Combined analysis of energy band diagram and equivalent circuit on nanocrystal solid
Shinya Kano, Masato Sasaki, and Minoru Fujii
Citation: Journal of Applied Physics 119, 215304 (2016); doi: 10.1063/1.4953216
View online: http://dx.doi.org/10.1063/1.4953216
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JOURNAL OF APPLIED PHYSICS 119, 215304 (2016)

Combined analysis of energy band diagram and equivalent circuiton nanocrystal solid
Shinya Kano,a) Masato Sasaki, and Minoru Fujiia)
Department of Electrical and Electronic Engineering, Graduate School of Engineering, Kobe University,
Rokkodai, Nada, Kobe 657-8501, Japan(Received 6 April 2016; accepted 21 May 2016; published online 3 June 2016)
We investigate a combined analysis of an energy band diagram and an equivalent circuit on
nanocrystal (NC) solids. We prepared a flat silicon-NC solid in order to carry out the analysis. An
energy band diagram of a NC solid is determined from DC transport properties. Current-voltage
characteristics, photocurrent measurements, and conductive atomic force microscopy images
indicate that a tunneling transport through a NC solid is dominant. Impedance spectroscopy gives
an equivalent circuit: a series of parallel resistor-capacitors corresponding to NC/metal and NC/NC
interfaces. The equivalent circuit also provides an evidence that the NC/NC interface mainly
dominates the carrier transport through NC solids. Tunneling barriers inside a NC solid can betaken into account in a combined capacitance. Evaluated circuit parameters coincide with simple
geometrical models of capacitances. As a result, impedance spectroscopy is also a useful techniqueto analyze semiconductor NC solids as well as usual DC transport. The analyses provide indispensable
information to implement NC solids into actual electronic devices. Published by AIP Publishing.
[http://dx.doi.org/10.1063/1.4953216]

I. INTRODUCTION

Semiconductor nanocrystal (NC) has been intensively
studied in the field of nanomaterials for decades due tothe unique electronic and opto-electronic properties.1–4 A
semiconductor NC solid has a potential application in solution–processed electronic devices, such as a photodetector,
field-effect transistor, and solar cell.5 Carrier transport of theNC solid in electronic devices can be strongly dependent on
the interface of semiconductor NCs. For example, tunnelingbarriers of a NC surface caused by a slight oxidation impede
the transport of charge carriers.6,7 Various mechanisms ofthe carrier transport through NCs have been proposed: forexample, variable range hopping,8,9 Fowler-Nordheim
tunneling,10 and Poole-Frenkel emission.11,12 Traditionally,
these analyses are based on carrier dynamics in energy band
diagrams.
Impedance spectroscopy has been recently used to findan equivalent electrical circuit of nanomaterial devices.13–16
Sugiyama et al. suggested an application of impedance
spectroscopy to Cu(In, Ga)Se2-based solar cells because anequivalent circuit can be affected by defects around materialinterfaces.16 Huh et al. evaluated a Schottky contact between
a SnO2 nanowire and a metal electrode by using equivalent
electrical circuit.13 However, limited research articles haveshown an impedance analysis on NC solids.12,17 In addition,correspondence of an equivalent electrical circuit to anenergy band diagram on a NC solid has not been shown. For
example, it is not clear how to treat tunneling barriers insidea NC solid as a capacitance quantitatively. It is meaningful
to understand the correspondence by using impedancea)Authors to whom correspondence should be addressed. Electronic
addresses: kano@eedept.kobe-u.ac.jp and fujii@eedept.kobe-u.ac.jp

0021-8979/2016/119(21)/215304/6/$30.00

spectroscopy in order to design electronic devices based on
NC solids.
In this article, we analyze electrical properties of NC
solids by using both an energy band diagram and an equivalent circuit. We form a flat solid by using a completely
dispersed NC colloid and investigate the current transport byusing these two viewpoints. An energy band diagram of theNC solid is discussed by analyzing DC-voltage characteristics and a photocurrent generation of NC solids. Impedance
spectroscopy is carried out to express the NC solid as anequivalent circuit. We discuss the validity of an estimatedequivalent circuit.
II. EXPERIMENTAL

