504, 2300 RA Leiden, The Netherlands
(Dated: December 28 2007 accepted for Phys. Rev. B)
Conducting submicron particles are well-suited as filler particles in non-conductingpolymer ma-
8
tricestoobtainaconductingcompositewithalowpercolationthreshold. Goingtonanometer-sized
0
filler particles imposes a restriction to the conductivity of the composite, due to the reduction of
0
the density of states involved in the hopping process between the particles, compared to its value
2
withinthecrystallites. Weshowhowthosemicroscopicparametersthatgovernthecharge-transport

n processes across many decades of length scales, can accurately and consistently be determined by
a a range of dielectric-spectroscopy techniques from a few Hz to infrared frequencies. The method,
J
which is suited for a variety of systems with restricted geometries, is applied to densely packed 7-

5 nm-sizedtin-oxidecrystallineparticleswithvariousdegreeofantimonydopingandthequantitative
1 results unambiguously show the role of the nanocrystal charging energy in limiting the hopping
process.
]
i
c PACSnumbers: 73.22.-f,72.80.Tm,73.63.Bd,77.84.Lf
s
-
l
r I. INTRODUCTION ping, for heterogeneity-induced enhanced ac response,
t
m forphonon-orphoton-assistednearest-neighborhopping,
and for the Drude response of individual nanocrystals.
. Small submicron particles are well-suited as fillers in
at non-conductingpolymermatricestoobtainaconducting Due to these quantitative results we can unambiguously
m determine also the role of the nanocrystal charging en-
composite with a percolation threshold (far) below 1 %.
ergy in limiting the hopping process. We apply the
- The low percolation threshold is due to the formation
d method to antimony-doped tin-oxide (ATO) crystallites
of airy aggregates of conducting particles, in which the
n of 7 nm diameter and to 100 nm sized crystallites of
particles are grown together by diffusion-limited cluster
o
Phthalcon-11. It turns out that in densely packed crys-
aggregation,creatinga networkwitha fractaldimension
c
[ around 1.7.1,2,3 These airy aggregates can be thought tallites ofATO, due to the stronginfluence of the charg-
ing energy on the DOS, σ at room temperature is four
of as conducting spheres forming a 3-dimensional per- dc
1 orders of magnitude lower than the dc conductivity ex-
colating network around the expected aggregate filling
v
trapolatedfromtheDrudeplasmafrequency(ω )ofthe
8 fraction 0.16. As a consequence of the fractal structure pD
crystallites-aresultwithobviousimplicationsforthede-
2 within the aggregates, the filler fraction of the particles
sign of conducting composites. The dielectric method is
2 at the percolation point is much lower. Even in case
2 the particles touch, the dc conductivity (σ ) of these well suited for a variety of systems with restricted ge-
dc
1. composites at high filling fractions turns out to be or- ometries, as we will illustrate by a short discussion of
phase-change materials10 and granular oxides.11.
