# Anderson Insulators in Self-Assembled Gold Nanoparticles Thin Films: Single Electron Hopping between Charge Puddles Originated from Disorder

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{n}S

_{2}-linked Au NPs, electrical behaviors ranging from insulating to metallic-like were discovered. Following Mott’s prediction, the insulating property presents when the carbon chain length is greater than a critical value n = 5. This strong impact of the molecule linkage on the charge conduction originates from the exponential increase of tunneling resistance, R

_{T}with the number of carbon atoms: ${R}_{T}\propto \mathrm{exp}\left(\beta n\right)$. The tunneling decay constant per carbon atom β ~ 1.0 ± 0.1 for alkanedithiol molecular junctions [10,11]. In general, the Mott–Hubbard theory states that the charge conduction is mainly governed by the coupling strength, which is quantified by the dimensionless tunneling conductance g = R

_{K}/R

_{T}. R

_{K}= h/e

^{2}~ 25.8 kΩ is the quantum resistance. In the strong-coupling regime g >> 1, the NP assembly is metallic [12], but, in the opposite case g << 1, it becomes an insulator. Some experiments demonstrate the possibilities to control coupling strength by other means, such as NP coverage and film thickness [13,14].

^{−7}to 10

^{2}. In particular, we present detailed results on AIs, including the temperature-dependent transport, magnetoresistance and device current modulated by a gate voltage. To begin with, let us review some important theoretical predictions on the Mott insulators (MIs).

#### 1.1. Hubbard Bandgap and Mott Insulators

_{m}~ 2.6 is the dielectric constant of the linkage molecule.

_{T}is comparable to R

_{K}. When g is not too large, the activation energy becomes

_{T}= R

_{K}. Owing to the existence of random charge offset in real NP assemblies, the energy cost for charge tunneling ranges between 0 and E

_{C}. Therefore the activation energy would be half of the value given by Equation (2) [21].

_{1}= E

_{a}/k

_{B}.

#### 1.2. The Cotunneling and Temperature-Dependent Resistance

_{j}, via the elastic cotunneling channel,

_{a}. $\delta $ is the electron energy level spacing in the NPs. With the same approach as Efros–Shklovskii (E–S) theory of VRH [26], the overall charge conduction should balance the charge hopping distance and energy cost, which arises from long-range Coulomb repulsion, $U\sim {e}^{2}/4\pi \kappa {\epsilon}_{0}{r}_{j}$. κ is the effective dielectric constant of the NP assembly. Therefore, the overall hopping probability should also include an Arrhenius factor, describing the thermal activation to overcome the energy U, and reads as

_{j}by varying r

_{j}, one obtains a most probable hopping range,

^{*}, (~T

^{−1/2}) so the resistance still follows the relation ln R ~ T

^{−1/2}well. If comparing Equations (5) and (10), one finds that the inelastic cotunneling dominates over the elastic one when ${l}_{in}>{l}_{el}$ or ${k}_{B}T~0.1\sqrt{E\delta}$.

## 2. Results and Discussion

#### 2.1. Temperature Dependent Resistance

_{x}s) are found, with β

_{x}~ 11.8 nm

^{−1}.

_{RT}: A high-R

_{RT}device tends to be insulator-like. Since the Au NP assemblies using our deposition scheme were not always closely packed and regular, R

_{RT}for a specific molecule modification varied from sample to sample. To clarify the role of R

_{RT}in MIT, we lowered the junction barrier height of insulating MOA devices by applying high dosage e-beam exposure. The exposure reduced the R

_{RT}of MOA devices at most five orders of magnitude and drove some devices across the critical point of MIT at R

_{RT}~ R

_{K}. As one can see in Figure 2c, e-beam exposed MOA devices may exhibit metallic (MOA M) or insulating (MOA MI) behaviors, evidently relating to their R

_{RT}. In addition to the interparticle coupling strength, the disorder, which is hardly determined by R

_{RT}, serves as another parameter governing the MIT in these NP assemblies.