Among various elements of semiconductor NCs, silicon(Si) NCs are recently investigated intensively because of thehigh compatibility to the complementary metal-oxide semiconductor (CMOS) technology.9,18–22 We have recently
developed a colloidal Si NC having heavily boron (B) andphosphorus (P) codoped shells.23–28 The codoped Si NC isdispersible in polar solvents without organic ligands becauseof a negative electrostatic potential induced on the surface.23
NC solids made from the colloids have very high flatness ina wide range, allowing us to study the transport properties indetail.29
Colloidal B and P codoped Si NCs were prepared using
a co-sputtering method described in our previous papers.23–25
Si-rich borophosphosilicate films were deposited on a stainless steel plate by co-sputtering Si, SiO2, B2O3, and P2O5.
After removal from the stainless steel plate, the films wereannealed in a N2 atmosphere at 1200  C for 30 min. Codoped
Si NCs were grown in the films by annealing. Then, the films
were mixed with hydrofluoric acid (46 wt. %) to etch SiO2

119, 215304-1

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215304-2

Kano, Sasaki, and Fujii

matrices and extract the Si NCs. The Si NCs were centrifuged and transferred into methanol. Concentration of colloidal Si NCs was 0.4–0.5 mg/ml. The surface of Si NCs was
slowly oxidized during storage in methanol. The thickness ofthe surface oxides is estimated to be 1 nm at most.26 Figure1(a) shows a photograph of finished colloidal Si-NC solution. Clear solution without any agglomerates can beobtained because the Si NC in the solution is well dispersed.
This dispersion is important to fabricate a flat NC solid on asubstrate. Figure 1(b) shows a transmission electron microscopy (TEM) image of codoped Si NCs annealed at 1200  C.
An average diameter of NCs is estimated to be D ¼ 7 nm in
our previous report.24
A fabrication method of a Si-NC solid is as follows.
Colloidal Si-NC solution was spin-coated on an indium-tinoxide (ITO, 90 nm)/glass substrate, which was cleaned up by
ultrasonication with alcohol and UV/O3 treatments. The
thickness of the spin-coated solids was measured by a stylus
profiler (DektakXT, BRUKER Corporation) and was 50 nm.
Aluminum (Al) electrodes (area A ¼ 1.1  106 m2, thickness t ¼ 100 nm) were thermally deposited on the solid. It isnoted that the Si-NC solids were not intentionally etched andsintered prior to the Al deposition, and thus the Si NCs do
not coalesce each other. Topography and local conductivityof the solids were obtained by a conductive atomic force
microscopy (AFM) using a rhodium (Rh)-coated cantileverat ambient condition (HITACHI High-Tech Science,E-sweep). Topographic and conductive images of the solids
were visualized by using Gwyddion.30
Figure 1(c) shows a schematic illustration of a device
structure under investigation. Current-voltage characteristics
were taken by a source measure unit (Keithley, 236) at

FIG. 1. (a) Photograph of colloidal Si NCs in methanol. (b) TEM image of
Si NCs. (c) Schematic illustration of a device under investigation. A bias
voltage is applied from ITO to Al electrode.

J. Appl. Phys. 119, 215304 (2016)

ambient condition. The accuracy of DC measurement was
0.3%. Photocurrent was obtained by illuminating the sample
with a 405-nm semiconductor laser module. The illuminatedarea was approximately 2–3 mm in diameter. Impedance
spectroscopy was carried out by using LCR meter (NF
Corporation, ZM2376). Frequency of AC voltage waschanged from 1 Hz to 1 MHz. An AC voltage was constant at
10 mV through all measurements. The accuracy of impedance was estimated to be several tens of % at most, which isgood enough to evaluate the equivalent circuit.
III. RESULTS AND DISCUSSION

Figure 2(a) shows a linear plot of a current-voltage characteristic of a device in Fig. 1(c). In a lower voltage region(1.0 < V < 0 V), a current is proportional to a bias voltage.
Since a very thin oxide layer surrounding NCs works as a tunneling barrier, this current can be attributed to a tunnelingphenomenon. It is known that a tunneling current through athin insulator is linearly proportional to a bias voltage in a small
voltage region.31 In a higher voltage region (V < 1.0 V), a
current increases rapidly.
Figure 2(b) shows a Fowler-Nordheim plot (ln(J/V2)
1/jVj) of the current-voltage characteristics. The dependence of a current density J versus V in a Fowler-Nordheim
tunneling model is described as32