0 ders of magnitude lower than of the bulk material, as
8 wasrecentlyillustratedforaparticularcrosslinkedepoxy
0 composite with filler particles of Phthalcon-11,4 Co ph-
:
v thalocyanine crystallites of 100 nm size, and explained II. CONDUCTIVITY IN CONDUCTING
i by purely structural arguments.5 POLYMER COMPOSITES AND GRANULAR
X
Whencrystallineparticleswithadiameteroflessthan METALS
r
10 nm instead of 100 nm are used, the small size of the
a
particles may impose another important restriction to Forrandomlyplacedconductingspheresinaninsulat-
the maximal possible composite conductivity, which is ing matrix, the relation between σ and the fraction φ
dc
due to the density of states (DOS) involved in the dc of spheres is known from percolation theory.1,12,13Above
conductivity through the network of particle contacts. thepercolationthresholdσ =σ |φ−φ |twhereφ ≈0.16
0 c c
Comparedtolargercrystallites,thisDOScanbestrongly is the percolation threshold, t ≈ 2.0 and σ is approxi-
0
reduced by the charging energy.6,7,8,9 matelyequaltotheconductivityofthespheres.3,14When
We show how those microscopic parameters, which the building blocks of the network are fractal aggregates
governthecharge-transportprocessacrossmanydecades instead of solid spheres, σ has to be replaced by the
0
oflengthscales,canaccuratelyandconsistentlybedeter- aggregate conductivity σ and depends on the particle
a
minedbyac(alternatingcurrent)dielectricspectroscopy conductivity and, via the non-linear relation σ /σ =
a p
from a few Hz to infrared frequencies. In particular we (φ /0.16)1+xwithx>0,ontherealpercolationthresh-
p,c
canaddressthe parametersfor Mottvariable-rangehop- old φ of the particles.5 The value of the exponent x is
p,c
2
relatedtotherandom-walkdimensionandthefractaldi- TheexperimentsofYuet al.6,20 onthinfilmsofhighly
mension d , and is maximally 2(d −1)/(3−d ). This monodispersedsemiconductingnanocrystalsofCdSeof6
f f f
shows that on purely geometrical grounds for a network nm diameter, slightly smaller than the ATO crystallites
with φ = 0.0055, at the highest filling fraction of the discussedhere, showedgoodagreementbetweenthe the-
p,c
aggregates σ will be three to four orders of magnitude oretical and experimental value of T , T and T .21
dc 0,ES crit A
lower than in the pure filler powder.5
As remarkedin the introduction, when nanosized par-
ticles are used as fillers, chargingenergies (and quantum B. Sub-THz and far-infrared regime
sizeeffects)imposeafurtherimportantrestrictiontothe
maximal possible composite conductivity.6,7 This effect At sufficiently low frequencies the conductivity will be
can be conveniently studied in densely packed powders frequencyindependentandequaltoitsdcvalue,because
of filler material by dielectric spectroscopy. the inhomogeneities are averaged out by the motion of
the charge carriers. The minimal length scale for homo-
geneity is referred to as L ,
hom
A. Parameters for the dc conductivity
L2 =σ k T/n e2(f /2D), (6)
hom dc B hom os
Intheohmicregime,ifthereisanon-negligibledensity where n is the density of the carriers involved in the
hom
of states aroundthe chemicalpotential, andthe temper- hoppingprocessattheborderofthehomogenousregime,
atureT ishighenoughthatalsotheCoulombinteraction and the onset frequency f for the frequency depen-
os
can be neglected (T >Tcrit), σdc will obey Mott’s equa- dence of σ is divided by 2D, with D the dimension of
tionfor conductionvia variable-rangehopping (VRH):15 the system.16
Athighenoughfrequencies,whenduringhalfaperiod
σdc ∝exp[−(T0,Mott/T)ν] (1) of the oscillation of the applied field electrons can hop
solelybetweennearest-neighbors,themajorcontribution
with ν =1/4 and
to the conductivity will be due to tunnelling between
k T ≈20/(g a3), (2) localized states at neighboring sites (the pair limit).22