#### 2.2. Mott Insulator and Cotunneling in Molecule Junctions

_{1}, a measure of Hubbard gap. Summarized in Table 1, T

_{1}falls in the range between 40 and 80 K, except some devices with R

_{RT}< R

_{K}. As we have mentioned, it is plausible to assume that R

_{RT}is a good estimation of tunneling resistance in the theoretical model. Consequently, the insulating devices with R

_{RT}> R

_{K}receive negligible charge fluctuation so T

_{1}~ E

_{C}/k

_{B}, which is similar in our devices with the same NP size. According to Abeles formula, E

_{C}/k

_{B}ranges from 50 K to 75 K depending on the value of interparticle distance s. In contrast, the devices with R

_{RT}close to R

_{K}would have a much smaller T

_{1}, typically smaller than 10 K, because of the quantum correction. In an earlier work, we already presented how T

_{1}depends on the tunneling resistance and found a good agreement with Equation (2) [19].

_{B}(~T

_{1}) = 50 K. Moreover, we should look if E-S VRH law is well obeyed with a temperature-dependent l

_{in}. Calculation on the hopping exponent (r

^{*}/l

_{in}) was performed by using parameters obtained in our experiments: a = 12 nm, g = 0.01, E/k

_{B}(~T

_{1}) = 50 K, κ = 21.5. The blue curve in Figure 4a shows temperature-dependent l

_{in}using Equation (10), while the green curve shows temperature-independent l

_{in}= 0.1 a. As a comparison, the function (r

^{*}/l

_{in}) = (T

_{2}/T)

^{1/2}with T

_{2}= 700 K is plotted with the red curve. Indeed, Equation (10) describes a slowly varying function of T, so the blue curve is highly linear in the temperature regime from 25 K to 1 K. In addition, all curves show similar slopes in this plot.

_{2}values for insulating devices. One could see that T

_{2}~ 1000 K for devices with R

_{RT}> R

_{K}. As discussed in the previous section, a large R

_{RT}guarantees a weakly coupled NP network and a standard MI. As such, the cotunneling length l

_{in}~ a according to Equation (10). An effective dielectric constant κ ~ 20 is much larger than that of the insulating matrix molecules κ

_{m}, owing to the charge screening effect from mobile charges within nearby NPs.

_{RT}follow the E–S VRH law but with a much lower T

_{2}value. The unusual low T

_{2}could result from a huge κ and/or a very large l

_{in}. The later seems unreasonable in the scope of charge tunneling in weakly coupled NPs; however, these low T

_{2}systems are on the threshold of strong coupling, so a new picture beyond inter-NP charge hopping is needed. If κ is unchanged, the cotunneling length is increased by two orders of magnitude, inferring the charge delocalized from a single NP but confined in a larger space l

_{in}~ 10–100 a. Such a confinement can be argued to originate from Anderson localization due to quantum interference. To confirm this, one should investigate the magnetic response of these devices.

#### 2.3. Magnetoresistance

_{e}and l

_{φ}are elastic and inelastic scattering lengths. $\mathsf{\Psi}$ is the digamma function. Figure 5a also presents the fitting result according to Equation (12) by using t = 30 nm, and a sheet resistivity of about 12 Ω. The characteristic length $\sqrt{{l}_{e}{l}_{\varphi}}$ ~ 300 nm, which gives a minimal inelastic scattering l

_{φ}~ 9 µm if one accounts for the upper bound of l

_{e}~ 2r = 12 nm. In contrast, the fitting result for Figure 5b gives a characteristic length $\sqrt{{l}_{e}{l}_{\varphi}}$ ~ 100 nm. The reduction of l

_{e}and l

_{φ}for the latter case is probably due to the electron–photon scattering at a higher temperature. Nevertheless, the MR indicates a large l

_{φ}and vigorous quantum inference for conduction electrons at T < 1 K.