1
jJj / jVj2 exp 
:
(1)
jVj
Because a straight line can be well fitted in jVj > 1.0 V
of Fig. 2(b), we clearly see a Fowler-Nordheim tunnelingprocess contributes to the current transport in this region.
The electric field in the tunneling barrier between Si NCs is
1 MV/cm at jVj ¼ 1.5 V if an applied voltage is divided inevery Si-NC shells (shell thickness 1 nm (Ref. 27)) equally.
This high electric field can be attributed to the FowlerNordheim tunneling regime.
Figure 3(a) shows a current response of a Si-NC solid
under light irradiation (k ¼ 405 nm, 3.3 mW) as a function oftime. Conductance of NC solids increases under the illumination because an electron and hole pair is generated inside
the Si NC. The increment of the conductivity by a light irradiation is much weaker at longer wavelength (786 nm) due
to smaller absorption of Si-NCs (Figure S1). A time constantof the device in this measurement is estimated to be about 3 s(Figure S2). The validity of the time constant will be
discussed later.
Figure 3(b) shows current-voltage characteristics as afunction of light intensity. In general, photocurrent Iph
increases with light intensity F following a power law as Iph/ Fc.7 Here, Iph is the difference between a current measured
under illumination and a dark current. Figure 3(c) describes
a relationship between c and V. As the applied voltage
increases, the value of c increases from 0.1 to unity overjVj > 1.0 V, which means excited carriers can be extracted
from the solids more efficiently. Because the voltage where
the Fowler-Nordheim tunneling appears is also jVj > 1.0 V,

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215304-3

Kano, Sasaki, and Fujii

J. Appl. Phys. 119, 215304 (2016)

FIG. 2. Current-voltage characteristics
of a device in Fig. 1(c) at room temperature. (a) Linear and (b) FowlerNordheim plots. Dashed lines are linear fit lines.

the Fowler-Nordheim tunneling possibly enhances the carrier
extraction in this voltage region.
Current images of a Si-NC solid by conductive AFM
support the tunneling mechanism of carrier transport.
Conductive AFM can obtain a local conductance mapping of
nanomaterial films.33–35 Figure 4(a) shows a local topographic image of a Si-NC solid. A surface roughness of thesolid is about 10 nm, corresponding to a length of a few number of NCs in series. This image supports the formation of aflat NC solid. Figures 4(b)–4(d) show a conductance mapping of the Si-NC solid by applying V ¼ 0, 6, and 10 V,
respectively. As increasing V, a conductance mapping
similar to Fig. 4(a) can be obtained. We can see a lower

topographic area gives higher conductance, which is a similar tendency of a polycrystalline NiO insulator.35 In addition,a small current less than 10 pA is observed on the top of a Si
NC solid in Fig. 4(d) although V ¼ 10 V is applied. This
image indicates that the intensity of current decays exponentially as increasing the thickness of a solid (I / expðadÞ,
where a is a decay constant and d is the thickness). Thistendency supports that a tunneling transport is dominant
through the Si-NC solid.
In Fig. 5, we summarize a band diagram of a Si-NC
solid. Valence band edge (5.0 eV) and conduction band edge
(4.0 eV) in 7-nm diameter of Si NCs have been estimated byusing photoelectron yield spectroscopy and valence band

FIG. 3. (a) Current response of Si-NC
solids with light irradiation (k ¼ 405 nm,
3.3 mW) at V ¼ 0.1 V. Colored andnon-colored regions correspond to the
response with and without the irradiation, respectively. (b) I—V characteristics with light irradiation. (c) Exponent
factor (Iph / Fc) calculated in each constant voltage.