B 0,Mott hop
Thisincoherentprocesscanbeeitherbyphonon-assisted
where adenotes the decaylengthofthe electrondensity, or photon-assisted hopping, where in the latter case the
k theBoltzmannconstant,andg thedensityofstates energydifferencebetweenthesitesissuppliedbyphotons
B hop
relevant in the hopping process.16 instead of phonons.16 The phonon-assisted contribution
For randomly packed spheres of radius R and spacing to the conductivity is given by
s the localization length a˜ will be enlarged,7,8,9,17,18 and π2 ω
can be approximated by7,17 σphonon(ω)= 192e2ωkBTL5hopgh2opln4 ωph , (7)
(cid:16) (cid:17)
a˜=(2R/s)a. (3) with Lhop ∼ a the decay length of the electronic state
outside the conducting particles, g the relevant DOS
hop
In the following we drop the tilde. at the Fermi energy E , and ω the phonon ‘attempt’
F ph
Below Tcrit the T dependence of the conductivity will frequency.16 This formula is valid when ω < ωph; at
bedominatedbyasoftCoulombgap,leadingtoso-called higherω,wherethecontributionofphonon-assistedhop-
Efros-Shklovskii(ES) VRH:19 ping to σ becomes constant, photon-assisted processes
usually take over, with a conductivity σ given by
photon
σ ∝exp[−(T /T)1/2]. (4)
dc 0,ES
π2 2I
σ (ω)= e2~ω2L5 g2 ln4 0 . (8)
In the ES VRH model in the dilute limit of a large dis- photon 6 hop hop (cid:18)~ω(cid:19)
tance between the particles T is given by
0,ES The energy k T in eq. (7) is in eq. (8) replaced by ~ω
B
T0,ES =2.8e2/(4πǫǫ0akB), (5) and the phonon attempt frequency ωph by 2I0, with I0
being the ‘overlap’pre-factorforthe energylevels oftwo
with e the electron charge, ǫ0 the vacuum dielectric neighboring sites. In analogy with ωph, I0/~ can be in-
constant, and ǫ the relative dielectric constant of the terpreted as the attempt frequency for photon-assisted
medium. Tcrit is given by Tcrit ∼ e4aghop/[kB(4πǫǫ0)2] hopping. Equation (8) is only valid when ω < I0/~. As
and (for s>R) the charging energy by ∼e2/(4πǫǫ R). in phonon-assistedhopping, σ passes over into a plateau
0
For densely packed small particles, at high temper- at high ω.
atures but still in the regime, where Coulomb interac-
tions are important (T ≤ T ), ES VRH behavior will
crit
evolve into nearest-neighbor hopping at a temperature C. Visible and optical response
T . AboveT theconductionisthermallyactivatedwith
A A
an activation energy ∆E of the order of the charging At high frequencies (for ATO in the infrared regime)
A
energy, and T ∼(∆E )2/T . the short period of the electromagnetic field will restrict
A A 0,ES
3
the motionofthe carriersto the nanocrystallite,andthe
-2
10
ddiieelleeccttrriicc rceosnpsotnasnetcǫh(aωr)ac=ter1iz−edωbp2yD/th[ωe(cωom+pilΓe)x]rwelialltibvee 1100--43 (a) (b)
Drude-like, with ωpD the Drude plasma resonance fre- -5
10
quency and Γ = 1/τ the damping rate. In practice the -6
) 10
constant 1 has to be replaced by ǫ∞ due to other contri- m -7
butions in this frequency regime, like the polarization of S/c 1100-8
the ion cores.23 The Drude plasma frequency is related ( -9
10
to the number of carriers per unit of volume n and -10
crys 10
the effective mass m∗ as -11
e 10
-12
ω2 =n e2/ǫ m∗. (9) 10-13
pD crys 0 e 10
-14
10
For damping rates comparable to the Drude plasma fre- -15
10
quency, the real plasma frequency (where the dielectric 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2
0.5 0.5 -1 -1
constant becomes zero) will be larger than ωpD. Γ is T (K ) T (K )
determined by the boundaries of the nanoparticle and
additional (ionized impurity) scattering: FIG.1: dcconductivityasfunctionofT foradenselypacked
powder of 7%-doped ATO. For T ≤ 50 K the data can be
Γ=1/τ =vF/Lcrys, (10) fitted byeq. (4) (a), while for T ≥50 K the T dependenceis
activated (b).
where1/L isthesumoftheinversesizeoftheparticle
crys
and the inverse phonon scattering length.