_{2}and R

_{RT}. Devices having large T

_{2}values are virtually not affected by the magnetic field. In contrast, extraordinary magnetic field responses were observed in devices having low T

_{2}values (typical < 10 K). Figure 6a illustrates the IV characteristic of a low T

_{2}MPA device at B = 0 and B = 8 T at T ~ 50 mK. In both cases, the IV presents sub-meV gap, of which the size clearly shrinks in the large field. When V > 0.05 mV, the non-linear IV curves follow a power law, $I\propto {V}^{\eta}$ as shown in Figure 6b. At zero field, the IV exponent η is ~5, while it reduces to ~2 in a large magnetic field. The quantitative change in IV curves for some insulators is best illustrated in Figure 6c, in which we summarize how the IV exponent η changes with the field. It is found that η can be reduced by one-half at most. A power law in IV curve is predicted by the cotunneling theory for weakly coupled NPs, and the exponent $\eta =2j-1$, where j is the number of junctions involved in the cotunneling process [30]. However, such a picture needs correction before applied to AIs, since the localization length for cotunneling may be very large, l

_{in}>> a.

^{*}, or the field simply enlarges the inter-junction distance. For the former one, η for all kinds of insulator should be changed, which is not true for MIs. The later one is meaningful only for AIs, in which “inter-junction” distance is the size of charge puddles, not of NPs.

_{RT}. Again, we can see a good correlation between MR and R

_{RT}: when R

_{RT}is smaller than ~100 Ω, the device presents a small positive MR, typical at the order of 0.2%, featuring the weak anti-localization in a disordered metal. When R

_{RT}is greater than ~10

^{5}Ω, we see very small MR except for two special cases. When R

_{RT}is in between, the device would present large negative MR. In most cases, the resistance can be reduced for one to two orders of magnitude at B = 8 T. This infers that the magnetic response is a good indication for telling the difference between an MI and an AI.

_{φ}, l

_{H}) > 2r. To meet the criteria in our case, low temperature and low magnetic field—B ~ 0.1 T are essential. As the magnetic field increases, the charge puddles should be gradually enlarged, and an external gate voltage can be employed to probe the size change. We will discuss the gate-voltage modulation in the following section.

#### 2.4. Single Electron Tunneling and Gate-Modulated Transport

_{b}characteristics at V

_{g}= 0 and −3 V. Here, V

_{b}and V

_{g}respectively denote the source–drain bias voltage and gate voltage. When V

_{g}= −3 V, the threshold voltage for charge conduction ~0.22 mV, and when V

_{g}= 0 V, the threshold voltage decreases maximally by ~0.1 mV. Although the magnetic field also reduces the Coulomb gap, here the gate modulation is much smaller than the magnetic change. One can see this by comparing IV

_{b}characteristics in zero field, B = 0.5 T and B = 8 T, as illustrated in Figure 8b. When the field is small, B = 0.5 T, the Coulomb gap is almost unchanged, but as we will see later, such a small field already alters the gate modulation into a very different pattern from that at zero fields.

_{b}as a function of V

_{b}and V

_{g}for such a typical device. Guided by the dot lines, we clearly see a regular modulation of the threshold voltage by V

_{g}with a period of ΔV

_{g}~ 6.5 V, presenting the signature of “Coulomb diamond”. In addition to the main structure, modulations with a smaller period are also seen, though the period is not clear to estimate. In a small field 0.5 T, the gate modulation changes to a very different pattern as shown in Figure 8d: both the gate-modulated threshold voltage and the period shrinks in the small field. We note that, when the field is increased to ~0.6 T, no gate modulation could be observed.

_{g}= 6.5 V to 1.0 V, giving an increase of gate capacitance from C

_{g}= e/ΔV

_{g}= 25 zF to 100 zF when the field rises to 0.5 T. At the same time, the gate modulated threshold voltage also reduces from ~0.1 mV to ~0.05 mV, suggesting that the single island charging energy is halved and the island capacitance is enlarged from C = e

^{2}/2E

_{C}~ 0.8 fF to 1.6 fF. Apparently, both increments of C and C

_{g}provide strong evidence of an enlarged SET island in the application of magnetic fields. In particular, the vanishing of gate modulation suggests that the SET island may be destroyed in large fields.

_{H}< 2r, the field-sensitive gate-modulation cannot exist in a field higher than this limit.