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215304-4

Kano, Sasaki, and Fujii

J. Appl. Phys. 119, 215304 (2016)

FIG. 4. (a) Topographic image and(b)–(d) current image of colloidal SiNC solid/ITO substrate observed byconductive AFM. Bias voltage V isapplied to ITO as (b) V ¼ 0 V, (c)V ¼ 6 V, and (d) V ¼ 10 V.

spectra.28 The small value of the band gap (1.0 eV) is dueto heavy doping of B and P in Si NCs.24 When no voltage isapplied (V ¼ 0 V, Fig. 5(a)), an offset voltage appears corresponding to the difference of work function between Al and
ITO. This offset voltage is expected to be 0.6 eV; however,
only 0.2 V of the offset voltage is observed in Fig. 2(a). Thisdifference is probably attributed to the trapped charge ininterface states between NC solids and metal electrodes.
When jVj < 1.0 V (low voltage regime, Fig. 5(c)),
carriers in metal are injected via simple tunneling process
through tunneling barriers of codoped Si-NC shells. As
jVj > 1.0 V (high voltage regime, Fig. 5(d)), the FowlerNordheim tunneling starts to appear through a conduction

band of the tunneling barrier. A higher voltage enhances amean free path of carriers, and thus, the carriers irradiated byphoton flux move more efficiently.
Here, we show an impedance spectroscopy of a Si-NC
solid. We discuss a relationship between the band diagram in
Fig. 5(a) and an equivalent circuit of the structure.
Figure 6(a) shows complex-plane impedance plots of adevice in Fig. 1(c) in ambient condition. Filled square andfilled circle express experimental and theoretically simulated
results, respectively. The inclined experimental curve indicates that a NC solid consists of both capacitance and resistance components, as shown in a previous report.36 Here, we
fit a theoretical impedance plot using an electrical circuit

FIG. 5. Band diagram of a Si-NC
solid. (a) Under V ¼ 0 V and (b) corresponding schematic illustration of thestructure. (c) Under V ¼ 0.8 V (lowvoltage regime), and (d) V ¼ 1.5 V(high voltage regime).

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215304-5

Kano, Sasaki, and Fujii

J. Appl. Phys. 119, 215304 (2016)

FIG. 6. (a) Impedance plots of a device
in Fig. 1(c) at ambient condition. A
DC bias VDC ¼ 0 V is applied to the device. (b) Equivalent circuit of a device
in Fig. 1(c). (c) Definition of eachjunction Ji (i ¼ 1, 2, and 3) in a schematic model.

model in Fig. 6(b): parallel resistor-capacitors 1–3 in seriescorresponding to ITO/NC, NC/NC, and NC/Al interfaces,
respectively (Fig. 6(c)). In this model, we consider a NC
solid as a continuum insulator, and then resistor-capacitor
networks are combined into a single parallel resistorcapacitor model.
Impedance of the circuit in Fig. 6(b) is described asfollows:
Z ¼ Z0 þ iZ 00
R1
R2
R3
¼
þ
þ
2
2
1 þ ðxC3 R3 Þ2
1 þ ðxC1 R1 Þ
1 þ ðxC2 R2 Þ
i

xC1 R211 þ ðxC1 R1 Þ2þ

xC2 R221 þ ðxC2 R2 Þ2þ!

xC3 R231 þ ðxC3 R3 Þ2;(2)

where x is an angular frequency, and Rj and Cj are resistanceand capacitance of jth junction (j ¼ 1, 2, and 3). Evaluatedparameters from Fig. 6(a) are shown in Table I.
The value of R2 is more than a hundred times larger thanthose of R1 and R3. This is due to serially connected 5–7 tunneling barriers between Si NCs as shown in Fig. 6(c). This
result indicates that tunneling barriers exist in a Si-NC solid.
The values of resistance R1 and R3 might be related to difference of Fermi levels between a NC solid and metals.
It is noted that the experimental difference between C1and C3 is possibly attributed to the morphology of electrode/NC junctions. In case of an Al/Si NC junction, the Al electrodecovers the surface of NCs because of the thermal evaporation
process as shown in Fig. 6(c). In contrast, the ITO electrode
under Si NCs ideally contacts on the top of NCs since the NCs
are just spin-coated. The capacitance value of C3 can beslightly larger than that of C1, and we assign the Al electrodeto the larger capacitance junction (J3) in Fig. 6(c).
Capacitances C1 and C3 between the NC solid and themetals are evaluated by using a mutual capacitance between

a Si NC in 7-nm diameter and a metal. Mutual capacitance
between a single sphere and an infinite conductive plane canbe evaluated as
C ¼ 4pr sinh a