III. EXPERIMENTDAALTAPROCEDURES AND sample (cmN−3) (Sσ/dccm) (ωs−pD1) (s−Γ1) (mm∗ee)
ATO-film28 2·1020 102 1015 8·1014 0.3
ITO-film29 1021 5·1014 3·1014 0.35
Measurements were performed on Sb-doped tin-oxide ATO-powder25 2·1021 10−6 1015 6·1014 0.27
nanoparticles with [Sb]/([Sn]+[Sb]) equal to 0, 2, 5, 7, ATO-powder(pw) 1021 10−2 4·1014 1014 0.3
9, and 13 at.%. The particles are monocrystalline and
spherical with diameters close to 7 nm.24 The diameter TABLE I: Transport parameters obtained for ATO and in-
ofthe 7%dopedcrystallite is 7.1 nm. Sb is incorporated dium tin oxide (ITO) (second row) at doping levels of 1020
in the casserite SnO lattice by replacing Sn4+. At the to1021 percm3. Theroom-temperaturedcconductivitiesare
2
doping level of 7%, Sb is mainly present as Sb5+, result- given in S/cm, the Drude frequencies and damping rate in
s−1, and the effective mass in free-electron masses. The first
inginn-typeconductivityoftheATOparticlesaccording
to Nutz et al..25 The amountof Sb3+ presentin the par- and second row are obtained for films of ATO28 and ITO29
resp., the last two rows contain the data on powders of 6%
ticles is negligibly small.24,26 doped ATO particles of Nu¨tz et al.25 and our data (labelled
The followed experimental procedures for the dc con-
aspwforpresentwork)onsampleswith7%Sbdoping. Fora
ductivity and dielectric measurements are described in bulkmaterialwithaDrudefrequencyof1015 Hz,ascattering
Refs. 5 and 27. The thickness of the samples was typi- timeof10−14 sandacarriermassof0.3meadcconductivity
callyafewmm. Thedcconductivitymeasurementswere is expected of 102 S/cm
performed in the dark under helium atmosphere.
TheT dependenceisgiveninFig.1,andthefrequency
dependence at temperatures down to 7 K in Fig. 2. All
data shown are for 7%-doped ATO. Similar results were
obtained at other doping levels, be it with different ab-
solute values. The data were taken in the ohmic regime. IV. ANALYSIS AND DISCUSSION
The infrared (IR) transmittance was measured on a
pellet of KBr mixed with a small amount of ATO. For
the IR-reflectance we used a precipitated film of ATO In the analysis we first show the procedure to extract
with a thickness of about 1 mm. The data are shown the parameter values from the data in the different fre-
in Fig. 3. For the analysis we also used the sub-THz quency regimes and to check their consistency. We also
transmittance and phase data (only shown in Fig. 4). makeacomparisontotheparametervaluesofPhthalcon-
InTableIwesummarizeourdataondenselypacked7- 11, for which the data are published elsewhere.27 Then
nm-sized ATO crystallites and compare them with mea- we concentrate on the density of states; the latter being
surements on doped tin oxide published in the litera- importantforthe dc conductivity. Subsequently,the im-
ture. The values of ω agree within a factor 2, while plicationsfortheuseoftheparticlesasfillersinnanocom-
pD
the spread in the scattering rates is larger. posites are discussed.
4
-1 from the packing density, we find a ∼ 2 nm, in good
10 agreement with the value calculated from T .
-2 0,ES
10 The onset of the frequency dependence of the conduc-
-3 tivity (see Fig. 2) signals that the carrier starts to feel
10
m) 10-4 371 K the inhomogeneity of the underlying structure. Using
c eq. (7), the typical length scale L at the onset can
S/ -5 123 K hom
10 be found. For 7%-doped ATO at 300 K, the onset fre-
’ ( -6 15 K quencyf =3·108Hzandσ =10−2S/cmgiveavalue
10 os dc
10-7 7.2 K of L2homnhom =9·107 cm−1.
The linear frequency dependence of the conductivity
-8
10 at 7 K in the double logarithmic plot of Fig. 2, is in
-9
10 agreementwithphonon-assistedtunneling,seeeq.(7). In
-10 therangeof10-100cm−1photon-assistedprocessestake
10 102 103 104 105 106 107 108 109 over.30 Applying eq. (7) to the conductivity data at 293
K and taking the usual value for the phonon frequency
f (Hz)
in solids ω = 1012 s−1,16,30 we find L5 g2 = 3·103
ph hop hop
eV−2cm−1, see Fig. 4.