_{b}curves. The best dI/dV

_{b}-plot we could get is presented in Figure 9b, in which the junction parameters are C

_{J}

_{1}+ C

_{J}

_{2}= 1.4 fF, R

_{J}

_{1}+ R

_{J}

_{2}= 100 kΩ, and C

_{J}

_{1}/C

_{J}

_{2}= R

_{J}

_{2}/R

_{J}

_{1}= 1.2. However, the calculation result would give a zero-threshold condition at V

_{g}= e/2C

_{g}, against the experimental observations. Then, the multi-island circuits were assumed in further trials. By using a series combination of three islands presented in Figure 9c and identical junction parameters, C

_{J}= 1.0 fF, R

_{J}= 25 kΩ, identical island-to-ground capacitance C

_{g}= 25 zF, and randomly distributed charge offsets, 0.2 e, −0.2 e and 0 e, we could obtain a dI/dV

_{b}-plot as illustrated in Figure 9d. This multi-island circuit could more reasonably explain the structure observed in Figure 9a.

_{b}characteristics as illustrated in Figure 10a were observed in some MPA AI devices when V

_{g}was largely biased. Such staircase structures would result in resonance-like peaks in the dynamical conductance dI/dV

_{b}curve as in Figure 10b Again, we found that these resonance peaks become smeared as the magnetic field is elevated up to 0.5 T. When B > 0.6 T, the resonance peak and gate modulation disappear, together with the shrinkage of the Coulomb gap. The magnetic field destroying gate modulation is roughly the same as that for the SET-like devices, suggesting that they have the same origin.

_{b}curves may have two possible origins: Coulomb staircase in a SET with asymmetric tunnel junctions, and the resonant tunneling in a quantum dot. The dynamical conductance as a function of V

_{b}and V

_{g}illustrated in Figure 10c is essential for the understanding of the underlying physics. To elucidate the origin, we calculated the device current based on an asymmetric SET circuit. By carefully choosing the model parameters, we may obtain some calculated IV

_{b}curve highly agreeing with the experimental data at a few V

_{g}values, but the calculated dI/dV

_{b}-plot shows structures of slanted Coulomb diamonds, a clear disagreement to the data shown in Figure 10c, where the resonance peaks move in two different directions that form symmetric diamonds. Therefore, the resonant tunneling likely explains the observed phenomenon. The discrete quantum levels may originate from the ultra low charge density and large wavelength comparable to the puddle size.

## 3. Materials and Methods

_{4}by tannic acid and were sized 12 nm in diameter [32]. We used 4 different molecules to modify the Au NP surface: from short to long, 3-mercaptopropionic acid (MPA), 6-mercaptohexadecanoic acid (MHA), 8-mercaptooctanoic acid (MOA), and 11-mercaptoundecanoic acid (MUA). The longer molecules would introduce a longer interparticle spacing when Au NPs are assembled into a two-dimensional film. Au NP colloidal solution (typically 5 mL with concentration ~3 × 10

^{12}cm

^{−3}) was added in a 30 mL centrifuge tube, together with the (3-aminopropyl)-trimethoxysilane(APTMS)- modified substrate laid on a support in the tube. After being centrifuged at 8500 g for 20 min, Au NPs were fully deposited on the SiO

_{2}/Si substrate, and we could directly get a multilayer Au NP film by gently pulling out the sample and drying it in the air. The thickness of the deposited Au NP film could be controlled by the total amount of Au NP in the solution and was further observed by the scanning electron microscopy. In this study, the typical thickness of the Au NP film is 2–3 NP layers. The details in deposition process, as well as thickness control, were described elsewhere [32]. Prior to NP deposition, 20 nm/50 nm Cr/Au electrodes were fabricated on the substrate by using e-beam or photolithography and lift-off technique.

_{T}. Electron microscopy imaging confirmed that the e-beam exposure did not change the NP network structure and the interparticle spacing s. R

_{T}for our samples was estimated from the monolayer sheet resistance at room temperature (RT). Because of the geometry of our device, the sheet resistance is on the same order of the device resistance. In addition, the e-beam bombardment enhances disorder strength of the exposed devices.