1
X

1
;
sinh
na
n¼1

(3)

where r is a radius of a sphere,  is a dielectric constant of a
medium, a is defined to be acosh(d/r), and d is a distance
between the center of a sphere and the surface of a conductive plane, respectively (Figure S3(a)).37 Combined capacitance of all NCs against a metal is estimated to be 91 nF at
maximum, which is 1.1–1.7 times larger than the experimental C1 and C3 (the detailed calculation is shown in the supplementary material).38 In our previous study, an experimental
fill factor of drop-coated NC solids was 35%.27 This difference between experimental and theoretical results of C1 andC3 is in an acceptable range.
Capacitance C2 inside the NC solid is evaluated by using
a mutual capacitance between two Si NCs in 7-nm diameter.
Mutual capacitance between two spheres in a radius r can be
expressed as,
C ¼ 4pr sinh b

1
X

1
;
sinh
2nb
n¼1

(4)

where b is defined to be acosh(d0 /2r), and d0 is a distance
between the center of each sphere (Figure S3(b)).37,39,40
Combined capacitance inside the solid is 4.0 nF (the detailed
calculation is shown in the supplementary material).38 Thisvalue is smaller than the experimental C2, but still within an
acceptable range by considering the crudeness of the model.
We also show that the impedance plot is not dependent
on a DC bias voltage and 405-nm light irradiation in Figs.
S4(a) and S4(b).38 All of the fitted parameters and evaluatedtime constant (si ¼ RiCi) are nearly constant in Figs.
S4(c)–S4(e). The time constant in the NC solid (s2) is around

TABLE I. Equivalent circuit parameters of an Al/Si-NC solid/ITO layered structure in Fig. 6(a).
Interface of ITO/Si-NC solid (J1)
R1
141 kXA Si-NC solid (J2)Interface of Si-NC solid/Al (J3)

C1

R2

C2

R3

C3

54.6 nF

101 MX

13.4 nF

890 kX

79.8 nF

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215304-6

Kano, Sasaki, and Fujii

3 s, which clearly corresponds to the decay constant of thephotocurrent in Fig. 3(a). These results imply that most of
VDC are applied to a Si-NC solid due to its high resistivity,
which dominates the carrier transport. As shown above, theequivalent circuit obtained by impedance spectroscopy easily provides us the similar information about a NC solid in
DC transport analysis.
IV. CONCLUSIONSIn conclusion, we analyze the electrical properties of
NC solids by using both an energy band diagram and an
equivalent circuit. DC transport of a NC solid indicates thatcharge carriers are tunneling through tunneling barriers inthe solid. Photocurrent generation is enhanced by applying ahigher electric field because of the Fowler-Nordheim tunneling process. Conductive AFM images of a NC solid showthat a current through the solid is strongly dependent on the
thickness, which also indicates the tunneling transport.
Impedance spectroscopy reveals the equivalent circuit of NC
solids between ITO and Al. It is a three series of parallel
resistor-capacitors corresponding to ITO/NC, NC/NC, and
NC/Al. The obtained circuit parameters are explained by asimple structural model. As a result, impedance spectroscopy
can be a useful technique to analyze semiconductor NC solids as well as usual DC transport. The combined analysis of
the energy band diagram and the equivalent circuit will be
indispensable to obtain information necessary to implement
NC solids into actual electronic devices.
ACKNOWLEDGMENTS

The authors thank Dr. K. Imakita for discussion
regarding the results. This work was partly supported by
KAKENHI (26886008 and 16H03828), 2014 JSPS Bilateral
Joint Research Projects (Japan-Czech Republic), Visegrad
Group (V4)-Japan Joint Research Project on Advanced
Materials “NaMSeN”. Measurements on a conductive AFM
and a stylus profiler were supported by “Nanotechnology
Platform Project (Nanotechnology Open Facilities in Osaka
University)” of Ministry of Education, Culture, Sports,
Science, and Technology, Japan [No. S-15-OS-0033].
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