FIG. 2: Frequency dependence of the conductivity for 7%-
doped ATO at various temperatures. In this double loga- Turning next to the high-frequency data presented in
rithmic plot the linear dependence at low temperatures is in Fig. 3, we performed a simple Drude analysis. The fit
agreement with eq. (7). (ωpD = 11000 cm−1 and τ = 3300 cm−1, together with
adielectricconstantof4.0)reproducesthemainfeatures
of the increase of the transmission and the level of the

1.0 reflectance(theoscillationsinthefittothetransmittance
Transmission (a) Reflectance (b) are an artefact because the effective ATO film thickness
0.8 data data of0.005mmismuchsmallerthantherealthicknessofthe
fit fit pressedKBrpellet). Thenumberofcarriersncrys of1021
cm−3 isdirectlyderivedfromtheDrudefrequencyandis
0.6
slightlylowerthanobtainedfromasimpleinterpretation

of the chemical composition. The bulk dc conductivity
0.4 calculated from the Drude parameters is 102 S/cm. The
fit parameters of the present samples are given in Table
0.2 I and agree well with the literature.
Fig. 4 shows the reconstructed conductivity of ATO
as function of frequency due to the processes discussed
0.0
0 100 200 300 400 0 100 200 300 400 above.

Energy (meV) For ATO the important values for the dc conductivity
can be deduced from the combination of variables that
we found from the previous analysis (i) L2 n = 9·
FIG. 3: Transmittance and reflectance of ATO with 7% Sb hom hom
doping in the infrared, after background correction. (a) For 107 cm−1, (ii) gh2opL5hop = 8 · 1043 J−2m−1 or 3 · 103
the transmission ATO is mixed with KBr and pressed into eV−2cm−1 forphoton-assistedhoppingand1042J−2m−1
a 0.57 mm thick pellet to have sufficient transparency. The for the phonon-fit to the data at 7 K, and (iii) n =
crys
oscillationsinthefittothetransmittanceareanartefact(the 1021 cm−3 and τ =10−14 s.
fit is based on the calculated effective ATO-film thickness of Using (iii) the ‘extrapolated’ dc conductivity is 102
0.005mm). (b)Likeforthesub-THzdata,thereflectiondata S/cm, a factor 104 larger than the found value of
aretakenfromprecipitatedfilmsof1mmthicktypically. The 10−2 S/cm. The estimated Fermi energy E is around
Drudefitisdiscussedinthetext. Thedatacomparewellwith F
2 eV, and from g(E) = (3/2)(n/E ) (valid for free elec-
thedata published in theliterature, see Table I. F
trons)wegetg(E )=5·1021 eV−1cm−3. From(ii)with
F
L =a of 3 nm, we find for g =3·1018 eV−1cm−3,
hop hop
a factor 103 lower than g(E ). Note that this is an av-
F
A. Procedure
erageddensity of states involvedin photon assisted hop-
ping. Duetothecurvatureofthedensityofstatesaround
Regarding the T dependence of σ (Fig. 1), the data the chemical potential, g(E) will be lower at lower ener-
canbefittedwithν =1(eq.(1))ifthefitisrestrictedto gies. Forexample,forthe phonon-fitat7Kghop isequal
T ≥ 50 K and with ν = 0.5 (eq. (4)) for T ≤ 50 K. The to 0.3·1018 eV−1cm−3.
ν = 1 fit gives an activation energy of 102 K, while the The values for τ and L can be used as a consis-
hom
exponent ν =0.5 at low T gives T =0.6·104 K.The tency check. The combination of the estimated Fermi
0,ES
localizationlengthfromT ,seeeq.(5),iscalculatedto velocityof0.8·108 cm/s,withthecrystallitesizeof7nm
0,ES
be a = 3 nm. Using eq. (3) and (2R/s) = 10 estimated and m∗ = 0.3m ,25 predicts a surface scattering rate of
e e
5
3 aresemiconducting crystalswithalownumberofcharge
m) 11002 carriers. All charges participating in the conductivity
1 within the crystal also contribute to the dc conductiv-
c 10
/ 0 ity. As for ATO the obtained conduction parameters for
S 10
-1 Phthalcon-11 from the dielectric scans give a consistent
( 10
y 10-2 picture.