## 4. Conclusions

_{T}~ R

_{K}in an NP assembly with finite disorder. In such a system, the charge is mobile between the NPs but confined in charge puddles originated by the quantum interference due to multiple scattering. The charge puddle is fragile to the application of a magnetic field, which destroys the time-reversal symmetry. Because the size of charge puddles is on the order of the device dimension, it is possible to modulate charge transport hopping between charge puddles with a gate voltage, similar to a SET or a quantum dot. The breakdown of the charge puddle in large magnetic field results in a dramatic change of the gate-voltage modulation.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**a**) the scanning electron micrograph of a 8-mercaptooctanoic acid (MOA) device; rectangles with a light color are the measurement electrodes; (

**b**) the transmission electron microscopy image of the synthesized Au nanoparticles deposited on a Si

_{3}N

_{4}membrane. Here, the capping molecules are 11-mercaptoundecanoic acid (MUA) molecules.

**Figure 2.**(

**a**) the schematic of a two-terminal device; (

**b**) the R vs. T curves for MUA, MOA, MHA and metallic MPA (MPA M) devices. The MPA M device shows metallic behavior while the others show insulating behavior; (

**c**) R(T) curves for various MPA and e-beam exposed MOA devices. MPA devices may have versatile R(T) behaviors: when R

_{RT}> R

_{K}, the device is an insulator (MPA MI). When R

_{RT}< R

_{K}, it can be metallic (MPA M) or insulator (MPA AI) due to different disorder strength. The e-beam exposure may reduce the device R

_{RT}and turn a MOA device into a metal.

**Figure 3.**R vs. 1/T plots for insulating MHA (

**a**), MOA and MUA (

**b**) devices. Each color represents individual device. At higher temperatures, the resistance follows the thermal activation property, namely, ln R ~ T

^{−1}. There is a slight deviation near room temperature, suggesting effect from thermal expansion.

**Figure 4.**(

**a**) hopping exponent (r

^{*}/l

_{in}) vs. T

^{−1/2}plots for temperature-dependent l

_{in}using Equation (10) (blue), and temperature-independent l

_{in}= 0.1 a (green). The red curve is a function plot, (r

^{*}/l

_{in}) = (T

_{2}/T)

^{1/2}with T

_{2}= 700 K. The slopes of all curves are very similar; (

**b**) a summary of T

_{2}values for insulating devices. Clearly, the devices can be divided into two categories. Devices with R

_{RT}larger than 10

^{5}Ω have T

_{2}larger than ~ 100 K.

**Figure 5.**Positive MR in metallic devices. (

**a**) an MPA metallic device in the parallel field at T ~ 100 mK; (

**b**) an MOA metallic device in the perpendicular field at T = 2 K. Red curves are fitting results by using Equations (11) and (12).

**Figure 6.**(

**a**) The IV characteristics of an MPA AI device at B = 0 (blue curve) and B = 8 T (red curve) at 50 mK; (

**b**) log-log plot of the IV curves in (

**a**). The IV exponent η reduces from ~5 to ~2 when B is elevated from 0 to 8 T; (

**c**) the change of IV exponent η from zero fields to the high field (MI at B = 9 T, AI at B = 8 T); (

**d**) a summary of MR as a function of RT resistance. Metallic devices exhibit a small, typical 0.2% positive MR at B ~ 3T (T < 100 mK). MIs do not present magnetic response at the high field of B = 9 T (T > 2 K). For AIs, the zero bias resistance may be reduced as much as 10 times by a large field, B = 8 T (T < 100 mK).

**Figure 7.**The formation of charge puddles in the AI and charge hopping between them. The red arrows show the perpendicular magnetic fields B. At the intermediate magnetic field l

_{H}< ~l

_{φ}, the size of the charge puddles becomes enlarged. When the field is so large that l

_{H}< 2r, the quantum interference is totally destroyed.