t -3
vi 10-4 293 K (0 - GHz)
ti 10 293 K (GHz - THz)
c -5
u 10-6 293 K (IR, Drude fit) B. Density of states
d 10 7.2 K (0 - GHz)
-7
n 10 . . . . ’homogeneous’
o -8 phonon ForATO,the differencesbetweenthe density ofstates
C 1100-9 photon involvedinthehoppingprocessghop =3·1018eV−1cm−3
-10 andtheDrudeconductionwithinthecrystallitesg(E )=
10 10-910-810-710-610-510-410-310-210-1100 101 102 103 5·1021 eV−1cm−3 are clearly significant. The resuFlt is
asanticipatedfromthe estimatedchargingenergy ofthe
Energy (meV)
order of 50 meV, and shows its importance for the dc
powder conductivity.
FIG.4: Thevariouscontributionstotheconductivityasfunc-
For Phthalcon-11 the very low number of carriers in-
tion of photon energy. At room temperature at the lowest
volved in the hopping process is similar to the number
frequencies the conductivity is dominated by charging ener-
ofcarriersthatdeterminestheDrudecontributioninthe
gies, at intermediate frequencies phonon and photon assisted
crystallites. Since the mean size of the particles is 20
hoppingprocesses describethefrequencydependence,andin
the infrared the Drude conductivity inside the crystallites is times larger than for ATO, the charging energies will be
seen. ofthe orderof3meV,andhence areexpectedtobe neg-
ligible at room temperature.
1014 s−1, in agreement with the found value of Γ. Next,
from g k T ∼ n eV−1cm−3 we now can estimate C. Implications
hop B hom
n at T =293 K as n =1017 cm−3. Using (i) and
hom hom
nhom we find Lhom = 0.3 µm. In polymer nanocomposites with building blocks
In short, the dielectric data of ATO allow a consis- formed by diffusion-limited cluster aggregation, the airy
tent picture of the conduction process. In these densely structure of the particle network gives a strong reduc-
packed crystallites the localization length is enhanced tion in conductivity of the composite compared to the
by a factor 10 and the density of states involved in the filler (for the Phthalcon11/polymer composite a factor
dc conductivity is more than a factor 103 smaller than 104).5 This effect can be compensated by using better
that in the conduction within the crystallites. The rel- conducting particles. Particles of ATO or ITO seem to
atively large length scale for homogeneity is indicative be well-suited as the material is known to be very well-
for the presence of aggregates. Indeed, like in Ketjen- conducting. In addition, ATO crystallites are relatively
Black,2 nanoparticles of ATO are known to form chem- easilyobtainedinsizesaround7nm,and‘whenproperly
ically bonded aggregates that survive the preparation dispersed can give polymer composites with a low per-
stage.24 Duetothenatureofthechemicalbond,thecon- colationthreshold.31 However,even if the filler nanopar-
ductivities between neighboring crystallites in and out- ticles in the composite touch, they will not be in better
side the aggregates are expected to be only slightly dif- contactthaninadenselypackedpowder. Asshownhere
ferent. Note that also the value of gcrys has to be seen forATO,forthesesmallcrystallitesthe DOSinvolvedin
as an average, as inhomogeneities in the doping of ATO σ isdramaticallyreducedduetotheshiftoftheenergy
dc
might be present as well.24,26 levels away from the Fermi level by Coulomb charging
For the studied Phthalcon-11 crystallites ωpD = 1013 effects. As a consequence, an additional four orders of
s−1 and τ = 10−13 s leading to ncrys = 2·1015 cm−3, magnitude in σdc are lost compared to the bulk value.
i.e. about 1 charge per crystallite.27 The other values Other systems where size restrictions are expected to
foundforPhthalcon-11are: (i)L2homnhom =2·106cm−1, be present might be conveniently studied in a similar
(ii) g2 L5 = 1043 J−2m−1 or 2.5·103 eV−2cm−1. In way. For example several chalcogenide alloys exhibit
hop hop
these organic crystals with such a low carrier density, a pronounced contrast between the optical absorption
the charge carriers can be seen as an electron gas with in the metastable rocksalt after the intense laser pulse
an energy scale set by kBT, and g(E) can be estimated and in the initial amorphous phase.10 As shown by ex-
from g(E)kBT ∼ncrys to be 1017 eV−1cm−3. This value tendedx-rayabsorptionfinestructurespectroscopy(EX-
ofg(E)istheupperlimitforghopandghom. Fromghom = AFS)theresistivechangeaftertheintenselaserrecording
1017 eV−1cm−3, we find a decay length a of 3 nm, as pulse goes together with a crystallizationprocess, where
expected from the packing. alsosmalldomainsareinherently present. Ourdielectric
The Phthalcon-11 parameters show that the crystals method might visualize to what extent the domain walls
6
after crystallizationlimit the conductivity andhavecon- sponse, for phonon- or photon-assisted nearest-neighbor
sequencesforthebandstructurecalculations. Ifthewalls hopping, and for the Drude response of individual
becomerealbarriersquantumsizeeffectsinthesmalldo- nanocrystals. For 7 nm antimony-doped tin-oxide parti-
mainswillinvalidatetheuseofperiodicboundarycondi- cles the analysisunambiguouslyquantifiedthe reduction
tions in the calculations. Also the glassybehavior in the of the density of states involved in the dc conduction
conductanceofdepositedindium-tinoxidesamplesinthe compared to the value extrapolated from the Drude re-
insulating regime,11 and of quench-condensed insulating sponse at infrared frequencies. Dielectric scans with a
granularmetals32 mightbefurtherclarifiedbytheuseof similar analysis will also be revealing in other systems
our dielectric approach and analysis. Scanning the fre- where size limitations are expected to play a role.
quency will reveal the evolution of the length scales and
DOS involved in the relaxation processes.
Acknowledgments
V. CONCLUSIONS It is a pleasure to acknowledge Roel van de Belt of
NanoSpecials(Geleen,The Netherlands),whomadethe
By combining data of sub-THz transmission with in- ATO samples available, and Matthias Wuttig from the
frared transmission and reflection we were able to ex- Physikalisches Institut of the RWTH Aachen University
plainthefullfrequencyresponseofdensely-packednano- in Germany for fruitful discussions about phase-change
sizedcrystallitesusingthe parametersfor Mottvariable- materials. This workforms partofthe researchprogram
rangehopping,forheterogeneity-inducedenhancedacre- of the Dutch Polymer Institute (DPI), project DPI435.
1 Fractals and Disordered Systems, eds. A. Bunde and S. 15 N. F. Mott, Philos. Mag. 19, 835 (1969).
Havlin (Springer-Verlag,Berlin, 1991). 16 H. B¨ottger and V. V. Bryksin, Hopping Conduction in
2 D. van der Putten, J. T. Moonen, H. B. Brom, J. C. M. Solids (Akademie-Verlag,Berlin, 1985).
Brokken-Zijp, and M. A. J. Michels, Phys. Rev. Lett. 69, 17 R.N´emethandB.Mu¨hlschlegel,Z.Phys.B70,159(1988).
494(1992);L.J.Adriaanse,J.A.Reedijk,P.A.A.Teunis- 18 I. P.Zvyagin, JETP Lett. 69, 932 (1999).
sen,H.B.Brom, M.A.J.Michels, andJ.C.M.Brokken- 19 B. I. Shklovskii and A.L. Efros, Electronic Properties of
Zijp, Phys.Rev.Lett. 78, 1755 (1997). Doped Semiconductors (Springer-Verlag, Berlin, 1984).
3 C. Grimaldi, T. Maeder, P. Ryser, and S. Str¨assler, Phys. 20 The energy levels are discrete and assumed to have an
Rev. B 68, 024207 (2003); C. Grimaldi and I. Balberg, overall homogeneous linewidth around 100 meV.