**Figure 8.**(

**a**) the IV

_{b}characteristics for an MPA “single electron transistor (SET)” device with back-gate voltage V

_{g}= 0 V and −3.0 V, which respectively give the maximum and minimum current; (

**b**) the IV

_{b}characteristics in various magnetic fields, B = 0 T, 0.5 T and 8 T at V

_{g}= 0 V. In a small magnetic field, the Coulomb gap is not affected much; (

**c**) intensity plots of dI/dV

_{b}as a function of V

_{g}and V

_{b}in zero fields. The structure similar to Coulomb diamonds signifies single electron tunneling in the device. From the plot, one can clearly determine a gate-modulation period ΔV

_{g}~ 6.5 V; (

**d**) in the field B = 0.5 T, the gate-modulation period shrinks to ΔV

_{g}~ 1.0 V.

**Figure 9.**(

**a**) the experimental data of dynamical conductance for a different MPA AI device; (

**b**) the calculated dynamical conductance of a single-island SET circuit; (

**c**,

**d**) the assumed three-island circuit and calculated dynamical conductance.

**Figure 10.**(

**a**) IV

_{b}characteristics for an MPA “quantum dot” device with different back-gate voltage V

_{g}= 9.5 V, 0 V and −9.5 V, at which it presents staircase structure; (

**b**) family of the dynamic conductance dI/dV

_{b}as a function of V

_{b}for another AI device at various magnetic fields. At low fields, dI/dV

_{b}shows resonance peaks. When the field is larger than 0.5 T, the resonance structures are smeared out; (

**c**) intensity plots of dI/dV

_{b}as a function of V

_{g}and V

_{b}. As a guide to the eye, the dot lines illustrate the Coulomb diamond structure in this plot.

Device | MUA | MOA | MHA | MPA MI | MOAe | MPA AI | MPA M |
---|---|---|---|---|---|---|---|

Carbon Number, n | 11 | 8 | 6 | 3 | 8 | 3 | 3 |

s (nm) | 1.88 | 1.51 | 1.27 | 0.90 | 1.51 | 0.90 | 0.90 |

R_{RT} (Ω) | 10^{7}–10^{9} | ~10^{7} | ~10^{6} | 10^{5}–10^{6} | 10^{1}–10^{7} | ~10^{3} | ~10^{3} |

T_{1} (K) | 43–80 | 40–52 | 45–56 | 32–49 | 17–50 | <5 | 0 |

T_{2} (K) | 700–1600 | 500–780 | 460–860 | 573–770 | 6–210 | 6–9 | 0 |

l_{in} (nm) | 2.4–4.9 | 4.2–4.7 | 6.8-7.7 | 8.3–11 | - | - | - |

κ | 6–28 | 17–22 | 6–15 | 5–10 | - | - | - |

E_{C} (meV) | 11.0 | 9.3 | 8.1 | 6.0 | - | - | - |

E_{a} (meV) | 10.6–10.7 | 9.2 | 6.3–7.4 | 4.6–5.2 | - | - | - |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Jiang, C.-W.; Ni, I.-C.; Hsieh, Y.-L.; Tzeng, S.-D.; Wu, C.-S.; Kuo, W.
Anderson Insulators in Self-Assembled Gold Nanoparticles Thin Films: Single Electron Hopping between Charge Puddles Originated from Disorder. *Materials* **2017**, *10*, 645.
https://doi.org/10.3390/ma10060645

**AMA Style**

Jiang C-W, Ni I-C, Hsieh Y-L, Tzeng S-D, Wu C-S, Kuo W.
Anderson Insulators in Self-Assembled Gold Nanoparticles Thin Films: Single Electron Hopping between Charge Puddles Originated from Disorder. *Materials*. 2017; 10(6):645.
https://doi.org/10.3390/ma10060645

**Chicago/Turabian Style**

Jiang, Cheng-Wei, I-Chih Ni, Yun-Lien Hsieh, Shien-Der Tzeng, Cen-Shawn Wu, and Watson Kuo.
2017. "Anderson Insulators in Self-Assembled Gold Nanoparticles Thin Films: Single Electron Hopping between Charge Puddles Originated from Disorder" *Materials* 10, no. 6: 645.
https://doi.org/10.3390/ma10060645