Phys. Rev.Lett. 96, 066602 (2006). 21 Duetoa differentgeometry, theactivation energies in the
4 J. C. M. Brokken-Zijp, D. P. de Bruijn, K. P. Datema, deposited films of CdSe nanoparticles found by I. Balberg
C. A. Emeis, A. H. Kramer, J. B. van Mechelen, and et al.,Phys. Rev.B 75, 153301 (2007), are larger.
A. J. Meruma, patents: US05319009, EP0064254A1, 22 M.PollakandT.M.Geballe,Phys.Rev.122,1742(1961).
WO09324562A1. 23 C.Kittel, Introduction to Solid State Physics(John Wiley
5 L. J. Huijbregts, H. B. Brom, J. C. M. Brokken-Zijp, M. and Sons, New York,7th edition 1996).
Kemerink, Z. Chen, M. de Goeje, M. Yuan, and M. A. J. 24 W.E. Kleinjan, J. C.M. Brokken-Zijp,R.vandeBelt, Z.
Michels, J. Phys.Chem. B 110, 23115 (2006). Chen, and G. de With, J. Mater. Res., accepted.
6 D.Yu,C.Wang,B.L.Wehrenberg,andP.Guyot-Sionnest, 25 T.Nu¨tz,U.zumFelde,andF.Haase,J.Chem.Phys.110,
Phys. Rev.Lett. 92, 216802 (2004). 12142 (1999).
7 J. Zhang and B. I. Shklovskii, Phys. Rev. B 70, 115317 26 C. McGinley, H. Borchert, M. Pflughoefft, S. Al Mous-
(2004). salami, A. R. B. de Castro, M. Haase, H. Weller, and T.
8 I. S. Beloborodov, A. V. Lopatin, and V. M. Vinokur, M¨oller, Phys.Rev.B 64, 245312 (2001).
Phys. Rev.B 72, 125121 (2005). 27 L. J. Huijbregts, H. B. Brom, J. C. M. Brokken-Zijp, M.
9 M.V.Feigelman andA.S.Ioselevich,JETP Lett.81,277 A. J. Michels, M. de Goeje, and M. Yuan, Phys. Stat.
(2005). Sol.(c) 3, 259 (2006); L. J. Huijbregts, H. B. Brom, J. C.
10 W. Welnic, S. Botti, L. Reining, and M. Wuttig, Phys. M. Brokken-Zijp, M. Yuan, and M. A. J. Michels, Phys.
Rev.Lett. 98, 236403 (2007); M. Wuttigand N. Yamada, Stat. Sol. (c), accepted.
Nature materials 6, 824 (2007). 28 E. Shanthi, V. Dutta, A. Banerjee, and K. L. Chopra, J.
11 M.Ben-Chorin,Z.Ovadyahu,andM.Pollak,Phys.Rev.B Appl.Phys. 51, 6243 (1980).
48,15025(1993);A.Vaknin,Z.Ovadyahu,andM.Pollak, 29 D. Mergel and Z. Qiao, J. Phys. D:Appl. Phys. 35, 794
Phys. Rev. Lett. 84, 3402 (2000); V. Orlyanchik and Z. (2002).
Ovadyahu,Phys. Rev.B 75, 174205 (2007). 30 J.A.Reedijk,L.J.Adriaanse,H.B.Brom,L.J.deJongh,
12 D. Stauffer and A. Aharony, Introduction to percolation and G. Schmid, Phys. Rev. B 57, R15116 (1998); J. A.
theory (Taylor & Francis, London,1985). Reedijk, H. C. F. Martens, H. B. Brom, and M. A. J.
13 J. P. Straley, J. Phys. C 9, 783 (1976), Phys. Rev. B 15, Michels, Phys. Rev.Lett. 83, 3904 (1999).
5733 (1977). 31 V. A. Soloukhin, J. C. M. Brokken-Zijp, and G. de With,
14 R. Zallen, The Physics of Amorphous Solids (Wiley and J. Pol. Sci. B 45, 2147 (2007).
Sons, New York,1983). 32 N.KurzweilandA.Frydman,Phys.Rev.B75,020202(R)
7
(2007).